How does Multi Body Dynamics software work for flexible joints?

Question

I need to model a "fishing rod" in 2 dimensions by joining several "rigid sticks" by flexible/elastic joints. The joints act as plate/torsion springs with different spring constants. The "fishing rod" will bend at these joints. The segments of the rod become lighter and the springs softer, towards the tip.

The only input data are the length and weight of each piece of rod and the respective spring constant by which it is attached to the previous one.

I wish to see how this object bends and moves when one end (the handle) is exposed to torque (i.e. how the fishing rod bends when you throw the bait. - There is no need to include torque around the object's own axis.)

(This is not a problem involving stress analysis by FEM. It is about movement in a time frame of seconds. I will later try to extend this model into biomechanics.)

I can follow (and have simulated) the Lagrange transform for the double pendulum, but for more than two segments, the Lagrange transform becomes "virtually impossible" to compute.

Multi body dynamics software can model multi linked objects (and many more things). See this video for a N-link pendulum simulation. How can the movements of such a complex object be computed!?

If I could just get a grip on how the "physics engine" for multiple, elastic/flexible joints work, in this kind of software, I could maybe use this to model the fishing rod.

Answer 1

As @BillGreene mentioned, this would be better modeled as a continuous elastic body. It would be something like a cantilever beam subject to its own weight and a force at the tip. You can use, Euler-Bernoulli beams for this, where the equations are as follows:

$$ \cfrac{\partial^2 }{\partial x^2}\left(EI\cfrac{\partial^2 w}{\partial x^2}\right) = - \mu\cfrac{\partial^2 w}{\partial t^2} + q(x)$$

Then, you can discretize your domain and obtain a time-integration scheme, you can use Finite Differences or Finite Elements for this. A common one is Verlet integration,

$$\vec x_{n+1}=2\vec x_n-\vec x_{n-1}+\vec a_n\,\Delta t^2$$

where you express your time stepping as a function of the past two time steps and the force in your nodes. You can see an implementation here.

On the other hand, I think that your idea might work. It will not be a correct representation of the physics of the problem, but I think that it might make sense. I was checking a couple of physics simulations software like Physion, and they do not have rotational springs. So, you will need to implement that part.

Regarding on how you do this, you can formulate the Lagrangian and then obtain the equations of motion. An example for the n-links pendulums is here. Where the author used Python and SymPy as part of the task.After this, you will, probably, find a pattern to assemble your stiffness and mass matrices and then you can use as many links as you want.

In all cases, you will need a way to compute the "forces" between the nodes in your system and a time-integration technique, that is commonly explicit (and symplectic).