Dynamical Behaviors of a Translating Liquid Crystal Elastomer Fiber in a Linear Temperature Field

Abstract

Liquid crystal elastomer (LCE) fiber with a fixed end in an inhomogeneous temperature field is capable of self-oscillating because of coupling between heat transfer and deformation, and the dynamics of a translating LCE fiber in an inhomogeneous temperature field are worth investigating to widen its applications. In this paper, we propose a theoretic constitutive model and the asymptotic relationship of a LCE fiber translating in a linear temperature field and investigate the dynamical behaviors of a corresponding fiber-mass system. In the three cases of the frame at rest, uniform, and accelerating translation, the fiber-mass system can still self-oscillate, which is determined by the combination of the heat-transfer characteristic time, the temperature gradient, and the thermal expansion coefficient. The self-oscillation is maintained by the energy input from the ambient linear temperature field to compensate for damping dissipation. Meanwhile, the amplitude and frequency of the self-oscillation are not affected by the translating frame for the three cases. Compared with the cases of the frame at rest, the translating frame can change the equilibrium position of the self-oscillation. The results are expected to provide some useful recommendations for the design and motion control in the fields of micro-robots, energy harvesters, and clinical surgical scenarios.

Keywords: constitutive model; dynamics; fiber; heat-driven; liquid crystal elastomer; translating.

Figure 1

Schematic model for a LCE fiber connected with a translating frame in a linear temperature field. (a) Reference state; (b) Current state. The original length of the fiber is $L$. $w_1\left(t\right)$, ${\dot{w}}_1\left(t\right)$, $w_2\left(t\right)$, and ${\dot{w}}_2\left(t\right)$ are the displacements and velocities of the translating frame and the free end of the LCE fiber. When the displacements and velocities of the two ends of the LCE fiber are given, the tension force of the fiber can be determined.

Figure 2

Schematic of the fiber-mass system translating in a linear temperature field. (a) Reference state; (b) Current state. A mass block with a mass of $m$ is attached to the free end of the LCE fiber connected to a frame that ignores gravity. The motion of the fiber-mass system is determined by the displacement, velocity, and acceleration of the mass connected to the end of the LCE fiber, as well as the displacement of the frame attached to the other end of the LCE fiber.

Figure 3

Variation of the displacement of the mass block with time during the damped free vibration of the system in a linear temperature field with a static frame. The vibration of the mass block stops in a static condition due to the damping consuming the energy.

Figure 4

(a) Time history and (b) phase trajectory of the displacement of the system consisting of fiber and mass in the case of the frame at rest in a linear temperature field. (c) Time history and (d) phase trajectory of the displacement of the system consisting of fiber and mass in the case of the frame at uniform translation in a linear steady temperature field. The red line represents ${\overline{w}}_1\left(\overline{t}\right)$ and the blue line represents ${\overline{w}}_2\left(\overline{t}\right)$. The equilibrium position of mass and the length of the LCE fiber depend on the translation of the frame.

Figure 5

(a) Time history and (b) phase trajectory of the displacement of the system consisting of fiber and mass in the case of the frame at uniformly accelerated translation in a linear temperature field. The red line represents ${\overline{w}}_1\left(\overline{t}\right)$ and the blue line represents ${\overline{w}}_2\left(\overline{t}\right)$. Whether the frame is at uniform translation or uniformly accelerated translation, the equilibrium position of the mass and the length of the LCE fiber vary with time because of the variation of contraction of the LCE fiber translating in the linear steady temperature field.

Figure 6

Variation of the displacement of the mass block with time during the self-sustained oscillation in a linear temperature field with a static frame. The LCE fiber in the linear temperature field can stretch or shorten continuously and eventually develop into a periodic self-sustained oscillation.

Figure 7

Variation of the displacement of the mass block with time in a linear temperature field with temperature gradient $\overline{\beta }=-1$. It is found that the mass block develops into stationary due to energy dissipation, which is much different from the self-oscillation for the case of $\overline{\beta }=1$.

Figure 8

(a) Time history of the displacement and (b) phase trajectory of the self-oscillation in the case of the frame at rest in a linear temperature field. (c) Time history of the displacement and (d) phase trajectory of the self-oscillation in the case of the frame at uniform translation in a linear temperature field. The equilibrium position of mass and the length of fiber depend on the translation of the frame.

Figure 9

(a) Time history of the displacement and (b) phase trajectory of the self-oscillation of the fiber-mass system in the case of the frame at uniformly accelerated translation in a linear temperature field. For the case of the frame at uniformly accelerated translation, the mass block can also self-oscillate periodically.

Figure 10

Time histories of the displacement of the fiber-mass system with a static frame in a linear temperature field for (a) finite characteristic times $\overline{\tau }=0.1$, (b) $\overline{\tau }=0.3$, and (c) $\overline{\tau }=0.7$. The amplitude first increases and then decreases obviously with the increasing characteristic time.

Figure 11

Variations of the displacement of the fiber-mass system with time for the case of a uniform translational frame in a linear temperature field for (a) finite characteristic times $\overline{\tau }=0.1$, (b) $\overline{\tau }=0.3$, and (c) $\overline{\tau }=0.7$. The amplitude first increases and then decreases obviously with the increasing characteristic time.

Figure 12

Time histories of the displacement of the fiber-mass system with uniformly accelerated translational frame in a linear temperature field for (a) finite characteristic times $\overline{\tau }=0.1$, (b) $\overline{\tau }=0.3$, and (c) $\overline{\tau }=0.7$. The amplitude first increases and then decreases obviously with the increasing characteristic time.