2. Viscous damping model#
2.1. 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.
For a simple oscillator with stiffness \(k\), mass \(m\), and viscous damping \(c\), the external force applied balances the three components: the elastic restoring force \(\mathit{ku}\), the damping force \(c\dot{u}\) and the inertial force \(m\ddot{u}\), and the inertial force, hence the dynamic equation in absolute motion:
For this viscous damping model, the energy dissipated during a pulsation cycle \(\omega\) is proportional to the vibratory speed \(-\omega {u}_{0}\text{sin}(\omegat )\) associated with the displacement \({u}_{0}\text{cos}(\omegat )\):
: label: eq-4
{E} _ {d} ^ {text {cycle}} = {int}}} = {int} _ {0} _ {0}text {sin}omegamathit {td}omegamathit {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 \({u}_{0}\text{cos}\omega t\) is:
: label: eq-5
{E} _ {p} ^ {text {max}} = {text {max}} = {int}} _ {int} _ {text {cos}omega tdleft ({u}left ({u}left ({u} _ {0} _ {0}0}text {cos}omega tright) =frac {1} {2} {mathit {ku}}omega tright) =frac {1} {2} {mathit {ku}}omega t {ku}}omega tleft ({u} left ({u}} left ({u}) _ {0} _ {0} ^ {right) 2}
For a cycle of pulsation \(\omega\) and sinusoidal displacement \({u}_{0}\text{cos}\omega t\), the loss coefficient is proportional to the frequency of the movement:
2.2. Harmonic oscillator with viscous damping#
The classical analysis of the undamped model associated with equation (), put in the form \(\left(k-m{\omega }^{2}\right)u=0\) gives us \({\omega }_{0}=\sqrt{\frac{k}{m}}\) the natural pulsation. The critical damping from which the differential equation () no longer has an oscillating solution is given by the formulas \({c}_{\text{critique}}=2\sqrt{\text{km}}=2m{\omega }_{0}=\frac{2k}{{\omega }_{0}}\), which makes it possible to give a numerical interpretation of the reduced damping, which is often expressed as a percentage of the critical damping:
2.2.1. Response to an outburst of excitement#
Starting from a static deformation \({u}_{\text{st}}=\frac{{f}_{0}}{k}\), a release (release of the system) produces a free oscillatory movement \({u}_{l}(t)\mathrm{=}{u}_{0}{e}^{\text{-}\xi {\omega }_{0}t}\mathrm{cos}{\omega }_{0}^{\text{'}}t\) which reveals the natural pulsation of the damped system \(\omega {\text{'}}_{0}\mathrm{=}{\omega }_{0}\sqrt{(1\mathrm{-}{\xi }^{2})}\).
Over time, the extreme amplitude \(({u}_{1},{u}_{2})\) decreases with each period of \({e}^{-\xi {\omega }_{0}T}={e}^{-2\pi \xi }={e}^{-\delta }\) where \(\delta\) is the logarithmic decrement such as \(\delta =2\pi \xi\).
2.2.2. Response to harmonic excitation#
The response to a harmonic excitation of the form \(f(t)={f}_{0}{e}^{j\omega t}\) is written with a forced response particular permanent solution \(u(t)={u}_{0}{e}^{\left(j\omega t-\phi \right)}\) which is written with the reduced pulsation \(\lambda =\frac{\omega }{{\omega }_{0}}\) \(\frac{{\mathit{ku}}_{0}}{{f}_{0}}=\frac{1}{1-{\lambda }^{2}+j2\xi \lambda }={H}_{v}\left(j\omega \right)\) where \({H}_{v}(j\omega )\) 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 (jomegaright) |=frac {1} {sqrt {{right) |=frac {1} {sqrt {{right] left (1- {lambda} ^ {2}right)} ^ {2}right)} ^ {2}} + {left (2xilambdaright)} ^ {2}}} reveals a dynamic amplification compared to the static answer:math: {u}} _ {text {st}}}.
This amplification is maximum for \(\lambda =\frac{{\omega }_{0}^{\text{'}}}{{\omega }_{0}}=\sqrt{\left(1-{\xi }^{2}\right)}\) and gives the value of the maximum displacement \(\frac{{u}_{0\text{max}}}{{u}_{\text{st}}}=\frac{1}{2\xi \sqrt{\left(1-{\xi }^{2}\right)}}\). If we observe the vibration speed \(\dot{u}(t)\mathrm{=}j\omega u(t)\), the vibration speed amplification is maximum for \(\lambda =\frac{{\omega }_{0}}{{\omega }_{0}}=1\) and the maximum amplitude of the speed is \({\dot{u}}_{0\text{max}}=\frac{1}{2\xi }=Q\), where \(Q\) 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.