Mechanical system with two degree of freedom

[ohmaley]

Homework Statement

1) Two cylinder can roll without sliding on a horizontal plan. they are connected throught the center by a spring of rigidity "k".

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They both share the same radius "R" and mass "m". The left cylinder has its mass around the perimeter and the right cylinder's mass is spread uniformly.

http://tinypic.com/r/16kpevc/7

a)Determine the free body diagram. Figure out the horizontal and rotational mouvements of each rolling bodies.

b) adding the cinematic equations (the Xi in function of the rotation angles θi), determine the differential equation system in x1 and x2.

c) Calculate both w and their associated modes.

Homework Equations

sum Fx = m*agx

sum Mg = Jg*α (alpha)

Inertia momentum of the two cyclinders:

Jg=(M*R^2)/2 for mass spread uniformly

Jg=m*R^2 for mass around perimeter

The Attempt at a Solution

I figured since there is no sliding -> X1=R* θ1 and X2=R" θ2.

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m1*ag=-k(x1-x2)

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m2*ag=k(x1-x2)

with x(t) = Rcos(wt), "R" being the amplitude

i am not sure how to proceed about this problem.

 

[lippyka]

Any help would be greatly appreciated. For the free body diagram, the forces acting on the cylinders are the gravitational force, the spring force due to the connection between the two cylinder, and the normal force. The gravitational force will act downwards, the spring force will be equal in magnitude but opposite in direction between the two cylinders, and the normal force will act perpendicular to the surface. For the equations of motion, we can use the equations of Newton's second law:sum Fx = m1*agx1 = -k(x1 - x2)sum Fx = m2*agx2 = k(x1 - x2)For the rotational equations, we can use the equation of angular momentum:sum Mg = Jg1*α1 = -k(θ1 - θ2)sum Mg = Jg2*α2 = k(θ1 - θ2)where Jg1 and Jg2 are the inertial moments of the cylinders.Then, by combining the equations of motion and angular momentum, we can find the differential equation system in x1 and x2.Finally, solving the differential equation system will give us the frequency and associated modes of the system.