Advanced FRF Based Determination of Structural Inertia Properties_whitepaper

Advanced FRF based determination of structural inertia properties

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Introduction

The challenge for today’s engineers is not only to develop a better product, but to bring it to the market quickly. In order to meet this challenge it is important to establish a close relationship between simulation data and experimental results, and rigid body characteristics are keys to speeding up the modeling and simulation process. Because it is not always possible to accurately calculate the rigid body properties of a complex structure, being able to estimate them from measured data and use them to validate and update a finite element model provides the relationship we need.This application note outlines the characteristics of rigid body properties and their importance. It describes how they can be estimated from modal measurement data and demonstrates how these estimates compare with calculated FE data.

Why inertia properties are needed

The determination of inertia properties; the center of gravity and the moments of inertia of a structure, is important if engineers are to meet the challenges of an effective design and development cycle. For a simple structure, these inertia properties can be calculated mathematically. For a complex structure however, constructed from several components composed of different materials, such as a car body, a truck, or a washing machine, a finite element model is required. To generate these models in itself, is not a trivial task.

Rigid body properties which describe the inertia properties of a structure can be used to complete or update a finite element model and serve as input for a simulation model. Rigid body modes which represent the mass behavior of a structure are an essential component of a realistic modal model over the low frequency range.

An understanding of inertia properties is essential when considering the assembly of a complex structure from its constituent components, such as the motor, gear box, drum, damper, of a washing machine. In order to predict the kinematic and dynamic behavior of the structure, a finite element model for each of these components would be required. The creation of all these finite element models is expensive, so in this case it is both effective and practical to carry out a modal test on the components to generate the rigid body properties and rigid body modes that can then be used for a multi-body dynamic calculation.

Rigid body properties and rigid body modes are also applicable in the context of coupling a small rigid component to a complex finite element model, such as coupling an exhaust to a car body for instance. The rigid body modes of the small component though rarely available from a classical experimental modal test are vital for the accuracy of a modal coupling calculation (modal synthesized method).

A further example of the important application of rigid body characteristics relates to vehicle engine and body mount systems. In order to optimize the powertrain mount, the vehicle is divided into three main subsystems; the powertrain, the body and the suspension. The powertrain itself is modeled as a rigid body system with external excitations, and its mounting system is modeled by its dynamic stiffness curves over the frequency range of interest. The body and suspension are represented by their measured FRF model. The model for the whole vehicle is then constructed by combining the three subsystem models using the substructuring method. In considering the powertrain as a rigid body system, inertia properties will be required to build up the equations of motion.


Experimental methods to determine inertia properties

Pendulum test

Traditionally, inertia properties were obtained experimentally by a somewhat laborious pendulum test that involved suspending the structure freely in different directions. This is not only difficult in the case of a complex structure such as a car body, it is also very time consuming.

FRF based method

An effective and accurate alternative to the pendulum test is a modal test. Inertia characteristics are determined by calculating rigid body properties based on the measured frequency response functions and mass line. The modal test can be either impact hammer or shaker excitation on the structure in free-free conditions. In most cases 6 excitation locations in single direction and between 8 and 12 response locations in 3 directions are sufficient. The test involves no special equipment and only limited measurement effort. The new LMS Test.Lab Rigid Body Properties workbook uses FRF measurements to determine inertia properties.

How to determine rigid body properties

Two essential requirements for this process are an accurate geometrical wire-frame model and the weight of the structure in kilograms. The structure then needs to be suspended in free-free conditions and frequency response functions measured using either hammer or shaker excitation. Mass line methods are used to compensate for the possible poor quality of the measured FRF’s at low frequencies, and to reduce the least square error. A frequency band which best represents the mass line is selected. Rigid body properties can then be calculated with a least square solution over this selected frequency band. Based on the defined geometry the resulting rigid body properties (center of gravity and moment of inertia), rigid body modes are synthesized using user defined frequency and damping values. The results can be validated either by observing the animation of rigid body modes or by comparing the rigid body properties to those of a finite element model.

Theoretical background

A mass-spring model of single degree of freedom is shown in Figure 1.

ground

ck

m

f(t)

x(t)

Figure 1: Mass-spring model of single degree of freedom

The equation of motion in the frequency domain is expressed by:

€

(−mω 2 + jωc + k) x(ω) = F (ω) in displacement format and

€

(m+ cjω

− kω 2 ) ˙ ̇ x (ω) = F (ω) in acceleration format.

It can also be expressed as inertance (acceleration/force) by:

€

˙ ̇ x (ω)F (ω)

= 1

(m+ cjω

− kω 2 ) .

The frequency response functions (compliance, mobility and inertance) are shown in Figure 1. At low frequencies, below the resonance frequency, i.e.

€

(ω << ωn) , the inertance is approximately equal to the asymptote shown in the lower graph in Figure 2. This asymptote is called

the stiffness line and is approximately equal to

€

≈ −ω 2

k .

At higher frequencies, above the resonance frequency, i.e.

€

(ω >> ωn), the inertance is approximately equal to the asymptote shown in the lower graph in Figure 1. This asymptote is called the mass line and is approximately

equal to

€

≈ 1m

.

For a single degree of freedom model, i.e. a rigid body structure, the resonance frequency depends on the mounting of the structure and the mass line that lies above this resonance frequency. At the resonance frequency, the structure vibrates as rigid body.


Mass line methods

At frequencies above the rigid body modes, the inertance level becomes constant and is related to the mass value. Determination of the mass line is required in order to calculate the rigid body properties. In practice however, a structure does not behave as a purely rigid body; flexible modes are also present during any classical modal test. In addition, the manner in which the structure is suspended influences the rigid body frequencies. It is of fundamental importance that the structure is suspended in free-free conditions to ensure that the rigid body motion has negligible effect on the flexible modes.

Figure 3: Frequency response function used for different mass line method

Figure 2: Different formats of a frequency response functions for a single degree of freedom system. Compliance (upper), mobility (middle) and inertance (lower)

Thus the identification of the mass line may not be easy, but three methods, that depend on the characteristics of the FRFs, can be used to distinguish it.

Unchanged FRF’sWhen rigid body modes and flexible modes are sufficiently separated, for an inertance FRF, the curve between the rigid body modes and the first flexible mode appears as a horizontal line. The frequency band this represents can be used for the mass line method to extract rigid body properties. The solid line (red curve) in Figure 3 shows rigid body modes that are quite separate from the flexible )modes.

Corrected FRF’sWhen insufficient bandwidth exists between the rigid body and the flexible modes, the measured FRF’s must be corrected by subtracting the flexible modes. Firstly the flexible modes are estimated from measured FRF’s using modal parameter estimators such as LMS PolyMAX for pole estimation and least square frequency domain (LSFD) for mode shape estimation. Then FRF’s are synthesized from these modes (without lower residual) in the selected frequency band and subtracted from the measured FRF. This results in corrected FRFs in which the mass line of the rigid body modes is no longer affected by the flexible modes. Such a corrected FRF is shown by the dashed line (green curve) in Figure 3.

Lower residualIn the situation where accurate measured FRF’s are not available in the frequency range between the rigid body and the flexible modes, lower residual terms can be used. Lower residual terms are estimated from the measured FRF’s using modal parameter estimators such as LMS PolyMAX for pole estimation and least square frequency domain (LSFD) solver for mode shape, upper and lower residual term estimation. Lower residual terms represent the influence of the modes below the frequency band used for modal parameter estimation. In the case where the frequency band used for modal analysis extends only to the first flexible modes, they therefore represent the influence of the rigid body modes.


More explanation on the mass line methods can be found in reference [4].

Synthesis of rigid body modes

Rigid body modes are necessary to have a complete modal model for either simulation or predictive purposes. The preferred solution is of course to measure them, but obtaining good quality data at the low frequencies, where rigid body modes occur, is not easy. The main reason for this is due to the difficulties in exciting these low frequencies resulting in noisy FRF’s in this range. In addition, the frequency resolution used to identify flexible modes is often insufficient to accurately estimate rigid body modes. An alternative is to synthesize mode shapes based on geometrical coordinates and the inertia properties and then add the rigid body mode shapes to the modal model.

A free-free structure has six rigid body modes, and for simplification, 3 translational and 3 rotational modes around the principal axes of inertia are synthesized. The residues depend on the total mass of the structure, the principal axes of inertia and the principal inertia values. Each mode has a mode shape vector and participation factor, a resonance frequency and damping value. The residue is expressed as the product of mode shape and participation factor. The equation of motion of a single degree of freedom system can be expressed as:

€

m˙ ̇ x + c˙ x + kx = F

Transforming into Laplace domain (s) this yields:

€

1 m

s2 + csm

+ km

= XF

€

1 ms2 + 2ξω0s + ω0

2 = XF where

€

ω0 = k m = 2π f0 with the natural frequency and

€

ξ = c2m k m .

The denominator has two poles:

€

s1 = −ξω0 + jω0 1−ξ 2

s2 = −ξω0 − jω0 1−ξ 2.

Partial fraction expansion yields:

€

A

s − −ξω0 + jω0 1−ξ 2( ) + B

s − −ξω0 − jω0 1−ξ 2( ) = 1 ms2 + 2ξω0s + ω0

2

€

A = −B = − j

2mω0 1−ξ 2

€

XF

= Ri

s − λ+ Ri

*

s − λ*

The displacement residue of a single degree of freedom system thus has an amplitude:

€

12mω0 1−ξ 2 and a phase of -90°.

Rotational rigid body modes are derived similarly, except that the rotational inertia should be translated to a linear mass. With the angular displacement and the moment, the angular equation of motion of a free-free inertia is

€

˙ ̇ γ M

= 1I

and

€

γM

= 1Iω 2

€

X rx

F .rF

= 1Iω 2

€

XF

= rF ⋅ rx

Iω 2

Replacing

€

1m

by

€

rF rx

I in the residue for a translational

mode gives us the residue for a rotational mode.

€

Ri = rF rx

2Iω0 1−ξ 2 .

Figure 4 show the distances

€

rF and

€

rX used in the moment applied around the Z axis (perpendicular to the X and Y axes) and a reference point in the X direction.

F

€

rF

€

rX

Y

X

€

rY

Figure 4: Distance rX and rF used for rotational residues


The frequency and critical damping ratio can be arbitrarily chosen. If the damping of the rigid body mode is chosen

to be small so that

€

1−ξ 2 is approximately 1, the residues become:

€

Rtranslational = 12mω0

= 12m⋅ 2π f0

Rrotational = rF rX

2Iω0

= r0r2I ⋅ 2π f0

Choosing a rigid body frequency that is too small can give rise to numerical problems, but if the frequency is too high, it will influence the calculated flexible modes, after a coupling calculation for example. Acceptable values for the rigid body frequencies lie between 1/10 and 1/100 below the first flexible mode and above 1/1000 of the highest flexible mode.

The calculated inertia properties can be controlled by synthesizing FRF’s (inertance) for each rigid body mode on a significant DOF as response and the DOF used to determine the participation factor as reference. In the upper frequency range, the FRF should have a value given by:

€

˙ ̇ X F

= 1m for a translational inertia and

€

˙ ̇ X F

= rF rX

I for a rotational inertia.

More detailed explanations on this can be found in reference [2]

How to guarantee the accuracy of inertia properties

The quality of experimental rigid body properties depends on the accuracy of the measurements. The points given below provide some practical considerations that can be implemented to realise the best possible results.

Suspend the structure in free-free boundary conditions. Use a very soft suspension that allows rigid body movements in all directions; translational and rotational. Measure the location of the measurement points as accurately as possible.

••

•

Calibrate the transducer as accurately as possible. Aim for high measurement accuracy in the lower frequency range. The frequency resolution must be very high, so use a small frequency step and a long measurement time. In general, at least 5 frequency lines in the selected frequency band used for rigid body estimation will be sufficient. Perform the modal FRF test using a single reference excitation. Use a hammer for excitation (impact test) given the number of excitation points. Use a frequency range of excitation that includes the first flexible mode. Because the frequency range of interest is low, use an appropriate hammer and hammer tip (soft rubber). Typically 6 different excitation DOFs are required (at least one time on each direction X, Y, and Z) Between 8 and 12 response points in 3 directions are required. Validate the consistency of the measured FRF’s, such as correct calibration, correct direction and sign of accelerometer, before starting the rigid body calculation. Choose the most appropriate mass line method based on the available FRF’s. Synthesize the rigid body modes with a low frequency (close to zero) and damping for further use in modeling and simulation.

LMS Test.Lab Rigid body properties calculator

Implementation

To use the LMS Test.Lab Rigid body properties calculator requires a set of FRF’s between excitation DOFs and response DOFs. A frequency band, which represents the mass line, is selected between the rigid body modes and flexible modes (as shown in Figure 5). These mass line values are used for calculating rigid body properties according to the selected mass line method. All the spectral lines of the frequency band are used as input for a global (least square) solution. More detailed explanations of the kinematic and dynamic equations used are available in reference [4].

••

•

•

•

•

•

•

•

•

•

Figure 5: Frequency band of the FRF’s represent the mass line which is between the rigid body modes and the first flexible/deformation mode.


Worksheet flow

The LMS Test.Lab rigid body properties calculator uses a frequency response function (FRF) to extract essential inertia properties such as the center of gravity, moments of inertia and principal axes of inertia. As with other members of Test.Lab product family, the software guides you through the different steps required to calculate rigid body properties and synthesize rigid body modes.

In the first “Data Selection” worksheet a selection of FRF’s is made for the analysis. Individual FRF’s and a sum of FRF’s can be visualized and inspected in a dedicated 2D display. The three mass line methods described above are available in addition to tools such as, rigid body correlation and ODS animation that provide immediate and firm data validation feedback before performing the analysis.

In a second “Calculate” worksheet, only one click is required to extract the rigid body properties and synthesize rigid body modes with respect to the defined geometry. The frequency and damping values are defined by the user. The center of gravity and the principal direction can also be visualised on the geometry model.

Assessing the performance of the rigid body properties calculator on an academic case

Validation

A “representative” academic model of an engine is used to validate the performance of the calculator. This structure was selected for a number of reasons; the analytical inertia properties are easy to calculate, the structure itself however is not trivially simple and it exhibits low frequency flexible modes which allow evaluation of the different mass line methods. Figure 6 shows the geometry of the structure.

FRF’s were measured by impact testing. In total there were 8 excitation points represented in Figure 7 and 10 response points measured in 3 directions represented in Figure 8. In both cases the arrows represent the direction of the references and responses.

Figure 6: Geometry of the “representative” academic structure of an engine.

Figure 7: Excitation directions on the “representative” academic structure of the engine.

Figure 8: Response directions on the “representative” academic structure of the engine.


Figure 10: Excitation directions of impact tests on the frame structure.

Figure 9 shows the sum of FRF’s (red, solid) and some of the individual FRF’s (dotted). A comparison of the results of the three mass line methods is shown in Table I. Compared to the FE results, all 3 methods give an excellent prediction, the lower residual method being the closest.

Assessing the performance of the LMS Test.Lab rigid body properties calculator on real life applications

Frame

In this investigation a more realistic frame structure is analysed. The inertia derived from an ANSYS finite element model with 2272 elements and 2347 nodes was used as the reference for the experimental rigid body analysis. An appropriate test setup was configured to measure 408 FRF’s with 12 excitation DOFs and 32 response DOFs as shown in Figure 10 and Figure 11.

The frame structure is excited at the four corner nodes in 3 directions.

Figure 11: Response directions of impact testes on the frame structure.

Figure 9: Sum of FRF’s (red, solid) and some typical FRF’s (dotted) of the “representative” academic structure of the engine

Table I : Comparison of the results of mass line methods and FE model results.


Figure 12 shows the sum of all the FRF’s (in red) and other significant FRF’s. The rigid body modes are sufficiently separate from the first flexible modes, and the 25 Hz to 35 Hz frequency band is used for the experimental rigid body properties calculation.

As the results in Table II show, all the mass line methods are close to the finite element result (ANSYS). More details on this test case can be found in reference [4].This test case also serves to illustrate the influence of the number of references and responses used in the modal test. An assessment of these factors can be made using the concept of rigid body correlation.

Rigid body correlation measures the correlation between the operational deflection shape and the rigid body modes in a selected band. While it is obvious that a minimum number of references and responses are required for any modal test, the quality of the measured FRF’s also influences the quality of the estimated rigid body properties. When there are more than 6 references and 24 responses (8 response locations in 3 directions), conditions defined as the minimum requirement, the quality of the FRF’s can be evaluated using rigid body correlation. Rigid body correlation for input should be higher than 90% and that for the output should be higher than 98%. The column with “6 ref 34 resp” (6 references and 34 responses, highlighted in yellow) in the Table III and the column with “12 ref 27 resp” (12 refenences and 27 responses, highlighted in yellow) in the Table IV show that the optimum values of the rigid body correlation are indeed dependent on number of references and responses. The column with “6 ref 27 resp” (6 referenced and 27 responses, highlighted in green) in the Table V shows the combination of the two.

Figure 12: Sum of FRF’s (red) and some significant FRF’s (dotted).

Table II: Comparison of the results of the three mass line methods and the ANSYS results on the frame structure.

Table III: Influence of reducing the number of reference on the frame structure.

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Exhaust

A second real life test case concerns an exhaust system. The purpose of this exercise was to include rigid body modes in a modal coupling calculation in which the exhaust is coupled to a car body. In this particular case a detailed finite element model of the exhaust system was available which allowed comparison between the experimental results and the finite element results. It should be recalled that even though rigid body modes are of primary importance concerning the accuracy of a modal coupling calculation (modal synthesis method), these rigid body modes are not usually available from a classical experimental modal test due to their low resonance frequency.

The exhaust system was suspended in free-free conditions, and a modal test carried out using a total of 10 excitation points and 10 response points in 3 directions. The excitation points were chosen such that each point excited at least 1 rigid body mode. All directions were excited with a hammer and the excitations points were distributed evenly over the exhaust system. To avoid mass loading effects generated by modification of the setup, the locations of the accelerometers were not moved between measurements, only the measure directions were changed.

Figure 13 shows the sum of all the FRF’s (in red) and one particular FRF. This clearly shows that a mass line is not easy to distinguish. In this situation the best policy is to use the Lower residual method. The poor quality of the mass line is due to the suspension of the test setup; it was not soft enough and so influences the rigid body movement of the structure on all the directions.It is clearly shown in Table VI that the lower residual method gives better results than the other two methods, especially for the moment of inertia.

A study made on the influence of number of reference and response on the rigid body calculations for this case study is described in reference [6]. This reference also highlights the importance of obtaining good quality measurements, assessing influences such as the modal parameter estimation method, the frequency band for modal analysis, the frequency band for rigid body properties calculation, the precision of the geometry wireframe and others.

Table IV: Influence of reducing the number of response on the frame structure.

Table V: Influence of reducing both number of reference and response on the frame structure.

Figure 13: Sum of all the FRFs (red) and a typical FRF (blue) of the exhaust.

Table VI: Comparison of the FE model results and the three mass line meth-ods on the exhaust.

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Rigid body modes of the exhaust

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Conclusion

This application note explains importance of and the concepts behind rigid body properties. It proves that based on classical FRF measurement, with only limited measurement effort, it is possible to perform a fast and accurate analysis which uses a least squares solution to determine inertia properties together with rigid body modes for simulation purpose such as modal-based substructuring. Several mass line methods are presented to compensate for the effects of the first flexible mode and eliminate the effects of poor quality FRF’s. A number of integrated validation tools, rigid body correlation, animation and coloring feedback, help the user to validate data used for rigid body properties calculation. The LMS Test.Lab Rigid body calculator workbook is suitable for a wide range of applications such as determination of the center of gravity of

a truck cabspoiler of a formula onethe drum of a washing machinea satellitea tractora tanka helicopterand others

Applications can also be extended to a simulation purpose such as multi-body dynamic simulation, modal coupling and modal-based modification.

••••••••

References

[1] J. Toivola, O. Nuutila. “Comparison of three methods for determining rigid body inertia properties from frequency response functions”. Tampere University of Technology, P.O. Box 589, SF-33101 Tampere, Finland.

[2] LMS international. “How to add rigid body modes to an existing modal model in CADA-X”. LMS international consulting reports, Leuven, Belgium, 1991.

[3] H. Okuzumi. “Identification of the rigid body characteristics of a powerplant by using experimental obtained transfer functions”. Central engineering laboratories, Nissan motor Co., Ltd., June 1991.

[4] W. Leurs, L. Gielen, M. Burghmans, B. Dierckx. “Calculation of rigid body properties from FRF data: pratical implementation and test cases”. Proc. of the 15th International Modal Analysis Conference, Tokyo, Japan, 1997.

[5] LMS international. LMS Test.Lab Modal Analysis Workbook – Modal Rigid Body Manual, Rev 7B, LMS international, Leuven, Belgium, 2006.

[6] LMS international. “Measuring rigid body properties of an exhaust system: A comparative analysis”. LMS international, Leuven, Belgium, 1998.

[7] R. Madjlesi, A. Khajepour, F. Ismail, M. Wybenga, B. Rice, J. Mihalic. “Optimization of engine mounting systems using experimental FRF vehicle model”. Int. J. Vehicle Noise and Vibration, Vol. 1, Nos. _ 2005.

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Advanced FRF Based Determination of Structural Inertia Properties_whitepaper

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