The design of the gear-and-shaft system shown requires that...

The design of the gear-and-shaft system shown requires that steel shafts of the same diameter be used for both $A B$ and $C D .$ It is further required that $\tau_{\max } \leq 60$ MPa and that the angle $\phi_{D}$ through which end $D$ of shaft $C D$ rotates not exceed $1.5^{\circ} .$ Knowing that $G=77.2 \mathrm{GPa},$ determine the required diameter of the shafts.

Key Concepts

Material Properties and Modulus of Rigidity

In the analysis of torsion, the material property known as the modulus of rigidity (or shear modulus) describes the material's ability to resist shear deformations. It relates the applied shear stress to the resulting strain and is a critical parameter in the calculation of the angle of twist, ensuring that the material’s inherent limitations are not exceeded in the design.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that influences its resistance to torsional deformation. For circular shafts, it is directly related to the diameter raised to the fourth power. This property plays a central role in both the determination of shear stress and the angle of twist under applied torques.

Angle of Twist

The angle of twist quantifies the shaft’s rotation under an applied torque over its length. It is dependent on the torque, the length of the shaft, the material’s modulus of rigidity, and the shaft's polar moment of inertia. This concept is important for ensuring that deformations remain within acceptable limits for proper operation and alignment of connected components.

Torsional Shear Stress

This concept involves the calculation of the shear stress that develops in a circular shaft when it is subjected to a torque. The maximum shear stress occurs at the outer surface of the shaft and is determined by the applied torque, the geometry of the shaft (specifically the diameter), and the distribution of material as captured by the polar moment of inertia. It is crucial for ensuring that the stress does not exceed the material’s allowable limits to prevent failure.

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