Synchronous behaviors of three coupled liquid crystal elastomer-based spring oscillators under linear temperature fields

Self-oscillating coupled systems possess the ability to actively absorb external environmental energy to sustain their motion. This quality endows them with autonomy and sustainability, making them have application value in the fields of synchronization and clustering, thereby furthering research and exploration in these domains. Building upon the foundation of thermal responsive liquid crystal elastomer-based (LCE-based) spring oscillators, a synchronous system comprising three LCE-based spring oscillators interconnected by springs is established. In this paper, the synchronization phenomenon is described, and the self-oscillation mechanism is revealed. The results indicate that by varying system parameters and initial conditions, three synchronization patterns emerge, namely, full synchronous mode, partial synchronous mode, and asynchronous mode. For strongly interacting systems, full synchronous mode always prevails, while for weak interactions, the adjustment of initial velocities in magnitude and direction yields the three synchronization patterns. Additionally, this study explores the impact of several system parameters, including LCE elasticity coefficient and spring elasticity coefficient, on the amplitude, frequency, and synchronous mode of the system. The findings in this paper can enhance our understanding of the synchronization behavior of multiple mutually coupled LCE-based spring oscillators, with promising applications in energy harvesting, soft robotics, signal monitoring, and various other fields.

Figure 1

Schematic diagram of self-oscillating coupled system. (a) Reference state; (b) prestretched state; (c) current state; (d) force analysis of the masses.

Figure 2

Three synchronous modes and two motion regimes of the coupled self-oscillating system, where the parameters are $\overline{K}=14, \overline{k}=6, \overline{\alpha }=-0.2, \overline{\beta }=0.6, {\overline{a}}_1=0.6$, and $\overline{\tau }=0.15$. (a) Static regime in full synchronous mode, and the parameter is set as ${\overline{a}}_0=0.2, v_1^0=0, v_2^0=0$, and $v_3^0=0$. (b) Self-oscillation regime in full synchronous mode, and the parameters are set as ${\overline{a}}_0=0.02, v_1^0=0, v_2^0=0$, and $v_3^0=0$. (c) Static regime in partial synchronous mode, and the parameters are set as ${\overline{a}}_0=0.2, v_1^0=0, v_2^0=0$, and $v_3^0=5$. (d) Self-oscillation regime in partial synchronous mode, and the parameters are set as ${\overline{a}}_0=0.02, v_1^0=0, v_2^0=0$, and $v_3^0=5$. (e) Static regime in asynchronous mode, and the parameters are set as ${\overline{a}}_0=0.2, v_1^0=1, v_2^0=0$, and $v_3^0=-5$. (f) Self-oscillation regime in asynchronous mode, and the parameters are set as ${\overline{a}}_0=0.02, v_1^0=1, v_2^0=0$, and $v_3^0=-5$. There exist three synchronous modes of the system, that is, full synchronous mode, partial synchronous mode, and asynchronous mode.

Figure 3

Mechanism of the self-oscillation. The parameters of the full synchronous mode are set as $\overline{K}=14, \overline{k}=6, \overline{\alpha }=-0.2, \overline{\beta }=0.6, {\overline{a}}_0=0.02, {\overline{a}}_1=0.6, \overline{\tau }=0.15, v_1^0=0, v_2^0=0$, and $v_3^0=0$. The parameters of the partial synchronous mode are set as $\overline{K}=14, \overline{k}=6, \overline{\alpha }=-0.2, \overline{\beta }=0.6, {\overline{a}}_0=0.02, {\overline{a}}_1=0.6, \overline{\tau }=0.15, v_1^0=5, v_2^0=0$, and $v_3^0=0$. The parameters of the asynchronous mode are set as $\overline{K}=14, \overline{k}=6, \overline{\alpha }=-0.2, \overline{\beta }=0.6, {\overline{a}}_0=0.02, {\overline{a}}_1=0.6, \overline{\tau }=0.15, v_1^0=1, v_2^0=0$, and $v_3^0=-5$. (a)–(c) Time history curve of full synchronous mode, partial synchronous mode, and asynchronous mode. (d)–(f) Curves of the tension of LCE fiber change over time in the full synchronous mode, partial synchronous mode, and asynchronous mode. (g)–(i) Curves of damping forces change with time in full synchronous mode, partial synchronous mode, and asynchronous mode. (j–(l) The relationship between displacement and the tension of LCE fibers in full synchronous mode, partial synchronous mode, and asynchronous mode. (m)–(o) The relationship between displacement and the damping forces in full synchronous mode, partial synchronous mode, and asynchronous mode. The energy absorbed by the fibers from the temperature field can offset damping dissipation, enabling the system to sustain self-oscillation.

Figure 4

The effect of the thermal expansion coefficient on the self-oscillation in full synchronous mode. (a) Variations of amplitude and frequency with different thermal expansion coefficients. (b) Limit cycles with $\overline{\alpha }=-0.1, \overline{\alpha }=-0.15$, and $\overline{\alpha }=-0.2$.

Figure 5

Synchronization modes of three coupled nonidentical self-oscillators with different thermal expansion coefficients. (a) Full synchronous mode. (b) Partial synchronous mode. (c) Asynchronous mode.

Figure 6

The effect of an LCE elastic coefficient on the self-oscillation in full synchronous mode. (a) Variations of amplitude and frequency with different LCE elastic coefficients. (b) Limit cycles with $\overline{K}=14, \overline{K}=10$, and $\overline{K}=6$.

Figure 7

The effect of spring elastic coefficient on the self-oscillation in full synchronous mode. (a) Variations of amplitude and frequency with different spring elastic coefficients. (b) Limit cycles with $\overline{k}=6$, and $\overline{k}=10$.

Figure 8

The effect of temperature gradient on the self-oscillation in full synchronous mode. (a) Variations of amplitude and frequency with different temperature gradient. (b) Limit cycles with $\overline{\beta }=0.4, \overline{\beta }=0.6$, and $\overline{\beta }=0.8$.

Figure 9

The effect of the first damping coefficient on the self-oscillation in full synchronous mode. (a) Variations of amplitude and frequency with different first damping coefficients. (b) Limit cycles with ${\overline{a}}_0=0.02, {\overline{a}}_0=0.06$, and ${\overline{a}}_0=0.1$.

Figure 10

The effect of the second damping coefficient on the self-oscillation in full synchronous mode. (a) Variations of amplitude and frequency with different second damping coefficients. (b) Limit cycles with ${\overline{a}}_1=0.3, {\overline{a}}_1=0.6$, and ${\overline{a}}_1=0.9$.

Figure 11

The effect of characteristic time on the self-oscillation in full synchronous mode. (a) Variations of amplitude and frequency with characteristic time. (b) Limit cycles with $\overline{\tau }=0.1, \overline{\tau }=0.15$, and $\overline{\tau }=0.2$.

Figure 12

The effect of coefficient of thermal expansion on the self-oscillation in partial synchronous mode. (a) Variations of amplitude and frequency with different coefficients of thermal expansion. (b) Limit cycles with $\overline{\alpha }=-0.1, \overline{\alpha }=-0.15$, and $\overline{\alpha }=-0.2$.

Figure 13

The effect of LCE elastic coefficient on the self-oscillation in partial synchronous mode. (a) Variations of amplitude and frequency with different LCE elastic coefficients. (b) Limit cycles with $\overline{K}=14, \overline{K}=10$, and $\overline{K}=6$.

Figure 14

The effect of spring elastic coefficient on the self-oscillation in partial synchronous mode. (a) Variations of amplitude and frequency with different spring elastic coefficients. (b) Limit cycles with $\overline{k}=6, \overline{k}=8$, and $\overline{k}=10$.

Figure 15

The effect of temperature gradient on the self-oscillation in partial synchronous mode. (a) Variations of amplitude and frequency with different temperature gradients. (b) Limit cycles with $\overline{\beta }=0.4, \overline{\beta }=0.6$, and $\overline{\beta }=0.8$.

Figure 16

The effect of the first damping coefficient on the self-oscillation in partial synchronous mode. (a) Variations of amplitude and frequency with different first damping coefficients. (b) Limit cycles with ${\overline{a}}_0=0.02, {\overline{a}}_0=0.06$, and ${\overline{a}}_0=0.1$.

Figure 17

The effect of the second damping coefficient on the self-oscillation in partial synchronous mode. (a) Variations of amplitude and frequency with different second damping coefficients. (b) Limit cycles with ${\overline{a}}_1=0.3, {\overline{a}}_1=0.6$, and ${\overline{a}}_1=0.9$.

Figure 18

The effect of characteristic time on the self-oscillation in partial synchronous mode. (a) Variations of amplitude and frequency with characteristic time. (b) Limit cycles with $\overline{\tau }=0.1, \overline{\tau }=0.15$, and $\overline{\tau }=0.2$.

Figure 19

The effect of thermal expansion coefficient on the self-oscillation in asynchronous mode. (a) Variations of amplitude and frequency with different thermal expansion coefficients. (b) Variations of phase difference with different thermal expansion coefficients.

Figure 20

The effect of LCE elastic coefficient on the self-oscillation in asynchronous mode. (a) Variations of amplitude and frequency with different LCE elastic coefficients. (b) Variations of phase difference with different LCE elastic coefficients.

Figure 21

The effect of spring elastic coefficient on the self-oscillation in asynchronous mode. (a) Variations of amplitude and frequency with different spring elastic coefficients. (b) Variations of phase difference with different spring elastic coefficients.

Figure 22

The effect of temperature gradient on the self-oscillation in asynchronous mode. (a) Variations of amplitude and frequency with different temperature gradients. (b) Variations of phase difference with different temperature gradients.

Figure 23

The effect of the first damping coefficient on the self-oscillation in asynchronous mode. (a) Variations of amplitude and frequency with different first damping coefficients. (b) Variations of phase difference with different first damping coefficients.

Figure 24

The effect of the second damping coefficient on the self-oscillation in asynchronous mode. (a) Variations of amplitude and frequency with different second damping coefficients. (b) Variations of phase difference with different second damping coefficients.

Figure 25

The effect of characteristic time on the self-oscillation in asynchronous mode. (a) Variations of amplitude and frequency with different characteristic times. (b) Variations of phase difference with different characteristic times.