Forced oscillations equation

Forced, or driven, oscillations are a type of oscillations in the presence of an external driving force acting on the oscillating system. In the case of an oscillating external force, two angular frequencies are associated with such a system:

1. the natural angular frequency of the system, which is the angular frequency the system would oscillate with if no external force were present,

2. the angular frequency of the external force driving the oscillations.

Such systems can undergo resonance if the angular frequency of the driving force is close to the natural angular frequency of the oscillator.

Links:

1. Physics LibreTexts, formula (15.7.1).

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

displacement

The displacement of the oscillating body from rest value as a function of time. See position.

Symbol:

x(t)

Latex:

\(x{\left(t \right)}\)

Dimension:

length

mass

The mass of the oscillating body.

Symbol:

m

Latex:

\(m\)

Dimension:

mass

natural_angular_frequency

The natural angular_frequency of the oscillator.

Symbol:

w_0

Latex:

\(\omega_{0}\)

Dimension:

angle/time

driving_force_amplitude

The amplitude of the driving force.

Symbol:

F

Latex:

\(F\)

Dimension:

force

driving_angular_frequency

The angular_frequency of the driving force.

Symbol:

w

Latex:

\(\omega\)

Dimension:

angle/time

driving_phase_lag

The phase_shift of the driving force.

Symbol:

phi

Latex:

\(\varphi\)

Dimension:

angle

law

Derivative(x(t), (t, 2)) + w_0^2 * x(t) = F / m * cos(w * t + phi)

Latex:

\[\frac{d^{2}}{d t^{2}} x{\left(t \right)} + \omega_{0}^{2} x{\left(t \right)} = \frac{F}{m} \cos{\left(\omega t + \varphi \right)}\]