a single link is driven by a motor and a gear train as shown

a single link is driven by a motor and a gear train as shown. The motor provides the torque input ,T_m. the inertia of the gears and the shafts are negligible. the free body diagram of the system is developed as shown. the coordinate system is defined such that positive theta_s is in the counterclockwise direction. Using the symbolic parameters and the following steps, derive the equation of motion of the system.
J_m=motor mass moment of inertia
L=link length
L_s=distance from center of mass to the joint
r1=gear 1 radius
r2=gear 2 radius
m=mass of link
g=gravity
theta_s=link angle
theta_m=motor angle
T_m=torque input from the motor

a. find the gear ratio N=theta_m/theta_s
b. find the moment produced by gravity about the center of rotation at the shafts center.
c. the link is modeled as a concentrated mass at the distance L_s. the equivalent mass moment of inertia of the system is found as J_eq=J_m+m(L_s/N)^2,apply the moment balance at the shafts center to develop the system equation in terms of theta_m.
d. linearize the system equation using small angle approximation (sintheta=theta)
e. find the kinematic relationship between the link angular position (theta_s) and the end position of the link, x_s.

Center of mass Gear 2 Gear l Shaft center ment at the Motor

Solution

ANS:

click on below link u got the ans.....

http://web.iku.edu.tr/courses/ee/ee670/ee670/lecture%20notes/ch02.pdf

http://www.iaa.ncku.edu.tw/~chiehli/course/control/Homeworks%202-10%20Dorf%2012ed.pdf