Exam 3 Review Guide

ST U D Y G U I D E F O R E X A M 3 M ASS-SPR I N G-D A M P E R M O D E LS, L A P L A C E 1. Suppose a 4-lb weight stretches a spring 6 inches beyond its natural length. An external force equal to 12 cos 8t is acting on the spring. If the weight is started in motion from its equilibrium position with an upward velocity of 4 ft/sec and there is no damping, set up a initial value problem whose solution will give an equation of the motion of the mass + spring.

2. (#9, Sec. 3.8) A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also attached to a viscous damper with a damping constant of 400 dyne-sec/cm. If the mass is pulled down an additional 2 cm and then released, set up an initial value problem to model the motion of the mass + spring.

Review Guide for Exam 3, Page 1 of 6

3. Suppose the solution of a vibrating model (unforced) is u (t ) a) Describe the type of motion of the system.

2 cos 4t

2 sin 4t .

b) Put the solution in the amplitude-phase angle form.

c) Identify the period and (circular) frequency of the motion.

d) If the mass attached to the spring is 4 kilograms, what is the spring constant of the spring?

4. Classify the type of motion of each equation below. a) u(t )

c1e

2t

b) u (t )

c1 e

u (t )

c1 e

c) d)

u (t )

cos 4t

3t

3t

c2 e

2t

sin 4t _____________________________

c 2 te

3t

__________________________________

c2 e

2t

__________________________________

c1 cos 2t

c 2 sin 2t

1

2

sin 4t ______________________________

5. Suppose a mass-spring-damper system is modeled by u ' ' 9u ' 14u 1 2 sin t , u(0) = 0, .18e 2t .2e 7 t .03sin t .02 cos t , u (0) = 1. If the specific solution is u(t ) identify the steady-state portion of the solution and rewrite it in amplitude-phase angle form.

Review Guide for Exam 3, Page 2 of 6

6. a) Convert f (t ) sketch.

t 2(t 1)u1 (t ) (t 1)u 2 (t ) u3 (t ) to a piece-wise function and

b) Find L(f(t))

dI V (t ) . If R = dt 2, L = 1, and V(t) = e 3t , use Laplace transforms to find the zero-state output of the circuit.

7. The current flow of a circuit can be modeled by the equation RI

Review Guide for Exam 3, Page 3 of 6

L

8. Use Laplace transforms to find the zero-state output of the mass-damper system modeled by u ' ' 2u ' u t 3 e t .

9. Find the Laplace or inverse Laplace transforms as indicated. a) L (3t cos 2t )

b) L (e 3t cos 2t )

c) L

1

s

3

2 2s 2

s

Review Guide for Exam 3, Page 4 of 6

9. continued t

cos 2 d

d) L 0

e) L e

3t

t

cos 2 d 0

f) L e 3t u 2 (t )

g) L 1

h) L 1

i) L 1

s

1

1

s 2

s 4

s 4s 5

2

e s2

Evaluate as a convolution.

2s

4s 5

Review Guide for Exam 3, Page 5 of 6

10. Find the Laplace transform of the following periodic function.

Review Guide for Exam 3, Page 6 of 6