By Ted Rose

1. An Approach to Properly Account for Structural Damping, Frequency-Dependent Stiffness/Damping, and to Use Complex Matrices in Transient Response By Ted Rose

2. Or (more simply)Some Uses for Fourier Transforms in Transient Analysis By Ted Rose

3. 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

4. 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

5. Fourier Transforms in Transient • The user interface is simple: • Set up your file for transient response • Change the solution to 108 or 111 • Add a FREQ command to CASE CONTROL • Add a FREQ1 entry to the BULK DATA • Use a constant DF = 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)

6. 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

7. Applied Load in Transient

8. Load after Fourier Transform Duration of TLOAD2 Is 1.0, therefore, DF=1./1.=1.

9. Load after Fourier Transform Poorly selected Input for FREQ1 – Although DF is 1.0, the Starting frequency is .5, Resulting in a poor transformation $ 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,

10. Compare the Results Original Load Good Fourier Transform Bad Fourier Transform

11. Structural Damping • Handled correctly, it forms a complex stiffness matrix [Ktotal] = [K](1+iG) + iSKeGe • Unfortunately, transient response does not allow complex matrices, so we must approximate structural damping using: [Btotal] = [B] + [K]G/W3 + SkeGe/W4 • Where w3 and w4 are the “dominant” frequency of response

12. 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

13. 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

14. Multi-Frequency Steady-State • As an example, let us look at a typical jet engine model with 3 rotating imbalances

15. Multi-Frequency Steady-State • All right, how about this model? Model courtesy of Pratt and Whitney

16. 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 Rotating in opposite direction $ 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

17. Multi-Frequency Steady-State