Consider the motor below Do the following Find the transfer

Consider the motor below. Do the following: Find the transfer function G(s) = X(s)/E_a (s). Ignore gravity and note that J_a and D_a are the inertia and damping of the motor, respectively, while J is the inertia of the gear. Simulate the system for an input e_a (t) = 24u(t) (i.e. a step of height 24). Estimate the final value to the input in b) using Final Value Theorem. Comment on the agreement of your calculations and the simulation in b). For the motor: J_a = 1 kg-m^2 D_a = 1 N-m-s/rad R_a = 1 Ohm K_b = 1 V-s/rad K_t = 1 N-m/A

Solution

for the transfer function we need to find the inverse if the matrix which is obtained by solving the dynamic equations and multipying with input.

assume that there are no losses,no inertia and perfectly rigid

The input gear on the left has radius r1 and N1 teeth. It is rotated by 1(t) due to a torque T1(t). What is the relationship between the rotation of the input gear and that of the output gear, 2(t)? Although, the angles will differ, the arc length through which both gears turn will be the same: r11 = r22

Therefore the relation between angles is as follows, 2 = (r1 /r2) 1

Since the number of teeth is proportional to the radius, then the following also holds, 2 = (N1/ N2) 1

This system can be modelled as follows, (Js^2( N1/ N2 )^2 + Ds ( N1 /N2)^ 2 + K ( N1/ N2)^ 2 ) 1(s) = T1(s)

the transfer function is obtained as

G(s) = 2(s)/ T1(s) = (N2/N1)/( Jes^ 2 + De s + K2)

by substituting the given values we will obtaine the required solution