A uniform disk of radius r and mass md rolls without slipping on a cylindrical s

Question

A uniform disk of radius r and mass md rolls without slipping on a cylindrical surface and is attached to a uniform slender bar AB of mass mb. The bar is attached to a spring of constant K and can rotate freely in the vertical plane about point A as shown in the figure . If the bar AB is displaced by small angle and released, determine

The energy of the system in terms of theta and theta . The equation of motion in terms of theta and theta . The equation of motion in terms of theta and theta Lagrange's equations. The natural frequency of vibration of the system.

Explanation / Answer

Let theta=Th and D(theta)/Dt= DTh and D(D(theta)/Dt)/Dt =DDTh

a) The total energy = energy of spring + rotational energy of slender rod and disc + kinetic energy of disc

= 1/2*k*(L/2*Th)2 + 1/2*(1/12*mb*L2)*(DTh2) + 1/2*(.5*md*r2)*(L/2r*DTh)2 + 1/2*md*(L/2*DTh)2

As total energy remains constant, differentiating the above equation we get,

[ k*Th*(L/2)2 + (1/12*mb*L2)*DDTh + (.5*md*r2)*( L/2r + 1 )2*DDTh + md*(L/2)2*DDTh ]*DTh = 0

or k/4*Th + 1/12*mb*DDTh + 1/2*(1/2+r/L)2*md*DDTh + 1/4*md*DDTh=0

or 3k*Th + ( mb + 6*(1/2+r/L)2*md + 3*md )*DDTh =0 ....b)

c) Natural frequency = [ 3k/( mb + 6*(1/2+r/L)2*md) + 3*md ].5

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