Nonlinear vibration of a conical PVC gel actuator under electrical excitation

1. Introduction

Polyvinyl chloride (PVC) gel is an electroactive polymer with low cost, large deformability, light weight, and fatigue resistance (Irfan et al., 2023; Tang et al., 2021; Wu et al., 2024). Numerous studies have investigated its properties from various perspectives, including composition and composites (Zhang et al., 2023), mechanical and damage modeling (Neubauer et al., 2022), and electromechanical responses (Wang et al., 2023; Yoon and Bae, 2024). In addition, it exhibits bidirectional electromechanical coupling, offering significant application prospects in intelligent sensing (e.g., Huang et al., 2023) and flexible actuation (e.g., Li et al., 2021; Motohashi et al., 2022). These applications extend to fields including microfluidics, wearable devices, and energy harvesting (Neubauer et al., 2022; Siekierka et al., 2023). The material’s lightweight and flexible nature makes it suitable for various applications, including bionic gecko adhesive grippers (He et al., 2022), bionic fish tails (Dong et al., 2022), and finger tremor correctors (Liu and Zhang, 2024), demonstrating its potential as an efficient novel actuation solution. Notably, the conical structure of PVC gel exhibits exceptional application value in flexible grasping and precise directional actuation owing to its unique morphology. Therefore, theoretical modeling of conical PVC gel structures is crucial for unlocking their full application potential.

The dynamic behavior of PVC gel under load is directly relevant to its applications, and has therefore attracted increasing research attention to the dynamic characteristics of PVC gel actuators. Current research trends primarily focus on three directions: theoretical modeling, performance optimization, and application exploration. Among these, theoretical modeling provides fundamental guidelines for application structures. Lei et al. conducted quasi-static and dynamic mechanical compression tests on PVC gel elastomers under different conditions, establishing the Yeoh hyperelastic model and a rate-dependent viscoelastic model that accurately describes the material’s static behavior (Lei et al., 2020). Du et al. developed a constrained theoretical model for circular planar PVC gel actuators, performed finite element verification, analyzed the vibration characteristics, and predicted the deformation under different bias voltages for various dibutyl adipate contents (Du et al., 2024). Li et al. adopted the Gent constitutive model to predict the deformation mechanism of fiber-constrained PVC gels, estimated their electrical responses under different pre-stretch ratios and voltage amplitudes, and validated the predictions experimentally (Li et al., 2023). Luo et al. developed a mathematical model for the vibration behavior of planar square PVC gel films and systematically analyzed their dynamic response characteristics with varying geometric parameters (Luo et al., 2024).

The Gent hyperelastic model (Gent, 1996) adopted in the constitutive framework of this study is a well-established phenomenological model for describing the large deformation behavior of rubber-like polymers. Unlike Gaussian-based models such as the Neo-Hookean and Yeoh models, the Gent model is rooted in non-Gaussian statistics and accounts for the finite extensibility of polymer chains, making it more physically realistic for materials approaching their stretch limit. The Gent model characterizes this finite extensibility through the parameter J m . In the limit J m → ∞ , the Gent model naturally reduces to the classical Neo-Hookean model. Originally proposed by Gent (1996), the model has since been widely adopted in the field of electroactive polymers. The material parameters ( μ , J m ) required by the Gent model are physically meaningful quantities that can be determined through standard experimental characterization procedures. Therefore, although the present study focuses on the theoretical modeling and nonlinear dynamic analysis of the conical configuration, the underlying constitutive framework rests on a solid theoretical foundation, ensuring the physical realizability of the proposed model.

The present work on conical PVC gel actuators is underpinned by broader advances in nonlinear dynamics. The concept of regulating system stiffness with multistable mechanisms has laid a solid foundation for vibration mitigation and energy conversion. Studies have shown that hybrid vibration isolators with negative stiffness springs can effectively attenuate low-frequency vibrations (Huang et al., 2025), and that magnetoelastic tunable bistable (Huang et al., 2024) and quad-stable (Zhang et al., 2024) energy converters can achieve broadband performance by facilitating interwell chaotic motions. Concurrently, the “energy shunt” perspective—which views vibration suppression in terms of mechanical energy transfer, dissipation, and conversion—has provided a unified methodology for designing and evaluating such nonlinear systems (Huang and Yang, 2023). Notably, these advances have been successfully translated into high-performance devices. Dynamic bistable electromagnetic harvesters have demonstrated substantially enhanced power output under ultra-low-frequency excitations (Fang et al., 2026), while absorbers with multi-stiffness combinations have achieved simultaneous vibration mitigation and energy harvesting by exploiting the synergistic effects of hybrid nonlinear oscillators (Huang, 2024). Furthermore, the applicability of nonlinear dynamics has been expanded to complex scenarios such as flow-induced vibration energy harvesting (Huang and Zhong, 2023). Most fundamentally for the present study, Huang and Yang (2021) thoroughly investigated the energy trapping and conversion performances of a multistable vibration absorber, establishing the core analytical framework of interwell dynamics and targeted energy transfer that directly informs our modeling of the conical PVC gel actuator.

Overall, theoretical research on PVC gel has been largely confined to planar structures with simple geometric shapes, while modeling and dynamic studies of conical structures remain unexplored in existing research. To address this gap, this paper establishes a nonlinear dynamic model for a conical PVC gel actuator based on the Euler-Lagrange equation. Using this model, we investigate the effects of four key parameters (damping ratio, voltage amplitude, pre-stretch ratio, and tensile limit) on the actuator’s vibration, along with the influence of spring stiffness. Bifurcation and Lyapunov exponent analyses are performed to determine the critical thresholds for state transitions under varying parameters, thereby providing a design basis and theoretical foundation for conical PVC gel actuators. This study not only provides a theoretical design basis and fundamental support for the development of conical PVC gel actuators, but also is expected to promote the technological advancement of related fields such as soft robotics, intelligent manufacturing, and wearable devices. Combined with advanced processing and material characterization techniques, the research results can also provide theoretical guidance for the development of PVC gel actuators with improved response speed, load capacity, and durability (Figure1).OPEN IN VIEWER

2. Physical model of three-dimensional conical actuator

Figure 2 illustrates the three typical states of the conical actuator. In the compressed state in Figure 2(a), the spring is fully compressed and the PVC gel film is radially pre-stretched. After release, the system reaches the initial equilibrium state in Figure 2(b), where the spring’s restoring force balances the elastic force of the PVC gel, establishing an initial cone height h 0 . When an external voltage ϕ is applied to the flexible electrodes (Figure 2(c)), charge Q accumulates on the electrodes. Owing to the incompressibility of the PVC gel, the film thickness decreases and its area expands, producing a vertical displacement Δ h and increasing the cone height to h = h 0 + Δ h .OPEN IN VIEWER

In this study, the Euler-Lagrange equation was used to establish the analytical model of the PVC gel cone structure to drive the system. Therefore, the kinetic control equation of PVC gel is established as follows:

∂ L ∂ q i − d d t ( ∂ L ∂ q ˙ i ) = F c , L = V − U

(1)

In equation (1), L is the Lagrange function, V is the kinetic energy of PVC gel system, U is the potential energy of PVC gel system, q i is the variable of PVC gel system-variable height h and electric displacement D, F c is the work of PVC gel system damping ratio, and c is the damping coefficient.

In this system, the mass of the mass block is much larger than that of the PVC gel film and the spring. Therefore, the kinetic energy V of the system can be expressed by equation (2).

V = 1 2 m ( d h d t ) 2

(2)

The potential energy U of the system is composed of the Helmholtz free energy of PVC gel, the elastic potential energy of the spring and the work done by the external excitation voltage. Therefore, we established the potential energy equation of the system as follows

U = π B 2 − A 2 H 0 λ p 2 W + 1 2 k h − h 0 2 − ϕ Q

(3)

In equation (3), W denotes the Helmholtz free energy density of PVC gel film, k denotes the stiffness of linear spring, h 0 denotes the initial height of PVC gel film, and h denotes the variable height of PVC gel film.

Because PVC gel is an ideal incompressible elastomer, the tensile deformation rate of PVC gel material in X, Y and Z directions is λ 1 λ 2 λ 3 = 1 . Therefore, the strain energy density function of PVC gel was described by the superelastic material Gent model in this study. The Helmholtz free energy density W is expressed as follows

W = − μ J m 2 log ( 1 − λ 1 2 + λ 2 2 + λ 1 − 2 λ 2 − 2 − 3 J m ) + D 2 2 ε

(4)

In equation (4), μ is the shear modulus of PVC gel material, J m is the tensile limit of PVC molecular chain, and ε is the dielectric constant of PVC gel. D is the electric displacement.

According to the geometric relationship, the axial tensile ratio λ 1 of the conical PVC gel structure actuator can be expressed as follows

λ 1 = h 2 + B − A 2 B − A λ p

(5)

The PVC gel film is fixed in the radial direction by a rigid fixed clamping device, so the radial stretching λ 2 = λ p . The accumulated charge on the electrode can be expressed as equation (2). Under the stimulation of external sinusoidal voltage, the amount of charge accumulated on the flexible electrode on the upper and lower surfaces of the PVC gel film is Q

Q = π ( A + B ) D ( h 2 + ( B − A ) 2 )

(6)

Thus, the Euler-Lagrange equations of variable height h and electric displacement D are shown in equations (7) and (8).

∂ L ∂ D − d d t ( ∂ L ∂ D ˙ ) = 0

(7)

∂ L ∂ h − d d t ( ∂ L ∂ h ˙ ) = − c h ˙

(8)

Substituting Formulas (1-6) into equations (7) and (8), equation (9) and (10) can be obtained. The equations are as follows.

D = A 2 − 2 A B + B 2 + h 2 ε λ p 2 ϕ ( B − A ) H 0

(9)

m h ¨ + c h ˙ + h − h 0 k − A + B h π ε λ p 2 ϕ 2 B − A H 0 + A + B hH 0 π B - A 1 − B − A 4 / B − A 2 + h 2 2 λ p 6 μ 1 − B − A 2 B − A 2 + h 2 λ p 4 + 2 λ p 2 + h 2 λ p 2 B − A 2 − 3 / J = 0

(10)

Equation (11) can be obtained by further simplifying equation (10).

y ¨ + ξ y ˙ + ( r y − h 0 ) K − y Φ 2 + J m y ( 1 − ( 1 + y 2 ) 2 λ p 6 ) ( 1 + y 2 ) λ p 2 ( 1 − ( 3 + J m ) ( 1 + y 2 ) λ p 4 + ( 2 + 3 y 2 + y 4 ) λ p 6 ) = 0

(11)

Equation (11) describes the vibration state of the conical PVC gel actuator. The fourth-order Runge-Kutta method was used to solve Eq. (11). To validate the numerical implementation, a reduced linear case was tested: setting Φ 0 = 0 , ξ = 0 , and taking J m → ∞ (the Neo-Hookean limit) reduces Eq. (11) to a conservative linear oscillator. The numerical solution reproduces the analytical sinusoidal waveform with constant amplitude and perfectly closed elliptical phase portrait, confirming negligible error.

To validate the proposed analytical model, a static force-displacement analysis was performed and compared with finite element simulations conducted in COMSOL Multiphysics. In the static equilibrium state, both acceleration and velocity are zero, and the reaction force F ( Δ h , λ p , K , Φ 0 ) on the top surface can be determined from equation (11). Figure 3 presents the force-displacement curves at three representative voltage levels ( Φ 0 = 0.1 , 0.2 , 0.3 ). When the voltage reaches Φ 0 = 0.3 , the curve intersects the zero-force baseline at three points y 1 , y 2 , and y 3 , corresponding to three equilibrium positions. The positive slopes at y 1 and y 3 indicate stable equilibria, while the negative slope at y 2 indicates an unstable equilibrium, demonstrating that the conical PVC gel actuator exhibits bistable mechanical behavior under specific driving conditions. The finite element results are shown as markers alongside the analytical solutions (solid lines) in Figure 3. The close agreement at all three voltage levels provides strong evidence for the accuracy of the derived analytical model (Table 1).OPEN IN VIEWER

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Table 1.

Structural and material parameters used in the model validation.

Symbol	Parameter	Value	Unit
A	Radius of the hard disc	10	mm
B	Circular radius of the PVC gel film	50	mm
h 0	Initial height	0	mm
J m	Tensile limit	50	-
K	Linear stiffness of the spring	1.5	-
λ p	Radial pre-stretching	1	-

Equation (11) is expressed in dimensionless form, which endows the model with generality. The geometric parameters A , B , and H 0 can be adjusted to represent different sizes and aspect ratios, while the material constants μ , J m , ε , and ξ can be determined experimentally for various PVC gel formulations. Moreover, the dimensionless formulation ensures that systems with different physical dimensions but identical dimensionless parameters exhibit qualitatively similar dynamic behavior. Hence, this framework provides a universal design guideline for conical PVC gel actuators across a wide range of geometries and material compositions.

Based on the above, we discuss the vibration of the conical actuator under variations of damping ratio ξ , spring stiffness K , voltage amplitude Φ 0 and pre-stretch λ p , and tensile limit J m .

3. Nonlinear vibration of the conical PVC gel actuator under internal and external factors

The PVC gel cone actuator will undergo large deformation under electromechanical coupling, so it has good driving performance (Acharya et al., 2023; Mehraeen et al., 2023; Shin et al., 2018). The factors affecting the driving behavior of the system mainly come from the material characteristics of the system itself and the external excitation conditions. Firstly, due to the different proportion of DBA molecules and PVC molecules in the formula and the different configuration process, PVC gel films with different damping ratio and tensile limits will be obtained in the production process of PVC gel materials, which further leads to the vibration performance of PVC gel cone actuators will be different. Secondly, the difference of external voltage amplitude, pre-stretched external force and linear spring stiffness will also change the nonlinear vibration behavior of PVC gel cone actuator. In this study, the sinusoidal voltage Φ 0 = 0.3 sin ( 0.5 t ) , the initial conditions are y ( 0 ) = 1 , y ˙ ( 0 ) = 0 .

4. Threshold analysis of PVC gel cone actuator under different parameter change process

The changes in internal factors and external conditions significantly alter the vibration state of the three-dimensional conical actuator system. To improve vibration stability and controllability for effective actuation, we analyze four key parameters individually: damping ratio ξ , voltage amplitude Φ 0 , pre-stretching ratio λ p , and tensile limit J m . Basic parameters are fixed as follows: ξ = 0.01 , Φ 0 = 0.3 , λ p = 1.2 , J m = 50 , and K = 1.5 . We investigate the bifurcation characteristics of each single parameter to explore periodic vibration state transitions, which are further verified by quantitative Lyapunov exponent analysis.

From the bifurcation of Figure 13(a), it can be seen that during the change of damping ratio in the range of 0-2, the diagram shows a multi-point distribution when the damping ratio is extremely small, reflecting the sensitive dependence on initial conditions characteristic of a near-conservative system, and then gradually tends to a single branch and reaches a stable state. At the same time, according to Figure 13(e), as the damping ratio increases, the Lyapunov exponent decreases from slightly greater than 0 to always less than 0. It should be noted that the slightly positive Lyapunov exponent observed at ξ → 0 reflects the sensitivity to initial conditions characteristic of a conservative Hamiltonian system, rather than the presence of a strange attractor indicative of dissipative chaos. For ξ > 0.1 , the system enters the overdamped regime, and the Lyapunov exponent is consistently non-positive, in agreement with the rapid convergence observed in Section 3.1. (Figure 4)OPEN IN VIEWER

As the damping ratio increases, the conical PVC gel actuator transitions to stable periodic vibration and eventually to damped motion. Figure 13(b) depicts the bifurcation trend as the voltage amplitude varies within the range of 0-2. Combined with the Lyapunov exponents shown in Figure 13(f), when the voltage amplitude is in the range of 0 ≤ Φ 0 ≤ 1.04 , the Lyapunov exponent is always less than or equal to 0. However, when Φ 0 = 0.93 , the Lyapunov exponent becomes positive, indicating the onset of weak chaotic motion at this time. Thus, the conical PVC gel actuator remains in a stable quasi-periodic vibration within this voltage amplitude range, which corresponds to the damped periodic responses shown in Figure 7(a)–(c) for Φ 0 = 0.5 .When Φ 0 approaches 1.04, the Lyapunov exponent becomes positive, indicating the onset of weak chaotic motion. This transitional state aligns with the irregular time-domain response and scattered Poincaré section observed in Figure 7(d)–(f) for Φ 0 = 1.0 . For Φ 0 > 1.04 , the bifurcation diagram exhibits significant divergence, and the Lyapunov exponent remains above zero, suggesting that the conical PVC gel actuator undergoes developed chaotic vibration. This is consistent with the highly erratic responses in Figure 7(g)–(i) for Φ 0 = 1.5 .

From the bifurcation diagram of Figure 13(c), it can be seen that with the increase of the pre-stretch λ p , the bifurcation phenomenon obviously appears when λ p = 0.3 . However, through Figure 13(g), we find that the Lyapunov exponent at λ p = 0.3 has a “Jump” phenomenon. Specifically, when λ p = 0.3 (corresponding to the chaotic vibration in Figure 10(a)–(c)), the Lyapunov exponent is positive, confirming chaotic motion. In the process of 0 ≤ λ p ≤ 0.6 , the conical PVC gel actuator is always in a chaotic vibration state. In the range 0.9 ≤ λ p ≤ 1 , the Lyapunov exponent oscillates around zero, indicating state transitions between periodic and chaotic vibrations. This is consistent with the small-amplitude chaotic vibration observed in Figure 10(d)–(f) for λ p = 1 .When λ p > 1 , the Lyapunov exponent stabilizes at zero, and the PVC gel actuator enters a stable quasi-periodic vibration state, as demonstrated in Figure 10(g)–(i) for λ p = 2 .

Building upon the analysis in Section 3.2, which showed that J m = 1 leads to chaotic behavior while J m = 50 , 100 yield stable damped periodic motion, we now extend the investigation to its full parameter range, including the critically small values that approach the material’s stretch limit. Figure 13(d) shows that when the tensile limit J m ≤ 1.5 , the points in the bifurcation diagram exhibit an irregular and randomly scattered distribution. However, when J m is extremely small, the scattered points in the bifurcation diagram are uniformly distributed along a short vertical line. In Figure 13(h), when the Lyapunov exponent J m ≤ 1.5 , the Lyapunov exponent diagram is generally greater than 0; yet, when J m is close to 0, local regions in the Lyapunov exponent diagram become less than 0. When the tensile limit J m > 1.5 , the bifurcation diagram and Lyapunov exponent diagram show that the actuator is in a stable damped periodic motion state. In summary, the influence of the tensile limit J m is governed by a critical threshold at J m ≈ 1.5 .For J m ≤ 1.5 , the system enters a chaotic regime. Within this regime, the dynamics are complex: when J m is moderately small (e.g., between 0 and 1.5), the Lyapunov exponent is predominantly positive, indicating chaotic vibration; however, as J m approaches zero, the Lyapunov exponent locally becomes negative, suggesting transient or localized stable phases amidst the overall unstable trend, which may correspond to a degraded material state prior to mechanical failure. For J m > 1.5 , which encompasses and extends beyond the representative values analyzed in Section 3.2, variations in J m have negligible impact, and the system maintains stable damped periodic vibrations. Thus, the conclusion in Section 3.2 that J m exhibits a threshold-dependent regulatory effect holds robustly within the stable operating regime J m > 1.5 , while the newly identified chaotic regime and its associated complex dynamics are confined to the sub-threshold regime J m ≤ 1.5 .Therefore, it indicates that when J m ≤ 1.5 , the tensile limit causes the conical structure actuator to be generally in a state of chaotic vibration; however, when J m is extremely small, the conical structure actuator will be in a state of locally stable vibration. When J m > 1.5 , changes in the tensile limit will not cause the actuator to undergo chaotic vibration, consistent with the energetic interpretation provided in Figure 6.

In summary, changes in damping ratio will never induce chaotic vibrations. At extremely small damping ratios, the system may exhibit quasi-periodic motion characteristic of a near-conservative system, but fully developed dissipative chaos does not occur under the examined parameter conditions. Whereas voltage amplitude and pre-stretching can, within certain ranges, trigger chaotic vibrations in PVC gel cone-structured actuators. When the tensile limit is at a relatively small value, chaotic vibrations occur; however, within the normal range ( J m > 1.5 ), the system maintains stable damped periodic motion, the actuator remains in a stable quasi-periodic vibration state.

To address the synergistic influence of Φ 0 and λ p on vibration stability, a two-parameter analysis was conducted. Figure 14(a) presents a three-dimensional bifurcation diagram in which Poincaré sections for seven representative λ p values are stacked along the pre-stretch axis. This visualization directly captures the evolution of the dynamical regime: for λ p = 0.1 , 0.4 0.7 , the Poincaré points transition from tight clusters at low Φ 0 to widely scattered distributions as Φ 0 increases, consistent with the chaotic responses identified in Figure 10(a)–(c) and the positive Lyapunov exponents in Figure 13(g) for small λ p . For λ p = 1 . 3 , 1.6 , 2.0 , the points remain densely clustered regardless of Φ 0 , in agreement with the stable damped periodic motion shown in Figure 10(g)–(i). The transition from dispersed to clustered points as λ p increases provides direct phase-space evidence that pre-stretch suppresses voltage-induced chaos.OPEN IN VIEWER

Figure 14(b) complements this picture with a quantitative stability map of the largest Lyapunov exponent. The map reveals distinct dominant mechanisms in different parameter regions. In the high-voltage regime ( Φ 0 > 1.2 ), the exponent is predominantly positive irrespective of λ p , indicating voltage-dominated chaos. This is consistent with the single-parameter bifurcation in Figure 13(b), where chaos sets in at Φ 0 ≈ 1.04 and persists thereafter. In the low pre-stretch regime ( λ p < 0 . 6 ), chaos appears even at moderate voltages, confirming that small λ p alone can destabilize the actuator, as observed in Figure 13(c) and (g). A narrow stable pocket (deep blue, L < 0 ) exists around λ p ≈ 0.8 , Φ 0 ≤ 0.8 . The closed L = 0 contour encloses a transitional domain ( λ p ∈ [ 0.6 − 1.8 ] ; Φ 0 ≤ 1.2 ) that contains this pocket and surrounding quasi-periodic zones.

Taken together, Figure 14 demonstrates that the chaotic threshold is not determined by any single parameter, but is a joint function of Φ 0 and λ p . This two-parameter analysis provides a clear design guideline for stable actuator operation.

5. Conclusion

Through the Euler-Lagrange equation, we calculated the kinetic energy and potential energy of the conical PVC gel actuator. The potential energy was mainly derived from the superelastic strain energy of the PVC gel itself and the voltage potential energy generated by the external voltage. The Helmholtz free energy density and the superelastic Gent model were used to describe the superelastic strain energy change of the PVC gel. Finally, the vibration deformation control equation in the z direction was obtained.

Through the analysis of different internal factors of PVC gel cone structure actuator, the dynamic behavior characteristics under various parameters were explored. The internal factors include damping ratio and tensile limit, and the change of these two parameters is determined by the solution configuration ratio of PVC gel itself. Specifically, it is found that the damping ratio parameter may induce quasi-periodic or high-periodic vibration when it is extremely small, but with the increase of damping ratio, the vibration attenuation of the conical PVC gel actuator becomes faster, and variations in damping ratio do not induce chaotic vibration. The tensile limit J m exhibits a critical threshold effect. When J m falls below a critical value (approximately 1.5 ), the actuator enters a failure regime characterized by complex and unstable dynamics. For J m above this threshold, its influence is negligible, ensuring stable operation. This highlights the practical necessity of maintaining J m above its critical value.

For external factors, voltage amplitude and pre-stretching force significantly influence the vibration of the conical PVC gel actuator, inducing mixed vibrations with distinct transition thresholds. A two-parameter stability analysis reveals that the chaotic threshold is a joint function of these two parameters: for Φ 0 > 1.2 , chaos is voltage-dominated; for λ p < 0 . 6 , chaos is pre-stretch-dominated. Adopting λ p ≥ 1.3 ensures stable periodic motion across the entire voltage range. In contrast, changes in spring stiffness K only alter the vibration frequency without affecting stability or periodicity. In summary, external mechanical forces and voltage exert a far greater influence on the actuator’s vibration than internal parameters of the PVC gel itself.

Therefore, appropriate external excitation conditions should be selected for stable actuation, and the identified critical thresholds provide direct guidelines for the design and safe operation of conical PVC gel actuators.