Chapter 3: Torsion¶
3.1 Concept¶
- Shaft（轴）：torsion is its primary deformation.
- Torsion（扭转）：扭矩作用下的变形，扭矩作用面与轴线垂直
- Angle of twist（扭转角 \(\varphi\)）：一个横截面相对于另一个横截面转过的角度
- Shearing strain（切应变 \(\gamma\)）：The change of a right angle between two straight lines.
Examples: - Steering rod - 丝锥
3.2 External/Internal Torque of A Driving Shaft¶
External torque of a driving shaft¶
\(W = Pt = m \cdot \omega \cdot t\)
\[\begin{equation} \tag{3-1} m = 9549 \cdot \frac{P}{n} \quad\mathrm{(N \cdot m)} \end{equation}\]
- \(P\): power（功率，kW）
- \(n\): speed（转速，r/min 或 rpm）
Internal torque and its diagram¶
Sign convention: 右手螺旋定则（右手拇指指向外法线正方向为正）
3.3 Torsion of the Thin-Walled Hollow Shafts¶
壁厚 \(t < 0.1D\)
Experiment
Before deformation:
After deformation:
- The circumference lines do not change
- The longitudinal lines are changed into slants
Conclusions: - 圆周线不变 - 所有纵线转过的角度都是 \(\gamma\) - 所有正方形变形成相同的平行四边形
取一个小体积元
- 没有正应力，只有剪应力 \(\tau\)
- 在同一个截面上，剪应力 \(\tau\) 处处相等（因为壁很薄，沿半径方向 \(z\) 的变化忽略不计），垂直于半径方向
Magnitude of the shear stress \(\tau\)¶
内力矩
\[\int_A \tau \cdot dA \cdot r_0 = T\]
\(\tau, r_0\) 为常数，得到
\[\tau \cdot r_0 \cdot 2\pi r_0 t = T\]
即
\[\begin{equation} \tag{3-2} \tau = \frac{T}{2\pi r_0^2 t} = \frac{T}{2A_0 t} \end{equation}\]
Theorem of conjugate shearing stresses¶
切应力互等定理
\[ \begin{aligned} \sum m_z &= 0 \\ \tau t dx \cdot dy &= \tau' t dy \cdot dx \end{aligned} \]
故
\[\boxed{\tau = \tau'}\]
Hooke's law of shear¶
几何关系：\(\gamma l = r \varphi\)
结合式 \eqref{3-2} 得到
3.5 Deformation of a Circular Shaft¶
Deformation in torsion¶
From the formula
\[\frac{d\varphi}{dx} = \frac{T}{GI_p}\]
\[\varphi = \int d\varphi = \int_0^l \frac{T}{GI_p} dx \overset{T = \text{const.}}{=} \frac{Tl}{GI_p}\]
Angle of twist per unit length \(\theta\)¶
\[\theta = \frac{d\varphi}{dx} = \frac{T}{GI_p}\]
\(GI_p\) 反映了材料抵抗扭转的能力（扭转刚度）
Rigidity Check
3.6 Statically Indeterminate Problems of Round Shafts¶
3.7 Stress & Deformation in Springs¶
\[\tau_{\max} = \tau_Q + \tau_T = \frac{Q}{A} + \frac{T}{W_t} = \frac{PD/2}{\pi d^3/16} + \frac{P}{\pi d^2/4} = \left(\frac{d}{2D} + 1\right) \frac{8PD}{\pi d^3}\]
3.8 Stress & Deformation of Non-Circular Shafts¶
- Free torsion: Warping of each cross section in torsion is not restricted and the magnitude is the same.（每个横截面翘曲的程度相同）
- Restricted torsion: 翘曲程度不同
- Shearing stress on the section of the rectangular rod
- 角点：切应力为0
- 形心
- 边中点：最大切应力
\[\tau_{\max} = \frac{|T|_{\max}}{W_t}, \quad W_t = \alpha hb^2\]
3.9 Stress of Opened and Closed Thin-Walled Rods¶
开口比闭口的更容易变形
- 开口
中心线上切应力为0
- 闭口
切应力均匀分布
\[T = \oint (\tau \delta ds) \rho = 2 \tau \delta \oint \frac{1}{2} \rho ds = 2 \tau \delta A\]
Example 3-8
