# 轉動慣量

## 概念

### 動能

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

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

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

## 慣性張量

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

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

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

### 導引

$\mathbf{L}_G=\int\ \mathbf{r}\times\mathbf{v}\ dm\,\!$

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

\begin{align} 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{align}\,\!

$L_{Gx}=\omega_x\int\ y^2+z^2\ dm - \omega_y\int\ xy\ dm - \omega_z\int\ xz\ dm \,\!$
$L_{Gy}= - \omega_x\int\ xy\ dm+\omega_y\int\ x^2+z^2\ dm - \omega_z \int\ yz\ dm \,\!$
$L_{Gz}= - \omega_x\int\ xz\ dm - \omega_y\int\ yz\ dm+\omega_z\int\ x^2+y^2\ dm\,\!$

$\mathbf{L}_G=\mathbf{I}_G\ \boldsymbol{\omega}\,\!$(4）

### 平行軸定理

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

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

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

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

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

b)依照方程式（3），

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

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

### 對於任意軸的轉動慣量

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

$I_{OQ}=\int\ (\eta_yz - \eta_zy)^2+(\eta_xz - \eta_zx)^2+(\eta_xy - \eta_yx)^2\ dm\,\!$

\begin{align} 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{align}\,\!

$I_{OQ}=\eta_x^2I_{xx}+\eta_y^2I_{yy}+\eta_z^2I_{zz}+2\eta_x\eta_yI_{xy}+2\eta_x\eta_zI_{xz}+2\eta_y\eta_zI_{yz}\,\!$(7）

### 主轉動慣量

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

$\mathbf{I} = \begin{vmatrix} I_{xx} - \lambda & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} - \lambda & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} - \lambda \end{vmatrix}\,\!$

$\omega_x^2+\omega_y^2+\omega_z^2=1\,\!$

$\mathbf{L}=(I_x\omega_x\;,\;I_y\omega_y\;,\;I_z\omega_z)\,\!$

### 動能

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

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

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

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

$K=\frac{1}{2}m\bar{v}^2+\frac{1}{2}(I_{xx}{\omega_x}^2+I_{yy}{\omega_y}^2+I_{zz}{\omega_z}^2+2I_{xy}\omega_x\omega_y+2I_{xz}\omega_x\omega_z+ 2I_{yz}\omega_y\omega_z)\,\!$(9）

$K=\frac{1}{2}m\bar{v}^2+\frac{1}{2}(I_{x}{\omega_x}^2+I_{y}{\omega_y}^2+I_{z}{\omega_z}^2)\,\!$(10）

## 計算

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

$I_{end} = \int r^2 dm = \lambda \int_{0}^{\ell} x^2 dx = \frac{m}{\ell}\ (\frac{1}{3} x^3)\bigg|_{0}^{\ell} = \frac{1}{3}\, m\ell^2$
$I_{end} = I_{CM} + M D^2 = \frac{1}{12}\, m\ell^2 + m(\frac{\ell}{2})^2 = \frac{1}{3}\, m\ell^2$

## 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 （英文）.
• Beer, Ferdinand; E. Russell Johnston, Jr., William E. Clausen (2004). Vector Mechanics for Engineers. 7th edition. USA: McGraw-Hill, ISBN 0-07-230492-8