HW 9, ME 740, Spring 2013

HW 9, ME 740, Spring 2013

Nasser M. Abbasi

September 24, 2017

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Let the left most spring stiffness be $k1,$ the middle spring stiffness be $k2$ and the rightmost spring stiffness be $k3$ . Let the mass of the left cylinder be $m1$ and the mass of the right cylinder be $m2$ .

The generalized coordinates are $θ1$ and $θ2$ . Using Lagrangian $L = T − V$ method we find

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Therefore

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and for $θ1$

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and for $θ2$

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Hence EQM are

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or

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we obtain the equations of motion

M X ′′ + kX = 0

( ) ( ) ( ) ( ) ( ) 2 32m1 0 θ¨1 2 k1 + k2 − k2 θ1 0 r 0 3m2 θ¨2 + r − k2 k2 + k3 θ2 = 0 2

For the special case when $k1 = k$ , $k2 = 2k,k3 = 3k$ and $m1 = m2 = m$ we obtain

( ) ( ) ( ) ( ) ( ) 2 3m 0 ¨θ1 2 3 − 2 θ1 0 r 20 3m ¨ + kr − 2 5 θ = 0 2 θ2 2