Anomalous amplitude-frequency dependence in a micromechanical resonator under synchronization

Abstract

It is well known that the oscillation frequency relates approximately quadratically with amplitude in a Duffing nonlinear oscillator while the frequency is independent of amplitude in a linear oscillator. In this article, the dynamics of a micromechanical oscillator during synchronization is studied and anomalous amplitude-frequency (a-f) dependence in a micromechanical resonator is observed. We theoretically and experimentally observed that in a linear resonator the amplitude is tuned quadratically with frequency while tuned linearly in a hardening as well as a softening nonlinear resonator, when the self-sustained resonator is synchronized to an external weak perturbation. Further investigation shows that the tunable range of the oscillation amplitude of a certain oscillator directly relies on the synchronization bandwidth, perturbation amplitude and frequency difference. The slope of the dependence can be tuned by phase delay in the feedback loop, while the feedback force dominantly determines the properties of the dependence from nonlinear relation to linear relation. This anomalous a-f effect provides a convenient technique for precise amplitude control.

Appendix

1.1 A. Arch c-c beam

According to Eq. (32), we can find that if the oscillator exhibits hardening spring effect, the amplitude variation has a positive correlation to the frequency variation, while negative correlation for the softening spring oscillator. To validate this effect, we introduce an arch-type beam resonator. For the softening spring oscillation case, an arch beam is used for experiments. The model of the arch beam can be expressed as,

$$\begin{aligned} \begin{aligned} \rho S \frac{\partial ^2Y}{\partial t_1^2}=&-EI\left( \frac{\partial ^4 Y}{\partial X^4}-\frac{\, \mathrm {d}^4 Y_0}{\, \mathrm {d}X^4}\right) \\&+\frac{EA}{2L}\frac{\partial ^2 Y}{\partial X^2}\int _{0}^{L}\left[ \left( \frac{\partial Y}{\partial X}\right) ^2-\left( \frac{\, \mathrm {d}Y_0}{\, \mathrm {d}X}\right) ^2\right] \, \mathrm {d}X \end{aligned} \end{aligned}$$

(35)

where the boundary conditions are,

$$\begin{aligned} Y=Y_0=0, \frac{\partial ^2 Y}{\partial X^2}=\frac{\, \mathrm {d}^2 Y_0}{\, \mathrm {d}X^2}=0, while X=0,L \end{aligned}$$

(36)

Introducing $W(X,t_1)=Y(X,t_1)-Y_0(X)$, Eq. (35) can be rewritten as,

$$\begin{aligned} \begin{aligned} \rho S \frac{\partial ^2W}{\partial t_1^2}=&-EI\frac{\partial ^4 W}{\partial X^4}+\frac{EA}{2L}\left[ \frac{\partial ^2 W}{\partial X^2}\right. \\&\left. +\frac{\, \mathrm {d}^2 Y_0}{\, \mathrm {d}X^2}\right] \int _{0}^{L}\left[ \left( \frac{\partial W}{\partial X}\right) ^2+2\frac{\partial W}{\partial X}\frac{\, \mathrm {d}Y_0}{\, \mathrm {d}X}\right] \, \mathrm {d}X \end{aligned} \end{aligned}$$

(37)

Redefining time $t=(t_1/L^2)\sqrt{EI/\rho S}$ and normalizing by $w=W/b$, $y_0=Y_0/b$, $x=X/L$ and $q=QL^4/EIb$,

$$\begin{aligned} \begin{aligned} \frac{\partial ^2w}{\partial t^2}=&-\frac{\partial ^4 w}{\partial x^4}+\frac{1}{2}[\frac{\partial ^2 w}{\partial x^2}+\frac{\, \mathrm {d}^2 y_0}{\, \mathrm {d}x^2}]\int _{0}^{1}\left[ \left( \frac{\partial w}{\partial x}\right) ^2\right. \\&\left. +2\frac{\partial w}{\partial x}\frac{\, \mathrm {d}y_0}{\, \mathrm {d}x}\right] \, \mathrm {d}x \end{aligned} \end{aligned}$$

(38)

To simplify this problem, we treat the beam as a sine-type arch,

$$\begin{aligned} y_0(x)=\lambda \sin (\pi x). \end{aligned}$$

(39)

By separating the variables,

$$\begin{aligned} w(x,t)=u(t)\sin (\pi x), \end{aligned}$$

(40)

In consideration of the boundary conditions, we obtain the free vibration governing equation

$$\begin{aligned} \frac{\, \mathrm {d}^2 u(t) }{\, \mathrm {d}t^2}+\omega _0^2 u(t)+k_2 u^2(t)+ k_3 u^3(t)=0 \end{aligned}$$

(41)

where $\omega _0^2=\pi ^4(1+0.5\lambda ^2)$, $k_2=0.75\lambda \pi ^4$, $k_3=0.25\pi ^4$. For a harmonic oscillator including second-order and third-order nonlinear terms, the nonlinear frequency can be expressed as $\omega =\omega _0+(\frac{3 k_3 }{8 \omega _0}-\frac{5 k_2^2 }{12 \omega _0^3})a^2$ [51]. Therefore, the criterion for hardening or softening effect is,

$$\begin{aligned} \varDelta =\frac{9}{4}\pi ^8(1-2\lambda ^2). \end{aligned}$$

(42)

For $\lambda ^2<0.5$, the oscillation exhibits a hardening effect. Otherwise, for $\lambda ^2>0.5$, the softening effect emerges. We design a micromechanical arch resonator shown in Fig. 9a, where the arch parameter $\lambda \approx 2$. Thus, the nonlinear coefficient $\frac{3 k_3 }{8 \omega _0}-\frac{5 k_2^2 }{12 \omega _0^3}=-\frac{7}{32}\frac{\pi ^4}{\omega _0}<0$. So the arch-beam exhibits a distinct softening effect in open-loop test.

1.2 B. Phase plots during synchronization

Fig. 10

Phase response of the linear oscillator during synchronization

Fig. 11

Phase responses of the nonlinear oscillator during synchronization

Figure 10 shows the phase response of the linear oscillation under synchronization, when the perturbation frequency $\varOmega _s$ is swept. During 22 s to 24.5 s, the oscillator is synchronized. Figure 11 shows the phase response of the nonlinear oscillation under synchronization,when the perturbation frequency $\varOmega _s$ is swept. During 35.2 s to 43.6 s, the nonlinear oscillator is synchronized. As the phase of the output signal of the function generator is fixed, the oscillation phase is fixed when synchronization occurs because of the phase locking effect.