$a_0$	first damping coefficient of the mass block
$a_1$	second damping coefficient of the mass block
$a_m$	accelerated velocity of the translational frame
$A$	initial amplitude of the free vibration of the mass block
$C$	uniform velocity of the translational frame
$F_{\mathrm{d}}\left({\dot{w}}_2\right)$	damping force of the mass block
$F_{\mathrm{L}}\left(t\right)$	tensional force of the translating LCE fiber
$g$	gravitational acceleration
$h$	heat exchange coefficient between the LCE fiber and the environment
$K$	spring constant of the LCE fiber
$L$	original length of the LCE fiber
$m$	mass of the mass block
$R$	radius of the thin LCE fiber
$t$	time
$T\left(X,t\right)$	temperature field of the LCE fiber
$T_{\mathrm{ext}}\left(x\right)$	environment temperature field
$T_{\mathrm{L}}$	environmental temperature at $x=L$
$T_{\mathrm{r}}$	reference temperature in reference state
$u\left(X,t\right)$	instantaneous displacement of a material point
$w_1\left(t\right)$	displacement of the translating frame
${\dot{w}}_1\left(t\right)$	velocity of the translating frame
$w_2\left(t\right)$	displacement of the free end of the LCE fiber
${\dot{w}}_2\left(t\right)$	velocity of the free end of the LCE fiber
$X$	Lagrangian coordinate of the LCE fiber
$x$	Eulerian coordinate
$x\left(X,t\right)$	instantaneous position of a material point
$\alpha$	thermal expansion coefficient of the LCE fiber
$\beta$	temperature gradient of the external temperature field
$\delta$	damping coefficient of the free vibration of the mass block
$\epsilon \left(X,t\right)$	one-dimensional strain of the LCE fiber
${\epsilon }_{\mathrm{T}}\left(X,t\right)$	thermally induced strain of the LCE fiber
$\theta$	initial phase angle of the free vibration
${\rho }_{\mathrm{c}}$	heat capacity of the LCE fiber per unit length
$\tau$	heat transfer characteristic time
${\omega }_0$	natural angular frequency of the free vibration