Wolfram Language™

Model a Hanging Chain

Find the position with minimal potential energy of a chain or cable of length suspended between two points. Set parameter values for the length of the chain , the left-end height , and the right-end height .

In:= `L = 4; a = 1; b = 3; `

Let be the height of the chain as a function of horizontal position, with .

In:= `xf = 1; nh = 201; h := xf/nh;`

Set up variables for the height of the chain .

In:= `varsy = Array[y, nh + 1, {0, nh}];`

Denote the slope at position by and set up variables for it.

In:= `varsm = Array[m, nh + 1, {0, nh}];`

Denote the partial potential energy from to by .

In:= `varsv = Array[v, nh + 1, {0, nh}];`

Denote the length of the chain at position by and set up variables for it.

In:= `varss = Array[s, nh + 1, {0, nh}];`

Join all variables.

In:= `vars = Join[varsm, varsy, varsv, varss];`

The objective is to minimize the total potential energy .

In:= `objfn = v[nh];`

Here are the boundary value constraints from the geometry.

In:= `bndcons = {y == a, y[nh] == b, v == 0, s == 0, s[nh] == L};`

Discretize the ODEs: , , .

In:= ```odecons = {Table[ y[j + 1] == y[j] + 0.5*h*(m[j] + m[j + 1]), {j, 0, nh - 1}], Table[v[j + 1] == v[j] + 0.5* h*(y[j]*Sqrt[1 + m[j]^2] + y[j + 1]*Sqrt[1 + m[j + 1]^2]), {j, 0, nh - 1}], Table[s[j + 1] == s[j] + 0.5*h*(Sqrt[1 + m[j]^2] + Sqrt[1 + m[j + 1]^2]), {j, 0, nh - 1}]};```

Choose initial points for the variables.

In:= ```tmin = If[b > a, 0.25 , 0.75]; init = Join[Table[4*Abs[b - a]*((k/nh) - tmin), {k, 0, nh}], Table[4*Abs[b - a]*(k/nh)*(0.5*(k/nh) - tmin) + a, {k, 0, nh}], Table[(4*Abs[b - a]*(k/nh)*(0.5*(k/nh) - tmin) + a)*4* Abs[b - a]*((k/nh) - tmin), {k, 0, nh}], Table[4*Abs[b - a]*((k/nh) - tmin), {k, 0, nh}]];```

Minimize the total potential energy, subject to the constraints.

In:= ```sol = FindMinimum[{objfn, Join[bndcons, odecons]}, Thread[{vars, init}]];```

Extract the solution points.

In:= `solpts = Table[{i h, y[i] /. sol[]}, {i, 0, nh}];`

Plot the position of the chain with minimal potential energy.

In:= `ListPlot[solpts, ImageSize -> Medium, PlotTheme -> "Marketing"]`
Out= Use FindFit to fit the result to the catenary curve.

In:= `catenary[t_] = c1 + (1/c2) Cosh[c2 (t - c3)];`
In:= `fitsol = FindFit[solpts, catenary[t], {c1, c2, c3}, {t}]`
Out= Plot the solution points together with the catenary curve.

In:= ```Show[Plot[catenary[t] /. fitsol, {t, 0, 1}, PlotStyle -> Directive[Green, Thickness[0.01]], ImageSize -> Medium], ListPlot[Take[solpts, 1 ;; nh ;; 5], PlotStyle -> PointSize[.02]]] ```
Out= 