106 J.R. Foley et al. While commonly used for a variety of test applications, accelerometers designed for high-shock environments generally utilize stiffer elements with correspondingly high resonant frequencies (>50 kHz) and extremely low damping ratios ( 0:01) that are effectively undamped [4]. The latest generation of shock accelerometers, developed and produced in the past decade, have increased the damping ratio significantly even though they are still relatively lightly damped ( 0:01). Representative commercially-available sensors include the Meggitt Model 727 and PCB Piezotronics Model 3991 [5]. The details of the damping mechanisms of these devices depend on the specific design. However, they are generally specified and modeled as possessing a constant damping ratio. This paper tests this hypothesis by estimating the damping ratios for sensors at varying levels of excitation. A nonlinear model is proposed and preliminary results are calculated and discussed using experimental data. 11.2 Analysis The accelerometer is modeled as a linear, damped, single degree of freedom (SDOF) system as shown in Fig. 11.1. The mathematical description is a one-dimensional simple harmonic oscillator where x is the motion of the spring within the sensor, u is the base motion, and z is the relative coordinate, i.e., z Dx u. The corresponding equation of motion for the system is then mRz CcPz Ckz D mRu; (11.1) where mis the mass, c is the damping, and k is the stiffness of the dynamic element [6]. A state-space model is appropriate to reduce the problem into a simple linear system of equations. We begin with the following two substitutions: z1 Dz and z2 D Pz : (11.2) It immediately follows that Pz1 Dz2: (11.3) Likewise, the governing equation of motion in terms of z1 and z2, solved for z¨2, is Pz2 D k m z1 c m z2 Ru: (11.4) The linear system of equations is then Pz1 Pz2 D 0 1 k m c m z1 z2 C 0 Ru : (11.5) Fig. 11.1 Schematic of the coordinate system and components of a simplified MEMS accelerometer with a dynamic seismic mass
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