• Obtain a state space model for the system shown in problem 3.9 of the text. The input for your model will be the torque Ti(t), and the output of the model will be the motion of the rack x1. Place the model in "A,B,C,D" vector-matrix form. To simplify the problem; (i) assume that the steady part of the applied torque is zero, (ii), replace the NLS (nonlinear spring) with a linear spring with coefficient k1, and (iii), replace the NLD (nonlinear damper) with a linear damper with coefficient b.
• Consider the input/output model obtained for problem 1 of homework 2. Place the model in state space format, using the applied torque Tm as the input, and the motion x1 as the output. Using a time step of 0.02 sec, compute the values of the states by hand for three time steps using Eulers method. Repeat the solution of the system using the Matlab function ode45, solving the system over the range 0.0 sec to 4 sec, using the same time step as for the Euler compuation above. After your computation, plot the values of the states versus time. Use the following values for the parameters in your model: m=1.2 kg, J=0.3 kg*m^2, r1=0.3 m, r2=0.35 m, B=0.5 N*s/m, b=2 N*s/m, k=20 N/m, and Tm=0, and the initial conditions x1(0)=0.3 m, and x1dot(0)=0.0 m/s.
• Simulate the state system for problem 3.9 in the text as qualified by problem 1 above. Use the Matlab function lsim to perform the simulation. For the linear replacement for the NLS, use a spring coefficient of k1=2 N*m. For the linear replacement for the NLD, use a damping coefficient of b=2 N*s/m. All other parameters should have the same values, however, the steady part of the input torque should be zero. Simulate the system over the time range 0 to 50 sec, using a time step of that will allow 500 steps of this period. Let Ti(t) be a step input of 15 N occuring at time t=0, and set the initial values of the states to zero for the simulation. Plot the motion of the mass (the state corresponding to x1) versus time.

Assignement due October 9 2000.