Name: elastic twist in a circular shaft
Uploaded: 2024-05-23T12:59:40.000Z
Duration: 1 min 13 s
Channel: JoVE

長さLの円形シャフトと、その自由端でトルクを受ける半径rの均一な断面を考えてみましょう。
シャフトの最大せん断ひずみは、ねじれ角度とシャフトの軸からの半径方向距離の両方に比例します。
弾性範囲では、このせん断ひずみは、印加トルク、半径方向距離、極慣性モーメント、および剛性係数で表すことができます。
これら 2 つの方程式を等価にすることで、弾性範囲内のねじれ角度の式を導き出します。この方程式は、トルクがその端の1つだけに適用される場合に、均一な断面を持つ均質なシャフトに適用されます。
シャフトが異なる場所でトルクを受ける場合、またはシャフトが異なる断面や材料を持つさまざまな部品で構成されている場合は、ねじれ角度を部品ごとに個別に考慮する必要があります。
ねじれの合計角度は、シャフトの各部分のすべての個々の値を合計するか、不均一な断面を持つシャフトの長さに沿って積分することによって計算されます。

ねじれ角 - 弾性範囲

Angle of Twist - Elastic Range

長さをLとし、一定の断面半径をrとする円筒形のシャフトを考えます。このシャフトは自由端でトルクを受けます。 シャフト内の最大せん断ひずみは、ねじれ角度とシャフト軸からの半径方向の距離に比例します。 シャフトが弾性的に振舞う場合、このせん断ひずみは、適用されるトルク、半径方向の距離、極慣性モーメント、剛性率などの変数を使用して表現できます。 これら２つの式を等しく設定することにより、弾性範囲内のねじれ角を表す式を立てることができます。


<img alt="Equation 1" src="/files/ftp_upload/15615/15615_Equation_1.svg" style="height: 30px;">

この方程式は、トルクがその末端にのみ作用する、均一で均質なシャフトに当てはまります。 ただし、シャフトがさまざまな点でトルクにさらされている場合、またはさまざまな断面や材質の異なる部品で構成されている場合は、ねじれ角を各セクションごとに個別に評価する必要があります。 各シャフトセグメントの個々の値をすべて合計すると、合計のねじれ角度が決まります。 あるいは、不均一な断面を持つシャフトの長さに沿って積分することによって計算することもできます。 このアプローチにより、さまざまな条件下でのシャフトの挙動を包括的に理解できます。

<img alt="Equation 2" src="/files/ftp_upload/15615/15615_Equation_2.svg" style="height: 46px;">

長さLの円形シャフトと、その自由端でトルクを受ける半径rの均一な断面を考えてみましょう。
最大せん断ひずみ...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Torsion

ねじれ

KEY TERMS AND CONCEPTS

主要用語と概念

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 shaft resists twisting. The modulus of rigidity describes how strongly the material resists shear deformation.

Setting the strain relation from twist equal to the elastic strain relation gives an expression for the angle of twist in the elastic range. This formula applies to a uniform, homogeneous shaft when torque is applied only at its ends. If torque acts at different points, or if the shaft has sections with different shapes or materials, the twist must be found for each section separately.

The total angle of twist is then the sum of the twist in each shaft segment. For a shaft with a non-uniform cross-section, the same result can be found by integrating along its length. This gives a complete way to analyze torsion in shafts under changing conditions.

Elastic Twist in a Circular Shaft

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 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

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...

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 torque creates shear strain and angle of twist in a circular shaft. See how elastic range formulas work for uniform shafts and segmented cases.

Elastic Twist in a Circular Shaft | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Elastic Twist in a Circular Shaft

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