Name: sizing a transmission shaft by torque
Uploaded: 2024-05-23T12:59:41.000Z
Duration: 1 min 16 s
Channel: JoVE

传动轴的设计规范要求包括轴所传递的功率及其旋转速度。
根据这些参数，通过确保轴在指定转速下传递所需功率时，材料所允许的最大剪切应力仍处于弹性极限范围内，来选择轴的材料和横截面尺寸。
现在，系统的功率取决于所施加的扭矩。通过重新整理各项，可计算出作用在轴上的扭矩。该公式同时考虑了轴的功率需求和旋转频率。
将计算得到的扭矩和最大允许应力代入弹性扭转公式，以确定轴半径的最小允许值。
对于实心圆轴，极惯性矩与轴半径的比值随轴半径的三次方变化。代入后可得到轴半径的最小所需值。
对于空心圆柱体，需计算轴的外半径允许的最小值。

传动轴设计

Design of Transmission Shafts

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传动轴的设计主要是由两个参数来进行决定的：传输的功率和转速。这些参数能够指导轴的材料和横截面尺寸的选择，以此来确保材料的最大剪切应力能够保持在弹性极限内，同时需要以给定的传递速度来配置所需的功率。系统的功率在本质上与施加到轴上的扭矩是相关的。施加到轴上的扭矩可以通过对各项进行重新配置来计算得出。
 
<img alt="Equation 1" src="/files/ftp_upload/15617/15617_Equation_1.svg" style="text-align: center; height: 38px;">
该计算过程同时考虑了功率的要求和轴的转速。一旦能够计算出扭矩和最大的许用应力，它们就会被纳入到弹性扭转公式中。该过程能够得出轴半径所允许的最小值。在含有实心圆轴的情况下，极惯性矩与轴半径之比会随着轴半径的立方而发生变化。通过代入该值来计算出轴半径所需的最小值。

<img alt="Equation 2" src="/files/ftp_upload/15617/15617_Equation_2.svg" style="text-align: center; height: 62px;">
对于空心圆柱轴来说，需要计算出轴外半径所允许的最小值。因此，传动轴的设计过程是在功率、速度、应力极限和物理尺寸之间经过仔细衡量后所得出的平衡结果。

<img alt="Equation 3" src="/files/ftp_upload/15617/15617_Equation_3.svg" style="text-align: center; height: 62px;">
</div>

传动轴的设计规格要求包括轴所传递的功率及其旋转速度。  
根据这些规格，……

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Torsion

扭转

KEY TERMS AND CONCEPTS

关键术语与概念

sizing a transmission shaft by torque

A transmission shaft is sized from the power it must carry and the speed it rotates at. These two design limits guide the choice of shaft material and cross-sectional size. The goal is to keep the maximum shearing stress within the elastic limit while still transmitting the needed power.

Power is directly linked to torque, which is the twisting force applied to the shaft. By rearranging the power relation, the torque can be found from the required power and the shaft speed. That torque value is then used with the maximum allowable stress in the elastic torsion formula.

The torsion calculation gives the minimum allowed shaft radius. For a solid circular shaft, the polar moment of inertia divided by the radius changes with the cube of the radius, so this value is used to find the minimum radius. For a hollow cylindrical shaft, the same process gives the minimum outer radius. In this way, shaft design balances power, speed, stress limits, and physical size.

Sizing a Transmission Shaft by Torque

stresses in a shaft

Torque and Shear in a Twisted Shaft

A shaft under torque develops internal shear forces as it twists. In this setup, equal and opposite torques act on the shaft PQ. To study the effect, a section is cut perpendicular to the shaft axis at an arbitrary point R. The free-body diagram of the QR segment shows the shearing forces applied by the PR portion. These forces act as the shaft resists twisting. Using equilibrium on the QR segment links the internal shear forces in the cut section to the internal torque. Here, r is the...

deformation in a circular shaft

Shear Strain in a Circular Shaft

Shear strain in a circular shaft describes how the shaft’s surface changes shape under torsion. A circular shaft keeps each cross-section flat and undistorted as it twists. Each section behaves like a rigid slab that rotates in place. To study the strain inside the shaft, consider a cylindrical section with length L and radius R. One end of this section is fixed. The radius at a point inside the section is labeled r. Before the load is applied, a small square element lies on the surface of...

circular shaft - stresses in linear range

Torsion Formula for Circular Shafts

Torsion in a circular shaft is analyzed when the applied torque stays within Hooke's law. In this linear range, the shaft does not undergo permanent deformation. The discussion uses shearing strain, shearing stress, and the modulus of rigidity to build the stress equation. The derivation also requires the moments of the small forces on a cross-section to equal the applied torque. When the shearing stress expression is rewritten, an integral term appears. This term represents the polar moment...

angle of twist - elastic range

Elastic Twist in a Circular Shaft

Elastic twist in a circular shaft describes how a cylindrical shaft rotates when torque is applied. The shaft has a length L and a uniform radius r. A torque at the free end creates shearing strain, which is greatest at points farther from the shaft axis. When the shaft stays within the elastic range, the shearing strain can be written in terms of the applied torque, radial distance, polar moment of inertia, and modulus of rigidity. The polar moment of inertia is a geometric measure of how the...

angle of twist: problem solving

Shaft Torsion Between Pulleys B and C

Shaft torsion between pulleys B and C is found by analyzing how torque twists an aluminum shaft. An electric motor applies 700 N·m of torque and produces stable rotation. The shaft carries pulleys B and C, which experience torques of 300 N·m and 400 N·m. The calculation starts with a section cut between pulleys B and C. A free-body diagram is used to study the cut cross-section. Because pulley B has an anticlockwise torque, the torque at the cut must balance it in the opposite direction. The...

design of transmission shafts

A transmission shaft is sized from the power it must carry and the speed it rotates at. These two design limits guide the choice of shaft material and cross-sectional size. The goal is to keep the maximum shearing stress within the elastic limit while still transmitting the needed power. Power is directly linked to torque, which is the twisting force applied to the shaft. By rearranging the power relation, the torque can be found from the required power and the shaft speed. That torque value...

stress concentrations in circular shafts

Why Shafts Develop Local Stress Peaks

Circular shafts can develop local stress peaks when torque is applied through real machine parts. The elastic torsion formula works best for a circular shaft with a uniform cross-section and rigid plates fixed to its ends. In practice, torque is often passed through flange couplings or gears, and these parts use keys in keyways to connect to the shaft. That loading method changes the stress pattern near the point where torque enters the shaft. The stress distribution no longer matches the...

plastic deformation in circular shafts

Torsional Failure and Plastic Strain in Shafts

Plastic deformation in circular shafts happens when torsional force pushes the material past its yield strength. At that point, the shaft does not return to its original shape. Instead, it keeps a permanent strain. To measure this change, the stress pattern inside the shaft must be understood. The maximum shearing stress is calculated first. Then a shearing-stress-strain diagram can be drawn to find the maximum shearing strain. Shearing strain changes linearly with the distance from the shaft...

circular shafts - elastoplastic materials

Torsion Limits in Circular Shafts

Circular shafts under torsion show a clear change from elastic behavior to plastic deformation as torque increases. In the elastic limit, stress rises in direct proportion to the distance from the shaft’s center. The outer material carries the highest stress first, and the shaft can still return to its original shape. At a critical stress level, the shaft begins to yield. This marks the maximum torque the shaft can carry without permanent deformation. Beyond this point, a plastic region forms...

residual stresses in circular shafts

Residual Stress After Shaft Torsion

Residual stress can remain in a circular shaft after torque causes plastic deformation. This happens in elastoplastic materials, which show both elastic and plastic behavior. The plastic change can come from high shearing stress or large strain. Residual stress is the internal stress that stays in the material after the external force is removed. In a shaft under torque, the angle of twist does not return fully to its original value. That lasting change shows that residual stress is still...

torsion of noncircular members

Torsion in Square and Rectangular Bars

Torsion in square and rectangular bars shows how shape changes the way a material twists. Circular shafts keep their cross-sections nearly unchanged because they are axis-symmetric. That symmetry helps the stress spread evenly, so the shaft can resist torsion without distorting. Square bars behave differently because they do not have the same axial symmetry. When twisted, their cross-sections distort, although the diagonals and the lines connecting the midpoints stay as exceptions. This means...

thin-walled hollow shafts

Shear Flow in Thin-Walled Shafts

Shear flow in a thin-walled hollow shaft describes how torsional force is carried through the wall. When a small segment of width dx is isolated, it is still in equilibrium, but it experiences torsional shearing forces at its ends. These forces come from the longitudinal shearing stress on the segment's minor surface multiplied by that surface area. The result is a uniform shear flow through the structure. To study the wall more closely, a small section with length ds is considered. The force...

key terms and concepts

Learn how transmission shaft size is set by power, speed, torque, and allowable shear stress. See how solid and hollow shafts are designed.

Sizing a Transmission Shaft by Torque | JoVE K12

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mechanical engineering

Sizing a Transmission Shaft by Torque

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