Displacement in resonant oscillations

When an oscillating external force is driving the oscillations of an oscillator, amplitude of oscillations is greatest when the angular frequency of the driving is equal to the natural frequency of the oscillator. This condition is called resonance.

Conditions:

1. Angular frequency of the driving force is equal to the natural frequency of the oscillator.

2. No damping is present in the system.

Notes:

1. The expression of the driving force has the form \(F \cos{\left( \omega t + \varphi \right)}\) where \(\omega\) is the angular frequency of its oscillations.

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

resonant_displacement

The displacement of resonant oscillations as a function of time. See position.

Symbol:

x(t)

Latex:

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

Dimension:

length

mass

The mass of the oscillator.

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_phase_lag

The phase_shift of the oscillations of the driving force.

Symbol:

phi

Latex:

\(\varphi\)

Dimension:

angle

law

x(t) = F / (2 * m * w_0) * t * sin(w_0 * t + phi)

Latex:

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