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¶
timeat whichdisplacementis 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_constantof 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_shiftof 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:
- \[d = a \exp{\left(- \lambda t \right)} \cos{\left(\omega_\text{d} t + \varphi \right)}\]