Displacement in critical damping

In the presence of a damping force in the oscillating system, the system’s behavior depends on the value of the damping ratio. When it is equal to \(1\), the system returns to equilibrium as quickly as possible without oscillating, but overshoot can occur if initial velocity is nonzero. This behavior is also called critical damping.

Conditions:

1. The system is critically damped, i.e. its damping ratio \(\zeta = 1\).

displacement

Displacement from rest, usually a function of time. See euclidean_distance.

Symbol:

d

Latex:

\(d\)

Dimension:

length

time

time at which displacement is measured.

Symbol:

t

Latex:

\(t\)

Dimension:

time

undamped_angular_frequency

angular_frequency of the undamped oscillator.

Symbol:

w

Latex:

\(\omega\)

Dimension:

angle/time

initial_position

Initial position of the oscillator.

Symbol:

x_0

Latex:

\(x_{0}\)

Dimension:

length

initial_speed

Initial speed of the oscillator.

Symbol:

v_0

Latex:

\(v_{0}\)

Dimension:

velocity

law

d = exp(-w * t) * (x_0 + (v_0 + x_0 * w) * t)

Latex:

\[d = \exp{\left(- \omega t \right)} \left(x_{0} + \left(v_{0} + x_{0} \omega\right) t\right)\]