Missing deflection of Simscape Flexible Beam with two fixed supports

I started to investigate the behavior of Flexible Bodies in Simscape. My goal is to model a flexible motor shaft, which is supported at both ends, with only one rotational degree of freedom around the shaft axis (Revolute Joint at each end of the shaft).

As a simplified model I skipped the rotation and used a bending beam with a weld joint at each end of the beam (red beam / beam B). This model is attached as a .slx file.The analytical solution of this beding case can bee seen in the two attached .png files. However, the numerical solution does not result in any deflection at all. The beam just behaves as it would be a rigid brick solid.

In contrast, the tyical support case of only one weld joint (blue beam / beam A) results in a deflection and approximates the analytical solution.

Is there a mistake in the support of beam B or am I missing some important modeling aspects?

Answer 1

[Arnav]

Hi @Jonas,

The issue here is that a beam that is fixed from both sides is statically indeterminate i.e. it is not possible to statically determine all forces and moments acting on the body. In this case, we cannot deduce the reaction moments exerted by the fixed ends statically. These moments can be deduced analytically by considering the bent beam to be in equilibrium with the forces and unknown moments.

A workaround would be to use a pin joint and a roller joint, and apply the fixed end moments externally. For this case, the fixed end moments is PL/8:

Image Source: https://en.wikipedia.org/wiki/Fixed_end_moment

Please see the attached model to understand how to do the above in Simscape.

The beam deflections of midpoints measured for the attached model are:

We can also see the shape of the beams in the Mechanics Explorer as:

You can see more about simulating a flexible shaft here:

https://www.mathworks.com/help/releases/R2023b/sdl/ug/flexible-shaft.html

You can learn more about simulating flexible bodies in general here:

https://www.mathworks.com/matlabcentral/answers/1570248-beam-deflection-in-simulink?s_tid=answers_rc1-2_p2_MLT