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: 

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

#. 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:**

#. `Physics LibreTexts, formula (15.7.1) <https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/15%3A_Oscillations/15.07%3A_Forced_Oscillations>`__.

.. py:currentmodule:: symplyphysics.laws.dynamics.forced_oscillations_equation

.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time`.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: displacement

    The displacement of the oscillating body from rest value as a function of :attr:`~time`.
    See :attr:`~symplyphysics.symbols.classical_mechanics.position`.

Symbol:
    :code:`x(t)`

Latex:
    :math:`x{\left(t \right)}`

Dimension:
    :code:`length`


.. py:data:: mass

    The :attr:`~symplyphysics.symbols.basic.mass` of the oscillating body.

Symbol:
    :code:`m`

Latex:
    :math:`m`

Dimension:
    :code:`mass`


.. py:data:: natural_angular_frequency

    The natural :attr:`~symplyphysics.symbols.classical_mechanics.angular_frequency` of the oscillator.

Symbol:
    :code:`w_0`

Latex:
    :math:`\omega_{0}`

Dimension:
    :code:`angle/time`


.. py:data:: driving_force_amplitude

    The amplitude of the driving :attr:`~symplyphysics.symbols.classical_mechanics.force`.

Symbol:
    :code:`F`

Latex:
    :math:`F`

Dimension:
    :code:`force`


.. py:data:: driving_angular_frequency

    The :attr:`~symplyphysics.symbols.classical_mechanics.angular_frequency` of the driving force.

Symbol:
    :code:`w`

Latex:
    :math:`\omega`

Dimension:
    :code:`angle/time`


.. py:data:: driving_phase_lag

    The :attr:`~symplyphysics.symbols.classical_mechanics.phase_shift` of the driving force.

Symbol:
    :code:`phi`

Latex:
    :math:`\varphi`

Dimension:
    :code:`angle`


.. py:data:: law

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


    Latex:
        .. math::
            \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)}