Generalized SDOF Systems

Generalized SDOF Systems - CE 573 Structural Dynamics...

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CE 573: Structural Dynamics Generalized SDOF Systems Theory Assume dynamic equilibrium for displacement ( , ): ( ) ( ) ( ) ( ) 0, with each force dependent on ( , ) except for the external forces. I D S E beam beam beam beam v x t F t F t F t F t v x t + + + = E.g.: inertial forces – 2 2 2 2 0 0 2 2 0 0 0 ( ) ( ) ( , ) ( , ); ( ) ( ) ( , ) ( , ); ( ) ( ) ( , ) ( , ) ( ) ( , ) ; ( ) ( , ) ( , ). L L I a a D beam a beam d L L L S c c E beam c beam b v v v v d d b b F t m x x t dx m x t F t c x x t dx c x t t t t t v F t k x v x t dx k v x t EI x x t dx F t f x t dx F x t x = − = − = − = + If we now give the beam a virtual displacement ( , ) v x t δ from dynamic equilibrium, we see that
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* ( ) ( ) ( ) ( ) *0 ( ) ( ) ( ) ( ) 0. I D S E beam beam beam beam I D S E beam beam beam beam total v F t F t F t F t v W t W t W t W t W δ δ δ δ δ δ δ + + + = + + + = 0 = 2 2 2 2 0 0 2 0 ( ) ( ) ( , )* ( , ) ( , )* ( , ); ( ) ( ) ( , )* ( , ) ( , )* ( , ); ( ) ( ) ( , )* ( , ) ( , )* ( , ) ( ) L I a a a beam a L D d d d beam d L S c c c beam c v v W t m x x t v x t dx m x t v x t t t v v W t c x x t v x t dx c x t v x t
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