An Approach to Properly Account for Structural Damping, Frequency-Dependent Stiffness/Damping, and

to Use Complex Matrices in Transient Response

By

Ted Rose

Or (more simply)Some Uses for Fourier Transforms

in Transient AnalysisBy

Ted Rose

Overview

• Transient Response analysis has a number of limitations– It requires an approximation be used to model

structural damping – It does not support frequency-dependent

elements– It does not allow complex matrices– Obtaining steady-state solutions to multiple

rotating imbalances can take very long

Fourier Transforms in Transient

• All of these limitations can be overcome by using Fourier Transforms– In 1995 Dean Bellinger presented a paper of

Fourier Transforms– His paper, plus the Application Note on Fourier

Transforms, provides the documentation on this approach

Fourier Transforms in Transient

• The user interface is simple:1. Set up your file for transient response

2. Change the solution to 108 or 111

3. Add a FREQ command to CASE CONTROL

4. Add a FREQ1 entry to the BULK DATA

• Use a constant F = 1/T Where T = the duration/period of the transient event

• Make sure that the duration/period of the load is correct (TLOAD1/2 duration is = T)

Fourier Transforms in Transient

• Verify the transformation by plotting the applied load (sample input in paper)

• Sample – three simultaneous sine inputs (1hz, 2hz, and 3hz) with a 1.0 second duration

freq1,99,1.,1.,2DLOAD,1,1.,1.,10,1.,20,1.,30$ T = 1.0TLOAD2,10,25,,,0.,1.,1.,-90.TLOAD2,20,25,,,0.,1.,2.,-90.TLOAD2,30,25,,,0.,1.,3.,-90.

$ T = 1.0TLOAD2,10,25,,,0.,1.,1.,-90.TLOAD2,20,25,,,0.,1.,2.,-90.TLOAD2,30,25,,,0.,1.,3.,-90.

Applied Load in Transient

freq1,99,1.,1.,2DLOAD,1,1.,1.,10,1.,20,1.,30$ T = 1.0TLOAD2,10,25,,,0.,1.,1.,-90.TLOAD2,20,25,,,0.,1.,2.,-90.TLOAD2,30,25,,,0.,1.,3.,-90.

Load after Fourier Transform$ DF = 1/Tfreq1,99,1.,1.,2DLOAD,1,1.,1.,10,1.,20,1.,30$ T = 1.0TLOAD2,10,25,,,0.,1.,1.,-90.TLOAD2,20,25,,,0.,1.,2.,-90.TLOAD2,30,25,,,0.,1.,3.,-90.DAREA,25,1,1,1.TSTEP,20,100,.01,

Duration of TLOAD2Is 1.0, therefore, F=1./1.=1.

Load after Fourier Transform

$ DF = 1/Tfreq1,99,.5,1.,3DLOAD,1,1.,1.,10,1.,20,1.,30$ T = 1.0TLOAD2,10,25,,,0.,1.,1.,-90.TLOAD2,20,25,,,0.,1.,2.,-90.TLOAD2,30,25,,,0.,1.,3.,-90.DAREA,25,1,1,1.TSTEP,20,100,.01,

$ wrong inputfreq1,99,.5,1.,3DLOAD,1,1.,1.,10,1.,20,1.,30$ T = 1.0TLOAD2,10,25,,,0.,1.,1.,-90.TLOAD2,20,25,,,0.,1.,2.,-90.TLOAD2,30,25,,,0.,1.,3.,-90.DAREA,25,1,1,1.TSTEP,20,100,.01,

Poorly selectedInput for FREQ1 –Although F is 1.0, the Starting frequency is .5,Resulting in a poor transformation

Compare the Results

Original Load Good Fourier Transform Bad Fourier Transform

Structural Damping

• Handled correctly, it forms a complex stiffness matrix

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

• Unfortunately, transient response does not allow complex matrices, so we must approximate structural damping using:

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

• Where w3 and w4 are the “dominant” frequency of response

Structural Damping

• If the actual response is at a frequency less than w3, the results have too little damping, if it is at a frequency greater than w3, the results have too much damping

• This means that unless you are performing a “steady-state” analysis, your damping will not be handled correctly

• Using Fourier Transforms allows you to apply structural damping properly

Multi-Frequency Steady-State

• Many structures (engines, compressors, etc) have multiple rotating bodies

• In many cases, they are not all rotating at the same frequency

• In order to handle this in conventional Transient analysis, it requires a very long integration interval to reach the steady-state response

• With Fourier transforms, it is easy to solve for the steady-state solution

Multi-Frequency Steady-State

• As an example, let us look at a typical jet engine model with 3 rotating imbalances

Multi-Frequency Steady-State

• All right, how about this model?

Model courtesy of Pratt and Whitney

Multi-Frequency Steady-State

• Although rotating imbalances in jet engines occur at much higher frequencies, for this example, I will use .5hz, 1.0hz, and 2.0hz

$ dynamic loading$dload,101,1.,1.,1002,1.,1003,1.,2002,1.,2003,1.,3002,1.,3003$tload2,1002,12,,,0.,10.,1.,-90.tload2,1003,13,,,0.,10.,1.,0.force,12,660001,,10.,,2.,force,13,660001,,10.,,,2.$tload2,2002,22,,,0.,10.,2.,90.tload2,2003,23,,,0.,10.,2.,0.force,22,670001,,10.,,4.,force,23,670001,,10.,,,4.$

tload2,3002,32,,,0.,10.,.5,0.tload2,3003,33,,,0.,10.,.5,90.force,32,680001,,10.,,1.,force,33,680001,,10.,,,1.$eigrl,10,,,10tabdmp1,1,crit,0.,.01,1000.,.01,endt$tstep,103,100,.02$$ set delta F=1/T$freq1,102,.5,.5,5

Rotating in opposite direction

Multi-Frequency Steady-State