Many rotating shafts and axles used for power transmission experience torsional vibration, particularly when the prime mover is a reciprocating engine. The shafts used in high‐speed machinery, especially those carrying heavy wheels, are subjected to dynamic torsional forces and vibration. A solid or hollow cylindrical rod of circular section undergoes torsional displacement or twisting such that each transverse section remains in its own plane when a torsional moment is applied. In this case the cross‐sections of the rod do not experience any motion parallel to the axis of the rod. However, if the cross‐section of the rod is not circular, the effect of a twist will be more involved. In this case the twist will be accompanied by a warping of normal cross‐sections. The torsional vibrations of uniform and nonuniform rods with circular cross‐section and rods with noncircular section are considered in this chapter. For noncircular sections, the equations of motion are derived using both the Saint‐Venant and the Timoshenko–Gere theories. The methods of determining the torsional rigidity of noncircular rods are presented using the Prandtl stress function and the Prandtl membrane analogy.
10.2 ELEMENTARY THEORY: EQUATION OF MOTION
10.2.1 Equilibrium Approach
Consider an element of a nonuniform circular shaft between two cross‐sections at x and , as shown in Fig. 10.1(a). Let denote the torque induced in the shaft at x and time ...