Self-Sustained Collective Motion of Two Joint Liquid Crystal Elastomer Spring Oscillator Powered by Steady Illumination

Abstract

For complex micro-active machines or micro-robotics, it is crucial to clarify the coupling and collective motion of their multiple self-oscillators. In this article, we construct two joint liquid crystal elastomer (LCE) spring oscillators connected by a spring and theoretically investigate their collective motion based on a well-established dynamic LCE model. The numerical calculations show that the coupled system has three steady synchronization modes: in-phase mode, anti-phase mode, and non-phase-locked mode, and the in-phase mode is more easily achieved than the anti-phase mode and the non-phase-locked mode. Meanwhile, the self-excited oscillation mechanism is elucidated by the competition between network that is achieved by the driving force and the damping dissipation. Furthermore, the phase diagram of three steady synchronization modes under different coupling stiffness and different initial states is given. The effects of several key physical quantities on the amplitude and frequency of the three synchronization modes are studied in detail, and the equivalent systems of in-phase mode and anti-phase mode are proposed. The study of the coupled LCE spring oscillators will deepen people's understanding of collective motion and has potential applications in the fields of micro-active machines and micro-robots with multiple coupled self-oscillators.

Keywords: collective motion; domain of attraction; liquid crystal elastomer; spring oscillator.

Figure 1

The dynamic model of the self-oscillation coupling system is composed of two identical LCE fibers and a spring. (a) Reference configuration. (b) Pre-stretched state. (c) Current configuration. Under uniform and constant illumination, under uniform and constant illumination, the self-oscillation of the spring oscillators can be triggered by the coupling between the light-driven contraction of the fibers and their movement.

Figure 2

Three steady synchronization modes of the self-oscillation coupling system. The other parameters are ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0.2$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=0.3$, and ${\tilde{\dot{u}}}_2^0=-0.3$. (a–c) Anti-phase synchronization mode (${\tilde{K}}_{\mathrm{s}}=0.05$ ). (d–f) Non-phase-locked mode (${\tilde{K}}_{\mathrm{s}}=0.15$ ). (g–i) In-phase synchronization mode (${\tilde{K}}_{\mathrm{s}}=0.5$ ).

Figure 3

Mechanism of the self-excited oscillation in anti-phase mode. The parameters are ${\tilde{K}}_{\mathrm{f}}=5$, ${\tilde{K}}_{\mathrm{s}}=0.1$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=0.3$, and ${\tilde{\dot{u}}}_2^0=-0.3$. The illuminated zone is represented by the shaded area. (a) Time histories of the number fractions of cis-isomers in the two LCE fibers. (b) Time histories of contraction strains. (c) The spring forces of the two LCE fibers (driving forces). (d) The dependence of the spring force of LCE fiber on the mass displacement. (e) The spring force of the spring (spring force); (f) The displacements of the two mass blocks.

Figure 4

Dependence of the synchronization mode on the coupling stiffness and initial state. The parameters are ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, and ${\tilde{\dot{u}}}_1^0=-0.3$. Where the purple area is the phase diagram of the in-phase mode, the yellow area is the phase diagram of the anti-phase mode, and the green area is the phase diagram of the anti-phase mode.

Figure 5

Effect of coupling stiffness on the self-oscillation for a constant synchronous mode. (a,b) Effect of coupling stiffness on the in-phase mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_2^0=-0.2$. (c,d) Effect of coupling stiffness on the anti-phase mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, $\tilde{I}=0.65$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_2^0=0.3$. (e–h) Effect of coupling stiffness on the non-phase-locked mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_2^0=0.25$. The coupling stiffness has no effect on the self-oscillation in in-phase mode and has an effect on the period and amplitude of the self-oscillation in anti-phase mode and non-phase-locked mode.

Figure 6

Effect of light intensity on self-oscillation for a constant synchronous mode. (a,b) Effect of light intensity on in-phase mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.3$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=-0.2$. (c,d) Effect of light intensity on the anti-phase mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.1$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.3$. (e–h) Effect of light intensity on the non-phase-locked mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $C_0=0.6$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.1$. Light intensity only affects the amplitudes of the three synchronization modes.

Figure 7

Effect of contraction coefficient on the self-oscillation for a constant synchronous mode. (a,b) Effect of three different contraction coefficients on self-oscillation in in-phase mode. The parameters are ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.6$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=-0.2$. (c,d) Effect of three different contraction coefficients on self-oscillation in anti-phase mode. The parameters are ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.6$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.3$. (e–h) Effect of the contraction coefficient on the self-oscillation in the non-phase-locked mode. The parameters are ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.6$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.1$. The contraction coefficient only affects the amplitude of the three synchronization modes.

Figure 8

Effect of the damping coefficient on self-oscillation for a constant synchronous mode. (a,b) Effect of three different damping coefficients on self-oscillation in in-phase mode. The parameters are ${\tilde{K}}_{\mathrm{f}}=5$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.6$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=-0.2$. (c,d) Effect of three different damping coefficients on the self-oscillation in anti-phase mode. The parameters are ${\tilde{K}}_{\mathrm{f}}=5$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.8$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.3$. (e–h) Effect of the damping coefficient on the self-oscillation in the non-phase-locked mode for ${\tilde{K}}_{\mathrm{f}}=5$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.8$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.1$. The damping coefficient only affects the amplitude of the three synchronization modes.

Figure 9

Effect of spring constant of LCE fiber on self-oscillation for a constant synchronous mode. (a,b) Effect of three different spring constants on self-oscillation in in-phase mode. The parameters are $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.3$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=-0.1$. (c,d) Effect of three different spring constants on self-oscillation in anti-phase mode. The parameters are $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.3$, $\tilde{I}=0.6$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.3$. (e–h) Effect of spring constant of the LCE fiber on the self-oscillation in non-phase-locked mode for $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.3$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, ${\tilde{\dot{u}}}_1^0=-0.3$, and ${\tilde{\dot{u}}}_1^0=0.2$. The spring constants of the LCE fiber have effects on the frequency and amplitude of the three synchronization modes.

Figure 10

Effect of the initial condition on self-oscillation for a constant synchronous mode. (a,b) Effect of initial condition on self-oscillation in in-phase mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, and ${\tilde{\dot{u}}}_1^0=-0.3$. (c,d) Effect of initial condition on the self-oscillation in anti-phase mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.05$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, and ${\tilde{\dot{u}}}_1^0=-0.3$. (e–h) Effect of initial condition on the self-oscillation in non-phase-locked mode for ${\tilde{K}}_{\mathrm{f}}=5$, $\tilde{c}=0.1$, ${\lambda }_{\mathrm{p}}=1.15$, ${\tilde{K}}_{\mathrm{s}}=0.2$, $\tilde{I}=0.5$, $C_0=0.7$, ${\tilde{u}}_1^0=0$, ${\tilde{u}}_2^0=0$, and ${\tilde{\dot{u}}}_1^0=-0.3$. The initial conditions affect the amplitude of the non-phase-locked mode and the change of the amplitude within a doubling period.

Figure 11

Equivalent systems of (a) in-phase mode and (b) anti-phase mode. In in-phase mode, the system is equivalent to a single oscillator with a constant force. In anti-phase mode, the system is equivalent to a single oscillator constrained by a fixed spring with half original length.