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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
… Definition & Purpose
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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
… Definition & Purpose
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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
… Definition & Purpose
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** Covered in this seminar
Spectrum Analysis
… Definition & Purpose
• 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
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f
u
Spectrum Analysis - Terminology & Concepts
… 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.
Spectrum Analysis - Terminology & Concepts
… 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.
Spectrum Analysis - Terminology & Concepts
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.
Spectrum Analysis - Terminology & Concepts
… 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
Spectrum Analysis - Terminology & Concepts
… 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.
Spectrum Analysis - Terminology & Concepts
… 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.
Spectrum Analysis - Terminology & Concepts
… 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
Spectrum Analysis - Terminology & Concepts
… 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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Response Spectrum Procedure
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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Response Spectrum Procedure
… 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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Response Spectrum Procedure
… Switch to Spectrum Analysis Type
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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Response Spectrum Procedure
Define the Response Spectrum
Build the model
Obtain the modal solution
Switch to spectrum analysis type
Define the response spectrum
• Settings: type of spectrum and excitation direction
• Table of spectral value versus frequency
• Mode combination method
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Response Spectrum Procedure
… 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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Response Spectrum Procedure
… Define the Response Spectrum
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.
Response Spectrum Procedure
… Define the Response Spectrum
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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.
Response Spectrum Procedure
… Define the Response Spectrum
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Response Spectrum Procedure
Solve and Review Results
Build the model
Obtain the modal solution
Switch to spectrum analysis type
Define the response spectrum
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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Response Spectrum Procedure
… 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
Obtain the modal solution
Switch to spectrum analysis type
Define the response spectrum
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.
Random Vibration Analysis
… 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.
Random Vibration Analysis
… Definition and Purpose
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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
Random Vibration Analysis
… Definition and Purpose
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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.
Random Vibration Analysis
… Definition and Purpose
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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
Random Vibration Analysis
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
Random Vibration Analysis
… 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.
Random Vibration Analysis
… 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
Obtain the modal solution
• Same procedure as a normal modal analysis.
• A few differences, discussed next.
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Random Vibrations
… 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.
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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
… Obtain the Modal Solution
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Random Vibrations 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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Random Vibrations
… Switch to Spectrum Analysis Type
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
… Switch to Spectrum Analysis Type
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Random Vibrations
Define and Apply the PSD Excitation
Build the model
Obtain the modal solution
Switch to spectrum analysis type
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
… Define and Apply the PSD Excitation
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PSD versus frequency table (continued)
• Graph the PSD table to verify the input.
Random Vibrations
… Define and Apply the PSD Excitation
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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
… Define and Apply the PSD Excitation
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Random Vibrations
… Define and Apply the PSD Excitation
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
Obtain the modal solution
Switch to spectrum analysis type
Define and apply the PSD excitation
• 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
Obtain the modal solution
Switch to spectrum analysis type
Define and apply the PSD excitation
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.
Random Vibrations- Review Results
… Review 1-Sigma Stresses
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Random Vibrations- Review Results
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.
Random Vibrations- Review Results
Review 1-Sigma Stresses
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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.
Random Vibrations- Review Results
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.
Random Vibrations- Review Results
… 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...
Random Vibrations- Review Results
… Response PSD
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3. Calculate and plot the response PSD.
– TimeHist Postpro > Calc Resp PSD...
– TimeHist Postpro > Graph Variables…
Random Vibrations- Review Results
… 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.
Random Vibrations- Review Results
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.
Random Vibrations- Review Results
… 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...
Random Vibrations- Review Results
… 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.
Random Vibrations- Review Results
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
Random Vibrations- Review Results
… 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
Random Vibrations- Review Results
… RISK & equivalent stress
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• When will it fail?
• What is the probability of failure?
t
a
X(t)
t
Random Vibrations- Review Results
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
Random Vibrations- Review Results
First Passage 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
Random Vibrations- Review Results
First Passage Failure
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Build the model
Obtain the modal solution
Switch to spectrum analysis type
Define and apply the PSD excitation
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 ).