Question

3) For the system shown in the figure, the input is the torque T(t) and the outputs are the linear displacements x(t) and the angular displacement θ(t). The equilibrium position corresponds to x 0 0. Note that there is viscous friction between the rack and the surface it slides on. Also, you may treat the small diameter shaft as massless and rigid. mr Clearly state all assumptions to be used for modeling this system. Draw the freebody diagrams. State your "presumptions" on the configuration of the states of the system. Be sure to label the magnitude and direction of the forces consistent with those presumptions write the kinematic equation relating x and θ. Use the freebody diagrams and Newton's laws to derive oneequation of motion (ODE) for the system. That equation will include i. (Note: the kinematic equation from part (c) and a little algebra will allow you to eliminate θ from the equations). Arrange the equation so that the highest derivative term appears on the left side of the equations and all other terms are on the right side of the equation a) b) c) d) e) Use the energy method to derive the same equation of motion (ODE) for the system. f What is the equivalent mass for this system?

Answer 1

Assumptions used for modeling are (i) Small diameter shaft is massless and rigid. (ii) There is no loss of work between rack and pinion. (iii) Spring is massless. (iv) At equilibrium position, spring is unstretched. Let the contact force between rack and pinionis f Free body diagram of the system is given as K (01 02) In a K(0 -) wi CV mT kx

Here, K is shaft torsional stiffness. Kinematic equations are given as 1,02 + Rfc-K(4-9-)-0 1,01 + K (a-92)+7-0 From the geometric condition, So, equation (3) becomes

And equation (2) becomes Substitute f in equation (1) K K 1,6 I,62 K This is the required equation of motion (ODE) for the system.