User`s guide
100
underdamped cases, there exists a certain level of damping at which the system will just fail to
overshoot and will not make a single oscillation. This case is called critical damping. The key
difference between critical damping and overdamping is that, in critical damping, the system
returns to equilibrium in the minimum amount of time.
The damping ratio expresses the level of damping in a system relative to critical
damping. For a damped harmonic oscillator with mass m, damping coefficient c, and spring
constant k, it can be defined as the ratio of the damping coefficient in the system's differential
equation to the critical damping coefficient:
where the system's equation of motion is
and the corresponding critical damping coefficient is
A common method for analyzing the damping of an underdamped oscillation is the
logarithmic decrement method, for which the following relationships apply.
where
is the amplitude of peak i (i is an integer counting each peak), n is the number of
cycles being considered, is the log decrement,
is the undamped natural frequency, and
is
the damped natural frequency. Both frequencies are in radiance per second. Note, it is assumed that
object oscillates about zero. If there is an offset in y, the
amplitude must be defined relative to that
offset.
According to Eq.25 and Eq.33, the equivalent stiffness and equivalent mass are expressed
as:
Eq.36
Eq.37
Eq.38
Eq.33
Eq.34
Eq.35