## Homework 6, due March 14, 2008.

Due Friday, March 14, 2008.
goodwine
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### Homework 6, due March 14, 2008.

1. Consider the inverted pendulum illustrated in the following figure (you may consider the mass to be an idealized point mass).
• 1. (5 points) Determine the equation of motion for the system and then determine the linear approximation to the system by assuming that the angle is small so that you may assume that sin(theta) is approximately theta and cos(theta) is approximately 1.
2. (5 points) Determine the transfer function from the input torque to the output angle.
3. (5 points) Determine the transfer function from the input torque to the output angular velocity.
4. Assume the torque is supplied by a d.c. motor, as illustrated in the following figure.
• • (5 points) Determine the transfer function from the applied voltage to the output angle.
• (5 points) Determine the transfer function from the applied voltage to the output angular velocity.
2. Consider the mass-pulley system illustrated in the following figure (assume the system is planar, i.e., no gravity).
• 1. (5 points) Determine the equation of motion for the system. Hint: in addition to using Newton's law on the mass and two pulleys, assume that the cord does not slip on the pulleys. That assumption will provide two equations: one relating the angle of pulley 1, the radius of pulley 1, the angle of pulley 2 and the radius of pulley 2 and another relating the angle and radius of pulley two and the displacement of the mass (these are known as kinematic constraints).
2. (5 points) Determine the transfer function from the torque to the position of the mass.
3. (5 points) Determine the transfer function from the torque to the velocity of the mass.
4. Assume the torque is supplied by a d.c. motor, as illustrated in the following figure.
• • (5 points) Determine the transfer function from the applied voltage to the position of the mass. Hint: you may want to (perhaps even need to) add a variable for the voltage drop across the capacitor and then add an equation relating the current through the capacitor to the voltage across it.
• (5 points) Determine the transfer function from the applied voltage to the velocity of the mass.
• (5 points) Determine the transfer function from the current in the circuit to the position of the mass.
• (5 points) Determine the transfer function from the current in the circuit to the velocity of the mass.
3. (20 points) Determine the transfer function from y(t) to x(t) for the system in the following figure.
• The system is a box which as a mass-spring-damper inside it. Assume that the box only moves vertically. The position of the box is described by the variable y(t) and the position of the mass relative to the bottom of the box is given by x(t). You may assume that x(t)=0 when the box is not accelerating.
Bill Goodwine, 376 Fitzpatrick
sdelaure

### Problem 3

Am I missing something big here, or is there really only one equation governing the system in Problem 3? The way I see it, there's really only 2 variables and 1 equation (in laplace space) after writing NII using the absolute acceleration (x**+y**) of the mass.[/code]