4.22 1. The differential equations for the suspension shown below with input displacement u(t) are 10 100z 10w 100w (a) Assume all initial conditions are zero and convert the equations above to Laplace domain. (b) Demonstrate that if you solve for W(s) in the first equation and then substitute this expression for W(s) into the 2nd equation, when you solve for Z(s) you get z(s)-b, 40000 330s + 30001 US) 340s2 +300s +3000 (c) What is the transfer function for this system? 92 (d) Consider the following MATLAB command and results >>roots(1 40 300 3000]) ans 33.7442 3.1279 8.8950i 3.1279-8.8950i What are the eigenvalues of this system? (e) What are the time constants of this system? (f) If the input u(t) is a unit step, the general form of the equation for z(t) can be shown to be (g) What is the final value of z(t) and how long will it take to get within 1% of this value? 2. Use the residue theorem to solve the following differential equation for y(t). 22y 44y 88 y(0)5 4. Use separation of variables to solve the differential equation in problem 2. 4.22 1. The differential equations for the suspension shown below with input displacement u(t) are 10 100z 10w 100w (a) Assume all initial conditions are zero and convert the equations above to Laplace domain. (b) Demonstrate that if you solve for W(s) in the first equation and then substitute this expression for W(s) into the 2nd equation, when you solve for Z(s) you get z(s)-b, 40000 330s + 30001 US) 340s2 +300s +3000 (c) What is the transfer function for this system? 92 (d) Consider the following MATLAB command and results >>roots(1 40 300 3000]) ans 33.7442 3.1279 8.8950i 3.1279-8.8950i What are the eigenvalues of this system? (e) What are the time constants of this system? (f) If the input u(t) is a unit step, the general form of the equation for z(t) can be shown to be (g) What is the final value of z(t) and how long will it take to get within 1% of this value? 2. Use the residue theorem to solve the following differential equation for y(t). 22y 44y 88 y(0)5 4. Use separation of variables to solve the differential equation in problem 2