Torsional Vibrations of a Conic Shaft with Opposite Tapers Carrying Arbitrary Concentrated Elements

Figure 1

The coordinate system for a P-taper conic shaft with diameter 𝑑_ℓ at its larger end, diameter 𝑑_𝑠 at its smaller end, and length 𝐿_shaft=𝐿=𝐿_ℓ−𝐿_𝑠.