chapter 36 - Vestibular Mechanics - KEMT FEI TUKE

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This equation can be nondimensionalized using the following nondimensional variables denoted by overbars © 2000 by CRC Press LLC (36.15) where Ω is a characteristic angular velocity of the canal. In terms of the nondimensional variables, the governing equation for endolymph flow velocity becomes where the nondimensional parameter ε is defined by The boundary conditions for this equation are as follows and the initial condition is – u ( – r,0) = 0. r v u r = t = t u a a R ⎛ ⎞ ⎜ ⎟ = ⎝ ⎠ 2 Ω ∂ ∂ + u ⎛ π⎞ ∂ ⎛ ∂ ⎞ ⎜ ⎟ ()=− t ( urdr) dt + ⎝ ⎠ ∫ ∫ ∂ ⎜r t r r ⎝ ∂ ⎟ ⎠ u t 1 2 1 α β 0 0 r 6 2Kπa = 2 ρβRv ∂u u( 1, t)= 0 ( 0, t )= 0 ∂ r 36.7 Semicircular Canal Frequency Response (36.16) (36.17) To examine the frequency response of the SCC, we will first get a solution to the nondimensional canal equation for the case of a step change in angular velocity of the skull. A step in angular velocity corresponds to an impulse in angular acceleration, and in dimensionless form this impulse is α()=− t Ωδ() t (36.18) where δ(t) is the unit impulse function and Ω is again a characteristic angular velocity of the canal. The nondimensional volumetric displacement is defined as t 1 φ= ∫ ( ) 0 0 ∫ urdr dt (36.19) the dimensional volumetric displacement is given by ∆V = (4πRΩa 4 /v)φ and the solution (for ε 1) is given by ( ) ∞ π β φ = ∑ λ 22 4 n= 1 π (36.20)
where λ n represents the roots of the equation J o(x) = 0, where J o is the Bessel function of zero order (λ 1 = 2.405, λ 2 = 5.520), and for infinite time φ = π/8β. For details of the solution see Van Buskirk and Grant [1987] and Van Buskirk and coworkers [1978]. A transfer function can be developed from the previous solution for a step change in angular velocity of the canal. The transfer function of mean angular displacement of endolymph θ related to ω, the angular velocity of the head, is © 2000 by CRC Press LLC ⎛ ⎞ ⎛ 2 θ ( 2π βλ ) ⎞ ⎜ ⎟ 1 ()= s ⎜s ⎟ ⎜ 1 ⎟ ω ⎜ ⎟ ⎛ ⎞ ⎝ 8 ⎜ ⎛ ⎞ ⎠ 1 1 ⎟ ⎜ ⎜s+ s ⎝ τ ⎟ ⎜ + ⎠ ⎝ τ ⎟ ⎟ ⎝ ⎠ ⎟ ⎠ L s (36.21) where τ L = 8ρvβR/Kπa 4 , τ s = a 2 /λ 2 1v, and s is the Laplace transform variable. The utility of the above transfer function is apparent when used to generate the frequency response of the system. The values for the various parameters are as follows: a = 0.15 mm, R = 3.2 mm, the dynamic viscosity of endolymph µ = 0.85 mPa·s (v = µ/ρ), ρ = 1000 kg/m 3 , β = 1.4π, and K = 3.4 GPa/m 3 . This produces values of the two time constants of τ L = 20.8 s and τ s = 0.00385 s. The frequency response of the system can be see in Fig. 36.6. The range of frequencies from 0.01 Hz to 30 Hz establishes the SCCs as angular velocity transducers of head motion. This range of frequencies is that encountered in FIGURE 36.6 Frequency response of the human semicircular canals for the transfer function of mean angular displacement of endolymph fluid θ related to angular velocity of the head w.
Page 1 and 2: Grant, W. “Vestibular Mechanics.
Page 3 and 4: Each SCC has a bulge called the amp
Page 5 and 6: The gel layer is treated as a Kelvi
Page 7 and 8: 36.4 Otolith Transfer Function A tr
Page 9 and 10: 36.5 Otolith Frequency Response ©
Page 11: FIGURE 36.5 Schematic structure of
Page 15: ρ o = density of the otoconial lay

chapter 36 - Vestibular Mechanics - KEMT FEI TUKE

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