例題 1Example 1
題目 Problem
The disk has a constant angular velocity \(p\) about its \(z\)-axis, and the yoke A has a constant angular velocity \(\omega_2\) about its shaft as shown. Simultaneously the entire assembly revolves about the fixed \(X\)-axis with a constant angular velocity \(\omega_1\). Determine the expression for the angular acceleration of the disk as the yoke brings it into the vertical plane in the position shown.
解答 Solution
$$\vec\omega = \omega_1\vec i + \omega_2\vec j + p\vec k$$
$$\begin{aligned}
\vec\alpha = \dot{\vec\omega} &= \omega_1\vec i\times\omega_2\vec j + \left(\omega_1\vec i + \omega_2\vec j\right)\times p\vec k \\
&= \omega_1\omega_2\,\vec k - \omega_1 p\,\vec j + \omega_2 p\,\vec i
\end{aligned}$$
- 三個角速度大小都固定，\(\dot{\vec\omega}\) 只來自「向量方向被外層轉動帶著轉」：\(\omega_2\vec j\) 被 \(\omega_1\vec i\) 帶動，\(p\vec k\) 被 \(\omega_1\vec i+\omega_2\vec j\) 帶動。
對照原始圖檔
