Viscous damping model
===============================



Physical definition of viscous damping
-----------------------------------------------

Conventional damping devices (for example, by rolling a viscous fluid through the holes of a piston driven by vibratory movement) deliver forces that are proportional to the speed of the movement and of the opposite sign. During a cycle, the work of these forces is positive: it is viscous damping.


.. image:: images/1000067C0000078B00000894872BA668D3E83E70.svg
    :width: 97
    :height: 110


.. _RefImage_1000067C0000078B00000894872BA668D3E83E70.svg:

For a simple oscillator with stiffness :math:`k`, mass :math:`m`, and viscous damping :math:`c`, the external force applied balances the three components: the elastic restoring force :math:`\mathit{ku}`, the damping force :math:`c\dot{u}` and the inertial force :math:`m\ddot{u}`, and the inertial force, hence the dynamic equation in absolute motion:


.. math::
 :label: eq-3

    m\ ddot {u} +c\ dot {u} +\ mathrm {ku} =f


For this viscous damping model, the energy dissipated during a pulsation cycle :math:`\omega` is proportional to the vibratory speed :math:`-\omega {u}_{0}\text{sin}(\omegat )` associated with the displacement :math:`{u}_{0}\text{cos}(\omegat )`:


.. math::
: label: eq-4

    {E} _ {d} ^ {\ text {cycle}} = {\ int}}} = {\ int} _ {0} _ {0}\ text {sin}\ omega\ mathit {td}\ omega\ mathit {td}\ left ({u} {td}\ left ({u} {td}}\ left ({u} _ {td}}\ left ({u} _ {td}}\ left ({u} _ {td}}\ left ({u} _ {td}\ left ({u} _ {td}}\ left ({u} _ {td}}\ left ({u} _ {td}}\ left ({u} _ {td}}\ left ({u} _ {td}}\ left ({u} _ {td}\ left


and the potential energy for a sine wave :math:`{u}_{0}\text{cos}\omega t` is:


.. math::
: label: eq-5

    {E} _ {p} ^ {\ text {max}} = {\ text {max}} = {\ int}} _ {\ int} _ {\ text {cos}\ omega td\ left ({u}\ left ({u}\ left ({u} _ {0} _ {0}\ 0}\ text {cos}\ omega t\ right) =\ frac {1} {2} {\ mathit {ku}}\ omega t\ right) =\ frac {1} {2} {\ mathit {ku}}\ omega t {ku}}\ omega t\ left ({u} left ({u}} left ({u}) _ {0} _ {0} ^ {right) 2}


For a cycle of pulsation :math:`\omega` and sinusoidal displacement :math:`{u}_{0}\text{cos}\omega t`, the loss coefficient is proportional to the frequency of the movement:


.. math::
 :label: eq-6

    \ eta =\ frac {c\ omega} {k}



.. _RefNumPara__1959_403351756:




Harmonic oscillator with viscous damping
--------------------------------------------------

The classical analysis of the undamped model associated with equation (), put in the form :math:`\left(k-m{\omega }^{2}\right)u=0` gives us :math:`{\omega }_{0}=\sqrt{\frac{k}{m}}` the natural pulsation. The critical damping from which the differential equation () no longer has an oscillating solution is given by the formulas :math:`{c}_{\text{critique}}=2\sqrt{\text{km}}=2m{\omega }_{0}=\frac{2k}{{\omega }_{0}}`, which makes it possible to give a numerical interpretation of the reduced damping, which is often expressed as a percentage of the critical damping:


.. math::
 :label: eq-7

    \ xi =\ frac {\ eta} {2} =\ frac {c} {{c} _ {\ text {critical}}}} =\ frac {c} {2m {\ omega} _ {0}}





Response to an outburst of excitement
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Starting from a static deformation :math:`{u}_{\text{st}}=\frac{{f}_{0}}{k}`, a release (release of the system) produces a free oscillatory movement :math:`{u}_{l}(t)\mathrm{=}{u}_{0}{e}^{\text{-}\xi {\omega }_{0}t}\mathrm{cos}{\omega }_{0}^{\text{'}}t` which reveals the natural pulsation of the damped system :math:`\omega {\text{'}}_{0}\mathrm{=}{\omega }_{0}\sqrt{(1\mathrm{-}{\xi }^{2})}`.

Over time, the extreme amplitude :math:`({u}_{1},{u}_{2})` decreases with each period of :math:`{e}^{-\xi {\omega }_{0}T}={e}^{-2\pi \xi }={e}^{-\delta }` where :math:`\delta` is the logarithmic decrement such as :math:`\delta =2\pi \xi`.




Response to harmonic excitation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The response to a harmonic excitation of the form :math:`f(t)={f}_{0}{e}^{j\omega t}` is written with a forced response particular permanent solution :math:`u(t)={u}_{0}{e}^{\left(j\omega t-\phi \right)}` which is written with the reduced pulsation :math:`\lambda =\frac{\omega }{{\omega }_{0}}` :math:`\frac{{\mathit{ku}}_{0}}{{f}_{0}}=\frac{1}{1-{\lambda }^{2}+j2\xi \lambda }={H}_{v}\left(j\omega \right)` where :math:`{H}_{v}(j\omega )` is the complex transfer function of a simple oscillator with viscous damping.

The answer modulus:math: `\ frac {{u} _ {{u} _ {0}} {{0}}} {{u}} _\ frac {{\ mathit {ku}} _ {0}} {{0}}} {{f}}} {{f} _ {0}}} =| {H} _ {0}}} =| {H} _ {v}\ left (j\ omega\ right) |=\ frac {1} {\ sqrt {{\ right) |=\ frac {1} {\ sqrt {{\ right] left (1- {\ lambda} ^ {2}\ right)} ^ {2}\ right)} ^ {2}} + {\ left (2\ xi\ lambda\ right)} ^ {2}}}` reveals a dynamic amplification compared to the static answer:math: `{u}} _ {\ text {st}}}`.

This amplification is maximum for :math:`\lambda =\frac{{\omega }_{0}^{\text{'}}}{{\omega }_{0}}=\sqrt{\left(1-{\xi }^{2}\right)}` and gives the value of the maximum displacement :math:`\frac{{u}_{0\text{max}}}{{u}_{\text{st}}}=\frac{1}{2\xi \sqrt{\left(1-{\xi }^{2}\right)}}`. If we observe the vibration speed :math:`\dot{u}(t)\mathrm{=}j\omega u(t)`, the vibration speed amplification is maximum for :math:`\lambda =\frac{{\omega }_{0}}{{\omega }_{0}}=1` and the maximum amplitude of the speed is :math:`{\dot{u}}_{0\text{max}}=\frac{1}{2\xi }=Q`, where :math:`Q` is the mechanical analogy of the electricians' overvoltage factor. These properties are at the origin of methods for measuring the damping characteristics of mechanical structures.