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MEM427_Lecture_7_Dynamic Problems

MEM427_Lecture_7_Dynamic Problems - Lecture7 DynamicSystems...

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MEM427 Introduction to Finite Element Method Lect re 7 Lecture 7 Finite Element Analysis of Dynamic Systems Chapter 7 Finite Element Analysis of Dynamic Systems 1 MEM427 Introduction to Finite Element Method FE Analysis of Dynamic Systems Modal Analysis : Determine the natural frequencies and natural modes of a structure Harmonic Response Analysis : Determine the cyclic response of a structure (typically using modal superposition method) due to a sustained cyclic load Transient Dynamic Analysis (Time history Analysis): 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 model (earthquake, etc.) Chapter 7 Finite Element Analysis of Dynamic Systems 2 MEM427 Introduction to Finite Element Method Analysis of Dynamic Systems E ti 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 m k ω Let   t f   t x   t f   t x Free Body Diagram     0 2 t x t x n n 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:   t B t A t i o Solution: x n n sin cos Constants A and B are determined by using the   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 by x 0 , then release it       t f t kx t x m   t f   t x   t f   t x Free Vibrations     m k t x t x 0 2   t x t x n cos 0 Chapter 7 Finite Element Analysis of Dynamic Systems 4 n n , 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 F t k t i   t kx k     t F t x t x sin 2 kx x m f sin m x m m m f n 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
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