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 Analysis By 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 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

Applied Load in Transient

Load after Fourier Transform Duration of TLOAD2 Is 1.0, therefore,  F=1./1.=1.

Load after Fourier Transform $ wrong input freq1,99,.5,1.,3 DLOAD,1,1.,1.,10,1.,20,1.,30 $ T = 1.0 TLOAD2,10,25,,,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 selected Input for FREQ1 – Although  F is 1.0, the Starting frequency is.5, Resulting in a poor transformation

Compare the Results Original LoadGood Fourier TransformBad Fourier Transform

Structural Damping Handled correctly, it forms a complex stiffness matrix [K total ] = [K](1+iG) + i  K e G e Unfortunately, transient response does not allow complex matrices, so we must approximate structural damping using: [B total ] = [B] + [K]G/W 3 +  k e G e /W 4 Where w 3 and w 4 are the “dominant” frequency of response

Structural Damping If the actual response is at a frequency less than w 3, the results have too little damping, if it is at a frequency greater than w 3, 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,,,10 tabdmp1,1,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