Question

Q 1- 08 Pts) Figure below is a diagram of a DC motor connected in parnllel to a current source i,. The torque and back-EMF constants of the motor are Ko K respectively, the motor resistance is R, also modeled as connected in parallel, the motor inertia is 1- (not shown), and the motor inductance is negligible. The motor load is an inertia J with compliance (stiffness) K and viscous friction coefficient b, and it is attached a gear pair with gear ratio N N (N and N are the numbers of teeth on input is the current i, and the output is the rotation angle θ of the 0m is the rotation angle of the motor Tm is the torque generated by the motor. vs : back emf. to the motor via gear pair). The system The torque is proportional to the armature current, so T -Ka, The induced voltage v, is directly proportional to the angular velocity d@dt, or v,- K, ded The equivalent inertia of the gear train referred to shaft inertia is Je Q 1.1- Derive the governing equations for the electrical part (2pts) Q 1.1- Derive the governing equations for the electrical part (2pts). 1.2- Derive the governing equations for the mechanical part (2pts) 1.3-Obtain the transfer function of the servomotor if the system input is the current i, and the output is the rotation angle 0 of the inertia.(4 pts)

Answer 1

rn 0 dlt rn S adt Ni he knoh that

SD From (S) and clt NI 2