Assignment #1 Read Appendix B and Chapter 1

1. Using the defining integral of the Laplace transform, derive the Laplace transform of:

(a) f(t) = u(t-4)

(b) x(t) = exp(-4t)u(t-2)

2. Using the Laplace transform tables, find the Laplace transform of:

(a) f(t) = -4texp(-3t)u(t)

(b) g(t) = 2tcos(3t)u(t)

(c) x(t) = 5cos(3t+30^o)u(t)

3. Find the inverse Laplace transform of:

(a) F(s) = 5/((s(s+1)(s+3))

(b) G(s) = (2s+1)/(s²+2s+10)

(c) H(s) = (s-30)/((s(s²+4s+29))

(d) X(s) = 2/(s²(s+1))

4. For each function of question 3 above,

(a) Which inverse Laplace transforms do not have final values?

(b) Find the final values for those functions that have one.

5. Given the differential equation: x´´(t)+ 5x´(t)+4x(t) = 3u(t) where ´ indicates differentiation with respect to t,

(a) Find x(t) for the case that all initial conditions are zero.

(b) Find x(t) for the case that x(0) = x´(0) = 1 Show that your solution yields the correct initial conditions.

6. (a) Write the differential equation for the mechanical system (a) below. There are no applied forces.

(b) A force f(t) is applied downward to mass M. Find the transfer function from the applied force to the displacement, x₁(t), of the Mass, X₁(s)/F(s) .

(c) Repeat (a) for the system (b) below.

(d) A force f(t) is applied downward to the mass M in (b). Find the transfer function X₁(s)/F(s) .

7. Shown below is the block diagram of the servo-control system for one of the joints of a robot.

(a) Find the plant transfer function q_L(s)/E_a(s) .

(b) Find the closed loop system transfer function q_L(s)/ q_c(s) .

(c) Find the transfer function from the system input q_c(s) to the motor armature voltage E_a(s) for the closed loop system.

PLEASE NOTE:  All assignments  are due by 4:00 pm on the due date in to the drop box in the B-C Infill