note4 - MEEN 364 Notes from Session on September 29, 2004...

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MEEN 364 Notes from Session on September 29, 2004 304 Fermier Hall, 6:00~7:00 pm 1) Following figure shows a system consisting of a gear of radius r and moment of inertia J, a rack of mass m, a linear spring of stiffness k and a torsional spring with stiffness K. (a) Obtain the governing differential equation of motion for the given system. (b) Represent the system in state-space form. Fig.4 Kinematics stage Notice that, the rack and gear move in synchronization and hence x and θ are related by the expression r x = . Thus, the system has just one-degree of freedom, represented by the displacement of rack, x (or the rotation of the gear ). This completes the kinematics stage. Kinetics stage (a) Taking as the chosen coordinate, The total kinetic energy of the system is ( ) 2 2 2 2 2 2 1 2 1 2 1 2 1 & & & & eq J mr J x m J T = + = + = (1) since, from geometry of motion, r x = . Similarly the total potential energy of the system is ( ) 2 2 2 2 2 2 1 2 1 2 1 2 1 eq K kr K kx K V = + = + = ( 2 ) Thus, the resulting system become equivalent to the one shown in the fig. 4(b) which is a simple rotational spring-inertia system. The governing equation of motion of this system is given by 0 = + eq eq K J & & ( 3 ) where 2 2 K and kr K mr J J eq eq + = + = 1
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MEEN 364 Notes from Session on September 29, 2004 304 Fermier Hall, 6:00~7:00 pm (b) Alternatively, taking x as the chosen coordinate, The total kinetic energy of the system is 2 2 2 2 2 2 1 2 1 2 1 2 1 x m x m r J x m J T eq & & & & = + = + = θ (4) Similarly the total potential energy of the system is 2 2 2 2 2 2 2 1 2 1 2 1 2 1 x k x k r K kx K V eq =
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note4 - MEEN 364 Notes from Session on September 29, 2004...

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