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:**

#. Angular frequency of the external force is strictly not equal to the natural angular frequency of the oscillator.
#. No damping is present in the system.

**Notes:**

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

#. `Physics LibreTexts, derivable from (15.7.2) and (15.7.3) <https://en.wikipedia.org/wiki/Force#Second_law>`__.

.. py:currentmodule:: symplyphysics.laws.dynamics.displacement_in_forced_non_resonant_oscillations

.. py:data:: time

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

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: displacement

    The particular solution of the forced oscillations equation that accounts for the
    oscillator's response to the driving force. See :attr:`~symplyphysics.symbols.classical_mechanics.position`.

Symbol:
    :code:`q(t)`

Latex:
    :math:`q{\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 external 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 external 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 oscillations of the external force.

Symbol:
    :code:`phi`

Latex:
    :math:`\varphi`

Dimension:
    :code:`angle`


.. py:data:: law

    .. only:: comment

        `displacement` is a sympy Symbol, therefore auto-generation of formulas is impossible

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

    Latex:
        .. math::
            q(t) = \frac{F}{m \left( \omega_0^2 - \omega^2 \right)} \cos{\left( \omega t + \varphi \right)}