Editing Vestibular system (section)

===Mechanics===

The mechanics of the semicircular canals can be described by a damped oscillator. If we designate the deflection of the cupula with <math>\theta</math>, and the head velocity with <math>\dot q</math>, the cupula deflection is approximately<ref name=":0">{{Cite journal |last1=Buskirk |first1=W. C. Van |last2=Watts |first2=R. G. |last3=Liu |first3=Y. K. |date=November 1976 |title=The fluid mechanics of the semicircular canals |url=https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/fluid-mechanics-of-the-semicircular-canals/A75DA3D0F571ECF009E1B85B8D755713 |journal=Journal of Fluid Mechanics |language=en |volume=78 |issue=1 |pages=87–98 |doi=10.1017/S0022112076002346 |bibcode=1976JFM....78...87V |issn=1469-7645|url-access=subscription }}</ref>

: <math>\theta (s) = \frac{\alpha s}{(T_1 s+1)(T_2 s+1)} \dot{q} (s)</math>

α is a proportionality factor, and ''s'' corresponds to the frequency. For fluid simulations, the endolymph has roughly the same density and viscosity as water. The cupula has the same density as endolymph,<ref name=":0" /> and it is a jelly mostly made of [[Polysaccharide|polysaccharides]] with [[Young's modulus]] <math>5.4\; \mathrm{Pa}</math>.<ref>{{Cite journal |last1=Selva |first1=Pierre |last2=Oman |first2=Charles M. |last3=Stone |first3=Howard A. |date=2010-04-30 |title=Mechanical properties and motion of the cupula of the human semicircular canal |url=https://www.medra.org/servlet/aliasResolver?alias=iospress&doi=10.3233/VES-2009-0359 |journal=Journal of Vestibular Research |volume=19 |issue=3–4 |pages=95–110 |doi=10.3233/VES-2009-0359|pmid=20448336 |url-access=subscription }}</ref> 

T<sub>1</sub> is the characteristic time required for the cupula to accelerate until it reaches terminal velocity, and T<sub>2</sub> is the characteristic time required for the cupula to relax back to neutral position. The cupula has a small inertia compared to the elastic force (due to the jelly) and the viscous force (due to the endolymph), so T<sub>1</sub> is very small compared to T<sub>2</sub>. For humans, the time constants T<sub>1</sub> and T<sub>2</sub> are approximately 5 ms and 20 s, respectively.<ref>{{Cite journal |last1=Xu |first1=Mingyu |last2=Tan |first2=Wenchang |date=2000-05-01 |title=The problem of fluid-dynamics in semicircular canal |url=https://doi.org/10.1007/BF02897143 |journal=Science in China Series A: Mathematics |language=en |volume=43 |issue=5 |pages=517–526 |doi=10.1007/BF02897143 |bibcode=2000ScChA..43..517X |issn=1862-2763|url-access=subscription }}</ref> As a result, for typical head movements, which cover the frequency range of 0.1&nbsp;Hz and 10&nbsp;Hz, the deflection of the cupula is approximately proportional to the head velocity. This is very useful since the velocity of the eyes must be opposite to the velocity of the head to maintain clear vision.