chapter 36 - Vestibular Mechanics - KEMT FEI TUKE

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with an initial condition of – v (0) = 0. For the gel layer © 2000 by CRC Press LLC ∂ + ( − ) ∂ ∂ ∂ − ⎡ ⎤ ⎛ v Vs ∂ ⎢ ⎥ = ⎜ u 1 R g x t ⎣⎢ t ⎦⎥ ⎜ ∂y ⎝ R w ∂ ∂ t = ∫ o t ⎞ ⎟ ⎟ ⎠ − (36.7) (36.8) with boundary conditions of – w(1, – t) = – v( – t) and – w(0, – t) = 0 and initial conditions of – w ( – y g,0) = 0 and – δg( – y g, 0) = 0. These equations can be solved numerically for the case of a step change in velocity of the skull. This solution is shown in Figure 36.3 for a step change in the velocity of the head and a step change in the acceleration of the head, and can be found in Grant and Cotton [1991]. FIGURE 36.3 This figure shows the time response of the otolith model for various values of the nondimensional parameters. Parts (a) through (d) show the response to a step change in the velocity of the head. This response can be thought of as a transient response since it is an impulse in acceleration. The step change has a nondimensional skull velocity of magnitude one. Parts (e) and (f) show the response to a step change in the acceleration of the skull or simulates a constant acceleration stimulus. Again the magnitude of the nondimensional step change is one. (a) Short time response for various values of the damping parameter M. Note the effect on maximum displacement produced by this parameter. (b) Long time response for various M values. Note the small undershoot for M = 1, no conditions of an overdamped system were put on the model. As the system is highly overdamped this set of nondimensional parameters (M = 1, R = 0.75, and ε = 0.2) would not be an acceptable solution. (c) Short time response for various values of ε. Note that this elastic parameter does not have much effect on the otoconial layer maximum displacement for a step change in velocity solution. This is not true for a step change in acceleration, see part (f). (d) Effect of the mass or density parameter R on response. R changes the maximum otoconial layer displacement. (e) Response to a step change in acceleration for various values of the damping parameter M. Note the overshoot for M = 1. (f) Acceleration step change time response for various values of the elastic parameter ε. Note that this parameter affects the final displacement of the system for this type of constant stimulus (compare with part (c)). (a) t ∫0 ⎛ ⎜ ∂w ⎜ ∂y ⎝ f g 0 1 ⎛ 2 ∂ w ⎞ 2 ∂ w ⎜ dt M ⎜ ⎟ + 2 ⎝ ∂y ⎟ 2 g ⎠ ∂y g ⎞ ⎛ ⎟ ⎜ ∂w dt − M ⎟ ⎜ ∂y ⎠ ⎝ g ⎞ ⎟ ⎟ ⎠ 1
36.4 Otolith Transfer Function A transfer function of otoconial layer deflection related to skull acceleration can be obtained from the governing equations. For details of this derivation see Grant and colleagues [1994]. Starting with the nondimensional fluid and gel layer equations, taking the Laplace transform with respect to time, and using the initial conditions give two ordinary differential equations. These equations can then be solved, using the boundary conditions. Taking the Laplace transform of the otoconial layer motion equation, combining with the two differential equation solutions, and integrating otoconial layer velocity to get deflection produces the transfer function for displacement re acceleration © 2000 by CRC Press LLC (b) (c) FIGURE 36.3 (continued)
Page 1 and 2: Grant, W. “Vestibular Mechanics.
Page 3 and 4: Each SCC has a bulge called the amp
Page 5: The gel layer is treated as a Kelvi
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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