Transverse vibration of nematic elastomer Timoshenko beams

Being a rubber-like liquid crystalline elastomer, a nematic elastomer (NE) is anisotropic viscoelastic, and displays dynamic soft elasticity. In this paper, the transverse vibration of a NE Timoshenko beam is studied based on the linear viscoelasticity theory of nematic elastomers. The governing equation of motion for the transverse vibration of a NE Timoshenko beam is derived. A complex modal analysis method is used to obtain the natural frequencies and decrement coefficients of NE beams. The influences of the nematic director rotation, the rubber relaxation time, and the director rotation time on the vibration characteristic of NE Timoshenko beams are discussed in detail. The sensitivity of the dynamic performance of NE beams to director initial angle and relaxation times provides a possibility of intelligent controlling of their dynamic performance.

Figure 1

Diagrammatic sketch of nematic elastomers: (a) microscopic picture: polymers are average spherical in the isotropic state (I) and elongated in the nematic state (N); the director $\mathbf{n}$ points along the long axis of the shape spheroid; (b) director rotation in prolate elastomers (the long axis of the spheroid points along the nematic director $\mathbf{n}$) toward the elongation diagonal direction; (c) director rotation in oblate elastomers (the long axis of the spheroid is perpendicular to the nematic director $\mathbf{n}$) toward the compression direction; in order to distinguish the director after rotation from the director before rotation, the initial state of the director is marked as ${\mathbf{n}}_0$; ${\varepsilon }_{nl}$ is the symmetric shear strain and $\delta \mathbf{n}$ is the variation of director orientation due to the shear strain.

Figure 2

The relationship between the renormalized shear modulus and the frequencies (a) for different ${\tau }_1/{\tau }_2$ with $r=3, {\tau }_2={10}^{-4}\mathrm{s}, {\tau }_R={10}^{-5}\mathrm{s}$; and (b) for different $r$ with ${\tau }_1={10}^{-2}\mathrm{s}, {\tau }_2={10}^{-4}\mathrm{s}, {\tau }_R={10}^{-5}\mathrm{s}$. The blue lines represent the real part of $C_{44}^R$ and the red lines represent the imaginary part of $C_{44}^R$.

Figure 3

The diagrammatic sketch of (a) a Timoshenko beam made of nematic elastomers; and (b) the undistorted director $\mathbf{n}$ having an initial angle $\theta$ with the $z$ axis ($\theta$ is positive while it is counterclockwise).

Figure 4

Comparison of (a) the dimensionless natural frequencies; (b) decrement coefficients obtained by the present model and those given in Ref. [27].

Figure 5

The effect of ${\tau }_1^{\prime }$ on the first-order (a) dimensionless frequencies and (b) decrement coefficients with $r=3, {\tau }_2=0.5\times {10}^{-4}\mathrm{s}, {\tau }_R={10}^{-5}\mathrm{s}$ (the cyan part corresponding to $r=1$; the color coding shows the corresponding values).

Figure 6

The effect of ${\tau }_1^{\prime }$ on the fourth-order (a) dimensionless frequencies and (b) decrement coefficients with $r=3, {\tau }_2=0.5\times {10}^{-4}\mathrm{s}, {\tau }_R={10}^{-5}\mathrm{s}$ (the cyan part corresponding to $r=1$; the color coding shows the corresponding values).

Figure 7

The effect of ${\tau }_2^{\prime }$ on the first-order (a) dimensionless frequencies and (b) decrement coefficients with $r=3, {\tau }_1={10}^{-2}\mathrm{s}, {\tau }_R={10}^{-5}\mathrm{s}$ (the cyan part corresponding to $r=1$; the color coding shows the corresponding values).

Figure 8

The effect of ${\tau }_2^{\prime }$ on the fourth-order (a) dimensionless frequencies and (b) decrement coefficients with $r=3, {\tau }_1={10}^{-2}\mathrm{s}, {\tau }_R={10}^{-5}\mathrm{s}$ (the cyan part corresponding to $r=1$; the color coding shows the corresponding values).

Figure 9

The effect of ${\tau }_R^{\prime }$ on the first-order (a) dimensionless frequencies and (b) decrement coefficients with $r=3, {\tau }_1={10}^{-2}\mathrm{s}, {\tau }_2=0.5\times {10}^{-4}\mathrm{s}$ (the cyan part corresponding to $r=1$; the color coding shows the corresponding values).

Figure 10

The effect of ${\tau }_R^{\prime }$ on the fourth-order (a) dimensionless frequencies and (b) decrement coefficients with $r=3, {\tau }_1={10}^{-2}\mathrm{s}, {\tau }_2=0.5\times {10}^{-4}\mathrm{s}$ (the cyan part corresponding to $r=1$; the color coding shows the corresponding values).

Figure 11

The first five dimensionless (a) natural frequencies and (b) decrement coefficients with different chain anisotropic parameter $r$ when ${\tau }_1={10}^{-2}\mathrm{s}, {\tau }_2=0.5\times {10}^{-4}\mathrm{s}, {\tau }_R={10}^{-5}\mathrm{s}$.

Figure 12

The variation of decrement coefficients of the first five eigenmodes with respect to natural frequencies for NE beams with differernt anisotropic parameter $r$.

Figure 13

(a) The real part and (b) the imaginary part of the first and fourth order modal functions of NE Timoshenko beam with hinged-hinged ends with ${\tau }_1={10}^{-2}\mathrm{s}, {\tau }_2={10}^{-4}\mathrm{s}, {\tau }_R={10}^{-6}\mathrm{s}$.

Figure 14

The effect of director angle on the real parts of (a) the first and (b) fourth order modal functions of NE Timoshenko beam with hinged-hinged ends with ${\tau }_1={10}^{-2}\mathrm{s}, {\tau }_2={10}^{-4}\mathrm{s}, {\tau }_R={10}^{-6}\mathrm{s}$; the color coding shows the corresponding values.