Question

Q1: Use the equivalent system method to derive the differential equation governing the free vibrations of the system of Figure below. Use x, the displacement of the mass center of the disk from the system's equilibrium position, as the generalized coordinate. The disk rolls without slipping, no slip occurs at the pulley, and the pulley is frictionless. Include an approximation for the inertia effects of the springs. Each spring has a mass ms. Use newton's method. D 1 2m Txo=22 cu
solve with newton's method

Answer 1

1) k In 2m Tre mindre Y xg bethe downward displacement of the block, Let @ be the clockwise angiela rolation of the pulley from the system's equilibrium position Also measured from equilibrium. Then, x B= 2.r eqn 0 = 22'. Eleminating e bet? these equations leads to XB since the disk rolls without slip, It's angular velocity is wo= 2 Yo The inertia eltect of each spring is approximated by placing a porticle of mass ms/3 at the location where the spring is attached to the system.

Is 3 To this end it is imogired that a particle of mass ms/ 3 is attached to the center of the disk and a particle of mass my attached to the block. 5. The tolaj kinetic energy of the system, including the linetic energies of the imagined attached particles is. Tama?+ *3W3+ LIp6+ (27) 2+ Ts+Ts, mit klāmas) (7.+163) 22 (2m)(2.3). mis (23)2 1 + T = 1/3 ( 12 12 ID 10+ m2 + m ) za 15 19 m) - + 틈 ms ea The equivalent mass Meq. = 1 ho The potential energy of Hoe system w cin arbitrary instant V = a + z^4}k (2) V = f(sk)z is comparison to the quadratic form of potential energy leads to keq = 5k

The work done by the viscous campers bel? two arbitrary, instanis is د کرے C(2*)d (2x) 16cide ..companison conto the general form of work done by a viscous comper leads to Ceq = 50. :. The differental equcetion governing Free vibration of the system !5. 19 Ip mt 22 mg) x + 5 x + 5kx=0