259355946-2009-CT-4140-Book-of-Slides.42

# 259355946-2009-CT-4140-Book-of-Slides.42 - 42 Vibrations of...

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161 Vibrations of a Fixed-Fixed String Subjected to Distributed Damping CT 4140 Structural Mechanics Faculty of Civil Engineering and Geosciences 2009 Delft, The Netherlands Consider the coordinate-related part: ( ) ( ) 2 2 0, 0 0 W W W W L c ω ′′ + = = = The general solution for W : ( ) ( ) ( ) sin cos , W x C x D x c β β β ω = + = Substituting this expression into the boundary conditions, we obtain: ( ) ( ) ( ) 0 0 0 0 sin 0 W D W L L β = = = = Thus the separation constant and the mode shapes are the same as in the undamped case: ( ) ( ) , 1,2,3,... the separation constant sin the mode shapes n n n n n n L n c n c L n W x C x β π ω β π β = = = = = Lecture 12 Vibrations of a String with Distributed Damping 162 CT 4140 Structural Mechanics Faculty of Civil Engineering and Geosciences 2009 Delft, The Netherlands The general solution and initial conditions: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 1 0 , exp sin cos sin 0, , , , d n n n n n n n n d t w x t n t A t B t x w x x w t x t x n ω ω β ϕ ψ ω ω = = = + = = = Substituting the general solution into the initial conditions, we obtain: ( ) ( ) ( ) ( ) ( ) 1 1 1 sin
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• Fall '09
• Andrei V. Metrikine
• Structural Engineering, Dynamics Of Structures, SDOF, MDOF, Vibration, Trigraph, Mode shape

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