Displacement in forced non-resonant oscillations

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, and (2) the angular frequency of the external force driving the oscillations.

Conditions:

1. Angular frequency of the external force is strictly not equal to the natural angular frequency of the oscillator.

2. No damping is present in the system.

Notes:

1. The external driving force has the form of \(f(t) = f_m \cos{\left( \omega t + \varphi \right)}\).

2. The complete expression of the displacement function can be found as the sum of the solution of simple harmonic motion equation and the particular solution presented here.

Links:

1. Physics LibreTexts, derivable from (15.7.2) and (15.7.3).

displacement

The particular solution of the forced oscillations equation that accounts for the oscillator’s response to the driving force.

Symbol:

q(t)

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 external driving force.

Symbol:

F

Latex:

\(F\)

Dimension:

force

driving_angular_frequency

The angular_frequency of the external driving force.

Symbol:

w

Latex:

\(\omega\)

Dimension:

angle/time

driving_phase_lag

The phase_shift of the oscillations of the external force.

Symbol:

phi

Latex:

\(\varphi\)

Dimension:

angle

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

law

q(t) = F / (m * (w0^2 - w^2)) * cos(w * t + phi)

Latex:

\[q(t) = \frac{F}{m \left( \omega_0^2 - \omega^2 \right)} \cos{\left( \omega t + \varphi \right)}\]