I am trying to create a model for simulating the dynamics of a rope by treating it like several connected pendulums. The goal is to solve the following system for $\vec\alpha$, which is a vector of all the angular accelerations $\alpha$ of all the rope segments. The system is:

$$C\vec\alpha = S\vec\omega - \frac{g}{L}\vec s$$

Where $\vec\alpha$ is the vector of unknowns, $\vec\omega$ is a vector of known angular velocities and $\frac{g}{L}$ is a constant. $C$ and $S$ are $m \times m$ matrices and $\vec s$ is a vector. They're defined as follows:

$$ C = \begin{pmatrix} 1& 0& 0& \cdots& 0\\ \cos(\theta_1-\theta_2)& 1& 0& \cdots& 0\\ \cos(\theta_1-\theta_3)& \cos(\theta_2-\theta_3)& 1& \cdots& 0\\ \vdots& \vdots& \vdots& \ddots& \vdots&\\ \cos(\theta_1-\theta_m)& \cos(\theta_2-\theta_m)& cos(\theta_3-\theta_m)& \cdots& 1 \end{pmatrix}$$

$S$ is the same as $C$, except every instance of $\cos$ is replaced with $\sin$.

$$ \vec s = \begin{pmatrix} \sin(\theta_1)\\ \sin(\theta_2)\\ \vdots\\ \sin(\theta_m)\\ \end{pmatrix}$$

Additionally, $\theta_n$ is a known angle for all integers $n$ in $[1,m]$.

The objective is to solve the given linear system for $\vec\alpha$. The obvious way to do so is to left-multiply by the inverse of $C$. Am I right in assuming this would yield the following?

$$ \vec \alpha = C^{-1}S\vec\omega-\frac{g}{L}C^{-1}S$$

I'm quite confident this is correct, but if it breaks some rule of matrix multiplication I've overlooked please tell me. Either way, I would like help with finding the inverse of $C$. Additionally, if possible, I would like help finding $C^{-1}S$ and $C^{-1}\vec s$. Any help is appreciated.


This is correct. However it would be much easier just doing substitution instead of using the inverse of a matrix.

Assume the right hand side vector is $(a_1, \dots, a_m)$. To do substitution, notice that the equations are:

$$\alpha_1=a_1\\ \cos (\theta_1-\theta_2)\alpha_1+\alpha_2=a_2\\ \cos (\theta_1-\theta_3)\alpha_1+\cos(\theta_2-\theta_3)\alpha_2+\alpha_3=a_3\\ \dots \dots$$

The first equation gives you the value of $\alpha_1$. Plugging this into the second equation gives you the value of $\alpha_2$. Plugging these two into the third equation gives you the value of $\alpha_3$. Proceeding like this, you can solve the whole system.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.