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Lecture8_2015

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  The Finite Element Method for the Analysis of Non-Linear and Dynamic Systems Prof. Dr. Eleni Chatzi Lecture 6 - 5 November, 2015 Institute of Structural Engineering Method of Finite Elements II 1  Introduction to Dynamic Analysis The very basics - Newton’s 2nd law of motion:   ( ) mx w k x = − Δ+    ( ) 0 mx w k x − + Δ+ =  ( ) 0 mx k k x mx kxk  − Δ+ Δ+ = + =     0 n  x x x xm ω  + = + =⇓     k  Unstretched position m Δ   m  Stretched position(static equilibrium) w mg  =    x  ,   x ( t ) =  Asinω n t  +  Bcosω n t ⇒ x ( t ) = ˙ x 0 ω n sinω n t  +  x 0 cosω n t where  ω n  = 2 πT   =   km   Institute of Structural Engineering Method of Finite Elements II 2  Introduction to Dynamic Analysis We have previously considered the equilibrium equations governingthe linear dynamic response of a system of finite elements: M¨U + C˙U + KU =  R where: M :  Mass Matrix K :  Stiffness Matrix U :  Displacements ˙U :  Velocities ¨U :  Accelerations F I  ( t ) + F D ( t ) + F E  ( t ) =  R ( t ) where: F I  ( t ) = M¨U  Inertial Force F D ( t ) = C˙U  Damping Force F E  ( t ) = KU  Internal Force Institute of Structural Engineering Method of Finite Elements II 3  Introduction to Dynamic Analysis When is Dynamic Analysis required? Whether a dynamic analysis is needed or not is generally up toengineering judgmentrequires understanding of the interaction between loading andstructural response!In general if the loading varies over time with frequencies higherthan the Eigen-frequencies of the structure  −→  dynamic analysis willbe required Institute of Structural Engineering Method of Finite Elements II 4
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