259355946-2009-CT-4140-Book-of-Slides.9

259355946-2009-CT-4140-Book-of-Slides.9 - 9 Resonance in 1...

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29 Resonance in 1 DOFS without Damping CT 4140 Structural Mechanics Faculty of Civil Engineering and Geosciences 2009 Delft, The Netherlands Frequency-domain representation Time-domain representation of resonance : of resonance 0 1 2 3 ω/ω n 0 1 2 3 4 5 A/x static 0 2 2 1 1 n F A k ω ω = 0 5 10 15 20 25 t [ s ] -20 -10 0 10 20 x ( t ) k / F 0 ( ) ( ) 0 sin 2 F x t t t k ω ω = Lecture 2 Recollection of knowledge regarding undamped 1 DOF systems 30 Vibrations of 1 DOFS without Damping under General Disturbing Force CT 4140 Structural Mechanics Faculty of Civil Engineering and Geosciences 2009 Delft, The Netherlands Equation of motion and initial conditions: ( ) ( ) ( ) 0 0 0 0 mx+kx= F t x x , x v = = && & Solution in the form of Duhamel’s integral: ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 1 cos sin sin t n n n n n v x t x t t F t t t dt m ω ω ω ω ω = + + The idea behind the Duhamel’s integral is representation of the force by a sequence of short pulses: t F t dt ( ) ( ) ( ) ( ) 1 sin , p n n x t F t t t dt t t m ω ω = > Lecture 2
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