RESPONSE OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM SUBJECTED TO A TERMINAL SAWTOOTH APPLIED FORCE

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1 RESPONSE OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM SUBJECTED TO A TERMINAL SAWTOOTH APPLIED FORCE By Tom Irvie Emil: tomirvie@ol.om Ferury 5, 6 Itroutio Coier the igle-egree-of-freeom ytem i Figure. F(t) x m k Figure. where m i the m i the viou mpig oeffiiet k i the tiffe x i the olute iplemet of the m F(t) i the pplie fore A free-oy igrm i how i Figure.

2 F(t) m kx x& Figure. Summtio of fore i the vertil iretio F m & x () m & x x& kx F(t) () mx && x& k x F(t) () && k x x x F(t) m & m m (4) By ovetio ( m) ξω (k m) ω where ω i the turl frequey i (rie), ξ i the mpig rtio. Sutitute the ovetio term ito equtio (5). & x ξωx& ω x F(t) (5) m

3 Termil Swtooth Pule Coier the pule give y equtio (6). F (t) Fˆ t T,, t T t > T (6) The equtio of motio eome ˆF t && x ξω x& ω x, m T t T (7) Now tke the Lple trform. { } ˆF t L && x ξω x& ω x L m T (8) X() x() x() & ξω X() ξωx() Fˆ ω X() (9) { } { } { } Fˆ ξω ω X() x() & ξω x() () { } { } Fˆ ξω ω X() x() & ξω x() ()

4 { } ˆ x() & ξω x() F X() ξω ω ξω ω () Let X () X () X f () () where { ξω } x() & x() X () ξω ω (4) Fˆ X f () ξω ω (5) Coier the eomitor term, ( ) ( ) ξω ω ξω ω ξω (6) ( ) ( ) ξω ω ξω ω ξ (7) Now efie the mpe turl frequey, ω ω ξ (8) Sutitute equtio (8) ito (7), ξω ( ) ω ξω ω (9) Sutitute equtio (9) ito (5). 4

5 x() & X () { ξω } x() ( ξω ) ω () Rerrge the term ito oveiet formt prior to the ivere Lple trform. X () ( ξω ) ( ξω ) x() ω x() & ( ξω ) x() ( ξω ) ω () X () ( ξω ) ( ξω ) ( ξω ) x() & x() ω x() ω ω ( ξω ) ω () Tke the ivere Lple trform uig Referee. x (t) x() exp ( ξω ) o( ω ) t t ( ξω ) x() & ω x() exp ( ξω ) x() ( ξω ) ( ω ) x() t i t t & x (t) exp x() o ω ( ξω ) i( ω ) t ( ω ) t () (4) Rell equtio (5). Fˆ X f () ξω ω (5) Exp ito prtil frtio uig Appeix A. (6) 5

6 ξω (7) ω (8) X () f ˆ F ξω ω ξω ξω ω 4 ω ξω ω ( ) (9) X () f ˆ F ξ ω ξ ξ ω ω ω ξω ω ( ) () X () f ˆ F ξ ω ξ ξ ω ω ω ξω ω ( ) () ( ξω ) ω ξω ω () X () f ˆ ( F ξ ω ξ ξ ) ω ω ω ( ξω ) ω () X () f ξ ω ( ξ) ˆF ξ ω ω ( ξω ) ω (4) 6

7 ( ) Fˆ ω ξ ξ ω ξ X f () [ ] ξ ω ( ξω ) ω (5) ω ( ) ξ ˆF ξ ω ξ X f () [ ] ξ ω ( ) ξω ω (6) ω ( ) ξ ˆF ξ ω ξ X f () [ ] ξ ( ) ω ξω ω (7) ˆ F ξ ω ξω X () ω f ξ ( ) ( ω ξω ω ξω ) ω (8) ˆ F ξ ω ξω X () ω f ξ ( ) ( ω ξω ω ξω ) ω (9) ˆ F ξ ω X () ξω ξ ω ω f ξ ( ) ( ω ξω ω ξω ) ω (4) 7

8 ˆ F ξ ω ξω ξ X f () ξ ω ( ) ( ω ξω ω ξω ) ω (4) X () f ˆ F ξ ξ ξω ξ ω ω ω ( ) ( ) ξω ω ξω ω (4) X () f ˆ F ξ ξ ξω ξ ω ω ( ) ( ) ω ξω ω ξω ω (4) Tke the ivere Lple trform uig Referee. The iplemet i ˆF ξ ξ x (t) t exp f ( t) o( t) ( ) i ( t) ξω ω ξ ω ω ω ω ω (44) ( ξω ) x() ( ξω ) ( ω ) x() t i t t & x (t) exp x() o ω ( ω ) (45) 8

9 The totl iplemet i ( ) x() & ξω x() x(t) exp( ξωt ) x() o( ω t) i ( ωt ) ω ˆF ξ ξ t exp ( t) o( t) ( ) i ( t ), ξω ω ξ ω ω ω ω ω t T (46) The olutio for t > T i the free virtio olutio. The totl veloity i ( ) x() & ξω x() x(t) & ξω exp( ξωt) x()o( ω t) i( ωt) ω { } ( ) ( ) & ( ) ( ) exp ξωt x() ωi ω t x() ξω x() o ωt ˆF ξ exp( t) o( t) ( ) i ( t) ξω ξω ω ξ ω ω ω ω ˆF ξω { ( ) exp t } i ( t) ω ξω ω ( ξ ) o( ωt ), ω t T (47) 9

10 ω ( ) x(t) & exp ξωt x() & o( ω t) [ ξx() & ω x() ] i ( ωt ) ω Fˆ ξω exp ( t) o ω ξω ( ω t) ( ξ ) i ( ωt ) ω ω ω, t T (48) ω x(t) exp( t) x() o( t) & ξω & ω [ ξx() & ω x() ] i ( ωt ) ω ˆF ξω exp( t ) o( t) ( ) ( ) i ( t) ξω ω ξ ξ ω ω ω, t T (49)

11 ω x(t) exp( t) x() o( t) & ξω & ω [ ξx() & ω x() ] i ( ωt ) ω ˆF ξω exp( t) o( t) i ( t ), ξω ω ω ω ω t T (5)

12 Exmple APPLIED FORCE TERMINAL SAWTOOTH PULSE (, N,. e ) 8 FORCE (N) Figure. TIME (SEC) DISPLACEMENT TIME HISTORY ( m5 kg, f 8 Hz, Q ) 5 DISP (mm) TIME (SEC) Figure 4. A igle-egree-of-freeom ytem i ujete to the pplie fore i Figure. The repoe i give i Figure 4. The hrteriti of the ytem re give i the plot title.

13 Exmple NON-DIMENSIONAL SRS APPLIED FORCE TERMINAL SAWTOOTH PULSE Q 5 Negtive Poitive K X F.5... DURATION PERIOD Figure 5. A o-imeiol iplemet SRS i give i Figure 5. The perio i the ivere of the turl frequey. Referee. T. Irvie, Tle of Lple Trform, Virtiot, T. Irvie, Prtil Frtio i Shok Virtio Alyi, Virtiot, 999.

14 4 APPENDIX A Prtil Frtio Expio (A-) [ ] [ ] (A-) (A-) [ ] [ ] [ ] (A-4) (A-5) (A-6)

15 5 (A-7) (A-8) (A-9) ( ) (A-) ( ) [ ] (A-)

16 6 [ ] (A-) (A-) (A-4) (A-5)