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Unformatted text preview: Vibrations Homework Problems ME 274 Problem VI2 A homogeneous disk having a mass of m and outer radius R is supported by a cable. This cable is attached to ground at end B and attached to a spring at end E. This spring has a stiffness of k with its upper end attached to ground at D. Let x represent the motion of the disk’s center of mass O (measured positively downward) with the spring being unstretched when x = 0. Assume that the cable does not slip on the disk and does not go slack at any time. a) Draw a free body diagram of the disk. b) Derive the single differential equation of motion (EOM) for the system in terms of the coordinate x and, at most, the following parameters: g , m , R and k . c) Let x st represent the value of x when the system is in static equilibrium. Recognizing that !! x = when the system is in static equilibrium, determine the value of x st in terms of the system parameters from your EOM in b). d) Let z represent the motion of the system as measured from its static equilibrium case; that is, z ! x ! x st . Substitute this definition along with x st from c) into your EOM from b) to find the EOM in terms of z. How does this EOM differ from the one that you found in b)? k VERTICAL PLANE g R O C A x + m B D E Vibrations Homework Problems ME 274 Problem VI3 A homogeneous disk having a mass of m and outer radius R is pinned to ground at its center O. This drum is in gearedcontact with block A. Block A, having a mass of 3m, is able to slide along a smooth horizontal surface and in such a way that the block does not slip in its contact with drum. Two springs, having stiffnesses of k and 2k, are attached between block A and ground, as shown in the figure below. Let ! represent the rotation of the drum with ! being measure positively counterclockwise and with ! = when the springs are unstretched. a) Draw individual free body diagrams of the drum and block. b) Derive the single differential equation of motion (EOM) for the system in terms of the coordinate ! and, at most, the following parameters: m , R and k . c) Based on your EOM from b) above, determine the natural frequency of free vibrations for the system. Express your answer in BOTH rad/sec AND cycles per second (Hz). d) The system is released with the springs unstretched with ! ! ( ) = " (CCW). Determine the response of the system, ! t ( ) , for t > 0. Use the following parameters in your analysis: m = 3 kg , k = 5600 N / meter , R = 0.25 meters and ! ¡ = ¡.4 rad / s . O C ! + m 3m smooth smooth no slip R k 2k A Vibrations Homework Problems ME 274 Problem VI5 A homogeneous disk having a mass of m and outer radius R rolls without slipping on a rough horizontal surface. A spring of stiffness k is connected between the center O of the disk and ground on the left side of the disk. A second spring (of stiffness 3k ) and a dashpot with damping constant c is connected between O and ground on the right side of the disk, as shown in the figure below. Let !...
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 Fall '09
 Force, Mass, single differential equation

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