Solitons in one-dimensional mechanical linkage

It has been observed that certain classical chains admit topologically protected zero-energy modes that are localized on the boundaries. The static features of such localized modes are captured by linearized equations of motion, but the dynamical features are governed by its nonlinearity. We study quasiperiodic solutions of nonlinear equations of motion of one-dimensional classical chains. Such quasi-periodic solutions correspond to periodic trajectories in the configuration space of the discrete systems, which allows us to define solitons without relying on a continuum theory. Furthermore, we study the dynamics of solitons in inhomogeneous systems by connecting two chains with distinct parameter sets, where transmission or reflection of solitons occurs at the boundary of the two chains.

Figure 1

Schematic of the linkage system. ${\theta }_i$ is the angle associated with the $i\mathrm{th}$ rotor, $a_i$ is the separation between the $i\mathrm{th}$ and $(i+1)\mathrm{th}$ pivots, $r_i$ is the length of the $i\mathrm{th}$ arm, and $l_i$ is the bond length between the adjacent mass points at the $i\mathrm{th}$ and the $(i+1)\mathrm{th}$ sites.

Figure 2

A single four-bar linkage with the arm lengths given by $r$ and the separation of pivots given by $a$. The length of the bond (green) connecting two mass points (blue) is given by $l({\theta }_1,{\theta }_2)$.

Figure 3

The configuration spaces a four-bar linkage and its dependence on the parameters.

Figure 4

Two sublinkages with different parameter sets are connected to make a hybrid linkage system. The above figure illustrates the case in which the left (right) sublinkage, marked in blue (red), is in the connected (disconnected) phase. $r_{1\left(2\right)}$ and $l_{1\left(2\right)}$ are the arm length and equilibrium bond length for the left (right) sublinkage. ${\theta }_i$ is the angle of the rotor at the $i\mathrm{th}$ site measured with respect to the vertical line, and the $i\mathrm{th}$ bond length is denoted by $l_i$. $l$ is the length of the bond connecting the left (blue) and right (red) sublinkages.

Figure 5

In type I, we have combined a connected-nonrepulsive phase ($a_1=1.5$, $r_1=1$, and ${\overline{\theta }}_i=2$ for $i=1,\cdots ,10$) and a disconnected-repulsive phase ($a_2=0.5$, $r_2=1$, and ${\overline{\theta }}_i=2$ for $i=1,\cdots ,10$) with spacing $a=2$. In type II, we have combined two disconnected-nonrepulsive phases ($a_1=1$, $r_1=2$, and ${\overline{\theta }}_i=2$ for $i=1,\cdots ,10$) and ($a_2=0.5$, $r_2=1$, and ${\overline{\theta }}_i=2$ for $i=1,\cdots ,10$) with spacing $a=2$. In type III, we have combined a disconnected-nonrepulsive phase ($a_1=1$, $r_1=2$, ${\overline{\theta }}_i=2$ for $i=1,\cdots ,10$) and a connected-repulsive phase ($a_2=0.5$, $r_2=1$, ${\overline{\theta }}_i=0.2$ for $i=1,\cdots ,10$) with spacing $a=2$. In all the cases, we took the mass and spring constant as $M=0.01$ and $K=10$, respectively. The graphs on the top show the profile of the initial angular velocities, $\dot{\theta }\left(0\right)$ vs site locations, given by Eq. (12) with the initial angles ${\overline{\theta }}_i$ for $i=1,\cdots ,20$ shown above for each case. The figures of the linkages show the time evolution of a soliton in each case.