Lecture05-Prof.Ju

Lecture05-Prof.Ju - CEE M237A / MAE M269A Lecture 5:...

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CEE M237A / MAE M269A Lecture 5: Formulation of MDOF Equation of Motion Professor J. Woody Ju
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Matrix Formulation of the MDOF Equation of Motion y Selection of DOF 2 a. Lumped mass method b. F.E.M. : c. Di G screte parameter system for eneralized displacement d m metho ±
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Matrix Formulation of the MDOF Equation of Motion (Cont’d) y Dynamic-Equilibrium Condition 3 () 11 1 1 22 2 2 At any point i: Four forces: For each node: i Ii Di Si IDS pt f f f f ff p t f p t ++= # ( ) f p t ±± ± ± --- applied load inertia damping spring (elastic)
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Matrix Formulation of the MDOF Equation of Motion (Cont’d) 4 () 11 1 2 21 3 3 1 ... , , 1,2,. .., DOF , where Similarly, for damping , where for intertia force , wh stiffness coefficients damping coefficie s re n e t Si ij j Di ij j Ii ij j S ij i N j N fk v fc k v fm v f kv kv kv k v ij N c = = = =+ + + + ⇒= = = ± ±± mass coefficie linear MDO ts n F ij mv c v m vk p t + += = ± ²² ²² ²² ²
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Matrix Formulation of the MDOF Equation of Motion (Cont’d) y Geometric Stiffness – Axial Force Effects 5 N ( ) () geometric forces In particular, , where geometric stiffness coef 0: buckling pro ficients effective stiffness matr bl wher em i x ! e IDS G Gi Gij j k Gij G ff f f p t fk v mv k cv k k v p t k k ++ = = ++− = = = = ± ±± ± ± ± ²² ² ³´µ´¶ ±± ±± ± ± ± ± ± ±
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Evaluation of Elastic Properties y Evaluation of Elastic Properties - 6 s f ± 1 Flexibility matrix: , Fk vF P = = ±± ± displacement () If statics,
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Lecture05-Prof.Ju - CEE M237A / MAE M269A Lecture 5:...

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