随着航天事业的发展, 现代航天器需要携带大量的液体燃料, 并且带有太阳能帆板、天线、机械臂等柔性附件, 多为刚-液-柔耦合系统。相关研究表明, 由于耦合效应的存在, 航天器系统存在着静止、周期运动、准周期运动和混沌运动等复杂的非线性现象, 且在不同的外激励参数下, 面内/外模态的稳态动力学行为会发生变化^[1-2]。

很多学者对刚柔耦合系统的稳定性进行了深入研究。Krishnaprasad等^[3]通过Poisson流形和简化方法得到刚柔耦合系统的Poisson括号, 并采用Poisson括号及系统的能量函数得到系统的运动方程, 通过能量-Casimir法对系统的稳定性进行分析, 得到非线性稳定性条件。基于Krishnaprasad等^[3]的研究, Posbergh等^[4]对带有柔性附件刚体的非线性稳定性分析进行详细推导, 得到系统自旋稳定的条件。Kane等^[5]讨论带有柔性附件刚体平动和转动相互耦合的情况, 对由科氏力造成的离心刚化效应进行研究, 并对结果进行数值仿真。Bloch^[6]针对由平面刚体和柔性附件组成的系统, 分别建立存在离心刚化和不存在离心刚化的两种模型, 并运用能量-动量法对两种情况的平衡点非线性稳定进行分析。

岳宝增等^[7]以及Ahmad等^[8]分别采用能量-Casimir法对部分充液航天器姿态运动的稳定性进行研究, 通过将液体晃动分别等效为质量弹簧模型和球摆模型, 建立充液航天器的力学模型, 并通过能量-Casimir法得到耦合系统的稳定性条件。杨旦旦等^[9]基于Lyapunov稳定性理论, 研究带轻质悬臂梁附件充液航天器的姿态机动控制问题, 将晃动液体用黏性力矩球摆模型等效, 利用Lyapunov稳定性理论得到姿态机动的稳定性判据, 并通过数值仿真验证了控制算法的有效性。

本文基于能量-Casimir法, 研究含有柔性附件充液航天器系统的稳定性。为简化起见, 将晃动液体燃料简化为质量-弹簧模型, 仅考虑液体燃料沿本体坐标系某一坐标轴方向的横向晃动。设定航天器的储液腔为椭球形, 将柔性附件简化为线性剪切梁。

1 航天器动力学建模

考虑如图 1所示的含有柔性附件和椭球形储液腔的刚体航天器。

设惯性坐标系原点为储液腔的几何中心O, 航天器刚体部分的质量(除燃料和柔性附件之外的质量)为${{m}_{\text{H}}}$, 选择点O为本体坐标系原点, 本体坐标系沿刚体航天器惯性主轴方向, 其中坐标轴的单位向量为$({{\mathit{\boldsymbol{e}}}_{1}}, \ {{\mathit{\boldsymbol{e}}}_{2}}, \ {{\mathit{\boldsymbol{e}}}_{3}})$, 本体坐标系相对于惯性坐标系中的角速度为Ω, 刚体航天器关于本体标架惯性矩阵为${{\mathit{\boldsymbol{J}}}_{\rm{H}}}=\rm{diag}({{j}_{11}}, \ {{j}_{22}}, \ {{j}_{33}})$。

液体燃料晃动的简化力学模型通过质量-弹簧模型描述, 如图 2所示。晃动质量为$\bar{m}$, 在本体坐标系中为${{\mathit{\boldsymbol{r}}}_{{\bar{m}}}}={{({{r}_{m}}, \ 0, \ {{a}_{1}})}^{\rm{T}}}$, 晃动质量静止位置记为${{\mathit{\boldsymbol{{r}'}}}_{{\bar{m}}}}={{(0, \ 0, \ {{a}_{1}})}^{\rm{T}}}$。不参与晃动燃料的质量为${{m}_{\text{F}}}$, 在本体坐标系为${{\mathit{\boldsymbol{r}}}_{\rm{F}}}={{(0, \ 0, \ {{a}_{2}})}^{\rm{T}}}$。对任意向量a=${{({{a}_{1}}, \ {{a}_{2}}, \ {{a}_{3}})}^{\text{T}}}$, 定义a的反对称矩阵S(a)为

$ \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{a}})=\left( \begin{matrix} 0 & -{{a}_{3}} & {{a}_{2}} \\ {{a}_{3}} & 0 & -{{a}_{1}} \\ -{{a}_{2}} & {{a}_{1}} & 0 \\ \end{matrix} \right)。 $

(1)

根据以上定义, 对于任意向量$\mathit{\boldsymbol{b}}={{({{b}_{1}}, \ {{b}_{2}}, \ {{b}_{3}})}^{\rm{T}}}$, 有$\mathit{\boldsymbol{a}}\times \mathit{\boldsymbol{b}}=\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{a}})\mathit{\boldsymbol{b}}$。航天器相对于本体坐标系的静止总质量为${{m}_{\text{S}}}={{m}_{\text{H}}}+{{m}_{\text{F}}}$。首先, 考虑晃动质量$\bar{m}$的动能为

$ \begin{align} & {{K}_{m}}=\frac{1}{2}\bar{m}(\mathit{\boldsymbol{ }}\!\!\mathit{\pmb{\Omega}}\!\!\rm{ }\times {{\mathit{\boldsymbol{r}}}_{{\bar{m}}}})\cdot (\mathit{\boldsymbol{ }}\!\!\mathit{\pmb{\Omega}}\!\!\rm{ }\times {{\mathit{\boldsymbol{r}}}_{{\bar{m}}}})+ \\ & \ \ \ \ \ \ \bar{m}(\mathit{\boldsymbol{ }}\!\!\mathit{\pmb{\Omega}}\!\!\rm{ }\times {{\mathit{\boldsymbol{r}}}_{{\bar{m}}}})\cdot {{{\mathit{\boldsymbol{\dot{r}}}}}_{{\bar{m}}}}+\frac{1}{2}\bar{m}{{{\mathit{\boldsymbol{\dot{r}}}}}_{{\bar{m}}}}\cdot {{{\mathit{\boldsymbol{\dot{r}}}}}_{{\bar{m}}}}\ 。 \\ \end{align} $

(2)

未晃动质量${{m}_{\text{F}}}$和航天器的刚体部分${{m}_{\text{H}}}$的动能分别表示为

$ {{K}_{\rm{F}}}=\frac{1}{2}{{m}_{\rm{F}}}(\mathit{\pmb{\Omega}}\times {{\mathit{\boldsymbol{r}}}_{\rm{F}}})\cdot (\mathit{\pmb{\Omega}}\times {{\mathit{\boldsymbol{r}}}_{\rm{F}}})=\frac{1}{2}{{m}_{\rm{F}}}{{\mathit{\boldsymbol{ }}\!\!\mathit{\pmb{\Omega}}\!\!\rm{ }}^{\rm{T}}}{{\mathit{\boldsymbol{S}}}^{\rm{T}}}({{\mathit{\boldsymbol{r}}}_{\rm{F}}})\mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{r}}}_{\rm{F}}})\mathit{\boldsymbol{ }}\!\!\mathit{\pmb{\Omega}}\!\!\rm{ }, $

(3)

$ {K_{\rm{H}}} = \frac{1}{2}{m_{\rm{H}}}(\mathit{\pmb{\Omega}} \times {\mathit{\boldsymbol{r}}_{\rm{H}}}) \cdot (\mathit{\pmb{\Omega}} \times {\mathit{\boldsymbol{r}}_{\rm{H}}}) = \frac{1}{2}{\mathit{\pmb{\Omega}} ^{\rm{T}}}{\mathit{\boldsymbol{J}}_{\rm{H}}}\mathit{\pmb{\Omega}} 。 $

(4)

根据式(2)~(4)可得到刚液耦合航天器系统的动能为

$ \begin{array}{l} K = {K_m} + {K_{\rm{H}}} + {K_{\rm{F}}} = \frac{1}{2}(\begin{array}{*{20}{c}} {{\mathit{\pmb{\Omega}} ^{\rm{T}}}}&{{{\mathit{\boldsymbol{\dot r}}}_{\bar m}}^{\rm{T}}} \end{array}) \cdot \\ \;\;\;\;\;\;\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})}&{\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})}\\ { - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})}&{\bar m\mathit{\boldsymbol{I}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} \mathit{\pmb{\Omega}} \\ {{{\mathit{\boldsymbol{\dot r}}}_{\bar m}}} \end{array}} \right), \end{array} $

(5)

其中, ${{\mathit{\boldsymbol{J}}}_{\rm{S}}}={{\mathit{\boldsymbol{J}}}_{\rm{H}}}+{{m}_{\rm{F}}}{{\mathit{\boldsymbol{S}}}^{\rm{T}}}({{\mathit{\boldsymbol{r}}}_{\rm{F}}})\mathit{\boldsymbol{S (}}{{\mathit{\boldsymbol{r}}}_{\rm{F}}})=\rm{diag}({{j}_{\rm{S}1}}, \ {{j}_{\rm{S}2}}, \ \ {{j}_{\rm{S}3}})$为${{m}_{\text{S}}}$相对于本体坐标系的惯性矩, ${{j}_{\text{S}1}}={{j}_{11}}+{{m}_{\text{F}}}a_{2}^{2}, \ {{j}_{\text{S2}}}={{j}_{22}}+{{m}_{\text{F}}}a_{2}^{2}, \ {{j}_{\text{S}3}}={{j}_{33}}$。

晃动模型的等效弹性力为

$ {{\mathit{\boldsymbol{f}}}_{\operatorname{int}}}=-k({{\mathit{\boldsymbol{r}}}_{{\bar{m}}}}-{{\mathit{\boldsymbol{{r}'}}}_{{\bar{m}}}})=-\partial P/\partial ({{\mathit{\boldsymbol{r}}}_{{\bar{m}}}}-{{\mathit{\boldsymbol{{r}'}}}_{{\bar{m}}}}), $

(6)

其中, k为等效弹性系数。因此, 弹性势能为$P=k ({{\mathrm{r}}_{{\mathbf{\bar{m}}}}}-{{\mathrm{{r}'}}_{{\mathbf{\bar{m}}}}})\cdot ({{\mathrm{r}}_{{\mathbf{\bar{m}}}}}-{{\mathrm{{r}'}}_{{\mathbf{\bar{m}}}}})/2$, 航天器系统刚体和液体部分的能量为动能和弹簧的弹性势能之和:

$ \begin{array}{l} {H_1} = K + P\\ \;\;\;\; = \frac{1}{2}(\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^{\rm{T}}}}&{{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}^{\rm{T}}} \end{array})\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})}&{\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})}\\ { - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})}&{\bar m\mathit{\boldsymbol{I}}} \end{array}} \right)\\ \;\;\;\;\;\;\;\left( {\begin{array}{*{20}{c}} {\boldsymbol{\varOmega}}\\ {{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}} \end{array}} \right) + \frac{1}{2}k({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}}) \cdot ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}})\;。 \end{array} $

(7)

下面将柔性附件简化为线性可伸展的剪切梁模型, 设定梁在静止状态下沿着本体坐标系的e₃轴方向, 柔性附件与刚体的连接点在本体坐标下为b=(0, 0, b)^T。令r₀为单位长度的剪切梁的质量, L为剪切梁的长度, 剪切梁在静止状态下点s∈[0, L]在梁发生小变形时对应的位置为r_b(s), 动量密度为s(s), 令K为剪切梁弹性系数的对角阵, 则剪切梁的能量函数为

$ {H_2} = \frac{1}{2}\int_0^L {\frac{{{{\left\| {\mathit{\boldsymbol{\sigma }}(s)} \right\|}^2}}}{{{\rho _0}}}} \;{\rm{d}}s + \frac{1}{2}\int_0^L {\mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}}} \;{\rm{d}}s。 $

(8)

剪切梁的边界方程为

$\left\{ \begin{array}{l} {\mathit{\boldsymbol{r}}_b}(0) = {\left[ {0,\;0,\;b} \right]^{\rm{T}}} = \mathit{\boldsymbol{b}},\\ {{\mathit{\boldsymbol{r'}}}_b}(L) = {\left[ {0,\;0,\;1} \right]^{\rm{T}}} = {\mathit{\boldsymbol{e}}_3}。 \end{array} \right.$

(9)

因此, 由式(7)和(8)可以得到耦合航天器系统的Hamilton函数为

$ \begin{array}{l} H{\rm{ = }}{H_1}{\rm{ + }}{H_2}\\ \;\;\;{\rm{ = }}\frac{1}{2}\left( {{\mathit{\pmb{\Omega}}^{\rm{T}}}\;\;\;\dot r_{\bar m}^{\rm{T}}} \right)\left( {\begin{array}{*{20}{c}} {{J_S} - \bar m\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_{\bar m}}} \right)}&{\bar m\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_{\bar m}}} \right)}\\ { - \bar m\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_{\bar m}}} \right)}&{\bar m\mathit{\boldsymbol{I}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} \mathit{\pmb{\Omega}}\\ {{{\dot r}_{\bar m}}} \end{array}} \right) + \\ \;\;\;\;\;\frac{1}{2}k\left( {{r_{\bar m}} - r{'_{\bar m}}} \right) \cdot \left( {{r_{\bar m}} - r{'_{\bar m}}} \right){\rm{ + }}\frac{1}{2}\int_0^L {\frac{{{{\left\| {\sigma \left( s \right)} \right\|}^2}}}{{{\rho _0}}}{\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}\int_0^L {\mathit{K}\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}}} {\rm{d}}s, \end{array} $

(10)

可推出刚-液耦合系统总的角动量及晃动质量线动量的表达式为

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{ \boldsymbol{\varPi} }} = {\mathit{\boldsymbol{J}}_{\rm{S}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \bar m{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \bar m{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}},\\ {\mathit{\boldsymbol{P}}_{\bar m}} = - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}\; \end{array} \right. $

(11)

因此, 考虑边界条件, 可得到系统的动力学方程为

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{ \boldsymbol{\dot \varPi} }} = \mathit{\boldsymbol{ \boldsymbol{\varPi} }} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - {\mathit{\boldsymbol{r}}_b}(L) \times \mathit{\boldsymbol{K}}{\mathit{\boldsymbol{e}}_3} + b{{\bf{e}}_3} \times \\ \;\;\;\;\;\;\mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} + \int_0^L {\left( {\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \times \mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}}} \right)} \;{\rm{d}}s,\\ {{\mathit{\boldsymbol{\dot r}}}_b} = \frac{\mathit{\boldsymbol{\sigma }}}{{{\rho _0}}} + \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_b},\\ {\bf{ \pmb{\mathsf{\dot σ}} }} = - \mathit{\boldsymbol{K}}\frac{{{{\rm{d}}^2}{\mathit{\boldsymbol{r}}_b}}}{{{\rm{d}}{s^2}}} - \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\bf{ \pmb{\mathsf{ σ}} }},\\ {{\mathit{\boldsymbol{\dot P}}}_{\bar m}} = {\mathit{\boldsymbol{P}}_{\bar m}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - k({{\bf{r}}_{{\bf{\bar m}}}} - {{{\bf{r'}}}_{{\bf{\bar m}}}})\;。 \end{array} \right. $

(12)

2 稳定性分析

考虑耦合系统不受到外力和外力矩, 则系统的能量和角动量为守恒量。可通过能量函数以及定义Casimir函数, 运用能量-Casimir法判断刚-液-柔系统的稳定性。令

$ C={{\left\| {{\mathit{\boldsymbol{J}}}_{\rm{S}}}\mathit{\boldsymbol{ }}\!\!\mathit{\pmb{\Omega}}\!\!\rm{ }-\bar{m}{{\mathit{\boldsymbol{r}}}_{{\bf{\bar{m}}}}}\times ({{\mathit{\boldsymbol{r}}}_{{\bf{\bar{m}}}}}\times \mathit{\boldsymbol{ }}\!\!\mathit{\pmb{\Omega}}\!\!\rm{ })+\bar{m}{{\mathit{\boldsymbol{r}}}_{{\bf{\bar{m}}}}}\times {{{\mathit{\boldsymbol{\dot{r}}}}}_{{\bf{\bar{m}}}}}+\int_{0}^{L}{{{\mathit{\boldsymbol{r}}}_{b}}\times \mathit{\boldsymbol{ }}\!\!\sigma\!\!\rm{ }\ }\rm{d}s \right\|}^{2}}, $

定义Casimir函数$\psi=\frac{1}{2}{{\psi }_{C}}(C)$以及Casimir函数的一阶导数和二阶导数分别为

$ {\psi }'=\frac{\partial {{\psi }_{C}}}{\partial C},{\psi }''=\frac{{{\partial }^{2}}{{\psi }_{C}}}{\partial {{C}^{2}}}。 $

因此能量函数和Casimir函数的和为

$ \begin{array}{c} H + \psi = \frac{1}{2}\bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \cdot (\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \cdot {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \;\;\frac{1}{2}\bar m{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \cdot {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \frac{1}{2}{\mathit{\boldsymbol{J}}_{\rm{S}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \frac{1}{2}k({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}}) \cdot ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}}) + \\ \;\frac{1}{2}\int_0^L {\frac{{{{\left\| {\mathit{\boldsymbol{\sigma }}(s)} \right\|}^2}}}{{{\rho _0}}}} {\rm{d}}s + \frac{1}{2}\int_0^L {\mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}}} {\rm{d}}s + \psi 。 \end{array} $

(13)

下面求函数H+ψ的一阶变分。定义

$ \left\{ \begin{array}{l} {f_1} = \frac{1}{2}\bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \cdot (\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \cdot \\ \;\;\;\;\;\;{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \frac{1}{2}\bar m{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \cdot {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \frac{1}{2}{J_{\rm{S}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \\ \;\;\;\;\;\;\frac{1}{2}k({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}}) \cdot ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}}),\\ {f_2} = \frac{1}{2}\int_0^L {\frac{{{{\left\| {\mathit{\boldsymbol{\sigma }}(s)} \right\|}^2}}}{{{\rho _0}}}} {\rm{d}}s,\\ {f_3} = \frac{1}{2}\int_0^L {\mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}}\;} {\rm{d}}s \end{array} \right. $

(14)

分别表示刚-液系统的能量函数及剪切梁的动能和势能函数。对以上各式进行变分, 则有

$ \begin{array}{c} {\rm{D}}({f_1}) = ( - \bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - \bar m({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \\ {\mathit{\boldsymbol{J}}_{\rm{S}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}) \cdot {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ (\bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ k({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}})) \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}, \end{array} $

(15)

$ \begin{array}{l} {\rm{D}}({f_2}) = \int_0^L {\frac{{\mathit{\boldsymbol{\sigma }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}}}{{{\rho _0}}}} {\rm{d}}s,\\ {\rm{D}}({f_3}) = \int_0^L {\mathit{\boldsymbol{K}}\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \frac{{\partial {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}}}{{\partial s}}} {\rm{d}}s。 \end{array} $

(16)

根据边界条件$\rm{ }\!\!\delta\!\!\rm{ }{{\mathit{\boldsymbol{r}}}_{b}}(0)=\rm{ }\!\!\delta\!\!\rm{ }{{\mathit{\boldsymbol{r}}}_{b}}(L)=0$, 则D (f₃)可表示为

$ \begin{array}{l} {\rm{D}}({f_3}) = \int_0^L {\mathit{\boldsymbol{K}}\left[ {\frac{\partial }{{\partial s}}\left( {\frac{{\partial {\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \delta {\mathit{\boldsymbol{r}}_b}} \right) - \frac{{{\partial ^2}{\mathit{\boldsymbol{r}}_b}}}{{\partial {s^2}}} \cdot \delta {\mathit{\boldsymbol{r}}_b}} \right]{\rm{d}}s} \\ \;\;\;\;\;\;\;\; = - \int_0^L {\mathit{\boldsymbol{K}}\frac{{{\partial ^2}{\mathit{\boldsymbol{r}}_b}}}{{\partial {s^2}}} \cdot \delta {\mathit{\boldsymbol{r}}_b}{\rm{d}}s} 。 \end{array} $

(17)

定义

$ \mathit{\boldsymbol{\alpha }} = {\mathit{\boldsymbol{J}}_{\rm{S}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \bar m{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \bar m{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times {\mathit{\boldsymbol{\dot r}}_{{\bf{\bar m}}}} + \int_0^L {{\mathit{\boldsymbol{r}}_b} \times \mathit{\boldsymbol{\sigma }}} {\rm{d}}s $

(18)

表示耦合航天器系统的总角动量, 则Casimir函数y的一阶变分为

$ \begin{array}{c} {\rm{D}}\psi (C) = \psi '\mathit{\boldsymbol{\alpha }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\alpha }}\\ = \psi '(\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}}\mathit{\boldsymbol{\alpha }} - \bar m(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \\ \;\;\psi '\bar m(\mathit{\boldsymbol{\alpha }} \times ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + (\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \times \\ \;\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{\alpha }} \times {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}) \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \psi '\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \cdot {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} - \\ \int_0^L {\psi '(\mathit{\boldsymbol{\alpha }} \times \mathit{\boldsymbol{\sigma }}) \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\psi '(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_b}) \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s。 \end{array} $

(19)

根据式(15)~(17)和(19)可以得到函数H+y的一阶变分:

$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \psi '\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \cdot {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\bar m(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;k({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - {{\mathit{\boldsymbol{r'}}}_{{\bf{\bar m}}}}) + \psi '\bar m(\mathit{\boldsymbol{\alpha }} \times ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \mathit{\boldsymbol{\alpha }} \times {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})) \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_0^L {\left( {\mathit{\boldsymbol{K}}\frac{{{\partial ^2}{\mathit{\boldsymbol{r}}_b}}}{{\partial {s^2}}} + \psi '(\mathit{\boldsymbol{\alpha }} \times \mathit{\boldsymbol{\sigma }})} \right) \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\int_0^L {\frac{{\mathit{\boldsymbol{\sigma }} \cdot \delta \mathit{\boldsymbol{\sigma }}}}{{{\rho _0}}}} {\rm{d}}s + \int_0^L {\psi '(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_b}) \cdot {\rm{ \mathsf{ δ} }}{\bf{ \pmb{\mathsf{ σ}} }}} {\rm{d}}s} \right)\;。 \end{array} $

(20)

设定系统的平衡点为$({{\mathit{\pmb{\Omega}}}^{e}}, \ \mathit{\boldsymbol{\dot{r}}}_{{\bf{\bar{m}}}}^{e}, \ \mathit{\boldsymbol{r}}_{{\bf{\bar{m}}}}^{e}, \ {{\mathit{\boldsymbol{ }}\!\!\sigma\!\!\rm{ }}^{e}}, \ \mathit{\boldsymbol{r}}_{b}^{e})$, 令

$ \begin{array}{l} {\mathit{\boldsymbol{\alpha }}^e} = {\mathit{\boldsymbol{J}}_{\rm{S}}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^e} - \bar m\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e \times (\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e \times {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^e}) + \\ \;\;\;\;\;\;\bar m\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e \times \mathit{\boldsymbol{\dot r}}_{{\bf{\bar m}}}^e + \int_0^L {\mathit{\boldsymbol{r}}_b^e \times {\mathit{\boldsymbol{\sigma }}^e}} {\rm{d}}s。 \end{array} $

(21)

在平衡点D (D+y)=0, 下面求二阶变分D²(H+y)。首先考虑能量函数H的二阶变分, 对式(15)进行一阶变分, 可得到f₁的二阶变分:

$ \begin{array}{c} {\rm{D}}({\rm{D}}{f_1}) = {{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}[\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}}]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}[ - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}[\bar m( - {\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \;\;{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \\ \;\;{\mathit{\boldsymbol{S}}^{\rm{T}}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}))]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + {{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[\bar m({\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}))]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[\bar m\mathit{\boldsymbol{I}}]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + {{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[ - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\bar m( - {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - {{\rm{ \mathsf{ δ} }}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - \\ \;\;{\mathit{\boldsymbol{S}}^{\rm{T}}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}))]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + k\mathit{\boldsymbol{I}}]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}\\ = (\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}){\mathit{\boldsymbol{M}}_1}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\\ {{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}} \right), \end{array} $

(22)

其中,

$ {\mathit{\boldsymbol{M}}_1} = \left( {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})}&{\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})}&{\bar m(\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{\bar m}}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{\bar m}}))}\\ { - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})}&{\bar m\mathit{\boldsymbol{I}}}&{\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})}\\ {\bar m(\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{\bar m}}) + \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{\bar m}}))}&{ - \bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})}&{ - \bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + k\mathit{\boldsymbol{I}}} \end{array}} \right)。 $

类似地, 根据式(15)和(17), 并考虑边界条件, 可得到f₂和f₃的二阶变分:

$ \left\{ \begin{array}{l} {\rm{D}}({\rm{D}}{f_2}) = \int_0^L {\frac{{{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}}}{{{\rho _0}}}} {\rm{d}}s,\\ {\rm{D}}({\rm{D}}{f_3}) = - \int_0^L {\mathit{\boldsymbol{K}}\frac{{{\partial ^2}{\rm{ \mathsf{ δ} }}{{\bf{r}}_b}}}{{\partial {s^2}}} \cdot {\rm{ \mathsf{ δ} }}{{\bf{r}}_b}{\rm{d}}s} = \int_0^L {\mathit{\boldsymbol{K}}\frac{{\partial {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \frac{{\partial {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}}}{{\partial s}}{\rm{d}}s。} \end{array} \right. $

(23)

设定K为对角阵, 可采用Pioncare类不等式, 对式(23)的下界进行估计:

$ \int_0^L {\mathit{\boldsymbol{K}}\frac{{\partial {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}}}{{\partial s}} \cdot \frac{{\partial {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}}}{{\partial s}}{\rm{d}}s} \ge c\int_0^L {\mathit{\boldsymbol{K}}{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b} \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s} , $

(24)

其中, $c={{\rm{\pi }}^2}/(4{L^2})$, 因此

$ \begin{array}{c} {{\rm{D}}^2}(H) = {\rm{D}}({\rm{D}}{f_1}) + {\rm{D}}({\rm{D}}{f_2}) + {\rm{D}}({\rm{D}}{f_3})\\ \ge {{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}[\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{J}}_S^{\rm{T}}]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}[ - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}[\bar m( - {\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \\ \;\;{\mathit{\boldsymbol{S}}^{\rm{T}}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}))]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[\bar m({\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}))]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[\bar m\mathit{\boldsymbol{I}}]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + {{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[ - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\bar m( - {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - \\ \;\;{\mathit{\boldsymbol{S}}^{\rm{T}}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}))]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + k\mathit{\boldsymbol{I}}]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}\frac{1}{{{\rho _0}}}\mathit{\boldsymbol{I}}{\rm{ \mathsf{ δ} }}{\bf{ \pmb{\mathsf{ σ}} }}} {\rm{d}}s + c\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}\mathit{\boldsymbol{K}}{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s。} \end{array} $

(25)

下面考虑表达式Casimir函数二阶变分, 对应的表达式为

$ \begin{array}{l} {{\rm{D}}^2}\psi (C) = (2\psi ''\mathit{\boldsymbol{a}} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{a}})\mathit{\boldsymbol{a}} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{a}} + \psi '{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{a}} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{a}} + \psi '\mathit{\boldsymbol{a}} \cdot {\rm{D}}({\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{a}})\\ \;\;\;\;\;\;\;\;\;\;\;\; = (2\psi ''\mathit{\boldsymbol{a}} \otimes \mathit{\boldsymbol{a}} + \psi '\mathit{\boldsymbol{I}})\;({\mathit{\boldsymbol{J}}_{\rm{S}}}{\rm{ \mathsf{ δ} }}\;\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \bar m{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \times ({\mathit{\boldsymbol{r}}_{\bar m}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\bar m{\mathit{\boldsymbol{r}}_{\bar m}} \times ({\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \bar m{\mathit{\boldsymbol{r}}_{\bar m}} \times ({\mathit{\boldsymbol{r}}_{\bar m}} \times {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\bar m{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \times {{\mathit{\boldsymbol{\dot r}}}_{\bar m}} + \bar m{\mathit{\boldsymbol{r}}_{\bar m}} \times {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{\bar m}} + \int_0^L {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} \times \mathit{\boldsymbol{\sigma }}{\rm{d}}s + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left. {\int_0^L {{\mathit{\boldsymbol{r}}_b}} \times {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}{\rm{d}}s} \right)^2} + \psi '\mathit{\boldsymbol{a}} \cdot ( - 2\bar m{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \times ({\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2\;\bar m{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \times ({\mathit{\boldsymbol{r}}_{\bar m}} \times {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - 2\bar m{\mathit{\boldsymbol{r}}_{\bar m}} \times ({\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \times {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2\;\bar m\delta {\mathit{\boldsymbol{r}}_{\bar m}} \times {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{\bar m}} + 2\int_0^L {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} \times {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}{\rm{d}}s), \end{array} $

(26)

展开式(26)右端的后两项, 可得

$ \begin{array}{c} - {{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}\bar m\psi '[{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\bar m\psi '[{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\psi '\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + {{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[\psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[2\psi '\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}[\psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s + \int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}[\psi '\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s。 \end{array} $

(27)

因此, 式(25)可重新表示为

$ \begin{array}{c} \;\;{{\rm{D}}^2}\psi (C)\\ = (2\psi ''\mathit{\boldsymbol{\alpha }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{\alpha }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\alpha }} + \psi '{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\alpha }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\alpha }} + \psi '\mathit{\boldsymbol{\alpha }} \cdot {\rm{D}}({\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\alpha }})\\ = (2\psi ''\mathit{\boldsymbol{\alpha }} \otimes \mathit{\boldsymbol{\alpha }} + \psi '\mathit{\boldsymbol{I}})({\mathit{\boldsymbol{J}}_{\rm{S}}}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \bar m{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \\ \;\;\bar m{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times ({\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \bar m{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times ({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \bar m{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times {{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \;\;\bar m{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \int_0^L {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b} \times \mathit{\boldsymbol{\sigma }}} {\rm{d}}s + \int_0^L {{\mathit{\boldsymbol{r}}_b} \times {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s{)^2} - \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}\bar m\psi '[{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} - \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\bar m\psi '[{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[\psi '\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + {{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}[\psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}[2\psi '\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}[\psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s + \int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\bf{ \pmb{\mathsf{ σ}} }}[\psi '\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s。 \end{array} $

(28)

由式(25)和(28)可得

$ \begin{array}{l} {{\rm{D}}^2}(H + \psi ) \ge (\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}){\mathit{\boldsymbol{R}}_1}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\\ {{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_0^L {(\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}} \end{array}){\mathit{\boldsymbol{T}}_1}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} \end{array}} \right)} {\rm{ d}}s + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2\psi ''\mathit{\boldsymbol{\alpha }} \otimes \mathit{\boldsymbol{\alpha }} + \psi '\mathit{\boldsymbol{I}})([{\mathit{\boldsymbol{J}}_{\rm{S}}} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s{)^2}, \end{array} $

(29)

其中,

$ \begin{array}{l} {\mathit{\boldsymbol{R}}_1} = \left( {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}} - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})}&{\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})}&{{\mathit{\boldsymbol{V}}_1}}\\ { - \bar m(\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}))}&{\bar m\mathit{\boldsymbol{I}}}&{\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})}\\ {{\mathit{\boldsymbol{V}}_2}}&{ - \bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})}&{\bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + k\mathit{\boldsymbol{I}} - 2\psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})} \end{array}} \right)\\ \;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{V}}_1} = \bar m(\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})) - \bar m\psi '( - \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})),\\ \;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{V}}_2} = \bar m(\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})) - \bar m\psi '( - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})),\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{T}}_1} = \left( {\begin{array}{*{20}{c}} {\frac{1}{{{\rho _0}}}\mathit{\boldsymbol{I}}}&{ - \psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})}\\ {\psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})}&{c\mathit{\boldsymbol{K}}} \end{array}} \right)。 \end{array} $

令$\mathit{\boldsymbol{P}}=(2{\psi ''_e}\mathit{\boldsymbol{\alpha }} \otimes \mathit{\boldsymbol{\alpha }} + {\psi '_e}\mathit{\boldsymbol{I}})$, 展开式(29)右端第三项, 可得

$ \begin{array}{l} \mathit{\boldsymbol{P}}{([{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})^2} + \\ \mathit{\boldsymbol{P}}{\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}{\bf{ \pmb{\mathsf{ σ}} }}} {\rm{d}}s} \right)^2} + 2\mathit{\boldsymbol{P}}([{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}) \cdot \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}{\bf{ \pmb{\mathsf{ σ}} }}} {\rm{d}}s} \right)。 \end{array} $

(30)

展开式(30)中的第一项, 可得

$ \begin{array}{l} \mathit{\boldsymbol{P}}{([{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})^2}\\ = (\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}){\mathit{\boldsymbol{R}}_2}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\\ {{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}} \right), \end{array} $

(31)

其中, 矩阵的R₂的表达式为

$ {\mathit{\boldsymbol{R}}_2} = \left( \begin{array}{c} [\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}} - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}[{\mathit{\boldsymbol{J}}_S} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]\\ \bar m[\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}} - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\\ [\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}} - \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})] \end{array} \right. $

$ \begin{array}{c} \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{P}}^{\rm{T}}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]\\ \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\\ \bar m{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})] \end{array} $

$ \left. \begin{array}{c} [{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}){\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - {\mathit{\boldsymbol{S}}^{\rm{T}}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m[{\mathit{\boldsymbol{J}}_S} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]\\ \bar m[{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}){\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - {\mathit{\boldsymbol{S}}^{\rm{T}}}\mathit{\boldsymbol{(}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\\ \bar m[{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}){\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - {\mathit{\boldsymbol{S}}^{\rm{T}}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})] \end{array} \right)\;。 $

令

$ \begin{array}{l} {\mathit{\boldsymbol{R}}_3} = {\mathit{\boldsymbol{R}}_1} + {\mathit{\boldsymbol{R}}_2} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{R}}_{11}}}&{{\mathit{\boldsymbol{R}}_{12}}}&{{\mathit{\boldsymbol{R}}_{13}}}\\ {{\mathit{\boldsymbol{R}}_{21}}}&{{\mathit{\boldsymbol{R}}_{22}}}&{{\mathit{\boldsymbol{R}}_{23}}}\\ {{\mathit{\boldsymbol{R}}_{31}}}&{{\mathit{\boldsymbol{R}}_{32}}}&{{\mathit{\boldsymbol{R}}_{33}}} \end{array}} \right]\; = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{R}}_{11}}}&0&0\\ 0&{{\mathit{\boldsymbol{R}}_{22}}}&0\\ 0&0&{{\mathit{\boldsymbol{R}}_{33}}} \end{array}} \right] + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} 0&{{\mathit{\boldsymbol{R}}_{12}}}&{{\mathit{\boldsymbol{R}}_{13}}}\\ {{\mathit{\boldsymbol{R}}_{21}}}&0&{{\mathit{\boldsymbol{R}}_{23}}}\\ {{\mathit{\boldsymbol{R}}_{31}}}&{{\mathit{\boldsymbol{R}}_{32}}}&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{R}}_{11}}}&0&0\\ 0&{{\mathit{\boldsymbol{R}}_{22}}}&0\\ 0&0&{{\mathit{\boldsymbol{R}}_{33}}} \end{array}} \right] + {\mathit{\boldsymbol{R}}_4}, \end{array} $

(32)

其中,

$ \begin{array}{l} {\mathit{\boldsymbol{R}}_{11}} = [\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})][{\mathit{\boldsymbol{P}}^{\rm{T}}}{\mathit{\boldsymbol{J}}_{\rm{S}}} - {\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{I}}],\\ {\mathit{\boldsymbol{R}}_{12}} = - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})[{\mathit{\boldsymbol{P}}^{\rm{T}}}{\mathit{\boldsymbol{J}}_{\rm{S}}} - {\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - \mathit{\boldsymbol{I}}],\\ {\mathit{\boldsymbol{R}}_{13}} = \bar m(\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})) - \\ \;\;\;\;\;\;\;\;\bar m\psi '( - \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})) + \\ \;\;\;\;\;\;\;\;\bar m[ - \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \\ \;\;\;\;\;\;\;\;\mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})],\\ {\mathit{\boldsymbol{R}}_{21}} = \bar m[\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}}{\mathit{\boldsymbol{P}}^{\rm{T}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{P}}^{\rm{T}}} - \mathit{\boldsymbol{I}}]\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}),\\ {\mathit{\boldsymbol{R}}_{22}} = \bar m\mathit{\boldsymbol{(I}} - \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})),\\ {\mathit{\boldsymbol{R}}_{23}} = \bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }}) + \bar m[ - \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}),\\ {\mathit{\boldsymbol{R}}_{31}} = \bar m(\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})) - \\ \;\;\;\;\;\;\;\bar m\psi '[ - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }} \times {\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})] + \\ \;\;\;\;\;\;\;[\mathit{\boldsymbol{J}}_{\rm{S}}^{\rm{T}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m[\mathit{\boldsymbol{S(}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})], \end{array} $

$ \begin{array}{l} {\mathit{\boldsymbol{R}}_{32}} = - \bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }}) - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})],\\ {\mathit{\boldsymbol{R}}_{33}} = - \bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + k\mathit{\boldsymbol{I}} - 2\psi '\bar m\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\alpha }})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \\ \;\;\;\;\;\;\;\bar m[ - \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) + \\ \;\;\;\;\;\;\;\mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\mathit{\boldsymbol{P}}^{\rm{T}}}\bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S(}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]\;。 \end{array} $

对于式(30)中的第3项, 有

$ \begin{array}{l} 2\mathit{\boldsymbol{P}}([{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ \bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}) \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)\\ = 2\mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + } \right.\\ \left. {\int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}{\bf{ \pmb{\mathsf{ σ}} }}} {\rm{d}}s} \right) + 2\mathit{\boldsymbol{P}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \cdot \\ \;\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right) + \\ 2\mathit{\boldsymbol{P}}\bar m[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \cdot \\ \;\;\;\;\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)\;。 \end{array} $

(33)

下面对式(33)中第一项进行处理, 合并该项及式(30)中δΩ的平方项, 得

$ \begin{array}{l} {{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}{\mathit{\boldsymbol{R}}_{11}}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + 2\mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}\mathit{\boldsymbol{)S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \\ \;\left( { - \int_0^L {\mathit{\boldsymbol{S(\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)\\ = \mathit{\boldsymbol{R}}_{11}^{\rm{T}}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + 2\mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_S} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)\;。 \end{array} $

(34)

令

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{R}}_{11}^{\rm{T}} = \mathit{\boldsymbol{M}}_1^{\rm{T}}{\mathit{\boldsymbol{M}}_1},\\ \mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})] = \mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{M}}_1}, \end{array} \right. $

(35)

则式(34)可表示为

$ \begin{array}{l} \mathit{\boldsymbol{R}}_{11}^{\rm{T}}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + 2\mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)\\ = \mathit{\boldsymbol{M}}_1^{\rm{T}}{\mathit{\boldsymbol{M}}_1}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + 2\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{M}}_1}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)\\ = {\left\| {{\mathit{\boldsymbol{M}}_1}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + {\mathit{\boldsymbol{N}}_1}\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)} \right\|^2} - \\ \mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1}\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right) \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right), \end{array} $

(36)

其中,

$ \begin{array}{l} \mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1} = \mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{M}}_1}{(\mathit{\boldsymbol{M}}_1^{\rm{T}}{\mathit{\boldsymbol{M}}_1})^{ - 1}}{(\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{M}}_1})^{\rm{T}}}\\ \;\;\;\;\;\;\;\;\; = \mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{(\mathit{\boldsymbol{R}}_{11}^{\rm{T}})^{ - 1}} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;{[\mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]]^{\rm{T}}}\;。 \end{array} $

(37)

类似地, 可对式(33)中第二项和第三项进行处理, 令

$ \left\{ \begin{array}{l} \mathit{\boldsymbol{R}}_{22}^{\rm{T}} = \mathit{\boldsymbol{M}}_2^{\rm{T}}{\mathit{\boldsymbol{M}}_2},\\ \bar m\mathit{\boldsymbol{PS}}({\mathit{\boldsymbol{r}}_{\bar m}}) = \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{M}}_2},\\ \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2} = \bar m\mathit{\boldsymbol{PS}}({\mathit{\boldsymbol{r}}_{\bar m}}) \cdot {(\mathit{\boldsymbol{R}}_{22}^{\rm{T}})^{ - 1}}\bar m{[\mathit{\boldsymbol{PS}}({\mathit{\boldsymbol{r}}_{\bar m}})]^{\rm{T}}},\\ \mathit{\boldsymbol{R}}_{33}^{\rm{T}} = \mathit{\boldsymbol{M}}_3^{\rm{T}}{\mathit{\boldsymbol{M}}_3},\\ \mathit{\boldsymbol{P}}[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{\mathit{\boldsymbol{\bar m}}}})] = \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{M}}_3},\\ \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3} = \bar m\mathit{\boldsymbol{P}}[\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{\bar m}})]{(\mathit{\boldsymbol{R}}_{33}^{\rm{T}})^{ - 1}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\bar m{[\mathit{\boldsymbol{P}}(\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{\bar m}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{\bar m}}))]^{\rm{T}}}, \end{array} \right. $

(38)

类似于式(36), 有

$ \begin{array}{l} \mathit{\boldsymbol{R}}_{22}^{\rm{T}}{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \cdot {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + 2\mathit{\boldsymbol{P}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \cdot \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}} {\rm{d}}s + } \right.\\ \left. {\int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}){\rm{ \mathsf{ δ} }}{\bf{ \pmb{\mathsf{ σ}} }}} {\rm{d}}s} \right)\\ = {\left\| {{\mathit{\boldsymbol{M}}_2}{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + {\mathit{\boldsymbol{N}}_2}\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)} \right\|^2} - \\ \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2}\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }})\mathit{\boldsymbol{\delta }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right) \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right),\\ \mathit{\boldsymbol{R}}_{33}^{\rm{T}}{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + 2\mathit{\boldsymbol{P}}\bar m[\mathit{\boldsymbol{S}}({{\bf{r}}_{{\bf{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) + \mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}) - \\ \mathit{\boldsymbol{S}}({{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} \cdot \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)\\ = {\left\| {{\mathit{\boldsymbol{M}}_3}{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}} + {\mathit{\boldsymbol{N}}_3}\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)} \right\|^2} - \\ \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3}\left( { - \int_0^L {S(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {S\mathit{\boldsymbol{(}}{\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right) \cdot \\ \left( { - \int_0^L {S(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}} {\rm{d}}s + \int_0^L {S({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right), \end{array} $

(39)

因此,

$ \begin{array}{l} \mathit{\boldsymbol{R}}_{11}^{\rm{T}}{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} + 2\mathit{\boldsymbol{P}}[{\mathit{\boldsymbol{J}}_{\rm{S}}} - \bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}})]{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right) + \mathit{\boldsymbol{R}}_{22}^{\rm{T}}{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \cdot {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} + \\ 2\mathit{\boldsymbol{P}}\bar m\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}){\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}} \cdot \left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_\mathit{\boldsymbol{b}}}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right) + \\ \mathit{\boldsymbol{P}}{\left( { - \int_0^L {\mathit{\boldsymbol{S}}\left( \mathit{\boldsymbol{\sigma }} \right){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s{\rm{ + }}\int_0^L {\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_b}} \right){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}{\rm{d}}s} } } \right)^2} + \mathit{\boldsymbol{R}}_{33}^{\rm{T}}{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\mathit{\boldsymbol{\bar m}}}} \cdot {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\mathit{\boldsymbol{\bar m}}}}{\rm{ + }}\\ 2\mathit{\boldsymbol{P}}\bar m\left[ {\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_{\mathit{\boldsymbol{\bar m}}}} \times \mathit{\boldsymbol{ \boldsymbol{\varOmega} }}} \right) + \mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_{\mathit{\boldsymbol{\bar m}}}}} \right)\mathit{\boldsymbol{S}}\left( \mathit{\boldsymbol{ \boldsymbol{\varOmega} }} \right) - \mathit{\boldsymbol{S}}\left( {{{\mathit{\boldsymbol{\dot r}}}_{\mathit{\boldsymbol{\bar m}}}}} \right)} \right]{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\mathit{\boldsymbol{\bar m}}}} \cdot \\ \left( { - \int_0^L {\mathit{\boldsymbol{S}}\left( \mathit{\boldsymbol{\sigma }} \right){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s{\rm{ + }}\int_0^L {\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_b}} \right){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}{\rm{d}}s} } } \right)\\ = {\left\| {{\mathit{\boldsymbol{M}}_1}{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{\mathit{\boldsymbol{\bar m}}}} + {\mathit{\boldsymbol{N}}_1}\left( { - \int_0^L {\mathit{\boldsymbol{S}}\left( \mathit{\boldsymbol{\sigma }} \right){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s{\rm{ + }}\int_0^L {\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_b}} \right){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}{\rm{d}}s} } } \right)} \right\|^2} + \\ \;\;{\left\| {{\mathit{\boldsymbol{M}}_2}{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{\mathit{\boldsymbol{\bar m}}}} + {\mathit{\boldsymbol{N}}_2}\left( { - \int_0^L {\mathit{\boldsymbol{S}}\left( \mathit{\boldsymbol{\sigma }} \right){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s{\rm{ + }}\int_0^L {\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_b}} \right){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}{\rm{d}}s} } } \right)} \right\|^2} + \\ \;\;{\left\| {{\mathit{\boldsymbol{M}}_3}{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{\mathit{\boldsymbol{\bar m}}}} + {\mathit{\boldsymbol{N}}_3}\left( { - \int_0^L {\mathit{\boldsymbol{S}}\left( \mathit{\boldsymbol{\sigma }} \right){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}{\rm{d}}s{\rm{ + }}\int_0^L {\mathit{\boldsymbol{S}}\left( {{\mathit{\boldsymbol{r}}_b}} \right){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}{\rm{d}}s} } } \right)} \right\|^2} - \\ \;\;\left( {\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1}{\rm{ + }}\mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2}{\rm{ + }}\mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3} - \mathit{\boldsymbol{P}}} \right).\\ \;\;\;{\left\| {\left( { - \int_0^L {\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} {\rm{d}}s + \int_0^L {\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}} {\rm{d}}s} \right)} \right\|^2}。 \end{array} $

(40)

式(40)右端前三项显然为正, 下面考虑右端第四项, 可表示为

$ \begin{array}{l} - \int_0^L {\int_0^L {\left[ {\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}(s)}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}(s)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} { - {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\sigma }}(s))}\\ {{\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_b}(s))} \end{array}} \right]} } \cdot \\ ((\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1} + \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2} + \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3}) - {\mathit{\boldsymbol{P}}^{\rm{T}}}) \cdot \\ \left[ {\begin{array}{*{20}{c}} { - \mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}(s))}&{\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}(p))} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}(p)}\\ {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}(p)} \end{array}} \right]\;{\rm{d}}s{\rm{d}}p\\ = - \int_0^L {\int_0^L {\left[ {\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}(s)}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}(s)} \end{array}} \right]\mathit{\boldsymbol{U}}\left[ {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}(p)}\\ {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}(p)} \end{array}} \right]} } \;{\rm{d}}s{\rm{d}}p,\\ = - \int_0^L {\int_0^L {\left[ {\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}(s)}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}(s)} \end{array}} \right]\mathit{\boldsymbol{U}}\left[ {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}(p)}\\ {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}(p)} \end{array}} \right]} } \;{\rm{d}}s{\rm{d}}p, \end{array} $

(41)

其中,

$ \begin{array}{l} \mathit{\boldsymbol{U}} = \left[ \begin{array}{l} {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\sigma }}(s))((\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1} + \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2} + \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3}) - {\mathit{\boldsymbol{P}}^{\rm{T}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}(p))\\ - {\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_b}(s))((\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1} + \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2} + \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3}) - {\mathit{\boldsymbol{P}}^{\rm{T}}})\mathit{\boldsymbol{S}}(\mathit{\boldsymbol{\sigma }}(p)) \end{array} \right.\\ \;\;\;\;\;\;\left. \begin{array}{l} - {\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\sigma }}(s))((\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1} + \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2} + \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3}) - {\mathit{\boldsymbol{P}}^{\rm{T}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}(p))\\ {\mathit{\boldsymbol{S}}^{\rm{T}}}({\mathit{\boldsymbol{r}}_b}(s))((\mathit{\boldsymbol{N}}_1^{\rm{T}}{\mathit{\boldsymbol{N}}_1} + \mathit{\boldsymbol{N}}_2^{\rm{T}}{\mathit{\boldsymbol{N}}_2} + \mathit{\boldsymbol{N}}_3^{\rm{T}}{\mathit{\boldsymbol{N}}_3}) - {\mathit{\boldsymbol{P}}^{\rm{T}}})\mathit{\boldsymbol{S}}({\mathit{\boldsymbol{r}}_b}(p)) \end{array} \right]\;。 \end{array} $

(42)

设定λ²为矩阵U的最大特征值, 令${\bar \lambda ^2}=\int_0^L {{\lambda ^2}(s){\rm{d}}s} $, 则根据式(41)可得

$ \begin{array}{l} - \int_0^L {\int_0^L {\left[ {\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}(s)}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}(s)} \end{array}} \right]\mathit{\boldsymbol{U}}\left[ {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}(p)}\\ {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}(p)} \end{array}} \right]} } \;{\rm{d}}s{\rm{d}}p\\ \ge - {{\bar \lambda }^2}\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}(s){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}(s){\rm{d}}s} - {{\bar \lambda }^2}\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}(s){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}(s){\rm{d}}s} , \end{array} $

(43)

因此,

$ \begin{array}{l} - {{\bar \lambda }^2}\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}(s){\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}(s){\rm{d}}s} - {{\bar \lambda }^2}\int_0^L {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}(s){\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}(s){\rm{d}}s} + \\ \int_0^L {(\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}} \end{array}){\mathit{\boldsymbol{T}}_1}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} \end{array}} \right)\;} {\rm{d}}s\\ = \int_0^L {(\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}} \end{array})\left( {\begin{array}{*{20}{c}} {\frac{1}{{{\rho _0}}}\mathit{\boldsymbol{I}} - {{\bar \lambda }^2}\mathit{\boldsymbol{I}}}&{ - \psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})}\\ {\psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})}&{c\mathit{\boldsymbol{K}} - {{\bar \lambda }^2}\mathit{\boldsymbol{I}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} \end{array}} \right)\;} {\rm{d}}s\\ = \int_0^L {(\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}} \end{array}){\mathit{\boldsymbol{T}}_2}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} \end{array}} \right)} \;{\rm{d}}s\;。 \end{array} $

(44)

因此, 式(29)可表示为

$ \begin{array}{l} {{\rm{D}}^2}(H + \psi ) \ge (\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}){\mathit{\boldsymbol{R}}_4}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}}\\ {{\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{{\bf{\bar m}}}}}\\ {\mathit{\boldsymbol{\delta }}{\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}}} \end{array}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_0^L {(\begin{array}{*{20}{c}} {{{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}}&{{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b}} \end{array}){\mathit{\boldsymbol{T}}_2}\left( {\begin{array}{*{20}{c}} {{\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}}\\ {{\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b}} \end{array}} \right)} \;{\rm{d}}s 。 \end{array} $

(45)

由于系统的平衡点为$({{\mathit{\pmb{\Omega}}}^e}, \mathit{\boldsymbol{\dot r}}_{{\bf{\bar m}}}^e, \mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e, {\mathit{\boldsymbol{\sigma }}^e}, \mathit{\boldsymbol{r}}_b^e)$, 且

$ \begin{array}{l} {\mathit{\boldsymbol{\alpha }}^e} = {\mathit{\boldsymbol{J}}_{\rm{S}}}{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^e} - \bar m\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e \times (\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e \times {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^e}) + \\ \;\;\;\;\;\;\bar m\mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e \times \mathit{\boldsymbol{\dot r}}_{{\bf{\bar m}}}^e + \int_0^L {\mathit{\boldsymbol{r}}_b^e \times {\mathit{\boldsymbol{\sigma }}^e}} {\rm{d}}s, \end{array} $

(46)

令$\mathit{\boldsymbol{R}}_4^e$和$\mathit{\boldsymbol{T}}_2^e$分别表示矩阵R₄和T₂在平衡点的值, ${\psi ''_e}$和${\psi '_e}$分别表示$\psi ''$和$\; \psi '$在平衡点的值, 则式(46)在平衡点的值为

$ \begin{array}{l} \;\;{{\rm{D}}^2}{(H + \psi )_{({\Omega ^e},\;\dot r_{\bar m}^e,\;r_{\bar m}^e,\;{\sigma ^e},\;r_b^e)}}\\ \ge ({{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\;\;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{{\mathit{\boldsymbol{\dot r}}}_{\bar m}}\;\;\;{\delta ^{\rm{T}}}{\mathit{\boldsymbol{r}}_{\bar m}})\mathit{\boldsymbol{R}}_4^e\left( \begin{array}{l} {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}\\ {\rm{ \mathsf{ δ} }}{{\mathit{\boldsymbol{\dot r}}}_{\bar m}}\\ {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_{\bar m}} \end{array} \right) + \\ \;\;\int_0^L {({{\rm{ \mathsf{ δ} }}^{\rm{T}}}\mathit{\boldsymbol{\sigma }}\;\;\;{{\rm{ \mathsf{ δ} }}^{\rm{T}}}{\mathit{\boldsymbol{r}}_b})} \mathit{\boldsymbol{T}}_2^e\left( \begin{array}{l} {\rm{ \mathsf{ δ} }}\mathit{\boldsymbol{\sigma }}\\ {\rm{ \mathsf{ δ} }}{\mathit{\boldsymbol{r}}_b} \end{array} \right)\;{\rm{d}}s。 \end{array} $

(47)

根据表达式(47), 可得如下定理。

定理  若矩阵R₄和T₂在平衡点$({\mathit{\pmb{\Omega}}^e}, \; \mathit{\boldsymbol{\dot r}}_{{\bf{\bar m}}}^e, \; \mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e, {\mathit{\boldsymbol{\sigma }}^e}, \mathit{\boldsymbol{r}}_b^e)$是正定的, 则刚-液-柔耦合航天器系统(式(10))是非线性稳定的, 其中,

$ {\mathit{\boldsymbol{R}}_4} = \left( {\begin{array}{*{20}{c}} 0&{{\mathit{\boldsymbol{R}}_{12}}}&{{\mathit{\boldsymbol{R}}_{13}}}\\ {{\mathit{\boldsymbol{R}}_{21}}}&0&{{\mathit{\boldsymbol{R}}_{23}}}\\ {{\mathit{\boldsymbol{R}}_{31}}}&{{\mathit{\boldsymbol{R}}_{32}}}&0 \end{array}} \right),\;\;{\mathit{\boldsymbol{T}}_2} = \left( {\begin{array}{*{20}{c}} {\frac{1}{{{\rho _0}}}\mathit{\boldsymbol{I}} - {{\bar \lambda }^2}\mathit{\boldsymbol{I}}}&{ - \psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})}\\ {\psi '{\mathit{\boldsymbol{S}}^{\rm{T}}}(\mathit{\boldsymbol{\alpha }})}&{c\mathit{\boldsymbol{K}} - {{\bar \lambda }^2}\mathit{\boldsymbol{I}}} \end{array}} \right)\;。 $

3 耦合系统平凡解和数值仿真 3.1 耦合系统平凡解

下面考虑刚-液-柔耦合航天器系统的平凡解(即系统绕着线性剪切梁的轴向进行旋转), 设定该情况下平衡点的形式为$({\mathit{\pmb{\Omega}}^e}, \; \mathit{\boldsymbol{\dot r}}_{{\bf{\bar m}}}^e, \; \mathit{\boldsymbol{r}}_{{\bf{\bar m}}}^e, {\mathit{\boldsymbol{\sigma }}^e}, \mathit{\boldsymbol{r}}_b^e)$, 系统的角速度为${\mathit{\pmb{\Omega}}^e}=\omega _3^e{\mathit{\boldsymbol{e}}_3}$, 表示角速度沿着第三惯性主轴方向。由于梁是未拉伸的, 则有$\mathit{\boldsymbol{r}}_b^e=(b + s){\mathit{\boldsymbol{e}}_3}, {\mathit{\boldsymbol{\sigma }}^e}={\bf{0}}, \; 0 \le s \le l\; .$晃动质量$\bar m$的平衡解为${r_m}=0, \; {\dot r_m}=0$, 对应于晃动质量块静止在平衡位置, 不发生运动。平衡点位置的角动量为${\mathit{\boldsymbol{\alpha }}^e}={j_{33}}\omega _3^e{\mathit{\boldsymbol{e}}_3}$, 根据上式可得到$\psi '$和$\psi ''$在平衡点的值为

$ \left\{ \begin{array}{l} \psi '(||{\mathit{\boldsymbol{\alpha }}^e}|{|^2}) = - \frac{{{\mathit{\boldsymbol{\alpha }}^e} \cdot {\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^e}}}{{||{\mathit{\boldsymbol{\alpha }}^e}|{|^2}}} = - \frac{1}{{{j_{33}}}},\;\\ \psi ''(||{\mathit{\boldsymbol{\alpha }}^e}|{|^2}) = \frac{{{\mathit{\boldsymbol{ \boldsymbol{\varOmega} }}^e} \cdot {\mathit{\boldsymbol{\alpha }}^e}}}{{2||{\mathit{\boldsymbol{\alpha }}^e}|{|^4}}} = \frac{1}{{2{{({j_{33}})}^3}{{(\omega _3^e)}^2}}}。 \end{array} \right. $

(48)

矩阵T₂在平衡点的表达式为

$ \mathit{\boldsymbol{T}}_2^e = \left( {\begin{array}{*{20}{c}} {\frac{1}{{{\rho _0}}} - {\lambda _1}\int_0^L {r_b^2{\rm{d}}s} }&0&0&0&{\omega _3^e}&0\\ 0&{\frac{1}{{{\rho _0}}} - {\lambda _2}\int_0^L {r_b^2{\rm{d}}s} }&0&{ - \omega _3^e}&0&0\\ 0&0&{\frac{1}{{{\rho _0}}}}&0&0&0\\ 0&{ - \omega _3^e}&0&{c{k_x}}&0&0\\ {\omega _3^e}&0&0&0&{c{k_y}}&0\\ 0&0&0&0&0&{c{k_z}} \end{array}} \right), $

(49)

其中,

$ \left\{ \begin{array}{l} {\lambda _1} = \frac{1}{{{j_{33}} - {j_{22}} - \bar ma_1^2 - {m_{\rm{F}}}a_2^2}} + \frac{{\bar ma_1^2}}{{{j_{33}}({j_{33}} - \bar ma_1^2)}} + \frac{{{{\bar m}^2}a_1^2{{(\omega _3^e)}^2}}}{{{j_{33}}(k{j_{33}} - {{\bar m}^2}a_1^2{{(\omega _3^e)}^2} - \bar m{{(\omega _3^e)}^2}{j_{33}})}},\\ {\lambda _2} = \frac{1}{{{j_{33}} - {j_{11}} - \bar ma_1^2 - {m_{\rm{F}}}a_2^2}} + \frac{{\bar ma_1^2}}{{{j_{33}}({j_{33}} - \bar ma_1^2)}} + \frac{{{{\bar m}^2}a_1^2{{(\omega _3^e)}^2}}}{{{j_{33}}(k{j_{33}} - {{\bar m}^2}a_1^2{{(\omega _3^e)}^2} - \bar m{{(\omega _3^e)}^2}{j_{33}})}}。 \end{array} \right. $

(50)

为使得矩阵$\mathit{\boldsymbol{T}}_2^e$为正定的, 则需要满足以下条件:

$ \left\{ \begin{array}{l} c{k_x}\left( {\frac{1}{{{\rho _0}}} - {\lambda _1}\int_0^L {r_b^2} {\rm{d}}s} \right) > {(\omega _3^e)^2},\;\\ c{k_y}\left( {\frac{1}{{{\rho _0}}} - {\lambda _2}\int_0^L {r_b^2{\rm{d}}s} } \right) > {(\omega _3^e)^2},\\ {\lambda _1} < \frac{1}{{{\rho _0}\int_0^L {r_b^2{\rm{d}}s} }},\;{\lambda _2} < \frac{1}{{{\rho _0}\int_0^L {r_b^2{\rm{d}}s} }}\;。 \end{array} \right. $

(51)

考虑矩阵R₄在平衡点的形式, 可得到矩阵R₄^e的各阶顺序主子式均为零, 因此, 矩阵R₄^e为半正定。因此, 若满足式(51), 则耦合航天器系统为非线性稳定的。式(51)的前两个条件是系统稳定自旋的条件, 与仅考虑刚体自旋稳定性的条件相比, 由于液体晃动和柔性附件的影响, 对应的转动惯量需要进行修正。当不考虑液体和柔性附件的影响时, 式(51)退化为刚体自旋稳定的条件。式(51)的后两个条件是对系统转速的限制, 表明刚体的角频率不能超过横截梁的修正特征频率。在不考虑液体晃动的情况下, 稳定性条件(式(51))与文献[6]中含有柔性附件的刚体稳定性条件一致。

3.2 数值仿真

选取刚体航天器的惯性矩阵的参数j₁₁=420 kg·m², j₂₂=385 kg·m², 系统的角速度为$\omega _3^e$; 未晃动燃料的质量为m_F=152.12 kg; 参与晃动的液体质量$\bar m$=60.92 kg, a₁=-0.88 m, a₂=-0.96 m; 弹簧的刚度系数为k=220.21N/m; 单位长度剪切梁的质量r₀=0.3768km/m; 梁的弹性系数为k_x=k_y=k_z=84 N/m; 柔性附件与刚体的连接点在本体坐标下为b=(0, 0, b)^T, b=1.428 m。根据稳定性条件(式(51)), 给出耦合航天器系统在参数空间中的稳定和不稳定区域, 其中参数为刚体航天器转动惯量j₃₃和梁的长度L。

图 3中角速度为$\omega _3^e=1.0\; {\rm{rad}}/{\rm{s}}$, 阴影部分为稳定区域。从图 3可看出, 航天器自旋轴的转动惯量增大会增加航天器姿态的稳定性, 而附件长度的增加会将降低航天器姿态的稳定性。

图 4给出角速度$\omega _3^e$的变化对系统稳定性的影响, 阴影部分为稳定区域。从图 4可以看出, 自旋角速度的增加会降低系统的稳定性。

下面考虑储液腔内液体燃料的变化, 即充液比的改变对耦合系统稳定性的影响。航天器刚体的惯性矩阵

$ {\mathit{\boldsymbol{J}}_{\rm{H}}} = \left[ {\begin{array}{*{20}{c}} {420}&0&0\\ 0&{385}&0\\ 0&0&{520} \end{array}} \right]{\rm{(kg}} \cdot {{\rm{m}}^2}), $

系统的角速度为$\omega _3^e$, 梁的密度为r=300kg/m³, 剪切梁的弹性系数为k_x=k_y=k_z=84N/m, 梁的长为L=6.4m, 半径为r=0.02m, b=1.428m, 梁的单位长度的质量为r₀=0.3768kg/m。设定储液腔的几何形状为球形, 半径R=0.4135m, 储液腔内液体燃料最大质量m_liquid=300kg, 储液腔内实际燃料质量为m_total=m_slosh+m_rest, 其中m_slosh, m_rest分别表示晃动液体质量和静止液体质量。令h=m_total/m_liquid表示储液腔的充液比(0≤h≤1)。通过文献[10-11]可以得到m_slosh/m_total, m_rest/m_total, a₁, a₂随充液比的变化, 结合等效晃动质量可得到等效模型弹簧的刚度。图 5给出了柔性附件和刚体航天器在参数不变的情况下, 0.1≤h≤0.9时稳定区域的分布。从图 5可以看出, 随着充液比的增加, 稳定性区域呈现先减小后增加的趋势。

4 结论

耦合航天器稳定性分析在航天器动力学与控制研究中起着重要作用。本文针对含有柔性附件的充液航天器系统进行稳定性分析。首先, 给出刚-液-柔耦合航天器的力学模型, 通过分析各个部分的能量函数, 得到系统的总能量函数和Casimir函数; 接着, 计算能量-Casimir函数的一阶变分, 得到耦合系统平衡点所满足的条件, 然后计算能量-Casimir函数的二阶变分, 得到耦合系统的非线性稳定性条件; 最后, 给出绕三轴稳定自旋的情况, 得到非线性稳定性条件, 并通过数值仿真验证了相关结论。研究结果显示, 刚体的转动惯量、剪切梁的长度以及储液腔的充液比对系统稳定性有较大的影响。