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For an electrical design, i am having a cable modeled as a perfect elastic static rod, fixed at both ends, as i have depicted how should it be for different given reference lengths.

I've studied some introductory chapters about solid mechanics, but at this moment i am confused on how should i model this in order to solve this computationally in i.e. matlab. for fixed lengths of cable.

I've wrote some large matlab routines (hard and i think rather useless to paste in here btw due to its failed approach) for dividing the cable into n pieces, each representing both an linear and angular spring, but at this moment they failed to converge.

I am trying to calculate the forces at each node both from linear (stretching) and angular (bending) (no twisting) springs as finite pieces (elements?), but for most cases (n bigger than 5) the solution explodes into infinity (?).

Anybody could present me a reference for implementing a numerical solution, with given initial conditions and reference length?

enter image description here

Here is a solution with n=5 for forces depicted in red. This case for some reason (my lack of knowledge) diverges. Most other solutions also diverges. The iteration is done simply by adapting the solution by a 1/10th of the (loosely calculated) force.

enter image description here

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    $\begingroup$ A quick question on terminology: a cable, by definition, can't sustain a bending moment; it simply hangs under its own weight in the shape of a catenary. A beam, in contrast, can sustain a bending moment. So are you modeling a cable or a beam? What constitutive equations are you using to describe the deflection? Please also add a free-body diagram of the system; this will be much more useful than images of the failed attempts. $\endgroup$
    – Chemomechanics
    Commented Mar 27, 2018 at 21:28
  • $\begingroup$ I mean an electrical cable, in particular a coaxial cable, you are right this is not a mechanical cable. Hence this is actually a rod, which can sustain bending, stretching (and twisting, not in this case). I am reviewing this link at this moment trying to figure how to do the calculations. I am just using a constant spring as constitutive for both stretching and bending. $\endgroup$
    – Brethlosze
    Commented Mar 27, 2018 at 21:56

3 Answers 3

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A partial answer concerning modeling:

You say you want a model "for given lengths of cable." And you also mention in a comment you are modeling coaxial cable. It sounds like you can neglect stretching for your model, and if the cable doesn't stretch, you don't need to include linear springs in the model as you did. Rigid rod elements joined by your "angular" springs should work as a model. More elements will give a better approximation.

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  • $\begingroup$ Yeah thats right, stretching part is about zero, so i am including it instead as a mathematical constraint?. I am not clear on how to include that given fixed length on this numerical computation. $\endgroup$
    – Brethlosze
    Commented Mar 28, 2018 at 2:11
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    $\begingroup$ Okay, if you want to include it, for a linear spring, stretching decreases as the spring constant increases, so you could set the spring constant very high. $\endgroup$
    – Tom B.
    Commented Mar 28, 2018 at 2:23
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    $\begingroup$ Also, I assume there are two nodes, one at each end, where the cable is fixed, right? $\endgroup$
    – Tom B.
    Commented Mar 28, 2018 at 2:41
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A typical way to approach this problem would be to use either the gradient descent algorithm or perhaps the conjugate gradient algorithm, using the system's total potential energy as the function to be minimized. Either of those two algorithms should converge to the solution nicely for this problem, if coded correctly. The Wikipedia articles I linked to above even have MATLAB code examples.

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  • $\begingroup$ I am now calculating the energy and the gradiend... i hope to be able to do it .... $\endgroup$
    – Brethlosze
    Commented Mar 27, 2018 at 22:00
  • $\begingroup$ Because i am starting with a linear compressed cable as initial condition, the energy will increase by moving a node along the tangent to the rod, and the energy will reduce by moving perpendicular to the rod. Hence i am starting the computation with a horseshoe saddle point... I never expected this to happen in this kind of problem, but this explains why the convergence is so odd. How should i suppose to compute the gradient then? The partial derivative approximation is giving me bad results. $\endgroup$
    – Brethlosze
    Commented Mar 28, 2018 at 15:53
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The final results are shown in here. The graphics shows possible cables from $(0,0)$ with $\theta=0$ to $(1,1)$ with $\theta=0$, with increasing lengths from $\sqrt2$ to $20$, by intervals of $0.5$.

enter image description here

The problem was formalized as: $$ \mathscr P) \ \ \min \sum_{i=1}^n \theta_i +\lambda \sum_{i=1}^n (l_i-l_0)^2 \\ l_i^2=x_i^2+y_i^2 \\ \theta_i=\mod(\text{arg}(x_{i+1},y_{i+1})-\text{arg}(x_{i},y_{i})-\pi,2\pi)+\pi\\ (x_1,y_1)=(0,0),\theta_1=0 \\ (x_n,y_n)=(1,1),\theta_n=0 \\ $$

The optimization was solved through Matlab fmincon, with initial solution $(x_i,y_i)$ the straight line from $(x_1,y_1)$ to $(x_n,y_n)$, with $\lambda=1$, and using only nonlinear equality constraints.

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