Shock Response Spectrum Analysis Via The Finite Element Method
SHOCK RESPONSE SPECTRUM ANALYSIS VIA THE FINITE ELEMENT METHODRevision CBy Tom IrvineEmail: tomirvine@aol.comNovember 19, 2010IntroductionThis report gives a method for determining the response of a multi-degree-of-freedomsystem to a base excitation shock, where the shock is defined in terms of a ShockResponse Spectrum (SRS).A finite element model is used to determine the normal modes and frequency responsefunction of a sample structure. Commercial finite element analysis software is used forthis purpose.The following steps are done outside the finite element software by using programswritten in C/C . The source code for these programs is available from the author byrequest. See also Appendix C.The impulse response function is calculated from the frequency response function via aninverse Fourier transform.A time history is synthesized to satisfy the SRS. The response time history of thestructure is then calculated via a convolution integral using the synthesized time historyand the impulse response function.This approach is referred to as the synthesis method in this report. An advantage of thismethod is that the impulse response function can be used for numerous time historyinputs. There is no need to rerun the finite element analysis for each input case.Sample StructureConsider a circuit board made from G10 material. The modulus of elasticity is 2.00E 06lbf/in 2. The dimensions of the circuit board are 2 in x 4 in x 0.063 in. The board is fixedat each corner.The board has a uniform mass distribution. The total mass is 0.115 lbm. This includesthe G10 board and the electronic components. Assume that the electronic componentsdo not add any stiffness.The circuit board is assumed to have an amplification factor of Q 10 for all modes.Normal ModesThe normal modes are analyzed via the finite element method using FEMAP andNE/Nastran software. The filename is SRS plate normal.nas.1
The undeformed model is shown in Figure 1. The model consists of 1624 plateelements and 1711 nodes. The natural frequencies for the first 20 modes are given inTable 1. The mode shapes for the first twelve modes are given in Figures 2 through 13,respectively.Figure 1. Finite Element Model of Circuit Board, UndeformedThe figure has node labels at three locations of interest. The node numbers are 480,523, and 1423.2
Table 1. Circuit Board Natural FrequenciesModefn 1.546E-022.39E-0423502.289E-090.00E 0034101.613E-090.00E 0046606.376E-034.06E-055779-5.592E-140.00E 0068931.469E-080.00E 00711162.073E-034.30E-06811410.000E 000.00E 00913221.804E-090.00E 001014161.177E-031.39E-061116871.149E-090.00E 001219164.433E-150.00E 00132142-5.880E-100.00E 001423483.373E-090.00E 00152402-2.177E-034.74E-061625370.000E 000.00E 001727170.000E 000.00E 00182724-1.254E-150.00E 001927840.000E 000.00E 00202805-6.125E-100.00E 00The modal participation factor does not have a unit.The modal effective mass has a unit of lbf sec 2/in.Further information about these parameters is given in Reference 1.3
Figure 2. Mode 1Figure 3. Mode 24
Figure 4. Mode 3Figure 5. Mode 45
Figure 6. Mode 5Figure 7. Mode 66
Figure 8. Mode 7Figure 9. Mode 87
Figure 10. Mode 9Figure 11. Mode 108
Figure 12. Mode 11Figure 13. Mode 129
Frequency Response FunctionsThe next step is to perform a frequency response analysis. The resulting frequencyresponse functions give the response to the base input. The response parameter maybe either displacement or acceleration. The base input is in terms of acceleration for thesample problem.This analysis is called “SOL SEMFREQ” in Nastran terminology. It is also referred to asa modal frequency analysis.The filename is SRS plate frf 4k.nas.The base function is a spectral function with a magnitude of unity, from 0 Hz to 4000 Hz.This represents acceleration. It is applied at the four corner nodes in the Z-axis, which isnormal to the plane of the board.The FEMAP software converts the base acceleration to an equivalent force. The force isapplied to a base mass which is much greater than the circuit board mass. The fourcorner nodes are attached to the base mass via rigid link elements, as shown in Figure14.The force and mass values are adjusted so that the desired spectral acceleration isapplied at the four corner nodes.The resulting frequency response function magnitudes for the three nodes of interest aregiven in Figures 15 through 17, respectively.Figure 14. Mode with Base Mass and Rigid Links10
FREQUENCY RESPONSE FUNCTIONNODE 480100MAGNITUDE ( Gout / Gin )1010.10.01101001000FREQUENCY (Hz)Figure 15. Edge, Midpoint along Length114000
FREQUENCY RESPONSE FUNCTIONNODE 523100MAGNITUDE ( Gout / Gin )1010.10.01101001000FREQUENCY (Hz)Figure 16. Edge, Midpoint along Width124000
FREQUENCY RESPONSE FUNCTIONNODE 1423100MAGNITUDE ( Gout / Gin )1010.10.011010010004000FREQUENCY (Hz)Figure 17. Center of BoardImpulse Response FunctionsThe impulse response function for each node is calculated by taking an inverse Fouriertransform of the complex frequency response function. The results for the three nodesof interest are shown in Figures 18 through 20, respectively.Note that the impulse response functions presented in this report consist of discretecoordinate pairs.Each response function must be divided by the total number ofcoordinate points. This is done during the convolution integration in this analysis.Thus, the plotted impulse response functions are not yet normalized by the number ofcoordinate points.13
ACCELERATION IMPULSE RESPONSE FUNCTIONNODE 4801500AMPLITUDE (Gout / Gin )10005000-500-1000-150000.10.2TIME (SEC)Figure 18. Edge, Midpoint along Length140.30.4
ACCELERATION IMPULSE RESPONSE FUNCTIONNODE 5231500AMPLITUDE (Gout / Gin)10005000-500-1000-150000.10.2TIME (SEC)Figure 19. Edge, Midpoint along Width150.30.4
ACCELERATION IMPULSE RESPONSE FUNCTIONNODE 14231500AMPLITUDE (Gout / Gin )10005000-500-1000-150000.10.2TIME (SEC)Figure 20. Center of Board160.30.4
SRS SynthesisThe SRS specification for the sample problem is given in Table 2. This is the same levelas MIL-STD-810E, Method 516.4, Crash Hazard for Ground Equipment. The test can beperformed using a shaker table.Table 2. SRS Q 200075A time history is synthesized to satisfy the specification, using the wavelet method inReference 2.The resulting time history is given in Figure 21. The corresponding positive and negativespectra closely match the specification as shown in Figure 22.The time history is not unique, however.17
SYNTHESIZED TIME HISTORY20ACCEL (G)100-10-2000.10.20.3TIME (SEC)Figure 21. Synthesized Time History to Satisfy SRS Specification180.4
SRS Q 10200Synthesis NegativeSynthesis PostiveSRS SpecificationPEAK ACCEL (G)100502010510100NATURAL FREQUENCY (Hz)Figure 22. Comparison of Spectra1910002000
Acceleration Response Time HistoriesThe response time history is calculate using the synthesized time history and theappropriate impulse response function via a convolution integral.The acceleration responses for the three nodes of interest are shown in Figures 23through 25, respectively. The peak values are summarized in Table 3.Table 3. Acceleration ResultsNodeLocationPeak AbsoluteAccel (G)480Edge, Midpoint along Length99.0523Edge, Midpoint along Width69.51423Center of Board99.720
ACCELERATION RESPONSE TIME HISTORY NODE 480150100ACCEL (G)500-50-100-15000.10.2TIME (SEC)Figure 23. Edge, Midpoint along Length210.30.4
ACCELERATION RESPONSE TIME HISTORY NODE 523150100ACCEL (G)500-50-100-15000.10.2TIME (SEC)Figure 24. Edge, Midpoint along Width220.30.4
ACCELERATION RESPONSE TIME HISTORY NODE 1423150100ACCEL (G)500-50-100-15000.10.20.30.4TIME (SEC)Figure 25. Center of BoardThis analysis is repeated for relative displacement in Appendix A.The results are compared with a separate method in Appendix B.References1. T. Irvine, Effective Modal Mass & Modal Participation Factors, Vibrationdata,2003.2. T. Irvine, Shock Response Spectrum Testing for Commercial Products, Rev C,Vibrationdata, 1999.3. T. Irvine, Shock Response of Multi-degree-of-freedom Systems, Rev C,Vibrationdata, 2003.23
APPENDIX ARELATIVE DISPLACEMENT ANALYSISFrequency Response FunctionsThe following frequency response functions relate the relative displacement to the baseinput acceleration.RELATIVE DISPLACEMENT FREQUENCY RESPONSE FUNCTIONNODE 480-210-3MAGNITUDE ( inch / Gin )10-410-510-610-710101001000FREQUENCY (Hz)Figure A-1. Edge, Midpoint along Length244000
RELATIVE DISPLACEMENT FREQUENCY RESPONSE FUNCTIONNODE 523-210-3MAGNITUDE ( inch / Gin )10-410-510-610-710101001000FREQUENCY (Hz)Figure A-2. Edge, Midpoint along Width254000
RELATIVE DISPLACEMENT FREQUENCY RESPONSE FUNCTIONNODE 1423-210-3MAGNITUDE ( inch / Gin )10-410-510-610-710101001000FREQUENCY (Hz)Figure A-3. Center of Board264000
Impulse Response FunctionsRELATIVE DISPLACEMENT IMPULSE RESPONSE FUNCTIONNODE 4800.2AMPLITUDE ( inch / Gin )0.10-0.1-0.200.10.2TIME (SEC)Figure A-4. Edge, Midpoint along Length270.30.4
RELATIVE DISPLACEMENT IMPULSE RESPONSE FUNCTIONNODE 5230.040.03AMPLITUDE ( inch / Gin )0.020.010-0.01-0.02-0.03-0.0400.10.2TIME (SEC)Figure A-5. Edge, Midpoint along Width280.30.4
RELATIVE DISPLACEMENT IMPULSE RESPONSE FUNCTIONNODE 14230.2AMPLITUDE ( inch / Gin )0.10-0.1-0.200.10.2TIME (SEC)Figure A-6. Center of Board290.30.4
Response Time HistoriesRELATIVE DISPLACEMENT RESPONSE NODE 4800.10REL DISP (INCH)0.050-0.05-0.1000.10.2TIME (SEC)Figure A-7. Edge, Midpoint along Length300.30.4
RELATIVE DISPLACEMENT RESPONSE NODE 5230.0040.003REL DISP ME (SEC)Figure A-8. Edge, Midpoint along Width310.30.4
RELATIVE DISPLACEMENT RESPONSE NODE 14230.10REL DISP (INCH)0.050-0.05-0.1000.10.20.3TIME (SEC)Figure A-9. Center of BoardTable 3. Relative Displacement ResultsNodeLocationPeak AbsoluteRelative Displacement(inch)480Edge, Midpoint along Length0.0562523Edge, Midpoint along Width0.00311423Center of Board0.0564320.4
APPENDIX BSquare Root of the Sum of the SquaresThe relative displacement results from Appendix A are compared with the results fromthe Square Root of the Sum of the Squares (SRSS). The SRSS is an approximationmethod, given in Reference 3.The SRSS equation for the relative displacement z i of node i is z i max N j 1 j q̂ i j D j , max 2(B-1)where jis the modal participation factor for mode jq̂ i j is the mass-normalized eigenvector coefficient for node i and mode jD j is the relative displacement of mode j regarded as a single-degree-of-freedom systemThe eigenvectors and participation factors are taken from the finite element analysis.The participation factors j are shown in Table 1.The relative displacement values D jacceleration by nSRS specification.are approximated by dividing the SRS2 at the corresponding natural frequency. Refer to Table 2 for theNow model the circuit board with three natural frequencies, at 134, 660, and 1116 Hz.These frequencies are chosen based on the frequency response functions and on themodal effective mass values. Refer to Table 1 and to Figures A-1 through A-3.The SRSS parameters are given in Tables B-1 through B-3.The SRSS results are given in Table B-4.33
Table B-1. Parameters for the f1 134 Hz CaseNodeLocation jq̂ i jD j (inch)480Edge, Midpoint along Length1.546E-028.33E 010.0408523Edge, Midpoint along Width1.546E-024.10E 000.04081423Center of Board1.546E-028.34E 010.0408Table B-2. Parameters for the f2 660 Hz CaseNodeLocation jq̂ i jD j (inch)480Edge, Midpoint along Length6.376E-03-8.48E 010.0017523Edge, Midpoint along Width6.376E-031.41E 020.00171423Center of Board6.376E-03-2.71E 010.0017Table B-3. Parameters for the f3 1116 Hz CaseNodeLocation jq̂ i jD j (inch)480Edge, Midpoint along Length2.073E-038.07E 010.0006523Edge, Midpoint along Width2.073E-03-4.21E 010.00061423Center of Board2.073E-03-1.28E 020.0006DifferenceTable B-4. Relative Displacement ResultsNodeLocationSRSS (inch)480Edge, Midpoint along Length0.0526SynthesisMethod (inch)0.0562523Edge, Midpoint along Width0.00300.00313.2%1423Center of Board0.05260.05646.7%6.4%The synthesis method results are taken from Appendix A. The difference is with respectto the synthesis results. The synthesis results are considered as nearly exact for thesynthesized pulse.Some of the error is due to the fact that the synthesized spectra tended to be slightlyhigher than the SRS specification as shown in Figure 22.34
APPENDIX CWeb PagesThe source and executable codes for the following pre and post-processing programsare taken from the following .vibrationdata.com/SRS.htmFRF GenerationThe FEA solver was: NEiNastran version 9.0.1.183The input file was: srs plate frf 4k.nas.The response file was: srs plate frf 4k.outThe response file was input to program: ne postprocess all.exeThis program generated a series of output files for each node of interest.Node 1423 was at the center of the plate.The output file of interest for this node was: 1423 complex accelH double.zzThe output file is a complex, double-sided frequency response function. The amplitudedimension is (G out / G in) as a function of frequency (Hz). Double-sided means that thefunction's upper frequency is equal to the sample rate, which is twice the Nyquistfrequency.The first z in the filename extension identifies the input axis. The second z identifies theresponse axis.Base Input SynthesisThe synthesized base input time history is: synthesis.txt (Figure 21)The time history was generated outside of the FEA software.As an aside, two examples of synthesis programs are:1. damped sine syn.exe2. wavelet synth.ese35
Acceleration ResponseThe 1423 complex accelH double.zz and synthesis.txt files were then applied toprogram: blast.exeThis program calculates a response time history from an input time history and a transferfunction where the transfer function is a complex Fourier transform. It uses theconvolution method.The output file was: 1423 accel.out (Figure 25)Relative Displacement ResponseThe relative displacement FRF file was: 1423 complex rdH double.zzThe amplitude dimension of the FRF is (inch out / G in).The 1423 complex rdH double.zz and synthesis.txt files were then applied to program:blast.exeThe resulting relative displacement time history had a peak value of 0.056 inches asreported in Table B-4.Alternate MethodThe synthesized time history can be applied directly to the FEA model for a "modaltransient analysis."The following program can be used to prepare the base input time history in a formatsuitable for Nastran-type programs: ne tabled2.exeNote that some FEA programs may have FRF capability but not modal transient.36
A finite element model is used to determine the normal modes and frequency response function of a sample structure. Commercial finite element analysis software is used for this purpose. The following steps are done outside the finite element software by using programs written in C/C .
We will begin with an overview of spectrum analysis. In this section, we will define spectrum analysis as well as present a brief introduction to the types of tests that are made with a spectrum analyzer. From there, we will learn about spectrum analyzers in term s of File Size: 1MBPage Count: 86Explore furtherSpectrum Analysis Basics, Part 1 - What is a Spectrum .blogs.keysight.comSpectrum Analysis Basics (AN150) Keysightwww.keysight.comSpectrum Analyzer : Working Principle, Classfication & Its .www.elprocus.comFundamentals of Spectrum Analysis - TU Delftqtwork.tudelft.nlRecommended to you b
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Mil 810E Shock Summary Report Mil 810E Shock Summary Report NV175 Mil 810E Shock Report Page 1 of 6 Date Issued: 17/11/05 A.Irwin Lambda UK Confidential Procedure I – Functional Shock i Objective Designed to represent a shock condition typical of that in operational use. The following conditions are taken
new termination shock and the velocity (V2) of the rarefac-tion wave or weak second shock that propagates down-stream after the interaction of an interplanetary shock with the termination shock. Consider a frame of reference A0 in which the initial termination shock S0 is static. We treat the plasma as an ideal gas whose ratio of specific heats .
Note that MIL-STD-1540C and MIL-STD-810E require this format for certain shock environments. The purpose of this report is to explain the shock response spectrum and to give a derivation of a calculation method. The calculation metho
shock. The random vibration specification is in the form of a power spectral density (PSD). The shock requirement is a shock response spectrum (SRS). Pyrotechnic-type SRS tests are often more difficult to configure and control, and are thus more expensive than shaker table PSD tests. Furthermore, some lower and even mid-level SRS specifications .
– Authored the original edition of the Spectrum Analysis Basics application note and contributed to subsequent editions – Helped launch the 8566/68 spectrum analyzers, marking the beginning of modern spectrum analysis, and the PSA Series spectrum analyzers that set new performance benchmarks in the industry when they were introduced
