The design of a machine element calls for a 40...

The design of a machine element calls for a 40 -mm-outer-diameter shaft to transmit $45 \mathrm{kW}$. (a) If the speed of rotation is $720 \mathrm{rpm}$ determine the maximum shearing stress in shaft $a .(b)$ If the speed of rotation can be increased $50 \%$ to $1080 \mathrm{rpm}$, determine the largest inner diameter of shaft $b$ for which the maximum shearing stress will be the same in each shaft.

Key Concepts

Polar Moment of Inertia for Circular Sections

The polar moment of inertia is a geometric property that measures a cross-section's resistance to torsional deformation. For circular shafts, it is computed differently for solid and hollow sections. In the case of a hollow circular shaft, the polar moment of inertia is determined by subtracting the inner diameter's contribution from the outer diameter's contribution. This factor is essential in calculating the material's capacity to withstand torque without excessive twisting.

Torsional Shear Stress

This concept involves the stress experienced by a material when subjected to a twisting (torsional) load. The maximum shear stress in a circular shaft is determined using the formula ? = Tc/J, where T is the applied torque, c is the outer radius of the shaft, and J is the polar moment of inertia of the shaft’s cross-sectional area. This relationship is central when analyzing component failure under torsion and ensuring that the material’s strength is not exceeded.

Power and Torque Relationship

In rotating machinery, power is transmitted by applying a torque at a certain angular velocity. The relationship is given by P = T?, where P is the power transmitted, T is the torque, and ? is the angular speed in radians per second. This principle allows engineers to calculate the necessary torque using the given power and rotational speed, which in turn is used for stress analysis in shafts.

Design Constraints and Comparative Shaft Analysis

This concept involves designing different shaft geometries to meet a specific performance criterion, such as maintaining the same maximum shear stress under different operating conditions (e.g., varying rotational speeds). The design often requires adjusting dimensions, like modifying the inner diameter, to balance the torsional capacity between shafts while transmitting the same power without exceeding allowable stress limits.

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