Synchronization of a Passive Oscillator and a Liquid Crystal Elastomer Self-Oscillator Powered by Steady Illumination

Abstract

Self-oscillators have the advantages of actively harvesting energy from external steady environment, autonomy, and portability, and can be adopted as an engine to drive additional working equipment. The synchronous behavior of self-oscillators and passive oscillators may have an important impact on their functions. In this paper, we construct a self-oscillating system composed of a passive oscillator and an active liquid crystal elastomer self-oscillator powered by steady illumination, and theoretically investigate the synchronization of two coupled oscillators. There exist three synchronous regimes of the two coupled oscillators: static, in-phase, and anti-phase. The mechanisms of self-oscillations in in-phase and anti-phase synchronous regimes are elucidated in detail by calculating several key physical parameters. In addition, the effects of spring constant, initial velocity, contraction coefficient, light intensity, and damping coefficient on the self-oscillations of two coupled oscillators are further investigated, and the critical conditions for triggering self-oscillations are obtained. Numerical calculations show that the synchronous regime of self-oscillations is mainly determined by the spring constant, and the amplitudes of self-oscillations of two oscillators increase with increasing contraction coefficient, light intensity, and spring constant, while decrease with increasing damping coefficient. This study deepens the understanding of synchronization between coupled oscillators and may provide new design ideas for energy harvesters, soft robotics, signal detection, active motors, and self-sustained machinery.

Keywords: liquid crystal elastomer; optically-responsive; passive oscillator; self-oscillator; synchronization.

Figure 1

Schematic of a self-oscillating system composed of a passive spring oscillator and an LCE active self-oscillator powered by steady illumination. Both the LCE fiber and spring are connected to each mass block. The two mass blocks are placed into a fluid, and the damping coefficient can be easily tuned by controlling the viscosity of the fluid. Under steady illumination, the two coupled oscillators may vibrate synchronously.

Figure 2

Time histories and domain of attraction of in-phase synchronous regime of light-powered LCE self-excited coupled oscillators for $${\overline{k}}_2=16$$. (a) $${\overline{u}}_1$$ vs. $$\overline{t}$$; (b) $${\overline{u}}_2$$ vs. $$\overline{t}$$; (c) domain of attraction. The two couple oscillators are in in-phase regime.

Figure 3

Time histories and domain of attraction of anti-phase synchronous regime of light-powered LCE self-excited coupled oscillators for $${\overline{k}}_2=4.5$$. (a) $${\overline{u}}_1$$ vs. $$\overline{t}$$; (b) $${\overline{u}}_2$$ vs. $$\overline{t}$$; (c) domain of attraction. The two couple oscillators are in anti-phase regime.

Figure 4

Time histories and domain of attraction of in-phase synchronous regime of light-powered LCE self-excited coupled oscillators for $${\overline{k}}_2=8$$. (a) $${\overline{u}}_1$$ vs. $$\overline{t}$$; (b) $${\overline{u}}_2$$ vs. $$\overline{t}$$; (c) domain of attraction. The two couple oscillators are in static regime.

Figure 5

Mechanism of self-excited oscillation in in-phase regime of $${\overline{k}}_2=16$$ in Figure 2. (a) $$\phi \left(\overline{X}=1\right)$$ vs. $$\overline{t}$$. (b) $$\left|\epsilon \left(\overline{X}=1\right)\right|$$ vs. $$\overline{t}$$. (c) $${\overline{F}}_L$$ vs. $$\overline{t}$$. (d) $${\overline{F}}_L$$ vs. $${\overline{u}}_1$$. (e) $${\overline{F}}_s$$ vs. $$\overline{t}$$. (f) $${\overline{F}}_s$$ vs. $${\overline{u}}_2$$. In Figure 3d,f, the area enclosed by the closed loop represents the net work done by the tensions of the LCE fiber and spring, which compensates for the damping dissipation to maintain the oscillations of two coupled oscillators.

Figure 6

Mechanism of self-excited oscillation in anti-phase regime of $${\overline{k}}_2=4.5$$ in Figure 3. (a) $$\phi \left(\overline{X}=1\right)$$ vs. $$\overline{t}$$. (b) $$\left|\epsilon \left(\overline{X}=1\right)\right|$$ vs. $$\overline{t}$$. (c) $${\overline{F}}_L$$ vs. $$\overline{t}$$. (d) $${\overline{F}}_L$$ vs. $${\overline{u}}_1$$. (e) $${\overline{F}}_s$$ vs. $$\overline{t}$$. (f) $${\overline{F}}_s$$ vs. $${\overline{u}}_2$$. In Figure 4d,f, the area enclosed by the closed loop represents the net work done by the tensions of LCE fiber and spring, which compensates for the damping dissipation to maintain the oscillations of two coupled oscillators.

Figure 7

The effect of spring constant $${\overline{k}}_2$$ on the self-oscillations of two coupled oscillators. In the computation, the other geometric and material parameters are given in Table 2. (a) Domain of attraction, and (c,e) limit cycles for in-phase regimes of $${\overline{k}}_2>9.5$$. (b) Domain of attraction, and (d,f) limit cycles for anti-phase regimes of $${\overline{k}}_2<7$$. The synchronous regime of self-oscillations of two oscillators is mainly determined by the spring constant.

Figure 8

The effect of damping coefficient $$\overline{c}$$ on self-oscillations of the two coupled oscillators. (a) Domain of attraction, and (c,e) limit cycles for in-phase regime of $${\overline{k}}_2=16$$. (b) Domain of attraction, and (d,f) limit cycles for anti-phase regime of $${\overline{k}}_2=4.5$$. In the computation, the other geometric and material parameters are given in Table 2. With the increase of $$\overline{c}$$, the amplitudes of self-oscillations of two coupled oscillators decrease.

Figure 9

The effect of contraction coefficient $$C_0$$ on self-oscillations of the two coupled oscillators. (a) Domain of attraction, and (c,e) limit cycles for in-phase regime of $${\overline{k}}_2=16$$. (b) Domain of attraction, and (d,f) limit cycles for anti-phase regime of $${\overline{k}}_2=4.5$$. In the computation, the other geometric and material parameters are given in Table 2. With the increase of $$C_0$$, the amplitudes of self-oscillations of two coupled oscillators increase.

Figure 10

The effect of light intensity $$\overline{I}$$ on self-oscillations of the two coupled oscillators. (a) Domain of attraction, and (c,e) limit cycles for in-phase regime of $${\overline{k}}_2=16$$. (b) Domain of attraction, and (d,f) limit cycles for anti-phase regime of $${\overline{k}}_2=4.5$$. In the computation, the other geometric and material parameters are given in Table 2. With the increase of $$\overline{I}$$, the amplitudes of self-oscillations of two coupled oscillators increase.

Figure 11

The effect of initial velocity $${\overline{\dot{u}}}_1^0$$ on self-oscillations of the two coupled oscillators. (a) Domain of attraction, and (c,e) limit cycles for in-phase regime of $${\overline{k}}_2=16$$. (b) Domain of attraction, and (d,f) limit cycles for anti-phase regime of $${\overline{k}}_2=4.5$$. In the computation, the other geometric and material parameters are given in Table 2. The initial velocity $${\overline{\dot{u}}}_1^0$$ does not affect the self-oscillations of two oscillators.

Figure 12

The effect of initial velocity $${\overline{\dot{u}}}_2^0$$ on self-oscillations of the two coupled oscillators. (a) Domain of attraction, and (c,e) limit cycles for in-phase regime of $${\overline{k}}_2=16$$. (b) Domain of attraction, and (d,f) limit cycles for anti-phase regime of $${\overline{k}}_2=4.5$$. In the computation, the other geometric and material parameters are given in Table 2. The initial velocity $${\overline{\dot{u}}}_2^0$$ does not affect the self-oscillations of two oscillators.