质量矩阵

${\displaystyle T={\frac {1}{2}}{\dot {q}}^{\mathrm {T} }M{\dot {q}}}$

${\displaystyle T\;=\;{\frac {1}{2}}m|v|^{2}\;=\;{\frac {1}{2}}v\cdot mv}$

${\displaystyle \mathbf {x} =[x_{1}\,x_{2}]^{\mathrm {T} }}$

${\displaystyle E=\sum _{i=1}^{2}{\frac {1}{2}}m_{i}{\dot {x_{i}}}^{2}}$

${\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}}$

${\displaystyle E={\frac {1}{2}}{\dot {\mathbf {x} }}^{\mathrm {T} }M{\dot {\mathbf {x} }}}$

${\displaystyle M_{j,j}=M_{j+1,j+1}=m_{i}}$

示例

二体线性系统

${\displaystyle q=[x_{1}\,x_{2}]^{\mathrm {T} }}$.

${\displaystyle T=\sum _{i=1}^{2}{\frac {1}{2}}m_{i}{\dot {x}}_{i}{}^{2}}$

${\displaystyle T={\frac {1}{2}}{\dot {q}}^{\mathrm {T} }M{\dot {q}}}$

${\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}}$

N体系统

${\displaystyle M=\mathrm {diag} [m_{1}I_{n_{1}},m_{2}I_{n_{2}},\cdots ,m_{N}I_{n_{N}}]}$

${\displaystyle M={\begin{bmatrix}m_{1}&\cdots &0&0&\cdots &0&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &m_{1}&0&\cdots &0&\cdots &0&\cdots &0\\0&\cdots &0&m_{2}&\cdots &0&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &0&0&\cdots &m_{2}&\cdots &0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &0&0&\cdots &0&\cdots &m_{N}&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots &\ddots &\vdots &\ddots &\vdots \\0&\cdots &0&0&\cdots &0&\cdots &0&\cdots &m_{N}\\\end{bmatrix}}}$

转动系统

${\displaystyle q=[x,y,\alpha ]}$

${\displaystyle {\begin{array}{ll}p_{1}=(x,y)+R(\cos \alpha ,\sin \alpha )&v_{1}=({\dot {x}},{\dot {y}})+R{\dot {\alpha }}(-\sin \alpha ,\cos \alpha )\\p_{2}=(x,y)-R(\cos \alpha ,\sin \alpha )&v_{2}=({\dot {x}},{\dot {y}})-R{\dot {\alpha }}(-\sin \alpha ,\cos \alpha )\end{array}}}$

${\displaystyle T=m{\dot {x}}^{2}+m{\dot {y}}^{2}+mR^{2}{\dot {\alpha }}^{2}+2Rd\cos \alpha {\dot {x}}{\dot {\alpha }}+2Rd\sin \alpha {\dot {y}}{\dot {\alpha }}}$

${\displaystyle T={\frac {1}{2}}{\dot {q}}^{\mathrm {T} }M{\dot {q}}}$

${\displaystyle M={\begin{bmatrix}m&0&Rd\cos \alpha \\0&m&Rd\sin \alpha \\Rd\cos \alpha &Rd\sin \alpha &R^{2}m\end{bmatrix}}}$

参考文献

1. ^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
2. ^ Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978 0 521 57572 0