chapter 36 - Vestibular Mechanics - KEMT FEI TUKE

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FIGURE 36.2 The free-body diagrams of each layer of the otolith with the forces that act on each layer. The interfaces are coupled by shear stresses of equal magnitude that act in opposite directions at each surface. The τ g shear stress acts between the gel-otoconial layer, and the τ f acts between the fluid-otoconial layer. The forces acting at these interfaces are the product of shear stress τ and area dA. The B x and W x forces are respectively the components of the buoyant and weight forces acting in the plane of the otoconial layer. See the nomenclature table for definitions of other variables. In the equation of motion for the endolymph fluid, the force τ fdA acts on the fluid, at the fluidotoconial layer interface. This shear stress τ f is responsible for driving the fluid flow. The linear Navier- Stokes equations for an incompressible fluid are used to describe this endolymph flow. Expressions for the pressure gradient, the flow velocity of the fluid measured with respect to an inertial reference frame, and the force due to gravity (body force) are substituted into the Navier-Stokes equation for flow in the x-direction yielding with boundary and initial conditions: u(0, t) = v(t); u(∞, t) = 0; u (y f, 0) = 0. © 2000 by CRC Press LLC 2 ∂u ∂ u =µ ∂ t ∂y ρ f f f (36.1)
The gel layer is treated as a Kelvin-Voight viscoelastic material where the gel shear stress has both an elastic component and a viscous component acting in parallel. This viscoelastic material model is substituted into the momentum equation, and the resulting gel layer equation of motion is © 2000 by CRC Press LLC (36.2) with boundary and initial conditions: w(b,t) = v(t); w(0,t) = 0; w(y g,0) = 0; δ g (y g,0) = 0. The elastic term in the equation is written in terms of the integral of velocity with respect to time, instead of displacement, so the equation is in terms of a single dependent variable, the velocity. The otoconial layer equation was developed using Newton’s second law of motion, equating the forces that act on the otoconial layer—fluid shear, gel shear, buoyancy, and weight—to the product of mass and inertial acceleration. The resulting otoconial layer equation is ( f ) ∂ ρobρo ρ v ∂ + − ∂ t with the initial condition v(0) = 0. ρ g w t G ∂ ∂ = ⎛ ∂ w ⎞ w ⎜ dt ⎜ ⎟ +µ g ⎝ ∂y ⎟ g ⎠ y g ∂ 2 2 2 2 ∂ 36.3 Nondimensionalization of the Motion Equations t ∫0 Vs u g x f ∂t y f − ⎛ ⎡ ⎤ ⎢ ⎥ =µ ⎜ ∂ ⎣⎢ ⎦⎥ ⎜ ∂ ⎝ ⎞ ⎟ G ⎟ ⎠ − ⎛ ⎜ ∂w ⎜ ∂y ⎝ ⎞ ⎛ ⎟ w dt +µ ⎜ ∂ g ⎟ ⎜ ∂y ⎠ ⎝ 0 y g y g f= 0 g= b y g= b (36.3) The equations of motion are then nondimensionalized to reduce the number of physical and dimensional parameters and combine them into some useful nondimensional numbers. The following nondimensional variables, which are indicated by overbars, are introduced into the motion equations: y f y y u v y t t u v w b b b V V w f g f = g = = V µ ⎛ ⎞ ⎜ ⎟ = = = 2 ⎝ ⎠ ρ o (36.4) Several nondimensional parameters occur naturally as a part of the nondimensionalization process. These parameters are ρ R = ρ f o 2 Gb µ ⎛ 2 ρ g b ⎞ o ρo = M = gx= ⎜ ⎟ g 2 µ µ f ⎝ Vµ f ⎠ f (36.5) These parameters represent the following: R is the density ratio, ε is a nondimensional elastic parameter, M is the viscosity ratio and represents a major portion of the system damping, and – g x is the nondimensional gravity. The governing equations of motion in nondimensional form are then as follows. For the endolymph fluid layer R u 2 ∂ ∂ u = ∂ t ∂ (36.6) with boundary conditions of – u (0, – t) = – v ( – t ) and – u (∞, – t) = 0 and initial conditions of – u ( – y f, 0) = 0. For the otoconial layer 2 y f ∫ t x ⎞ ⎟ ⎟ ⎠
Page 1 and 2: Grant, W. “Vestibular Mechanics.
Page 3: Each SCC has a bulge called the amp
Page 7 and 8: 36.4 Otolith Transfer Function A tr
Page 9 and 10: 36.5 Otolith Frequency Response ©
Page 11 and 12: FIGURE 36.5 Schematic structure of
Page 13 and 14: where λ n represents the roots of
Page 15: ρ o = density of the otoconial lay

chapter 36 - Vestibular Mechanics - KEMT FEI TUKE

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