Efficient dynamics modeling and analysis for probabilistic uncertain beam systems with geometric nonlinearity and thermal coupling effect

Abstract

An adaptive sparse polynomial chaos (PC) transfer matrix method (TMM) is proposed to study the dynamics modeling and analysis of probabilistic uncertain beam systems with geometric nonlinearity and thermal coupling effect. Based on the floating reference frame and the nonlinear elasticity theory, the dynamics equations of flexible beam with geometric nonlinearity and thermal coupling effect are deduced firstly, and then, its transfer equations and transfer matrices and that of the involved hinge elements are established, which can be used to assemble the overall transfer equation of beam system easily according to the system’s topological structure. For a deterministic system, the dynamics of system can be obtained by solving the coupled transfer equation and transient heat conduction equation easily. The proposed TMM doesn’t need the global differential–algebraic equation of system and greatly reduces the involved matrix order and calculation cost, so it is very useful for improving the valuation efficiency of the design of experiment in the subsequent uncertain analysis. When considering the stochastic parameters, above equations are all uncertain equations. Based on the hyperbolic basis truncation scheme, the least angle regression algorithm, and the posteriori cross-validation error estimation method, a nonintrusive adaptive sparse PC expansion method is adopted for uncertain analysis, which only needs a small number of model evaluations and could further reduce the calculation cost of uncertain analysis. The numerical simulation is presented to validate the correctness, feasibility, and computational efficiency of this method. By deducing new transfer equations of related body/hinge, this research can be extended to the dynamics modeling and analysis for general rigid–flexible–thermal coupling multibody systems with probabilistic uncertainties.

Appendix

$$ \begin{aligned} & {\mathbf{M}}_{{e_{k} }} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{{e_{k} ,11}} } & {{\mathbf{M}}_{{e_{k} ,12}} } & {{\mathbf{M}}_{{e_{k} ,13}} } \\ {{\mathbf{M}}_{{e_{k} ,12}}^{{\text{T}}} } & {{\mathbf{M}}_{{e_{k} ,22}} } & {{\mathbf{M}}_{{e_{k} ,23}} } \\ {{\mathbf{M}}_{{e_{k} ,13}}^{{\text{T}}} } & {{\mathbf{M}}_{{e_{k} ,23}}^{{\text{T}}} } & {{\mathbf{M}}_{{e_{k} ,33}} } \\ \end{array} } \right],\quad {\mathbf{Q}}_{{in,e_{k} }} { = }\left[ {\begin{array}{*{20}c} {{\mathbf{Q}}_{{in,e_{k} 1}} } \\ {{\mathbf{Q}}_{{in,e_{k} 2}} } \\ {{\mathbf{Q}}_{{in,e_{k} 3}} } \\ \end{array} } \right],\quad {\mathbf{M}}_{{e_{k} ,11}} = \int_{{V_{{e_{k} }} }} {\rho {\mathbf{I}}{\text{d}}V} = {\mathbf{D}}_{1} \\ & {\mathbf{M}}_{{e_{k} ,12}} = - {\mathbf{A}}_{If} ({\tilde{\mathbf{D}}}_{2} + \sum\limits_{j = 1}^{M} {{\tilde{\mathbf{D}}}_{3,j} \eta_{j} } + \underline{{{\tilde{\mathbf{h}}}_{{1}} {{\varvec{\upeta}}}^{{\text{T}}} {\mathbf{D}}_{10}^{{}} {{\varvec{\upeta}}}}} ){\mathbf{A}}_{If}^{{\text{T}}} ,\quad {\mathbf{M}}_{{e_{k} ,13}} = {\mathbf{A}}_{If} ({\mathbf{D}}_{3} + \underline{{2{\mathbf{h}}_{{1}} {{\varvec{\upeta}}}^{{\text{T}}} {\mathbf{D}}_{10} }} ) \\ & {\mathbf{M}}_{{e_{k} ,22}} = {\mathbf{A}}_{If} \left( {{\mathbf{D}}_{7} + \sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{8,j} + {\mathbf{D}}_{8,j}^{{\text{T}}} } \right)\eta_{j} } + \sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {{\mathbf{D}}_{9,ij} \eta_{i} \eta_{j} } { + }\underline{{\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{11,ij} \eta_{i} \eta_{j} } \right){\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} + } } \left[ {\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{11,ij} \eta_{i} \eta_{j} } \right){\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} } } } \right]^{{\text{T}}} }} } } \right){\mathbf{A}}_{If}^{{\text{T}}} \\ & {\mathbf{M}}_{{e_{k} ,23}} = {\mathbf{A}}_{If} \left( {{\mathbf{D}}_{4} + \sum\limits_{j = 1}^{M} {{\mathbf{D}}_{5,j} \eta_{j} } + \underline{{2\sum\limits_{j = 1}^{M} {{\mathbf{D}}_{12,j} \eta_{j} } { + }\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\tilde{\mathbf{h}}}_{{1}} {\mathbf{D}}_{{{20,}ij}}^{{\text{T}}} + 2{\mathbf{D}}_{{{23,}ij}}^{{\text{T}}} } \right)\eta_{i} \eta_{j} } } }} } \right) \\ & {\mathbf{M}}_{{e_{k} ,33}} = {\mathbf{D}}_{6} { + }\underline{{{2}\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{{5{0},j}} + {\mathbf{D}}_{{5{0},j}}^{{\text{T}}} } \right)\eta_{j} } { + }4\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {{\mathbf{D}}_{{5{2},ij}} \eta_{i} \eta_{j} } } }} \\ \end{aligned} $$

$$ {\mathbf{Q}}_{{in,e_{k} 1}} = - {\mathbf{A}}_{If} {\tilde{\boldsymbol{\omega }}}_{If} \left( {{\tilde{\mathbf{D}}}_{2}^{{\text{T}}} + \sum\limits_{j = 1}^{M} {{\tilde{\mathbf{D}}}_{3,j}^{{\text{T}}} \eta_{j} } + \underline{{{\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} {{\varvec{\upeta}}}^{{\text{T}}} {\mathbf{D}}_{10} {{\varvec{\upeta}}}}} } \right){{\varvec{\upomega}}}_{If} - 2{\mathbf{A}}_{If} \left( {\left( {\sum\limits_{j = 1}^{M} {{\tilde{\mathbf{D}}}_{3,j}^{{\text{T}}} \dot{\eta }_{j} + } \underline{{2{\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} {{\varvec{\upeta}}}^{{\text{T}}} {\mathbf{D}}_{10} {\dot{\boldsymbol{\eta }}}}} } \right){{\varvec{\upomega}}}_{If} + \underline{{{\mathbf{h}}_{{1}} {\dot{\boldsymbol{\eta }}}^{{\text{T}}} {\mathbf{D}}_{10} {\dot{\boldsymbol{\eta }}}}} } \right) $$

$$ \begin{aligned} {\mathbf{Q}}_{{in,e_{k} {2}}} & = - 2{\mathbf{A}}_{If} \left( {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{8,j} + \frac{1}{2}{\mathbf{D}}_{13,j} } \right)\dot{\eta }_{j} + \sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {{\mathbf{D}}_{9,ij} \eta_{i} } \dot{\eta }_{j} + \underline{{2\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{11,ij}^{{}} \eta_{i} \dot{\eta }_{j} } \right){\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} } } }} } } } \right){{\varvec{\upomega}}}_{If} \\ & \quad - {\mathbf{A}}_{If} \sum\limits_{i = 1}^{3} {\omega_{If,i} \left( {{\mathbf{D}}_{14,i} + \sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{15,ij} - {\mathbf{D}}_{15,ij}^{{\text{T}}} } \right)\eta_{j} } + \sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{M} {{\mathbf{D}}_{16,ijk} \eta_{j} \eta_{k} } } } \right)} {{\varvec{\upomega}}}_{If} - \underline{{2{\mathbf{A}}_{If} \sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{11,ij}^{{}} \dot{\eta }_{i} \dot{\eta }_{j} } \right){\mathbf{h}}_{{1}} } } }} \\ & \quad - \underline{{{\mathbf{A}}_{If} \sum\limits_{i = 1}^{3} {\omega_{If,i} \left( {\sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{M} {\left( {{\mathbf{D}}_{11,jk}^{{}} \eta_{j} \eta_{k} } \right){\tilde{\mathbf{h}}}_{i} {\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} } + {\tilde{\mathbf{h}}}_{{1}} {\tilde{\mathbf{h}}}_{i} \sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{M} {\left( {{\mathbf{D}}_{11,jk}^{{\text{T}}} \eta_{j} \eta_{k} } \right)} } } } \right)} {{\varvec{\upomega}}}_{If} }} \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{Q}}_{{in,e_{k} {3}}} & = - \left( {\sum\limits_{j = 1}^{M} {{\mathbf{D}}_{17,j} \dot{\eta }_{j} } + \sum\limits_{i = 1}^{3} {\omega_{If,i} } \left( {{\mathbf{D}}_{18,i} + \sum\limits_{j = 1}^{M} {{\mathbf{D}}_{19,ij} \eta_{j} } + \underline{{\sum\limits_{k = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{20,jk}^{{}} \eta_{j} \eta_{k} } \right){\tilde{\mathbf{h}}}_{i} {\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} } } }} } \right)} \right){{\varvec{\upomega}}}_{If} \\ & \quad - \underline{{\left( {2\sum\limits_{i = 1}^{3} {\omega_{If,i} } \left( {\sum\limits_{j = 1}^{M} {{\mathbf{D}}_{21,ij} \eta_{j} } { + }\sum\limits_{j = 1}^{M} {\sum\limits_{k = 1}^{M} {{\mathbf{D}}_{22,ijk} \eta_{j} } \eta_{k} } } \right)} \right){{\varvec{\upomega}}}_{If} - 2\sum\limits_{k = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{20,jk}^{{}} \dot{\eta }_{j} \dot{\eta }_{k} } \right){\mathbf{h}}_{{1}} } } }} \\ & \quad - \left( {2\sum\limits_{j = 1}^{M} {{\mathbf{D}}_{5,j}^{{\text{T}}} \dot{\eta }_{j} } + \underline{{4\left( {\sum\limits_{k = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{20,jk}^{{}} \dot{\eta }_{j} \eta_{k} } \right){\tilde{\mathbf{h}}}_{{1}}^{{\text{T}}} } } + \sum\limits_{k = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{23,jk} \eta_{j} \dot{\eta }_{k} } \right)} } } \right)}} } \right){{\varvec{\upomega}}}_{If} - \underline{{4\sum\limits_{l = 1}^{M} {\sum\limits_{i = 1}^{M} {\sum\limits_{j = 1}^{M} {\left( {{\mathbf{D}}_{51,lij} \eta_{l} \dot{\eta }_{i} \dot{\eta }_{j} } \right)} } } }} \\ \end{aligned} $$

$$ \begin{aligned} & {\mathbf{D}}_{2} = \int_{{V_{{e_{k} }} }} {\rho {\mathbf{l}}_{{O_{2} p}} {\text{d}}V} ,\,\,\,{\mathbf{D}}_{3,j} = \int_{{V_{{e_{k} }} }} {\rho {\mathbf{N}}_{{pu,{\text{col}}_{j} }} {\text{d}}V} ,\,\,\,{\mathbf{D}}_{3} = \left[ {\begin{array}{*{20}c} {{\mathbf{D}}_{3,1} } & {{\mathbf{D}}_{3,2} } & \cdots & {{\mathbf{D}}_{3,M} } \\ \end{array} } \right] \\ & {\mathbf{D}}_{4} = \int_{{V_{{e_{k} }} }} {\left( {\rho {\tilde{\mathbf{l}}}_{{O_{2} ,p}} {\mathbf{N}}_{pu} + {\mathbf{J}}_{p} {\mathbf{N}}_{p\theta } } \right)} {\text{d}}V,\,\,\,{\mathbf{D}}_{5,j} = \int_{{V_{{e_{k} }} }} \rho {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{j} }} {\mathbf{N}}_{pu} {\text{d}}V,\,\,\,{\mathbf{D}}_{6} = \int_{{V_{{e_{k} }} }} {\left( {\rho {\mathbf{N}}_{pu}^{{\text{T}}} {\mathbf{N}}_{pu} + {\mathbf{N}}_{p\theta }^{{\text{T}}} {\mathbf{J}}_{p} {\mathbf{N}}_{p\theta } } \right)} {\text{d}}V \\ & {\mathbf{D}}_{7} = \int_{{V_{{e_{k} }} }} {\left( {\rho {\tilde{\mathbf{l}}}_{{O_{2} ,p}} {\tilde{\mathbf{l}}}_{{O_{2} P}}^{{\text{T}}} + {\mathbf{J}}_{p} } \right)} {\text{d}}V,\,\,\,{\mathbf{D}}_{8,j} = \int_{{V_{{e_{k} }} }} {\rho {\tilde{\mathbf{l}}}_{{O_{2} ,p}} } {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{j} }}^{{\text{T}}} {\text{d}}V,\,\,\,{\mathbf{D}}_{9,ij} = \int_{{V_{{e_{k} }} }} {\rho {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{i} }} } {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{j} }}^{{\text{T}}} {\text{d}}V \\ & {\mathbf{D}}_{10} = \int_{{V_{{e_{k} }} }} {\rho {{\varvec{\Gamma}}}{\text{d}}V,\begin{array}{*{20}c} {} \\ \end{array} } {\mathbf{D}}_{11,ij} = \int_{{V_{{e_{k} }} }} {\rho {{\varvec{\Gamma}}}_{ij} } {\tilde{\mathbf{l}}}_{{O_{2} ,p}} {\text{d}}V,\,\,\,{\mathbf{D}}_{12,j} = \int_{{V_{{e_{k} }} }} {\rho {\tilde{\mathbf{l}}}_{{O_{2} p}} {\mathbf{h}}_{1} {{\varvec{\Gamma}}}_{j}^{{\text{T}}} } {\text{d}}V,\,\,\,{\mathbf{D}}_{13,j} = \int_{{V_{{e_{k} }} }} {{\mathbf{J}}_{p} {\tilde{\mathbf{N}}}_{{p\theta ,{\text{col}}_{j} }}^{{\text{T}}} } {\text{d}}V \\ & {\mathbf{D}}_{14,j} = \int_{{V_{{e_{k} }} }} {\rho {\tilde{\mathbf{l}}}_{{O_{2} p}} {\tilde{\mathbf{h}}}_{j} {\tilde{\mathbf{l}}}_{{O_{2} P}}^{{\text{T}}} } {\text{d}}V,\,\,\,{\mathbf{D}}_{15,ij} = \int_{{V_{{e_{k} }} }} {\rho {\tilde{\mathbf{l}}}_{{O_{2} p}} {\tilde{\mathbf{h}}}_{i} {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{j} }}^{{\text{T}}} } {\text{d}}V,\,\,\,{\mathbf{D}}_{16,ijk} = \int_{{V_{{e_{k} }} }} {\rho {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{j} }} {\tilde{\mathbf{h}}}_{i} {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{k} }}^{{\text{T}}} } {\text{d}}V \\ & {\mathbf{D}}_{17,j} = \int_{{V_{{e_{k} }} }} {{\mathbf{N}}_{p\theta }^{{\text{T}}} {\mathbf{J}}_{p} {\tilde{\mathbf{N}}}_{{p\theta ,{\text{col}}_{j} }}^{{\text{T}}} } {\text{d}}V,\,\,\,{\mathbf{D}}_{18,j} = \int_{{V_{{e_{k} }} }} {\rho {\mathbf{N}}_{pu}^{{\text{T}}} {\tilde{\mathbf{h}}}_{j} {\tilde{\mathbf{l}}}_{{O_{2} p}}^{{\text{T}}} } {\text{d}}V,\,\,\,{\mathbf{D}}_{19,ij} = \int_{{V_{{e_{k} }} }} {\rho {\mathbf{N}}_{pu}^{{\text{T}}} {\tilde{\mathbf{h}}}_{i} {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{j} }}^{{\text{T}}} } {\text{d}}V \\ & {\mathbf{D}}_{20,ij} = \int_{{V_{{e_{k} }} }} \rho {{\varvec{\Gamma}}}_{ij} {\mathbf{N}}_{pu}^{{\text{T}}} {\text{d}}V,\,\,\,{\mathbf{D}}_{21,ij} = \int_{{V_{{e_{k} }} }} \rho {{\varvec{\Gamma}}}_{j}^{{}} {\mathbf{h}}_{1}^{{\text{T}}} {\tilde{\mathbf{h}}}_{i} {\tilde{\mathbf{l}}}_{{O_{2} p}}^{{\text{T}}} {\text{d}}V,\,\,\,{\mathbf{D}}_{22,ijk} = \int_{{V_{{e_{k} }} }} \rho {{\varvec{\Gamma}}}_{j}^{{}} {\mathbf{h}}_{1}^{{\text{T}}} {\tilde{\mathbf{h}}}_{i} {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{k} }}^{{\text{T}}} {\text{d}}V \\ & {\mathbf{D}}_{23,jk} = \int_{{V_{{e_{k} }} }} \rho {{\varvec{\Gamma}}}_{j}^{{}} {\mathbf{h}}_{1}^{{\text{T}}} {\tilde{\mathbf{N}}}_{{pu,{\text{col}}_{k} }}^{{\text{T}}} {\text{d}}V,\,\,\,{\mathbf{D}}_{50,j} = \int_{{V_{{e_{k} }} }} {\rho {{\varvec{\Gamma}}}_{j} {\mathbf{h}}_{1}^{{\text{T}}} {\mathbf{N}}_{pu} } {\text{d}}V,\,\,\,{\mathbf{D}}_{51,li} = \left[ {\begin{array}{*{20}c} {{\mathbf{D}}_{51,li1} } & {{\mathbf{D}}_{51,li2} } & \cdots & {{\mathbf{D}}_{51,liM} } \\ \end{array} } \right] \\ & {\mathbf{D}}_{51,lij} = \int_{{V_{{e_{k} }} }} {\rho {{\varvec{\Gamma}}}_{l} {{\varvec{\Gamma}}}_{ij} } {\text{d}}V,\,\,\,{\mathbf{D}}_{{5{2},ij}} = \int_{{V_{{e_{k} }} }} {\rho {{\varvec{\Gamma}}}_{j} {{\varvec{\Gamma}}}_{i}^{{\text{T}}} } {\text{d}}V,\,\,\,{\mathbf{h}}_{2} = \left[ {0,1,0} \right]^{{\text{T}}} ,\,\,\,{\mathbf{h}}_{3} { = }\left[ {0,0,1} \right]^{{\text{T}}} \\ \end{aligned} $$

(A.1)

It must be pointed out that, in this study, the product quantities higher than second-order related to the deformed generalized coordinates are ignored, which have little effect on the calculation accuracy in most cases.