A steel shaft must transmit 150 kW at a speed of 360 rpm . Knowing that G=77.2 GPa, design a sol

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A steel shaft must transmit 150 kW at a speed of 360 rpm ....

A steel shaft must transmit $150 \mathrm{kW}$ at a speed of $360 \mathrm{rpm} .$ Knowing that $G=77.2 \mathrm{GPa}$, design a solid shaft so that the maximum shearing stress will not exceed $50 \mathrm{MPa}$ and the angle of twist in a $2.5-\mathrm{m}$ length must not exceed $3^{\circ}$.

Key Concepts

Angle of Twist Constraint

The angle of twist is the measure of the rotational displacement along the length of a shaft subject to a torque. Design constraints often require that the twist remain within certain limits to avoid misalignment or excessive deformation at the component level. This concept involves using the relationship between applied torque, shaft length, shear modulus, and polar moment of inertia to ensure acceptable performance.

Material Properties

Materials such as steel have specific properties like the shear modulus that define their response to applied loads. The shear modulus, in particular, is crucial in calculating both shear stress and angle of twist when a material is subjected to torsion. It represents the stiffness of the material in shear and informs how the material will perform under operational conditions.

Shear Stress Constraint

This concept involves ensuring that the induced shear stress due to the applied torque does not exceed the maximum allowable limit for the material. Controlling shear stress is crucial in preventing material failure, and it is directly calculated from the applied torque and the polar moment of inertia.

Power and Torque Relationship

This concept relates to how mechanical power, rotational speed, and torque are interconnected. In rotating systems, the power transmitted by the shaft is directly related to the torque applied and the angular velocity. Accurately calculating torque from a given power output is essential for shaft design because it governs subsequent design requirements such as stress and deflection.

Torsion in Circular Shafts

Torsion theory describes how shafts behave under applied torques. When a torque is applied to a circular shaft, it produces a distribution of shear stresses and results in an angular twist along the length of the shaft. Understanding torsion is critical when designing shafts to ensure they do not experience excessive deformation or stress that could lead to failure.

Polar Moment of Inertia

The polar moment of inertia is a geometric property that quantifies a cross-section's resistance to torsional deformation. For circular shafts, the polar moment of inertia depends on the diameter and is a key component in calculating shear stress and angle of twist. It provides a measure of how a cross-section's area is distributed with respect to an axis.

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