CALCULATING TRANSFER FUNCTIONS FROM NORMAL MODES Revision F

Transcription

1 CALCULATING TRANSFER FUNCTIONS FROM NORMAL MODES Revision F By Tom Ivine tom@vibationdata.com Januay, 04 Vaiables F f N H j (f ) i Excitation fequency Natual fequency fo mode Total degees-of-feedom The steady state displacement at coodinate i due to a hamonic foce excitation only at coodinate j Damping atio fo mode i Mass-nomalized eigenvecto fo physical coodinate i and mode numbe Excitation fequency (ad/sec) Natual fequency (ad/sec) fo mode Receptance The steady-state displacement at coodinate i due to a hamonic foce excitation only at coodinate j is: N i j Hi j(f ) ĵ () whee f / f () ĵ (3)

2 Note that the phase angle is typically epesented as the angle by which foce leads displacement. In tems of a C++ o Matlab type equation, the phase angle would be Phase = - atan(imag(h), eal(h)) (4) Note that both the phase and the tansfe function vay with fequency. A moe fomal equation is imag Hi j(f ) Phase(f ) actan eal Hi j(f ) (5) Mobility The steady-state velocity at coodinate i due to a hamonic foce excitation only at coodinate j is Ĥ i j (f ) j N i j ĵ (6) Acceleance The steady-state acceleation at coodinate i due to a hamonic foce excitation only at coodinate j is N i j H ~ i j(f ) ĵ (7)

3 3 Relative Displacement Conside two tanslational degees-of-feedom i and j. A foce is applied at degee-offeedom k. The steady-state elative displacement tansfe function R ij between i and j due to an applied foce at k is N k j N k i jk ik i j ĵ ĵ (f ) H (f ) H R (8) N k j i i j ĵ R (9) The steady-state elative displacement tansfe function R ij between i and j due to an applied foce at k is N k j N k i jk ik i j ĵ ĵ (f ) H (f ) H R (0)

4 4 N k j i N k j i i j j R () Refeence. R. Caig & A. Kudila, Fundamentals of Stuctual Dynamics, Second Edition, Wiley, New Jesey, 006.

5 APPENDIX A EXAMPLE m x k Conside the system in Figue A-. Assign the values in Table A-. Table A-. Paametes Vaiable Value Unit m.0 lbf sec^/in k 3.946e+05 lbf/in Damping Ratio y 5

6 PHASE (deg) TRANSFER FUNCTION PHASE H FREQUENCY (Hz) Figue A-. This is the phase angle by which the foce leads the esponse. 6

7 Disp/Foce (in/lbf) TRANSFER FUNCTION MAGNITUDE H FREQUENCY (Hz) Figue A-. 7

8 APPENDIX B EXAMPLE Nomal Modes Analysis m x k 3 k x m k y Figue B-. Conside the system in Figue B-. Assign the values in Table B-. Table B-. Paametes Vaiable Value Unit m 3.0 lbf sec^/in m.0 lbf sec^/in k 400,000 lbf/in k 300,000 lbf/in k 3 00,000 lbf/in Futhemoe, assume. Each mode has a damping value of 5%.. Zeo initial conditions 8

9 The homogeneous, undamped poblem is z 500,000 z 00,000 00,000 z 0 400,000 z 0 (B-3) The eigenvalue poblem is 500,000 00,000 00,000 q 0 400,000 q 0 (B-4) The analysis is pefomed using Matlab scipt: tansfe_fom_modes.m >> tansfe_fom_modes tansfe_fom_modes.m ve.4 June 4, 00 by Tom Ivine This pogam calculates a tansfe function (displacement/foce) fo each degee-of-feedom in a system based on the mode shapes, natual fequencies, and damping atios. Select input method =mass & stiffness matices =natual fequencies and mass-nomalized eigenvectos Select output metic =displacement/foce =velocity/foce 3=acceleation/foce Ente the mass matix name: mm mass = Divide mass by 386? =yes =no 9

10 Ente the stiffness matix name: kk stiffness = Natual Fequencies (Hz) Modes Shapes (column fomat) QE =

11 Disp/Foce (in/lbf) 0-4 Tansfe Function Magnitude H Fequency(Hz) Figue B-. The cuve is the steady-state displacement at coodinate due to a hamonic foce excitation only at coodinate.

12 Figue B-3.

13 Disp/Foce (in/lbf) 0-4 Tansfe Function Magnitude H Fequency(Hz) Figue B-4. The cuve is the steady-state displacement at coodinate due to a hamonic foce excitation only at coodinate. Due to ecipocity, it is also the steady-state displacement at coodinate due to a hamonic foce excitation only at coodinate. 3

14 Figue B-5. 4

15 Disp/Foce (in/lbf) 0-4 Tansfe Function Magnitude H Figue B Fequency(Hz) The cuve is the steady-state displacement at coodinate due to a hamonic foce excitation only at coodinate. 5

16 Figue B-7. 6