Modeling Bifurcation-Driven Self-Rotation and Pendulum in a Light-Powered LCE Fiber Engine

Abstract

Self-oscillating systems are capable of transforming ambient energy directly into mechanical output, and exploring novel designs is of great value for energy harvesters, actuators, and engine applications. The inspiration for this study is drawn from the four-stroke engine; we designed a new self-rotating engine formed by a turnplate, a hinge, and an LCE fiber, operating with steady illumination applied. To analyze its rotation dynamics, a nonlinear theoretical framework was formulated constructed with the dynamic LCE model as a framework. The central discovery is that the light-driven LCE engine can operate in three distinct states under steady illumination—static equilibrium, pendulum-like oscillation and sustained self-rotation—switching between them through a supercritical Hopf bifurcation. The persistence of both the pendulum and rotary motions stems from an energy balance in which the positive work produced by photo-induced contraction of the LCE fiber is exactly offset by damping dissipation, while oscillation amplitude and rotation frequency are strongly governed by light intensity, contraction coefficient, damping coefficient, spring constant and turntable radius. Compared with many previously reported self-oscillating designs, the present self-rotating engine is distinctive for its lightweight and simple configuration, tunable size, and rapid operation. These features enable compact integration and broaden its potential applications in micro-scale systems and devices. The advancement in artificial muscles, medical instruments and micro sensors is strongly promoted by this, making it possible to create devices that are both smaller in size and superior in functionality.

1. Introduction

Self-sustained oscillations are rhythmic phenomena initiated by steady environmental stimuli [1], capable of maintaining oscillatory motion by continuously uptake energy from these external sources, without the need for intricate controllers or portable batteries [2,3,4,5,6]. Given that self-oscillating systems reduce reliance on external energy sources and complex controls, their complexity is significantly reduced, thus paving the way for a variety of innovative applications, including portability [7,8,9,10,11,12,13]. Additionally, the frequency of self-oscillation is predominantly governed by the system’s intrinsic parameters, with minimal sensitivity to starting conditions, thus enhancing the system’s robustness [14,15]. Because of the exceptional merits associated with self-oscillation, such systems are increasingly regarded as promising options in a wide range of practical applications, incorporating active machine functions [16,17,18,19,20,21,22,23,24], autonomous robotics [25], energy-absorbing devices [26,27,28], motors [29], and more.

Taking externally triggered materials, exemplified by liquid crystal elastomers (LCEs), as the foundation [30,31], ionic gels [32], and hydrogels [33,34,35], a broad spectrum of self-oscillating systems has arisen in the past few years. Specifically, many efforts have been dedicated to constructing diverse self-sustained motion patterns, including vibration [36,37,38], bending [39,40], ejecting [41], rolling [42,43], self-wobbling [44], torsion [45], self-striking [46], shrinking [47], swimming [48], swinging [16,49], twisting [50,51], jumping [52,53], chaos [54,55] and the synchronization phenomena observed in coupled self-oscillators [56]. To offset the energy dissipation caused by system damping, such self-oscillating systems typically rely on specific mechanisms to harness energy from external sources [1]. A number of feedback mechanisms were formulated to support energy recovery for different types of externally triggered materials [16,17,57], exemplified by the interaction between chemical reactions and large deformations [32], self-shading mechanisms [16,17,58], and the involvement of multiple coupled processes connecting droplet evaporation and motion [59]. These mechanisms often involve nonlinear interactions integrating several processes through self-oscillatory feedback mechanisms [57,58,59].

Within the externally triggered materials that form self-excited oscillatory systems, LCE reveals distinct benefits. through embedding anisotropic rod-like liquid crystal molecules into flexible long-chain polymers, LCE is formed [60]. Upon exposure of liquid crystal monomer molecules to external perturbations like light [21,22,23,24], heat [61], electricity [62,63,64], and magnetism [65], rotation or phase transitions occur, leading to modifications in their configurations and consequently causing macroscopic deformations [66]. Among these external stimuli, optical stimulation is the preferred choice due to its ability to enable remote and non-contact operation. At the same time, the intensity, wavelength, and polarization direction of the light can be conveniently and precisely adjusted. Moreover, optical stimulation offers notable advantages, including plentiful light availability and a clean surrounding environment. The photo-responsive LCE displays quick responsiveness, pronounced inherent deformation, and reversible strain characteristics. These unique properties enable feedback to be induced through various methods, causing light-driven self-sustained oscillations [67,68,69], which has facilitated the widespread deployment of LCE-based light-driven self-oscillating systems [67,68,69,70].

Although LCE-based self-oscillatory systems have attracted considerable interest, having also facilitated the design of numerous analogous systems, it is still essential to develop self-oscillating systems exhibiting a wider range of dynamic behaviors, designed to meet the needs of different applied functions. Classified as an internal combustion engine, the four-stroke engine is characterized by its completion of a full operating cycle using four piston strokes, consisting mainly of components like the spark plug, piston, connecting rod, and crankshaft. It sequentially goes through intake, compression, combustion, and exhaust phase concludes the entire operation. The piston undergoes bidirectional motion from the highest stroke end to the lowest stroke end, completing four strokes, while the crankshaft undergoes two full rotations, as illustrated in Figure 1a. Drawing inspiration from Knežević’s work [71] and the four-stroke engine, a prototype self-rotating engine founded on photo-responsive materials is proposed in this work. Composed of an LCE fiber, a turntable, and a hinge, this engine can self-rotate when exposed to steady illumination applied. Unlike the zero-energy-mode engine proposed by Knežević [72], the LCE engine we propose exhibits unstable self-rotation, with its angular velocity varying periodically. This study seeks to introduce a conceptual light-powered self-rotating engine based on LCE, explore its dynamic properties, and offer insights for its practical use in engineering applications. This self-rotating system offers advantages featuring properties of structural minimalism, scalable size, fast actuation, and reduced weight [73,74,75]. It is expected to offer a wider array of design concepts, applicable to various fields such as soft robotics, energy harvesting systems, and medical instruments.

In studying LCE fiber behavior, various mathematical models have been developed. Each has strengths and weaknesses: linear models suit small deformations but lack detail on large deformations and photo-induced changes; nonlinear models better handle large deformations but oversimplify material properties and photo-induced mechanisms; photo-induced models focus on light-driven changes but assume uniform conditions that may not reflect reality; and multiphysics models, while comprehensive, are complex and challenging to parameterize. Validation through experiments is crucial for refining these models. Future work should emphasize multiscale modeling, experimental validation, and simulation of complex conditions to enhance model accuracy and optimize LCE fiber applications.

The following sections are arranged in this way: Section 2 starts by presenting a nonlinear mechanical behavior model to describe the behavior of the self-rotating LCE-based engine with steady illumination applied. Based on the recognized dynamic LCE theory, this approach formulates the associated ruling equations. In Section 3, the three approach modes of the LCE-based engine with steady illumination applied are presented, followed by an elaborate analysis of the self-rotation. The discussion in Section 4 focuses on the examination of the Hopf bifurcation conditions, which distinguish the self-rotation, static, and pendulum states, and examines how various parameters of the system govern how the frequency and amplitude of self-rotation behave. Section 5 centers on outlining the key findings and examining both the applications and theoretical implications of the results.

2. Model and Formulation

In the following section, we introduce a conceptual self-rotating engine constructed from LCE, whose design is inspired by the four-stroke engine illustrated in Figure 1a and Knežević’s study [71], and it operates with steady illumination applied. Built with a hinge, an LCE fiber, and a turnplate, the LCE-based engine can achieve oscillation and continuous rotation when exposed to steady illumination applied. Considering the coupling between optical and mechanical effects, we develop a dynamic modeling approach for the LCE-based self-rotating engine operating with steady illumination applied, aiming to characterize its motion features and evolutionary behavior. The primary focus of this work involves analyzing the rotational behavior governed by the self-rotating engine, formulating the light-induced torque, and modeling the tension within the LCE fiber. In addition, the ruling equations are transformed through nondimensionalization, and approach strategies for the variable-coefficient differential equations are discussed to provide a more comprehensive description of the system’s dynamic behavior.

2.1. Dynamics of the LCE-Based Engine

Figure 1 illustrates the concept of a light-powered, self-rotating LCE-based engine operating in two working regimes; the system is mainly built from three components—a turnplate, an LCE fiber, and a hinge. At point O, the fixed end, at one terminal, the fiber is clamped in place, while the opposite end is fastened to point P on the hinge. This hinge, in turn, is mechanically coupled to a turnplate of radius R, which is supported by a central shaft C that serves as the axis of rotation. In this study, the employed LCE fiber is prepared from either thiol–acrylate formulations or siloxane-based networks, both of which are known for their ability to undergo exceptionally large elastic deformations [76]. Such material choices ensure that the actuator can sustain repeated stretching and recovery without significant loss of performance. As displayed in Figure 1b, the LCE fiber in the reference configuration carries no mechanical load and retains its undeformed length. Within this state, the azobenzene liquid crystal embedded in the fiber adopt a linear trans conformation oriented with respect to the axis. The corresponding primary configuration is given in Figure 1d, where the highlighted yellow sector designates the region subjected to light exposure. At the onset, when departing from the standard state, the LCE-based engine is assigned a primitive rotational velocity. This preliminary input acts as the driving seed that sets the turnplate into motion, enabling the subsequent self-sustained dynamics. When rotation begins, the LCE fiber lies outside the light-exposed area, and its azobenzene units remain in the trans configuration (Figure 1d). At this stage, once the turnplate arrives at the position defined by the critical rotational angle $0.37\pi \le \theta \le 0.6\pi$ and $1.3\pi \le \theta \le 1.72\pi$, under illumination, the LCE fiber contracts as the embedded azobenzene mesogens shift from an extended transform to a bent cis configuration (Figure 1f). This molecular rearrangement serves as the microscopic origin of the observed macroscopic deformation.

For rotational angles smaller than the threshold value ($0\le \theta \le \mathsf{\pi }$), the restoring force from the stretched LCE fiber generates a torque that drives the turnplate counterclockwise, tending to restore it toward its reference position. Once the rotational displacement of the turnplate surpasses the specified critical angle ($1.3\pi \le \theta \le 1.72\pi$), the fiber enters the illuminated region and begins to experience continuous light exposure. When subjected to continuous light exposure, the stretched LCE fiber develops an increasing internal force together with a growing torque, which promotes further clockwise motion of the turnplate. At this stage, the azobenzene liquid crystal embedded in the fiber is subjected to a light-induced structural change, switching from their elongated trans configuration to a bent cis form, as can be seen from Figure 1f. Once the turnplate reaches the threshold angle, the residual inertia drives it to keep rotating even without additional external input. With steady illumination applied, the turnplate eventually shows continuous periodic rotational behavior.

After adjusting different basic parameters, a pendulum state will be exhibited by the LCE-based engine, beginning from its primitive state, the LCE-based engine proceeds until the turnplate arrives at the critical angle ($0.37\pi \le \theta \le 0.6\pi$), the LCE fiber begins to obtain steady illumination applied. Once subjected to steady illumination, the corresponding rotational moment and the tension in the LCE fiber increase, and there is a trend of counterclockwise rotation of the turntable, at this time the corresponding the molecular configuration of azobenzene liquid crystals alters, shifting from trans toward bent cis, as shown in Figure 1h. The speed of the turnplate under the action of the force slowly decreased to zero, and began to rotate in accordance with the counterclockwise, finally, the LCE-based engine in the steady illumination for periodic movement.

As illustrated in Figure 1, the turnplate experiences both a driving torque $M_{\mathrm{F}}(t)$ and a resistive damping torque $M_{\mathrm{d}}(t)$. To simplify the analysis, the damping effect $M_{\mathrm{d}}(t)$ is modeled as being linearly dependent on the angular velocity of rotation. On the basis of the preceding discussion, the motion of the self-rotating engine can be expressed through a nonlinear differential equation, expressed as follows:

$$J{\displaystyle \frac{{\mathrm{d}}^2\theta (t)}{\mathrm{d}{t}^2}}=M_{\mathrm{F}}(t)-\beta {\displaystyle \frac{\mathrm{d}\theta (t)}{\mathrm{d}t},}$$

(1)

here, $\beta$ denotes the damping coefficient, while J represents the turnplate’s inertia with respect to rotation around center C. The corresponding starting conditions can then be specified, as follows:

$$\theta (t)={\theta }_0,{\displaystyle \frac{\mathrm{d}\theta (t)}{\mathrm{d}t}}=\stackrel{·}{{\theta }_0}\mathrm{at} t=0$$

(2)

The torque appearing $M_{\mathrm{F}}(t)$ in Equation (1) is a direct consequence of the fiber tension in the LCE, and its quantitative description is provided in the next section.

2.2. Rotational Moment

The rotational moment $M_{\mathrm{F}}(t)$ is primarily governed by the tensile $F(t)$ response of the LCE fiber, and its exact form varies as determined by the turnplate’s angular position. Specifically, when the plate rotates across the two critical angle ranges $2n\pi \le \theta \le \pi +2n\pi$ and $\pi +2n\pi \le \theta \le 2\pi +2n\pi$, distinct analytical representations for the rotational moment $M_{\mathrm{F}}(t)$ must be applied. To facilitate the theoretical derivation, the analysis is carried out under a set of simplifying premises. The LCE fiber is regarded as massless so that its inertial contribution can be neglected. Its cross-sectional geometry is assumed to remain unchanged when stretched, and any frictional effects along the fiber are omitted. Moreover, the pulling force is considered to be uniformly distributed across the entire fiber length.

For $2n\pi \le \theta \le \pi +2n\pi$, the rotational moment $M_{\mathrm{F}}(t)$ is strongly associated with the tension $F(t)$. As depicted in Figure 1e,h, the pulling force within the fiber is assumed to act strictly along its longitudinal axis. In developing the governing formulation, reference is made to the well-established theoretical framework describing the rigid-body spinning motion around an immovable axis [75,76], the rotational moment $M_{\mathrm{F}}(t)$ in $2n\pi \le \theta \le \pi +2n\pi$ is obtained in the following form:

$$M_{\mathrm{F}}(t)=-F(t)d.$$

(3)

For $\pi +2n\pi \le \theta \le 2\pi +2n\pi$, as indicated in Figure 1f,i, one may determine the rotational moment conveniently by applying the subsequent formula:

$$M_{\mathrm{F}}(t)=F(t)d.$$

(4)

The fiber of LCE $L(t)$ extends over a given length, under both illuminated and non-illuminated conditions, can be readily derived and is given as follows:

$$L(t)=\sqrt{R^2+{(L_0+R)}^2-2R(R+L_0)\cos \theta ,}$$

(5)

consequently, the expression for the arm of moment d can be regarded as straightforwardly derived in the following form:

$$d={\displaystyle \frac{(L_0+R)R\sin \theta }{L(t),}}$$

(6)

2.3. Tension of the LCE Fiber

To evaluate the rotational moment $M_{\mathrm{F}}(t)$ described in Equations (3) and (4), the pulling force within the LCE fiber must first be identified. In the modeling process, both the mass and frictional resistance of the fiber are neglected for simplification. Furthermore, the tension $F(t)$ is treated as linearly dependent on the elastic strain ${\epsilon }_{\mathrm{e}}$, as expressed below:

$$F(t)=k{L}_0{\epsilon }_{\mathrm{e}}(t),$$

(7)

parameter k stands for the spring constant attributed to the LCE fiber. It must be highlighted that the elastic strain ${\epsilon }_{\mathrm{e}}(t)$ is homogeneous in the LCE fiber, whereas both the overall strain ${\epsilon }_{\mathrm{tot}}$ and the light-powered contraction ${\epsilon }_{\mathrm{L}}$ are inhomogeneous. For the purpose of examining the inhomogeneous deformation, the Lagrangian arc coordinate system X is established in the reference nominal state (Figure 1b) along with introducing the Eulerian arc coordinate framework x under the current configuration (Figure 1e,h). As the turnplate rotates, the instantaneous placement of a given material point along the LCE fiber is describable by the mapping x = x (X, t), in this formulation, x is regarded as the Lagrangian coordinate and t corresponds to time where denotes the reference coordinate and is time. Subsequently, the overall strain in the LCE fiber together with the photonic-induced contraction can be formulated as ${\epsilon }_{\mathrm{tot}}(X,t)$ for one expression and as ${\epsilon }_{\mathrm{L}}(X,t)$ for the other. In order to reduce complexity, when the deformation is small, the elastic strain ${\epsilon }_{\mathrm{e}}(t)$ may be represented as a superposition of the light-driven contraction ${\epsilon }_{\mathrm{L}}(X,t)$ and the overall strain ${\epsilon }_{\mathrm{tot}}(X,t)$, namely ${\epsilon }_{\mathrm{tot}}(t)={\epsilon }_{\mathrm{e}}(t)+{\epsilon }_{\mathrm{L}}(t)$. Accordingly, the expression for the tension may be reformulated as shown below:

$$F(t)=k{L}_0\left[{\epsilon }_{\mathrm{tot}}(t)-{\epsilon }_{\mathrm{L}}(t)\right],$$

(8)

${\epsilon }_{\mathrm{L}}(t)$ is regarded as a linear dependence on the fraction of cis $\varphi (t)$ molecules by number contained in the LCE fiber, namely:

$${\epsilon }_{\mathrm{L}}(X,t)=-C_0\varphi (t),$$

(9)

$C_0$ denotes the contraction coefficient. For convenience, the overall strain can therefore be characterized mathematically in the following form: ${\epsilon }_{\mathrm{tot}}(t)={\displaystyle \frac{L(t)-L_0}{L_0}}$.

Accordingly, the expression for the tension $F(t)$ given in Equation (8) may be reformulated as described below:

$$F(t)=k\left\{\left[L(t)-L_0\right]+C_0{L}_0\varphi (t)\right\}.$$

(10)

2.4. Dynamic LCE Model

The quantity representing the number fraction $\varphi (t)$ of cis in Equation (10), the dynamic framework for LCE behavior originally developed by Finkelmann and collaborators [60,77] is adopted as the theoretical basis. Yu and colleagues [78] reported that when LCEs are exposed to ultraviolet light or laser irradiation with wavelengths below 400 nm, the embedded azobenzene units undergo a photoinduced transition from the trans to cis configuration. The number fraction $\varphi (t)$ of cis is governed by three competing processes: the thermal promotion of molecules from the trans to cis the spontaneous thermal back-relaxation from trans to cis, and the optical excitation that drives the trans to cis conversion. Compared with the optical excitation, the thermal transition from trans to cis is usually insignificant and can be safely ignored [77,79]. Hence, the number fraction $\varphi (t)$ of cis is typically formulated through the following governing relation:

$${\displaystyle \frac{\partial \varphi (t)}{\partial t}}={\eta }_0{I}_0\left[1-\varphi (t)\right]-{\tau }_0^{-1}\varphi (t),$$

(11)

this ${\tau }_0$ refers to the thermal recovery time during the trans to cis process, the parameter $I_0$ corresponds to the light intensity, whereas the constant ${\eta }_0$ characterizes light absorption. After accounting for the starting conditions, solving Equation (11) yields:

$$\varphi (t)={\displaystyle \frac{{\eta }_0{\tau }_0{I}_0}{{\eta }_0{\tau }_0{I}_0+1}}+({\varphi }_0-{\displaystyle \frac{{\eta }_0{\tau }_0{I}_0}{{\eta }_0{\tau }_0{I}_0+1}})\exp [-{\displaystyle \frac{t}{{\tau }_0}}({\eta }_0{\tau }_0{I}_0+1)],$$

(12)

the quantity ${\varphi }_0$ is used to describe the fraction of cis molecules by number at the initial time $t=0$. By incorporating the starting number fraction, i.e., ${\varphi }_0=0$, Equation (12) may be reduced to the following expression:

$$\varphi (t)={\displaystyle \frac{{\eta }_0{\tau }_0{I}_0}{{\eta }_0{\tau }_0{I}_0+1}}\{1-\exp [-{\displaystyle \frac{t_1}{{\tau }_0}}({\eta }_0{\tau }_0{I}_0+1)]\}.$$

(13)

Upon the LCE fiber’s entry into the illuminated section, the ratio of cis can be rigorously formulated as follows:

$$\varphi (t)={\displaystyle \frac{{\eta }_0{\tau }_0{I}_0}{{\eta }_0{\tau }_0{I}_0+1}}+({\varphi }_{\mathrm{dark}}-{\displaystyle \frac{{\eta }_0{\tau }_0{I}_0}{{\eta }_0{\tau }_0{I}_0+1}})\exp [-{\displaystyle \frac{t_2}{{\tau }_0}}({\eta }_0{\tau }_0{I}_0+1)].$$

(14)

In the absence of illumination, the expression given in Equation (12) can be reformulated in an alternative form as follows:

$$\varphi (t)={\varphi }_{\mathrm{illum}}\exp (-{\displaystyle \frac{t_3}{{\tau }_0}}).$$

(15)

For Equations (13)–(15), $t_1$, $t_2$ and $t_3$ are the durations of the current state, respectively, here, ${\varphi }_{\mathrm{dark}}$ and ${\varphi }_{\mathrm{illum}}$ correspond to the instantaneous proportions of cis that emerge when the fiber transitions from darkness into exposure, and conversely when it moves from the illuminated region back into the shaded region.

2.5. Nondimensionalization

To ease the analysis, dimensionless quantities are introduced in the manner shown below: $\overline{t}=t/{\tau }_0$, $\overline{L}=L/L_0$, $\overline{R}=R/L_0$, $\overline{X}=X/L_0$, $\overline{x}=x/L_0$, $\overline{d}=d/L_0$, ${\overline{M}}_{\mathrm{F}}=M_{\mathrm{F}}{\tau }_0^2/J$, $\overline{F}=F{L}_0{\tau }_0^2/J$, $\overline{\beta }=\beta {\tau }_0/J$, $\overline{k}=k{\tau }_0^2{L}_0^2/J$, $\overline{I}=I{\eta }_0{\tau }_0$, ${\overline{I}}_0=I_0{\eta }_0{\tau }_0$. The system mitigates thermal lag or delays in LCE contraction through the nondimensionalization of the time delay ${\tau }_0$. In practice, ${\tau }_0$ depends on the fiber diameter and thermal conductivity. By reducing the fiber diameter and increasing the thermal conductivity, the time delay can be effectively minimized, thereby enhancing the system’s response speed. As a next step, Equations (1)–(6), as well as Equations (10), (11) and (13)–(15), can be, respectively, expressed as follows.

The nondimensional form of the governing dynamic equation describing the behavior of the self-rotating engine, it is derived through the procedure outlined below. For example, transform Equation (1) from the form $M_{\mathrm{F}}={\overline{M}}_{\mathrm{F}}J/{\tau }_0^2$ to the form ${\overline{M}}_{\mathrm{F}}=M_{\mathrm{F}}{\tau }_0^2/J$, transform $\beta =\overline{\beta }J/{\tau }_0$ to the form $\overline{\beta }=\beta {\tau }_0/J$, substituting these into the equation yields the following result:

$${\displaystyle \frac{{\mathrm{d}}^2\theta (\overline{t})}{\mathrm{d}{\overline{t}}^2}}={\overline{M}}_{\mathrm{F}}(\overline{t})-\overline{\beta }{\displaystyle \frac{\mathrm{d}\theta (\overline{t})}{\mathrm{d}\overline{t}}},$$

(16)

transform Equation (2) from the form $t=\overline{t}{\tau }_0$ to the form $\overline{t}=t/{\tau }_0$, substituting these into the equation yields the following result:

$$\theta (\overline{t})={\overline{\theta }}_0,{\displaystyle \frac{\mathrm{d}\theta (\overline{t})}{\mathrm{d}t}}={\stackrel{·}{\overline{\theta }}}_0\mathrm{at} \overline{t}=0,$$

(17)

similarly, the remaining equations can be derived.

The term ${\overline{M}}_{\mathrm{F}}(\overline{t})$ is determined in a pair of situations in the following form: for $2n\pi \le \theta \le \pi +2n\pi$,

$${\overline{M}}_F(\overline{t})=-\overline{F}(\overline{t})\overline{d},$$

(18)

and for $\pi +2n\pi \le \theta \le 2\pi +2n\pi$,

$${\overline{M}}_F(\overline{t})=\overline{F}(\overline{t})\overline{d}.$$

(19)

The current fiber length $\overline{L}(\overline{t})$ for non-illuminated and light-exposed state is described by the following expression:

$$\overline{L}(\overline{t})=\sqrt{{\overline{R}}^2+{(1+\overline{R})}^2-2\overline{R}(\overline{R}+1)\cos \theta },$$

(20)

and the current moment arm d is as follows:

$$\overline{d}={\displaystyle \frac{(1+\overline{R})\overline{R}\sin \theta }{\overline{L}(\overline{t})}},$$

(21)

with $\overline{F}(\overline{t})$ denoting

$$\overline{F}(\overline{t})=\overline{k}\left\{[\overline{L}(\overline{t})-1]+C_0\varphi (\overline{t})\right\},$$

(22)

and $\varphi \left(\overline{t}\right)$ can be calculated from

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

(23)

The result follows from the steps presented in the subsequent discussion:

At $t=0$,

$$\varphi (\overline{t})={\displaystyle \frac{{\overline{I}}_0}{{\overline{I}}_0+1}}\{1-\exp [-{\overline{t}}_1({\overline{I}}_0+1)]\}.$$

(24)

In the illuminated region,

$$\varphi (\overline{t})={\displaystyle \frac{{\overline{I}}_0}{{\overline{I}}_0+1}}+({\varphi }_{\mathrm{dark}}-{\displaystyle \frac{{\overline{I}}_0}{{\overline{I}}_0+1}})\exp [-{\overline{t}}_2({\overline{I}}_0+1)].$$

(25)

Within the shaded region,

$$\varphi (\overline{t})={\varphi }_{\mathrm{illum}}\exp (-{\overline{t}}_3).$$

(26)

Equations (16), (25) and (26) characterize the operation of the LCE-fiber-based self-rotating engine with steady illumination applied, where the angular displacement of the engine is directly coupled with the time-varying number fraction of cis. The Runge-Kutta method was employed to solve the nonlinear differential equations with time-dependent coefficients due to its high accuracy, stability, and computational efficiency. This method is particularly suitable for such equations, ensuring high precision of the solution with small step sizes. After verifying convergence, a time step $h$ of 0.001 was chosen to ensure the numerical solution is both accurate and stable. Numerical experiments with different time steps were also conducted, showing that while smaller time steps yield more precise solutions, they increase computational costs. Thus, the selected time step of 0.001 achieves a good balance between accuracy and computational efficiency. Given the number fraction ${\varphi }_{\mathrm{i}}$ of cis and the turnplate’s position ${\theta }_{\mathrm{i}}$ at time ${\overline{t}}_{\mathrm{i}}$, the instantaneous tension ${\overline{F}}_{\mathrm{i}}$ of the LCE fiber together with the corresponding rotational moment ${\overline{M}}_{\mathrm{Fi}}$ are obtained through Equations (18)–(22). The angular position of the turnplate ${\theta }_{\mathrm{i}+1}$ at a given time ${\overline{t}}_{\mathrm{i}+1}$ is determined from Equation (16). The LCE fiber’s cis number fraction may be evaluated through Equation (23).

In the Matlab program, the average computation time to obtain the numerical solution is 10 s. Computation time is mainly influenced by the time step and system parameters. For example, reducing the time step from 0.001 to 0.0005 doubles the computation time while halving the relative error from 0.1% to 0.05%. Error analysis shows that the relative error of the solution is 0.1%, mainly due to numerical truncation error and time step choice. A time step of 0.001 balances accuracy and efficiency well.

3. Three Motion Regimes and Mechanism of Self-Rotation

This part of this study analyzes the solutions of the governing Equations (16) and (23) to identify three representative motion patterns of the LCE-based engine: the pendulum regime, the equilibrium state, and a sustained self-rotating state. Following this classification, particular emphasis is placed on clarifying the underlying mechanism responsible for the emergence of self-rotation.

3.1. Two Motion Regimes

An investigation is carried out regarding the efficiency of the LCE engine operating with steady illumination applied, one needs to first specify representative the distinguishing indicators of the nondimensional parameters used in the theoretical model. Drawing upon reported experimental data [16,80,81], the key material properties together with the relevant geometric dimensions are summarized in

Table 1. Material properties and geometric parameters.

Parameter	Definition	Value	Unit
$\mathsf{\beta }$	Damping coefficient	0~0.001	$\mathrm{mg}·{\mathrm{mm}}^2/\mathrm{s}$
$C_0$	Contraction coefficient [81]	0~0.3	/
${\mathsf{\eta }}_0$	Light-absorption constant	0.0003	${\mathrm{m}}^2/\mathrm{s}·\mathrm{W}$
${\mathsf{\tau }}_0$	Trans-to-cis thermal relaxation time [82]	1~100	ms
J	Moment inertia of turnplate	0~1	$\mathrm{mg}·{\mathrm{mm}}^2$
$I_0$	Light intensity	0~16.67	$\mathrm{KW}/{\mathrm{m}}^2$
$L_0$	Original length of LCE fiber	1	m
k	Spring constant of LCE fiber [83]	9.5	N/m
R	Radius of turnplate	0.05	m

; at the same time, their corresponding nondimensionalized forms are presented in

Table 2. Nondimensional parameters.

Parameter	${\theta }_0$	${\stackrel{·}{\overline{\theta }}}_0$	$\overline{\beta }$	$\overline{I_0}$	$C_0$	$\overline{k}$	$\overline{R}$
Value	0~10	0~10	0~0.1	0~0.5	0~0.3	0~30	0~0.05

for clarity.

By integrating Equations (18)–(26) numerically, one can track the temporal evolution and corresponding phase portraits of the self-rotating motion; included in them are the cases for ${\overline{I}}_0=0.05$, ${\overline{I}}_0=0.3$ and ${\overline{I}}_0=0.5$, which are depicted in Figure 2. During the computation, the remaining parameters are assigned as $C_0=0.3$, $\overline{k}=30$, $\overline{\beta }=0.02$, ${\theta }_0=0$, $\overline{R}=0.05$, and ${\stackrel{·}{\overline{\theta }}}_0=2$. For the case of ${\overline{I}}_0=0.05$, the LCE engine, influenced by damping loss, shows a transient growth in its rotational response but gradually loses energy over time. Eventually, the LCE-based engine ultimately comes to rest at its equilibrium configuration, a regime that is termed the static state (Figure 2a,b). For the case of ${\overline{I}}_0=0.3$, the LCE engine departs from rest and progressively develops into a stable oscillatory motion. With time, both the angular frequency and amplitude converge to steady values, and steady illumination applied the system manifests as a sustained periodic swinging behavior, referred to here as the pendulum regime (Figure 2c,d). For the situation of ${\overline{I}}_0=0.5$, the frequency of oscillation of the LCE-driven engine gradually converges toward a steady value, and the device ultimately establishes a persistent clockwise motion, a state that we identify which is identified as the self-rotation regime (Figure 2e,f,). Such a dynamic response indicates that the system goes through a supercritical Hopf bifurcation, a process portrayed earlier in Section 4. In what follows, the underlying mechanisms governing both pendulum-like motion and sustained self-rotation is to be analyzed with further depth.

3.2. Mechanism of Pendulum

To shed light on the underlying dynamics of the pendulum-like motion, Figure 3 plots the evolution of several representative physical variables of the LCE-based engine, corresponding to the typical scenario illustrated in Figure 2c,d. Figure 3a displays the temporal evolution of the fraction of cis $\varphi$ molecules by number, which demonstrates a periodic fluctuation over time. Once the LCE fiber reaches the illuminated region, the number fraction $\varphi$ rises progressively until it reaches the maximum value, whereas in the dark zone it shows a continuous decline. Figure 3b and Figure 3c illustrate, respectively, the temporal evolution of the LCE fiber’s length change $\Delta \overline{L}$ and the illumination-induced tension $\overline{F}$. Both signals display a clear periodicity, which originates from the cyclic fluctuation in the cis $\varphi$ number fraction. Figure 3d illustrates the temporal variation in the rotational moment ${\overline{M}}_{\mathrm{F}}$, while Figure 3e highlights its dependence on the angular position $\theta$. As indicated in Equations (18), (19) and (21), the rotational moment ${\overline{M}}_{\mathrm{F}}$ arises from the combined effect of fiber tension $\overline{F}$ and the instantaneous angular position $\theta$. Moreover, the closed loop formed in Figure 3e corresponds to the total net work (positive) contributed by the LCE fiber, whose value is quantitatively estimated as 0.00102. Figure 3f illustrates how the damping moment ${\overline{M}}_{\mathrm{d}}$ varies with angular position $\theta$, where the enclosed loop of the curve quantifies the energy damping loss, calculated to be approximately 0.00102. Interestingly, this dissipated energy is found to be equal in magnitude to the total net work (positive) generated by the LCE fiber. Such an exact balance suggests that the sustained self-rotation of the system stems from the interplay between energy supplied by the fiber and the energy damping loss.

3.3. Mechanism of Self-Rotation

To better uncover the driving principle of self-rotation, Figure 4 presents several representative physical responses of the LCE engine under the characteristic scenario illustrated in Figure 2e,f. In particular, Figure 4a plots the temporal evolution of the fraction of cis $\varphi$ molecules by number, clearly revealing its oscillatory nature and highlighting how molecular-scale transitions translate into a periodic macroscopic effect over time. Once the illuminated region is reached by the LCE fiber, the fraction of cis $\varphi$ molecules by number increases toward a maximum, whereas within the light-free domain it steadily decreases. Figure 4b and Figure 4c illustrate, respectively, the temporal evolution of the LCE fiber’s length change $\Delta \overline{L}$ and the illumination-induced tension $\overline{F}$. Both signals display a clear periodicity, which originates from the cyclic fluctuation in the cis $\varphi$ number fraction. Figure 4d curve describing how the rotational moment ${\overline{M}}_{\mathrm{F}}$ evolves with time. Figure 4e correlates this torque with the angular position $\theta$. According to Equations (18), (19) and (21), the torque is determined through the combined effects of fiber tension $\overline{F}$ and angular position $\theta$. In Figure 4e, the closed loop formed by the curve quantifies the net positive work output of the LCE fiber, with numerical evaluation yielding a value of 0.07391. Correspondingly, Figure 4f correlation of the angular position $\theta$ with the damping moment ${\overline{M}}_{\mathrm{d}}$. The region bounded by its trajectory characterizes the energy damping loss, which is likewise computed to be 0.07391. The numerical analysis shows that the positive mechanical work generated by the LCE fiber is quantitatively equal to the energy damping loss. This equivalence indicates that the sustained self-rotation of the system is essentially governed by the delicate balance between energy input from the LCE fiber and the opposing damping loss.

The self-sustained motion of the LCE engine over a full cycle is sequentially illustrated in Figure 5. The upper and lower rows correspond to the pendulum and self-rotation regimes presented in Figure 2c,d and Figure 2e,f, respectively. The sequence begins with the LCE fiber in the dark region, stretched and dominated by trans to cis, which generates a restoring torque. As the turnplate rotates into the illuminated zone, light absorption triggers a trans to cis transition, inducing fiber contraction and producing a driving torque that propels the motion. The system’s inertia then carries the turnplate through the lit area and back into the dark, where the fiber re-stretches during the thermal cis to trans recovery, completing the energy cycle. This continuous process of photo-induced power stroke and passive recovery visually demonstrates the dynamic balance between energy input and dissipation that underlies both oscillatory and rotational limit cycles.

4. Influences of System Parameters on the LCE-Based Engine

By sweeping a single control parameter, we observe a sharp threshold: below it, the system remains quiescent, while above it, a finite-amplitude self-oscillation emerges whose amplitude grows continuously with the parameter— the hallmark of a supercritical Hopf bifurcation. This scenario is verified numerically by integrating Equation (16) with a Runge–Kutta scheme and plotting the resulting amplitude curve.

Referring to the earlier-described self-rotation model, six scaled system quantities are defined—namely, $C_0$, $\overline{k}$, ${\overline{I}}_0$, $\overline{\beta }$, $\overline{R}$, and ${\stackrel{·}{\overline{\theta }}}_0$—are involved. Our LCE-based engine’s self-rotation mechanism is ideal for real-time energy harvesting to power microsensors in implantable devices. It provides steady mechanical energy that can be converted into electricity. Future work focuses on optimizing prototypes, materials, and efficiency. The present section is devoted to examining how these parameters affect the occurrence of Hopf bifurcation together with the self-rotation’s amplitude and frequency that follows.

4.1. Influence of Light Intensity

Figure 6 describes the influence of light intensity ${\overline{I}}_0$ on the LCE-based engine. In performing the calculations, we assign $C_0=0.3$, $\overline{k}=30$, ${\stackrel{·}{\overline{\theta }}}_0=2$, $\overline{\beta }=0.02$, $\overline{R}=0.05$, and ${\overline{\theta }}_0=0$. A threshold value of nearly 0.31 is identified, at which the system switches from pendulum-like oscillation to self-rotation. It can be inferred from this result that the LCE-driven engine experiences a supercritical Hopf bifurcation once the light intensity attains the threshold value. Whenever ${\overline{I}}_0\le 0.11$, the engine remains fixed at steady-state position, corresponding to the equilibrium state. In contrast, the onset of self-rotation occurs once ${\overline{I}}_0\ge 0.31$, while the pendulum regime is activated when $0.11<{\overline{I}}_0<0.31$ falls within the range. Figure 6a within the pendulum regime, the plotted curves demonstrate the emergence of limit cycles, Figure 6b the plot illustrates the limit-cycle behavior observed during the self-rotation regime, and Figure 6c illustrates how light intensity ${\overline{I}}_0$ influences the amplitude as well as the frequency. Both the frequency and amplitude exhibit a pronounced growth with increasing light intensity. The self-rotation arises from the absorption of optical energy and its subsequent conversion into kinetic and potential energy. With stronger light intensity, a greater portion of the energy undergoes conversion, which in turn shortens the time span of self-rotation during a single cycle.

4.2. Influence of Contraction Coefficient

The effect of the contraction coefficient $C_0$ on the LCE-based engine is depicted in Figure 7. In performing the calculations, we assign ${\overline{I}}_0=0.5$, $\overline{k}=30$, ${\stackrel{·}{\overline{\theta }}}_0=2$, $\overline{\beta }=0.02$, $\overline{R}=0.05$, and ${\overline{\theta }}_0=0$. In a similar manner, the supercritical Hopf bifurcation is linked to a boundary contraction coefficient $C_0$ of about 0.215. The LCE-based engine triggers a transition between pendulum and self-rotation when $C_0\approx 0.215$. Whenever $C_0\le 0.105$, the engine settles into steady-state position, which corresponds to the equilibrium state. In contrast, the onset of the self-rotation regime can occur once $C_0\ge 0.215$, and pendulum regime can be triggered when $0.105<C_0<0.215$. Figure 7a illustrates the limit cycles corresponding to various damping constants within the pendulum regime, Figure 7b illustrates the limit-cycle behavior observed during the self-rotation regime, and Figure 7c illustrates how the contraction coefficient $C_0$ affects amplitude and frequency. As the contraction coefficient $C_0$ becomes larger, the amplitude together with the frequency tends to increase. Equation (9) indicates that a rise in the contraction coefficient enhances the light-driven contraction, thereby increasing the absorbed optical energy. Enhancing the contraction coefficient of the LCE material promotes a more efficient conversion of optical energy into kinetic and potential energy.

The hinge is key to transitioning from pendulum-like oscillation to continuous self-rotation. It connects the LCE fiber to the turntable and converts the fiber’s photo-induced tension into rotational torque. Acting as a pivot, the hinge allows the turntable to oscillate or rotate freely. When system parameters such as light intensity, damping, and contraction coefficient exceed critical values, the hinge ensures smooth energy transfer, helping the system overcome energy loss and achieve continuous rotation. The hinge’s geometry, particularly the moment-arm length, determines the torque magnitude, affecting bifurcation thresholds and rotational stability. Given that its role is similar to the effects of light intensity and contraction coefficient, it will not be discussed separately.

4.3. Influence of Damping Coefficient

Figure 8 illustrates the effect of the damping coefficient $\overline{\beta }$ on the LCE-based engine. Figure 8a illustrates the limit cycles corresponding to various damping constants within the pendulum regime, and Figure 8b illustrates the limit cycles observed in the self-rotation regime; Figure 8c, on the other hand, depicts the frequency and amplitude as functions of the damping coefficient $\overline{\beta }$. During the computational analysis, the remaining parameters are assigned as follows ${\overline{I}}_0=0.5$, $\overline{k}=30$, ${\stackrel{·}{\overline{\theta }}}_0=2$, $C_0=0.3$, $\overline{R}=0.05$, and ${\overline{\theta }}_0=0$. The supercritical Hopf bifurcation occurs at a threshold of the damping coefficient $\overline{\beta }$, which is found to be approximately 0.026. When $\overline{\beta }\ge 0.06$, because the damping-induced energy loss becomes excessive, while the external environment fails to supply enough input energy to offset this loss, the system eventually stabilizes in equilibrium state. In contrast, when $\overline{\beta }<0.026$ is satisfied, the self-rotation regime may arise. For $0.026<\overline{\beta }<0.06$, the pendulum regime can be triggered. In the self-rotation regime, a larger damping coefficient is associated with a lower oscillation frequency, indicating a negative dependence between these two parameters. For pendulum regime, an enhanced damping coefficient results in a reduction in the amplitude, and a trend that rises and then falls in the frequency. A larger damping coefficient causes more energy dissipation, reducing frequency and extending the self-rotation period.

4.4. Influence of Spring Constant

Figure 9 illustrates how the spring constant $\overline{k}$ governs the self-rotation dynamics of the LCE-based engine. In Figure 9a, the limit cycles corresponding to the pendulum regime under various spring constants are illustrated, Figure 9b shows this diagram elucidates the limit cycles observed in the self-rotation regime, and in Figure 9c, the amplitude together with the frequency is represented as dependent on the spring constant $\overline{k}$. In performing the calculations, we assign the other parameters as ${\overline{I}}_0=0.5$, $\overline{\beta }=0.02$, ${\stackrel{·}{\overline{\theta }}}_0=2$, $C_0=0.3$, $\overline{R}=0.05$, and ${\overline{\theta }}_0=0$. For the supercritical Hopf bifurcation, the spring constant $\overline{k}$ reaches a threshold near 13.5. When $\overline{k}\le 8$, a reduction in the tension of the LCE fiber decreases the net work performed by the rotational moment, which is insufficient to counteract the damping loss and thus fails to sustain self-rotation. Figure 9c shows that amplitude and frequency rise with the spring constant $\overline{k}$. From the previous analysis, a larger spring constant increases the rotational moment of the LCE fiber and tension, thereby raising the net work per cycle. Hence, enhancing the spring constant $\overline{k}$ improves the conversion efficiency enabling the transformation of light into kinetic and potential energy for engineering use.

4.5. Influence of Turnplate Radius

Figure 10 illustrates how the turnplate radius $\overline{R}$ affects the LCE-based engine’s self-rotation. In Figure 10a, the limit cycles corresponding to various turnplate radii are illustrated, whereas Figure 10b depicts how frequency varies with the turnplate radius. In performing the calculations, we assign ${\overline{I}}_0=0.2$, $\overline{\beta }=0.03$, ${\stackrel{·}{\overline{\theta }}}_0=1.3$, $C_0=0.3$, $\overline{k}=40$, and ${\overline{\theta }}_0=0$. The supercritical Hopf bifurcation arises when the turnplate radius $\overline{R}$ reaches a threshold near 0.01. When $\overline{R}\le 0.01$, in the case of a diminished turnplate radius, the rotational moment and its net work decrease, which cannot offset the damping dissipation, thus preventing sustained self-rotation. Figure 10b shows that the self-rotation frequency ${\overline{M}}_{\mathrm{F}}$ rises with the turnplate radius $\overline{R}$. From Equations (18)–(21), the rotational moment is also seen to grow with radius. Thus, enlarging the turnplate radius $\overline{R}$ enhances the self-rotation performance of the LCE engine with steady illumination applied.

4.6. Influence of Initial Velocity

Through the combination of Figure 11, the role of initial velocity ${\stackrel{·}{\overline{\theta }}}_0$ in controlling the self-rotation of the LCE-based engine is clarified. For this computation, the remaining parameters are assigned as $C_0=0.3$, $\overline{k}=30$, ${\overline{I}}_0=0.5$, $\overline{\beta }=0.02$, $\overline{R}=0.05$, and ${\overline{\theta }}_0=0$. As shown in Figure 11a, the pendulum regime exhibits a limit cycle that is illustrated for distinct initial velocities, and Figure 11b shows the self-rotation regime exhibits a limit cycle that is illustrated for distinct initial velocities, while Figure 11c depicts how amplitude and frequency vary with the initial velocity ${\stackrel{·}{\overline{\theta }}}_0$. A changeover between the pendulum state and self-rotation occurs at a critical velocity of about 0.74. The self-rotation regime is activated at ${\stackrel{·}{\overline{\theta }}}_0\ge 0.74$, and the pendulum regime is triggered at $0.15<{\stackrel{·}{\overline{\theta }}}_0<0.74$. It can be distinctly identified that the limit cycle, oscillation amplitude, and frequency prove insensitive to the initial velocity. Since the transformation process ensures equivalence between kinetic and potential energy, the starting conditions ${\overline{\theta }}_0$ and ${\stackrel{·}{\overline{\theta }}}_0$ exert no influence on either frequency or amplitude. This independence is regarded as an intrinsic feature of self-oscillation [42].

In the presence of environmental fluctuations (temperature, vibration, or non-uniform illumination), the stability of the oscillatory motion is influenced by several factors. Temperature variations can affect the contraction coefficient and elastic modulus of the LCE fiber, altering the frequency and amplitude of the oscillation. However, LCE fibers generally exhibit stable performance within a certain temperature range, so the oscillatory motion is expected to remain stable under moderate temperature fluctuations.

The single-fiber model can be expanded to a multi-fiber configuration, with fibers anchored at various positions on a turnplate or concentric rings. This enhances torque, reduces critical light intensity, increases amplitude, and introduces complex bifurcations and synchronization. Experimentally, fiber arrays can be fabricated via micro-extrusion or multi-nozzle 3D printing, enabling programmable activation sequences and expanding applications in soft robotics and biomedical microrobotics.

5. Conclusions

Self-oscillating systems are capable of harvesting ambient energy and persistently transforming it into mechanical output. The creation of additional self-oscillating designs paves the way for their broad utilization for engineering applications. Inspired by the concept of the four-stroke engine, we propose a streamlined theoretical model of a self-rotating engine under steady light, constructed from an LCE fiber, a turnplate, and a hinge. By incorporating the dynamic model of LCE, we establish a theoretical scheme for an LCE-driven engine exposed to steady illumination, aiming to explore its self-oscillatory characteristics. Simulation results reveal that this engine undergoes a supercritical Hopf bifurcation, shifting from the pendulum mode to the self-rotation mode. The self-rotation can be ascribed to the photo-induced contraction of the LCE fiber. Because the interaction of optical energy input with damping loss takes place, the system exhibits a sustained periodic cycle.

In addition, the oscillation frequency and rotational amplitude are mainly governed by specific parameters of the system. A rise in these parameters, such as contraction coefficient $C_0$, light intensity ${\overline{I}}_0$ and spring constant $\overline{k}$, is able to enhance the self-rotation frequency. From another perspective, a rise in the damping coefficient $\overline{\beta }$ results in the reduction in the self-rotation frequency, and for the pendulum regime, an elevated damping coefficient $\overline{\beta }$ results in a reduction in amplitude, and a trend that rises and then falls in the frequency. For the turnplate radius $\overline{R}$, when the turnplate radius $\overline{R}$ grows in the pendulum regime, a trend that rises and then falls in the frequency, and for the pendulum regime, when the turnplate radius $\overline{R}$ grows, a drop in amplitude occurs simultaneously with a rise in frequency. The self-rotating LCE-based engine, as predicted in this work, demonstrates qualitative behavior that corresponds closely to physical expectations. However, in practice, numerous intricate influences may cause quantitative discrepancies from the idealized case. Such influences involve the inertia of the LCE fiber, viscoelastic effects, nonlinear material responses, bending rigidity, and friction among the fiber, hinge, and turnplate. Therefore, incorporating these elements is essential for faithfully describing the real dynamics and performance pertaining to system.

The design can boost torque by increasing fiber count and size, optimizing material properties, and enhance amplitude by adjusting system parameters and structures. Future work includes building multi-fiber prototypes, experimental optimization, and verifying scalability in real applications.

Our study focuses on theoretical modeling and numerical calculation, and does not account for the influence of ambient temperature or humidity on the photothermal response of the LCE. We consider these factors to have a quantitative, rather than qualitative, impact on the system behavior. While they may shift the thresholds of bifurcation or modify the amplitude and frequency of oscillations, they are not expected to alter the fundamental mechanism of self-rotation or the occurrence of the supercritical Hopf bifurcation. Future work could incorporate environmental variables to refine the model and improve its predictive accuracy under realistic conditions.

Future studies should experimentally verify the self-rotation behavior by constructing a prototype of the LCE-based engine. For example, the engine could serve as a light-driven propeller in microrobots, where its lightweight design enhances agility and its tunable size allows it to fit specific anatomical constraints, such as blood vessel diameters. This not only boosts the functionality of microrobots but also opens new possibilities for minimally invasive medical procedures.