Damped harmonic oscillator equation
===================================

Assuming there is a damping force acting on an oscillating body that is linearly proportional
to the body's velocity, we can write a differential equation for the body's position. We're
assuming the body only moves in one direction.

**Links:**

#. `Physics LibreTexts, similar equation 15.6.2 <https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/15%3A_Oscillations/15.06%3A_Damped_Oscillations>`__.

.. py:currentmodule:: symplyphysics.definitions.damped_harmonic_oscillator_equation

.. py:data:: time

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

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: displacement

    Displacement of the oscillating body as a function of time. See :attr:`~symplyphysics.symbols.classical_mechanics.euclidean_distance`.

Symbol:
    :code:`x(t)`

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

Dimension:
    :code:`length`


.. py:data:: undamped_angular_frequency

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

Symbol:
    :code:`w`

Latex:
    :math:`\omega`

Dimension:
    :code:`angle/time`


.. py:data:: damping_ratio

    :attr:`~symplyphysics.symbols.classical_mechanics.damping_ratio`, which critically determines the behavior of the system.

Symbol:
    :code:`zeta`

Latex:
    :math:`\zeta`

Dimension:
    :code:`dimensionless`


.. py:data:: definition

    :code:`Derivative(x(t), (t, 2)) + 2 * zeta * w * Derivative(x(t), t) + w^2 * x(t) = 0`


    Latex:
        .. math::
            \frac{d^{2}}{d t^{2}} x{\left(t \right)} + 2 \zeta \omega \frac{d}{d t} x{\left(t \right)} + \omega^{2} x{\left(t \right)} = 0