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Diagram of a double ended tuning fork resonator. The voltage of the center electrode can be used to control the nature of nonlinearities from electrostatic forces. The resonance mode shape is also shown on the top left.
Experimentally measured—(a) resonant frequency shift due to change in (spring softening) shown using the transmission s-parameter () measurements and (b) amplitude-frequency dependence due to electrostatic softening nonlinearities.
Sketch of the dependence of coefficient on the for both cases. The parabolic dependence from 2nd order nonlinearities and the independent coefficient form the 3rd order nonlinearities are sketched for case 2.
(a) Measured resonant frequency shift at low drive amplitudes, due to spring softening for both configurations. (b) Measured coefficient for both cases. Parabolic dependence of is not observed at high in case (2). The figure also plots the theoretically predicted values of for each case using solid lines, using the expressions in Table I (the contribution from structural nonlinearities is neglected in the theoretical predictions).
Analytical results for case (1) and case (2) , .
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