Spectrum Analysis Module 6. Training Manual January 30, 2001 Inventory #001447 6-2 Module 6 Spectrum Analysis A. Define a spectrum analysis and its purpose.

Spectrum Analysis

Module 6

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Module 6

Spectrum Analysis

A. Define a spectrum analysis and its purpose.

B. Understand the underlying concepts and terminology.

C. Learn how to do a response spectrum analysis.

D. Guidelines for spectrum analysis.

E. Random Vibration Analysis

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Spectrum Analysis

A. Definition & Purpose

What is spectrum analysis?

• A technique to compute a structure’s response to transient excitations that contain many frequencies.

• Excitations could be from sources such as earthquakes, aircraft noise/ flight history, missile launches.

• A spectrum is a representation of a load’s time history in the frequency domain.

• This is also referred to as response spectrum.

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Acceleration vs. time Acceleration spectrum (G vs. Hz)

Spectrum Analysis

… Definition & Purpose

El Centro Earthquake ( 1940 )

A structure subject to the El Centro earthquake can be analyzed using either a Transient analysis or spectrum analysis.

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• Spectrum analysis follows a modal analysis.

• Computes the maximum response of the structure to a given spectrum at each natural frequency. This maximum response is computed as scale factor*mode shape.

• These maximum responses are then combined to give a total response of the structure.

Spectrum Analysis


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• An alternative is to perform a transient analysis.

• Transient analysis is generally more time consuming, especially when a number of components and load conditions have to be considered.

• However, transient analysis is more accurate.

• In spectrum analysis the focus is to get the maximum response quickly, and some information is lost (phase).

Spectrum Analysis


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• Used in the design of:

– Nuclear power plants (buildings and components)

– Airborne Electronic equipment (aircraft / missile)

– Spacecraft components

– Aircraft components

– Any structure or component that is subjected to seismic or other erratic loads

– Building frames and bridges

Spectrum Analysis


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** Covered in this seminar

Spectrum Analysis


• ANSYS allows four types of spectrum analysis:

• Single-point response spectrum**

– A single response spectrum excites all specified points in the model.

• Multi-point response spectrum **

– Different response spectra excite different points in the model.

• Dynamic design analysis method (DDAM)

– A specific type of spectrum defined by the U.S. Naval Research Laboratory to evaluate shock resistance of shipboard equipment.

• Power Spectral Density (PSD)**

– A probabilistic approach used in random vibration analysis.

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Spectrum Analysis

B. Terminology & Concepts

Topics covered:

• Definition of a spectrum

• How a response spectrum is used to calculate a structure’s response to the excitation

– Participation factor

– Mode coefficient

– Mode combination

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Spectrum Analysis - Terminology & Concepts

Definition of spectrum

What is a spectrum?

• A curve representing the maximum response of an idealized system to an excitation. The response may be acceleration, velocity, displacement, or force.

• Consider, for example, four single-DOF spring-mass systems mounted on a shaker table. Their frequencies are f1, f2, f3, and f4, with f1 < f2 < f3 < f4.

1 2 3 4

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• If the shaker table is excited at frequency f1 and the displacement response of the four systems is recorded, it will look as shown on the right.

• Now add a second excitation of frequency f3 and record the displacement response. Systems 1 and 3 will each reach their peak response.

• If now a general excitation containing several frequencies is applied and only the peak responses are recorded, we might get the curve shown. This curve is the spectrum, specifically a response spectrum.

f

u

f

u

f

u


… Definition of spectrum

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• Thus a response spectrum is an envelope of the maximum responses of a number of single DOF systems to a given excitation.

• Input to a spectrum analysis consists of a response spectrum curve and a direction of excitation.


… Definition of spectrum

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• Spectrum analysis follows a modal analysis in which natural frequencies and mode shapes have been computed.

• In doing a spectrum analysis you will encounter three new terms:

– Participation factor

– Mode coefficient

– Mode combination

• We will define these three terms along with the general outline of how a spectrum analysis is done.


Approach

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• For each mode of the structure, a participation factor i is calculated in the excitation direction.

• The participation factor is a function of the mode shape and the direction of excitation.

• This is a measure of how much a mode will contribute to the deflections (and hence stresses) in the direction of excitation.


… Approach - Participation factor

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• For example, consider the cantilever beam shown.

• If an excitation is applied in Y direction, mode 1 will have the highest PF and mode 2 a lower PF. Mode 3 will have zero PF.

• If the excitation is in the X direction, then modes 1 and 2 will have zero PF, whereas mode 3 will have a high PF.

m o d e 3

m o d e 22

m o d e 1

Y

X


… Approach - Participation factor

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• The mode coefficient is the “scale factor” used to multiply the mode shapes to get the maximum response.

• The mode coefficient Ai for each mode is Ai = Sii *

Si is the response spectrum value at frequency i

i is the participation factor for mode i

• The maximum modal response is then computed as

{U}i max = Ai {i

*A different formula is used for acceleration, velocity and force spectra; see the ANSYS Theory Manual.


… Approach - Mode coefficient

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• Once the maximum response at each mode is known for a given response spectrum, these need to be combined in some way to get the total response.

• The simplest combination is to add all the maximum modal responses. However, it is highly unlikely that all the maximum modal responses will occur at the same time.

• Several standard combination methods are published in the literature. Usually each industry’s regulating authority recommends or enforces a technique most suitable for that industry.


… Approach - Mode combination

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• Six different combination methods are available in the ANSYS program:

– Complete Quadratic Combination (CQC) method

– Grouping Method (GRP)

– Double Sum method (DSUM)

– Square Root of the Sum of the Squares (SRSS) method

– Naval Research Laboratory (NRL) sum method (DDAM)

– Power Spectral Density method


… Approach - Mode combination

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• We will discuss the procedure for a single-point response spectrum analysis.

• In the following discussion, we will use the term “response spectrum” to mean single-point response spectrum.

• To learn about multi-point response spectrum and DDAM, please refer to the ANSYS Structural Analysis Guide.

Spectrum Analysis

… Terminology & Concepts

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Training ManualC. Procedure

Five main steps:

• Build the model

• Obtain the modal solution

• Switch to spectrum analysis type

• Define the response spectrum

• Solve and review results

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Response Spectrum Procedure

… Obtain the Modal Solution

• Mode extraction:

– Only valid methods are Block Lanczos, subspace, or reduced.

– Block Lanczos strongly recommended

– Extract enough modes to cover the spectrum’s frequency content.

– Expand all modes. Only expanded modes can be used for the spectrum solution.

• Loads and BC’s: For a base excitation, be sure to constrain the appropriate DOFs.

• Files: The .mode file contains the eigenvectors and is needed for the spectrum solution.

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Switch to Spectrum Analysis Type

Build the model

Obtain the modal solution

Switch to spectrum analysis type

• Exit and re-enter Solution

• New analysis: Spectrum

• Analysis options: Discussed next

• Damping: Discussed next

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… Switch to Spectrum Analysis Type

Analysis options

• Type of spectrum: Single point

• Number of modes: If 0 or blank, all expanded modes are used for solution.

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Damping

• Available forms of damping are:

– Beta (stiffness) damping

– Constant damping ratio. Can be material dependent but only if specified as a material property* in the modal step.

– Frequency dependent damping ratio (modal damping)

• Some form of damping must be specified for the CQC mode combination method.

*Material property DAMP in this case is damping ratio, not beta damping.

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Define the Response Spectrum

Build the model



Define the response spectrum

• Settings: type of spectrum and excitation direction

• Table of spectral value versus frequency

• Mode combination method

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… Define the Response Spectrum

Settings:

• Type of spectrum

– Seismic or force (not PSD)

– Seismic spectra - automatically applied at the base

– Force spectrum - manually applied at desired nodes as a force

• Excitation direction (global Cartesian)

– Specified by a unit vector for seismic spectra: 1,0,0 means X; 0,1,0 means Y; 0,0,1 means Z.

– Implied by FX, FY, or FZ labels for force spectrum.

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Spectral value vs frequency table

• First define frequency table. Up to 20 points are allowed.

• Then define corresponding spectral values.

– Specify damping ratio only for multiple spectral curves.

– For a force spectrum, the spectral values can be scaled by the applied force value.

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Mode combination method

• Determines how the individual modal responses are combined.

• Five methods are available:

– CQC (Complete Quadratic Combination)

– GRP (Grouping)

– DSUM (Double Sum)

– SRSS (Square Root of Sum of Squares)

– NRLSUM (Naval Research Laboratory Sum)

Which method you choose typically depends on company or government standards being followed.



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Mode combinations (continued)

• The significance threshold allows you to include only significant modes in the mode combination. It is the ratio of the mode coefficient of a mode to the maximum mode coefficient. Use a zero value to include all modes.

• Type of output allows calculation of different response quantities: displacement, velocity, or acceleration.



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Solve and Review Results

Build the model




Solve and review results

• Solve the current load step.

• Mode combination calculations are written as POST1 commands to the .mcom file.

• Review results: discussed next.

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… Solve and Review Results

Review results:

• Enter POST1 (general postprocessor).

• Perform mode combinations

– Commands to do this are written to .mcom file during solution.

– Read the file jobname.mcom using Utility Menu > File > Read Input from...

• Review deformed shape.

• Plot and list stresses and strains.

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Training ManualResponse Spectrum Analysis Procedure

Build the model




Solve and review results

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Training ManualD. Spectrum Analysis Guidelines

• Modal analysis

– Make sure you extract and expand enough modes in the modal analysis to cover the frequency range of interest.

– For example, if the spectrum extends from 1 to 1000 Hz, a rule of thumb is to extract and expand modes up to 1500 Hz.

– Block Lanczos extraction technique recommended

– Use Lagrange multiplier (accurate) method if large numbers of constraint equations are present.

– If you have material dependent damping ratio, this should be specified in the modal analysis.

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Training Manual… Spectrum Analysis Guidelines

• Spectrum analysis

– Remember that no results file is written in a spectrum analysis. Instead the instructions for mode combination are written to jobname.mcom.

– Most combination methods involve squaring operations causing the component stresses to lose their signs. Hence deriving equivalent or principal stresses from these unsigned components will be non-conservative and incorrect.

– If equivalent or principal stresses and strains are of interest then you need to issue the command SUMTYPE,PRIN ( General Postprocessor > Load Case > Stress Option …) before reading in jobname.mcom. This causes direct operation on derived quantities leading to more conservative results.

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• Spectrum analysis

– During the spectrum analysis the effective mass for each mode as well as the sum of all the effective mass is printed out.

– For a lumped mass system the sum of the effective masses should approach the total mass of the structure as the number of modes used in the spectrum analysis is increased.

– The total effective mass is an indicator of whether enough modes are included in the spectrum analysis.

***** RESPONSE SPECTRUM CALCULATION SUMMARYCUMULATIVE

MODE FREQUENCY SV PARTIC.FACTOR MODE COEF. M.C. RATIO EFFECTIVE MASS MASS FRACTION

1 2.37E-04 10 -1.18E-20 -5.34E-14 0 1.40E-40 3.07E-382 474 21.099 6.22E-02 1.48E-07 1 3.87E-03 0.851323 1182 10 1.14E-15 2.07E-22 0 1.30E-30 0.851324 1182 10 3.42E-16 6.20E-23 0 1.17E-31 0.851325 1881 10 -5.08E-16 -3.64E-23 0 2.58E-31 0.851326 2361 10 3.52E-11 1.60E-18 0 1.24E-21 0.851327 2361 10 -2.60E-02 -1.18E-09 0.007981 6.76E-04 18 3044 10 -4.39E-13 -1.20E-20 0 1.93E-25 19 3044 10 1.27E-12 3.48E-20 0 1.62E-24 110 4011 10 5.08E-12 8.00E-20 0 2.58E-23 1

SUM OF EFFECTIVE MASSES 4.55E-03

… Spectrum Analysis Guidelines

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Training ManualE. Workshop - Response Spectrum Analysis

• In this workshop, you will determine the response of a workbench table to a response spectrum excitation.

• See your Dynamics Workshop supplement for details. (Response Spectrum Workshop - Workbench Table, Page W-40. ).

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Training ManualF. Random Vibration Analysis

Topics covered:

• Definition and purpose

• Overview of ANSYS capabilities

• ANSYS procedure

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Random Vibration Analysis

Definition and Purpose

What is random vibration analysis?

– A spectrum analysis technique based on probability and statistics.

– Meant for loads such as acceleration loads in a rocket launch that produce different time histories during every launch .

Reference: Random vibrations in mechanical systems by Crandall & Mark

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• Transient analysis is not an option since the time history is not deterministic.

• Instead, using statistics the sample time histories are converted to Power Spectral Density function (PSD), a statistical representation of the load time history.


… Definition and Purpose

Image from “Random Vibrations Theory and Practice” by Wirsching, Paez and Ortiz.

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What is a PSD?

• A PSD records the mean square value of the excitation and response as a function of frequency.

– The area under a PSD curve is the variance of the response (square of the standard deviation).

– The units used in PSD is mean square/Hz (e.g. an acceleration PSD will have units of G2/Hz).

– The quantity represented by PSD may be displacement, velocity, acceleration, force, or pressure.



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• Typical applications include

– Aircraft electronic packaging

– Airframe parts under atmospheric loading

– Blast deflectors

– Laser guidance systems

– Stable optical platform for telescopes

– Seismic loading of large structures



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Input:

– The structure’s natural frequencies and mode shapes

– The PSD curve (explained next)

Output:

– 1 displacements and stresses that can be used for fatigue life prediction.

– Response PSD curves that show the frequency content of any output quantity ( RPSD ).

– Undocumented (FPAS and RISK ) life prediction capability.



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• Loading:

– Base or nodal excitation

– Single-point excitation

• e.g. Single PSD excitation applied to all ground nodes

– Multi-point (i.e., multi-spectra) excitation

• Uncorrelated

• Partially correlated

• Fully correlated

– Partial correlation in terms of spatial coordinates

– Partial correlation in terms of a traveling wave


Overview of ANSYS Capabilities

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• Solution:

– Relative or absolute 1 output

– Option for calculating 1 forces/stresses etc.

– Solution for complete structure i.e., results can be contoured.

– Output in form of 1 displacements, velocities or accelerations


… Overview of ANSYS Capabilities

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• Postprocessing:

– 1results can be contoured like any other analysis.

– Response PSD can be computed for any result quantity ( e.g. stress or nodal force at a node of an element) or cross response spectra can be computed between any two quantities (RPSD).

• This enables the user to look at the frequency content of output.

– Covariance between any two quantities can be computed (CVAR).

– Undocumented commands RISK and FPAS allow user to compute equivalent stress / predict life.


… Overview of ANSYS Capabilities

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Training ManualRandom Vibrations Procedure

Six main steps:

• Build the model

• Obtain the modal solution

• Switch to spectrum analysis type

• Define and apply the PSD excitation

• Solve

• Review results

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Random Vibrations

Build the Model

Model

• Same considerations as a modal analysis.

• Linear elements and materials only. Nonlinearities are ignored.

• Remember density! Also, if material-dependent damping is present, it must be defined in this step.

• See also Modeling Considerations in Module 1.

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Random Vibrations

Obtain the Modal Solution

Build the model


• Same procedure as a normal modal analysis.

• A few differences, discussed next.

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Random Vibrations


• Mode extraction:

– Only valid methods are Block Lanczos, subspace, or reduced.

– Block Lanczos strongly recommended

– Extract enough modes to cover the spectrum’s frequency content.

– Expand all modes. Only expanded modes can be used for the spectrum solution.

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• Loads and BC’s:

– For a base excitation, be sure to constrain the appropriate DOFs.

– For a pressure PSD, apply the pressures on desired surfaces in this step.

• Files: The .mode file contains the eigenvectors and is needed for the spectrum solution.

Random Vibrations


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Random Vibrations Switch to Spectrum Analysis Type

Build the model



• Exit and re-enter Solution

• New analysis: Spectrum

• Analysis options: Discussed next

• Damping: Discussed next

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Random Vibrations


Analysis options

• Type of spectrum: PSD

• Number of modes: If 0 or blank, all expanded modes are used for solution.

• Element calculations: can be ON only if they were ON in the modal step.

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Damping

• All four forms are available.

– Alpha (mass) damping

– Beta (stiffness) damping

– Constant damping ratio

– Frequency dependent damping ratio (modal damping)

• If no damping is specified, ANSYS uses a 1% constant damping ratio as default.

Random Vibrations


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Random Vibrations

Define and Apply the PSD Excitation

Build the model



Define and apply the PSD excitation

• Specify PSD settings

• Define PSD versus frequency table

• Apply excitation at desired nodes

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Random Vibrations

… Define and Apply the PSD Excitation

PSD settings

• Spectrum type (units)

– Acceleration (normal units or g2/Hz)

– Velocity

– Displacement

– Force

– Pressure

• Table number defaults to 1. Used for multiple PSD curves.

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PSD versus frequency table

• Specify table number (usually 1).

• Then enter frequency and PSD value pairs.

Random Vibrations


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PSD versus frequency table (continued)

• Graph the PSD table to verify the input.

Random Vibrations


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Apply the PSD

• Procedure depends on the type of PSD.

• Acceleration, velocity, or displacement PSD:

– These are base excitations and can be applied only at previously constrained nodes.

– Apply as a constraint in UX, UY, or UZ (excitation direction) with a value of 1.0.

Pick nodes...

Random Vibrations


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Random Vibrations


Apply the PSD (cont'd.)

• Force PSD

– Nodal excitation

– Apply as a force in FX, FY, or FZ (excitation direction) with a value of 1.0 (or desired scale factor).

• Pressure PSD

– Requires pressure to be applied in the modal step.

– Use the load vector (calculated during modal solution) to apply the pressure PSD excitation.

– Set value to 1.0 or desired scale factor.

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Random Vibrations

Solve

Build the model




• Solve

• Activate PSD mode combination method

• Specify items to be calculated*

• Calculate participation factors*

• Initiate PSD solution*

*Discussed next

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Random Vibrations

… Solve

Items to be calculated:

• Default is to calculate the displacement solution (including stresses and strains) relative to base excitation.

• Velocity and acceleration solutions are also available, relative to base or absolute.

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Random Vibrations

… Solve

Calculate participation factors:

• Must be done for each PSD table defined.

• Specify base or nodal excitation.

Initiate PSD solution:

• Results are written to the .rst file.

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Random Vibrations

Review Results

Build the model




Solve

Review results

• Plot and list 1 quantities (POST1)

• Generate a response PSD (POST26)

• Calculate covariance between two quantities (POST26)

• Life prediction

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Random Vibrations- Review Results

Review 1-Sigma Stresses

• Random vibration results are 1 quantities: 1 displacements, 1 stresses, etc.

• All quantities assume a Gaussian (normal) distribution with zero mean.

• For example, a maximum displacement of Umax = 0.15 indicates a 68% probability (1) that Umax will be 0.15 or less. It also indicates:

– a 95% probability (2) that Umax will be 0.15x2 = 0.3 or less.

– a 98% probability (3) that Umax will be 0.15x3 = 0.45 or less.

123

Gaussian (normal)

Distribution

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To review 1 displacements & stresses:

• Enter POST1 (General Postproc).

• Read results from load step 3, which is where 1 results are stored on the results file.

– Note: 1 velocities and 1 accelerations, if requested, are stored in load steps 4 and 5, respectively always.

• Then plot and list the desired quantities.


… Review 1-Sigma Stresses

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1 results are typically used for:

• Fatigue calculations

– In PSD analyses, the average frequency of excitation (number of cycles/second) is given by 1 velocity / 1 displacement.

– Using normal distribution the stress level is at 1 68% of the time, at 2 27% of the time (95-68), and at 3 3% of the time (98-95).

– Knowing the above two quantities, fatigue life can be predicted using usual S-N diagram procedures.



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Response PSD

• Gives engineers an idea of how a response quantity (stress, for example) varies with frequency.

• Results file contains 1 values, which is the square root of the area under the PSD curve.

• POST26, the time-history postprocessor, is used to calculate response PSD.


Response PSD

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To calculate response PSD

1. Enter POST26 and first store the frequency vector.

– You can use 1 to 10 additional data points on either side of a natural frequency for a smoother frequency curve. Default is 5.

– Variable 1 is automatically assigned to the frequency vector.


… Response PSD

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2. Identify results quantities for which response PSD is to be calculated.

– TimeHist Postpro > Define Variables...

– Can be any nodal or element result item.

Choose category, then pick node...


… Response PSD

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3. Calculate and plot the response PSD.

– TimeHist Postpro > Calc Resp PSD...

– TimeHist Postpro > Graph Variables…


… Response PSD

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Covariance

• Covariance represents the correlation between two quantities.

• Can be calculated between any two response quantities; for example, stress at two different points in the model.

• POST26, the time-history postprocessor, is used to calculate covariance.


Covariance

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To calculate covariance:

1. Reset or exit and re-enter POST26.

2. Identify the two response quantities for which covariance is to be calculated.


… Covariance

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3. Calculate and retrieve the covariance.

– TimeHist Postpro > Calc Covariance...

– Use *GET to retrieve the covariance: *GET,COVAR,VARI,#,EXTREM,CVAR -or- Utility Menu > Parameters > Get Scalar Data...


… Covariance

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• In PSD analysis only component stresses are valid (the mode combinations work only on component stresses).

• Since the component stresses are 1statistical quantities, equivalent stress and principal stresses cannot be computed in the usual way.

• The RISK command can calculate equivalent stress by Monte Carlo simulation. RISK can also be used for life prediction.


RISK & equivalent stress

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• Sample uses of RISK command

– CASE 1

INPUT: Location of interest and Design strength (or mean and standard deviation of strength)

OUTPUT: Safety margin, Probability of failure, PDF, CDF

– CASE 2

INPUT: Location of interest and Acceptable probability of failure

OUTPUT: Required design strength, PDF, CDF


… RISK & equivalent stress

PDF - probability density function

CDF- Cumulative probability density function

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• RISK analysis sample output

RISK COMMAND WAS ISSUED FOR NODE 50 OF ELEMENT 28

SIX ESOL COMMANDS WITH VARIABLE NUMBERS 3 THRU 8 ARE CREATED

THE COVARIANCE MATRIX OF CARTESIAN STRESS RESPONSES IS COMPUTED BELOW:

SX SY SZ SXY SXZ SYZ

0.1665E+07 0.2156E+07 0.0000E+00 -0.5234E+05 0.0000E+00 0.0000E+00

0.2156E+07 0.2791E+07 0.0000E+00 -0.6776E+05 0.0000E+00 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

-0.5234E+05 -0.6776E+05 0.0000E+00 1645. 0.0000E+00 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

A DETERMINISTIC DESIGN STRENGTH = 4687.5 WAS SPECIFIED

STATISTICS BASED ON THE SAFETY MARGIN OF VON MISES STRESS WILL BE COMPUTED

MEAN OF THE SAFETY MARGIN = 3452.3

STANDARD DEVIATION = 900.41

COEFFICIENT OF SKEWNESS = -1.0480

COEFFICIENT OF KURTOSIS = 3.9813

COMPUTED SAFETY INDEX = 3.8342

COMPUTED PROBABILITY OF FAILURE = 0.20000E-02

COEFFICIENT OF VARIATION FOR COMPUTED PROBABILITY OF FAILURE = 0.70640E-01

BASED ON 95% CONFIDENCE THE PROBABILITY OF FAILURE IS LESS THAN = 0.22324E-02


… RISK & equivalent stress

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• When will it fail?

• What is the probability of failure?

t

a

X(t)

t


First Passage Failure

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• The FPAS command can be used to estimate first passage failure

• FPAS Works with displacement and component stresses

• Sample uses of FPAS command:

– CASE 1

INPUT: location , max. allowable value, desired probability of failure

OUTPUT: Statistical average frequency, life in seconds

– CASE 2

INPUT: Location, maximum allowable value, time to failure

OUTPUT: Statistical average frequency, probability of failure



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• Typical output for First Passage Failure

– fpas,ttfa,4,3,1,6.0,0.001

– COMPUTED STATISTICAL AVERAGE FREQUENCY IS 285.36

– THE EXPECTED NUMBER OF POSITIVE CROSSING OF THRESHOLD VALUE 1.0906

– PER UNIT TIME IS 0.69170E-06

– BASED ON FAILURE PROBABILITY OF 0.10000E-02 THE TIME TO FAILURE IS 1445.7

– TIME TO FAILURE IS 0.144571E+04



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Build the model




Solve

Review results

Random Vibrations

Procedure

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Training ManualG. Workshop – Random Vibration (PSD)

• In this workshop, you will determine the displacements and stresses in a model airplane wing due to an acceleration PSD.

• See your Dynamics Workshop supplement for details (Random Vibration Workshop - Model Airplane Wing , Page W-43 ).

Spectrum Analysis Module 6. Training Manual January 30, 2001 Inventory #001447 6-2 Module 6 Spectrum Analysis A. Define a spectrum analysis and its purpose.

Documents

response spectrum analysis

types of spectrum analysis

single response spectrum

transient analysis

given spectrum

spectrum analysismodule

modal analysis

multipoint response