Discussion and hints:
Shown below is an animation of the results of a simulation of the motion corresponding to an UNDERDAMPED system. The response is oscillatory, however, the amplitude of the response decays away at an exponential rate. One of your tasks in solving this problem is to determine if the system is underdamped, critically damped or overdamped.
The derivation of the dynamical equation of motion (EOM) for a system is a straight-forward application of what we have learned from Chapter 5 in using the Newton-Euler equations. The goal in deriving the EOM is to end up with a single differential equation in terms of a single dependent variable that describes the motion of the system. Here in this problem, we want our EOM to be in terms of x(t).
Recall the following four-step plan outline in the lecture book and discussed in lecture:
Step 1: FBDs
Draw an FBD of the block.
Step 2: Kinetics (Newton/Euler)
Write down the Newton/Euler equations for the block.
Step 3: Kinematics
Do you need any additional kinematics here?
Step 4: EOM
Your EOM should come directly from what you do in Step 2.
For this problem, you need to:
- Put your EOM in "standard form." From this, identify the undamped natural frequency, the damping ratio and the damped natural frequency in terms of the system parameters of m, c and k.
- If the system is underdamped (ζ < 1), then the solution can be written in terms of the decaying oscillation response derived in the lecture book and in class.
- You need to enforce the initial conditions on the problem (x(0) and dx/dt(0)) to determine the two response coefficients.