MEM427_Lecture_7_Dynamic Problems

# MEM427_Lecture_7_Dyn - Lecture7 DynamicSystems ModalAnalysis: :D

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MEM427 Introduction to Finite Element Method ct re Lecture 7 Finite Element Analysis of Dynamic Systems Chapter 7 Finite Element Analysis of Dynamic Systems 1 MEM427 Introduction to Finite Element Method E Analysis of Dynamic Systems FE Analysis of Dynamic Systems Modal Analysis : Determine the natural frequencies and natural modes of a structure Harmonic Response Analysis : Determine the cyclic response f a structure (typically using modal superposition method) of a structure (typically using modal superposition method) due to a sustained cyclic load Transient Dynamic Analysis (Time history Analysis): etermine the dynamic response of a structure (typically using Determine the dynamic response of a structure (typically using direct integration methods) under the action of any general time dependent loads Spectrum Analysis : Modal analysis results are used with a known spectrum to calculate displacements and stresses in the odel (earthquake, etc.) Chapter 7 Finite Element Analysis of Dynamic Systems 2 model (earthquake, etc.) MEM427 Introduction to Finite Element Method Analysis of Dynamic Systems t i f ti One degree of freedom (mass spring) systems:    t x m t f t kx F Equation of motion:  t kx k t f t kx t x m Free Vibrations: f ( t ) = 0 m x m m   0 t kx t x m et t f t x t f t x Free Body Diagram 0 2 t x t x n m k ω n Let Chapter 7 Finite Element Analysis of Dynamic Systems 3 t B t A t x n n sin cos Solution: MEM427 Introduction to Finite Element Method Analysis of Dynamic Systems Free vibration of one degree of freedom systems: lution:   t B t A t x n n sin cos Constants A and B are determined by using the Solution: t kx k initial conditions (I.C.) Example : Pull the mass down m x m m 0 0 , 0 : I.C. 0 x x x p by x 0 , then release it t f t kx t x m t f t x t f t x Free Vibrations 2   t x t x cos Chapter 7 Finite Element Analysis of Dynamic Systems 4     m k t x t x n n , 0 n 0 http://en.wikipedia.org/wiki/Harmonic_oscillator

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MEM427 Introduction to Finite Element Method Analysis of Dynamic Systems One degree of freedom systems: Forced Vibrations ( f (t) 0)  t F t f f sin Example: t kx k F n 2     t F t kx t x m f sin m x m m     t m t x t x f n sin Solution: t f t x t f t x     t f t kx t x m t A t x n f n sin ) ( 1 2 Chapter 7 Finite Element Analysis of Dynamic Systems 5 MEM427 Introduction to Finite Element Method Analysis of Dynamic Systems Two degrees of freedom systems: Equations of Motion 1 1 x k t f x k x k x m t f x k x k k x m 2 2 2 1 2 2 2 1 2 2 1 2 1 1 1 1 k 1 m 1 1 x m t f x k k k x m 1 1 2 2 1 1 1 0 1
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## This note was uploaded on 11/14/2010 for the course MEM 427 taught by Professor Tein-mintan during the Fall '10 term at Drexel.

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MEM427_Lecture_7_Dyn - Lecture7 DynamicSystems ModalAnalysis: :D

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