FREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revisio B By Tom Irvie Email: tomirvie@aol.com February, 005 Derivatio of the Equatio of Motio Cosier a sigle-egree-of-freeom system. m x k c where m c k x is the mass is the viscous ampig coefficiet is the stiffess is the absolute isplacemet of the mass Note that the ouble-ot eotes acceleratio. The free-boy iagram is m x k x cx Summatio of forces i the vertical irectio F mx (A-)
mx cx kx (A-) mx cx kx 0 (A-3) Divie through by m, By covetio, x c m x k m x 0 (A-4) (c / m) (k / m) where is the atural frequecy i (raias/sec), is the ampig ratio. By substitutio, x x x 0 (A-5) Now take the Laplace trasform. { x x x} { 0} (A-6) s X() s sx() 0 x ( 0) sx() s x() 0 Xs () 0 (A-7) { } { } { } s s X() s x ( 0) s x( 0) 0 (A-8) { } { } s s X() s x ( 0) s x( 0) (A-9)
{ } x( 0) s x( 0) Xs () s s (A-0) Cosier the eomiator of equatio (A-0), ( ) ( ) s s s (A-) ( ) ( ) s s s (A-) Now efie the ampe atural frequecy, (A-3) Substitute equatio (A-3) ito (A-), ( ) s s s (A-4) { } ( s ) x( 0) s x( 0) Xs () Xs () ( s ) ( s ) x ( ) ( s ) x( 0) ( 0) x( 0) (A-5) (A-6) Xs () ( s ) ( s ) x( 0) ( ) x ( 0) x( 0) ( s ) (A-7) 3
Oscillatory Motio Now take the iverse Laplace trasform usig staar tables. Assume that <. This case is referre to as oscillatory motio. The resultig isplacemet is exp A alterate form is ( t) [ ] cos( t) ( ) si < exp The velocity is exp exp exp exp ( t), ( t) { [ ] cos( t) [ ( ) ] si( t) }, < ( t) { [ ] cos( t) [ ( ) ] si( t) } ( t) { [ ] si( t) [ ( ) ] cos( t) }, < ( t) [ ] cos( t) [ ( ) ] si( t) ( t) { [ ] si( t) [ ( ) ] cos( t) }, < (B-) (B-) (B-3) (B-4) x (t) exp ( t) cos( t) [ ] [ ( ) ] si( t), < (B-5) 4
x (t) exp ( t) cos( t) si( t), < (B-6) x (t) exp ( t) cos( t) si( t), < (B-7) x (t) exp ( t) cos( t) [ ( ) ] si( t), < (B-8) exp ( t) cos( t) [ ] si( t), < (B-9) exp( t) cos( t) [ ] si( t), < (B-0) 5
Critically Dampe Motio Recall, (C-) Cosier the special case where The ampe atural frequecy chages to (C-) 0 (C-3) This case is referre to as critically ampe motio. Substitute equatios (C-) a (C-3) ito equatio (A-6), X(s) X(s) ( s ) ( s ) ( s ) s ( s ) (C-4) (C-5) The resultig isplacemet is fou via a iverse Laplace trasformatio. ( t) {[ ] [ ] t }, exp (C-6) 6
No-oscillatory Motio Now cosier the special case where > (D-) Recall equatio (A-0), restate here as equatio (D-). { s } X(s) s s (D-) Solve for the roots of the eomiator. s, ( ) 4 ± (D-3) s, ± (D-4) s, ± (D-5) Note that s s (D-6) s s (D-7) s s (D-8) 7
Equatio (D-) ca be rewritte as X(s) s X(s) { s } [ s s ][ s s ] [ ] [ s s ][ s s ] (D-9) (D-0) Equatio (D-0) ca be expae i terms of partial fractios usig the followig equatio from Referece. αs β β αλ ασ β (D-) ( s λ)( s σ) σ λ s λ s σ The expasio is performe i equatio (D-). s [ ] [ s s ][ s s ] s s [ ] s s [ ] s s s s (D-) X(s) s s [ ] s s [ ] s s s s (D-3) 8
9 Take the iverse Laplace trasform. { } [ ] [ ] s B s A where t) Bexp(s t) A exp(s s s (D-4) Apply the appropriate terms to equatio (D-4). [ ] [ ] B A where t Bexp t A exp (D-5) Simplify B A where t Bexp t A exp (D-6)
Simplify agai, A exp t Bexp t where A B (D-7) Cotrol Theory The trasfer fuctio eomiator forms the characteristic equatio, whe it is set to zero. The roots of the characteristic equatio are calle poles a have a crucial importace. The system is stable if the real part of each root is egative. The roots of the trasfer fuctio umerator are calle the zeros. Agai, the trasfer fuctio for the sigle-egree-of-freeom subjecte to free vibratio is { } x( 0) s x( 0) Xs () s s (E-) A alterative form is { } ( s ) x( 0) s x( 0) Xs () (E-) 0
The characteristic equatio is thus (E-3) ( s ) 0 The poles are thus s ± j (E-4) Or s j ± (E-5) The system is stable as log as > 0. State Space Moel The goverig seco-orer ODE ca be reuce to a pair of first-orer ODEs. x x x 0 (F-) Let x x (F-) x x (F-3) x x x 0 (F-4) The resultig pairs are x x (F-5) x x x (F-6)
The pair of equatios ca be expresse i matrix form as x 0 x x x (F-7) Solve for the eigevalues of he coefficiet matrix. 0 λ et 0 λ (F-8) λ et 0 λ (F-9) λ λ 0 (F-0) The eigevalues are λ ± j (F-) The eigevalues are the same as the poles. The complete solutio for equatio (F-7) is give i Referece. Refereces. T. Irvie, Partial Fractios i Shock a Vibratio Aalysis, Vibratioata Publicatios, 999.. T. Irvie, The State Space Metho for Solvig Shock a Vibratio Problems, Vibratioata Publicatios, 005.