# 运动学

## 質點運動學

### 位置、參考系

${\displaystyle \mathbf {r} =(x,y,z)}$

${\displaystyle |\mathbf {r} |={\sqrt {x^{\ 2}+y^{\ 2}+z^{\ 2}}}}$

### 路徑、路徑距離、位移

${\displaystyle \mathbf {r} _{P/Q}=\mathbf {r} _{P}-\mathbf {r} _{Q}=(x_{P}-x_{Q},y_{P}-y_{Q},z_{P}-z_{Q})}$

${\displaystyle s=\int _{t_{1}}^{t_{2}}|\mathrm {d} \mathbf {r} |=\int _{t_{1}}^{t_{2}}\mathrm {d} s=\int _{t_{1}}^{t_{2}}{\sqrt {\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}}}=\int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} z}{\mathrm {d} t}}\right)^{2}}}\;\mathrm {d} t}$

### 速度、加速度

${\displaystyle {\overline {\mathbf {v} }}={\frac {\Delta \mathbf {r} }{\Delta t}}}$

${\displaystyle \mathbf {v} \ {\stackrel {def}{=}}\ \lim _{\Delta t\rightarrow 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}}$

${\displaystyle v=|\mathbf {v} |=\left|{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right|={\frac {\mathrm {d} s}{\mathrm {d} t}}}$

${\displaystyle {\overline {\mathbf {a} }}={\frac {\Delta \mathbf {v} }{\Delta t}}}$

${\displaystyle \mathbf {a} \ {\stackrel {def}{=}}\ \lim _{\Delta t\rightarrow 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}}$

${\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\int _{t_{0}}^{t}\mathbf {a} (t)\ \mathrm {d} t}$
{\displaystyle {\begin{aligned}\mathbf {r} (t)&=\mathbf {r} _{0}+\int _{t_{0}}^{t}\mathbf {v} (t)\ \mathrm {d} t\\&=\mathbf {r} _{0}+\mathbf {v} _{0}t+\int _{t_{0}}^{t}\left[\int _{t_{0}}^{t}\mathbf {a} (t)\mathrm {d} t\right]\ \mathrm {d} t\\\end{aligned}}}

### 相對運動

${\displaystyle \mathbf {r} _{P/G}=\mathbf {r} _{P}-\mathbf {r} _{G}}$
${\displaystyle \mathbf {r} _{Q/G}=\mathbf {r} _{Q}-\mathbf {r} _{G}}$

${\displaystyle \mathbf {r} _{P/Q}=\mathbf {r} _{P}-\mathbf {r} _{Q}=\mathbf {r} _{P}-\mathbf {r} _{G}-\mathbf {r} _{Q}+\mathbf {r} _{G}=\mathbf {r} _{P/G}-\mathbf {r} _{Q/G}}$

${\displaystyle \mathbf {r} _{P/G}=\mathbf {r} _{P/Q}+\mathbf {r} _{Q/G}}$

${\displaystyle \mathbf {v} _{P/Q}=\mathbf {v} _{P}-\mathbf {v} _{Q}}$

${\displaystyle \mathbf {a} _{P/Q}=\mathbf {a} _{P}-\mathbf {a} _{Q}}$

### 直线运动

${\displaystyle {\overline {v}}={\frac {\Delta x}{\Delta t}}}$
${\displaystyle v=\lim _{{\Delta t}\to 0}{\frac {\Delta x}{\Delta t}}}$

${\displaystyle {\overline {a}}={\frac {\Delta v}{\Delta t}}}$
${\displaystyle a=\lim _{{\Delta t}\to 0}{\frac {\Delta v}{\Delta t}}}$

${\displaystyle v(t)={\frac {\mathrm {d} x}{\mathrm {d} t}}}$
${\displaystyle a(t)={\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}}$

${\displaystyle x_{f}=x_{i}+vt}$

${\displaystyle x_{f}-x_{i}=v_{i}t+{\frac {1}{2}}at^{2}}$
${\displaystyle x_{f}-x_{i}={\frac {1}{2}}(v_{f}+v_{i})t}$
${\displaystyle v_{f}=v_{i}+at}$
${\displaystyle v_{f}^{2}=v_{i}^{2}+2a(x_{f}-x_{i})}$

#### 实例：等加速直线运动

${\displaystyle x_{f}=x_{i}+v_{i}t+{\frac {1}{2}}at^{2}}$

${\displaystyle 0=v_{i}t+{\frac {1}{2}}at^{2}=t(v_{i}+{\frac {1}{2}}at)}$

${\displaystyle t=-{\frac {2v_{i}}{a}}=-{\frac {2*50}{-9.81}}=10.2\ s}$

${\displaystyle v_{f}^{2}=v_{i}^{2}+2a(x_{f}-x_{i})}$

${\displaystyle x_{f}={\frac {v_{f}^{2}-v_{i}^{2}}{2a}}+x_{i}={\frac {0-50^{2}}{2*-9.81}}+0=127.55\ m}$

${\displaystyle v_{f}={\sqrt {v_{i}^{2}+2a(x_{f}-x_{i})}}={\sqrt {0^{2}+2(-9.81)(0-127.55)}}=50\ m/s}$

### 曲线运动

${\displaystyle \mathbf {v} =\lim _{{\Delta t}\to 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}}$

${\displaystyle \Delta t\to 0}$  极限得到的速度向量，正切曲线于質點的位置。

${\displaystyle v={\begin{vmatrix}\mathbf {v} \end{vmatrix}}=\lim _{{\Delta t}\to 0}{\frac {\Delta s}{\Delta t}}={\frac {\mathrm {d} s}{\mathrm {d} t}}}$

${\displaystyle \mathbf {a} =\lim _{{\Delta t}\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}}$

#### 直角坐标系

${\displaystyle \mathbf {r} =x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}$
${\displaystyle \mathbf {v} =v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}}$
${\displaystyle \mathbf {a} =a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}}$

${\displaystyle v_{x}={\frac {\mathrm {d} x}{\mathrm {d} t}}\ \ ,\qquad \qquad v_{y}={\frac {\mathrm {d} y}{\mathrm {d} t}}\ \ ,\qquad \qquad v_{z}={\frac {\mathrm {d} z}{\mathrm {d} t}}}$
${\displaystyle a_{x}={\frac {\mathrm {d} v_{x}}{\mathrm {d} t}}\ ,\qquad \qquad a_{y}={\frac {\mathrm {d} v_{y}}{\mathrm {d} t}}\ ,\qquad \qquad a_{z}={\frac {\mathrm {d} v_{z}}{\mathrm {d} t}}}$

#### 极坐标系

${\displaystyle \mathbf {r} =r{\hat {\mathbf {r} }}}$
${\displaystyle \mathbf {v} ={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}}}$
${\displaystyle \mathbf {a} =({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}}$

${\displaystyle \alpha ={\dot {\omega }}={\ddot {\theta }}}$

${\displaystyle \theta _{f}-\theta _{i}=\omega _{i}t+{\frac {1}{2}}\alpha t^{2}}$
${\displaystyle \theta _{f}-\theta _{i}={\frac {1}{2}}(\omega _{f}+\omega _{i})t}$
${\displaystyle \omega _{f}=\omega _{i}+\alpha t}$
${\displaystyle \omega _{f}^{2}=\omega _{i}^{2}+2\alpha (\theta _{f}-\theta _{i})}$

#### 实例：等加速曲线运动

${\displaystyle \Delta x=x_{f}-x_{i}=v_{i}\cos(\Phi )\ t+{\frac {1}{2}}at^{2}=v_{i}\cos(\Phi )\ t}$

${\displaystyle 0=v_{i}\sin(\Phi )\ t+{\frac {1}{2}}at^{2}=t(v_{i}\sin(\Phi )+{\frac {1}{2}}at)}$

${\displaystyle \Delta x=v_{i}\cos(\Phi )\left({\frac {-2v_{i}\sin(\Phi )}{a}}\right)=-{\frac {v_{i}^{2}\sin 2(\Phi )}{a}}=220.70\ m}$

### 二維旋轉參考系

#### 單位向量的時間變化率

${\displaystyle {\hat {\mathbf {e} }}_{x}={\hat {\mathbf {x} }}}$
${\displaystyle {\hat {\mathbf {e} }}_{y}={\hat {\mathbf {y} }}}$

${\displaystyle {\hat {\mathbf {e} }}_{x}=\cos(\omega t){\hat {\mathbf {x} }}+\sin(\omega t){\hat {\mathbf {y} }}}$
${\displaystyle {\hat {\mathbf {e} }}_{y}=-\sin(\omega t){\hat {\mathbf {x} }}+\cos(\omega t){\hat {\mathbf {y} }}}$

${\displaystyle {\frac {\mathrm {d} {\hat {\mathbf {e} }}_{x}}{\mathrm {d} t}}=-\omega \sin(\omega t){\hat {\mathbf {x} }}+\omega \cos(\omega t){\hat {\mathbf {y} }}=\omega {\hat {\mathbf {e} }}_{y}}$
${\displaystyle {\frac {\mathrm {d} {\hat {\mathbf {e} }}_{y}}{\mathrm {d} t}}=-\omega \cos(\omega t){\hat {\mathbf {x} }}-\omega \sin(\omega t){\hat {\mathbf {y} }}=-\omega {\hat {\mathbf {e} }}_{x}}$

{\displaystyle {\begin{aligned}\left({\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}\right)&={\dot {f}}_{x}{\hat {\mathbf {x} }}+{\dot {f}}_{y}{\hat {\mathbf {y} }}\\&={\dot {F}}_{x}{\hat {\mathbf {e} }}_{x}+{\dot {F}}_{y}{\hat {\mathbf {e} }}_{y}+F_{x}\left({\frac {\mathrm {d} {\hat {\mathbf {e} }}_{x}}{\mathrm {d} t}}\right)+F_{y}\left({\frac {\mathrm {d} {\hat {\mathbf {e} }}_{y}}{\mathrm {d} t}}\right)\\&={\dot {F}}_{x}{\hat {\mathbf {e} }}_{x}+{\dot {F}}_{y}{\hat {\mathbf {e} }}_{y}+F_{x}\omega {\hat {\mathbf {e} }}_{y}-F_{y}\omega {\hat {\mathbf {e} }}_{x}\\&={\dot {F}}_{x}{\hat {\mathbf {e} }}_{x}+{\dot {F}}_{y}{\hat {\mathbf {e} }}_{y}+{\boldsymbol {\omega }}\times \mathbf {F} \\\end{aligned}}}

${\displaystyle \left({\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}\right)_{\mathrm {space} }=\left({\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}\right)_{\mathrm {rotate} }+{\boldsymbol {\omega }}\times \mathbf {F} }$

${\displaystyle \left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {space} }=\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {rotate} }+{\boldsymbol {\omega }}\times }$

#### 位置、速度、加速度

${\displaystyle \mathbf {r} =x_{S}\ {\hat {\mathbf {x} }}+y_{S}\ {\hat {\mathbf {y} }}}$

${\displaystyle \mathbf {r} =x_{R}\ {\hat {\mathbf {e} }}_{x}+y_{R}\ {\hat {\mathbf {e} }}_{y}}$

${\displaystyle \mathbf {v} \ {\stackrel {def}{=}}\ \left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {space} }=\left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {rotate} }+{\boldsymbol {\omega }}\times \mathbf {r} =\mathbf {v} _{R}+{\boldsymbol {\omega }}\times \mathbf {r} }$

{\displaystyle {\begin{aligned}\mathbf {a} &\ {\stackrel {def}{=}}\ \left({\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\right)_{\mathrm {space} }\\&=\left({\frac {\mathrm {d} \mathbf {v} _{R}}{\mathrm {d} t}}\right)_{\mathrm {space} }+\left({\frac {\mathrm {d} ({\boldsymbol {\omega }}\times \mathbf {r} )}{\mathrm {d} t}}\right)_{\mathrm {space} }\\&=\left({\frac {\mathrm {d} \mathbf {v} _{R}}{\mathrm {d} t}}\right)_{\mathrm {space} }+\left({\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\right)_{\mathrm {space} }\times \mathbf {r} +{\boldsymbol {\omega }}\times \left({\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\right)_{\mathrm {space} }\\&=\left({\frac {\mathrm {d} \mathbf {v} _{R}}{\mathrm {d} t}}\right)_{\mathrm {space} }+{\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} \\\end{aligned}}}

${\displaystyle \left({\frac {\mathrm {d} \mathbf {v} _{R}}{\mathrm {d} t}}\right)_{\mathrm {space} }=\left({\frac {\mathrm {d} \mathbf {v} _{R}}{\mathrm {d} t}}\right)_{\mathrm {rotate} }+{\boldsymbol {\omega }}\times \mathbf {v} _{R}=\mathbf {a} _{R}+{\boldsymbol {\omega }}\times \mathbf {v} _{R}}$

${\displaystyle \mathbf {a} =\mathbf {a} _{R}+2{\boldsymbol {\omega }}\times \mathbf {v} _{R}+{\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} )}$

## 刚体运动学

1. 剛體的「位置」：挑選剛體內部一點G來代表整個剛體，通常會設定物體的質心形心為這一點。從空間參考系S觀測，點G的位置就是整個剛體在空間的位置。位置可以應用向量的概念來表示：向量的起點為參考系S的原點，終點為點G。
2. 剛體的取向：描述剛體取向的方法有好幾種，包括方向餘弦歐拉角四元數等等。這些方法設定一個附體參考系B的取向（相對於空間參考系S）。附體參考系是固定於剛體的參考系。相對於剛體，附體參考系的取向固定不變。由於剛體可能會呈加速度運動，所以附體參考系可能不是慣性參考系。空間參考系是某設定慣性參考系，例如，在觀測飛機的飛行運動時，附著於飛機場控制塔的參考系可以設定為空間參考系，而附著於飛機的參考系則可設定為附體參考系。

### 沙勒定理

${\displaystyle \mathbf {r} _{P}=\mathbf {r} _{G}+\mathbf {r} _{P/G}}$

### 向量的時間變化率

${\displaystyle \left({\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}\right)_{\mathrm {space} }=\left({\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}\right)_{\mathrm {body} }+{\boldsymbol {\omega }}\times \mathbf {F} }$

${\displaystyle \left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {space} }=\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {body} }+{\boldsymbol {\omega }}\times }$

### 運動學方程式

${\displaystyle \mathbf {r} _{P}=\mathbf {r} _{G}+\mathbf {r} _{P/G}}$

${\displaystyle \mathbf {v} _{P}=\mathbf {v} _{G}+\mathbf {v} _{P/G}}$

${\displaystyle \mathbf {v} _{P/G}=\left({\frac {\mathrm {d} \mathbf {r} _{P/G}}{\mathrm {d} t}}\right)_{\mathrm {body} }+{\boldsymbol {\omega }}\times \mathbf {r} _{P/G}={\boldsymbol {\omega }}\times \mathbf {r} _{P/G}}$

${\displaystyle \mathbf {v} _{P}=\mathbf {v} _{G}+{\boldsymbol {\omega }}\times \mathbf {r} _{P/G}}$

${\displaystyle \mathbf {a} _{P}=\mathbf {a} _{G}+\mathbf {a} _{P/G}}$

${\displaystyle \mathbf {a} _{P/G}=\left({\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}\right)_{\mathrm {space} }\times \mathbf {r} _{P/G}+{\boldsymbol {\omega }}\times \mathbf {v} _{P/G}={\boldsymbol {\alpha }}\times \mathbf {r} _{P/G}+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} _{P/G})}$

## 運動約束

「運動約束」指的是一個動態系統的運動必須符合的約束條件。以下列出一些例子：

### 純滾動

${\displaystyle \mathbf {v} _{cm}={\boldsymbol {\omega }}\times \mathbf {r} _{cm/O}}$

${\displaystyle v=r_{cm/O}\omega }$

### 無伸縮性繩子

• 單擺：將一根無伸縮性繩子的一端固定，另外一端繫住一個錘。這就形成了一個簡單擺。在基礎動力學裏，簡單擺問題研究錘的擺動運動跟繩子長度、錘重量之間的關係。
• 溜溜球：在兩片圓盤之間連結的捲軸，繫著一根無伸縮性繩子。這就是古今中外、廣為流行的溜溜球玩具。
• 懸鏈線：將無伸縮性繩子的兩端分別固定於兩點，由於均勻引力作用於繩子的每一部份而形成的曲線形狀稱為懸鏈線[6]

## 參考文獻

1. ^ Beer, Ferdinand; Johnston, Jr., E. Russ, Vector Mechanics for Engineers:dynamics 7th: pp. 602, 2004, ISBN 007230492s 请检查|isbn=值 (帮助)
2. Whittaker, Edmund. A treatise on the analytical dynamics of particles and rigid bodies; with an introduction to the problem of three bodies. Cambridge University Press. 1917: chapter 1. 引用错误：带有name属性“Whittaker_1917”的<ref>标签用不同内容定义了多次
3. ^ R. Douglas Gregory. Classical mechanics: an undergraduate text. Cambridge: Cambridge University. 2006: pp. 475–476. ISBN 0521826780.
4. ^ Lorenzo Sciavicco, Bruno Siciliano. §2.4.2 Roll-pitch-yaw angles. Modelling and control of robot manipulators 2nd. Springer. 2000: 32. ISBN 1852332212.
5. ^ William Thomson Kelvin & Peter Guthrie Tait. Elements of Natural Philosophy. Cambridge University Press. 1894: 4. ISBN 1573929840.
6. ^ Morris Kline. Mathematical Thought from Ancient to Modern Times. Oxford University Press. 1990. ISBN 0195061365.
• Moon, Francis C. The Machines of Leonardo Da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century. Springer. 2007. ISBN 9781402055980.