Response Spectrum & Random Vibration Analyses
Acoustics ACTx R160 1
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March 13, 2015
Response Spectrum
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March 13, 2015
What is Response Spectrum Analysis • A response spectrum analysis is used in place of a timehistory analysis to determine the response of structures to random or time-dependent loading conditions such as: – – – – –
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earthquakes, wind loads, ocean wave loads, jet engine thrust, rocket motor vibrations, and so on.
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... What is Response Spectrum Analysis • The most accurate solution is to run a long transient analysis. – “Large” means many DOF. “Long” means many time points. – In many cases, this would take too much time and compute resources.
• Instead of solving the (1) large model and (2) long transient together, it can be desirable to approximate the maximum response quickly: – It uses the results of a modal analysis with a known spectrum to calculate displacements and stresses in the model. – The spectrum is a graph of peak response versus modal frequency that captures the peak response of each mode of the structure to a time-history loads. – The last step is to combine the modal results using the mode coefficients based on the spectrum. 4
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... What is Response Spectrum Analysis • Idea: solve the (1) large model and (2) long transient separately and combine the results. Large model Long transient
Large model Long transient
Large model Mode extraction ↓ Mode shapes
Full solution Slow, accurate 5
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Small model Long transient ↓ Response spectrum
Combined solution Fast, approximate
Generating the Response Spectrum • Response Spectrum: • A response spectrum is a plot of the maximum response of linear one-DOF systems to a given time-history input. • The abscissa of the plot is the natural frequencies of the systems, the ordinates is the maximum response: • • •
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Displacement Velocity Acceleration.
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Types of Analyses • There are two types of Response Spectrum Analysis available: – Single-Point Response Spectrum (SPRS)
- Multi-Point Response Spectrum (MPRS)
Note: For FSI analyses only SPRS is supported. 7
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Participation Factor, • The participation factor is a measure of the response of the structure at a given natural frequency. • represents how much each mode will contribute to the deflections and stresses in a particular direction. For FSI models, the left eigenvectors are used to calculate the participation factors.
w M K f 0 2 i
Modal Analysis
frequency
mode shape
spectrum value
participation factor
mode coefficient
response
1
w1
{f}1
S1
1
A1
{R}1
2
w2
{f}2
S2
2
A2
{R}2
3
w3
{f}3
S3
3
A3
{R}3
…
…
…
…
…
…
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i fL Ti M D
mode
…
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i
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Effective Mass For FSI Case •
Recall: printed in the modal output file is the effective mass.
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For a standard case without fluid interaction the effective mass for the ith mode is: M eff ,i
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i2
f Ti M f i
i2 , if f i M f i 1
For FSI case where the unsymmetric solver is used, the left and right eigenvectors are used and the effective mass is expressed as:
M eff ,i Ds K s fRs i T
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T
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March 13, 2015
fL Ti M D wi2
Recommended Solution Procedure •
The recommended solution method is generally specified by your design code. – combination method – rigid response method
ω1
ω2 ω3
ω4 ω5 ω 6
Note: missing mass effects are supported. 10
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ω7
ω8
Typical Application of Response Spectrum A common way to assess fluid filled tanks at nuclear facilities for earthquake survival is to use a response spectrum analysis. With the acoustic sloshing capability, a response spectrum analysis of the tank can include the effect of the fluid in an inexpensive earthquake analysis.
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Random Vibration
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What is Random Vibration Analysis • Random vibration analysis is another spectral method
• The purpose of a random vibration analysis is to determine some statistical properties of a structural response, normally the standard deviation (1) of a displacement, force, or stress. • (1) is used to determine fatigue life of a structure
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Definition and Purpose •
We have already seen sinusoidal vibration (free and forced) • This is vibration at one predominant frequency
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A more common type of vibration is random vibration • This is vibration at many frequencies at the same time
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Definition and Purpose •
Many common processes result in random vibration • • • •
Parts on a manufacturing line Vehicles travelling on a roadway Airplanes flying or taxiing Spacecraft during launch
Courtesy: NASA
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These random vibrations contain all frequencies at all times
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The amplitudes at these frequencies vary randomly with time. –
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We need some way of describing and quantifying this excitation.
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Definition and Purpose •
If the amplitude is constantly changing, how can a random excitation be evaluated?
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Key observation: at a given frequency, the amplitude of the excitation does constantly change, but for many processes, its average value tends to remain relatively constant. – This gives us the ability to easily characterize a random excitation.
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Assumptions & Restrictions The structure has:
• no random properties • no time varying stiffness, damping, or mass • no time varying forces, displacement, pressures, temperatures, etc •
applied light damping – damping forces are much smaller than inertial and elastic forces
The random process is:
• stationary (does not change with time) •
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– the response will also be a stationary random process ergodic (one sample tells us everything about the random process)
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Typical Application of Random Vibration A fluid container experiences random excitation during transport on the back of a truck. It would be a typical use of random vibration analysis on a vibro-acoustic model to determine the fatigue life of the container due to the random stresses caused by the road surface roughness.
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Application to FluidStructure Interaction Models
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Unsymmetric Coupled Formulation Response Spectrum analysis using acoustic element is supported for unsymmetric coupled formulation only. So in the upstream analysis please selected «Program Controlled Coupled » as Acoustic-Structural Coupled Body Options:
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Setup • Setup a response spectrum/random vibration analysis in the schematic by linking a modal system to a response spectrum/random vibration system at the solution level.
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Upstream Unsymmetric Modal For fluid-structure interaction models, unsymmetric eigensolver must be used. Right and left eigenmodes are written into Jobname.MODE & Jobname.LMODE which are necessary for the downstream mode superposition. To perform the modal analysis activating the output of right and left eigenmodes insert a “Unsymmetric Mode Extraction For Downstream Spectrum Analysis” from “Analysis Settings” menu of the Acoustics ACT extension.
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March 13, 2015
Upstream Unsymmetric Modal The “Unsymmetric Mode Extraction For Downstream Spectrum Analysis” object is used to switch the eigensolver to unsymmetric and request to output right and left eigenmodes (modopt,unsym,,,,real,,both MAPDL command). Natively Mechanical doesn’t support mode superposition with unsymmetric matrices so if you choose the unsymmetric solver directly in the Analysis Settings of the modal analysis the downstream analysis will be invalid. To avoid this, leave the Solver Type to Program Controlled in the modal analysis settings and the “Unsymmetric Mode Extraction For Downstream Spectrum Analysis” will switch it to unsymmetric solver internally:
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Downstream Spectrum Insert “FSI Spectrum Analysis” object from “Analysis Settings” menu of the Acoustics ACT extension in order to use right and left modes in the downstream spectrum analysis:
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The excitation must be applied at fixed degrees of freedom (excitation at fixed pressure isn’t supported).
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If response strain/stress is of interest, then the modal strain and the modal stress need to be determined in the modal analysis.
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March 13, 2015
