Theory

A mass-spring-damper rotational system was used to model the movement of the arm from a position of 90 degrees at the elbow to a position of 0 degrees at the elbow. The basic equation of motion for this system is

Where J is the moment of interia, B is the damping constant, and K is the spring constant The Laplace Transform of the equation of motion is

and with some manipulation,

Letting B/J = 2*alpha and K/J = the natural frequency, yields

The response of the system is characterized by the polynomial in the denominator. An overdamped response would have a polynomial that factors into two real, negative roots. A critically damped response would have a polynomial that factors into one double root. An underdamped response would have a polynomial with two complex roots. A stable system would have roots, complex or real, that are negative. An unstable system would have roots that are positive. It can be seen by taking the inverse Laplace trasform that the decaying exponential is of the form e^-(alpha*t) so the time constant is equal to 1/alpha. The natural frequency can be found by taking the inverse of the period.