# 轉動慣量

I

lbf·ft·s2

${\displaystyle I={\frac {L}{\omega }}}$

## 相关概念

### 定轴转动动能

${\displaystyle v=\omega r}$
${\displaystyle m={\frac {I}{r^{2}}}}$

${\displaystyle K={\frac {1}{2}}\left({\frac {I}{r^{2}}}\right)(\omega r)^{2}}$

${\displaystyle K={\frac {1}{2}}I\omega ^{2}}$

## 慣性張量

${\displaystyle \mathbf {I} ={\begin{bmatrix}I_{xx}&I_{xy}&I_{xz}\\I_{yx}&I_{yy}&I_{yz}\\I_{zx}&I_{zy}&I_{zz}\end{bmatrix}}\,\!}$ （1）

${\displaystyle I_{xx}\ {\stackrel {\mathrm {def} }{=}}\ \int \ (y^{2}+z^{2})\ dm\,\!}$
${\displaystyle I_{yy}\ {\stackrel {\mathrm {def} }{=}}\ \int \ (x^{2}+z^{2})\ dm\,\!}$ （2）
${\displaystyle I_{zz}\ {\stackrel {\mathrm {def} }{=}}\ \int \ (x^{2}+y^{2})\ dm\,\!}$

${\displaystyle I_{xy}=I_{yx}\ {\stackrel {\mathrm {def} }{=}}\ -\int \ xy\ dm\,\!}$
${\displaystyle I_{xz}=I_{zx}\ {\stackrel {\mathrm {def} }{=}}\ -\int \ xz\ dm\,\!}$ （3）
${\displaystyle I_{yz}=I_{zy}\ {\stackrel {\mathrm {def} }{=}}\ -\int \ yz\ dm\,\!}$

### 導引

${\displaystyle \mathbf {L} _{G}=\int \ \mathbf {r} \times \mathbf {v} \ dm\,\!}$

${\displaystyle \mathbf {L} _{G}=\int \ \mathbf {r} \times ({\boldsymbol {\omega }}\times \mathbf {r} )\ dm\,\!}$

{\displaystyle {\begin{aligned}L_{Gx}&=\int \ y({\boldsymbol {\omega }}\times \mathbf {r} )_{z}-z({\boldsymbol {\omega }}\times \mathbf {r} )_{y}\ dm\\&=\int \ y\omega _{x}y-y\omega _{y}x+z\omega _{x}z-z\omega _{z}x\ dm\\&=\int \ \omega _{x}(y^{2}+z^{2})-\omega _{y}xy-\omega _{z}xz\ dm\\&=\omega _{x}\int \ (y^{2}+z^{2})\ dm-\omega _{y}\int \ xy\ dm-\omega _{z}\int \ xz\ dm\ .\end{aligned}}\,\!}

${\displaystyle L_{Gx}=\omega _{x}\int \ (y^{2}+z^{2})\ dm-\omega _{y}\int \ xy\ dm-\omega _{z}\int \ xz\ dm\,\!}$
${\displaystyle L_{Gy}=-\omega _{x}\int \ xy\ dm+\omega _{y}\int \ (x^{2}+z^{2})\ dm-\omega _{z}\int \ yz\ dm\,\!}$
${\displaystyle L_{Gz}=-\omega _{x}\int \ xz\ dm-\omega _{y}\int \ yz\ dm+\omega _{z}\int \ (x^{2}+y^{2})\ dm\,\!}$

${\displaystyle \mathbf {L} _{G}=\mathbf {I} _{G}\ {\boldsymbol {\omega }}\,\!}$ （4）

### 平行軸定理

${\displaystyle I_{xx}=I_{G,xx}+m({\bar {y}}^{2}+{\bar {z}}^{2})\,\!}$
${\displaystyle I_{yy}=I_{G,yy}+m({\bar {x}}^{2}+{\bar {z}}^{2})\,\!}$ （5）
${\displaystyle I_{zz}=I_{G,zz}+m({\bar {x}}^{2}+{\bar {y}}^{2})\,\!}$
${\displaystyle I_{xy}=I_{yx}=I_{G,xy}-m{\bar {x}}{\bar {y}}\,\!}$
${\displaystyle I_{xz}=I_{zx}=I_{G,xz}-m{\bar {x}}{\bar {z}}\,\!}$ （6）
${\displaystyle I_{yz}=I_{zy}=I_{G,yz}-m{\bar {y}}{\bar {z}}\,\!}$

a)參考圖B，讓${\displaystyle (x\,',\ y\,',\ z\,')\,\!}$ ${\displaystyle (x,\ y,\ z)\,\!}$ 分別為微小質量${\displaystyle dm\,\!}$ 對質心${\displaystyle G}$ 與原點${\displaystyle O}$ 的相對位置：

${\displaystyle y=y\,'+{\bar {y}}\,\!}$ ${\displaystyle z=z\,'+{\bar {z}}\,\!}$

${\displaystyle I_{G,xx}=\int \ (y\,'\,^{2}+z\,'\,^{2})\ dm\,\!}$
${\displaystyle I_{xx}=\int \ (y^{2}+z^{2})\ dm\,\!}$

{\displaystyle {\begin{aligned}I_{xx}&=\int \ [(y\,'+{\bar {y}})^{2}+(z\,'+{\bar {z}})^{2}]\ dm\\&=I_{G,xx}+m({\bar {y}}^{2}+{\bar {z}}^{2})\ .\\\end{aligned}}\,\!}

b)依照方程式（3），

${\displaystyle I_{G,xy}=-\int \ x\,'y\,'\ dm\,\!}$
${\displaystyle I_{xy}=-\int \ xy\ dm\,\!}$

{\displaystyle {\begin{aligned}I_{xy}&=-\int \ (x\,'+{\bar {x}})(y\,'+{\bar {y}})\ dm\\&=I_{G,xy}-m{\bar {x}}{\bar {y}}\ .\\\end{aligned}}\,\!}

### 對於任意軸的轉動慣量

${\displaystyle I_{OQ}\ =\int \ \rho ^{2}\ dm\ =\ \int \ \left|{\boldsymbol {\eta }}\times \mathbf {r} \right|^{2}\ dm\,\!}$

${\displaystyle I_{OQ}=\int \ [(\eta _{y}z-\eta _{z}y)^{2}+(\eta _{x}z-\eta _{z}x)^{2}+(\eta _{x}y-\eta _{y}x)^{2}]\ dm\,\!}$

{\displaystyle {\begin{aligned}I_{OQ}=&\eta _{x}^{2}\int \ (y^{2}+z^{2})\ dm+\eta _{y}^{2}\int \ (x^{2}+z^{2})\ dm+\eta _{z}^{2}\int \ (x^{2}+y^{2})\ dm\\&-2\eta _{x}\eta _{y}\int \ xy\ dm-2\eta _{x}\eta _{z}\int \ xz\ dm-2\eta _{y}\eta _{z}\int \ yz\ dm\ .\\\end{aligned}}\,\!}

${\displaystyle I_{OQ}=\eta _{x}^{2}I_{xx}+\eta _{y}^{2}I_{yy}+\eta _{z}^{2}I_{zz}+2\eta _{x}\eta _{y}I_{xy}+2\eta _{x}\eta _{z}I_{xz}+2\eta _{y}\eta _{z}I_{yz}\,\!}$ （7）

### 主轉動慣量

${\displaystyle \mathbf {I} \ {\boldsymbol {\omega }}=\lambda \;{\boldsymbol {\omega }}\,\!}$ （8）

${\displaystyle \det {(\mathbf {I} -\left[{\begin{matrix}1&0&0\\0&1&0\\0&0&1\end{matrix}}\right]\lambda )}={\begin{vmatrix}I_{xx}-\lambda &I_{xy}&I_{xz}\\I_{yx}&I_{yy}-\lambda &I_{yz}\\I_{zx}&I_{zy}&I_{zz}-\lambda \end{vmatrix}}\,\!=0}$

${\displaystyle \omega _{x}^{2}+\omega _{y}^{2}+\omega _{z}^{2}=1\,\!}$

${\displaystyle \mathbf {L} =(I_{x}\omega _{x}\;,\;I_{y}\omega _{y}\;,\;I_{z}\omega _{z})\,\!}$

### 動能

${\displaystyle K={\frac {1}{2}}m{\bar {v}}^{2}+{\frac {1}{2}}\int \ v^{2}\ dm\,\!}$

${\displaystyle K\,\!'={\frac {1}{2}}\int \ ({\boldsymbol {\omega }}\times \mathbf {r} )\cdot ({\boldsymbol {\omega }}\times \mathbf {r} )\ dm\,\!}$

${\displaystyle K\,\!'={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \int \ \mathbf {r} \times ({\boldsymbol {\omega }}\times \mathbf {r} )\ dm={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {L} \,\!}$

${\displaystyle K\,\!'={\frac {1}{2}}{\boldsymbol {\omega }}^{\operatorname {T} }\ \mathbf {I} \ {\boldsymbol {\omega }}\,\!}$

${\displaystyle K={\frac {1}{2}}m{\bar {v}}^{2}+{\frac {1}{2}}(I_{xx}{\omega _{x}}^{2}+I_{yy}{\omega _{y}}^{2}+I_{zz}{\omega _{z}}^{2}+2I_{xy}\omega _{x}\omega _{y}+2I_{xz}\omega _{x}\omega _{z}+2I_{yz}\omega _{y}\omega _{z})\,\!}$ （9）

${\displaystyle K={\frac {1}{2}}m{\bar {v}}^{2}+{\frac {1}{2}}(I_{x}{\omega _{x}}^{2}+I_{y}{\omega _{y}}^{2}+I_{z}{\omega _{z}}^{2})\,\!}$ （10）

## 計算範例

${\displaystyle {\begin{smallmatrix}m=\lambda x\end{smallmatrix}}}$
${\displaystyle {\begin{smallmatrix}dm=\lambda dx\end{smallmatrix}}}$
${\displaystyle I_{\text{CM}}=\int r^{2}dm=\lambda \int _{-\ell /2}^{\ell /2}x^{2}dx={\frac {m}{\ell }}\ \left({\frac {1}{3}}x^{3}\right){\bigg |}_{-\ell /2}^{\ell /2}={\frac {1}{12}}\,m\ell ^{2}}$

${\displaystyle I_{\text{end}}=\int r^{2}dm=\lambda \int _{0}^{\ell }x^{2}dx={\frac {m}{\ell }}\ \left({\frac {1}{3}}x^{3}\right){\bigg |}_{0}^{\ell }={\frac {1}{3}}\,m\ell ^{2}}$
${\displaystyle I_{\text{end}}=I_{\text{CM}}+MD^{2}={\frac {1}{12}}\,m\ell ^{2}+m\left({\frac {\ell }{2}}\right)^{2}={\frac {1}{3}}\,m\ell ^{2}}$

## 參考文獻

1. ^ 普通物理学（修订版，化学数学专业用）。汪昭义主编。华东师范大学出版社.P81.三、转动惯量.ISBN 978-7-5617-0444-8/N·018
2. ^ O'Nan, Michael. Linear Algebra. USA: Harcourt Brace Jovanovich, Inc. 1971: pp。361. ISBN 0-15-518558-6 （英语）.
• Beer, Ferdinand; E. Russell Johnston, Jr., William E. Clausen (2004). Vector Mechanics for Engineers. 7th edition. USA: McGraw-Hill, ISBN 978-0-07-230492-3