ME450 Homework 2 Solution (Penn State University)
In the mechanical system shown in Figure 1, both springs are fully relaxed when the lever is horizontal. Furthermore, the equivalent viscous damping associated with the angular velocity of the lever is denoted by B.
- Making free body diagrams, derive differential equations of motion.
Select state variables, and derive state variable equations for this system. Using these state equations, construct the simulation diagram without using any differentiator.
- Linearize the state variable equations in the small vicinity of the normal operating point at which
the lever is horizontal. Treating Fi and as the input and output respectively, obtain the
linearized input-output differential equation. Also, obtain the transfer function of the system.
- Obtain the state space model directly from the input-output differential equation.
- For the following values of the system parameters, make reasonable simplifying assumptions to
reduce the order of the model to 2. Justify your assumptions.
m 2kg. , k 18 N / m , B 0.5 N sec m / rad , 1.0 m , and a 0.5 m 11
m2 20kg., k2 108N/m
Find undamped natural frequency, damped natural frequency and damping ratio.
Obtain the step response for the simplified model via Laplace transformation. Plot the response and find the maximum overshoot.
5. Determine and plot the response of the simplified model when the input command is as shown in Figure 2. Comment on the results.