Analytical modeling of an inclined folded-beam spring used in micromechanical resonator devices

Abstract

Accurate estimation of the mechanical behavior of springs is crucial for the proper design of Microelectormechanical systems (MEMS). The main objective of this study is to derive a closed-form equation for the calculation of the stiffness of an inclined spring in the form of folded beams. The energy-based method was used to calculate the displacements of a folded beam that was fixed at one end and giuded at the other end. The analytical model was then compared with the finite element method using ANSYS for different inclination angles of the folded beam spring, showing good agreement. The angle on inclination has changed from zero to 180 degress, and stiffness of folded beam is detemined. The derived expressions of the compliances were checked for the case of serpentine springs with inclination angle of zero, and different length ratios against the literature. It is found that neglecting small lengths for calculating the stiffness of the folded beam spring is not justified. The influential geometrical parameters, including different lengths of the spring and inclination angle of the spring, on the stiffness are studied. It is found that the angle of inclination of the principal axes of spring constants depends on the geometrical parameters, and the angle of inclination has more effect on the stiffness of a folded beam spring than the number of folds.

Design and Demonstration of a Microelectromechanical System Single-Ring Resonator with Inner Ring-Shaped Spring Supports for Inertial Sensors

“…Figures 11-13 depict the frequency response of the ring due to an applied voltage o 5 + 0.1 sin 𝑤𝑡 V to electrodes 2 and 4. As seen, the radial displacement of a point on th ring at 𝜃 = 45 degrees is zero, which implies that the mode shape is n = 2 sense mode o vibration [31]. As seen, in rings made from silicon <111> and <100>, the major resonance frequency is in the sense mode of n = 2, while for rings of <110>, it is in drive mode.…”

“…Figures 8-10 depict the frequency response of the ring due to an applied voltage of 5 + 0.1 sin wt (V) to electrode number 2 and 5 − 0.1 sin wt (V) to electrode number 4. As seen, the radial displacement of a point on the ring at θ = 45 degrees is zero, which implies that the mode shape is a n = 2 sense mode of vibration [31].…”

“…Figures 11-13 depict the frequency response of the ring due to an applied voltage of 5 + 0.1 sin wt V to electrodes 2 and 4. As seen, the radial displacement of a point on the ring at θ = 45 degrees is zero, which implies that the mode shape is n = 2 sense mode of vibration [31]. Table 4 depict the summary of dynamic analyses of the ring resonator made from different types of silicon.…”

“…Calculation of the stiffness of spokes is crucial for the determination of natural frequency. For folded beams with any number of beams and the orientation of the spoke when fixed at one end and free to slide at the other end in any direction, Rahbar Ranji et al [31] gave equations for the calculation of stiffness. For curved beams fixed at one end free to slide at the other end, Rahbar Ranji et al [30] gave the following equation for the calculation of stiffness:…”