Problem Statement:
Three damped free vibrations, overdamped, critical damped, and underdamped, are described in Exercises in Chapter 6 Functions in the book C for Engineers and Scientists: An Interpretive Approach (by Harry H. Cheng, published by McGraw-Hill, 2009).

The details for the programs using QuickAnimation, presented below, are decribed in QuickAnimation User's Guide.

Animation
Each of these three programs vibration.c, vibration2.ch, and vibration3.c uses Ch QuickAnimation for quick animation of these three damped vibrations below. One may consider the motion of the damped free vibrations below as an elevator approaching a stop. It would be very uncomfortable to ride if it were underdamped, and very slow to ride if it were overdamped. Critical damping provides the fastest and smoothest ride.

Plotting
The program vibration.cpp uses Ch plotting capabilities to generate a plot shown below for distances of these three damped vibrations. You can see the oscillation in underdamped free vibration clearly.

Display
The program position.c uses Ch QuickAnimation to display the positions shown below for three vibrations when time t is 2 seconds.