Mode shift detection of coupled resonators through parametric resonance and its application to mass sensing

Abstract

The eigenmode shift in weakly coupled resonators is used for ultra-sensitive micro-mass measurements. For operation in viscous environments, positive velocity feedback is often applied to compensate for the damping effect on the resonators. In the feedback, the real-time sensing of the displacement or velocity of the resonators is essential and in downsizing, a relatively large noise produced in a specific sensor for the velocity or displacement of the resonator may degrade the detection of the mode shift. In this study, we propose using parametric resonance to detect the mode shift without feedback control. Applying the harmonic balance method to an analytical model of coupled cantilevers, we clarify the unstable parameter region in which the resonators are parametrically excited with the first or second eigenmode. We also experimentally confirm without a special environment such as high vacuum, i.e., in air, that the parametrically excited coupled cantilevers produce a highly sensitive mode shift depending on the added mass. The sensitivity of the proposed method is increased by a factor of about ${{10}^{3}}$ in the first mode and ${{10}^{2}}$ in the second mode compared with the sensitivity in the conventional method based on the natural frequency shift of a single resonator.

Appendices

Appendix

A: The sensitivity depending on the position of added mass

The dependency of the mode shift on the position of added mass is theoretically examined under the external excitation [17]. The closer the mass is to the tip, the higher the sensitivity of mass measurement becomes. This result is experimentally shown also in the case under the parametric resonance as follows. Figure 6 shows the experimental results using the system mentioned in Sect. 3.1. The ordinate and abscissa are normalized by the total length of the beam and the eignmode shift in the case when the mass is added at the tip, respectively.

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Eigenmode shift of the parametrically excited coupled cantilevers depending on the position of added mass.

B: Validity of the three-term harmonic balance

Comparing with the stability boundary numerically obtained by 4th-Runge–Kutta method, we employ the three harmonic functions. For $\gamma =0.8$, the details are as follows.

Figure 7 shows the stability boundaries where the red, blue and green lines stand for the results by the one-term, two-term, and three-term harmonic balance methods, respectively. The numerically obtained results are shown by the plots; the circle and cross denote the combination of the excitation amplitude and frequency for the stable and unstable trivial steady states, respectively. The stability boundary obtained by the three-term harmonic balance method agrees with that by the numerical simulation. Therefore, we employ the three terms in the harmonic balance method.

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Stability boundaries of the parametric resonance with the first mode obtained by the one-, two-, and three-term harmonic balance methods. The circle and cross denote the stability of the trivial steady state numerically obtained by 4th-order Runge–Kutta method.

C: Experimental identification of the damping coefficient

The non-dimensional damping coefficient $\gamma$ of Eq. (2) is experimentally identified as follows.

We consider the state where one of the two cantilevers can oscillate and the other one is fixed. The equation governing the bending vibration can be expressed as

$$\begin{aligned} \rho A \frac{\partial ^{2} v}{\partial t^{2}}+c \frac{\partial v}{\partial t}+E I \frac{\partial ^{4} v}{\partial x^{4}}=0, \end{aligned}$$

(14)

where $\rho$ is the cantilever density, A is the cross-sectional area, c is the damping coefficient, E is the Young’s modulus of the constitutive material, and I is the moment of inertia of the cross section. We introduce the representative length (total length) L and the representative time $T^{\prime }=L^2 \sqrt{\rho A / E I}/\omega$, where $\omega$ is the minimum value satisfying the following equation:

$$\begin{aligned} 1+\cos \omega ^{2} \cosh \omega ^{2}=0. \end{aligned}$$

(15)

Then, Eq. (14) is rewritten as

$$\begin{aligned} \ddot{v}+2 \gamma \dot{v}+\frac{1}{\omega ^{2}} v^{\prime \prime \prime \prime }=0, \end{aligned}$$

(16)

where $\gamma =c L^{2} /(2 \omega \sqrt{\rho A E I})$. The dots and primes denote the partial derivative with respect to the nondimensional time $t^*=t/T$ and ordinate, $x^*=x/L$, respectively. Substituting $v=U(t^{*}) \phi {(x^{*})}$ into Eq. (16) yields

$$\begin{aligned} \ddot{U}+2 \gamma \dot{U}+U=0, \end{aligned}$$

(17)

$$\begin{aligned} \phi ^{\prime \prime \prime \prime }-\omega ^{2} \phi =0. \end{aligned}$$

(18)

The following boundary conditions are imposed on the equation of $\phi$:

$$\begin{aligned} \phi (0)=0, \end{aligned}$$

(19)

$$\begin{aligned} \phi ^{\prime }(0)=0, \end{aligned}$$

(20)

$$\begin{aligned} \phi ^{\prime \prime }(1)=0, \end{aligned}$$

(21)

$$\begin{aligned} \phi ^{\prime \prime \prime }(1)=0. \end{aligned}$$

(22)

In the case of $0<\gamma <1$, the solution of Eq. (17) is

$$\begin{aligned} U =U_{0} e^{-\gamma t^{*}} \cos t^{*}. \end{aligned}$$

(23)

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Experimental results at the excitation frequencies that are in the neighborhood of twice the natural frequency of the first and second modes. Time histories of the cantilevers without the mass in a the first mode and b the second mode. FFT analyses for the time history of the cantilevers without the mass in c the first mode and d the second mode. The predominant frequency spectrums are at c 6.450 Hz and d 6.460 Hz, respectively. FFT analyses for the time history of the vibrator in e the first mode and f the second mode. The predominant frequency spectrums are e 0.116 mm at 12.90 Hz and f 0.121 mm at 12.92 Hz, respectively.

or in the dimensional form

$$\begin{aligned} U =U_{0} e^{-\gamma \Omega t} \cos \Omega t, \end{aligned}$$

(24)

where $\Omega =1 / T^{\prime }$. Letting the maximum displacement be $t_{n}$ and the maximum one after the one period be $t_{n+1}$, $\gamma$ is expressed by using the logarithmic decrement as

$$\begin{aligned} \gamma =\frac{1}{2 \pi } \log \frac{U(t_{n})}{U(t_{n+1})}. \end{aligned}$$

(25)

Since Eq. (17) corresponds to the dimensionless equation of motion of the simple pendulum model in the manuscript, we can obtain $\gamma$ in Eq. (2) from the experimental free oscillation of the cantilever by using Eq. (24).

D: Experimental results in the case when the excitation amplitude is much lower

We conducted the experiments using the coupled cantilevers whose first and second natural frequencies are 6.450 Hz and 6.460 Hz, respectively. In Fig. 8, we show the experimental results in the case when the excitation amplitude is much lower, i,e., $a_e=0.12$ mm. Also in this case, the first or second mode is selectively resonated depending on the parametric excitation frequency.

E: Schematic and experimental diagrams of the mode shapes

In the first and second modes, the two cantilevers oscillate in phase and out of phase, respectively. They are schematically described in Fig. 9a and b. Figure 10a and b shows experimentally obtained first and second modes under the parametric excitation, respectively.

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Schematic diagram of the mode shapes. a First mode. b Second mode.

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Experimentally obtained oscillations. a First mode. b Second mode.