Chapter 8. Model of the Accelerometer 8.1 The static model 8.2 The dynamic model 8.3 Sensor System simulation
8.2.1 Basic equations 8.2.2 Resonant frequency 8.2.3 Squeeze-film damping 8.2 The dynamic model 8.2.3.1 Normal motion 8.2.3.2 Tilt motion 8.2.3.3 Combined normal and tilt motion 8.2.3.4 The effect of large amplitude 8.2.3.5 Continuum limits 8.2.3.6 Noise behavior of squeeze-film damper 8.2.3.7 Recommended use of the squeeze-film damper
8.2.1 Basic equations In order to predict the behavior of the device under a dynamic acceleration, the model need to be expanded to the dynamic model. Basic Equations: where M, I θx,i θy : mass and moment of inertia around x-axis and y-axis f z, f θx, f θy damping of the device in z-direction, around x-axis and y-axis. K z, K θx, K θy : stiffness of the mass movement in z-direction and rotational movement around x-axis and y-axis.
The Moment of Inertia The moment of inertia with respect to a line l is given by: 3 3 ρ silicon = 2.33 10 kgm
8.2.2 Resonant Frequency The maximum obtainable bandwidth is determined by the resonant frequency: These resonant frequencies should be as high as possible, which can be realized by maximizing K θx and K θy. These can be achieved by placing the beams far away from the center of the mass.
ANASYS v.s. Analytical Resonant Frequency The difference between the resonant frequencies of the translational and the rotational modes is not very much. In order to predict the behavior of the device under a dynamic acceleration, the damping factors f z, f qx, f qy need to be determined.
8.2.3 Squeeze-film damping The gas that is present between the seismic mass and the encapsulation, provides a damping for the accelerometer. Reynolds equation:
Assumptions of Reynolds Equation (1) The mean free path of the gas molecules is much smaller than the thickness of the film => The film can be treated as a continuum. (2) Laminar flow. (3) (4) Ref[2.9] Langlois et. al. Isothermal Squeeze Films, Quarterly of Applied Mathematics, vol.xx, no. 2 July 1962, pp.131-150
Further Assumptions for Linear Reynolds Equation Reynolds equation: (2-52) (5) the variation of the plate spacing and the pressure deviation from ambient P a are small: (2-53) Linear Reynolds equation: (2-54)
Assumptions to Solve Linear Reynolds Equation (6) Assume δp = P a ψ, η = δh / h 0, τ = ωt, X = x/a, Y = y/a (2-55) y b x Squeeze number σ: (2-56) a (7) Assume a small periodic displacement η = ε cos τ (2-57)
(8) Solve by separation of variables, by assuming: (2-58) (2-59) where ψ o : out-of-phase dimensionless relative pressure distributions over the plate surface. represents damping behavior of the gas-film. ψ l : in-phase dimensionless relative pressure distributions over the plate surface. represents spring behavior of the gas-film. (9) Series solution form: (2-60) Boundary conditions:
(10) When the solution ψ is found: δp = P a ψ (2-61) where F o, M o,θx and M o, θ y : represents damping force and moments. F l, M l,θx and M l, θ y : represents spring force and moments.
8.2.3.1 Normal Motion When the accelerometer is subjected to a normal acceleration, the mass moves in a vertical direction without rotation. In this case, the film thickness is constant over the plate (independent of the location): (2-62) η = h-h 0 / h 0 = ε cos τ (2-63) Solutions: (2-60) (2-64)
δp = P a ψ (2-60) (2-61) (2-65) Thus, a vertical movement of the mass results only in vertical forces. The series solution converges rather fast, so that only a few terms are necessary.
(1) For small squeeze numbers (low frequencies): F o depends linearly on the squeeze number (frequency) and thus on the velocity of the plate, so that the film acts as a pure damper. The air can move out the gap of the plate, resulting a damped movement due to the viscosity (2) For higher squeeze numbers: The relationship becomes non-linear. (3) Cut-off squeeze number s c : The damping force and spring force are equal in size. Using only one term approximation from series solutions: Plate aspect ratio β = 1 => σ c =2π 2 =19.7 v.s. 21.6 in Fig2-22 (2-66) F l (spring force) becomes larger than the F o (damping force). The film no longer damps the movement of the plate, but acts as a spring. The air is compressed and can t move out the gap of the plate, resulting a spring movement due to the compressibility effect. (4) Approximate cut-off frequency w c : sub (2-56) in Eq. (2-66) (2-67)
Assuming the frequency 1 Hz, s =5.4x10-4 << s c = p 2 (1+1/b 2 )=19.7 distribution over the plate: The correct factor =0.42 for β=1 - The maximum pressure occurs in the center of the plate. - For this low-frequency (1Hz), the spring force can be neglected compared to the damping force. - The damping force can be approximated by neglecting the σ term in Eq(2-65): (2-65) (2-68) - Eq(2-68) is known as the incompressible solution since it can be obtained by simply disregarding the time-derivative of the pressure in Eq(2-54): (2-54) Where v z is the velocity of plate in the vertical position (2-69)
8.2.3.2 Tilt Motion When the accelerometer is subjected to a lateral acceleration, the mass rotates around the x or y-axis. In this case, the film thickness depends on the location (x,y) according to: (2-70) (2-59) (2-71)
Solutions: δp = P a ψ
Since a mn1, b mn1, a mn4 and b mn4 are zero, the pressure distribution is anti-symmetric. Therefore, integration over the surface results in no damping or spring forces (F o =F 1 =0) acting on the total plate. There are damping and spring moments because of the anti-symmetric pressure distribution. The tilt modes are uncoupled; a rotation around the x-axis results only in a damping or spring moment around x-axis and similar for the rotations around the y-axis.
For a moving square plate β =1 (1) For small squeeze numbers (low frequencies): damping moment M 0, θx depends linearly on the squeeze number (frequency) and thus on the angular velocity of the plate, so that the film acts as a pure damper. (2) For higher squeeze numbers: The relationship becomes non-linear. spring moment M l, θx becomes larger than the damping moment M 0, θx :. The film no longer damps the tilt movement of the plate, but acts as a spring. (3) Cut-off squeeze number s c : (4) Approximate cut-off frequency w c : sub Eq. (2-56) in Eq. (2-74) The damping force and spring force are equal in size. One term approximation from series solutions: (2-75) Plate aspect ratio β = 1 => σ c,θx =5π 2 = 49.3 v.s. 61 in Fig2-25 (2-74)
2.2.3.3 combined normal and tilt motion (2-60) Since a mnl, b mnl are affected by a vertical movement of the plate. a mn2, b mn2 are affected by a tilt movement around the x-axis. a mn3, b mn3 are affected by a tilt movement around the y-axis. Therefore The pressure distributions of each mode can be added in case of a combination of vertical and angular velocities.
Normal Motion Combined normal + tilt motion Tilt Motion
8.2.3.4 The effect of large amplitude The previous sections assumed the deflections of the plate to be small compared to the nominal gap (δh<<h 0 or ε << 1). When the plate deflections becomes larger (assume small squeeze number): (1) The squeeze film theory results are no longer valid. (2) with increasing amplitude, the spring and damping coefficients of the gas film increase, resulting non-linear behavior of the film. (3) Due to the non-linearity, an undesired effect can result in the accelerometer response. (4) This undesired effect can be prevented by assuring a small plate displacement, as is achieved in the servo-accelerometer.
8.2.3.5 Continuum limits (1) K n < 0.01 - The film can be treated as a continuum. (2) At higher K n (by either a low-pressure or by a small gap) - Slip flow - The gas flow can be modeled by a modified Reynolds equation, which, in case of small pressure deviations results in an effective viscosity: - f(k n ) >0 which increases with increasing Knudsen number. Thus the decrease the effective viscosity. => σ (3) For the air at the atmosphere pressure => mean free path λ=63nm, ho=4µm => K n =0.016, µ eff =0.93 µ => the effect of slip flow is only of minor importance. - In order to reduce the damping for a given geometry, the pressure inside the film can be reduced, so that the Knudsen number increase and effective viscosity decreases.
8.2.3.6 Noise behavior of the squeeze-film damper (1) In electrical circuit theory, it is well known that electronic dampers (resistors) give rise to a white thermal noise current spectrum: where k B is Boltzman constant (=1.38x10 23 J/K), T is absolute temperature and R is the resistor value. (2) Analog to this case, the mechanical damper reveals a white noise: where f gas-damping is the coefficient relating the damping force F to the velocity v by F= f gas-damping v. -The higher this coefficient, the larger the damping will be, and also, the more noise the device will exhibit. - In case of accelerometer, the noise can not be distinguished from an actual acceleration, thus setting a lower limit to the minimum detectable acceleration.
8.2.3.7 Recommended use of the squeeze-film damper Since the main purpose of the gas film is to provide damping of the device, the spring behavior of the gas film should be avoided, i.e. the frequency of operation should be below the cut off frequency of the damping mechanism. Optimally, the design is such that the effect of the spring behavior of the gasfilm is beyond the bandwidth of the system, so that the film can simply be regarded as a damper. To avoid a non-linear behavior of the gas-film, the mass deflections should be small compared to the niminal gap, which can be realized by using servo feedback mechanism.