ES 205 Homework 1 Summer 2016 1.
[Class 1] Assume the beams shown have negligible mass. a.
What is the equivalent stiffness of the system given a force applied to the bottom of spring 2 as shown? b. If the bottom of spring 2 is fixed, and the force is applied at the tip of the shorter beam as shown below, then what is the equivalent stiffness of the new system? 2.
[Class 2] The equation for a forced Van der Pol oscillator is given by 1
, where is the forcing function and , , and are constant coefficients. a. Select an appropriate model form (transfer function or simulation diagram) to study how the coefficient effects the system response. Explain the reasons for your choice of model form. b. Put the model in your chosen form. 3.
[Class 2] Consider the mechanical system illustrated below. The corresponding equations of motion are provided. Equations of motion: ℓ
a.
Find the transfer function from 1 ℓ
b. Use Maple to check your solution. to Θ
by hand. Show you work. 2 4.
[Class 2] An electrical system is depicted below, and the corresponding modeling equations are provided. Use Maple to obtain the transfer function relating the input voltage to the output voltage . Modeling equations: (1)
(2)
2 (3)
3 (4)
4 (5)
5 (6)
6 (7)
p (8)
n (9)
1 (10)
(10) p (11)
(11) n (12) out (5)
(2)
0 (6)
(7)
(8)
(9)
1 (4)
0 Unknowns (1)
(3)
Eq. (12)
5.
[Class 3] Find the general form of the step response (t) for a system whose transfer function is given by, 10
5
2
5
. 6.
[Class 3] Find the general form of the step response (t) for a system whose transfer function is given by, 3
25
2
. 7
7.
[Class 3‐4] The transfer function for robot arm controlled using proportional control is Θ
, where = 8 kg m2 is the robot’s moment of inertia, = 8 Nm s is the joint damping, and the control parameter. a.
= 4 Nm is Compute the poles of the system. b. Sketch the step response. Don’t worry about specific values, just capture what the response will look like. c. What is the effect of decreasing on the system response? What happens when = 2 Nm?