A torque T of 675 N - m is applied to a shaft as shown in...

A torque $T$ of 675 N - m is applied to a shaft as shown in Fig. P.14.53. The shaft is attached to two rubber grommets, which in turn are attached to fixed disks. Each gives a linear, elastic torsional spring constant $K_T=550 \mathrm{~N}$-m/deg. If $G=1$ $\times 10^{11} \mathrm{~Pa}$, find the twist of end $A$ and the relative twist between support 1 and support 2 . Figure P.14.53.

Key Concepts

Relative Angular Displacement (Superposition of Twists)

When a system consists of multiple elements (like a shaft and attached torsional springs), the overall angular displacement of a point can be determined by combining the individual twists from each component. The net twist at an end, or the relative twist between supports, is found by superposing the contributions from the shaft and the elastic supports. This principle is essential in analyzing mechanisms where different segments deform independently under the same load, and then their effects are summed to yield the total deformation.

Shear Modulus and Polar Moment of Inertia

The shear modulus (G) is a material property that measures its rigidity (how resistant it is to shear deformation). In torsion analysis, it is used along with the polar moment of inertia (J) of the cross-section of a shaft to determine the torsional rigidity (GJ). This rigidity directly influences how much a shaft twists under an applied torque, as described by the formula ? = TL/(GJ), where ? is the angle of twist, T is the applied torque, and L is the length of the shaft.

Torque in Torsion

This concept involves the twisting action produced by a force acting at a distance from the axis of rotation. In torsional analysis, torque (or moment) is applied to a shaft which then experiences a twist. The relationship between the applied torque, the material’s ability to resist twist (i.e., its torsional rigidity), and the resulting angular deflection is fundamental in analyzing shafts and structural members under twisting loads.

Torsional Stiffness (Torsional Spring Constant)

Torsional stiffness refers to the resistance offered by an element to rotation about its axis. It is defined as the ratio of the applied torque to the resulting angular displacement, often expressed in units like N-m per degree or N-m per radian. In problems where supports (or other features) can flex, they are sometimes modeled as torsional springs. This simplification allows one to analyze the effect of their elastic behavior in the overall torsional response of the system.

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