Equilibrium of system of piecewise linear springs

This problem describes a mechanical system to find a hanging chain consisting of $N$ links. The first and last nodes are fixed. Each node has a weight $m_j$ hanging from it. The problem is to minimize the energy of the system (see [5,15]):

$$\begin{array}{rcl} \mbox{min}& \sum_j m_j g y_j+\frac{k}{2}\V... ...,a_2),& \\ & (x_N,y_N)=(b_1,b_2),& \\ & t\geq 0,& \end{array}$$ (3-10)

where $g$ is the acceleration due to gravity, $k$ is the stiffness constant of the springs, $l_0$ is the rest length of each spring, and $t_j$ is an upper bound on the spring energy of the $j$-th spring. Consider $\Vert t\Vert^2\leq 2uv$, $u=1$, $v\geq 0$, $s_j=t_j+l_0,$, $e_j=x_j-x_{j-1}$, $f_j=y_j-y_{j-1}$, $j=1:N$. Then $\sum_j f_j=b_2-a_2$ and $\sum_j e_j=b_1-a_1$. The objective function becomes $\sum_j m_j g y_j+kv.$ Furthermore,

$$\begin{array}{rcl} \sum_j m_j g y_j+kv&=&g\sum_jm_j(\sum_{i=... ...um_{i=1}^N(\sum_{j=i}^Nm_j)f_j+g\sum_{j=1}^Na_2+kv. \end{array}$$

The linear and cone constraints are

$$\begin{array}{rcl} \sum f_j=b_2-a_2, &\\ \sum e_j=b_1-a_1,... ...j^2,&j=1:N,\\ \Vert t\Vert^2\leq 2uv,& u,v\geq 0. \end{array}$$

Example 6. The data is the same as in springs.mod in [16]. We have $k=100,$ $l_o=2\sqrt{(b_1-a_1)^2+(b_2-a_2)^2}/N,$ $g=9.8,$ $m=1,$ $(a_1,a_2)=(0,0),$ and $(b_1,b_2)=(2,-1)$. We use $N=10,20,40,60$ as test cases.