This is something kind of explained with propellers and turbine blades but I'd love some good explanations. Thank you!
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it depends on what is causing the angular acceleration. In the case of a propeller, the torque of the shaft would be distributed as shear stress in a cantilever beam, however not linear. The torque distribution in a propeller blade can be modeled using blade theory. Blade theory divides the blade into small elements and integrates the forces, and stresses, and deformations over the length of the blade. depending on the pitch, cord, distance from the center, etc. It is usually done by an FEM program.
If the angular acceleration of the bar is by an external force in the most basic case ignoring air resistance the bar can be modeled as a beam loaded with,
$$d w= d(r^2d m)$$
- w= load
- r= distance from the shaft
- m=mass of element
as load and concentrate force accounted for as the $$T= F*r$$
- T= torque
- F= force
- r= distance from center of rotation
For momentary stresses and deflections, all the inertia and static components should be added as static load. Usually, a Lagrangian integration by parts can be utilized as a rough first estimate. Mathematically it can be similar to a foundation beam loaded with $ w(x)$ on a flexible soile.
