Identification method for backbone curve of cantilever beam using van der Pol-type self-excited oscillation

Abstract

This study presents an experimental method for identification of the backbone curves of cantilevers using the nonlinear dynamics of a van der Pol oscillator. The backbone curve characterizes the nonlinear stiffness and nonlinear inertia of the resonator, so it is important to identify this curve experimentally to realize high-sensitivity and high-accuracy sensing resonators. Unlike the conventional method based on the frequency response under external excitation, the proposed method based on self-excited oscillation enables direct backbone curve identification, because the effect of the viscous environment is eliminated under the linear velocity feedback condition. In this research, the method proposed for discrete systems is extended to give an identification method for continuum systems such as cantilever beams. The actuation is given with respect to both the linear and nonlinear feedbacks so that the system behaves as a van der Pol oscillator with a stable steady-state amplitude. By varying the nonlinear feedback gain, we can produce the self-excited oscillation experimentally with various steady-state amplitudes. Then, using the relationship between these steady-state amplitudes and the corresponding experimentally measured response frequencies, we can detect the backbone curve while varying the nonlinear feedback gain. The efficiency of the proposed method is determined by identifying the backbone curves of a macrocantilever with a tip mass and a macrocantilever subjected to atomic forces, which are representative sources of hardening and softening cubic nonlinearities, respectively.

Appendices

Nondimensionless equation of motion

The nondimensionless equation of motion is derived as:

$$\begin{aligned}&\rho A \frac{\partial ^2}{\partial t^2} \left( v + \eta \right) + c \frac{\partial v}{\partial t} \nonumber \\ {}&\quad + EI \left[ \frac{ \partial ^4 v }{ \partial s^4 } + \left( \frac{ \partial ^2 v }{ \partial s^2 } \right) ^3 + 4\frac{ \partial v }{ \partial s } \frac{ \partial ^2 v }{ \partial s^2 } \frac{ \partial ^3 v }{ \partial s^3 }\right. \nonumber \\ {}&\qquad \left. + \left( \frac{ \partial v }{ \partial s } \right) ^2 \frac{ \partial ^4 v }{ \partial s^4 } \right] \nonumber \\ {}&\quad + \rho A \frac{\partial v}{\partial s} \int _l^s \left. \left( \frac{ \partial ^2 v }{ \partial s \partial t }\right) ^2 + \frac{ \partial v }{\partial s}\frac{ \partial ^3 v }{ \partial s^2 \partial t } \right. \text{ d }s \nonumber \\ {}&\quad + \rho A \frac{\partial ^2 v}{\partial s^2} \int _0^s \int _l^s \left( \frac{ \partial ^2 v }{ \partial s \partial t }\right) ^2 + \frac{ \partial v }{\partial s}\frac{ \partial ^3 v }{ \partial s^2 \partial t } \text{ d }s \text{ d }s \nonumber \\ {}&\quad - m_t \frac{\partial ^2v}{\partial s^2} \int _0^l \left. \left( \frac{ \partial ^2 v }{ \partial s \partial t }\right) ^2 \right. \left. + \frac{ \partial v }{\partial s}\frac{ \partial ^3 v }{ \partial s^2 \partial t } \text{ d }s \right. = 0. \end{aligned}$$

(45)

The associated boundary conditions are:

$$\begin{aligned}&\left. v\right| _{s=0} = \left. \frac{\partial v}{\partial s}\right| _{s=0} = 0, \end{aligned}$$

(46)

$$\begin{aligned}&\left. \frac{\partial ^2 v}{\partial s^2}\right| _{s=l} = 0, \end{aligned}$$

(47)

$$\begin{aligned}&\left. EI\left[ \frac{\partial v ^3}{\partial s^3}\left( 1 +\left( \frac{\partial v}{\partial s} \right) ^2 \right) \right] \right| _{s=l} \nonumber \\&\quad - m_t \left. \frac{\partial ^2v}{\partial s^2}\right| _{s=l} \int _0^l \left. \left( \frac{ \partial ^2 v }{ \partial s \partial t }\right) ^2 \frac{\partial ^2 v}{ \partial s^2 }\right. \nonumber \\&\qquad \left. + \frac{ \partial v }{\partial s}\frac{ \partial ^3 v }{ \partial s^2 \partial t } \right. \hbox {d}s \nonumber \\&\quad - m_t\frac{\partial ^2}{\partial t^2} \left( v+\eta \right) \left| _{s=l}\right. \nonumber \\&\qquad \left. - k_1v\right| _{s=l} -k_3v\left| _{s=l}^3 = 0. \right. \end{aligned}$$

(48)

In Eq. (45), the equivalent damping effect, i.e., $c\frac{\partial v}{\partial t}$, is introduced.

Change of self-excited mode depending on the cut-off frequency of the filter

In sec. 4.1, we use the low-pass filter in order to cut the noise and the self-excited oscillations higher than the second mode. Unlike the single-degree-of-freedom system in our previous study [27], continuum systems such as a cantilever in the present study can be self-excited with any natural frequencies under the velocity feedback. In fact, as mentioned below, it is experimentally shown that the self-excited oscillation is produced with the second mode depending on the cut-off frequency of the filter. The experimental system when a relatively long cantilever beam made from phosphor bronze was used is shown in Fig. 9a; the dimensions are 0.216 $\times$ 0.02 $\times$ 0.002 m. The position of the laser displacement sensor is the same as Fig. 4. Figure 9b shows the frequency spectrum when the self-excited oscillation is produced by our feedback control. The red line shows that the cantilever is self-excited with the first natural frequency when the cut-off frequency is 6 Hz below the first natural frequency. On the other hand, when the cut-off frequency is set to be 18 Hz above the second natural frequency, the cantilever is not self-excited with the first mode but the second one.

Fig. 9

Change of self-excited mode depending on the cut-off frequency of the filter. a Cantilever whose natural frequency is set to be low in order to observe the higher mode oscillation. b Response frequency when the cut-off frequency is 6 Hz. The self-excited oscillation is produced with the first natural frequency. c The response frequency when the cut-off frequency is 18 Hz. The self-excited oscillation is produced with the second natural frequency

Fig. 10

Construction of the verification system. A disk is attached at the tip of the cantilever, which is same as that shown in Fig. 4, and means that the cantilever is subjected to relatively high viscous damping from the water. The mass and radius of this disk are 20 $\times 10^{-3}$ kg and 25 $\times 10^{-3}$ m, respectively

Fig. 11

Comparison of the free vibrations. The blue and red lines show the time histories when the disk is in air and when it is immersed in water, respectively. The response in the water decays more rapidly because of the relatively large dissipative force of the water

Fig. 12

Experimental frequency responses and backbone curves. The blue and red markers show the backbone curves obtained using our proposed method when the disk is in air and when it is immersed in water, respectively

Identification in a viscous damping environment

In this appendix, we show that the proposed method can identify the backbone curve independently of the magnitude of viscous damping in the environment through comparison of the backbone curve obtained for a specimen that was subject to fluid viscosity with the corresponding curve obtained for the same specimen in air.

Figure 10 shows the additional system required to validate the effect of the viscosity on the results. A thin disk is attached at the tip to increase the damping of the cantilever. We performed the experiment under two conditions: the first was the low damping condition, i.e., where the disk was set in air, and the second was the relatively high damping condition, i.e., where the disk was immersed in water. The comparison of the free decay responses is shown in Fig. 11, where the blue and red lines show the responses under the low and high damping conditions, respectively. The decay characteristics are different, because the water provides a larger dissipative force.

Next, we show the backbone curve obtained using the proposed method in Fig. 12, where the blue and red markers represent the experimental results obtained using the proposed method under the air and water conditions, respectively. The figure confirms that the backbone curve obtained in water shows quantitative agreement with the backbone curve obtained in air. Because of the power limitations of the piezo-driver used, the maximum amplitude of the results obtained in water is smaller than that of the results obtained in air. The small difference between the backbone curve in air and that obtained in water may be caused by the change in the added mass in the viscous fluid. Although the added mass in water differs from that in air, our proposed method cannot compensate for the change in the added mass, because it only compensates for the change in the viscous damping.