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2001-06

An Approach to Properly Account for Structural Damping,

Frequency-Dependent Stiffness/Damping, and to Use ComplexMatrices in Transient Response

Presented at the MSC.Software Corporation 2001 AerospaceConference and Technology Showcase

by: Ted Rose

Manager, MSC.Nastran Training and Support

MSC.Software Corporation

Abstract:Transient response analysis in MSC.Nastran is performed using real arithmetic. Structural damping

requires a complex stiffness matrix to properly account for its effects. Unfortunately, these two facts mean

that when performing transient response analysis in MSC.Nastran, structural damping must beapproximated.

Frequency-dependent elements (such as the CBUSH) are available in Frequency Response solutions, but

must be linear in Transient Response.

Adding a complex matrix into a transient response solution is not allowed, meaning that the user mustcome up with a way to "convert" their complex matrices into equivalent real matrices.

This paper provides an approach to properly account for structural damping, frequency-dependentelements, and to include complex matrices in transient response by using the Fourier Transform approach.

In addition, the approach in this paper allows for a simple solution of a structure with multiple harmonic

inputs acting simultaneously (for example a car, which has multiple rotating bodies, each with a differentsteady-state frequency, acting simultaneously. A list of some of these is:

a) engine crankshaft

b) camshaft(s)

c) the wheels,

d) driveshaft

e) alternator

f) power steering pump

and many more. Solving for a steady-state solution of this model would require a transient solution

integrating over an extended period of time. Using Fourier transformations and frequency response can doit quickly and efficiently.

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Introduction:

This paper uses standard features of MSC.Nastran to apply Fourier Transforms during a solution for

transient response analysis. By doing this, the program will be able to solve steady-state problems with

multiple simultaneous input frequencies, problems with structural damping (since transient responseanalysis in MSC.Nastran uses real arithmetic and structural damping requires a complex stiffness matrix,

there is a conflict whenever you wish to apply structural damping in a transient response solution),

problems including frequency-dependent elements, and problems with complex matrices. The approach is

based on the information provided in Dean Bellingers 1995 paper1 . In his paper, he describes the approach

in MSC.Nastran to use a Fourier Transform to perform a transient response solution in the frequency

domain. Since the frequency response solutions in MSC.Nastran use complex arithmetic and support

frequency-dependent elements, the above items are handled correctly in the solution.

Getting the Load Right (Selecting the Frequencies for theFourier Transform)

As mentioned in Dean Bellingers paper1 and the Application Note on Fourier Transforms inMSC.Nastran

4, performing a Fourier Transformation is very simple to do. Simply set up the file for a

transient response solution, then change the solution to SOL 108 (direct frequency response) or SOL 111

(modal frequency response) and add a FREQ case control command with the associated FREQi entries in

the bulk data section. The proper selection of t he frequencies is the hard part. The following (in addition to

the information in the two references) will help.

I talked to Dean and got the following recommendations from him:

1) Use FREQ1 entries to define the loading frequencies.

2) Use a constant DF (maximum value = 1/T , where T = the duration of the transient event)

when defining the frequencies. This is easiest on the FREQ1

3) Set T2 on the TLOAD2 entry to T (if you are using a TLOAD2).

4) IF YOU DO NOT HAVE A TSTEP (inverse Fourier Transform), set the A field on theDAREA entry to 2*DF*transient_DAREA (DF = frequency step size on the FREQ1,

transient_DAREA = EXCITEID on the DAREA entry in the transient analysis).

In selecting the frequencies, it is very important to be sure that you are able to represent the loading

properly when performing the Fourier transformation, so I recommend that you make up a simple model

first to verify the loading and Fourier transformation. The simple model is a single point with a mass of 1.0

on it (no stiffness or damping). The acceleration response will be equal to the input loading, so this is agood model use to verify the method.

To demonstrate this, I will use a simple loading, which is a time-decaying 1.0 hz sine wave. Although this

loading is very simple to define in the time domain (TLOAD2 entry), it is a very complicated loading in the

frequency domain, as there will be contributions from many frequencies.

The input file to verify the input loading for transient response is:

SOL 109CENDTITLE = TRANSIENT RESPONSE - PLOT INPUT LOADING$DLOAD = 10TSTEP = 20DISP(PLOT)=ALLACCEL = ALLOLOAD = ALL

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BEGIN BULKTLOAD2,10,25,,,0.,10.,1.,-90.,.5DAREA,25,1,1,1.TSTEP,20,1000.,.01,GRID,1CONM2,100,1,,1.ASET1,1,1

ENDDATA

In this file, T is 20., so we will use DF of .05 in the frequency response file. The file converted into a

frequency response run to use the Fourier transformation is:

SOL 108

diag 8,15,56c omp i l e s ed f r e q a l t e r ' f r l g . * p p f ' me ss ag e / / ' ' $ me s sa ge / / ' l o ad i n g t r a ns f o r me d i n t o t h e f r e qu en cy d oma i n ' $ me ss ag e / / ' ' $ ma t p r n pp f / / $ t a b p r t f r l / / / / / 1 $ CENDTITLE = FREQUENCY RESPONSE - PLOT INPUT LOADING$DLOAD = 10TSTEP = 20FREQ=99DISP(phase,PLOT)=ALLACCEL(phase) = ALLOLOAD = ALLOUTPUT (XYPLOT)$XGRID=YESYGRID=YESXYPLOT ACCE / 1(T1)BEGIN BULKf r e q1 , 9 9 , . 05 , . 0 5 , 6 0

TLOAD2,10,25,,,0.,10.,1.,-90.,-.5DAREA,25,1,1,1.TSTEP,20,1000,.01,GRID,1,,,,,,3456Spcoff,1,1CONM2,100,1,,1.$$ small spring for limitation 41260$celas2,1111,.00001,1,2ENDDATA

The DMAP in the above file is to print out the loading transformed into the frequency domain (NOTE: ifyou remove the TSTEP case control command, the output will be in the frequency domain). Plots from the

transient run and the frequency response run follow.

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Figure 1. Time History of loading from the transient run

Figure 2. Time history of load from frequency response run

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Figure 3. Plot of the loading in the frequency domain (Fourier transformation)

Figures 1 and 2 compare well, indicating that the Fourier transformation worked well for this example.

Figure 3 shows the loading converted into the frequency domain (the second file without the TSTEP case

control request).

NOTE: There is no checking in the program to determine whether your Fourier Transform is valid. This

step is up to you!

Multiple Simultaneous Harmonic Loads

When you have multiple simultaneous harmonic loads, which are not all acting at the same frequency,

performing a solution requires either a very long transient integration (to get the steady-state solution) or

using Fourier Transforms and frequency response analysis.

For this situation, I will use a simple model with multiple loads. The model is a cantilever beam with three

simultaneous harmonic loads. All of the loads will represent rotating imbalances and each will have a

different frequency associated with it. This might compare to a turbine with a transmission. This structure

has multiple components, each rotating at a different rate.

In the example, the 3 input loads will be out-of-phase with each other and one will be rotating in the

opposite direction of the others. This is accomplished by using the phase angle on the TLOAD2 entries.

Each load consists of a force (mr2 where m= mass, r= radius, = rotational frequency) which acts in the

radial direction as the mass is rotating. In order to describe this for the program, two TLOAD2 entries will

be used for each rotating imbalance with FORCE entries to provide the two components of the load ( for

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this model Y and Z), which will be 90 degrees out of phase. A single DLOAD entry will be used to applyall of the components simultaneously.

The model used for this is a simple cantilever beam. It is shown below . The three imbalanced loads are on

GRIDs 3, 6, and 9.

.

Figure 4 - Cantilever Beam Model

For this example, the loads are defined as rotating at 1.0, 2.0, and 3.0hz. The input file for a transientsolution is shown below:

$ multiple_rotating_imbalances_trans.dat - cantilever beam model$ with multiple imbalances - transient solution until steady-state$$ for this model use modal transient with 1% of critical damping on$ each modesol 112cendtitle = cantilever beam model with rotating imbalancessubtitle = transient analysis$spc = 1

dload = 1000method = 10disp(plot)=alltstep = 2000sdamp = 1$OUTPUT (XYPLOT)XGRID=YESYGRID=YESxtitle applied loadXYPLOT OLOAD / 9(T2)xtitle acceleration

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XYPLOT ACCE / 9(T2)xmin 15.xmax 20.xtitle applied load 15 to 20 secXYPLOT OLOAD / 9(T2)xtitle acceleration 15 to 20 secXYPLOT ACCE / 9(T2)begin bulk

$ modelinclude 'cantbeam.dat'$$ multiple rotating imbalances$param,resvec,yesdload,1000,1.,1.,1002,1.,1003,1.,2002,1.,2003,1.,3002,1.,3003$tload2,1002,12,,,0.,100.,1.,-90.tload2,1003,13,,,0.,100.,1.,0.force,12,9,,10.,,1.,force,13,9,,10.,,,1.$tload2,2002,22,,,0.,100.,2.,90.tload2,2003,23,,,0.,100.,2.,0.force,22,6,,10.,,1.,

force,23,6,,10.,,,1.$tload2,3002,32,,,0.,100.,.5,0.tload2,3003,33,,,0.,100.,.5,90.force,32,3,,10.,,1.,force,33,3,,10.,,,1.$eigrl,10,,,10tabdmp1,1,crit,0.,.01,10000.,.01,endt$tstep,2000,20000,.001,10enddata

In this file, modal transient response (SOL 112) is used with residual vectors to obtain a solution. The

following file is used to do the same thing in a modal frequency response (SOL 111) run:

$ multiple_rotating_imbalances_freq.dat - cantilever beam model$ with multiple imbalances - transient solution until steady-state$sol 111cendtitle = cantilever beam model with rotating imbalancessubtitle = frequency response analysis$spc = 1dload = 1000method = 10disp(plot)=alltstep = 2000freq = 500sdamp = 1

OUTPUT (XYPLOT)XGRID=YESYGRID=YESxtitle applied loadXYPLOT OLOAD / 9(T2)xtitle accelerationXYPLOT ACCE / 9(T2)begin bulkinclude 'cantbeam.dat'$$ multiple rotating imbalances$

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param,resvec,yesdload,1000,1.,1.,1002,1.,1003,1.,2002,1.,2003,1.,3002,1.,3003$tload2,1002,12,,,0.,10.,1.,-90.tload2,1003,13,,,0.,10.,1.,0.force,12,9,,10.,,1.,force,13,9,,10.,,,1.

$tload2,2002,22,,,0.,10.,2.,90.tload2,2003,23,,,0.,10.,2.,0.force,22,6,,10.,,1.,force,23,6,,10.,,,1.$tload2,3002,32,,,0.,10.,.5,0.tload2,3003,33,,,0.,10.,.5,90.force,32,3,,10.,,1.,force,33,3,,10.,,,1.$eigrl,10,,,10tabdmp1,1,crit,0.,.01,1000.,.01,endttstep,2000,5000,.001freq1,500,.1,.1,19

enddata

In these runs, the Y-direction acceleration for GRID 9 is plotted vs time. The following plot is the completetime history from the SOL 112 run.

Figure 5 - Complete Time History of Transient Response with 3 Rotating Imbalances

From this plot, it is apparent that the structure has reached a steady-state response by 10 seconds into theintegration. For safety, the following plot shows the last 5 seconds of the time history.

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Figure 6 - Steady State Response from Transient Response with 3 Rotating Imbalances

For comparison, here is the plot from the SOL 111 (modal frequency response) run.

Figure 7 - Steady State Response from Frequency Response with 3 Rotating Imbalances

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From the above, it becomes obvious that we can use frequency response with transient loads to get thesteady-state response for a structure undergoing multiple simultaneous oscillating loads.

Structural Damping

Handled properly, structural damping is included in the equations of motion as complex terms in thestiffness matrix. In MSC.Nastran, structural damping is defined using material damping Ge on the MATi

entries or PARAM,G to get overall structural damping. This is described in references 2 and 3.

When properly accounted for, these terms are added into the stiffness matrix as follows:

[Ktotal] = [K](1+iG) + iKeGe

Where:

[K] = stiffness due to elements +K2GG matrix

G = overall structural damping coefficient PARAM,G

Ge = structural damping value from MATi entries

Ke = element stiffness matrix

Which gives the equation of motion:

[M]{a} + [B]{v} + [Ktotal]{u} = {p}

where:

{a} = acceleration as a function of time

{v} = velocity as a function of time

{u} = displacement as a function of time

{p} = force as a function of time

[M] = Mass matrix[B] = viscous damping matrix

In frequency response the above equation is transformed into the frequency domain and solved.

Since transient response analysis does not include complex terms, the structural damping has to be

converted into an equivalent viscous damping. Details of this conversion (and an explanation of how it

works) are presented in the dynamics users guide2.

For transient analysis, the viscous damping matrix is:

[Btotal] = [B] + [K]G/W3 + keGe/W4

Where W3 and W4 are user-provided parameters. They represent the dominant frequency of the response

in radians per second. If the response is a steady state response and the user-provided parameters W3 and

W4 are set to the response frequency, this approximation is exact. Unfortunately, in transient response,there is normally no single dominant frequency of response and the above is simply an approximation,

which may or may not be reasonable.

Whenever structural damping is used in a transient response, you have introduced an approximation in the

damping, which is hard to measure. This approximation might result in unacceptable results without any

indication. If the structural damping could be properly handled in transient response, this approximation

could be avoided.

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There are two obvious solutions to this problem;

1) use complex arithmetic in transient response

2) transform the loading from the time domain into the frequency domain and solve using frequency

response

Since using complex arithmetic in transient response would require major DMAP or re-writing of the

program (not to add, this would still not provide support for frequency-dependent elements), this paper

deals with the second solution.

Using the model with 3 rotating imbalances, I will remove the modal damping (1% of critical damping) and

use structural damping. In an "ideal" world, at resonance, the ratio between the critical damping ratio andthe structural damping is:

= g/2where:

= critical damping ratiog = structural damping coefficient

Using this relationship, applying g=.02 should give the same damping at resonance. Unfortunately, the

loads are not being applied at resonance, so the relationship will not be exact for this problem. In order to

demonstrate the potential for error in applying structural damping in transient response, I will apply an

imbalanced load at the natural frequency of the first mode (6.3hz), and to exaggerate the potential for error,

I will set PARAM,W3 to 1256.6 (200hz) by "mistake". This will result in the damping being much lower

than it should be in the transient solution. PARAM,G,.02 will be added to both input files (transient andfrequency response).

$ structural_damping_freq.dat - cantilever beam model$ with a single load a resonance for mode 1$$ for this model use Structural damping - G=.02$sol 111cendtitle = cantilever beam model with structural dampingsubtitle = frequency response analysis

spc = 1dload = 1000method = 10disp(plot)=alltstep = 2000freq = 500OUTPUT (XYPLOT)XGRID=YESYGRID=YESxtitle applied loadXYPLOT OLOAD / 9(T2)xtitle displacementXYPLOT DISP / 9(T2)begin bulk$ modelinclude 'cantbeam.dat'$

$ structural damping$param,g,.02$param,resvec,yes$$ load at 6.3 hz$dload,1000,1.,1.,1002,1.,1003$tload2,1002,12,,,0.,10.,6.3,-90.tload2,1003,13,,,0.,10.,6.3,0.

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force,12,9,,10.,,1.,force,13,9,,10.,,,1.$eigrl,10,,,10tabdmp1,1,crit,0.,.01,1000.,.01,endt$tstep,2000,5000,.001,10

freq1,500,.1,.1,69enddata

Note that this run has a single steady-state harmonic load, so it can be run as frequency response using

RLOAD1 entries. Replacing the TLOAD2 entries by RLOAD1 entries and loading only at 6.3 hz gives a

steady state response with a maximum amplitude of 12.13797 at GRID 9, while the run above gives

12.13799. The difference is in the 7th significant figure. Setting PARAM.W3,1256.6 (200hz) gives a

maximum response in SOL 112 of 81.66032, which is noticeably different. NOTE that this difference is

caused mainly by the incorrect value of W3. As the response should be at the loading frequency, W3

should be set to 39.584 (6.3hz). The purpose of this example was simply to demonstrate how wrong the

answers could be if a transient response has the wrong value for W3 when structural damping is used. The

fact that the loading is at a resonant frequency of the model exaggerates the difference.

Frequency-dependent ElementsMSC.Nastran has several frequency-dependent elements, including the CBUSH. In frequency response, theproperties of these elements are updated to the proper values for each loading frequency. In transient

response, these elements use the nominal properties provided by the user. Therefore, it is up to the user to

select an acceptable set of properties for the frequency-dependent elements, which might lead to incorrect

results. Using a Fourier Transform and solving the problem in the frequency domain allows the program to

use the "correct" properties at each frequency.

To demonstrate this, I will add the following bushing element, which has frequency-dependent stiffness,

into the sample files used for the multiple rotating imbalances.

CBUSH,100,100,9,,,,,0PBUSH,100,K,10.,10.,10.PBUSHT,100,K,102,102,102

TABLED1,102,,.5,5.,1.,10.,2.,15.,endt

Adding this element into the model will use the "nominal" stiffness of 10.0 for the transient run, but will

use a stiffness of 5.0 when the load is at .5hz, 10.0 when the load is at 1.0hz, and 15. when the load is at

2.0hz in the frequency response run.

The resulting XYPLOTs for acceleration at steady state are shown below. The one on the left is from the

frequency response (fourier transform) run :and the one on the right is from the transient solution. The

results from the transient response run are approximately 10% higher at the peaks, indicating that

accounting for the frequency-depenent stiffness terms made a noticeable difference in the solution.

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Figure 8 - Frequency Response (fourier transform) and transient results including a frequency-dependent

spring with multiple rotating imbalances.

Matrices with Complex terms

This one is easy, simply select the matrices using the K2PP, M2PP, B2PP, or TF Case Control commands

and the program will include them in the solution. Note that the "G" matrices (K2GG, etc) must be

symmetric and real, while the "P" matrices do not have this requirement. In transient response analysis,complex terms are not allowed in the matrices, so the inclusion of complex coefficients would result in a

FATAL message, while frequency response uses complex arithmetic, so complex coefficients are allowed.

Summary and Conclusions.

This paper expands on the information presented by Bellinger in his 1995 paper. In this paper, I show how

to apply fourier transforms for specific cases and demonstrate how to include structural damping,frequency-dependent elements, and complex terms in matrices properly into a transient solution

References

1. Bellinger, Dean, Dynamic Analysis by the Fourier Transform Method in MSC/NASTRAN,

Proceedings 1995 MSC World Users Conference.

2. Sitton, Grant,MSC/NASTRAN V69Basic Dynamic Analysis Users Guide, MSC.Software, Los

Angeles, Ca, 1998

3. Herting, Dave,MSC/NASTRAN Advanced Dynamics User's Guide, MSC.Software, Los Angeles, Ca,1997

4. Fourier Transform Behavior and Usage in MSC/NASTRAN, Application Note, November, 1985,MSC/NASTRAN Application Manual, MSC Software, Los Angeles, Ca