Computer Assignment 2 Solutions

# Computer Assignment 2 Solutions - ME 240 Introduction to...

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1 ME 240: Introduction to Dynamics and Vibrations Mechanical Engineering Department University of Michigan Fall 2008 Computer Assignment #2 Solutions Problem 1 a) A free body diagram of the device of mass m . b) ma F x m y x c y x k ) ( ) ( ky y c kx x c x m Plug in ) cos( 0 0 t A y and ) sin( 0 t A y into the previous expression to obtain: ) sin( ) cos( 0 0 0 t kA t A c kx x c x m . The natural frequency of the motion of the device in the absence of the environmental vibrations is m k n . c) For state variables x x 1 , x x 2 . The state equations for the device of mass m are: 2 1 x x m t kA t A c kx cx x ) sin( ) cos( 0 0 0 1 2 2 . All MATLAB code for problems 2-4 are appended to the end of the solutions. x(t) m   ) ( ) ( t y t x c   ) ( ) ( t y t x k

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2 Problem 2 a) b) The steady state amplitude of the motion of the device of mass m (x ss ) is 1.3253 mm. Problem 3 a) 0 2 4 6 8 10 12 14 16 18 20 -1.5 -1 -0.5 0 0.5 1 1.5 x 10 -3 time (s) position (m) Position of the mass vs time 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 / n x ss /A x ss /A vs 0 / n
3 b) The value of the ratio of ω 0 n , which leads to the largest steady state amplitude for the motion of the device of mass m is 0.98425. Acceptable answers include 1, 0.99 and 0.98. Problem 4 a) b) The damping constant has negligible effect on the response of the system when ω 0 n < 0.4. When 0.4 < ω 0 n < 1.4 the plot indicates that the larger the damping constant the lower the steady state amplitude of the response of the system (the largest effect of the damping constant is approximately where ω 0 n = 1). When ω 0 n ≈ 1.4, the effect of the damping constant changes, for greater values of ω 0 n , larger damping constants lead to greater steady state amplitudes of the response of the system. MATLAB Code for Problems 2-4

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Computer Assignment 2 Solutions - ME 240 Introduction to...

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