Displacement in underdamping¶

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 less than \(1\), the system oscillates with a slightly different frequency than in the undamped case, and its amplitude decreasing to zero. This behavior is also known as underdamping.

Conditions:

The system is underdamped, 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

scaling_coefficient¶: Scaling coefficient to be found using initial conditions.

Symbol:: a
Latex:: \(a\)
Dimension:: length

exponential_decay_constant¶: exponential_decay_constant of the oscillator.

Symbol:: lambda
Latex:: \(\lambda\)
Dimension:: 1/time

damped_angular_frequency¶: Damped angular frequency of the oscillator. See angular_frequency.

Symbol:: w_d
Latex:: \(\omega_\text{d}\)
Dimension:: angle/time

phase_shift¶: phase_shift of the oscillations, i.e. the phase at \(t = 0\).

Symbol:: phi
Latex:: \(\varphi\)
Dimension:: angle

law¶

d = a * exp(-lambda * t) * cos(w_d * t + phi)

Latex:: \[d = a \exp{\left(- \lambda t \right)} \cos{\left(\omega_\text{d} t + \varphi \right)}\]