惯性矩

Moments of Inertia

面积惯性矩

Moment of Inertia for Areas

$$\begin{array}{rl}{I}_x& ={\int }_A{y}^2\mathrm{d}A\\ I_y& ={\int }_A{x}^2\mathrm{d}A\\ J_O& =I_x+I_y\end{array}$$

面积平行轴定理

Parallel-Axis Theorem for an area

$$\begin{array}{rl}{I}_x& ={\overline{I}}_{x^{\prime }}+A{d}_y^2\\ I_y& ={\overline{I}}_{y^{\prime }}+A{d}_x^2\\ J_O& ={\overline{J}}_C+A{d}^2\end{array}$$

面积回转半径

Radius of Gyration of an Area

$$\begin{array}{rl}{k}_x& =\sqrt{\frac{I_x}{A}}\\ k_y& =\sqrt{\frac{I_y}{A}}\\ k_O& =\sqrt{\frac{J_O}{A}}\end{array}$$

面积惯性积

Product of Inertia for an Area

$$I_{xy}={\int }_{A}xy\mathrm{d}A$$

Parallel-Axis Theorem

$$I_{xy}={\overline{I}}_{x^{\prime }{y}^{\prime }}+A\mathrm{d}x\mathrm{d}y$$

关于倾斜轴的面积惯性矩

Moments of Inertia for an Area about Inclined Axes

$$\begin{array}{rl}{I}_u& =\frac{I_x+I_y}{2}+\frac{I_x-I_y}{2}\cos 2\theta -I_{xy}\sin 2\theta \\ I_v& =\frac{I_x+I_y}{2}-\frac{I_x-I_y}{2}\cos 2\theta +I_{xy}\sin 2\theta \\ I_{uv}& =\frac{I_x-I_y}{2}\sin 2\theta +I_{xy}\cos 2\theta \end{array}$$

主惯性轴

Principal Moments of Inertia

$$\tan 2{\theta }_p=\frac{-I_{xy}}{(I_x-I_y)/2}$$

最大/最小惯性矩

$$I_{\text{min}}^{\text{max}}=\frac{I_x+I_y}{2}\pm \sqrt{{\left(\frac{I_x-I_y}{2}\right)}^2+I_{xy}^2}$$

莫尔圆

Mohr's Circle for Memoents of Inertia