Geometry-Flexible Liquid Crystal Elastomer Self-Oscillator Enabled by Light Feedback Routing

Abstract

Self-oscillators convert constant external stimuli into sustained mechanical work, offering potential for applications such as soft robotics, energy absorption, and mechanical logic. However, the effective design of a light-driven self-oscillation system is challenging due to geometrically constrained deformation modes and the inherent rigidity of rectilinear light propagation paths. Notably, the mirror-reflected optical feedback loop decouples the feedback mechanism from geometric constraints imposed by deformation modes, enabling dynamic coupling independent of structural geometry. In this study, we introduce a geometry-flexible light feedback loop to drive a liquid crystal elastomer (LCE) self-oscillator. The system comprises an optically responsive LCE fiber, a spring, a mirror, and a perforated plate. By integrating the dynamic photon propagation path in light feedback routing with the dynamic deformation model of the LCE, we develop a dynamic theoretical model of the oscillator under constant illumination. Numerical simulations reveal two distinct patterns: static equilibrium and self-oscillation. Self-oscillation is generated by the light-induced contraction of LCE fiber segments illuminated by reflected light. Crucially, mirror-reflected light enables localized deformations anywhere along the fiber to contribute to global displacement feedback, thereby transcending the constraints of geometric deformation modes. This capability transcends the limitations posed by constrained geometric deformation modes, enabling adaptable control of the optical feedback loop through simple geometric alterations. This innovative approach circumvents the need for intricate structural feedback designs and separate energy harvesters, as well as actuator systems.

1. Introduction

Self-oscillation refers to self-sustained, periodic motions exhibited by active materials or structures, driven by a non-oscillatory but constant energy source [1,2,3,4]. Such motions arise without relying on alternating external signals or repeated triggers; instead, they are maintained through steady power input and internal feedback loops [4]. This autonomy offers both inherent robustness and structural simplicity, with the oscillation frequency being predominantly determined by the system’s intrinsic properties rather than the characteristics of the external energy supply [2,3]. These advantages make self-oscillations particularly suitable for applications including soft robotics [5,6,7,8,9,10], energy dissipation devices [11,12,13], transport systems [14], tunable optics [15], and logic computing [16,17,18].

Significant efforts have been devoted to realizing self-sustained oscillations by using diverse stimulus-responsive materials, such as thermally activated polymers [5,19,20], hydrogels [21,22,23,24,25], and liquid crystal elastomers (LCEs) [26,27,28,29,30,31,32]. By harnessing these materials, a broad spectrum of autonomous oscillatory patterns has been developed, including crawling [33], swimming [34], galloping [35], vibration [36,37,38,39,40], rolling [41,42,43,44], sliding [45], twisting [15,46], swinging [47,48,49,50], buckling [51], rocking [52], rotation [53,54,55], peeling [56], ejecting [57], jumping [58,59,60], stirring [61], propelling [62], synchronization [63,64], and combined multi-mode oscillations [65,66,67,68,69,70,71]. These engineered nonlinearities effectively convert a constant external energy supply into continuous periodic motion.

Self-sustained motion systems based on stimulus-responsive materials require the establishment of positive and negative feedback loops between the material’s geometric deformation and the constant energy field. A variety of feedback mechanisms have been developed to achieve such self-oscillations. These can be broadly classified based on the interaction motion pattern between the constant energy field and the active material. Key categories include the self-shadowing effect [15,72,73], where the material’s motion periodically modulates its exposure to the stimulus; the non-uniform environmental field mechanism [74,75,76,77], which uses predefined spatial variations in stimulus intensity to trigger different deformation; and the self-regulating field mechanism [78,79], which operates via preprogrammed temporal cycles or adaptive energy release pathways. Each strategy offers a distinct route to maintain continuous motion under constant external conditions. Generally, a key challenge in the design of these systems is developing a feedback loop that sustains the system’s self-oscillatory motion. This is particularly true for light-driven self-oscillation systems based on the self-shadowing effect, which often rely on geometric deformation modes—such as bending and stretching—of the structure to intercept light along its propagation path [39,41,54,70]. However, the design of such systems is frequently constrained by two main factors: the limited types of deformation modes available to the geometry and the simplicity of rectilinear light propagation paths. Consequently, the light interception effect induced by geometric deformation is often insufficient, posing significant challenges in designing self-oscillating systems.

Recently, a self-oscillating system via a mirror-reflected light-driven feedback loop of two coupled hydrogels was reported [79]. In the experiments, the light-driven nano-functionalized hydrogel system demonstrated temperature oscillations via a negative feedback loop incorporating controlled time delays, as shown in Figure 1. Crucially, the light feedback loop, facilitated by mirror-reflected light, decouples the feedback mechanism from the geometric constraints imposed by deformation modes. Inspired by this mirror reflection approach, we introduce a geometry-flexible LCE self-oscillator enabled by light feedback routing. The system consists of an optically responsive LCE fiber, a spring, a mirror, and a perforated plate. Using a dynamic LCE model, we introduce a nonlinear model for the self-oscillator under constant light exposure. Simulations reveal two distinct motion patterns: static equilibrium and self-sustained oscillation. Self-oscillation originates from the contraction of the material points of the LCE fibers along the light path, with continuous motion being sustained by the interaction between light energy and damping dissipation. We investigate the conditions necessary to trigger self-oscillation and examine key parameters that influence its frequency and amplitude. Unlike typical self-oscillation systems, the LCE self-oscillator’s light-path configuration is dynamically regulated through mirror reflection, creating an oscillation-linked light routing that is influenced by its geometric structure. This design overcomes the constraints of limited deformation modes, enabling flexible control of the optical feedback loop through simple geometric deformations.

This work is organized as follows: In Section 2, the nonlinear dynamic model of a geometry-flexible LCE self-oscillator under constant light exposure is formulated based on the dynamic LCE model. In Section 3, the dynamics of the LCE self-oscillator under constant light are numerically analyzed. Two distinct motion patterns are identified, and the detailed mechanics of the self-oscillation process are elucidated. Section 4 investigates the conditions required to trigger self-oscillation and examines the effects of various system parameters on the oscillation frequency and amplitude. This paper concludes with a brief summary in Section 5.

2. Theoretical Model and Formulation

In this section, we develop a theoretical model for the geometry-flexible LCE self-oscillator under constant illumination. This model encompasses the dynamic governing equations of the oscillator, the tension within the LCE fiber, the temporal evolution of cis-isomer concentration, the photon propagation path in light feedback routing, and the corresponding nondimensionalization.

2.1. Dynamic Governing Equations of the Self-Oscillator

Figure 2 illustrates a geometry-flexible LCE self-oscillator enabled by light feedback routing. The system consists of a light-sensitive LCE fiber, a spring, a mirror, and a perforated plate, where the aperture regions are indicated in white and the non-aperture regions are shown in green. Depicted in Figure 2a, the spring’s left end is attached to the stationary left endpoint, $P_1$, while its right end joins the perforated plate; this plate’s right side is bonded to a light-sensitive LCE fiber, and the fiber’s right extremity is fastened to the fixed right endpoint, $P_2$. Under reference conditions, the nematic LCE material is in a stress-free state with an initial length $L_0$, and light-responsive molecules like azobenzene are oriented uniformly along the fiber’s axis. It is a recognized characteristic that LCE fibers undergo contraction when exposed to light and revert to their original state in darkness [80,81,82,83].

A steady slender band-like incident light beam aligned orthogonally to the $P_1{P}_2$ axis traverses the aperture regions in the plate and is reflected by a mirror onto the light-sensitive LCE fiber, as illustrated in Figure 2b. Upon exposure to reflected light, the light-sensitive liquid crystal molecules undergo a conformational shift from an elongated trans state to a curled cis configuration, causing the illuminated segments of the LCE fiber to shrink. Consequently, the perforated plate moves rightward along with the LCE fiber, resulting in a progressive increase in the restorative force of the spring. As the plate gradually moves rightward, the location of the illuminated segments changes; simultaneously, the span of the light beam passing through the aperture regions may diminish, leading to a reduction in the length of LCE fiber, and the contraction induced by light in segments that transition from being illuminated by the reflected light to being unilluminated gradually decreases. Under the influence of the growing restorative force, the plate decelerates, passes through the equilibrium position, and continues moving leftward due to its inertia. Accordingly, both the size and placement of the illuminated segment change inversely compared with the rightward motion phase, causing material points on the LCE fiber to alternate between illuminated and non-illuminated states. As a result, the perforated plate exhibits sustained cyclic oscillations around its static equilibrium position under constant light irradiation.

In order to examine the movement of the perforated plate, a coordinate system is established with the origin $O$ fixed at the midpoint $O_1$ of the plate’s left end in the reference state, and $u\left(t\right)$ denotes the instantaneous displacement of the perforated plate at time $t$. Additionally, let $L\left(t\right)$ represent the length of the LCE fiber over time, as shown in Figure 2f. During oscillation, the plate is subjected to the tension $F_{\mathrm{s}}\left(t\right)$ from the LCE fiber, the restorative force $F_{\mathrm{k}}\left(t\right)$ of the spring, and the damping force $F_{\mathrm{d}}\left(t\right)$, as illustrated in Figure 2f. Notably, the damping force $F_{\mathrm{d}}\left(t\right)$ exhibits a complex relationship with the system’s velocity in real-world scenarios. However, the linear damping model can provide a simple yet reasonable description of energy dissipation at low velocities. To focus on the dynamics of the self-oscillator enabled by light feedback routing, we assume that the damping force is proportional to the plate’s velocity $\dot{u}\left(t\right)$ and acts in the opposite direction, i.e., $F_{\mathrm{d}}(t)=c\dot{u}\left(t\right)$. Thus, the dynamic governing equation of the plate can be provided as follows:

$$m\ddot{u}\left(t\right)=F_{\mathrm{s}}(t)-F_{\mathrm{k}}(t)-c\dot{u}\left(t\right),$$

(1)

where $m$ signifies the mass of the plate, $c$ indicates the damping coefficient, and $\ddot{u}\left(t\right)$ corresponds to the acceleration. The system’s initial state is defined by

$$u\left(t\right)=u_0, \dot{u}\left(t\right)={\dot{u}}_0 \mathrm{at} t=0.$$

(2)

By employing linear elasticity theory, the force applied by the spring is expressed as

$$F_{\mathrm{k}}(t)=Ku(t),$$

(3)

where $K$ represents the stiffness constant of the spring.

2.2. Tension of the LCE Fiber

To determine the tension $F_{\mathrm{s}}\left(t\right)$ of the LCE fiber in Equation (1), the following assumptions are made in this study: (1) The mass of the LCE fiber can be neglected. (2) Both the tension $F_{\mathrm{s}}\left(t\right)$ and the elastic strain ${\epsilon }_{\mathrm{e}}\left(t\right)$ of the LCE fiber are homogeneous. (3) It is assumed that the LCE fiber undergoes small deformations. (4) The tension $F_{\mathrm{s}}\left(t\right)$ is proportional to the elastic strain ${\epsilon }_{\mathrm{e}}\left(t\right)$. According to the linear relationship between $F_{\mathrm{s}}\left(t\right)$ and ${\epsilon }_{\mathrm{e}}\left(t\right)$, the tension of the LCE fiber can be given as

$$F_{\mathrm{s}}\left(t\right)=k{L}_0{\epsilon }_{\mathrm{e}}\left(t\right),$$

(4)

where $k$ denotes the elastic constant of the LCE fiber. It is important to highlight that the elastic coefficient $k$ of the LCE fiber varies as a function of the cis-isomer concentration $\phi \left(X,t\right)$. Typically, it tends to decrease gradually as $\phi \left(X,t\right)$ increases, after which it remains relatively stable over a certain concentration range [84]. In fact, considering that the variation in elastic coefficient $k$ with cis-isomer concentration $\phi \left(X,t\right)$ presents a significant challenge in solving this self-oscillating system, in this study, we focus on the dynamic behavior of the geometry-flexible LCE self-oscillator enabled by light feedback routing. For the sake of simplicity, we assume that the elastic constant remains constant and does not fluctuate with cis-isomer concentration $\phi \left(X,t\right)$. Notably, when considering a bulk LCE without neglecting its mass, both the tension $F_{\mathrm{s}}\left(t\right)$ and the elastic strain ${\epsilon }_{\mathrm{e}}\left(t\right)$ of the LCE fiber are inhomogeneous. This, in turn, introduces inaccuracies in the numerical results of the model. However, these inaccuracies only have a quantitative effect on the LCE self-oscillator, not a qualitative one. To focus on the dynamic behavior of the geometry-flexible LCE self-oscillator enabled by light feedback routing, the LCE fiber in this study is assumed to be thin, allowing its mass to be neglected.

The elastic strain ${\epsilon }_{\mathrm{e}}\left(t\right)$ is presumed to be homogeneous along the LCE fiber, while both the total strain ${\epsilon }_{\mathrm{tot}}$ and the light-induced contraction ${\epsilon }_{\mathrm{L}}$ exhibit spatial variation. To address this inhomogeneous deformation, the Lagrangian arc-coordinate $X$ defined in the reference configuration and the Eulerian arc-coordinate $x_1$ in the current state are employed, as shown in Figure 2a. The instantaneous spatial position of a material point $X$ is described by $x_1=x_1\left(X,t\right)$. Consequently, the total strain and light-driven contraction are formulated as ${\epsilon }_{\mathrm{tot}}\left(X,t\right)$ and ${\epsilon }_{\mathrm{L}}\left(X,t\right)$, respectively. Under the assumption of small deformation, the total strain ${\epsilon }_{\mathrm{tot}}\left(X,t\right)$ is approximated as the linear superposition of the elastic strain ${\epsilon }_{\mathrm{e}}\left(t\right)$ and the light-driven contraction, i.e., ${\epsilon }_{\mathrm{tot}}\left(X,t\right)={\epsilon }_{\mathrm{e}}\left(t\right)-{\epsilon }_{\mathrm{L}}\left(X,t\right)$, leading to a revised expression for tension as

$$F_{\mathrm{s}}\left(t\right)=k{L}_0\left[{\epsilon }_{\mathrm{tot}}\left(X,t\right)+{\epsilon }_{\mathrm{L}}\left(X,t\right)\right],$$

(5)

where ${\epsilon }_{\mathrm{L}}\left(X,t\right)$ is taken to have a proportional relationship with the cis-isomer concentration $\phi \left(X,t\right)$, i.e.,

$${\epsilon }_{\mathrm{L}}\left(X,t\right)=C_0\phi \left(X,t\right),$$

(6)

where $C_0$ denotes the contraction constant.

For simplicity, the total strain is characterized as ${\epsilon }_{\mathrm{tot}}\left(X,t\right)=\lambda \left(X,t\right)-1$, with the deformation gradient $\lambda \left(X,t\right)$ being defined by

$$\lambda \left(X,t\right)={\displaystyle \frac{\mathrm{d}{x}_1\left(X,t\right)}{\mathrm{d}X}}.$$

(7)

Accordingly, $F_{\mathrm{s}}\left(t\right)$ in Equation (5) is reconfigured to

$$F_{\mathrm{s}}\left(t\right)=k{L}_0\left[\lambda \left(X,t\right)-1+C_0\phi \left(X,t\right)\right].$$

(8)

Subsequently, by performing integration on both sides of Equation (8) across the range from $0$ to $L_0$, the result is

$$F_{\mathrm{s}}\left(t\right)=k\left[L\left(t\right)-L_0+C_0{\displaystyle \underset{0}{\overset{L_0}{\int }}\phi \left(X,t\right)}\mathrm{d}X\right].$$

(9)

It is apparent that $u\left(t\right)=-\left[L\left(t\right)-L_0\right]$; Equation (9) can be restated as:

$$F_{\mathrm{s}}\left(t\right)=k\left[-u\left(t\right)+C_0{\displaystyle \underset{0}{\overset{L_0}{\int }}\phi \left(X,t\right)}\mathrm{d}X\right].$$

(10)

2.3. The Evolution of Cis-Isomer Concentration

To determine the cis-isomer concentration $\phi \left(X,t\right)$ for the LCE fiber referenced in Equation (10), the widely recognized dynamic model for LCEs introduced by Finkelmann and colleagues is applied [85,86]. Research conducted by Yu and others indicated that the transition from trans to cis states in LCEs can be triggered by ultraviolet light or laser sources having wavelengths below 400 nm [87]. The cis-isomer concentration $\phi \left(X,t\right)$ is affected by thermal activation from trans to cis, thermal relaxation from cis to trans, and photoinduced relaxation from trans to cis. Thermal activation is typically regarded as insignificant compared with light-induced activation [85,88]; thus the cis-isomer concentration $\phi \left(X,t\right)$ is generally modeled by the differential equation

$${\displaystyle \frac{\mathsf{\partial }\phi \left(X,t\right)}{\mathsf{\partial }t}}={\eta }_{0}I\left(X,t\right)\left[1-\phi \left(X,t\right)\right]-{\tau }_0^{-1}\phi \left(X,t\right),$$

(11)

where ${\eta }_0$ represents the light absorption coefficient, ${\tau }_0$ denotes the thermal relaxation duration for cis-to-trans reversion, and $I\left(X,t\right)$ is the instantaneous light intensity. Notably, the thermal relaxation duration of an LCE fiber is closely related to its cross-sectional size: a thicker fiber exhibits a longer relaxation time, whereas a thinner fiber relaxes faster. To match the dynamic feedback of the LCE self-oscillator, a fiber with a shorter thermal relaxation duration is preferred. Specifically, an LCE fiber with a diameter of 100–500 μm is recommended to ensure that light-induced contraction and elastic recovery can respond rapidly to displacement changes, thereby maintaining stable self-oscillation. Moreover, an excessively thin individual fiber results in insufficient tension in the LCE fiber, which in turn prevents the net work done by $F_{\mathrm{s}}\left(t\right)$ from compensating for the damping dissipation in the system. To address this, the system can employ multiple thin fibers arranged in parallel to simultaneously achieve both rapid thermal relaxation and sufficient fiber tension.

To address Equation (11), the instantaneous spatial coordinate $x_1\left(X,t\right)$ should be computed to determine whether the material point $X$ is illuminated by the reflected light or non-illuminated. By integrating Equations (8) and (10), the deformation gradient $\lambda \left(X,t\right)$ can be expressed as

$$\lambda \left(X,t\right)={\displaystyle \frac{1}{L_0}}\left[-u\left(t\right)+C_0{\displaystyle \underset{0}{\overset{L_0}{\int }}\phi \left(X,t\right)}\mathrm{d}X\right]+1-C_0\phi \left(X,t\right).$$

(12)

From Equations (7) and (12), we can obtain

$$\mathrm{d}{x}_1\left(X,t\right)=\left\{{\displaystyle \frac{1}{L_0}}\left[-u\left(t\right)+C_0{\displaystyle \underset{0}{\overset{L_0}{\int }}\phi \left(X,t\right)}\mathrm{d}X\right]+1-C_0\phi \left(X,t\right)\right\}\mathrm{d}X.$$

(13)

By performing integration on both sides of Equation (13) from $0$ to $X$, we arrive at

$$x_1\left(X,t\right)={\displaystyle \frac{X}{L_0}}\left[-u\left(t\right)+C_0{\displaystyle \underset{0}{\overset{L_0}{\int }}\phi \left(X,t\right)}\mathrm{d}X\right]+X-C_0{\displaystyle \underset{0}{\overset{X}{\int }}\phi \left(X,t\right)}\mathrm{d}X$$

(14)

The value of $x_1$ derived from Equation (14) enables the assessment of light exposure conditions for material points along the LCE fiber, distinguishing between illuminated and dark regions.

2.4. Photon Propagation Path in Light Feedback Routing

Determining whether a material point in the LCE is illuminated requires not only knowledge of its instantaneous position $x_1\left(X,t\right)$ but also the time-dependent position of the mirror-reflected light. This section will investigate the photon propagation path in the light feedback routing. In this study, the perforated plate contains an array of $N$ rhombus-aperture regions with side length $b$ and an interior angle $2\theta$, as shown in Figure 2d. By neglecting the height of the narrow beam, a geometric analysis reveals that the aperture regions width $d$ at the location perpendicular to the light beam on the plate can be expressed as:

$$d=2\left(b\cos \theta -Z\right)\tan \theta .$$

(15)

$Z$ represents the distance between the lines $P_1{P}_2$ and $O_1{O}_2$, as shown in Figure 2c. The coordinates $x_{\mathrm{A}n}$ and $x_{\mathrm{B}n}$ of the starting and ending points of the $n\mathrm{th}$ aperture regions on the plate along the line $P_1{P}_2$ can be expressed as

$$x_{\mathrm{A}n}\left(t\right)=Z\tan \theta +u\left(t\right)+2\left(n-1\right)b\sin \theta ,$$

(16)

$$x_{\mathrm{B}n}\left(t\right)=d+x_{\mathrm{A}n}\left(t\right).$$

(17)

where $n=1, 2,\dots , N$.

The width of the band-like light beam is denoted by $D$ (as shown in Figure 2e), with its left endpoint aligned with the left edge of the plate. Let $x_{\mathrm{L}}$ represent the coordinates of the photon emitted from the light source; so we have $0\le x_{\mathrm{L}}\le D$. Through geometric analysis, the coordinates $x_{\mathrm{LP}}$ of the photon reflected by the mirror from the point $x_{\mathrm{L}}$ to the line $P_1{P}_2$ can be expressed as

$$x_{\mathrm{LP}}=\left(Y+x_{\mathrm{L}}\tan \gamma \right)\tan 2\gamma .$$

(18)

where $\gamma$ is the angle between the mirror plane and the plate plane and $Y$ represents the distance from the origin $O$ to the mirror, as shown in Figure 2e.

It is worth mentioning that during the movement of the perforated plate, some photons pass through the aperture regions in the plate, while others are blocked by the plate. The photons that can pass through the aperture regions in the plate should satisfy the following condition:

$$x_{\mathrm{A}n}\le x_{\mathrm{L}}\le x_{\mathrm{B}n}.$$

(19)

By solving the system of Equations (18) and (19), the set $\mathsf{\Omega }$ of regions on the $P_1{P}_2$ line that are irradiated by the reflected photons can be determined. When the coordinates of a material point on the LCE fiber satisfy the condition $x\left(X,t\right)\in \mathsf{\Omega }$, the point is irradiated; otherwise, it is not. It is worth mentioning that according to the geometric relationship, we have:

$$x\left(X,t\right)=x_1\left(X,t\right)+2nb\sin \theta .$$

(20)

By substituting into Equation (14), the instantaneous spatial coordinate $x\left(X,t\right)$ of Equation (20) can be rewritten as:

$$x\left(X,t\right)=2Nb\sin \theta +{\displaystyle \frac{X}{L_0}}\left[-u\left(t\right)+C_0{\displaystyle \underset{0}{\overset{L_0}{\int }}\phi \left(X,t\right)}\mathrm{d}X\right]+X-C_0{\displaystyle \underset{0}{\overset{X}{\int }}\phi \left(X,t\right)}\mathrm{d}X.$$

(21)

Equation (21) distinguishes whether the material points along the LCE fiber are illuminated by the reflected light. When $x\left(X,t\right)\in \mathsf{\Omega }$, the material points of the LCE fiber are illuminated by the reflected light, and they are in the illuminated state; otherwise, they are in the dark state. It is worth noting that during the oscillation process of the perforated plate, there is an alternating appearance of aperture regions and non-aperture regions in the path of the incident light beam. Whether the incident light beam can propagate to the mirror depends on whether it encounters the aperture or non-aperture regions of the plate along its path. Consequently, the perforated plate functions as a binary switch that controls whether the incident light beam can reach the mirror and further regulates whether the light reflecting off the mirror illuminates specific points on the LCE fiber. In practical scenarios, the intensity of the reflected light received at the illuminated spots on the LCE fiber exhibits a non-uniform and complex spatial distribution due to factors such as beam spreading and variations in the propagation path. To focus on the self-oscillating behavior enabled by light feedback routing, this study simplifies the light intensity by assuming that the intensity received by the illuminated points on the LCE fiber is uniform. While this simplification neglects spatial heterogeneity, it is believed to have a quantitative rather than qualitative impact on the system’s self-oscillation. Thus, the instantaneous light intensity $I\left(X,t\right)$ in Equation (11) can be defined as $I\left(X,t\right)=I_0$ for $x\left(X,t\right)\in \mathsf{\Omega }$ and $I\left(X,t\right)=I_0$ for those in darkness. Notably, holes of various shapes, such as circular ones, are equally suitable for the self-oscillator design presented in this study. The choice of the shape only results in a quantitative impact on the set $\mathsf{\Omega }$, without affecting the qualitative dynamic behavior of self-oscillation. For simplicity, this study employs diamond-shaped holes to illustrate the working mechanism of the geometry-flexible LCE self-oscillator enabled by light feedback routing.

2.5. Nondimensionalization

For convenience, we introduce the following dimensionless quantities: $\overline{t}=t/{\tau }_0$, $\overline{K}=K{\tau }_0^2/m$, $\overline{k}=k{\tau }_0^2/m$, $\overline{c}=c{\tau }_0/m$, ${\overline{I}}_0=I_0{\eta }_0{\tau }_0$, $\overline{b}=b/L_0$, $\overline{D}=D/L_0$, $\overline{Z}=Z/L_0$, $\overline{Y}=Y/L_0$, $\overline{u}=u/L_0$, $\overline{\dot{u}}=\dot{u}{\tau }_0/L_0$, $\overline{\ddot{u}}=\ddot{u}{\tau }_0^2/L_0$, ${\overline{\dot{u}}}_0={\dot{u}}_0{\tau }_0/L_0$, ${\overline{\ddot{u}}}_0={\ddot{u}}_0{\tau }_0^2/L_0$, $\overline{X}=X/L_0$, $\overline{x}=x/L_0$, ${\overline{x}}_{\mathrm{L}}=x_{\mathrm{L}}/L_0$, ${\overline{x}}_{\mathrm{LP}}=x_{\mathrm{LP}}/L_0$, ${\overline{x}}_{\mathrm{A}n}=x_{\mathrm{A}n}/L_0$, ${\overline{x}}_{\mathrm{B}n}=x_{\mathrm{B}n}/L_0$, $\overline{d}=d/L_0$, ${\overline{F}}_s=F_s{\tau }_0^2/m{L}_0$, ${\overline{F}}_k=F_k{\tau }_0^2/m{L}_0$, ${\overline{F}}_d=F_d{\tau }_0^2/m{L}_0$ and $\overline{I}=I{\eta }_0{\tau }_0$. The dynamic control equation for the plate in Equation (1) can be rewritten as

$$\overline{\ddot{u}}\left(\overline{t}\right)={\overline{F}}_{\mathrm{s}}\left(\overline{t}\right)-{\overline{F}}_{\mathrm{k}}\left(\overline{t}\right)-\overline{c}\overline{\dot{u}}\left(\overline{t}\right),$$

(22)

with the initial conditions given as:

$$\overline{u}\left(\overline{t}\right)={\overline{u}}_0, \overline{\dot{u}}\left(\overline{t}\right)={\overline{\dot{u}}}_0 \mathrm{at} \overline{t}=0.$$

(23)

The restorative force of the spring described by Equation (3) can be expressed as

$${\overline{F}}_{\mathrm{k}}\left(\overline{t}\right)=\overline{K}\overline{u}\left(\overline{t}\right),$$

(24)

and the tension of the LCE fiber described by Equation (10) can be expressed as:

$${\overline{F}}_{\mathrm{s}}\left(\overline{t}\right)=\overline{k}\left[-\overline{u}\left(\overline{t}\right)+C_0{\displaystyle \underset{0}{\overset{1}{\int }}\phi \left(\overline{X},\overline{t}\right)}\mathrm{d}\overline{X}\right].$$

(25)

The cis-isomer concentration of Equation (11) can be rewritten as:

$${\displaystyle \frac{\mathsf{\partial }\phi \left(\overline{X},\overline{t}\right)}{\mathsf{\partial }\overline{t}}}=\overline{I}\left(\overline{X},\overline{t}\right)\left[1-\phi \left(\overline{X},\overline{t}\right)\right]-\phi \left(\overline{X},\overline{t}\right).$$

(26)

The coordinates of the photon reflected by the mirror can be rewritten as:

$${\overline{x}}_{\mathrm{LP}}=\left(\overline{Y}+{\overline{x}}_{\mathrm{L}}\tan \gamma \right)\tan 2\gamma .$$

(27)

${\overline{x}}_{\mathrm{L}}$ should satisfy the conditions

$$0\le {\overline{x}}_{\mathrm{L}}\le \overline{D} \mathrm{and} {\overline{x}}_{\mathrm{A}n}\le {\overline{x}}_{\mathrm{L}}\le {\overline{x}}_{\mathrm{B}n},$$

(28)

with

$${\overline{x}}_{\mathrm{A}n}\left(\overline{t}\right)=\overline{Z}\tan \theta +\overline{u}\left(\overline{t}\right)+2\left(n-1\right)\overline{b}\sin \theta ,$$

(29)

$${\overline{x}}_{\mathrm{B}n}\left(\overline{t}\right)=\overline{d}+{\overline{x}}_{\mathrm{A}n}\left(\overline{t}\right),$$

(30)

$$\overline{d}=2\left(\overline{b}\cos \theta -\overline{Z}\right)\tan \theta ,$$

(31)

and $n=1, 2,\dots , N$.

The instantaneous spatial coordinate $\overline{x}\left(\overline{X},\overline{t}\right)$ of Equation (21) can be expressed as:

$$\overline{x}\left(\overline{X},\overline{t}\right)=2N\overline{b}\sin \theta +\overline{X}\left[-\overline{u}\left(\overline{t}\right)+C_0{\displaystyle \underset{0}{\overset{1}{\int }}\phi \left(\overline{X},\overline{t}\right)}\mathrm{d}\overline{X}\right]+\overline{X}-C_0{\displaystyle \underset{0}{\overset{\overline{X}}{\int }}\phi \left(\overline{X},\overline{t}\right)}\mathrm{d}\overline{X}.$$

(32)

The dynamics of the geometry-flexible LCE self-oscillator activated by light feedback routing are dictated by Equations (22) and (26), wherein the temporally varying cis-isomer concentration interacts with the instantaneous displacement of the self-oscillating system. To resolve these intricate differential equations with variable coefficients, a fourth-order Runge–Kutta algorithm is implemented within the MATLAB R2024a computational environment. At a specific time instance ${\overline{t}}_{\mathrm{i}}$, using the known cis-isomer concentration ${\phi }_{\mathrm{i}}\left(\overline{X},{\overline{t}}_{\mathrm{i}}\right)$ and the displacement ${\overline{u}}_{\mathrm{i}}$, the current restorative force ${\overline{F}}_{\mathrm{ki}}$ and tension ${\overline{F}}_{\mathrm{si}}$ are computed via Equations (24) and (25). Then the displacement ${\overline{u}}_{\mathrm{i}+1}$ at the subsequent time step ${\overline{t}}_{\mathrm{i}+1}$ is then derived from Equation (22). Concurrently, the current displacement ${\overline{u}}_{\mathrm{i}}$ facilitates the estimation of the light intensity distribution ${\overline{I}}_{\mathrm{i}}\left(\overline{X},{\overline{t}}_{\mathrm{i}}\right)$ through Equations (27)–(32), thereby enabling the computation of the updated cis-isomer concentration ${\phi }_{\mathrm{i}+1}\left(\overline{X},{\overline{t}}_{\mathrm{i}+1}\right)$ via Equation (26). Through iterative computation across successive time steps, the dynamical behavior of the geometry-flexible LCE self-oscillator is numerically simulated for specified parameters: $C_0$, $\theta$, $\gamma$, $\overline{K}$, $\overline{k}$, $\overline{c}$, ${\overline{I}}_0$, $\overline{b}$, $\overline{D}$, $\overline{Z}$, $\overline{Y}$, $N$, ${\overline{u}}_0$ and ${\overline{\dot{u}}}_0$. It is worth mentioning that the time cost of the iterative computation is minimal. Using a time step of $\Delta \overline{t}=0.001$, running the MATLAB R2024a program on a standard office laptop to obtain a set of the self-oscillator’s temporal evolution takes only approximately 20 s.

3. Two Motion Patterns and Mechanism of Self-Oscillation

Based on solving governing Equations (22) and (26), this section examines the two distinct motion patterns of the geometry-flexible LCE self-oscillator, the static equilibrium pattern and self-sustained oscillation pattern, offering a mechanistic analysis of the self-oscillating behavior.

3.1. Two Motion Patterns

To investigate the dynamics of the geometry-flexible LCE self-oscillator activated by light feedback routing, it is essential to establish the characteristic values of the non-dimensional parameters within the model. Drawing upon prior experimental studies [89,90,91,92,93], standard material properties and geometric parameters are provided in

Table A1. Material properties and geometric parameters.

Parameter	Definition	Value	Units
$C_0$	Contraction coefficient	0~0.3	/
${\tau }_0$	trans-to-cis thermal relaxation time	1~100	ms
$I_0$	Light intensity	0~20	kW/m2
${\eta }_0$	Light-absorption constant	0.0003	m2/(s∙W)
$b$	Side length of rhombus-aperture regions	0~10	mm
$D$	Width of band-like light beam	0~10	mm
$Z$	Distance between the lines $P_1{P}_2$ and $O_1{O}_2$	0~10	mm
$Y$	Distance from origin $O$ to mirror	0~10	mm
$L_0$	Length of LCE fiber	0~10	cm
$K$	Stiffness constant of the spring	1~10	N/m
$k$	Elastic constant of the LCE fiber	1~10	N/m
$m$	Mass of the plate	0.01~10	g
$c$	Damping coefficient	0~0.001	mg∙mm2/s
$\theta$	Interior angle of rhombus-aperture regions	0~$\mathsf{\pi }/2$
$\gamma$	Angle between mirror and plate	0~$\mathsf{\pi }/4$
$\overline{K}$	Dimensionless stiffness constant of the spring	0~50
$\overline{k}$	Dimensionless elastic constant of the LCE fiber	0~100
${\overline{I}}_0$	Dimensionless light intensity	0~1
$\overline{c}$	Dimensionless damping coefficient	0~0.1
$\overline{b}$	Dimensionless side length of rhombus-aperture regions	0~1
$\overline{D}$	Dimensionless width of band-like light beam	0~2
$\overline{Z}$	Dimensionless distance between the lines $P_1{P}_2$ and $O_1{O}_2$	0~2
$\overline{Y}$	Dimensionless distance from origin $O$ to mirror	0~2

of Appendix A, while the corresponding dimensionless coefficients are summarized in

Table A2. Dimensionless parameters.

Parameter	Definition	Value
$\overline{K}$	Dimensionless stiffness constant of the spring	0~50
$\overline{k}$	Dimensionless elastic constant of the LCE fiber	0~100
${\overline{I}}_0$	Dimensionless light intensity	0~1
$\overline{c}$	Dimensionless damping coefficient	0~0.1
$\overline{b}$	Dimensionless side length of rhombus-aperture regions	0~1
$\overline{D}$	Dimensionless width of band-like light beam	0~2
$\overline{Z}$	Dimensionless distance between the lines $P_1{P}_2$ and $O_1{O}_2$	0~2
$\overline{Y}$	Dimensionless distance from origin $O$ to mirror	0~2

of Appendix A. These parameter values will subsequently be utilized to analyze the behavior of the geometry-flexible LCE self-oscillator activated by optical feedback routing.

Based on Equations (22) and (26), the temporal evolution and phase portraits of the LCE self-oscillator are derived, with specific cases corresponding to ${\overline{I}}_0=0$ and ${\overline{I}}_0=0.8$, as depicted in Figure 3. The remaining parameters are configured as follows: $C_0=0.3$, $\theta =\mathsf{\pi }/4$, $\gamma =\mathsf{\pi }/6$, $\overline{K}=20$, $\overline{k}=45$, $\overline{c}=0.045$, $\overline{b}=0.2$, $\overline{D}=1.3$, $\overline{Z}=0.05$, $\overline{Y}=0.6$, $N=5$, ${\overline{u}}_0=0.05$ and ${\overline{\dot{u}}}_0=0$. For the scenario where ${\overline{I}}_0=0$, as illustrated in Figure 3a,b, the displacement amplitude $\overline{u}$ demonstrates a progressive reduction over time owing to damping-induced energy dissipation, eventually leading the plate to stabilize at an equilibrium position—a behavior classified as the static equilibrium pattern. Conversely, under the condition ${\overline{I}}_0=0.8$, presented in Figure 3c,d, the plate initiates motion from an initial position, with its oscillation amplitude gradually amplifying until reaching a steady magnitude. With constant light exposure, the plate—activated via optical feedback mechanisms—sustains continuous periodic oscillation, thereby defining the self-oscillation pattern.

3.2. Mechanism of Self-Oscillation

To explore the mechanism behind the self-oscillation behavior triggered by light feedback routing, Figure 4 displays key physical quantities of the LCE self-oscillating system under the typical conditions presented in Figure 3c,d. Figure 4a depicts the variation over time in the cis-isomer concentration $\phi \left(\overline{X},\overline{t}\right)$ at representative material points $\overline{X}=0.3$, $\overline{X}=0.5$ and $\overline{X}=0.7$ on the LCE fiber, demonstrating a clear periodic pattern. When a material point enters the zone illuminated by the reflected light beam (taking $\overline{X}=0.3$ as an example), $\phi$ progressively rises over time, as indicated by segment A-B (or C-D) in Figure 4a, whereas it gradually declines in the dark, as represented by segment B-C (or D-A). As shown in Figure 4b, the variation in the length of the LCE fiber solely due to light exposure, denoted by $\Delta {\overline{L}}_{\mathrm{L}}=C_0{\displaystyle \underset{0}{\overset{1}{\int }}\phi \left(\overline{X},\overline{t}\right)}\mathrm{d}\overline{X}$, is presented as a function of time. Owing to the periodic fluctuation in cis-isomer concentration $\phi \left(\overline{X},\overline{t}\right)$, $\Delta {\overline{L}}_{\mathrm{L}}$ also displays periodic variations, which consequently causes the equivalent driving force ${\overline{F}}_{\mathrm{tot}}={\overline{F}}_{\mathrm{s}}-{\overline{F}}_{\mathrm{k}}$ to exhibit periodic behavior as well.

Figure 4c further illustrates the correlation between ${\overline{F}}_{\mathrm{tot}}$ and displacement $\overline{u}$ under the conditions corresponding to Figure 3c,d. This relationship forms a clockwise closed loop, and the area enclosed by this loop corresponds to the net work done by ${\overline{F}}_{\mathrm{tot}}$. This net work is calculated to be $0.012$ and serves as the energy source for the system’s self-oscillation. Similarly, the damping force ${\overline{F}}_{\mathrm{d}}$ also varies periodically over the motion cycle. Figure 3d depicts how ${\overline{F}}_{\mathrm{d}}$ changes with $\overline{u}$ throughout one complete self-oscillation period, forming a counterclockwise closed circle. The area within this closed loop represents the energy dissipated by ${\overline{F}}_{\mathrm{d}}$, which is calculated to also be $0.012$. The damping dissipation is balanced by the net work done by ${\overline{F}}_{\mathrm{d}}$, and such an energy balance enables the system to maintain continuous self-oscillation.

4. Parameter Analysis

In the aforementioned mechanical model of the self-oscillator, there are fourteen dimensionless system parameters: $\gamma$, $\overline{b}$, $\overline{D}$, $\overline{Z}$, $\overline{Y}$, $\theta$, $N$, $C_0$, ${\overline{I}}_0$, $\overline{K}$, $\overline{k}$, $\overline{c}$, ${\overline{u}}_0$ and ${\overline{\dot{u}}}_0$. In this section, the influence of these system parameters on the triggering conditions, frequency, and amplitude of self-oscillation is studied in detail. Herein, the dimensionless oscillation frequency and amplitude are denoted by $f$ and $A$, respectively. In general, the initial conditions ${\overline{u}}_0$ and ${\overline{\dot{u}}}_0$ have no effect on $f$ and $A$. And as the damping coefficient $\overline{c}$ increases, $f$ nearly remains constant, whereas $A$ exhibits a decreasing trend. In subsequent calculations, these initial conditions are fixed at $\overline{c}=0.045$, ${\overline{u}}_0=0.05$ and ${\overline{\dot{u}}}_0=0$.

4.1. Influence of Elastic Constant $\overline{k}$ and Stiffness Constant $\overline{K}$

Figure 5 illustrates the influence of the elastic constant $\overline{k}$ of the LCE fiber and the stiffness constant $\overline{K}$ of the spring on self-oscillation. The following parameter values are adopted in the calculations: $C_0=0.3$, ${\overline{I}}_0=0.8$, $\gamma =0.53$, $\overline{Y}=0.55$, $\theta =0.52$, $\overline{b}=0.2$, $\overline{Z}=0.05$, $\overline{D}=1.0$ and $N=5$. Figure 5a shows the variation in amplitude $A$ with the elastic constant $\overline{k}$ for three values of the stiffness constant: $\overline{K}=25$, $\overline{K}=35$ and $\overline{K}=45$. For each $\overline{K}$, there exists a distinct critical value of $\overline{k}$ required to trigger self-oscillation. When $\overline{k}$ is below this critical value, the LCE fiber and perforated plate system reaches a static equilibrium state. In contrast, when $\overline{k}$ exceeds the critical value, the system can sustain self-oscillation. This occurs because, for a small $\overline{k}$, the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ is too weak to provide sufficient net work to overcome damping losses, thereby preventing sustained oscillation. As the elastic constant $\overline{k}$ increases, the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ grows under the same light-induced contraction. Consequently, more energy is input into the system, and when the elastic constant $k$ exceeds a critical value, the energy compensates for damping losses, allowing the system to sustain self-oscillation. Furthermore, Figure 5a indicates that as $\overline{K}$ increases, the critical $\overline{k}$ required to initiate self-oscillation also increases. This is because a larger spring stiffness $\overline{K}$ resists the deformation of the LCE material, reducing both the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ and the net work it can supply.

In the self-oscillation pattern, when $\overline{K}$ is held constant, the amplitude $A$ increases significantly with $\overline{k}$, as shown in Figure 5a. This behavior arises because a larger value of $\overline{k}$ results in a stronger equivalent driving force ${\overline{F}}_{\mathrm{tot}}$, which performs more net work on the system and thus leads to a larger $A$. Conversely, for a fixed value of $\overline{k}$, the $A$ decreases as $\overline{K}$ increases. This reduction occurs because the spring resists the deformation of the LCE material. Figure 5b depicts the frequency $f$ of self-oscillation as a function of $\overline{k}$ and $\overline{K}$. It is observed that $f$ increases steadily with $\overline{k}$ for a given $\overline{K}$, and similarly shows a significant upward trend with the increase in $\overline{K}$. This behavior can be attributed to the fact that a higher $\overline{k}$ enables the LCE fiber to recover its deformation more rapidly, thereby increasing the oscillation frequency. Likewise, a larger $\overline{K}$ allows the spring to restore its elastic deformation more quickly. Thus, increasing either $\overline{k}$ or $\overline{K}$ contributes to a higher oscillation frequency.

4.2. Influence of Light Intensity ${\overline{I}}_0$ and Contraction Coefficient $C_0$

Figure 6 demonstrates the effects of both light intensity ${\overline{I}}_0$ and contraction coefficient $C_0$ on self-oscillation. The numerical analysis employs the following fixed parameters: $\overline{K}=25$, $\overline{k}=45$, $\gamma =0.53$, $\overline{Y}=0.55$, $\theta =0.52$, $\overline{b}=0.2$, $\overline{Z}=0.05$, $\overline{D}=1.0$ and $N=5$. As shown in Figure 6a, the amplitude $A$ varies with light intensity ${\overline{I}}_0$ for three contraction coefficient values: $C_0=0.24$, $C_0=0.27$ and $C_0=0.3$. For each $C_0$, a distinct critical light intensity ${\overline{I}}_0$ is necessary to initiate self-oscillation. Below this threshold, the LCE fiber–perforated plate system reaches a static equilibrium. Once ${\overline{I}}_0$ exceeds the critical value, self-oscillation occurs. This is attributed to the fact that at lower ${\overline{I}}_0$, the light-induced contraction of the LCE fiber is insufficient, resulting in a net work done by the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ that cannot overcome the energy dissipation due to damping. Moreover, Figure 6a indicates that a higher $C_0$ reduces the critical ${\overline{I}}_0$ required to trigger self-oscillation. This occurs because a larger $C_0$ enhances the light-induced contraction under the same ${\overline{I}}_0$, thereby reducing critical light intensity ${\overline{I}}_0$.

Figure 6a also illustrates the influence of the parameters ${\overline{I}}_0$ and $C_0$ on the oscillation amplitude $A$ within the self-oscillation pattern. When $C_0$ is held constant, $A$ exhibits a pronounced increase with higher values of ${\overline{I}}_0$. This enhancement can be attributed to the stronger light-induced contraction of the LCE fiber under more intense illumination, which raises the magnitude of the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ and consequently, the net work performed during each oscillation cycle. Moreover, Figure 6a indicates that at a fixed light intensity ${\overline{I}}_0$, an increase in $C_0$ also results in larger $A$. This behavior occurs because a higher value of $C_0$ amplifies the system’s response in a manner similar to increasing the illumination intensity ${\overline{I}}_0$, thereby further promoting the oscillatory motion. Furthermore, Figure 6b depicts the frequency $f$ of self-oscillation as a function of ${\overline{I}}_0$ and $C_0$. It is observed that $f$ remains constant with increases in ${\overline{I}}_0$ and $C_0$. This observation can be attributed to the fact that the frequency of self-oscillation is predominantly governed by the system’s intrinsic mechanical parameters.

4.3. Influence of the Angle $\gamma$ and Distance $\overline{Y}$

Figure 7 demonstrates how the angle $\gamma$ and the distance $\overline{Y}$ influence the self-oscillation behavior. The simulations were conducted using the parameter values $\overline{D}=1.0$, $\overline{Z}=0.05$, $\theta =0.52$, $\overline{b}=0.2$, $N=5$, $C_0=0.3$, ${\overline{I}}_0=0.8$, $\overline{K}=25$ and $\overline{k}=45$. In Figure 7a, the impact of the distance $\overline{Y}$—measured from the origin $O$ to the mirror—on the oscillation amplitude is shown for three values of $\gamma$: $0.525$, $0.53$ and $0.535$. For each angle $\gamma$, there exists a specific critical distance $\overline{Y}$, beyond which self-oscillation begins. When it is below this threshold, the fiber–plate system settles into a stationary equilibrium. Once it surpasses the critical value, self-oscillation initiates. This behavior arises because at smaller mirror distances $\overline{Y}$, the length of the LCE fiber illuminated by reflected light decreases, leading to insufficient light-driven contraction. As a result, the net work produced by the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ cannot overcome the energy loss from damping. Additionally, Figure 7a reveals that a larger $\gamma$ lowers the critical $\overline{Y}$ needed to initiate self-oscillation. This is because increasing $\gamma$ extends the illuminated portion of the LCE fiber under a fixed $\overline{Y}$, which in turn allows the critical $\overline{Y}$ to be appropriately reduced.

For a fixed value of $\gamma$, the amplitude $A$ increases markedly with greater $\overline{Y}$, as shown in Figure 7a. This growth results from the enhanced light-driven contraction when the mirror is placed farther away, which strengthens the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ and the associated net work, thereby boosting the oscillation amplitude. Moreover, Figure 7a indicates that a larger $\gamma$ also raises the amplitude $A$ at any given $\overline{Y}$, as increasing $\gamma$ extends the illuminated portion of the LCE fiber—an effect comparable to increasing the mirror distance $\overline{Y}$. Furthermore, Figure 7b illustrates how the self-oscillation frequency $f$ changes with $\gamma$ and $\overline{Y}$. It is found that it remains unchanged regardless of variations in $\gamma$ and $\overline{Y}$. This consistency occurs because alterations in these geometric parameters do not affect the intrinsic mechanical properties of the system, which ultimately determine the self-oscillation frequency.

4.4. Influence of Width $\overline{D}$ and Distance $\overline{Z}$

Figure 8 illustrates the influence of the width $\overline{D}$ of band-like light beam and the distance $\overline{Z}$ between the lines $P_1{P}_2$ and $O_1{O}_2$ on self-oscillation. In the calculations, we set $\gamma =0.53$, $\overline{Y}=0.55$, $\theta =0.52$, $\overline{b}=0.2$, $N=5$, $C_0=0.3$, ${\overline{I}}_0=0.8$, $\overline{K}=25$ and $\overline{k}=45$. Figure 8a shows the influence of the width $\overline{D}$ on the amplitude $A$ for $\overline{Z}=0.04$, $\overline{Z}=0.05$ and $\overline{Z}=0.06$. For each distance $\overline{Z}$, there exists a specific critical width $\overline{D}$. When it is below this threshold, the fiber–plate system settles into a stationary equilibrium. Once it surpasses the critical value, self-oscillation initiates. This is because a narrower width $\overline{D}$ means that less light is reflected onto the LCE fiber, and consequently it induces less light-induced contraction; as a result, the net work performed by the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ is insufficient to compensate for energy dissipation, leaving the system in a static equilibrium state. It can also be observed that as $\overline{Z}$ increases, the critical width $\overline{D}$ exhibits an increasing trend. The reason for this behavior is that a larger $\overline{Z}$ reduces the portion of light that can be reflected by the mirror onto the LCE fiber, as more light is blocked by the non-aperture regions of the plate. As a result, a wider critical width $\overline{D}$ is required to compensate for the reduced illuminated length and maintain sufficient light-induced actuation in the LCE fiber.

In the self-oscillation pattern, $A$ increases as $\overline{D}$ increases. This trend is due to the fact that the illuminated length of the LCE fiber grows with $\overline{D}$. This induces a greater overall light-induced contraction in the LCE fiber and a consequent rise in the net work performed by the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$. In contrast, $A$ gradually decreases as $\overline{Z}$ rises, which can be explained by the reduced proportion of light transmitted through the aperture regions, thereby weakening the light-induced contraction of the fiber. Observing Figure 8b, the computed results indicate that the frequency $f$ remains nearly constant across different values of $\overline{D}$ and $\overline{Z}$. This consistency occurs because the system’s intrinsic mechanical properties—governed by the mass–spring–damper subsystem—are not affected by these geometric or optical parameters.

4.5. Influence of Width $\overline{D}$ and Number $N$ of Apertures

Figure 9 illustrates the influence of the width $\overline{D}$ of the band-like light beam and the number $N$ of the aperture regions on self-oscillation. In the calculations, we set $\gamma =0.53$, $\overline{Y}=0.55$, $\theta =0.52$, $\overline{b}=0.2$, $\overline{Z}=0.05$, $C_0=0.3$, ${\overline{I}}_0=0.8$, $\overline{K}=25$ and $\overline{k}=45$. Figure 9a shows the influence of the width $\overline{D}$ on the amplitude $A$ for $N=3$, $N=4$ and $N=5$. For each number $N$, there exists a specific critical width $\overline{D}$ that must be exceeded for self-oscillation to begin. It can also be observed that as $N$ increases, the critical width $\overline{D}$ exhibits an increasing trend. The reason for this behavior is that with a larger $N$, the spatial position of the LCE fiber shifts. When $\overline{D}$ is kept constant, a portion of the LCE fiber remains unexposed to light. Therefore, increasing $\overline{D}$ is necessary to expose a longer segment of the LCE fiber. In the self-oscillation pattern, $A$ decreases as $N$ increases. This can be explained by the reduced irradiated length and the light-induced contraction of the LCE fiber. As shown in Figure 9b, the calculated results indicate that the frequency $f$ remains constant across different values of $\overline{D}$ and $N$. This can also be explained by the fact that $\overline{D}$ and $N$ do not alter the intrinsic mechanical properties of the system.

4.6. Influence of Interior Angle $\theta$ and Side Length $\overline{b}$

Figure 10 illustrates the influence of the interior angle $\theta$ and side length $\overline{b}$ on self-oscillation. In the calculations, the parameters are set to $\gamma =0.53$, $\overline{Y}=0.55$, $\overline{D}=1.0$, $\overline{Z}=0.05$, $N=5$, $C_0=0.3$, ${\overline{I}}_0=0.8$, $\overline{K}=25$ and $\overline{k}=45$. Figure 10a shows the effect of the side length $\overline{b}$ on the amplitude $A$ for $\theta =0.5$, $\theta =0.55$ and $\theta =0.6$. For the interior angle $\theta$, there exists a specific critical value of side length $\overline{b}$. When $\overline{b}$ is below this threshold, self-oscillation initiates. Once $\overline{b}$ exceeds the threshold, the fiber–plate system reaches a stationary equilibrium. This occurs because when $\overline{b}$ is too large, the right segment of the LCE fiber gradually moves out of the region illuminated by the reflected light, resulting in less light-induced contraction. As a consequence, the net work performed by the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$ is insufficient to compensate for energy dissipation, leaving the system in a static equilibrium state. It can also be observed that as the interior angle $\theta$ increases, the critical side length $\overline{b}$ tends to decrease. This behavior arises because an increase in $\theta$ alters the length of the LCE fiber exposed to the reflected light in a manner similar to an increase in $\overline{b}$. To ensure sufficient light-induced contraction in the LCE fiber under a larger $\theta$, a smaller $\overline{b}$ is required.

In the self-oscillation pattern, for a given interior angle $\theta$, the amplitude $A$ first increases and then decreases as $\overline{b}$ increases. This trend is attributed to the fact that for a fixed illuminated region, a larger $\overline{b}$ allows more light to pass through the aperture regions, thereby increasing the total light-induced contraction of the LCE fiber and the net work done by the equivalent driving force ${\overline{F}}_{\mathrm{tot}}$. However, when $\overline{b}$ becomes too large, the right segment of the LCE fiber remains outside the illuminated region of reflected light, leading to a reduction in both the total light-induced contraction and the net work performed by ${\overline{F}}_{\mathrm{tot}}$. Similarly, for a fixed side length $\overline{b}$, $A$ also first increases and then decreases as $\overline{b}$ increases, as shown in Figure 10a. Since variations in angle $\theta$ and side length $\overline{b}$ do not affect the intrinsic mechanical properties of the system, the oscillation frequency remains unchanged, as demonstrated in Figure 10b.

5. Conclusions

Self-oscillation systems typically rely on nonlinear feedback mechanisms, such as self-shadowing, to sustain oscillations. However, designing a feedback loop that is congruent with the motion patterns presents a significant challenge, primarily due to two constraints: the limited deformation modes available in the geometry and the straightforward nature of rectilinear light propagation paths. To overcome this challenge, we propose a geometry-flexible LCE self-oscillator enabled by light feedback routing. The system consists of an optically responsive LCE fiber, a spring, a mirror, and a perforated plate. Using a dynamic LCE model, we present a nonlinear model for the self-oscillator under constant light exposure. Simulations reveal two distinct motion patterns: static equilibrium and self-oscillation. The self-oscillation arises from the contraction of the material points of the LCE fibers within the light path, with continuous motion being maintained by light energy input and damping dissipation.

Furthermore, the frequency $f$ and amplitude $A$ of self-oscillation primarily depend on several system parameters, i.e., $\gamma$, $\overline{b}$, $\overline{D}$, $\overline{Z}$, $\overline{Y}$, $\theta$, $N$, $C_0$, ${\overline{I}}_0$, $\overline{K}$ and $\overline{k}$. Increasing the system parameters $\overline{k}$, ${\overline{I}}_0$, $C_0$, $\gamma$, $\overline{Y}$ and $\overline{D}$ leads to an increase in the amplitude of self-oscillation. In contrast, increasing $\overline{K}$, $\overline{Z}$ or $N$ results in a decrease in amplitude. The amplitude initially rises and then declines as $\overline{b}$ and $\theta$ increase. The frequency of self-oscillation increases with increases in $\overline{K}$ and $\overline{k}$, while it remains nearly unaffected by variations in $\overline{b}$, $\overline{D}$, $\overline{Z}$, $\overline{Y}$, $\theta$, $N$, $C_0$ and ${\overline{I}}_0$.

It should be noted that the current model has several limitations: the effect of nematic-order parameter variation on the elastic modulus of the LCE is neglected; a linear stress–strain relationship is assumed; and material fatigue effects are not considered. These simplifications may lead to quantitative discrepancies between predictions and practical observations. Although experimental validation remains technically challenging under current infrastructure constraints, the proposed light feedback configuration allows deformation at any material point to contribute to the oscillator’s motion, creating a geometry-flexible feedback loop. This capability overcomes the limitations of constrained geometric deformation modes, allowing for adaptable control of the optical feedback loop through straightforward geometric modifications. This approach offers new avenues for self-sustaining systems and may inspire innovative applications in soft robotics, energy harvesting, and logic gates.