Reference no: EM132374805

Motion Control Homework - Laplace Transform Review and Solve Differential Equations

Note: Print out this problem sheet and attach your solution sheets to this problem sheet. Staple them and turn in.

Problem 1: a) Use Laplace transform table to find the Laplace transform of the following time functions and tell the item number used in the table.

f(t) = 3t + cos(2t)

f(t) = 2 + 3e^-3t

f(t) = 1 - e^-tsin(2t)

b) Find the inverse Laplace transform of the following Laplace transform. Tell the item number used in the table

F(s) = 1/s³ F(s) = 5/(s²+25) F(s) = (s+1)/((s+1)²+16)

c) Find the Laplace transforms of the following differential equations (note: don't have to solve the diff eq, just find the transform). Again, please list which table items you used.

d²y/dt² + 12 dy/dt + 16 y = 32cos(5t)

d³y/dt² + 12 dy/dt = u(t)

Problem 2: Use partial fractional expansion to find the inverse Laplace transform of the following Laplace transform. Please do the problem by hand and show all of your work. You may use Matlab to check that your final answer is correct.

F(s) = 5/(s²+6s+9) F(s) = (s+5)/(s²+5s+6)

Problem 3: a) Given the mass-damper-spring system below, write a differential equation to model the system.

b) If m = 1kg, fv = 8 N-s/m, K = 15 N/m, and input force f(t) = 2u(t), find the time function of displacement x(t) if all initial conditions are zero.

c) Plot the output x(t). Titled "hw3-mass-damper-spring system-1-your name". Also label your output x(t).

d) If m = 1kg, fv = 2 N-s/m, K = 1 N/m, and input force f(t) = 2u(t), find the time function of displacement x(t) if all initial conditions are zero.

e) Plot the output x(t). Titled "hw3-mass-damper-spring system-2-your name". Also label your output x(t).

f) If m = 1kg, fv = 2 N-s/m, K = 9 N/m, and input force f(t) = 2u(t), find the time function of displacement x(t) if all initial conditions are zero.

g) Plot the output x(t). Titled "hw3-mass-damper-spring system-3-your name". Also label your output x(t).

Problem 4: For the physical system below, use differential equation to model it and solve d.e to find the time function of output displacement θ(t), with input T(t). Use W = 80 lbs, the length, L = 5 ft, D = 0.1 lb.s/rad and g=32.2 ft/s². Plot the system response with title "hw3-robotarm system-your name" and labels. Print out the plot.

A) T(t) = u(t), assume θ(t) = 0^o.

B) T(t) = δ(t), assume the initial condition, θ(t)=30^o.