Mechanical springs are essential components that enable to store elastic energy and to determine the natural frequency of resonators. In mechanical watches, spiral springs are used for both of these functions. Spiral springs are compact, they are compatible with rotational motion, and their main advantage is their linear response. This linearity ensures a constant driving torque and a constant natural frequency of the system. In literature there is a single model for the mechanical response of spiral springs, which is derived by assuming their stiffness is similar to that of straight beams with the same overall length. This model is suitable for spiral springs with many turns (n>5). However, in many applications, and especially in microelectromechanical systems (MEMS), spiral springs with a low number of turns (n<1) are used. In this case the prevalent model is inadequate, and designers are forced to use finite elements simulations to predict the stiffness of the spiral spring.
We have derived a new analytic model of the response of spiral springs, and we demonstrate its excellent predictive capabilities by comparing it to finite elements simulations and to experimental measurements. Computing the stiffness of spiral springs with the new model is ~104 time faster than numerical computation using finite elements.
However, there are many potential advantages for using nonlinear springs in resonators. Nonlinear mechanical springs can be easily implemented, but designing them to have a specific functional form of nonlinear stiffness is prohibitively complicated. To produce springs with a specific functional form of a nonlinear response, we use electrostatic transducers – MEMS comb-drive actuators with a tailored distribution of finger lengths. This also enables simple tuning of the amplitude of the nonlinear stiffness. We present a new methodology for designing electromechanical springs in which the restoring force f=x^n is a single integer power of motion x.
