Self-excited oscillation produced by a phase shift: linear and nonlinear instabilities

Abstract

Self-excited oscillation is a method to make a resonator vibrate at its natural frequency. This is widely used in vibration sensors such as mass sensors, atomic force microscopes and stiffness sensors. Self-excited oscillation is mainly produced by positive velocity feedback and time-delayed displacement feedback including phase-shifted displacement feedback. In this paper, we consider phase-shifted displacement feedback using a phase shifter. We perform nonlinear analysis to clarify the finite steady-state amplitude of the self-excited oscillation by considering nonlinear damping without a Laplace transform in the frequency domain, and formulate the effect of the phase shifter using an ordinary differential equation. We apply the method of multiple scales to the third-order ordinary differential equations expressing the coupling between the resonator and the phase shifter. This analytically reveals the parameter range of the phase shifter that produces self-excited oscillation in the resonator and the steady-state amplitude depending on the phase shift. We conduct an experiment using a cantilever as the resonator and produce self-excited oscillation via the feedback signal based on a phase shifter in a digital computer. The theoretically predicted characteristics of the self-excited oscillation agree well with the experimental ones.

Appendices

Appendix A

We show the method to experimentally identify the nondimensional coefficients of linear and cubic nonlinear damping via the free vibration of the cantilever beam as an experimentally obtained time history in Fig. 11. Letting $x_e$ be 0, we obtain the equation governing the free vibration as

$$\begin{aligned} \ddot{x}^*+\gamma {\dot{x}}^*+x^*+\gamma _3{\dot{x}}^{*3}=0, \end{aligned}$$

(A.1)

where the dot denotes the derivative with respect to nondimensional time $t^*$. The parameters $\gamma$, $\gamma _3$ and nondimensional time $t^*$ have the same definitions as in Sect. 2. Then, Eqs. (12), (14), (17) and (21) are simplified, respectively, as

$$\begin{aligned}&O(\epsilon ^{\frac{1}{2}}) :\quad {D_0^2}x_{1}+x_{1}=0, \end{aligned}$$

(A.2)

$$\begin{aligned}&O(\epsilon ^{\frac{3}{2}}) :\quad {D_0^2}x_{3}+x_{3}=-2D_0D_1x_{1}-{\hat{\gamma }}D_0x_{1}-{\hat{\gamma }}_3\left( D_0x_{1}\right) ^3, \end{aligned}$$

(A.3)

$$\begin{aligned}&x_{1}=A(t_1)\text{ e}^{i{t_0}}+CC, \end{aligned}$$

(A.4)

$$\begin{aligned}&D_1A+\frac{1}{2}{\hat{\gamma }}A+\frac{3}{2}{\hat{\gamma }}_3|A|^2A=0. \end{aligned}$$

(A.5)

Substituting Eqs. (18) and (A.4) into Eq. (A.5) and separating the resulting equation into the real and imaginary parts yields

We obtain the following functions of nondimensional amplitude a and $\phi$ with respect to time $t^*$ based on Eq. (A.6):

$$\begin{aligned}&a=\frac{1}{\sqrt{(\frac{1}{a(0)^2}+\frac{3\gamma _3}{4\gamma })\text{ e}^{\gamma t^*}-\frac{3\gamma _3}{4\gamma }}}, \end{aligned}$$

(A.8)

$$\begin{aligned}&\phi =\phi _0, \end{aligned}$$

(A.9)

where a(0) and $\phi _0$ denote the initial amplitude and the integral constant determined by the initial condition, respectively. The solution of $x^*$ obtained through Eqs. (8), (18), (A.4), (A.8) and (A.9) is

$$\begin{aligned} x^*=a\cos {(t^*+\phi _0)}+O(\epsilon ^{\frac{3}{2}}). \end{aligned}$$

(A.10)

Using the envelope theoretically obtained from Eq. (A.8) and the peak of each period of the experimentally obtained time history in Fig. 11, we identify $\gamma$ and $\gamma _3$. When the amplitude of the free vibration is small, the cubic nonlinear damping is extremely small compared with the linear damping. Therefore, we can neglect the nonlinear damping in the time history as shown in Fig. 11b. In this case, we can calculate $\gamma$ to be 0.0022 from the average of the logarithmic decrements [29], which are obtained from the 17 peaks in Fig. 11b. However, when the amplitude of the free vibration is comparatively large as shown in Fig. 11c, there is the effect of nonlinear damping. We use the time history from 0 s to 100 s in Fig. 11c to determine the coefficient of nonlinear damping $\gamma _3$; the number of peaks is $n=17$. We set the value of every peak in Fig. 11c as $p_i$ ($i=1,2,\cdots , n$) successively from 0 s to 100 s. Substituting the moment $t^*$ corresponding to each $p_i$ and $\gamma =0.0022$ into Eq. (A.8), we can obtain the function $a_i(\gamma _3)$ corresponding to $p_i$. We calculate $F(\gamma _3)=\sum _{i=1}^{n}\{p_i-a_i(\gamma _3)\}^2$ and determine $\gamma _3$ so that $\frac{\partial F(\gamma _3)}{\partial \gamma _3}=0$. As a result, $\gamma$ and $\gamma _3$ are experimentally identified as shown in Table 1. The black dashed line in Fig. 11a shows the envelope obtained from Eq. (A.8) using the values of $\gamma$ and $\gamma _3$ obtained above.

Appendix B

We investigate the sensitivity of the Hopf bifurcation points $\alpha _{cr1}$ and $\alpha _{cr2}$ to the uncertainties in $\beta$ and $\gamma$ using Eq. (26). Let the unknown exact values of $\beta$ and $\gamma$ be $\beta _0$ and $\gamma _0$. Then, the uncertainties in $\beta$ and $\gamma$ are defined as $\varDelta \beta =\beta -\beta _0$ and $\varDelta \gamma =\gamma -\gamma _0$. The Taylor expansions of $\alpha _{cr1}$ and $\alpha _{cr2}$ with respect to the uncertainties, $\varDelta \beta$ and $\varDelta \gamma$, are as follows:

$$\begin{aligned} \alpha _{cr1}(\gamma ,\beta )&= \alpha _{cr10}-\frac{\beta }{\gamma ^2}\left( 1-\frac{1}{\sqrt{1-c^2}}\right) \varDelta \gamma \nonumber \\&\qquad \!\!\! +\frac{1}{\gamma }\left( 1-\frac{1}{\sqrt{1-c^2}}\right) \varDelta \beta +O(\varDelta \gamma ^2,\varDelta \beta ^2), \end{aligned}$$

(B.1)

$$\begin{aligned} \alpha _{cr2}(\gamma ,\beta )&=\alpha _{cr20}-\frac{\beta }{\gamma ^2}\left( 1+\frac{1}{\sqrt{1-c^2}}\right) \varDelta \gamma \nonumber \\&\qquad \!\!\! +\frac{1}{\gamma }\left( 1+\frac{1}{\sqrt{1-c^2}}\right) \varDelta \beta +O(\varDelta \gamma ^2,\varDelta \beta ^2), \end{aligned}$$

(B.2)

where c=$\gamma /\beta$, and $\alpha _{cr10}$ and $\alpha _{cr20}$ are two unknown exact values of $\alpha _{cr}$ for two Hopf bifurcation points corresponding to $\beta _{0}$ and $\gamma _{0}$. The coefficients of $\varDelta \beta$ and $\varDelta \gamma$ are the sensitivities of the Hopf bifurcation points to the uncertainties in $\beta$ and $\gamma$. Substituting the values of $\beta$ and $\gamma$ in Table 1 into Eqs. (B.1) and (B.2) yields

$$\begin{aligned}&\alpha _{cr1}-\alpha _{cr10}=128\varDelta \gamma -60\varDelta \beta \end{aligned}$$

(B.3)

$$\begin{aligned}&\alpha _{cr2}-\alpha _{cr20}=2070\varDelta \gamma +969\varDelta \beta . \end{aligned}$$

(B.4)

We can see by comparing the coefficients of $\varDelta \gamma$ and $\varDelta \beta$ in Eqs. (B.3) and (B.4) that $\alpha _{cr2}$ is more affected by these uncertainties than $\alpha _{cr1}$. This indicates that the deviation between the theoretical and experimental bifurcation points at $\alpha _{cr2}$ is larger than that at $\alpha _{cr1}$, as observed in the experiment (Fig. 5). Therefore, there is a relatively large discrepancy between the deviations near $\alpha _{cr2}$.