19.5
扭转角度 - 解决问题
当电动机向铝轴施加 700 N·m 的扭矩，便会发生稳定的旋转。两个滑轮 B 和 C 分别承受了 300 N·m 和 400 N·m 的扭矩。其中的刚性模量为 25 GPa。通过了解每个滑轮的长度和直径后，便可以计算出两个滑轮之间的扭转角。首先，在滑轮 B 和 C 之间的截面进行切割，并使用受力分析…
电动机对铝制轴施加 700 N·m 的扭矩，使其以恒定速度旋转。皮带轮 B 和 C 所受的扭矩分别为 300 N·m 和 400 N·m。材料的剪切模量为 25 GPa。若已知各段轴的长度和直径，计算皮带轮 B 与 C 之间的扭转角。
首先，在滑轮 B 和 C 之间进行切割，并分析该切割截面的受力图。
作用在皮带轮B上的力矩为逆时针方向。因此，根据平衡原理，轴在截面处的力矩将与皮带轮B处的力矩大小相等、方向相反。
接下来，计算截面处的极惯性矩，该值与轴半径的四次方成正比。
代入所有已知参数后，即可确定滑轮 B 与 C 之间的扭转角。所得角度单位为弧度，可转换为度。
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Q1: How do you calculate the angle of twist between two pulleys on a shaft?
To calculate the angle of twist between pulleys, make a cut between them and analyze the cut cross-section using a free-body diagram. Apply equilibrium principles to find the internal torque, then calculate the polar moment of inertia, which is proportional to the fourth power of the shaft radius. Substitute all known parameters—torque, shaft length, diameter, and modulus of rigidity—into the angle of twist formula to determine the result in radians, then convert to degrees if needed.
Q2: What role does the polar moment of inertia play in torsion problems?
The polar moment of inertia is a geometric property that quantifies a shaft's resistance to twisting. It is proportional to the fourth power of the shaft radius, meaning small increases in diameter significantly increase resistance to torsion. This parameter is essential in the angle of twist equation; larger polar moments result in smaller twist angles for the same applied torque and material properties.
Q3: Why is the modulus of rigidity important when solving torsion problems?
The modulus of rigidity, also called the shear modulus, measures a material's resistance to shear deformation under torsional loading. It directly appears in the angle of twist formula; materials with higher modulus values resist twisting better and produce smaller twist angles. For the aluminum shaft in this problem, the modulus of rigidity of 25 GPa determines how much the shaft twists under the applied torques.
Q4: How does the equilibrium principle apply to finding internal torque at a cut section?
When a cut is made between two pulleys, the equilibrium principle states that internal torques must balance external torques. If pulley B exerts an anticlockwise torque, the internal torque at the cut cross-section must be equal in magnitude but opposite in direction to maintain equilibrium. This equilibrium torque is then used in the angle of twist calculation for that shaft section.
Q5: What is the relationship between shaft diameter and resistance to twisting?
Shaft diameter dramatically affects resistance to twisting because the polar moment of inertia depends on the fourth power of the radius. Doubling the diameter increases the polar moment by a factor of 16, substantially reducing the angle of twist for the same applied torque. This relationship makes diameter a critical design parameter in transmission shaft applications.
Q6: Why must the angle of twist result be converted from radians to degrees?
The angle of twist formula naturally produces results in radians because it uses standard engineering equations based on radian measure. Converting to degrees makes the result more intuitive for practical interpretation and communication. Both units represent the same physical rotation; degrees simply provide a more familiar scale for engineers and technicians reviewing the calculations.
Q7: How do multiple pulleys affect torque distribution along a shaft?
Multiple pulleys create different torque values in different shaft sections. In this problem, the motor applies 700 N-m, but pulleys B and C extract 300 N-m and 400 N-m respectively. Each section between pulleys experiences different internal torques, requiring separate free-body diagram analysis and angle of twist calculations for each segment to fully characterize the shaft's deformation.