One of two hollow drive shafts of a cruise ship is 125 ft...

One of two hollow drive shafts of a cruise ship is 125 ft long, and its outer and inner diameters are 16 in. and 8 in., respectively. The shaft is made of a steel for which $\tau_{\text {all }}=8500$ psi and $G=11.2 \times$ $10^{6}$ psi. Knowing that the maximum speed of rotation of the shaft is $165 \mathrm{rpm}$, determine ( $a$ ) the maximum power that can be transmitted by the one shaft to its propeller, (b) the corresponding angle of twist of the shaft.

Key Concepts

Unit Conversion and Consistency

Unit conversion is essential in engineering calculations to ensure consistency across all physical quantities. It involves converting measurements into a coherent system of units—such as converting inches to feet or rpm to radians per second—so that formulas and calculations yield correct results. Maintaining consistent units is vital for accurately applying engineering principles, especially in complex problems involving multiple physical dimensions.

Power Transmission in Rotational Systems

Power transmission in rotating shafts is determined by both the torque being transmitted and the angular velocity. The relationship P = T*?, where P is power, T is torque, and ? is angular velocity, connects the mechanical forces in the system with the rate of energy transfer. This concept is critical for ensuring that a shaft is capable of delivering the necessary power without failure.

Torsion in Circular Shafts

This concept involves understanding how torques applied to a circular shaft produce shear stresses. A key aspect is the relation ? = T*c / J, where ? is the shear stress, T is the applied torque, c is the outer radius of the shaft, and J is the polar moment of inertia. This relationship is essential for determining whether the shaft will operate within its material limits under torsional loading.

Polar Moment of Inertia for Hollow Circular Sections

The polar moment of inertia (J) represents the shaft’s resistance to twisting. For hollow circular shafts, J is calculated using the difference between the moments of inertia of the outer and inner circles. Accurately determining J is crucial because it directly affects the shear stress distribution and the angle of twist when a torque is applied.

Angle of Twist

The angle of twist is the measure of the angular deformation of a shaft due to an applied torque. It is calculated using the formula ? = T*L/(J*G), where T is the torque, L is the length of the shaft, J is the polar moment of inertia, and G is the modulus of rigidity. This concept is key in assessing how much the shaft will rotate or distort under load, which can impact the alignment and performance of connected components.

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