例題 3Example 3
題目 Problem
The test chamber of a flight simulator consists of a drum which spins about the a-a axis at a constant angular rate \(p\) relative to the cylindrical housing A. The housing in turn is mounted on transverse horizontal bearings and rotates about the axis b-b at the constant rate \(\dot\theta\). The entire assembly is made to rotate about the fixed vertical \(z\)-axis at the constant rate \(\Omega\). Write the expression for the angular acceleration \(\alpha\) of the spinning drum during the compounded motion for a given value of \(\theta\).
解答 Solution
$$\vec\omega = \Omega\vec k + \dot\theta\vec j + p\,\vec e_p \qquad \vec e_p = -\cos\theta\,\vec i + \sin\theta\,\vec k$$
$$\begin{aligned}
\vec\alpha = \dot{\vec\omega} &= \Omega\vec k\times\dot\theta\vec j + \left(\Omega\vec k + \dot\theta\vec j\right)\times p\,\vec e_p \\
&= \Omega\vec k\times\dot\theta\vec j + \left(\Omega\vec k + \dot\theta\vec j\right)\times\left(-p\cos\theta\,\vec i + p\sin\theta\,\vec k\right) \\
&= \dot\theta\left(p\sin\theta - \Omega\right)\vec i - p\,\Omega\cos\theta\,\vec j + p\,\dot\theta\cos\theta\,\vec k
\end{aligned}$$
- \(\vec e_p\) 是 a-a 軸方向的單位向量。
對照原始圖檔
