Improve mass matrix for better accuracy of dynamic simulations

[stfnp]

The results of comparing VirtualBow's beam elements with reference solutions (GXBeam) for static/dynamic example problems are already pretty close, often almost identical. Definitely good enough for our purposes.

There is however one systematic discrepancy in the dynamic results that could still be improved. The plots below show some bending shapes of a cantilever beam with sinusoidal forces applied to the beam tip. On the left, the beam axis passes through the cross-section center, while on the right the cross-sections are offset from the beam axis:

While the shapes in the centered case are nearly identical with the reference solution, there is a small but visible error present in the off-center case. Comparisons of static results do not show such a difference between centered and off-center, so the elastic forces are probably not the reason. I suspect that the error is due to the very simple lumped mass matrix that VirtualBow currently uses. It doesn't seem to account for the cross-sections's center of mass offset particularly well.

The big question is then if a lumped mass matrix can even capture these effects. While it can (and already attempts to) account for the higher moment of inertia of the off-center cross-sections, it definitely can't account for the introduced coupling between rotational and translational degrees of freedom, since a lumped mass matrix is by definition diagonal and doesn't contain any coupling terms.

So here are some of the possibilities:

• Improve the lumped mass matrix, which might not be possible as mentioned above

• The next step up from a lumped mass matrix would be a non-diagonal but still constant mass matrix. The mass matrix of a Timoshenko beam element with linear shape functions is apparently a popular choice for corotational beam elements (example), maybe that would perform better. Also look into the constant mass matrix option in GXBeam and how that works.

• Using a consistent mass matrix derived from the exact kinematics of the beam elements is out of the question, since this would be a significant complication for only a slight increase in accuracy that is probably not even noticeable in a bow simulation.