FREQUENCY RESPONSE FUNCTION ANALYSIS OF A MULTI-DEGREE-OF-FREEDOM SYSTEM WITH ENFORCED MOTION

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1 REQUENCY RESPONSE UNCON ANALYSS O A UL-DEGREE-O-REEDO SYSE WH ENORCED OON By om rvine tomirvine@aol.com August 6, Variables f Q u u u f i i ass matri Stiness matri Applie forces orces at riven noes orces at free noes entity matri ransformation matri Eigenvector matri Displacement vector Displacements at riven noes Displacements at free noes Ecitation frequency Natural frequency for moe i Damping ratio for moe i

2 he equation of motion for a multi-egree-of-freeom system is [ ][u] [][u] () u [ u] () uf Partition the matrices an vectors as follos f f u uf f f u uf f () he equations of motions for enforce isplacement an acceleration are given in Appenices A an B, respectively. Create a transformation matri such that u u uf u (4) (5) f f u u f f u u f (6) Premultiply by, f f u u f f u u f (7)

3 APPENDX A Enforce Acceleration Again, the partitione equation of motion is f f u u f f u u f (A-) ransform the equation of motion to uncouple the stiness matri so that the resulting stiness matri is ˆ ˆ (A-) f f (A-) f f f (A-4) f f f f (A-5)

4 f f f f f f f f (A-6) (A-7) Let (A-8) f f f f ˆ ˆ (A-9) f (A-) f (A-) f (A-) (A-) 4

5 5 f f ˆ (A-4) ˆ (A-5) f f (A-6) By similarity, the transforme mass matri is f f f f (A-7) f ˆ (A-8) f f ˆ (A-9) f f ˆ (A-) ˆ u u ˆ ˆ u u (A-)

6 u u ˆ u ˆ (A-) he equation of motion is thus u ˆ u ˆ u (A-) No solve the generalize eigenvalue problem. Let Q be the eigenvector matri. Let u Q (A-4) Q ˆ Q ˆ u (A-5) Premultiply by Q to ecouple the equations of motion. Q Q Q ˆ Q Q ˆ u (A-6) Assume that the applie forces are all zero. hus the only ecitation is the enforce acceleration. Q Q Q ˆ Q Q u (A-7) 6

7 Assume that the applie forces are all zero. hus the only ecitation is the enforce acceleration. Q Q Q ˆ Q Q u (A-8) ~ Q Q (A-9) ~ Q ˆ Q (A-) he ecouple equations of motion ith an ae moal amping matri C ~ are ~ C ~ ~ Q u (A-) Note that ~ is the ientity matri is a iagonal matri C ~ is a iagonal matri containing the terms ii ~ is a iagonal matri containing the terms i hus, i ii, i i, i Q,iu (A-) Perform a steay-state analysis. Represent the enforce motion via a ourier transform. u Û ep( jt) (A-) Assume N ep(j t) (A-4) 7

8 By substitution, Û ep(j t) - N ep(j t) j N ep(j t) N ep(j t) Q,i ii,i i,i,i (A-5) - N,i jii N,i i N,i Q,i Û (A-6) j ii i N,i Q,i Û (A-7) i jii N,i Q,i Û (A-8) N,i i Q i j i Û (A-9) he final acceleration frequency response functions are foun via U Q N (A-4) U U Uf U (A-4) f (A-4) 8

9 APPENDX B Enforce Acceleration Eample Equation of otion m m y m igure B-. Consier the system in igure B-. Assign the values in able B-. able B-. Parameters Variable m Value. g m. g m. g *, N/m, N/m, N/m he value of m is arbitrary because its motion ill be enforce. urthermore, assume that each moe has a amping value of 5%. 9

10 he folloing equations of motion are erive in Appeni C for the system in igure B-. m m m (B-),,,, 5,,,, 4, (B-) R Analysis he analysis is performe via atlab script: enforce_acceleration_frf.m. >> enforce_acceleration_frf enforce_acceleration_frf.m ver. August 6, by om rvine Enter the units system =English =metric Assume symmetric mass an stiness matrices. mass unit = g stiness unit = N/m Select file input metho =file preloae into atlab =Ecel file ass atri Enter the matri name: mmm Stiness atri Enter the matri name: nput atrices

11 mass = sti = Select moal amping input metho =uniform amping for all moes =amping vector Enter amping ratio.5 number of ofs = Enter the enforce acceleration of = =.e+5 * Natural requencies No. f(hz). 4.9e

12 oes Shapes (column format) oeshapes = =.. = = Natural requencies No. f(hz) oes Shapes (column format) oeshapes = Participation actors part =.756.8

13 RANS (G/G) RANSSSBLY AGNUDE of of of... REQUENCY (Hz) igure B-.

14 APPENDX C ( - ) m m ( - ) ( - ) m (- ) (- ) - m ( ) m m ( - ) ( - ) m (- ) (- ) ( - ) ( - ) m ( ) ( - ) ) m ( - ) ) m ( - ) ( - ) m (- ) (- ) m 4