Exploring Nonlinear Regime in Commercial Quartz Tuning Forks for Clocks, Robotics and Sensorics

Timekeeping has always been an essential task for humanity. As society developed, it needed to define time more precisely, up to hours, milli-, micro-, and nano-seconds… Thus, the clock was invented. An essential part of every clock is a reference oscillator that defines time intervals with the needed precision. In a mechanical clock, a pendulum performs the reference function. As a pendulum swings with a second period, it activates a ratchet system at each oscillation that moves the clock one second forward. The more precise tasks require references based on different physical principles. In addition, the need for precise timekeeping has expanded to a broad range of devices, so almost every electronic device needs a clock. Every watch, computer, smartphone, TV, appliance, and radio set needs a clock. Each of these clocks needs a reference oscillator, for which quartz crystal resonators are typically used. Often, these resonators are shaped as miniature tuning forks whose resonant frequency is tuned precisely to 215 = 32768 Hz. Thus, by dividing this frequency 15 times by 2, one obtains precise 1-second long time intervals further counted by the electronics. The performance of such tuning fork resonators is the subject of our study.<br/><br/>The clock's precise work requires the reference oscillator's precise frequency, which is possible only if the tuning fork operates in a linear regime. Commercial tuning forks can be easily driven from a linear to a nonlinear regime, where the frequency depends on the driving amplitude, reducing the clock’s precision. Meanwhile, the nonlinear regime is undesirable for the clock, but some other applications may benefit. For example, the quartz tuning forks may be used as sensors of various physical quantities, like mass or gas pressure. In these applications, the nonlinear regime of the tuning fork makes their sensitivity dramatically higher. Also, many fundamental physical aspects of the nonlinear regime and the transition between linear and nonlinear regimes remain unclear. Our goal is to shed more light on the problem.<br/><br/>We began our study by exploring the transition from linear to non-linear regime using a software-defined network analyzer. The network analyzer excites the fork to vibrate and records its amplitude and phase concerning the excitation signal. We have found that the resonance frequency of these forks depends on the excitation amplitude, even at relatively small levels (~ 0.05 V). As the excitation signal increases, the resonant frequency shift becomes progressively more significant. The measured resonant frequency depends on the direction of the frequency scan, exhibiting the classical behavior of a nonlinear oscillator characterized by a bifurcation region with the possibility of chaotic transitions between the branches of the resonant curve. Should this bifurcation appear when the oscillator is installed in a clock, it would introduce uncertainty in its measurements. However, if the fork operates as a sensor, the bifurcation makes it extremely sensitive to minor environmental changes. Our study shows that the behavior of this resonator can be modeled using the Duffing equation. Solving the Duffing equation for this oscillator numerically, one can determine the equation coefficients, such as spring and Duffing constants.<br/><br/>To gain deeper insight into the nonlinearity phenomenon, we are building an atomic force microscopy (AFM) system. This system will allow us to measure displacement at a particular point and map the tuning fork deformation as it vibrates. This work is in progress.