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Unformatted text preview: Assigned on 28.12.2009 Monday Due date 04.01.2010 Monday, beginning of the lecture ME429 Fall 2009 HW8 You are expected to provide clear explanation of each step in your solution, units, well annotated scaled plots (title, axis labels, units, ..), not random hand sketches, source code attached to your solution if you use a software package. ! 1) A simple car model can be expressed as a two‐degree‐of‐freedom system consisting of a beam supported by two springs and two dashpots as shown below: I, m G x(t) θ( t ) Equilibrium position k c y1(t) a L b k c y2(t) Derive the equations of motion for this system by using Lagrange’s equations with x and θ as the generalized coordinates. Note that y1(t) and y2(t) are inputs to the system. Assume small oscillations. 2) Derive the equations of motion of the system shown below by using Lagrange’s equations with x and θ as the generalized coordinates. Note that x(t) is measured form equilibrium position. k m x(t) θ(t ) m 3) Two of the eigenvectors of a vibrating system as given as 0.2754946 0.6916979 1 0.3994672 , 2 0.2974301 0.4490562 0.3389320 a) Show that these vectors are orthogonal with respect to the following mass matrix: 1 0 0 M 0 2 0 kg 0 0 3 b) Find the remaining eigenvector. c) Using the eigenvectors and the stiffness matrix given below, find the natural frequencies of the system without forming the eigenvalue problem. 6 4 0 K 4 10 0 N/m 0 0 6 ...
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