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Unformatted text preview: ME 563 HOMEWORK # 5 SOLUTIONS Fall 2010 PROBLEM 1: You are given the lumped parameter dynamic differential equations of motion for a two degreeof freedom model of an automobile suspension system for small rotations, with M =1,800 kg, L 1 +L 2 =3.6 m, L 1 =1.4 m, K 1 =42 kN/m, and K 2 =48 kN/m and a radius of gyration of R =1.4 m. Determine the modal frequencies, undamped natural frequencies of oscillation, and the modal vectors. The modal frequencies for this system can be found from the corresponding eigenvalue problem, which is given by: which can be solved in MATLAB to find the following eigenvalues, modal frequencies, undamped natural frequencies and eigenvectors (modal vectors): Draw schematics of the two modal vectors . Schematics of these two modal vectors are shown below. 2 Also give the forms for the two principal modes of vibration. The forms for the two principal modes of vibration each contain a temporal principal coordinate and a spatial modal vector. These forms are given below: Why are these modes adequate for describing general motions of the suspension system? These two modes are adequate because the principle of superposition holds for linear vibrating systems. When the differential equations of motion are rederived in terms of the motion at point P instead of CM, which can be done relatively easily by substituting x p =xL 1 θ , although the modal vectors are different due to the coordinate transformation, the modal frequencies are identical. The reason that the natural frequencies do not change when the coordinates are changed is because the natural frequencies are properties of the system and do not depend on the coordinates selected. natural frequencies are properties of the system and do not depend on the coordinates selected....
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 Fall '09

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