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145 CT 4140 Structural Mechanics Faculty of Civil Engineering and Geosciences 2009 Delft, The Netherlands Forced Vibrations of a Fixed-Fixed String under General Forcing Multiplying the above equation by sin( β m x) and integrating from 0 to L, we obtain: ( ) ( ) ( ) 2 2 1 2 0 2 , sin L n n n n n d q x t x dx Q t dt AL ω β ρ Ψ + Ψ = = The general solution to the above equation reads: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 1 0 cos sin sin t n n n n n n n n n d t t t Q t d dt ω ω τ ω τ τ ω ω Ψ Ψ = Ψ + + Thus, we found the following expression that satisfies the equation of motion and the boundary conditions: ( ) ( ) ( ) ( ) 1 , sin with defined above n n n n w t x t x t β = = Ψ Ψ To find the unknown constants, we will use the initial conditions. Lecture 11 Forced Vibrations of a Fixed-Fixed String: General Forcing 146 CT 4140 Structural Mechanics Faculty of Civil Engineering and Geosciences 2009 Delft, The Netherlands Forced Vibrations of a Fixed-Fixed String under General Forcing Substituting the above solution into the initial conditions, we obtain: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 0 1 0, 0,
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