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# hwk02 - ME563 – Fall 2011 Purdue University West...

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Unformatted text preview: ME563 – Fall 2011 Purdue University West Lafayette, IN Homework Set No. 2 Assignment date: Friday, September 2 Due date: Friday, September 9 Please attach this cover sheet to your completed homework assignment. Name PUID _________________________________________ Problem 2.1 __________________ Problem 2.2 __________________ Problem 2.3 __________________ TOTAL __________________ ME 563 – Fall 2011 Homework Prob. 2.1 A homogeneous wheel having a radius of R and mass of M rolls without slipping on a horizontal surface. A massless, inextensible cable is attached to the center of mass G of the wheel and to a block B (having a mass of m). The cable is pulled over a massless pulley. A spring of stiffness k is attached between G and a fixed wall. The coordinate x describes the position of G, with x being zero when the spring is unstretched. a) Use Lagrange’s equations to develop the EOM for this single-DOF system using the generalized coordinate of x. Assume that the cable remains taut for all motion. b) Using this EOM, determine the value of x corresponding to the equilibrium ˙x position of the system. (Note: The system is in equilibrium when x = ˙˙ = 0 .) x homogeneous wheel M k G massless pulley massless, inextensible cable R no slip B m g ME 563 – Fall 2011 Homework Prob. 2.2 A thin, homogenous bar having a mass of M and length of L is pinned to ground at point O. A particle P of mass m is free to slide on the smooth surface of the bar. A spring of stiffness k and unstretched length of R0 is attached between pin O and particle P. Let r be the radial distance from O to P and θ be the rotation of the bar from a fixed vertical line. a) Use Lagrange’s equations to develop the EOM’s for this two-DOF system using the generalized coordinates of r and θ. Recall that the potential energy stored in a spring is related to the square of the stretch in the spring, where the stretch is equal to the difference between its actual length and the unstretched length. Also, in writing down the velocity vector of P, you might need to review the polar kinematics expression for velocity. b) Using these EOM’s, determine the equilibrium values for r and θ. (Note: ˙ r ˙ ˙˙ The system is in equilibrium when r = ˙˙ = θ = θ = 0 ). r O L k θ m P g M ME 563 – Fall 2011 Homework Prob. 2.3 A system is made up of a homogeneous wheel (mass m and radius R), block A (mass of 3m) and block B (mass of 2m). The wheel can roll without slipping on A. Springs are attached between A and ground, between B and ground and between B and the centroid G of the wheel, with stiffnesses of k, 2k and 3k, respectively. The absolute coordinate x 1 describes the position of A, the relative coordinate x 2 describes the position of B relative to A and θ is the rotation of the wheel measured from a vertical line. All springs are unstretched when x 1 = x 2 = θ = 0 . A horizontal force F acts at the centroid G of the wheel. a) Use Lagrange’s equations to develop the EOMs for this three-DOF system using the generalized coordinates of: x 1 , x 2 and θ. b) Write these EOMs in the matrix-vector form of: x [ M ] ˙˙ + [K ] x = where x = f { x, . Identify the components of the mass and stiffness 1 x2 , θ } T matrices [M] and [K], as well as the components of the forcing vector f. x2 2k θ B 2m 3k m smooth R k A x1 F G G 3m no slip ...
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