Question

3.10 (Motor drive) Consider a system consisting of a motor driving two masses that are connected by a torsional spring, as shown in the diagram below. Motor This system can represent a motor with a flexible shaft that drives a load. Assuming that the motor delivers a torque that is proportional to the current I, the dynamics of the system can be described by the equations dt dt (3.43) dtdt where Ф1 and P2 are the angles of the two masses. Ф dOJdt are their velocities. J; represents moments of inertia, c is the damping coefficient, k represents the shaft stiffness, ki is the torque constant for the motor, and Td is the disturbance torque applied at the end of the shaft. Similar equations are obtained for a robot with flexible arms and for the arms of DVD . and optical disk drives Derive a state space model for the system by introducing the (normalized) state is the undamped natural frequency of the system when the control signal is zero.

Answer 1

From the description of the question, the state variables are defined.

$x_1=\varphi _1$, $x_2=\varphi _2$, $x_3=\omega _1/\omega _0$ ,  $x_4=\omega _2/\omega _0$

Also given$\frac{d\varphi _1}{dt}=\omega _1$ (1)

$\frac{d\varphi _2}{dt}=\omega _2$ (2)

From (1), $\frac{dx _1}{dt}=\omega _1(\frac{\omega _0}{\omega _0})=\omega _0x_3$ (3)

From (2), $\frac{dx _2}{dt}=\omega _2(\frac{\omega _0}{\omega _0})=\omega _0x_4$ (4)

From above equation we can write,

$\frac{d^2x _1}{dt}=\omega _0\frac{dx_3}{dt}$ (5)

$\frac{d^2x _2}{dt}=\omega _0\frac{dx_4}{dt}$ (6)

Given dynamic equations are written in terms of state variables as

$\frac{dx_3}{dt}=-\frac{c}{J_1}x_3+\frac{c}{J_1}x_4-\frac{k}{J_1\omega _0}x_1+\frac{k}{J_1\omega _0}x_2+\frac{k}{J_1\omega _0}I$ (7)

$\frac{dx_4}{dt}=-\frac{c}{J_2}x_4+\frac{c}{J_2}x_3+\frac{k}{J_2\omega _0}x_1-\frac{k}{J_2\omega _0}x_2+\frac{1}{J_2\omega _0}T_d$ (8)

From (3), (4), (7) and (8), we can obtain state space representation where the matrices are given below

A=[0 0 wo 0;0 0 0 wo;-k/(J1wo) k/(J1wo) -c/J1 c/J1);k/(J2wo) -k/(J2wo) c/J2 -c/J2)]

B=[0 0;0 0;k/(J1wo) 0;0 1/(J2wo)]