Theory LMS proprietary information: reproduction or distribution of this document requires permission in writing from LMS Estimation of modal parameters.doc Category: Analysis and design Topic: Estimation of modal parameters Estimation of modal parameters A modal analysis provides a set of modal parameters that characterize the dynamic behavior of a structure. These modal parameters form the modal model and Figure 2-1 illustrates the process of arriving at the modal parameters. Figure 2-1 Derivation of modal parameters If a structure exists on which measurements can be made, then it can be assumed that a parametric model can be defined that describes that data. The starting point is usually a set of measured data - most commonly frequency response functions (FRFs), or the time domain equivalent, impulse responses (IRs). For IRs the relation between modal parameters and the measurements is expressed in Equation 2-1.
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Theory
LMS proprietary information: reproduction or distribution
of this document requires permission in writing from LMS
Estimation of modal parameters.doc
Category: Analysis and design
Topic: Estimation of modal parameters
Estimation of modal parameters
A modal analysis provides a set of modal parameters that characterize the dynamic behavior of a structure. These modal parameters form the modal model and Figure 2-1 illustrates the process of arriving at the modal parameters.
Figure 2-1 Derivation of modal parameters
If a structure exists on which measurements can be made, then it can be assumed that a parametric model can be defined that describes that data. The starting point is usually a set of measured data - most commonly frequency response functions (FRFs), or the time domain equivalent, impulse responses (IRs). For IRs the relation between modal parameters and the measurements is expressed in Equation 2-1.
The corresponding relation for FRFs is given in Equation 2-2.
where
hij (t) = IR between the response (or output) degree of freedom i and the reference (or input) DOF j
hij (jω) = FRF between the response DOF i and reference DOF j
N = number of modes of vibration that contribute to the structure’s dynamic response within the frequency range under consideration
r ijk = residue value for mode k
λk = pole value for mode k.
* designates complex conjugate.
The pole value can be expressed as shown in Equations 2-3 and 2-4.
where
ωdk = the damped natural frequency of mode k
δk = the damping factor of mode k
or
where
ωnk = the undamped natural frequency of mode k
ζk = damping ratio of mode
Equation 2-5 shows that the residue can be proven to be the product of three terms
where
vik = the mode shape coefficient at response DOF i of mode k
vjk = the mode shape coefficient at reference DOF j of mode k
ak = a complex scaling constant, whose value is determined by the scaling of the mode shapes
Note that the mode shape coefficients can be either real (normal mode shapes) or complex. If the mode shapes are real, the scaling constant can be expressed as,
where
mk = the modal mass of mode k
The poles, natural frequencies (damped and undamped), damping factors or ratios, mode shapes, and residues are commonly referred to as modal parameters (parameters of the modes of the structure).
The fundamental problem of parameter estimation consists of adjusting (estimating) the parameters in the model, so that the data predicted by the model approximate (or curve-fit) the measured data as closely as possible. Modal parameters can be estimated using a number of techniques. These techniques are discussed in the following sections.
A note about units
The frequency and damping values have a dimension of 1/time, and are therefore stored in Hz.
The residues, as appearing in Equation 2-1 of 2-2, have the same dimension as the measurement data. As an aside, it is important to note that residues have a dimension. Residues are composed of a product of mode shape coefficients and a scaling constant, (Equation 2-5). The mode shape coefficients by themselves do not have any dimension, nor absolute (or scaled) magnitude. Dimension, and therefore units will be viewed as attributes of the scaling constant.
Finally, for multiple input analysis, the residues are written in factored form as the product of mode shapes with modal participation factors. Again, the product of the factors has a dimension and absolute magnitude. Formally, the mode shape coefficients will again be considered as without dimension and therefore units will be viewed as attributes of the residues.
Types of analysis
The section discusses some general principles to be considered when performing a modal analysis. These topics include -
• Using single or multiple degree of freedom methods in section 15.2.1
• Making local or global estimates in section 15.2.2
• Using multiple input analysis in section 15.2.3
• Using time or frequency domain analysis in section 15.2.4
• Special conditions which apply when performing vibro-acoustic analysis in section 15.2.5
The specific parameters estimation techniques are described in section 15.3
Single or multiple degree of freedom method
If, in a given frequency band, only one mode is assumed to be important, then the parameters of this mode can be determined separately. This assumption is sometimes called the single degree of freedom (sDOF) assumption.
Figure 2-2 The single degree of freedom assumption
Under this assumption, the FRF equation 2-2 can be simplified to equation 2-7. This is assuming the data to have the dimension of displacement over force.
It is possible to compensate for the modes in the neighborhood of this band, by introducing so called upper and lower residual terms into the equation.
where
urij = upper residual term (residual stiffness) used to approximate modes at frequencies above wmax.
lrij = lower residual term (residual mass) used to approximate modes at frequencies below wmin
Upper and lower residuals are illustrated in Figure 2-3.
Figure 2-3 Upper and lower residuals
Equation 2-7 can be further simplified by neglecting the complex conjugate term, and so becomes
Single degree of freedom methods The single DOF assumption forms the basis for parameter estimation techniques such as Peak picking, Mode picking and Circle fitting.
Multiple degree of freedom methods The sDOF assumption is valid only if the modes of the system are well decoupled. In general this may not be the case. It then becomes necessary to approximate the data with a model that includes terms for several modes. The parameters of several modes are then estimated simultaneously with so-called multiple degree of freedom methods.
Local or global estimates
If you recall the time domain relationship between modal parameters and measurement functions,
you will see that the pole values lk are independent of both the response and the reference DOFs. In other words the pole value lk is a characteristic of the system and should be found in any function that is measured on the structure. When applying parameter estimation techniques, one of two strategies can be employed; making local or global estimates.
Local estimates Global estimates Each data record hij is analyzed individually, and a potentially different estimate of the pole value lk is found each time.
All the data records are analyzed simultaneously in order to estimate the structure’s characteristics.
Analyzing data in this manner produces as many estimates of each pole as there are data records. It is then left to the user to decide which estimate is the best or to somehow calculate the best average of all the estimates.
With this approach, a unique estimate of the pole values lk is obtained. Such estimates are therefore called global estimates.
Peak picking and Circle fitting are techniques that calculate local estimates of pole values.
The Least Squares Complex Exponential, Complex Mode Indicator Function and Direct Parameter Identification methods allow you to obtain global estimates of structure characteristics.
Multiple input analysis
Assume that data is available between Ni input DOFs and No output DOFs. The expression for each of the individual data records ( equation 2-10) can then be rewritten in matrix form for all the data records.
where
[H] = (No,Ni) matrix with hij as elements
[Rk] = (No,Ni) matrix with rijk as elements
Equation 2-5 can be used to express the residue matrix in factored form,
where
Vk = No vector (column) with mode shape coefficients at the output DOFs
[Vr]k = Ni vector (row) with mode shape coefficients at the input DOFs
If DOFs i and j are both output and input DOFs then the above equation implies Maxwell Betti reciprocity,
This assumption is not essential however since the residue matrix can be expressed in a more general form,
where bLck is a vector (row) with Ni coefficients that express the participation of the mode k in response data relative to different input DOFs.
These coefficients are called modal participation factors therefore. Note that if reciprocity is assumed then the modal participation factors are proportional to the mode shape coefficient at the input DOFs.
Using the factored form of the residue matrix, equation 2-11 can be written as,
If just the data between any output DOF and all input DOFs are considered then
where
[H] i = Ni vector of data between output DOF i and all input DOFs.
It is essential in the model of equation 2-16 that both the poles and the modal participation factors are independent of the output DOF. In other words in this formulation the characteristics become -
A multiple input modal parameter estimation technique is one that analyses data relative to several inputs simultaneously to estimate the characteristics expressed by equation 2-17 (i.e. both the pole values and the modal participation factors). The basis for these techniques is a model expressed by equation 2-16.
The identification of modal participation factors is essential for decoupling highly coupled or even repeated roots. To illustrate this consider a structure that has two modes with pole values l1 and l2 very close to each other. Neglecting the other modes and the complex conjugate terms, the response data relative to the input DOF j can be expressed as
or since
The latter equation shows that in the response data relative to an input DOF j, a combination of the coupled modes is observed and not the individual modes. The combination coefficients for the modes are the modal participation factors l1j and l2j.
The response data relative to another input DOF l, is expressed by an equation similar to equation 2-19.
The only difference between these last two equations is the modal participation factors l1l and l2l. If they are linearly independent of the modal participation factors for input i, then the modes will appear in a different combination in the response data relative to input l. As a multiple input parameter estimation technique analyses data relative to several inputs simultaneously, and the modal participation factors are identified, then it is possible to detect highly coupled or repeated modes.
Time vs frequency domain implementation
Using digital signal processing methods, only samples of a continuous function are available. For modal parameter estimation the sampled data consist most frequently of FRF measurements. Normally these are taken at equally spaced frequency lines. Testing techniques such as stepped sine excitation allow you to measure data at unequally spaced frequency lines.
For modal parameter estimation applications with the data measured in the frequency domain, introducing the sampled nature of the data transforms the equation for the model to -
where
hij,n = samples of data in measured range.
wn = sampled value of frequency in measured range.
A frequency domain parameter estimation method uses data directly in the frequency domain to estimate modal parameters. It is therefore irrelevant whether the frequency lines are equally spaced or not. They are based directly on the model expressed by equation 2-21.
If the data are sampled at equally spaced frequency lines, then the FRF can be transformed back to the time domain to obtain a corresponding Impulse Response (IR). A Fast Fourier Transform (FFT) algorithm is used for this transformation but the restriction on the number of frequency lines being
equal to a power of 2 (e.g. 32, 64, 128...) is no longer valid. After transformation, a series of equally spaced samples of corresponding impulse response functions is obtained. A time domain parameter estimation technique allows you to analyze such equally spaced time samples to estimate modal parameters.
In practice, a variety of conditions mean that the frequency band over which data is analyzed is smaller than the full measurement band. This is illustrated in Figure 2-4.
Figure 2-4 Analysis frequency band vs. measurement band
The analysis frequency band includes only three modes whereas the measurement band includes five. If the data is transformed from frequency to time domain, then the time increment between samples will be determined by the analysis frequency band and not the measurement band. If the frequency band of analysis is bounded by wmax and wmin then t is determined from
By substituting sampled time for continuous time
or
where
Time domain parameter estimation methods are based on the model defined by equation 2-24. They analyze hij,n to estimate zk. lk is then calculated from equation 2-25. Note however that this calculation
is not unique since
This implies that no poles outside the frequency band 2p /Dt can be identified. In other words, with a time domain parameter estimation method, all estimated poles are to be found in the frequency band of analysis (w min, w max). This may cause problems in estimating modal parameters if the data in the frequency band of analysis is strongly influenced by modes outside this band (residual effects). Since with frequency domain methods lk is estimated directly, no such limitation arises. A frequency domain technique may therefore sometimes be preferred over a time domain technique for analyzing data over a narrow frequency band, where residual effects are important.
Vibro-acoustic modal analysis
Coupling between the structural dynamic behavior of a system and its interior acoustical characteristics can have an important impact in many applications. Based on combined vibrational and acoustical measurements with respect to acoustical or structural excitation, a mixed vibro-acoustical analysis can be performed.
The finite element equation of motion is used to derive the equations describing the vibro-acoustical behavior:
with
MS, CS, KS the structural mass, damping and stiffness matrices
f the externally applied forces
lp the acoustical pressure loading vectors
In the fluidum the indirect acoustical formulation states:
with
Mf, Cf, Kf matrices describing the pressure-volume acceleration
ω2lf the acoustical pressure loading vectors
Combining these equations with
and rewriting the formulations results in the description of the vibro-acoustical coupled system:
This represents a second order model formulation of the vibro-acoustical behavior which is clearly non-symmetrical.
The above equation also reflects the vibro-acoustical reciprocity principle which can be expressed as:
Most of the multiple input - multiple output modal parameter algorithms do not require symmetry. So the non-symmetry of the basic set of equations and hence the modal description does not pose a problem in obtaining valid modal frequencies, damping factors and mode shapes.
Structural excitation can be substituted for acoustical excitation. The modal models derived from both are compatible but differ in a scaling factor per mode due to the special non-symmetry of the set of equations. To go from the structural formulation to the acoustical formulation a scaling factor which is the squared eigenvalue of the corresponding mode is required. This is fully explained in the paper ‘Vibro-acoustical Modal Analysis : Reciprocity, Model Symmetry and Model Validity’ by K. Wyckaert and F. Augusztinovicz.
Parameter estimation methods
A summary of different methods and their applications is given in Table 2.1.
Method Application DOF Domain Estimates Inputs
Peak picking frequency, damping
single freq local single
Mode picking mode shapes single freq local single
Circle fitting frequency damping mode shapes
single freq local single
Complex Mode Indicator Function
frequency damping mode shapes
multi freq global single or multiple
Least Squares Complex Exponential
frequency damping modal participation factors
multi time global single or multiple
Least Squares Frequency Domain
mode shapes multi freq global single or multiple
Frequency domain Direct Parameter identification
frequency damping modal participation factors
multi freq global single or multiple
Table 2.1 Parameter estimation methods and application
Selection of a method
A guide on which parameter estimation techniques method to adopt is outlined below. Details on all the methods are given in the following sections.
SDOF Single degree of freedom curve fitters are rough and ready and will give you a quick impression of the most dominant modes (frequency damping and mode shapes) influencing a structure under test. As such they are useful in checking the measurement setup and can help assess:
• whether all the transducers are working and correctly calibrated;
• whether the accelerometers are correctly labelled with their node and direction;
• whether all the nodes are instrumented.
For this purpose it is recommended to identify real modes since these are the easiest to interpret when displayed. The circle fitter gives the most accurate estimates of the SDOF techniques, but may create large errors on nodal points of the mode shapes.
Complex MIF This method can be used in the same way as the SDOF techniques to give you an idea of the most dominant modes and check the test setup. It has the advantage that multiple input FRFs can be used and the mode shape estimates are of a higher quality. Furthermore, it can extract a modal model that includes the most dominant modes in a particular frequency band.
Time domain MDOF This is the most general purpose parameter estimation technique that is probably the standard tool used in modal analysis. It provides a complete and accurate modal model from MIMO FRFs. Its major weakness seems to be when analyzing heavily damped systems where the damping is greater than 5% such as in the case of a fully equipped car.
Frequency domain MDOF The Frequency Domain Direct Parameter technique provides similar results to the Time domain technique described above, in terms of accuracy but is generally slower. It is weak when dealing with lightly damped systems (damping less than 0.3%) but fortunately performs better on heavily damped ones, thus complementing the other MDOF technique. Since it operates in the frequency domain it is able to analyze FRFs with an unequally spaced frequency axis.
Peak picking
Peak picking is a single DOF method to make local estimates of frequency and damping. The method is based on the observation that the system response goes through an extremum in the neighborhood of the natural frequencies.
For example, on a frequency response function (FRF) the real part will be zero around the natural frequency (minimum coincident part), the imaginary part will be maximal (peak quadrature) and the amplitude will also be maximal (peak amplitude). The frequency value where this extremum is observed is called the resonant frequency wr and is a good estimate of the natural frequency of the mode wnk for lightly damped systems.
A corresponding estimate of the damping can be found with the 3dB rule. The frequency values w1 and w2,on both sides of the peak of the FRF at which the amplitude is half the peak amplitude (3dB down) are introduced in the formula in equation 3.1 to yield the critical damping ratio. The method is also illustrated in Figure 2-5 below. w1 and w2 are also called half power points.
Figure 2-5 Half power (3 dB) method for damping estimates
Since the curve fitter locates the resonance frequency on a spectral line, significant errors can be introduced if the FRF has a low frequency resolution and the peaks of modes fall between two spectral lines. This can be compensated for by extrapolating the slopes on either side of the picked line to determine the amplitude of the FRF more precisely.
It may be necessary to deal with the situation when one of the half power points is not found. This may arise when the frequency of one mode is close to that of another mode, or it is near to the ends of the measured frequency range.
Note! Peak picking is a single DOF method: it is therefore only suitable for data with well separated modes.
As this method yields local estimates, it requires only one data record to obtain frequency and damping values for all modes. However, if several data records are available, it may be that different records identify different modes.
Mode picking
If you assume that the modes are uncoupled and lightly damped, the modal amplitude can be computed from the peak quadrature or peak amplitude of the FRF. With this assumption, the data in the neighborhood of the resonant frequency can be approximated by
(see also equation 2-7)
The amplitude is maximum at the resonant frequency. However for lightly damped modes, the resonant frequency, natural frequency and damped natural frequency are all approximately the same. Therefore, the amplitude at resonance or the modal amplitude is found at wn which is equal to wdk.
By substituting ωdk for ωn in equation 2-34 the modal amplitude is given by
Note that from the modal amplitude a residue or mode shape estimate is obtained by multiplying by the modal damping.
To use the Mode picking method you must have an estimate of ωdk. This estimate can be obtained with the Peak picking method (see section 15.3.1) or other techniques.
The Mode Picking method is obviously quite sensitive to frequency shifts in the data. If for example the resonant frequency of a mode in a data record is shifted a few spectral lines with respect to the frequency that is used as resonant frequency for that mode, then the modal amplitude would be erroneously picked. To accommodate situations where frequency shifts occur, you need to specify an allowed frequency shift around the resonant frequencies ωdk that are used to calculate the modal amplitudes. Rather than picking the modal amplitude at the resonant frequencies the method now scans a band around each modal frequency for each data record. The maximum amplitude in this band is used to determine the modal amplitude and thus the mode shape coefficient.
Mode picking allows you to make a very quick determination of a modal model. The accuracy of this model however depends on how well the assumptions of the methods were applicable to the data.
Circle fitting
The Circle fitting method is based on estimating a circle in the complex plane through data points in a band around a selected mode. The method was originally developed by Kennedy and Pancu for lightly damped systems under the single DOF assumption. In the band around a mode, the data can be approximately described by
Making an abstraction of the indices i, j and k, introducing complex notation for the residue, and approximating the complex conjugate term by a complex constant, equation 2-36 transforms to
It can be demonstrated that the modal parameters in this expression can be derived from the coefficients of a circle that is fitted to the data in the complex plane, as shown in Figure 2-6.
Figure 2-6 Relation between circle fitting parameters and modal parameters
The natural frequency wd is determined by the maximum angular spacing method where the natural frequency is assumed to occur at the point of maximum rate of change of angle between data points in the complex plane.
Having determined the natural frequency and assuming a lightly damped system, the damping is given by equation 2-38.
The complex residue U + jV is determined from the diameter of the circle d, and the phase g as illustrated in Figure 2-6.
Circle fitting is a basic sDOF parameter estimation method. It can be used to obtain frequency, damping and mode shape estimates. The method is fast, but should really be used interactively to obtain the best possible results.
Complex mode indicator function
The Complex Mode Indicator Function method allows you to identify a modal model for a mechanical system where multiple reference FRFs were measured. The method provides a quick and easy way of determining the number of modes in a system and of detecting the presence of repeated roots. This information can then be used as a basis for more sophisticated multiple input techniques such as LSCE or FDPI. However in cases where modes are well excited and obvious it can yield sufficiently accurate estimates of modal parameters.
The FRF matrix of a system with No (output) and Ni (input) degrees of freedom can be expressed as follows
Or in matrix form as
where
[ H (w) ]= the FRF matrix of size Ni by No
[ F ] = the mode shape matrix of size No by 2N
Q r = the scaling factor for the rth mode
l r = the system pole value for the rth mode
[ L ] T = the transposed modal participation factor matrix of size Ni by 2N
Taking the singular value decomposition of the FRF matrix at each spectral line results in
where
[ U ]= the left singular matrix corresponding to the matrix of mode shape vectors
[ S ]= the diagonal singular value matrix
[ V ]= the right singular matrix corresponding to the matrix of modal participation vectors
In comparing equations 2-42 and 2-43, the mode shape and modal participation vectors in equation 2-42 are, through the singular value decomposition, scaled to be unitary vectors and the mass matrix in equation 2-43 is assumed to be an identity matrix, so that the orthogonality of modal vectors is still satisfied.
For any one mode, the natural frequency is the one where the maximum singular value occurs.
The Complex Mode Indicator Function is defined as the eigenvalues solved from the normal matrix, which is formed from the FRF matrix ( [ H ] H [ H ]) at each spectral line.
where
mk(w)= the kth eigenvalue of the normal FRF matrix at frequency w
sk(w)= the kth singular value of the FRF matrix at frequency w
Ni= the number of inputs
In practice the [ H ]H [ H ] matrix is calculated at each spectral line and the eigenvalues are obtained. The CMIF is a plot of these values on a log scale as a function of frequency. The same number of CMIFs as there are references can be obtained. Distinct peaks indicate modes and their corresponding frequency, the damped natural frequency of the mode. This is illustrated in Figure 2-7.
Peaks in the CMIF function can be searched for automatically whilst taking into account criteria that are used to eliminate spurious peaks due to noise or measurement errors.
Figure 2-7 Example of a CMIF showing selected frequencies
When the frequencies have been selected, equations 2-43 and 2-44 can be used to yield the complex conjugate of the modal participation factors [ V ], and the as yet unscaled mode shape vectors [ U ].
The unscaled mode shape vectors and the modal participation factors are used to generate an enhanced FRF for each mode (r), defined by
Since the mode shape vectors and modal participation factors are normalized to unitary vectors by the singular value decomposition, the enhanced FRF is actually the decoupled single mode response function
A single degree of freedom method (such as the circle fitter technique) can now be applied to improve the accuracy of the natural frequency estimate and then to extract damping values and the scaling factor for the mode shape.
log
.001
.01
.1
1CMIF
frequency
frequency
amp
Figure 2-8 Example of a CMIF and the corresponding enhanced FRF
One CMIF can be calculated for each reference DOF. They can be sorted in terms of the magnitude of the eigenvalues. They can all be plotted as a function of frequency as shown in the example in Figure 2-9.
Figure 2-9 Example of first and second order CMIFs
Cross checking and tracking
At any one frequency these functions will indicate how many significant independent phenomena are taking place as well as their relative importance. At a resonance, at least one CMIF will peak implying that at least one mode is active. At a different frequency however it may be that a different mode has increased its influence and is the major contributor to the response. Between resonances, a cross over point can occur where the contribution of two modes are equal. This can result in a higher order CMIF exhibiting peaks if they are sorted as shown in Figure 2-9 and in the effect of one CMIF exhibiting a dip at the same time as a lower order function is exhibiting a peak.
A check on peaks in the second order CMIF functions can be made to determine whether or not they are due to the cross over effect or a genuine pole of second order. This is done by calculating the MAC matrix using data on either side of the frequency of interest.
MAC (1a,1b) MAC (2a,1b)
MAC (1a,2b) MAC (2a,2b)
Where a and b represent the frequencies and 1 and 2 the CMIF functions. CMIF 1 contains the larger values and CMIF 2 the smaller ones
.
CMIF_1
CMIF_2
a b When this MAC matrix approximates a unity matrix then the peak in CMIF_2 represents a resonance peak. The mode is not changing between fre quencies a and b .
≅ 0 ≅0 ≅ 1
≅1
CMIF_1
CMIF_2
a b
When this MAC matrix is anti diago nal then the peak in CMIF_2 repre sents a cross over point. The mode is switching between frequencies a and b
≅0≅0 ≅ 1
≅1 Peak picking can be facilitated by using tracked CMIFs. This alters the display of the CMIFs for when the mode shapes represented by the two CMIFs are switched, the CMIFs are also switched. This is determined by the cross over check described above.
An example of the tracked versions of the CMIFs illustrated in Figure 2-9 is shown below.
log
.001
.01
.1
1CMIF
frequency Figure 2-10 Example of first and second order tracked CMIFs
Least squares complex exponential
The Least Squares Complex Exponential method allows you to estimate values of modal frequency and damping for several modes simultaneously. Since all the data is analyzed simultaneously, global estimates are obtained.
To understand how the method works, recall the expression for an impulse response (IR) given below
It can be seen from this expression that the pole values l k are not a function of a particular response (output) or reference (input) DOF. In other words the pole values are global (rather than local) characteristics of the structure. They are the same for any measured FRF on the structure. It should therefore be possible to use all the available data measured on the system to identify global estimates simultaneously.
This method can be used with single and multiple inputs.
Model for continuous data
A particular problem when trying to work with equation 2-48 to achieve the above objective is that it contains residues rijk which do depend on the response and reference DOFs. It is therefore essential to define another parametric model for the data hij, in which the coefficients are independent of response and reference DOFs and can be used to identify estimates for l k.. It can be proved that such a model takes the form of a linear differential equation of order 2N with constant real coefficients
Indeed, equation 2-48 expresses the data as a linear superposition of a set of 2N damped complex exponentials occurring in complex conjugate pairs. Such complex exponentials can be viewed as the characteristic solutions of a linear differential equation with constant real coefficients
The impulse response, being a linear superposition of characteristic solutions, is by itself also a characteristic solution. Therefore equation 2-49 is valid if the coefficients are such that
Turning the reasoning around therefore, one could first try to estimate the coefficients in equation 2-49 using all available data. Estimates of the complex exponential coefficients lk can then be found by solving equation 2-51.
Model for sampled data
Measured data is however sampled, not continuous. So rather than working from equation 2-48 it is necessary to work with
Instead of damped complex exponentials, the characteristics are now power series with base numbers zk.
Following a similar reasoning to that explained above for continuous data it can be proved that the sampled data is the solution of a linear finite difference equation with constant real coefficients of order 2N (instead of a differential equation as for continuous data).
The characteristics zk and therefore the poles lk can be found by solving,
Practical implementation of the method
The Least Squares Complex Exponential is a method that estimates the coefficients in equation 2-53 using data measured on the system.
In principle any data record hij,n can be used. Applying the method to just a single data record at a time will result in local estimates of the poles.
To estimate the coefficients in equation 2-53 in a least squares sense the equations for all possible time points and all possible response and reference DOFs are to be solved simultaneously as indicated in equation 2-55. This equation system will be greatly overdetermined. To find the least squares solution the normal equations technique can be applied so that the final solution is calculated from a compact equation with a square coefficient matrix, equation 2-56. The coefficient matrix in this equation is called a covariance matrix.
where
Nt = last available time sample
N0 = number of response DOFs
Ni = number of input DOFs
We can write this in a simpler manner
The coefficients in the covariance matrix are defined as
Building this covariance matrix is the first stage in applying the Least Squares Complex Exponential method. This phase is usually the most time consuming since all the available data is used to build the inner products expressed by equation 2-57.
Note that after solving equation 2-56 all that is required to calculate the estimates of modal frequency and damping is to substitute the estimated coefficients in equation 2-54 and to solve for zk.
Determining the optimum number of modes
The solution of equation 2-56 results in least squares estimates of the coefficients in the model expressed by equation 2-53. It is also possible therefore to calculate the corresponding least squares error. This error is of importance in determining the minimum number of modes in the data.
In the preceding discussion it has been assumed that N modes are present in the data. However, the number of modes contained in the data is in fact unknown. It is preferable that this should be
determined by the method itself. Using the Least Squares Complex Exponential method, this can be achieved by observing the evolution of the least squares error on the solutions of equation 2-56 as a function of the number of assumed modes.
To do this, an equation like equation 2-56 is initially created, assuming a number of modes N that is sufficiently large. A subset of such an equation is then taken to solve for the coefficients of a model that describes just one mode
The corresponding least squares error is represented by e1.
When 2 modes are assumed in the data then the sub set to be solved is
With corresponding least squares error e2, and so on. Now if a model is assumed with a number of modes equal to the number of modes that is present in the data then the corresponding least squares error should be significantly smaller than the error for models with fewer modes.
A diagram that plots the least squares error for increasing number of modes is called the least squares error chart. Figure 2-11 shows a typical diagram if data is analyzed for a system with 4 modes (and 4 modes are observable from the data!).
Noise on the data may cause the error diagram to show a significant drop at a certain number of modes, followed by a continued decrease of the error as the number of modes is increased. The problem now is to determine how many extra modes, or so called computational modes, are to be considered to compensate for the noise on the data so that the best estimates of modal frequency and damping can be obtained. This problem is also illustrated in Figure 2-11.
Figure 2-11 Least squares error diagram, system with 4 modes
To determine the optimal number of modes you could try to compare frequency and damping estimates that are calculated from models with various number of modes. Physical intuition would lead you to expect that estimates of frequency and damping corresponding to true structural modes, should recur (in approximately the same place) as the number of modes is increased. Computational modes will not reappear with identical frequency and damping. A diagram that shows the evolution of frequency and damping as the number of modes is increased is called a Stabilization diagram. The optimal number of modes that can be calculated for use can then be seen, as those modes for which the frequency and damping values of the physical modes do not change significantly. In other words, those which have stabilized.
of
d
s
v
sssss
ss
sss
s
ss
ss
sss
s
ss
ssss
fv
f
d
fv
f
fv
d
ff
fv
dnu
mbe
r of m
odes
ampl
itude
frequency Figure 2-12 A stabilization diagram
Example
Let two data records be measured on a system, both shown in Figure 2-13.
Figure 2-13 Example least squares complex exponential
Let four data samples be measured of which the values are listed in the Table below.
Consider a model for 1 mode (N=1). Equations 2-55 and 2-56 become respectively
The solution is therefore a1=0, a2=1. Now equation 2-54 is used to calculate zk and so lk,
The frequency and damping values follow from
The solution indicates a mode with a period 4Dt and zero damping. This is compatible with the trend of the cursor as shown in Figure 2-13.
Multiple input least squares complex exponential
The Least Squares Complex Exponential method, described above, uses all data measured on a structure to estimate global estimates of modal frequency and damping. In principle, data relative to several reference DOFs can be used. However the model used by the previous method does not take specific advantage of this.
The multiple input Least Squares Complex Exponential, (or polyreference), is an extension of the Least Squares Complex Exponential that does allow consistent simultaneous analysis of data relative to several reference DOFs. The method computes global estimates of frequency and damping and also of modal participation factors. Modal participation factors are terms which express the participation of modes in the system response as a function of the reference (or input) DOF (see section 15.2.3). The simultaneous estimation of frequency, damping and modal participation factors means that highly coupled, even repeated modes can be identified.
The basis for the Multiple Input Least Squares Complex Exponential method is the model of the data introduced in section 15.2.3 equation 2-16.
where
[H]i = Ni vector (row) of IRs between output DOF i and all input DOFs
[L]k = vector of modal participation factor for mode k. If Ni reference DOFs are assumed then bLck is of dimension Ni
vik = is the mode shape coefficient at response DOF i for mode k
Note that in this model, frequency, damping and modal participation factors are independent of the particular response DOF. It should therefore be possible to estimate these coefficients using all the available data simultaneously.
Model for sampled data
The model expressed by equation 2-58 is not directly suitable for global estimation of frequency, damping and modal participation factors as it still contains the mode shape coefficients that are dependant on the response DOF. Therefore a more suitable model must be derived.
Introducing firstly the sampled nature of the data, equation 2-58 is rewritten as,
It can be proved that if the data can be described by equation 2-59, it can also be described by the following model
if the following conditions are fulfilled
(The proof of this follows from basic calculus along the same lines as for Least Squares Complex Exponential in section 15.3.5).
Equation 2-60 represents, in matrix notation, a coupled set of Ni finite difference equations with constant coefficients. The coefficients A1 . . . Ap are therefore matrices of dimension (Ni Ni).
The condition expressed by equation 2-61 states that the terms [ Lk] and zkn are characteristic
solutions of this system of finite difference equations. As equation 2-59 is a superposition of 2N of such terms, it is essential that the number of characteristic solutions of this system of equations pNi at least equals 2N as expressed by equation 2-62.
Note finally, that if data for each reference DOF is treated individually, i.e. Ni = 1, then equation 2-60 and 2-61 simplify to equations 2-53 and 2-54. Thus the least squares complex exponential method is a special case of the multiple input least squares complex exponential method.
Practical implementation of the method
To estimate the coefficients in equation 2-60 in a least squares sense the equations for all possible time points and all possible response DOFs are to be solved simultaneously, as indicated by equation 2-63. A least squares solution is found, for example using the normal equations method, from equation 2-64. The coefficient matrix in this equation is again in the form of a covariance matrix,
where
Nt = the last available time sample
N0 = the number of response DOFs
The order (p) of the finite difference equation is related to the number of modes in the data by equation 2-62. It is preferable that this be determined by the method itself. As the coefficients of the finite difference equation are solved for in a least squares sense, this can be done by observing the least squares error as a function of the assumed order. As an order is reached such that the model can describe as many modes as are present in the data, the error should drop considerably.
Due to the condition expressed by equation 2-62 there is no linear relation between the number of modes that can be described by the model and the order of the model. The relation between the number of modes, the order of the model and the number of reference DOFs is listed in Table 2.2. It can be seen that a model of order 8 can describe 11 or 12 modes if data for 3 inputs are analyzed simultaneously. In the error diagrams therefore the same least squares error is shown for 11 and 12 modes.
As for the Least Squares Complex Exponential method, a stabilization diagram can again be created to determine the optimal number of modes. As well as comparing frequency and damping values calculated from models of consecutive order it is now also possible to compare the stabilization of modal participation factors. In section 15.2.3, the modal participation factors were shown to be proportional to the mode shape coefficients at the reference DOFs. They also represent a physical characteristic of the structure like the frequency and damping. Therefore, the values corresponding to structural modes should also stabilize as the order of the model is increased. This additional criterion
adds much to the readability of the stabilization diagram and to the ability to distinguish computational modes from physical modes
Additionally, the modal participation factors can be used by themselves to identify physical modes. If they are normalized with respect to the largest, the values should all be approximately real, in phase or in anti-phase, for structural modes.
N Ni=1 Ni=2 Ni=3 Ni=4 Ni=5 Ni=6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 51 52 54 56 58 60 62 64
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
1 2 2 3 4 4 5 6 6 7 8 8 9 10 10 11 12 12 13 14 14 15 16 16 17 18 18 19 20 20 21 22
1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16
1 1 2 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 8 9 9 10 10 10 11 11 12 12 12 13 13
1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11
Table 2.2 Relation between modal order (tabulated), number of modes (N) and number of reference DOFs (Ni)
Example
To clarify the method, consider again the example discussed on page 34. Let the example system satisfy reciprocity so that h12 is also equal to h21. The vector [h12 h21] then represents the data between response DOF 1 and reference DOFs 1 and 2.
Considering a model for 1 mode (so p= 1, as Ni= 2) equations 2-55 and 2-56 become respectively
The resulting matrix polynomial is therefore
and the solutions of this eigenvalue problem are
Notice that the solution for the frequency and damping is the same as found with the Least Squares Complex Exponential (see page 35). In addition you also find an estimate of the modal participation factors. For this example they indicate that there should be a phase difference of 90_ in the system response between excitation from reference DOFs 1 and 2 as h11 is a cosine, and h12 a sine. This estimate seems to be correct.
Least squares frequency domain
The Least Squares Frequency Domain method is a multiple DOF technique to estimate residues, or mode shape coefficients. The method requires that frequency and damping values have already been estimated. It can be used with single or multiple inputs.
Consider the model expressed by equation 2-66
If estimates of the modal frequency and damping are available, then the residues appear linearly as unknowns in this model.
To estimate the residues, equation 2-66 is transformed back to the frequency domain. Assuming sampled data therefore
where
urij = an upper residual term used to approximate modes at frequencies above wmax
lrij = an lower residual term used to approximate modes at frequencies below wmin
These are illustrated in Figure 2-3. Note that the residues as well as lower and upper residuals are local characteristics; in other words, they depend on the particular response and reference DOF.
The Least Squares Frequency Domain method is based on the model expressed by equation 2-67. Least squares estimates of residues, lower and upper residuals are calculated by analyzing all data values in a selected frequency range.
Multiple input least squares frequency domain
The multiple input Least Squares Frequency Domain method is a multiple DOF technique to estimate mode shapes. The method analyses data relative to several reference DOFs simultaneously to estimate mode shape coefficients that are independent of reference DOFs.
Consider the model expressed by equation 2-58,
If estimates of frequency, damping and modal participation factors are available, then the mode shape coefficients appear linearly as the only unknowns in this model. Furthermore, they are only dependent on the response DOF (and not on the reference DOF) so that data relative to several reference DOFs can be analyzed simultaneously.
To estimate the residues, equation 2-68 is transformed to the frequency domain. Adding residual terms and assuming sampled data results in
where
[UR] i = upper residuals between response DOF i and all reference DOFs, vector of dimension Ni
[LR] i = lower residuals between response DOF i and all reference DOFs, vector of dimension Ni
The multiple input LSFD method is based on equation 2-69.
Frequency domain direct parameter identification
The Frequency domain Direct Parameter Identification (FDPI) technique allows you to estimate the natural frequencies, damping values and mode shapes of several modes simultaneously. If data relative to several references are available, a multiple input analysis will also extract values for the modal participation factors. In this case, the FDPI technique offers the same capabilities as the LSCE time domain method.
Theoretical background
The basis of the FDPI method is the second order differential equation for mechanical structures
When transformed into the frequency domain, this equation can be reformulated in terms of measured FRFs
where
ω= frequency variable
A1 = M-1 C, mass modified damping matrix ( No by No)
A0 = M-1 K, mass modified stiffness matrix ( No by No)
H(w) = matrix of FRF’s (No by Ni )
B0, B1 are the force distribution matrices (No by Ni )
Note that for the single input case, the H(w) matrix becomes a column vector of frequency dependent FRF’s.
Equation 2-71 is valid for every discrete frequency value w. When these equations are assembled for all available FRFs, including multiple input - multiple output test cases, the unknown matrix coefficients A0, A1, B0, and B1 can be estimated from the measurement data H(w). Equation 2-71 thus means that the measurement data H(w) can be described by a second order linear model with constant matrix coefficients. From the identified matrices, the system’s poles and mode shapes can be estimated via an eigenvalue and eigenvector decomposition of the system matrix.
This will yield the diagonal matrix [L] of poles and a matrix Y of eigenvectors. It will become clear from the following section, that the matrix Y thus obtained is not equal to the matrix of mode shapes, although it is related to it.
In a final step, the modal participation factors are estimated from another least squares problem, using the obtained [L] and Y matrices.
Data reduction
Prior to estimating the system matrix, all available data are condensed via a projection on their principal components. For all response stations, a maximum of Nm principal components are first calculated and then analyzed. The obtained matrix Y represents the modal matrix for this set of fictitious response stations.
The data reduction procedure offers the following advantages
• the calculation time is drastically decreased for the estimation of model parameters. This is especially important for the calculation of least squares error charts and stabilization diagrams.
• the number of contributing modes is more easily determined from the singular value analysis.
Residual correction terms
The FDPI technique operates directly on frequency domain data. It is therefore capable of taking into account the effects of modes outside the frequency band of analysis. This feature significantly improves the analysis results when modes below or above the selected band influence the data set. In the case where both upper and lower residual terms are included in the model, equation 2-71 becomes
The presence of these residual terms will influence the estimates for frequency, damping and mode shapes (as well as the modal participation factors for multiple input analysis).
Determining the optimum number of modes
As with the Least Squares Complex Exponential (LSCE) method, a least squares error chart can be built to determine the optimal number of modes in the selected frequency band. Because of the principal component projection, this chart may look somewhat different. For small models, only the first (most important) principal data are used, and the global error will decrease drastically. As more and more principal components are included by estimating more modes, their information becomes less important, which may distort the least squares error chart.
A more reliable tool for estimating the optimal number of nodes for the FDPI technique is the singular values diagram. As an alternative to the error diagram, and to some extent to the stabilization diagram too, the rank of the calculated covariance matrix can be determined. The rank of the matrix is also a
good indication of the optimal number of modes to be used in the analysis. The rank of the matrix can be determined using a singular value decomposition. A diagram showing the normalized singular values in ascending order is called a singular values diagram: the rank of the matrix is determined at the point where the singular values become significantly smaller compared to the previous values.
When building a stabilization diagram, (see LSCE method page 33), the same data are described by models of increasing order. An updating procedure is implemented to save calculation time.
Pseudo-DOFs for small measurement sets
Due to the type of identification algorithm, the FDPI technique can only estimate as many modes in the model as there are measurement Degrees of Freedom. This means that normally
However, using a similar approach as for the time domain LSCE method, it is possible to create so-called “pseudo-” Degrees of Freedom from the measurements that are available, thus generating enough “new” measurements to allow a full identification on as few as one measurement.
Mode shape estimation
Using the reduced mode shapes Y for the principal responses, and the transformation matrix between the principal and physical responses, the FDPI algorithm allows you to identify the complete mode shapes of the system by expanding the reduced Y matrix.
This mode shape expansion offers several advantages :-
• it is very fast (no least squares solution required as for the LSFD method)
• it identifies a mode shape vector as a global direction in the modal space, rather than estimating its elements one by one via mutually independent least squares problems.
If the mode shape expansion method is not employed then the LSFD technique is used to estimate mode shapes.
Normal modes
From the meaning of the matrices [A0] and [A1] and the eigenvalue problem (2-72), it is possible to estimate damped (generally complex) mode shapes Y, or undamped real normal modes.
Normal modes can be identified via the FDPI technique by solving an eigenvalue problem for the reduced mass and stiffness matrices only
This eigenvalue problem is very much related to the one that is solved by FEM software packages that ignore the damping contribution in a system. This is an entirely different approach to the one that is used to estimate real modes via the LSFD technique. The latter technique estimates the real-valued
mode shape coefficients that curve-fit the data set in a best least squares sense (proportional damping assumed), while the FDPI method uses an FEM-like approach.
Damping values are computed by applying a circle-fitter to enhanced FRFs for each mode. The enhanced FRFs are calculated by projecting the principal FRFs on the reduced mode shapes.
Maximum likelihood method
A multi-variable frequency-domain maximum likelihood (ML) estimator is proposed to identify the modal parameters together with their confidence intervals. The solver is robust to errors in the non-parametric noise model and can handle measurements with a large dynamical range.
Although the LSCE-LSFD approach has proven to be useful in solving many vibration problems, the method has some drawbacks:
• the polyreference LSCE estimator does not always work well when the number of references (inputs) is larger than 3 for example
• the frequencies should be uniformly distributed
• the method is not able to handle noisy measurements properly, which can result in unclear stabilization plots and
• the method does not deliver confidence intervals on the estimated modal parameters.
Theoretical aspects
A scalar matrix-fraction description – better known as a common-denominator model – will be used. The Frequency Response Function (FRF) between output o and input i is modeled as
for i = 1, . . . . . , Ni and o = 1, . . . . . , No
with
the numerator polynomial between output o and input i
and
the common-denominator polynomial.
The polynomial basis functions Ωj (ωf) are given by for a discrete-tim model (with Ts the sampling period). The complex-valued coefficients Aj and Boij are the parameters to be estimated. The approach used to optimize the computation speed and memory requirements will first be explained for the Least Squares Solver and then these results will be extrapolated to the ML estimator.
The Least-Squares Solver
Replacing the model Hoi ωf in equation 2-75 by the measured FRF Hoi ωf gives, after multiplication with the denominator polynomial,
for i = 1, . . . . . , Ni 0 = 1, . . . . . , No and f= 1, . . . . . , Nf
Note that equation 2-76 can be multiplied with a weighting function Woi(wf). The quality of the estimate can often be improved by using an adequate weighting function.
As the elements in equation 2-76 are linear in the parameters, they can be reformulated as
The (complex) Jacobian matrix J of this least-squares problem
has Nf No Ni rows and (n+1)(No Ni +1) columns (with Nf >> n, where n is the order of the polynomials). Because every element in equation 2-76 has been weighted with Woi(ωf), the Xk’s in equation 2-77 can all be different.
The Maximum-Likelihood Solver
Using referenced measurements (e.g., FRF data) makes it easier to get global estimates from measurements that were obtained by roving the sensors over the structure under test (which is a common practice in experimental modal analysis). Because of this, the FRFs will be used here as primary data instead of the input/output spectra (i.e. non-referenced data). However, one should take care that the FRFs are not contaminated by systematic errors.
The ML equations
Assuming the different FRFs to be uncorrelated, the (negative) log-likelihood function reduces to
The ML estimate of q=[ B1T.?.?.?.? BNoNi
T AT ]T is given by minimizing equation 278. This can be
done by means of a Gauss-Newton optimization algorithm, which takes advantage of the quadratic form of the cost function (2-78). The Gauss-Newton iterations are given by -
Deriving confidence interval
The covariance matrix of the ML estimate θML is usually close to the corresponding Cramér-Rav Lower Bound (CRLB) : covθML≥CRLB
A good approximation of this Cramér-Rav Lower Bound is given by
with Jm the Jacobian matrix evaluated in the last iteration step of the Gauss-Newton algorithm. As one is mainly interested in the uncertainty on the resonance frequencies and damping ratios, only the covariance matrix of the denominator coefficients is in fact required.
Hence, it is not necessary to invert the full matrix to obtain the uncertainty on the poles (or on the resonance frequencies and the damping ratios).
Calculation of static compensation modes
Modal synthesis can be used to couple substructures together in low frequency ranges. For this, modal models for each of the substructures are required as separate disconnected items. However the
results of this coupling may be less than optimal due to truncation errors. Truncation errors arise because only a limited number of modes are taken into account.
To improve the results, both static and dynamic compensation terms can be used.
Truncation errors can be approximated by a quadratic function using a taylor expansion. It has been shown that there is a good correspondence between the real truncation error and the quadratic estimation.
Static compensation terms can be derived from direct (driving point) FRFs. These static compensation terms are calculated using the upper residual terms which were obtained while fitting the FRF matrix of the coupling points (driving points and cross terms). The upper residual terms are converted by means of a singular value compensation into regular mode shapes and participation factors. These mode shapes and participation factors can be used afterwards for modal substructuring, in addition to the regular modes of the two substructures.
Frequency of the static compensation modes
The frequency of the static compensation modes (w0) must be significantly higher than the frequency band of the modes, which are taken into account during the substructuring calculations. The upper limit of the frequency band used in modal substructuring is defined by the frequency of the upper residual (wupper residual)
The Singular Value decomposition (SVD)
In order to calculate the static compensation terms, a singular value decomposition has to be applied on the upper residual term matrix. This is obtained by putting all upper residual terms together in one big matrix.
The mode shape values of the static compensation mode (Y) are related to the left singular vector, the singular value, and the frequency value (ω0)
The participation factor values (L) can be derived from the mode shape values (ψ)
Theory
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Operational modal analysis.doc
Category: Analysis and design
Topic: Operationel modal analysis
Why operational modal analysis?
Traditional modal model identification methods and procedures are based on forced excitation laboratory tests during which Frequency Response Functions (FRFs) are measured. However, the real loading conditions to which a structure is subjected often differs considerably from those used in laboratory testing. Since all real-world systems are to a certain extent non-linear, the models obtained under real loading will be linearized for much more representative working points. Additionally, environmental influences on system behavior (such as pre-stress of suspensions, load-induced stiffening and aero-elastic interaction) will be taken into account.
In many cases, such as small excitation of off-shore platforms or traffic/wind excitation of civil constructions, forced excitation tests are very difficult, if not impossible, to conduct, at least with standard testing equipment. In such situations operational data are often the only ones available.
It is also the case that large in-operation data sets are measured anyway, for level verification, operating field shape analysis and other purposes. Hence, extending classical operating data analysis procedures with modal parameter identification capabilities will allow a better exploitation of these data.
Finally, the availability of in-operation established models opens the way for in situ model-based diagnosis and damage detection. Hence, a considerable interest exists in extracting valid models directly from operating data.
Traditional processing of operational data
An accepted way of dealing with operational analysis in industry is based on a peak-picking technique applied to the auto-and crosspowers of the operational responses. Such processing results in the so-called “Running Mode Analysis”. By selecting the peaks in the spectra, approximate estimates for the resonance frequencies and operational deflection shapes can be obtained. These shapes can then be compared to or even decomposed into the laboratory modal results.
Correlation of the operating data set with the modal database measured in the lab allows an assessment the modes which are dominant for a particular operating condition. In case of partially correlated inputs (e.g. road analysis), principal component techniques are employed to decompose the multi-reference problem into subsets of single reference problems, which can be analyzed in parallel. These decomposed sets of data can be fed to an animation program, to interpret the operational deflection shapes for each principal component as a function of frequency.
The auto-and crosspower peak-picking method requires considerable engineering skill to select the peaks which correspond to system resonances. In addition, no information about the damping of the modes is obtained and the operational deflections shapes may differ significantly from the real mode shapes in case of closely spaced modes. Pre-knowledge of a modal model derived from FRF measurements in the lab is often indispensable to successfully perform a conventional operational (running modes) analysis.
Curve-fitting techniques therefore, which allow modal parameters to be extracted directly from the operational data would be of a great use for the engineer. Such techniques would identify the dominant modes excited under driving conditions and this information might even be used to improve some traditional FRF tests in the laboratory.
Using Operational modal analysis
The purpose of this procedure is to extract modal frequencies, damping and mode shapes from data taken under operating conditions. This means that under the influence of its natural excitation such as airflow around the structure (e.g. wind turbines, aeroplanes, helicopters), road input, liquid flow (in pipes), road traffic (e.g. bridges), internal excitation (rotating machinery).
Theoretically, one could consider the case where the input forces are measured in such conditions which means that conventional FRF processing and modal analysis techniques could be used. However the Operational modal analysis software is aimed specifically at applications where the inputs can not be measured, and works when only responses such as accelerations signals are available. The ideal situation is when the input has a flat spectrum.
Three methods are discussed, all of which use time domain correlation functions. These auto- and cross-correlation functions can be calculated directly from raw time data or be derived from measured auto- and cross powers by an inverse FFT processing.
Theoretical aspects
This section describes the mathematical background to the methods used to identify modal parameters from operational data.
Over recent years, several modal parameter estimation techniques have been proposed and studied for modal parameter extraction from output-only data. These include - Auto-Regressive Moving Averaging models (ARMA), - Natural Excitation Technique (NExT) - Stochastic subspace methods.
The Natural Excitation Technique (NExT) The underlying principle of the NExT technique is that correlation functions between the responses can be expressed as a sum of decaying sinusoids. Each decaying sinusoid has a damped natural frequency and damping ratio that is identical to the one of the corresponding structural mode. Consequently, conventional modal parameter techniques such as polyreference Least-Squares Complex Exponential (LSCE) can be used for output-only system identification.
Stochastic subspace methods With the subspace approach, first a reduced set of system states is derived, and then a state space model is identified. From the state space model, the modal parameters are derived. The terminology
“subspace” comes mainly from the control theory – it is a “family name” which groups methods that use Singular Values Decomposition in the identification process. Two subspace techniques, referred to as the Balanced Realization (BR) and the Canonical Variate Analysis (CVA) are provided.
Stochastic substate identification methods
The following stochastic discrete time state space model is considered
where xk represents the state vector of dimension n, yk is the output vector of dimension Nresp wk, vk, are zero-mean, white vector sequences, which represent the process noise and measurement noise respectively.
For p and q large enough, the matrices [A] and [C] are respectively the state space matrix and the output matrix. Along with this model, the observability matrix [Op] of order p and the controllability matrix [Cq] of order q are defined :
where [G]=E[ xk+1ykT ] and E[.] denotes the expectation operator. The matrices [Op] and [Cq] are assumed to be of rank 2Nm, where Nm is the number of system modes.
The dynamics of the system are completely characterized by the eigenvalues and the observed parts of the eigenvectors of the [A] matrix. The eigenvalue decomposition of [A] is given by
Complex eigenvectors and eigenvalues in equation 3-3 always appear in complex conjugate pairs. The discrete eigenvalues λr on the diagonal of [Λ] can be transformed into continuous eigenvalues or system poles ∝r by using the following equation
where sr is the damping factor and wr the damped natural frequency of the r-th mode.
The damping ratio ξr of the r-th mode is given by
The mode shape yr of the r-th mode at the sensor locations are the observed parts of the system eigenvectors fr of [F], given by the following equation
The extracted mode shapes can not be mass-normalized as this requires the measurement of the input force.
The stochastic realization problem
The problem considered here is the estimation of the matrices [A] and [C] in equation 3-2, up to a similarity transformation, using only the output measurements yk . This problem is known as the stochastic realization problem and has been addressed by many researchers from the control departments as well as the statistics community [ 4, 5 and 6].
Two correlation-driven subspace algorithms are briefly discussed below, known as the Balanced Realization (BR) and the Canonical Variate Analysis (CVA).
Given a sequence of correlations
where ykref is a vector containing Nref outputs serving as references.
For p ≥ q, let [Hp,q] be the following block-Hankel matrix :
Direct computations of the [Rk] from the model equations lead to the following factorization property
Let [W1] and [W2] be two user-defined invertible weighting matrices of size pNresp and qNresp, respectively. Pre-and post multiplying the Hankel matrix with [W1] and [W2] and performing a SVD decomposition on the weighted Hankel matrix gives the following
where [S1] contains n non-zero singular values in decreasing order, the n columns of [U1] are the corresponding left singular vectors and the n columns of [V1] are the corresponding right singular vectors.
On the other hand, the factorization property of the weighted Hankel matrix results in
From equations 3-10 and 3-11, it can be easily seen that the observability matrix can be recovered, up to a similarity transformation, as
The system matrices are then estimated, up to similarity transformation, using the shift structure of [Op]. So,
and [A] is computed as the solution of
where [Op-1] is the matrix obtained by deleting the last block row of [Op] and [Op-1
↑] is the upper shifted matrix by one block row.
Different choices of weighting will lead to different stochastic subspace identification methods. Two particular choices for the weighting matrices give rise to the Balance Realization and the Canonical Variate Analysis methods.
Balanced Realization (BR)
So no weighting is involved.
Canonical Variate Analysis (CVA)
CVA requires that all responses are serving as references, so yk=ykref. Consequently, the correlation matrix [Rk] given by equation 3-7 is square. Define then the following Toeplitz matrices
Let the full-rank factorization of [ℜ+] and [ℜ-] be
In case of CVA, the weighting is as follows
With this weighting, the singular values in equation 3-10 correspond to the so-called canonical angles. A physical interpretation of the CVA weighting is that the system modes are balanced in terms of energy. Modes which are less well excited in operational conditions might be better identified.
Practical implementation of correlation-driven stochastic subspace methods
Equation 3-10 only holds for ‘true’ block-Hankel matrices and for a finite order system. In practice, the system has a larger, possibly infinite order and the Hankel and Toeplitz matrices in equations 3-8 and 3-16 will be filled up with ‘empirical’ correlations, which are computed as follows :
where M is the number of data samples.
Although equation 3-19 is a preferred estimator for the correlation functions as no leakage errors are made and as it can also be used for non-stationary data, the evaluation of equation (19) in the time domain is not really efficient in computational effort. A faster estimator for the correlation functions can be implemented by taking the inverse FFT of auto-and crosspower spectra which are calculated on the basis of the FFT and segment averaging. This however assumes stationary signals and time windowing (e.g. Hanning) is needed to avoid leakage.
The SVD decomposition of the weighted empirical Hankel matrix will then result in the following
Identification of a model with order n is done by truncating the singular values, so by keeping [S1]. The observability matrix is then approximated by
As the model order is typically unknown, inspection of the singular values might help the engineer to select n such that sn & sn+1 In practice, this criteria is not often of great use as no significant drop in the singular values can be observed. Other techniques such as stabilization diagrams are then needed in order to find the correct model order.
The remaining steps of the algorithm are similar to those described in equations 3-11 to 3-18, where theoretical quantities are replaced with empirical ones.
Natural Excitation Techniques
Subtitled : The Polyreference Least Squares Complex Exponential (LSCE) method applied to auto-and crosscorrelation functions
Polyreference LSCE applied to impulse response functions is a well-known technique in conventional modal analysis, yielding global estimates of poles and the modal participation factors [7]. It has been shown that, under the assumption that the system is excited by stationary white noise, correlation functions between the response signals can also be expressed as a sum of decaying sinusoids [ 8 ].
Each decaying sinusoid has a damped natural frequency and damping ratio that is identical to that of a corresponding structural mode. Consequently, the classical modal parameter techniques using impulse repines functions as input, like Polyreference LSCE, Eigensystem Realization Algorithm (ERA) and Ibrahim Time Domain are also appropriate to extract the modal parameters from response-only data measured under operational conditions.
This technique is also referred to as NExT, standing for Natural Excitation Technique. An interesting remark is that the ERA method applied to correlation functions instead of impulse response functions is basically the same as the Balanced Realization method.
Mathematically, the Polyreference LSCE will decompose the correlation functions as a sum of decaying sinusoids. So,
where λr=e∝r ∆t and Lr is a column vector of Nref constant multipliers which are constant for all response stations for the r-th mode. Note that in conventional modal analysis, these constant multipliers are the modal participation factors.
The combinations of complex exponential and constant multipliers, λrLrT or λr*LrT* are a solution of the following matrix finite difference equation of order t
where [F1]...[Ft] are coefficient matrices with dimension Nref x Nref.
In case the system has Nm physical modes, the order t in equation 3-24 should be theoretically equal to 2Nm/Nref in order to find the 2Nm characteristic poles. In practice, over specification of the model order will be needed.
Since the correlation functions are a linear combination of the characteristic solutions of equation 3-24, λrLrT or λr*LrT*, they are also a solution of that equation. Hence,
Equation 3-25 which uses all response stations simultaneously enables a global least squares estimate of the coefficient matrices [F1]... [Ft]... The overdetermination is also achieved by considering all available or selected time intervals. Once the coefficient matrices are known, equation 3-24 can be reformulated into a generalized eigenvalue problem resulting in Nreft eigenvalues λr, yielding estimates for the system poles ∝r and the corresponding left eigenvectors Lr
T .
The selection of outputs which function as references have to be chosen in such a way that they contain all of the relevant modal information. In fact, the selection of output-reference channels is similar to choosing the input-reference locations in a traditional modal test.
Extraction of mode shapes in a second step and model validation
Contrary to the stochastic subspace methods, the Polyreference LSCE does not yield the mode shapes. So, a second step is needed to extract the mode shapes using the identified modal frequencies and modal damping ratios. For output-only data, it has been shown [ 9 ] that this can be done by fitting the auto-and crosspower spectra between the responses and the responses serving as references :
where Xmn(jω) is the crosspower between m-th response station and the n-th response station serving as a reference.
In case of autopowers (m=n), Armn equals Br
mn. The residue Armn is proportional to the m-th component
of the mode shape yr and the residue Brmn is proportional to the n-th component of the mode shape
yr. Consequently, by fitting the crosspowers between all response stations and one reference station, the complete mode shape can be derived.
The power spectra fitting step offers the advantage that not all responses should be included in the time-domain parameter extraction scheme and that consequently, mode shapes of a large number of response stations can be easily processed by consecutively fitting the spectra. Additionally, it provides a graphical quality check by overlaying the actual test data with the synthesized data. In comparison with modal FRF synthesis, it can be observed in equation 3-26 that two additional terms as function of -jω need to be included for a correct synthesis of the auto-and crosspowers which are assumed to be estimated on the basis of the FFT and segment averaging. If Xmn(jω) would not be calculated with the FFT segment averaging approach, but as the FFT of the correlation function between response m and response n estimated using equation 3-19, the last 2 terms in equation 3-26 can be neglected.
Selection of the modal parameter identification method
This section discusses the criteria for selecting a particular method
LSCE - LSFD This classical Least Squares Complex Exponential method is adapted to work on Auto-correlation and Cross-correlation instead of FRFs or Impulse Response functions.
A subset of the response functions can be selected as references in the computation of the cross power functions. The responses chosen as references should contain all of the relevant modal information, as is required for the input-reference locations in a traditional modal test.
Mode shapes are identified in a secondary process using the Least Squares Frequency Domain procedure. For the theoretical background on this method see section 16.2.2
BR (Balanced Reduction) This is one of the “subspace” techniques which identifies frequency, damping and mode shapes.
A subset of the response functions can be selected as references. These are used in the computation of the cross power functions from the original time domain data.
This method is useful in identifying the most dominant modes occurring under operational conditions.
CVA (Canonical Variate Analysis) This is the second of the “subspace” techniques which identifies frequency damping and mode shapes.
In this case all the response functions must be selected as references which are used in the computation of the cross power functions from the original time domain data. Thus this method requires more computational effort but this algorithm will give “equal importance” to all modes and can identify modes which are not well excited under operational conditions. For the theoretical background on subspace methods see section 16.2.1.
References
[1] LMS CADA-X Running modes manual, 1997.
[2] Otte D., Development and Evaluation of Singular Value Analysis Methodologies for Studying Multivariate Noise and Vibration Problems, PhD K.U.Leuven, 1994.
[3] Otte D., Van de Ponseele P., Leuridan J., Operational Deflection Shapes in Multisource Environments, Proc. 8th International Modal Analysis Conference, p. 413-421, Florida, 1990.
[4] Abdelghani M., Basseville M., Benveniste A., “In-operation Damage Monitoring and Diagnostics of Vibrating Structures, with Applications to Offshore Structures and Rotating Machinery”, Proc. of IMAC XV, Orlando, 1997.
[5] Desai U.B., Debajyoti P., Kirkpatrick R.D., “A realization approach to stochastic model reduction”, Int. J. Control, Vol. 42, No. 4, pp. 821-838, 1985.
[6] Kung S., “A new identification and model reduction algorithm via singular value decomposition”, Proc. 12th Asilomar Conf. Circuits, Systems and Computers, pp. 705-714, Pacific Groves, 1978.
[7] Brown D., Allemang R., Zimmerman R., and Mergeay, M., “Parameter Estimation Techniques for Modal Analysis”, SAE Paper 790221, pp. 19, 1979.
[8] James G.H. III, Carne T.G., and Laufer J.P., “The Natural Excitation Technique (NExT) for Modal Parameter Extraction from Operating Structures, the international Journal of Analytical and Experimental Modal Analysis”, Vol. 10, no 4, pp. 260-277, 1995.
[9] Hermans L., Van der Auweraer H., “On the Use of Auto-and Cross-correlation functions to extract modal parameters from output-only data, Proc. of the 6th International conference on Recent Advances in Structural Dynamics, Work in progress Paper, 1997.
[10] Van der Auweraer H., Wyckaert K., Hendricx W., “From Sound Quality to the Engineering of Solutions for NVH Problems: Case Studies”, Acustica/Acta Acustica, Vol. 83, N° 5, pp. 796-804, 1997.
[11] Wyckaert K., Van der Auweraer H., Hendricx W., “Correlation of Acoustical Modal Analysis with Operating Data for Road Noise Problems”, Proc. 3rd International Congress on Air- and Structure-Borne Sound and Vibration, Montreal (CND), June 13-15, 1994, pp. 931-940, 1994.
[12] Wyckaert K., Hendricx W., “Transmission Path Analysis in View of Active Cancellation of Road Induced Noise in Automotive Vehicles”, 3rd International Congress on Air- and Structure-Borne Sound and Vibration, Montreal (CND), June 13-15, 1994, pp. 1437-1445, 1994.
[13] Van der Auweraer H., Ishaque K., Leuridan J., “Signal Processing and System Identification Techniques for Flutter Test Data Analysis”, Proc. 15th Int. Seminar of Modal Analysis, K.U.Leuven, pp. 517-538, Leuven, 1990.
[14] Van der Auweraer H, Guillaume P., “A Maximum Likelihood Parameter Estimation Technique to Analyse Multiple Input/Multiple Output Flutter Test Data”, AGARD Structures and Materials Panel Specialists’ Meeting on Advanced Aeroservoelastic Testing and Data Analysis, Paper no 12, May, 1995.
[15] Van der Auweraer H., Leuridan J., Pintelon R., Schoukens J., “A Frequency Domain Maximum Likelihood Identification Scheme with application to Flight Flutter Data Analysis”, Proc. 8-th IMAC, pp. 1252-1261, Kissimmee, 1990.
Theory
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of this document requires permission in writing from LMS
Running modes analysis.doc
Category: Analysis and design
Topic: Running modes analysis
Running mode analysis
The aim of modal analysis is to identify a modal model that describes the dynamic behavior of a (mechanical) system. This behavior is identified by means of the transfer function measured between any two degrees of freedom of the system.
The outcome of a modal analysis therefore is the estimated modal parameters of the system, which are the natural frequencies (ωn), damping ratios (δ) and scaled mode shapes (Vik).
One of the most common ways of estimating the modal parameters is based upon the measurement of FRFs between one or more input (reference DOFs) and all response DOFs of interest. These measurements are made under well defined and controlled conditions, where all input and output signals are measured and no unknown forces (external or internal) are acting on the system.
The modal model is (ideally) valid under any circumstances; that is to say, whatever the frequency contents, level or nature of the acting forces. This makes modal analysis a very powerful tool, and the modal model (once identified) can be used in a number of ways, such as trouble shooting, forced response prediction, sensitivity analysis or modification prediction.
For many reasons, a complete modal analysis can be impracticable. It may be that the cost of the test setup is too high, the measurement object (e.g. a prototype) cannot be made available for the period of time required to perform a modal analysis, or it is found to be simply impossible to isolate the object from all the forces acting on the system and excite it artificially.
In this case, it is possible to take measurements of the system while it is operating. A number of output signals can be measured (one at each response DOF), while the system is operating under stationary conditions. This provides a set of measurements (Xi(ω)) as a function of frequency.
The measured quantity Xi(ω) at DOF i can be any number of things: displacement, acceleration, voltage, angular position or acceleration, for example. It is however measured for one particular operating condition, with an unknown level or nature of the acting forces or inputs.
If you are interested in a particular phenomenon at a well defined frequency, it is very often most helpful to see what the output levels are at that frequency for each measurement DOF. So you might, for example, want to know what the harmonic motion of measurement point 13 is at 85.6 Hz, or perhaps its level of acceleration. These values can then be assembled in a vector X, having one element for each of the measurement DOFs.
Animating the system’s wire frame model can lead to a better understanding of these phenomena. This makes it possible to show each motion (or acceleration) level at the corresponding DOF, in a cyclic manner. Because of the external resemblance of the animated representation of the vector quantity X with the mode shape vector V, the vector X is called a running mode, or an operational deflection shape.
These running modes must be interpreted entirely differently from modal modes. Running modes only reflect the cyclic motion of each DOF under specific operational conditions, and at a specific frequency. Using a modal model based on displacement/force frequency response functions H, the displacement running mode X can be described as follows.
where,
i = the DOF counter
wp = the particular angular frequency
Fj(w) = the force input spectrum at DOF j
m = the number of acting forces
The above equation clearly shows that running modes:
• can be identified at any of the measured frequencies ωp, whereas a modal mode has a fixed natural frequency determined by the structural characteristics of the system (mass, size, Young’s modulus, etc.).
• depend on the level and nature of the acting force(s).
• depend on the structural characteristics of the system, through its FRF behavior.
• depend on the frequency contents of each of the acting forces : if F3(ωp) happens to be zero at ωp, it will not contribute to the running mode x(ωp).
• will be dominant at structural resonances (ωp ≅ λk), but also at peaks in the acting force spectra.
Measuring running modes
Ideally, all response spectra for a running mode analysis would be acquired:
• simultaneously
• in a short period of time in which the operating conditions of the test object remain constant
• with signals having a high signal to noise ratio, so that no averaging is required.
In practice, the number of acquisition channels on the measurement system limits the number of response signals which can be measured simultaneously, and so different sets of responses have to be measured at different periods of time. Additionally, if a relatively high level of noise is present on the signals, an averaging procedure may be necessary during the acquisition of the response signals.
Because of varying operation conditions, it is usual to choose a specific response DOF as a reference station and then measure the responses relative to this reference. If the operating conditions then change slightly from one measurement to the next, this will hopefully affect all response signals in the same way and the change will be cancelled out because of the relative nature of the measurements. This procedure also guarantees a fixed phase relationship between the different response signals, using the phase of the reference signal as a reference.
The two measured functions available for running mode analysis are: transmissibility functions and crosspower spectra.
Transmissibility functions
When the response signals are related to the reference by simply dividing each response signal frequency spectrum by the reference frequency spectrum, the result is the transmissibility function (T)
where j is the reference station.
When averaging is involved, transmissibilities can be calculated from measured crosspower and autopower spectra.
The transmissibility function represents the complex ratio (amplitude and phase) between two spectra. A peak in this function may thus be caused either by a peak in the numerator crosspower (i.e. a structural resonance or peak in the excitation spectrum), or a zero (anti-resonance) in the denominator autopower spectrum. As resonance peaks will occur at the same frequencies for cross and autopower spectra, while antiresonances do not, the denominator zeros will cause more peaks in Tij. Resonance peaks tend to cancel each other out.
In the case of Frequency Response Functions (acceleration over force), different estimators (H1, H2, HV) can be used to estimate the transmissibility functions. In practice, the difference between these
different methods of estimating Tij (w) is small when the coherence function is high (near 100 %). When estimating the transmissibility functions from Equation 4-4 above, the coherence function (g) can also be calculated using the following equation.
The coherence function expresses the linear relationship between both response signals of the measured system. This coherence function is expected to be high, since both responses are caused by the same acting forces. In practice, however, it can be low for the same reasons as those affecting the measurement of FRFs, that is to say due to low signal to noise ratio for one or both of the signals, bad signal conditioning, etc.
Another interesting reason why the coherence between two measured signals may be low, can be derived from equation 4-1, when it is substituted in equation 4-3. The linear relationship (and hence the coherence) will vary as a function of the weighting factors Fj(w), this can be because of changing operating conditions during the averaging process for example. High coherence function values in the frequency regions of interest therefore indicate both a high quality of the measurement signals and stationary operating conditions.
Absolutely scaled running mode coefficients for each DOF i can be obtained by multiplying the transmissibility spectra by the RMS value of the reference autopower spectrum.
When the measured autopower spectrum has units of displacement squared, the scaled running mode will be expressed in units of displacement (for example, meters, or inches), if the transmissibility functions themselves are dimensionless. Displacement running modes can be converted to velocity or accelerations by simply multiplying by jw or (jw)2. For a certain value of w (say wo), the following relationships apply.
Crosspower spectra
When it can be assumed that the operating conditions are not going to change while measuring all response signals, then it is possible to measure just crosspower spectra between each response DOF i and a certain reference DOF j
where * denotes the complex conjugate.
Compared to transmissibility functions, crosspower functions have the advantage that peaks clearly indicate high response levels (which may still be caused by a structural resonance or a peak in the acting force spectrum). This technique is especially useful when all the response signals are measured simultaneously by a multi-channel measurement system. In this case, the operating conditions are indeed the same for all response DOFs.
Absolutely scaled running modes can, in this case, be obtained again by means of the autopower spectrum of the reference station j
When displacements were measured, the running mode coefficient will have units of displacement. Equations 4-8 and 4-9 can be used to derive velocity or acceleration values.
Identification and scaling of running modes
Unlike modal modes, a running mode can be identified at any arbitrary frequency of the measured spectra.
Simple peak picking and mode picking methods can be used to extract the sampled values, corresponding to a certain spectral line from the measured spectra. They can then be scaled, and assembled into a vector which can be listed, or animated using a 3D wire frame model of the measured object. For a measurement blocksize of 1024 (512 spectral lines), it is thus possible to identify 512 running modes - or even more when interpolating between the spectral lines.
Note! There is no such quantity as damping defined for a running mode. Similarly other modal parameter concepts such as residues or modal participation factors have no meaning for running mode analysis.
Scaling of running modes
It is possible to scale the identified running modes to values with absolute meaning.
The scaling of running modes coefficients that have been determined using peak picking methods, depends upon the nature of the measurement data (e.g. transmissibilities, or autopowers).
Several ways of scaling running modes can be considered
• If transmissibility spectra were measured, then scaling can be performed using the reference autopower spectrum, as described in equation 4-6.
• If crosspowers were measured, then equation 4-11 can be applied to scale the running modes, again using the reference autopower spectrum.
• It is possible to convert between displacement, velocity and acceleration coefficients using equations 4-7, 4-8 and 4-9 where it is possible to integrate or differentiate once or twice.
• A number of running modes can be scaled manually, by entering a complex scale factor. Each individual mode shape coefficient will be multiplied with this scaling factor.
• Finally, a very general scaling mechanism can be used to scale a number of running modes using a spectrum. Individual running mode coefficients will be multiplied by the (possibly complex) value of the spectrum block, belonging to the spectral line that corresponds to the frequency of that particular mode.
Each one of the above scaling methods may change and influence the units of the scaled running mode. The scaling factor’s units will be incorporated into the mode shape coefficient units, which were initially obtained from the measurement data.
Interpretation of results
A set of functions exists, that are designed to assess the validity of modes. These include the functions of Modal Scale Factor, Modal Assurance Criterion and Modal decomposition.
Modal Scale Factors and Modal Assurance Criterion
Both the Modal Scale Factor and Modal Assurance Criterion are mathematical tools used to compare two vectors of equal length. They can be used to compare running and modal, mode shape information.
The Modal Scale Factor between columns l and j of mode shape k or MSFjlo is the ratio between two vectors. Although this ratio should be independent of the row index i (the response station), a least squares estimate has to be computed for it when more than one output station coefficient is available.
where V jk is the jth column of [Vk].
The corresponding Modal Assurance Criterion expresses the degree of confidence in this calculation, which is obtained using equation 4-13.
If a linear relationship exists between the two complex vectors V jk and V lk, then the MSF is the corresponding proportionality constant between them, and the MAC value will be near to one. If they are linearly independent, the MAC value will be small (near zero), and the MSF not very meaningful.
Modal Scale Factors and Modal Assurance Criterion values can be used to compare an obtained modal model with the accepted running modes. The MAC values for corresponding modeshapes should be near 100 % and the MSF between corresponding vectors should be close to unity. When multiple inputs are used, the MSF can be calculated for each input, while the corresponding MAC will be the same for all of them.
Modal decomposition
When a modal model for the same DOFs is available for a measured object, it is possible to compare modal and running modes and to track down resonance phenomena causing a particular running mode to become predominant. This is termed Modal decomposition. By using a decomposition of each running mode in a linear combination of the modal modes, it becomes clear whether or not a running mode originates primarily from a resonance phenomenon.
The modal modes form what is termed the ‘basis’ group of modes. The running modes are in a separate group that is to be decomposed. The following formula applies.
Where Xi is the ith mode of the group to be decomposed (running modes) Vi is the ith mode of the basis group (modal modes) ai are the scaling coefficients needed to satisfy the above equation.
The scaling coefficients are rescaled relative to the maximum value.
The “Rest” is expressed as a relative error
Note! Take care when interpreting these values since resemblance of the modal and the running mode may purely be coincidental. A running mode at 56 Hz will have no connection with a modal mode at 200 Hz even if they look alike.
Theory
LMS proprietary information: reproduction or distribution
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Modal validation.doc
Category: Analysis and design
Topic: Modal validation
Introduction
A number of means are available to validate the accuracy of modal models of frequencies, damping values, mode shapes and modal participation factors. These tools are -
• Modal Scale Factors between modes and corresponding correlation factors (Modal Assurance Criterion, MAC) described in section 18.2.
• Mode participation described in section 18.3.
• Reciprocity between inputs and outputs, described in section 18.4.
• Generalized modal parameters, described in section 18.5. (Scaling)
• Mode complexity, described in section 18.6.
• Modal Phase Collinearity and Mean Phase Deviation indices, described in section 18.7.
• Comparison of modal models described in section 18.8.
• Mode Indicator Functions, described in section 18.9.
• Summation of FRF data in the Index table, described in section 18.10.
• Synthesis of FRFs described in section 18.11
Some validation procedures allow you to convert the complex mode shape vectors to normalized ones. Normalized mode shapes are obtained from the amplitudes of the complex mode shape coefficients after a rotation over their weighted mean phase angle in the complex plane.
MSF and MAC
Modal Scale Factors and Modal Assurance Criterion
The FRF between input j and output i on a structure can be written in partial fraction expansion form as
The matrix of FRFs is then expressed as
where [RK] represents the matrix of residues. When Maxwell’s reciprocity principle holds for the tested structure this residue matrix is symmetric and can be rewritten as
The ratio between two residue elements on the same row i but on two different columns j and l can be computed as
This ratio MSFjlk is called the Modal Scale Factor between columns l and j of mode k. Although this ratio should be independent of the row index i (the response station), a least squares estimate has to be computed for it when more than one output station residue coefficient is available
where R jk is the jth column of [Rk].
The corresponding Modal Assurance Criterion expresses a degree of confidence for this calculation :
If a linear relationship exists between the two complex vectors R jk and R lk the MSF is the corresponding proportionality constant between them and the MAC value will be near to one. If they are linearly independent, the MAC value will be small (near zero), and the MSF not very meaningful.
In a more general way, the MAC concept can be applied on two arbitrary complex vectors. This is useful in comparing two arbitrary scaled mode shape vectors since similar mode shapes have a high MAC value.
Modal Scale Factors and Modal Assurance Criterion values can be used to compare two modal models obtained from two different modal parameter estimation processes on the same test data for example. When comparing mode shapes, the MAC values for corresponding modes should be near 100 % and the MSF between corresponding residue vectors (mode shapes, scaled by the modal participation factors) should be close to unity. When multiple inputs were used, this MSF can be calculated for each input while the corresponding MAC will be the same for all of them.
A second application for the MAC value is derived from the orthogonality of mode shape vectors when weighted by the mass matrix:
where mk represents the modal mass for mode k.
Even when no exact mass matrix is available, it can usually be assumed to be almost diagonal with more or less equal elements. In this case, the calculation of the MAC value between two different modes is approximately equivalent to checking their orthogonality.
For more specific information on using MSF and MAC for interpreting results in a running mode analysis see section 17.4.
Mode participation
The relative importance of different modes in a certain frequency band can be investigated using the concept of modal participation. For each mode, the sum of all residue values for a specific reference expresses that mode’s contribution to the response. At the same time these sums can be added over all references, to evaluate the importance of each mode.
Note! These evaluations are only meaningful when the same response and reference stations are included for all modes.
When a comparison is made of the residue sums for one mode at all the references, it evaluates the reference point selection for that mode. The reference with the highest residue sum is the best one to excite that mode.
When these sums are added together for all references, the importance of the modes themselves is evaluated. The mode with the highest result is the most important one.
Finally the sums of residues can be added for all modes. Comparison of these results between different inputs allows you to evaluate the selection of reference stations in a global sense for all modes.
Reciprocity between inputs and outputs
Reciprocity is one of the fundamental assumptions of modal analysis theory. This section discusses the reciprocity of FRFs and the reciprocity of the modal model.
Reciprocity of FRFs
Reciprocity of FRFs means that –
measuring the response at DOF i while exciting at DOF j is the same as measuring the response at DOF j while exciting at DOF i
This is expressed mathematically as -
This means that the FRF matrix is symmetric. Note that this property is inherently assumed when performing hammer impact testing to measure FRFs or impulse responses.
Reciprocity in the modal model
Using the modal model for the FRF matrix
it becomes clear that, when this matrix is symmetric, the role of mode shape vectors and modal participation vectors can be switched. Making an abstraction of the absolute scaling of residues, this property can be expressed as follows.
For a reciprocal test structure, the modal participation factors should be proportional to the mode shape coefficients at the input stations.
Using this proportionality between mode shapes and modal participation factors, reciprocity can be checked for each mode when data for more than one input station has been used for the modal parameter estimation.
If reciprocity exists then it is possible to correctly synthesize the transfer function between any pair of response and reference DOFs. This is done by computing a scaling factor between the driving point mode shape and the modal participation factor. This same scaling factor is then used as a reference to derive the necessary participation factor from the available mode shape coefficient.
If reciprocity is not satisfied then really only the transfer functions between the measured response and reference DOFs can be correctly synthesized. If reciprocity is required then it can be imposed on the model, and a number of options are available to calculate the proportionality factor needed to do this.
1 Select one driving point for each mode. The best choice in this case is the one with the largest driving point residue, since it is the one that best excites and is observed from that input DOF.
2 Select one specific driving point for all modes. Other participation factors are disregarded for scaling.
3 Compute a reciprocal scale factor (RSF) using a least squares average of all the driving point data as defined by the following formula for n driving points.
where
vi = the mode shape coefficient
li = the modal participation factor
Generalized modal parameters
This section deals with mode shape scaling and generalized parameters (modal mass).
The residue rij,k between locations i and j for mode k can be written as the product of a scaling factor ak (which is independent of the location) and the modal vector components in both locations. If the structure is proportionally damped, the modal vectors of the structure are real whereas the residues are purely imaginary. As a consequence, the scaling factor ak, is also purely imaginary.
Equation 5-1 can then be rewritten as
where
mk = modal mass of mode k wdk = damped natural frequency of mode k
=ωnk 1-ζk2
zk = the critical damping ratio of mode k wnk = the undamped natural frequency of mode k
At this point, it should be pointed out that equation 5-11 contains N more parameters than equation 5-1, i.e. one more parameter per mode. This is due to the fact that residues are scaled quantities
whereas the modal vectors are determined within a scaling factor only. In equation 5-11 the modal mass values play the role of the scaling constants. It is clear that the value of the modal mass depends on the scaling scheme that was used to obtain the numerical values of the modal vector amplitudes.
When the residues of a proportionally damped structure are known, equations 5-10 and 5-11 can therefore be used to compute the modal mass and the modal vector amplitudes once a scaling method is proposed. Indeed residues, modal vectors and modal mass are related by following equation
To compute the amplitudes of one modal vector and the corresponding modal mass from a set of residues with respect to a given input location j you need one additional equation since the set of equations that can be written for all output locations i in the form of equation 5-12 is undetermined. Therefore N equations in N +1 unknowns are obtained. This last equation will actually determine the scaling of the modal vector.
Note that an eigenvector determines only a direction in the state space and has no absolutely scaled amplitude, while a residue has a magnitude with physical meaning. The scaling of the eigenvectors will determine the modal mass. Modal stiffness is determined as the modal mass multiplied by the natural frequency squared. Modal damping is twice the modal mass multiplied by the natural frequency and the damping ratio.
• Unity mass In this case the mode shapes and participation factors are scaled such that the modal mass (mk) in equation 5-12 is equal to 1.
• Unity stiffness In this case the mode shapes and participation factors are scaled such that the modal stiffness (kk= mkwk
2 ) is scaled to 1.
• Unity modal A In this case the mode shapes and participation factors are scaled such that the scaling factor (ak) is scaled to 1. This scaling factor is independent of the DOFs.
• Unity length In this case the mode shapes and participation factors are scaled such that the squared norm of the vector vik is scaled to unity.
• Unity maximum
In this case the mode shapes and participation factors are scaled such that the vector vik is scaled to 1 where i is the DOF with the largest mode shape amplitude.
• Unity component In this case the mode shapes and participation factors are scaled such that the vector vik is scaled to 1 where i is any DOF selected by the user.
Mode complexity
When a mass is added to a mechanical structure at a certain measurement point then the damped natural frequencies for all modes will shift downwards. This theoretical characteristic forms the basis of a criterion for the evaluation of estimated mode shape vectors.
For each response station, the sensitivity of each natural frequency to a mass increase at that station can be calculated and should be negative. A quantity called the “Mode Overcomplexity Value” (MOV) is defined as the (weighted) percentage of the response stations for which a mass addition indeed decreases the natural frequency for a specific mode,
where
wi is the weighting factor = 1 for unweighted calculations = |vik|2 for weighted calculations
aik = 1 if the k th frequency sensitivity to a mass addition in point i is negative = 0 otherwise
This MOV index should be high (near 100 %) for high quality modes. If this index is low the considered mode shape vector is either computational or wrongly estimated. It is called “overcomplex”, which means that the phase angle of some modal coefficients exceeds a reasonable limit.
However if this MOV is low for all modes for a specific input station (say, below 10%), this might indicate that the excitation force direction was wrongly entered while measuring the FRFs for that input station. This error may be corrected by changing the signs of the modal participation factors for all modes for that particular input.
Modal phase collinearity
For lightly or proportionally damped structures, the estimated mode shapes should be purely normal. This means that the phase angle between two different complex mode shape coefficients of the same mode (i.e. for two different response stations) should be either 0_, 180_ or -180_. An indicator called the ‘‘Modal Phase Collinearity” (MPC) index expresses the linear functional relationship between the real and the imaginary parts of the unscaled mode shape vector.
This index should be high (near 100%) for real normal modes. A low MPC index indicates a rather complex mode, either because of local damping elements in the tested structure or because of an erroneous measurement or analysis procedure.
Mean phase deviation
Another indicator for the complexity of unscaled mode shape vectors is the Mean Phase Deviation (MPD). This index is the statistical variance of the phase angles for each mode shape coefficient from their mean value, and indicates the phase scatter of a mode shape. This MPD value should be low (near 0_) for real normal modes.
Comparison of models
When you have two groups of modes representing the same modal space then you can compare the two groups. The comparison concerns the damped frequencies, the damping values, the modal phase collinearities and the MAC values of the two groups. This is a useful way of comparing sets of modes generated from the same data but using different estimation techniques for example.
Mode indicator functions
Mode Indicator Functions (MIFs) are frequency domain functions that exhibit local minima at the natural frequencies of real normal modes. The number of MIFs that can be computed for a given data set equals the number of input locations that are available. The so-called primary MIF will exhibit a local minimum at each of the structure’s natural frequencies. The secondary MIF will have local minima only in the case of repeated roots. Depending on the number of input locations for which data is available, higher order MIFs can be computed to determine the multiplicity of the repeated root. So a root with a multiplicity of four will cause a minimum in the first, second, third and fourth MIF for example. An example of a MIF is shown below.
Given a structure’s FRF matrix [H], describing its input-output characteristics and a force vector, F, the output or response X can be computed from the following equation
Removing the brackets from the notation, equation 5-14 can be split into real and imaginary parts
For real normal modes, the structural response must lag the excitation forces by 90_. Therefore, when the structure is excited at the correct frequency according to one of these modes (modal tuning) the contribution of the real part of the response vector X to its total length must become minimal. Mathematically this can be formulated in the following minimisation problem
Substituting the expression for the real and imaginary parts of the response 5-15 in this expression yields
The solution of equation 5-17 reduces to finding minima of the frequency functions that are built from eigenvalues. The following eigenvalue problem is formulated at each spectral line under investigation
The square matrices Hrt Hr and Hi
t Hi have as many rows and columns as the number of input or reference locations that were used to create them (i.e. the number of columns of the FRF matrix that were measured). The primary Mode Indicator Function is now constructed from the smallest eigenvalue of expression 5-18 at each spectral line. It exhibits noticeable local minima at the frequencies where real normal modes exist. A second MIF can be constructed using the second smallest eigenvalue of 5-18 for each spectral line. It will contain noticeable local minima if the structure has repeated modes. This can be repeated for all other eigenvalues of equation 5-18. The number of functions that can be constructed is equal to the number of eigenvalues, which is the same as the number of input stations. From these functions, you can then deduce the multiplicity of each of the normal modes.
Summation of FRFs
An important indication of the accuracy of the natural frequency estimates is their coincidence with resonance peaks in the FRF measurements. These resonance peaks can be enhanced by a summation of all available data, either by real or imaginary parts.
Graphically comparing this summation of FRFs with values of the natural frequencies of modes in a display module can be useful. Problems like missing modes, erroneous frequency estimates or shifting resonances because of mass loading by the transducers can easily be detected this way.
Synthesis of FRFs
The FRFs that you have obtained from a modal model can be synthesized in a number of ways. Scaled mode shapes (i.e. mode shapes and modal participation factors) have to be available for at least one input station for which a mode shape coefficient is also available. Using the Maxwell-Betti reciprocity principle between inputs and outputs (section 18.4) it is however possible to calculate the FRF between any two measurement stations.
Correlation and errors
It is also possible to assess correlation and error values relating to the measured and synthesized FRFs.
The correlation is the normalized complex product of the synthesized and measured values.
with
Si = the complex value of the synthesized FRF at spectral line i
Mi = the complex value of the measured FRF at spectral line i
The LS error is the least square difference normalized to the synthesized values.
A listing of FRFs where the correlation is lower than a specified percentage and which exhibit an error higher than a specified percentage provides useful information on the quality if the synthesized FRF.
Theory
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Design.doc
Category: Analysis and design
Topic: Design
Using the modal model for modal design
Correctly scaled mode shapes are an absolute pre-requisite of the correct application of the design procedures described here.
The dynamic behavior of a structure can be fully described and modelled therefore if the poles (lk), and the residues rijk for each mode k and each pair of response and reference DOFs i and j are known. In practise however the modal model is often defined by the poles (frequency and damping values) and the residues for only one (or a few) reference station(s) j. The question now arises as to how this limited modal model can be used for the prediction of responses when forces are acting on a degree of freedom for which residues are not readily available. The residues required between any two degrees of freedom can be derived as follows.
For a linear structure which obeys the Maxwell-Betti reciprocity principle between inputs and outputs, the FRF between two DOFs i and j can be obtained by exciting the structure at DOF j and measuring the response at DOF i, or by exciting at DOF i and measuring the response at j:
In other words, the FRF matrix for a reciprocal structure is symmetric.
Under these circumstances, the residue for each mode k between two response DOFs m and n can be obtained from the ones between each of them and the available set of residues for reference j:
where
rmjk is the known residue between DOFs m and j
rnjk is the known residue between DOFs n and j
nmk is the unknown mode shape coefficient at response DOF m
nnk is the unknown mode shape coefficient at response DOF n
njk is the unknown mode shape coefficient at response DOF j
The required residue is then –
where rjjk is the known driving point residue.
The starting point for modal synthesis applications is the available modal model for the structure to be modified or for each of the substructures to be assembled.
It is important however that some conditions are met.
• In order to be able to scale the included mode shapes correctly, they must include driving point coefficients.
• Mode shape coefficients need only be available for the Degrees Of Freedom which are affected by the structural changes.
The information used to obtain this scaling are: poles, (unscaled) mode shapes and modal participation factors for a number of reference stations. The required scaled mode shape coefficients can be obtained from this information as follows -
For Ni points for which output data are also available (i.e. driving points), a vector of complex modal participation factors Lkj for each mode k can be built:
The corresponding unscaled mode shape coefficients Wik are assembled in a column vector Wk
The residues Rk are defined as the product of mode shapes and modal participation factors :
The scaled mode shapes Vk, used in the theoretical derivation of the previous chapter are related to the unscaled mode shapes Wk via a complex scaling factor ak for each mode :
From the definition of residues these mode shapes are scaled such that
or from equation 7-8
In the special case where only one input is considered, i.e. only one set of residues is available, the scaling factor becomes –
The scaling of equation 7-8 actually converts the generally valid modal model of mode shape vectors W and modal participation factors L to a model of scaled mode shape vectors V, in which the modal participation factors are absorbed via equation 7-10. Obviously some information is lost by removing the scaling factors L from the model, and as a consequence, the resulting model is only valid for reciprocal structures with a symmetric FRF matrix. The calculation of the scaling factor k according to equation 7-10 is in fact the best compromise in a least squares sense to approximate a non-reciprocal modal model by a reduced reciprocal one.
Sensitivity
An experimental modal analysis of a structure results in a dynamic model in terms of modal parameters. The qualitative information contained in this model can be used to identify dynamic
problems for example by animation of the mode shapes. Through physical insight and expertise structural modifications can be proposed to overcome specific dynamic problems.
For structures with complex dynamic behavior, predictions about the effect of physical changes on modal parameters are usually very difficult - if not impossible - to make. When unsatisfactory dynamic behavior is detected or suspected the designer can use trial and error procedures to try out a number of modifications, but there is no guarantee that any of these attempts will yield satisfying results. On the other hand numerical techniques can be employed which use the quantitative results of a modal test to evaluate the effects of structural changes.
These structural changes can be imposed by modifying the physical characteristics of the structure in terms of its inertia, stiffness and damping. A Sensitivity analysis allows you to see how changes in these physical characteristics affect particular modes at various points on the structure. It computes only the sensitivity of the modal model to structural alterations, and does not involve actually applying any changes. A Sensitivity analysis provides you with the means of determining the points where such modifications will have most effect.
Mathematical background to sensitivity analysis
Determining the sensitivity of a DOF to various parameters involves (in a mathematical sense) evaluating the partial derivatives of the eigen properties of a matrix with respect to its individual elements.
Modal parameters are related to the Frequency Response Function as follows.
The partial derivative of this equation to a physical parameter P, can be computed as follows
P can be a mass at one DOF or damping or stiffness between a pair of DOFs.
The dynamic stiffness matrix Q is given by
where M is the mass matrix C is the damping matrix K is the stiffness matrix
c is a subscript denoting that only elements in the matrices that are affected by P will be considered.
Using this equation and the theory of adjoined matrices, equation 7-13 can be rewritten in the form
Using equation 7-12 equation 7-15 becomes
Splitting up equation 7-16 into partial fractions, and identifying the corresponding terms of equation 7-13, gives the sensitivities for frequency (7-17) and mode shape (7-18).
So from equations 7-17 and 7-18, the residues rick and rcjk for each DOF c that is influenced by the structural change are required in order to calculate the sensitivity to that change. Even if not all the residues are available, the Maxwell-Betti reciprocity principle can be used to calculate the required values. The residue rick to be derived for any reference DOF c when the residues for DOFs i and c are available for an arbitrary reference j on condition that the driving point residue rjjk is also available. The driving point residue is also required if the mode shapes are to be correctly scaled.
From the general formula of equation 7-18, it is now possible to calculate the sensitivity value of a mode shape coefficient for DOF i when a structural change is considered for the parameter P, which will affect DOFs a and b. The corresponding scaled mode shape coefficients for each mode in the modal model are required. From the definition of the dynamic stiffness matrix Q, the three specific cases of P being a mass, a linear spring (stiffness) or a viscous damper can be considered.
Mass
This is the case where P is a mass at a specific DOF a. Equations 7-17 and 7-18 are then simplified to
Stiffness
This is the case where P is a linear spring between DOFs a and b. Equations 7-17 and 7-18 are then simplified to
Note that if DOF b is a fixed point (“ground”) then nbm = nbk = 0
Damping
This is the case where P is a viscous damper between DOFs a and b. Equations 7-17 and 7-18 then become
The imaginary parts of equation 7-19, 7-21 and 7-23 are used to compute the sensitivities of the damped natural frequencies. The corresponding real parts express the sensitivities of damping factors or exponential decay rates.
Modification prediction
This section describes the use of a dynamics modification theory to predict the effect of structural modifications on a mechanical structure’s modal parameters. These modifications can take the form of local mass, stiffness and/or damping, FEM-like rod, truss, beam or plate reinforcements. In addition to local modifications, a substructure assembly theory allows you to predict the modal model for a structure that consists of an assembly of substructures.
Modification prediction allows you to evaluate:
• the effect of structural modifications
• the effect of any number and type of connections between any number of substructures (only if installed)
• the dynamics of small scale models, built up from lumped mass-spring-dash pot elements
Such an analysis avoids time consuming experimental trial and error procedures of modifying prototypes or scale models of mechanical structures, measuring and analyzing the dynamic behavior and evaluating the effects of these modifications.
Mathematical background
The starting point for the structural modification and substructure theory is the modal model described in section 15.1.
The first section of this theoretical background deals with the coupling and modification of substructures using flexible coupling and general viscous damping. It continues with the cases of rigid coupling and flexible coupling with proportional damping.
Modal models for the assembly of substructures with flexible coupling and viscous damping
Modal models of substructures Consider two structures, 1 and 2. They obey the following equations of motion in the Laplace domain :
The matrices Mi , Ci and Ki are the mass, damping and stiffness matrices of the structure 1 or 2 corresponding to the subscript i. General viscous damping is allowed. The system matrices are symmetric. The displacement vectors are x1 and x2, and the force vectors f1 and f2 respectively.
The modal parameters for substructure 1 will first be derived in a general way. For substructure 2 the same method can be used but will not be entirely repeated.
The transformation to decouple the equations of motion can be found by adding a set of dummy equations (Duncan’s method) :
The system equations for substructure 1 become :
The matrices A1 and B1 are diagonalized by the transformation matrix V1, the matrix of eigenvectors of substructure 1. The corresponding eigenvalues are stored in the diagonal matrix L1. Due to the addition of equation 7-27 there are twice as many eigenvalues as there are degrees of freedom. They appear in complex conjugate pairs.
The matrices A1 and B1 are diagonalized by post- and pre-multiplication by the eigenvector matrix V1 and its transpose :
The matrix of eigenvectors V1 defines a coordinate transformation from physical co-ordinates y1 to modal coordinates q1 :
Using expressions 7-29 and 7-30 in the equation of motion 7-28 after pre-multiplication with the transpose of V1 and substitution with expression 7-31 one obtains the equations of motion in modal coordinates for substructure 1 :
It can be seen that the equations of motion in modal space are uncoupled.
The same procedure can be repeated for substructure 2, yielding a diagonal eigenvalue matrix Λ2
and an eigenvector matrix V2 . The eigenvector matrix V2 defines a transformation to modal coordinates q2. The equations of motion for substructure 2 in modal space are :
Substructure assembly The system matrices of both substructures can be merged to give a structure composed of two dynamically independent substructures. For this assembled structure one can easily derive the modal parameters since they are the same as those of the two substructures but gathered in one eigenvalue matrix and one eigenvector matrix.
More explicitly this substructuring yields the following system matrices :
which yields as equation :
It can be verified that the matrices of equation 7-35 are diagonalized by the eigenvector matrix V composed as follows :
and that the eigenvalue diagonal matrix is :
This yields a transformation to modal coordinates :
An expression of the type of equation 7-33 using the eigenvector and eigenvalue matrices, yields :
A close look at the matrix of eigenvectors V shows that the two substructures 1 and 2 are still dynamically independent. Indeed, any force at any point of one substructure will not induce any motion at any point of the other substructure.
The two substructures can now be connected with flexible connections modelled as springs and dampers. With the connection matrices Kc and Cc equation 7-35 becomes:
The system matrices of the connected substructures will no longer be diagonalized by the transformation matrix V as the unconnected substructures were. This is due to the introduction of the connection stiffness and/or damping values.
Modification of structures Before decoupling the equations of motion of the connected substructures a number of modifications to each substructure can be added. Let the structural modifications be gathered in the modification matrices –
These changes can be brought together in system matrices for the modifications:
It is clear from the matrices of previous expression that the modifications are not coupling the substructures, they are only modifying each substructure separately.
When the modifications of expression 7-42 are added to the system equation of the connected structure (Eqn. 7-40), one obtains the final equation in physical coordinates
Uncoupling the equations of motion Using the coordinate transformation of the original unconnected substructures (expression 7-36) and premultiplying with Vt, one derives a new set of equations of motion in modal coordinates :
The matrices Am and Bm for the modified structure can again be diagonalized by a general eigenvalue decomposition. When the new eigenvalues and eigenvectors are represented by L’ and W, one has :
Consider then the transformation :
Substituting equation 7-44 and premultiplying with Wt yields :
The transformation matrices V and W can be combined in one matrix Vi as –
which then gives the following transformation equation :
Equation 7-48 is the transformation between physical coordinates and modal coordinates of the connected and modified substructures. With this coordinate transformation the uncoupled equations of motion are :
The natural frequencies and the damping factors can be found as the imaginary resp. the real part of the eigenvalues in Λ . The mode shapes are the columns of the matrix Vi.
Flexible coupling with proportional damping
The theory discussed above relates to flexible coupling with general viscous damping. In this section we consider the case of zero and proportional damping.
Recall the general equation of motion for viscous damping
Zero damping In case of no damping : [C] = [0], next eigenvalue problem is to be solved with eigenvalues: -wr
2 and with eigenvectors : yr.
This system has purely imaginary poles, occurring in complex conjugate pairs.
The modal vectors are real, called normal modes (phase: +/ - 180_).
The equation of motion can be diagonalized, based on the orthogonality of the modal vectors. Transformation to modal coordinates leads to an equation of motion, with diagonal system matrices, being the modal mass and modal stiffness matrices:
Propotional damping In case of proportional damping, the damping system matrix is a linear combination of the mass system matrix and the stiffness system matrix:
This leads to the next equation of motion:
The eigenvalues are related to the complex poles
The complex poles are solved from the real eigenvalues (-ωn) and the damping factors (α, β). When more than two original modes are taken into account (in practical cases, this is always the case), the damping factors can solved in a least squares way from the modal masses, modal stiffnesses and modal damping factors.
Modal synthesis
Only mass and stiffness coupling modifications, ∆M, ∆K and not damping coupling modifications can be applied. The equation of motion of the coupled system are
In modal space:
Where :
The eigenvalues and eigenvectors of this equation, back-transformed from modal to physical space, are the modal parameters of the coupled system.
In case of proportional damping, the complex poles can be solved from the eigenvalues and the proportional damping factors: α and β.
The option to use proportional damping is provided when modes are predicted. It reduces the computation time when dealing with large structures with numerous modifications and mode shapes containing a lot of DOFs. At least two original modes must be used in order to determine a and b.
Rigid coupling
The above theory relates to flexible coupling, but it is also possible to place constraints on DOFs connecting substructures to create rigid coupling between them, or to constrain a single DOF, thus fixing it rigidly to ‘ground’. In this case the restrained DOFs will have zero displacement.
Constraints on the physical degrees of freedom are
Performing a modal transformation:
yields constraints in modal space:
The modal coordinates are split up into dependent modal coordinates qd an independent modal coordinates qi. The constraint matrix [T] is also split up
The choice of the dependent modal coordinates has to be made to lead to a non-singular [Td].
This leads to the new eigenvalue problem:
When the eigenvalues and the eigenvectors with the independent modal coordinates qi are solved, the dependent modal coordinates qd of the eigenvectors can be calculated. In a last step, the mode shapes in physical coordinates are found by the inverse modal transformation.
Constraints can be defined in the same way as other structural modifications.
Implementation of Modification prediction
This section discusses some of the more practical aspects of performing modification prediction. This process allows you to compute the natural frequencies, damping values and scaled mode shapes for a modified mechanical structure which is possibly build up from a number of substructures.
Retrieval of the modal model
The starting point for modal synthesis applications is the available modal model for the structure to be modified or for each of the substructures to be assembled.
All modal parameters (natural frequencies, damping values, and scaled mode shapes) have to be available for the calculation procedure. It is important however that some conditions are met -
1 Driving point coefficients In order to be able to scale the included mode shapes correctly, they must include driving point coefficients. This means that for at least one record of the modal participation factor table, the force input (reference) identifier should match with a record of the mode shape table for the same mode and neither one of them should be equal to zero. Note that this driving point Degree Of Freedom can be different for each of the included modes.
2 Matching DOFs for modes and modifications Mode shape coefficients need only be available for the Degrees Of Freedom which are affected by the structural changes. This means those for which mass, stiffness or damping modifications are to be considered or to which structural elements are to be attached. Moreover, it is perfectly possible to use incomplete mode shape vectors missing some coefficients for irrelevant Degrees Of Freedom.
To obtain correct results, the modal model should include all structural modes to accurately describe the dynamic response for the frequency band of interest. This aspect is especially important when an experimental modal model was obtained from a set of FRFs relative to only one reference station, which happened not to excite some structural modes. This may arise if the reference station was located on or near a nodal point for these modes. In this case the modal model may be well suited to describe the measured FRFs but not the dynamic behavior of the structure as such.
A similar problem occurs for out-of-band effects caused by the presence of modes above or below the frequency band of experimental modal parameter estimation. Some of the frequency domain techniques for estimating mode shape coefficients allow correction terms (residual masses and flexibilities) to compensate for these residual effects. Using these corrections it is often possible to curve-fit the measurement data fairly accurately. Unfortunately, these residual terms cannot be scaled correctly for other reference stations as is done for the mode shape coefficients in the previous sections. They cannot therefore be included in the calculations. For this reason, it is advisable to use a sufficiently large modal model i.e. one with at least one mode below and one mode above the frequency band of interest.
When using a modal model for a limited frequency band it is possible that important structural modifications would generate modes with a natural frequency outside the range of this frequency band. Since the original modal models are not valid at these frequencies, the predicted results will not be very reliable. It is therefore advisable to either include all modes for the frequency band of the resulting modal model or to keep the structural modifications small enough to avoid these problems. In any case you should not attach too much confidence to modes with natural frequencies outside the frequency band of the original modal model.
Included mode shapes are correctly scaled. To obtain correctly scaled mode shapes, the original mode shapes should be scaled in a consistent unit set which respects the consistency of physical quantities: poles, response engineering units per Volt, etc.... A correct calibration of measurement signal transducers and acquisition equipment is required to attach any absolute scaling values to the obtained results.
Definition of modifications to the model
At each of the available Degrees Of Freedom of the modal model you can define one or more local modifications to influence the dynamic behavior of the mechanical structure. The structure can also be modified by the addition of complete substructures for which modal models exist and by the use of constraints providing rigid coupling.
Mass modifications
A point mass can be added to a node on the structure. To add a mass modification you simply have to specify the node and the mass.
Stiffness modifications
A stiffness connection (spring) can be added between any two Degrees Of Freedom of the structure.
To add a stiffness modification you have to specify, the DOFs between which the stiffness is to be applied and the stiffness value.
Note that stiffness (with mass) can also be added to a structure through the addition of a truss or a rod.
Damping modifications
A damping element (dashpot) can be added between any two Degrees Of Freedom of the structure.
To add a stiffness modification you have to specify the DOFs between which the damping is to be applied, and the damping value.
Note that damping can also be added to the structure through the addition of a tuned absorber.
Truss elements
A truss element can be defined as a doubly hinged rod between two points. Forces located at the ends of the truss element (nodal forces) are directed along the axis of the rod. Since trusses are modelled with hinges at the end, they cannot withstand transversal forces. Bending and torsion moments cannot be transmitted from one element to the next. It provides a means of adding stiffness and mass between two points by the addition of a connection for which you know the physical characteristics.
To add a truss element you have to specify; The nodes between which the truss is to be fixed and the physical characteristics of the truss.
A truss element is characterized by its - cross sectional area A - material’s Young’s modulus of elasticity E - mass density d These must all expressed in the active unit system.
A truss element between two nodes is translated into elementary mass and stiffness modifications. The longitudinal stiffness is related to a 6 by 6 stiffness matrix for 6 Degrees Of Freedom (3 for each node). This matrix is obtained by projecting the longitudinal stiffness along each of the 3 coordinate axes.
Rod elements
A rod element can be added between any two separate nodes on the structure. Rods are modelled with hinges at their ends so (modal) forces acting on the ends are directed along the axis of the rod. Bending and torsion moments cannot be transmitted from one element to the next. In effect it provides a means of adding stiffness and mass between two points by the addition of a connection for which you know the mass and the stiffness.
To add a rod element you have to specify the nodes between which the rod is to be fixed and the physical characteristics of the rod. A rod element is characterized by its - longitudinal stiffness Kij - its mass M.
The longitudinal stiffness is related to a 6 by 6 stiffness matrix for 6 Degrees Of Freedom (3 for each node). This matrix is obtained by projecting the longitudinal stiffness along each of the 3 co-ordinate axes. The mass M is divided into two equal parts at both ends of the rod.
Beam elements
A beam element is an element that can transfer translational forces and moments of bending and torsion.
To add a beam element you have to specify the following parameters which are illustrated below :
• the two end nodes (n1, n2)
• the area of its cross section (A)
• the material’s Young’s modulus (E)
• the material’s mass density (m)
• the material’s shear modulus (G)
• the moment of inertia for bending in two planes (Ip, Ib)
• the moment of inertia for torsion (It)
• a reference node to define the orientation of the moments of inertia for bending (r)
The reference node together with the two end nodes defines the so-called reference plane. The moments of inertia for bending are defined in two directions : Ib for bending in the reference plane Ip for bending in a plane perpendicular to the reference plane
The 2 end nodes have six Degrees Of Freedom each: 3 translational and 3 rotations. A beam element can therefore transmit six forces to another beam element: 3 translational forces and 3 moments. For end nodes that are not connected to another beam only the translational forces can be transmitted as for example in the case for a stand-alone beam. In the same way, beams that are positioned on a straight line (colinear beams) will not be subjected to torsion.
Plate membrane elements
A plate membrane element is a two dimensional quadrilateral element capable of transferring both bending forces (perpendicular to the plane of the plate) and membrane forces (in the same plane as the plate).
To add a plate element you have to specify the following parameters which are illustrated below :
• The name of the plate
• The four corner nodes c1, c2, c3, and c4
• The plate thickness (t) expressed in the appropriate user unit
• The number of divisions along the first side, between c1 and c2 (a)
• The number of divisions along the second side, between c2 and c3 (b)
• The connection nodes n1, n2 and n3
• Material properties of the plate i.e. Young’s Modulus (E), Poisson’s ratio (g), mass density (m).
These must all be expressed in the appropriate unit.
When a plate is defined with a and b divisions along its two sides, a mesh of (a x b) rectangles is created as shown in the diagram. As the corner nodes already exist this means that ((a+1).(b+1) - 4) new nodes are generated.
If there are connection nodes defined then the mesh point situated closest to a connection node is replaced by that node.
• the mesh elements should not deviate too much from a rectangular form, i.e. each corner angle should be [900
• the mesh elements should be approximately square, i.e. the ratio of length/width should be [ 1
• the plate should not be too thick, i.e. the ratio of length/thickness should be u 5
Each of the corner nodes of the mesh elements has 6 Degrees Of Freedom - 3 translations and 3 rotations - and so can transmit six forces to another mesh element. This is also the case between elements of different plate membranes, as long as they are connected either at a corner or at a common connection node.
Tuned absorbers
A tuned absorber is a single Degree Of Freedom system consisting of a rigid mass which is connected by a spring and a dashpot to a more complex structure.
m
a
The parameters m, k and c of this SDOF system are designed such that the motion of the coupling point in the direction of this absorber is decreased (damped) as much as possible for a certain frequency, typically at resonance.
The plate so defined should comply with the following conditions (1 )
(1) The calculation of the mass and stiffness matrices of a plate membrane described here is based on the plate theory of Mindlin.
If the motion of the coupling point in the direction of the absorber is designated by xa and the frequency to be damped by f (= w/2p) then the following formulae apply for the equations of motion of m (xr is the relative displacement between the absorber’s mass and the attachment point).
When this equation is solved for xr :
The force acting on the attachment point is –
From equations 7-73 and 7-74
This force can be imagined as being generated by the inertia of an equivalent mass meq, which is rigidly attached to the attachment point :
It can be shown that if no damping is used (c=0) the mass and stiffness of the absorber can be designed such that the vibration of the attachment point is eliminated entirely (xa = 0). This happens if the natural frequency of the absorber equals the forcing frequency w.
The most practical application of a tuned absorber is the reduction of vibration levels at a resonance frequency wn. In this case, the absorber’s own natural frequency for optimal tuning is –
where m is the ratio between the absorber’s mass and the “equivalent” mass of the system at resonance :
An optimal damping ratio for the absorber is then obtained from :
From equations 7-78, 7-79 and 7-80 the physical parameters m, c and k of the attached absorber can be computed if the following values are known.
meq the equivalent mass (see further)
wn the target frequency of tuning, natural frequency of mode to be tuned
m the absorber’s mass to be specified by user
The equivalent mass of the system for a certain mode can be obtained as follows:
where
Vi is the scaled mode shape coefficient of the mode to be tuned at the attachment point
wd is the damped natural frequency of the mode to be tuned.
Constraints
Physical constraints can be defined between separate DOFs or between one DOF and itself.
Defining a constraint between two separate DOFs, applies a rigid coupling between them. Defining a constraint between a DOF and itself effectively fixes it to ‘ground’.
Modification prediction calculation
Once the required modifications have been defined the modification prediction calculation process can be started.
For the simplified case of two substructures which are possibly modified (symbol D) and connected to each other (subscript c), the following procedure is followed to predict the modal model of the resulting structure :
1. Retrieve the modal models for each substructure. Build the diagonal matrices [Λ1] and [Λ2] of poles and the (possibly complex) modal matrices V1 and V2 of scaled mode shapes.
2. Join both modal models into the global matrices [Λ] (equation 7-37) and V (7-36).
3. Define the connecting elements (springs and dash pots) between both substructures. This yields matrices Ac and Bc (equation 7-40).
4. Define the necessary modifications and join them into matrices ∆A and ∆B (equation 7-42).
5. Use the modal matrix V to transform the connection and modification matrices to the modal space.
6. Add the diagonalized matrices in modal space (equation 7-44) to yield the system matrix of the resulting structure.
7. Calculate the modal model via an eigenvalue and eigenvector decomposition of the resulting system matrix. This yields the complex poles (natural frequencies and damping factors) and the mode shapes.
Numerical problems
The eigenvalue problem mentioned above that is to be solved for the modified system, can be subject to numerical problems. These can arise from two sources.
• The presence of unbalanced structural modifications, such as those introducing large amounts of stiffness to simulate a fixation or local heavy dampers.
• A wide range of original natural frequencies. This can occur especially when rigid body modes of free-free systems (virtually at 0Hz) are imported from an FE code and mixed with flexible modes at high frequencies. More specifically in this case it is the ratio of the highest to lowest natural frequency that is the relevant factor.
In practice these numerical problems are manifested in the modified modal model by unrealistic modal parameters or missing modes. While it is impossible to eliminate such problems, they can be reported during the modification prediction calculation.
The criterion used in this respect is the condition number of the system matrix. The system matrix is the one whose eigenvalues and eigenvectors yield the modal parameters. If this condition number exceeds a certain (critical) value this is reported to the user. The critical value used has been established by empirical tests and is by default set to 1e+8.
Units of scaling
In order to obtain correct modification prediction results, it is absolutely necessary to maintain a correct scaling of the original modal model using a consistent unit set.
The scaled mode shapes of the original structure have a physical dimension related to the measurement data from which they were extracted by modal parameter estimation techniques. Since this modal model is a valid description for the relation between input forces and response
displacements, the applied modifications should be defined in a unit set which is consistent for these quantities. The same rule applies to the interpretation of the resulting modal model.
Erroneous results are bound to occur when the original mode shape vectors are not scaled correctly. This might arise because of the incorrect definition of the reference point for the data (wrong driving point residue), not using the correct transducer sensitivity or calibration factors for the experimental FRFs (force as well as response transducers), or the use of an inconsistent unit set during the modal test or analysis phase. These errors may cause an entirely wrong transformation of the applied physical modifications to the modal space and a small mass modification for example may grow out of proportion because of this bad scaling.
Example of the application of a beam element
The following example will illustrate the procedure. Suppose the dynamic behavior of an isotropic plate is to be influenced by a rib fixed to the plate as shown below.
main plate
I cross section beam
elem
elem
elem
elem
1
2
3
4 nodes
12
34
5
12
34
5
The procedure becomes :
1 Discretization of the rib into 4 beam elements, interconnected at nodes corresponding to measurement points of the experimental analysis.
2 Definition or calculation of the following physical parameters.
A = cross section of the beam
It = moment of inertia for torsion
Ib = moment of inertia for bending in the reference plane, defined by the nodes n1, n2 and r
E = Young's modulus of elasticity
G = shear modulus
L = length of the beam
1, 2, 3 = orientation of the local beam reference system in the global system. This information is derived from the position of the three nodes n1, n2 and r as shown in Figure 7-1.
m = material's mass density
From the geometrical properties of the beam the user can calculate the cross sectional area and the different moments of inertia. Tables listing characteristics of various types can be found.
3 Construction of the element matrices for each beam element. An element stiffness (full) and mass (diagonal) matrix can be built from the relations between the 6 forces and 6 Degrees Of Freedom at each end node (U1, V1, W1, f1, q1, and y1 for node 1, u2, v2, w2, f2, q2 and y2 for node 2)
4 Assembly of the element matrices as shown below
5 Perform a static condensation (see below) of the rotational DOFs.
6 Add the condensed matrices to the system matrices and continue the calculation procedure as for other (lumped) modifications.
Remarks :
• The element matrices of a beam model must be assembled before condensation and addition to the system matrices to allow moments to be transmitted between different elements.
• It is important to keep in mind that the basic assumption in beam-bending analysis is that a plane section originally normal to the neutral axis remains plane during deformation. This assumption is true provided that the ratio of beam length to beam height is greater than 2. Furthermore, shear effects do not contribute to the elements of the stiffness matrix.
• Care should be taken with the input of moments of inertia. In the example stated above the distance between the axis of the plate and the axis of the beam must be taken into account.
Static condensation
Static condensation in a dynamic analysis is based upon the assumption that the mass at some Degrees Of Freedom can be neglected without a significant loss of accuracy on the dynamic model in the frequency range of interest. More explicitly, for the beam elements in the application of interest consider the rotational Degrees Of Freedom to be without mass. The assembled mass and stiffness matrices of the entire beam can then be partitioned as follows,
where
T refers to the translational DOFs
R refers to the rotational DOFs.
The modal parameters describing the dynamic behavior of this structure are then obtained by solving following eigenvalue problem,
From the bottom half of equation 7-83 a relation between the translational and the rotational DOFs is derived.
which can be solved to express the rotational DOFs in terms of the translational ones,
Introduction of equation 7-85 into equation 7-83 yields :
with
The matrices [KT] and [MTT] of equation 7-86 are used to dynamically model the beam structure. The model will only be valid in the frequency range where the mass effects of the rotational DOFs are negligible. Mass effects only contribute significantly to the dynamic behavior around and above those resonances where they are capable of storing a considerable amount of kinetic energy.
Note that [KT] as expressed in equation 7-87 can only be computed if [KRR] is non-singular. The stiffness matrix is singular if rigid body motion is possible. The rigid body mode of a beam along its longitudinal axis is not naturally eliminated by constraining its three translational DOFs so causing in general a first order singularity. With such configurations it will not be possible to store torsional deformation energy in the beam therefore the corresponding off-diagonal elements of the assembled stiffness matrix can be neglected and the diagonal elements made relatively small. In this way the matrix becomes invertible and the predicted dynamic behavior will reflect the inability to store torsional deformation energy in the beam. This operation will, however, not be necessary when the beam is two or three dimensional, as in such cases, rigid body motion through rotation around one of the axes is no longer possible.
Forced response
Experimental modal analysis results in a dynamic model described by the modal parameters, damped natural frequency, exponential decay rate and scaled mode shapes (residues). These modal parameters provide valuable insight into the dynamic behavior of a structure. Problem areas can be identified by animating the mode shapes and the relative importance of the mode shapes can be assessed by comparing their amplitudes.
In most cases however the designer is less interested in dynamic characteristics themselves than in knowing how the structure is going to behave under normal operating conditions. The important points to determine are -
• what will happen under dynamic loading conditions ?
• which of the natural frequencies will dominate the response ?
• which points will exhibit large deformations ?
• how will the structure will deform at particular frequencies ?
The natural frequency of the modes of vibration which seem to be the most important parameters in the modal model may well not dominate the response if conditions are such that they are not excited.
The Forced response functions enable you to answer these questions by determining the response of the modal model to known force spectra.
Mathematical background for forced response
The structure’s modal model forms the input for the computation of its dynamic response and is the starting point for the forced response analysis.
The equations of motion of a linear, time invariant mechanical structure are expressed in the frequency domain as follows:
where X(w) is the response spectra vector (N0 by 1) [H(w)] is the Frequency Response Function matrix (N0 by N0) F(w) is the applied force spectra vector (N0 by 1).
These quantities are complex-valued functions of the frequency variable w and are valid for every value of w for which these functions are known.
When the response at one specific degree of freedom (DOF), say i, is needed the above equations become:
This means that the response at DOF i can be written as a linear combination of the applied forces, each weighted by the corresponding FRF between input DOF j and output DOF i. These frequency dependent weighting factors describe the dynamic flexibility between two degrees of freedom i and j of a mechanical structure.
When the modal model for that structure is available, e.g. from modal test data or finite element calculations, the FRF can be modelled as given by
Using equation 7-89, it is now possible to predict the dynamic response at DOF i when the structure is subjected to a number of simultaneous loads at DOFs j for which scaled mode shape coefficients (residues) are also available in the modal model.
Even if not all the residues are available, the Maxwell-Betti reciprocity principle can be used to calculate the required values. Equation 7-4 allows the residue rick to be derived for any reference DOF c when the residues for DOFs i and c are available for an arbitrary reference j on condition that the driving point residue rjjk is also available. The driving point residue is also required if the mode shapes are to be correctly scaled.
Equation 7-91 represents the response at all DOFs to all forces with a contribution from all modes. The contribution of each mode is given by –
The response for each DOF then taking into account the contribution of each mode is then given by
Theory
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Geometry concepts.doc
Category: Analysis and design
Topic: Geometry concepts
The geometry of a test structure
A geometrical representation of a test structure is necessary for the display and animation of mode shapes, and for the implementation of design modifications. This chapter discusses the basics regarding the geometry definition of a model for a test structure.
The most important part of the model is the nodes. These define the points where measurements will be taken on the structure, and the points where the mode shape deformation are calculated. It is common practice to defined connections or edges between specific nodes to form a wire frame model of the structure. In addition surfaces can be defined, that aid in the visual representation of the structure.
Figure 8-1 A wire frame model of a structure
Note that the definition of nodes and meshes for acoustic measurements are described in the “Acoustic” documentation.
Nodes
A node is defined by its location and its orientation.
Location
The location of a node in the 3D space is defined by a set of 3 real numbers known as the coordinates. Coordinates are always defined relative to a reference coordinate system.
The reference coordinates are normally shown along with the model in the the display window. The origin of the global coordinate system is the origin of the 3D space that contains the test structure and the global symmetry of the structure should be considered when defining this.
The reference coordinate system can be either Cartesian, cylindrical or spherical.
x
y
Z
Q
F
r
x
y
Z
x
yz
Right handed Cartesian
Cylindrical Spherical
x
y
Z
r
Q
z
Figure 8-2 Coordinate systems
So as an example, the same node defined in each of the coordinate systems would appear as follows.
Cartesian X Y Z 1 1 1
Cylindrical r Θ Z √2 45° 1
Spherical r Θ Φ √3 45° 55°
Orientation
Nodal orientation is defined using a Cartesian coordinate system. In many applications the orientation of the node defines the measurement directions.
x
z
y
Figure 8-3 Nodal coordinate system
The origin of the nodal coordinate system coincides with the node’s location. If the principal axes of the nodal coordinate system are not coincident with the measurement directions, in either a positive or a negative sense, then the difference must be defined with Euler angles.
Euler angles
Three Euler angles are used to define the orientation of a one coordinate system, relative to a reference coordinate system with the same origin.
Θxy
The first angle, Θxy (Euler XY) is a rotation about the Zr axis of the reference system. (Positive from the Zr axis to Yr axis). This generates a first intermediate system indicated by a single quote ‘ on the axis labels.
Θ
The second angle, Θxz (Euler XZ) is a rotation about the y’ axis of the first intermediate system. (Positive from the x’ axis to z’ axis). This generates a second intermediate system indicated by two quotes “ on the axis labels.
Θ
Finally the third angle, Θyz (Euler YZ) is a rotation about the x” axis of the second intermediate system. (Positive from the y” axis to z” axis). This last orientation generates the desired new coordinate system orientation.
Degrees Of Freedom (DOFs)
The Degrees Of Freedom of a node represent the directions in which a node is free to move. Each node therefore has a maximum of 7 Degrees Of Freedom; 3 translational, 3 rotational and a scalar DOF(Sc).
Y
Z
X
RX RY
RZ scalar
Figure 8-4 Degrees of freedom
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