Reduced-order modeling of geometrically nonlinear structures. Part I: A low-order elimination technique

Abstract

A general methodology for reduced-order modeling of geometrically nonlinear structures is proposed. This approach is built upon equivalent elimination of low-order nonlinear terms by employing a key concept termed ‘passive patterns’, defined to be essential dynamic features of nonlinear structures, produced by, namely slaved to, the active mode via low-order nonlinear effects. Thus, besides the active/dominant structural mode, passive patterns are regarded as secondary ‘energy-containing’ features. Their asymptotic construction procedure is explicitly presented in a weakly nonlinear framework. It is pointed out that both active mode and passive patterns are non-trivial dynamic features of nonlinear structures, and reduction errors of routine Galerkin truncation method are due to incomplete characterization of these passive patterns. Through elimination of their low-order sources (like nonlinearity), the proposed technique fully captures the passive patterns in an equivalent manner and thus leads to refined reduced-order models (ROMs) of the nonlinear structures, while its connection with other existing nonlinear reduction methods is detailed in Part II (Guo and Rega in Nonlinear Dyn, 2023) via an expanded theoretical correspondence.

Appendices

Appendix A

For the active modal coordinate $p_{m} \left( t \right) = A_{m} \left( t \right)e^{{{\text{i}}\,\omega_{m} \,t}} + cc.$, we have

$$ \begin{aligned} \frac{{{\text{d}}p_{m} }}{{{\text{d}}t}} & = {\text{i}}\omega_{m} A_{m} e^{{{\text{i}}\,\omega_{m} \,t}} + \frac{{{\text{d}}A_{m} }}{{{\text{d}}t}} \cdot e^{{{\text{i}}\,\omega_{m} \,t}} + cc., \\ \frac{{{\text{d}}^{2} p_{m} }}{{{\text{d}}t^{2} }} & = - \omega_{m}^{2} p_{m} + \left( {2{\text{i}}\omega_{m} \frac{{{\text{d}}A_{m} }}{{{\text{d}}t}} \cdot e^{{{\text{i}}\,\omega_{m} \,t}} + cc.} \right) = - \omega_{m}^{2} p_{m} + \varepsilon g_{\left( 1 \right)} + \varepsilon^{2} g_{\left( 2 \right)} { + } \cdots \\ \end{aligned} $$

(53)

with $O\left( {{{{\text{d}}A_{m} } \mathord{\left/ {\vphantom {{{\text{d}}A_{m} } {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}}} \right) \ll 1$.

The near-harmonic (slowly modulated) pattern coordinate $P_{{\Omega_{k} }} \left( t \right) \ll p_{m} \left( t \right)$ is denoted by $P_{{\Omega_{k} }} \left( t \right) = G_{{\Omega_{k} }} \left( t \right)e^{{{\text{i}}\,\Omega_{k} \,t}} + cc.$, with its derivatives given by

$$ \begin{aligned} \frac{{{\text{d}}P_{{\Omega_{k} }} }}{{{\text{d}}t}} & = {\text{i}}\Omega_{k} G_{{\Omega_{k} }} e^{{{\text{i}}\,\Omega_{k} \,t}} + \frac{{{\text{d}}G_{{\Omega_{k} }} }}{{{\text{d}}t}} \cdot e^{{{\text{i}}\,\Omega_{k} \,t}} + cc.,\quad \\ \frac{{{\text{d}}^{2} P_{{\Omega_{k} }} }}{{{\text{d}}t^{2} }} & = - \Omega_{k}^{2} P_{{\Omega_{k} }} + \left( {2{\text{i}}\Omega_{k} \frac{{{\text{d}}G_{{\Omega_{k} }} }}{{{\text{d}}t}} \cdot e^{{{\text{i}}\,\Omega_{k} \,t}} + cc.} \right) = - \Omega_{k}^{2} P_{{\Omega_{k} }} + \varepsilon h_{{\Omega_{k} \left( 1 \right)}} + \varepsilon^{2} h_{{\Omega_{k} \left( 2 \right)}} { + } \cdots \\ \end{aligned} $$

(54)

where $O\left( {{{{\text{d}}G_{{\Omega_{k} }} } \mathord{\left/ {\vphantom {{{\text{d}}G_{{\Omega_{k} }} } {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}}} \right) \ll 1$ as $P_{{\Omega_{k} }} \left( t \right)$ is slowly modulated due to nonlinear effects. Note that, although this slow modulation effect is neglected in the current work, it is important for developing more general high-order version of elimination method and will thus be taken into proper consideration in subsequent works.

Appendix B

For the second-order differentiation operator acting on $\hat{H}_{2}^{2}$, i.e., ${{{\hat{\text{d}}}^{2} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}^{2} } {{\hat{\text{d}}}t^{2} }}} \right. \kern-0pt} {{\hat{\text{d}}}t^{2} }}:\;\;\hat{H}_{2}^{2} \to \hat{H}_{2}^{2}$, which is spanned by two monomial bases $\{ e_{1} = p_{m}^{2} ,\;\;e_{2} = \dot{p}_{m}^{2} \}$, its matrix representation $D^{(2)}$ is

$$ \frac{{{\hat{\text{d}}}^{2} }}{{{\hat{\text{d}}}t^{2} }}\left[ {e_{1} ,\;e_{2} } \right] = \left[ {e_{1} ,\;e_{2} } \right]\;\underbrace {{\left[ {\begin{array}{*{20}c} { - 2\omega_{m}^{2} } & {2\omega_{m}^{4} } \\ 2 & { - 2\omega_{m}^{2} } \\ \end{array} } \right]}}_{{D^{(2)} }} $$

(55)

if noting

$$ \begin{aligned} {{{\hat{\text{d}}}e_{1} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}e_{1} } {{\hat{\text{d}}}t}}} \right. \kern-0pt} {{\hat{\text{d}}}t}} & = {{{\text{d}}p_{m}^{2} } \mathord{\left/ {\vphantom {{{\text{d}}p_{m}^{2} } {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}} = 2p_{m} \dot{p}_{m} ,\quad {{{\hat{\text{d}}}^{2} e_{1} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}^{2} e_{1} } {{\hat{\text{d}}}t^{2} }}} \right. \kern-0pt} {{\hat{\text{d}}}t^{2} }} = 2\dot{p}_{m}^{2} - 2\omega_{m}^{2} p_{m}^{2} \\ {{{\hat{\text{d}}}e_{2} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}e_{2} } {{\hat{\text{d}}}t}}} \right. \kern-0pt} {{\hat{\text{d}}}t}} & = {{{\text{d}}\dot{p}_{m}^{2} } \mathord{\left/ {\vphantom {{{\text{d}}\dot{p}_{m}^{2} } {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}} = - 2\omega_{m}^{2} p_{m} \dot{p}_{m}^{{}} ,\quad {{{\hat{\text{d}}}^{2} e_{2} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}^{2} e_{2} } {{\hat{\text{d}}}t^{2} }}} \right. \kern-0pt} {{\hat{\text{d}}}t^{2} }} = 2\omega_{m}^{4} p_{m}^{2} - 2\omega_{m}^{2} \dot{p}_{m}^{2} \\ \end{aligned} $$

(56)

where ${{{\text{d}}^{2} p_{m} } \mathord{\left/ {\vphantom {{{\text{d}}^{2} p_{m} } {{\text{d}}t^{2} }}} \right. \kern-0pt} {{\text{d}}t^{2} }} \approx - \omega_{m}^{2} p_{m}$ and ${{{\hat{\text{d}}}} \mathord{\left/ {\vphantom {{{\hat{\text{d}}}} {{\hat{\text{d}}}t{ = }{\partial \mathord{\left/ {\vphantom {\partial {\partial p_{m} \cdot \dot{p}_{m} }}} \right. \kern-0pt} {\partial p_{m} \cdot \dot{p}_{m} }}}}} \right. \kern-0pt} {{\hat{\text{d}}}t{ = }{\partial \mathord{\left/ {\vphantom {\partial {\partial p_{m} \cdot \dot{p}_{m} }}} \right. \kern-0pt} {\partial p_{m} \cdot \dot{p}_{m} }}}}{ + }{\partial \mathord{\left/ {\vphantom {\partial {\partial \dot{p}_{m} }}} \right. \kern-0pt} {\partial \dot{p}_{m} }} \cdot \left( { - \omega_{m}^{2} p_{m} } \right)$ have been used.

For the second-order differentiation operator acting on $\hat{H}_{2}^{3}$, i.e., ${{{\hat{\text{d}}}^{2} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}^{2} } {{\hat{\text{d}}}t^{2} }}} \right. \kern-0pt} {{\hat{\text{d}}}t^{2} }}:\;\;\hat{H}_{2}^{3} \to \hat{H}_{2}^{3}$, which is spanned by two monomial bases $\{ e_{1} = p_{m}^{3} ,\;\;e_{2} = \dot{p}_{m}^{2} p_{m} \}$, its matrix representation $D^{(3)}$ is

$$ \frac{{{\hat{\text{d}}}^{2} }}{{{\hat{\text{d}}}t^{2} }}\left[ {e_{1} ,\;e_{2} } \right] = \left[ {e_{1} ,\;e_{2} } \right]\;\underbrace {{\left[ {\begin{array}{*{20}c} { - 3\omega_{m}^{2} } & {2\omega_{m}^{4} } \\ 6 & { - 7\omega_{m}^{2} } \\ \end{array} } \right]}}_{{D^{(3)} }} $$

(57)

if noting

$$ \begin{aligned} {{{\hat{\text{d}}}e_{1} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}e_{1} } {{\hat{\text{d}}}t}}} \right. \kern-0pt} {{\hat{\text{d}}}t}} & = {{{\text{d}}p_{m}^{3} } \mathord{\left/ {\vphantom {{{\text{d}}p_{m}^{3} } {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}} = 3p_{m}^{2} \dot{p}_{m} ,\quad {{{\hat{\text{d}}}^{2} e_{1} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}^{2} e_{1} } {{\hat{\text{d}}}t^{2} }}} \right. \kern-0pt} {{\hat{\text{d}}}t^{2} }} = 6p_{m} \dot{p}_{m}^{2} - 3\omega_{m}^{2} p_{m}^{3} \\ {{{\hat{\text{d}}}e_{2} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}e_{2} } {{\hat{\text{d}}}t}}} \right. \kern-0pt} {{\hat{\text{d}}}t}} & = {{{\text{d}}\dot{p}_{m}^{2} p_{m} } \mathord{\left/ {\vphantom {{{\text{d}}\dot{p}_{m}^{2} p_{m} } {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}} = - 2\omega_{m}^{2} p_{m}^{2} \dot{p}_{m} + \dot{p}_{m}^{3} ,\\&{{{\hat{\text{d}}}^{2} e_{2} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}^{2} e_{2} } {{\hat{\text{d}}}t^{2} }}} \right. \kern-0pt} {{\hat{\text{d}}}t^{2} }} = 2\omega_{m}^{4} p_{m}^{3} - 7\omega_{m}^{2} p_{m} \dot{p}_{m}^{2} \\ \end{aligned} $$

(58)

where ${{{\text{d}}^{2} p_{m} } \mathord{\left/ {\vphantom {{{\text{d}}^{2} p_{m} } {{\text{d}}t^{2} }}} \right. \kern-0pt} {{\text{d}}t^{2} }} \approx - \omega_{m}^{2} p_{m}$ and ${{{\hat{\text{d}}}} \mathord{\left/ {\vphantom {{{\hat{\text{d}}}} {{\hat{\text{d}}}t{ = }{\partial \mathord{\left/ {\vphantom {\partial {\partial p_{m} \cdot \dot{p}_{m} }}} \right. \kern-0pt} {\partial p_{m} \cdot \dot{p}_{m} }}}}} \right. \kern-0pt} {{\hat{\text{d}}}t{ = }{\partial \mathord{\left/ {\vphantom {\partial {\partial p_{m} \cdot \dot{p}_{m} }}} \right. \kern-0pt} {\partial p_{m} \cdot \dot{p}_{m} }}}}{ + }{\partial \mathord{\left/ {\vphantom {\partial {\partial \dot{p}_{m} }}} \right. \kern-0pt} {\partial \dot{p}_{m} }} \cdot \left( { - \omega_{m}^{2} p_{m} } \right)$ have been used.

One more remark is regarding the two subspaces $\hat{H}_{2}^{2} \subset H_{2}^{2} ,\;\;\hat{H}_{2}^{3} \subset H_{2}^{3}$. Following dynamical system literature [54], a first-order state space formulation consisting of displacement and velocity fields can be used, e.g., $X = \left[ {w,\;\dot{w}} \right]^{T}$, and one has

$$ \begin{aligned} H_{2}^{2} & = {\text{span}}\left\{ {p_{m}^{2} \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right], \;\dot{p}_{m}^{2} \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right],\; p_{m} \dot{p}_{m} \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right],\; p_{m}^{2} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right],\; \dot{p}_{m}^{2} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right],p_{m} \dot{p}_{m} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right]} \right\}; \\ H_{2}^{3} & = {\text{span}}\left\{ {p_{m}^{3} \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right],\; p_{m} \dot{p}_{m}^{2} \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right],\;\dot{p}_{m}^{3} \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right],\; p_{m}^{2} \dot{p}_{m} \left[ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right],\; p_{m}^{3} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right],\;p_{m} \dot{p}_{m}^{2} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right],\; \dot{p}_{m}^{3} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right],\;p_{m}^{2} \dot{p}_{m} \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right]} \right\} \\ \end{aligned} $$

(59)

By contrast, for the current second-order oscillator formulation, only displacement field occurs in the homological equations, say, Eqs. (18) or (24), meaning that only the first three (four) monomial bases in Eq. (59) are required for describing nonlinear displacement in quadratic (cubic) problems, respectively,

$$ \begin{aligned} \overline{H}_{2}^{2} = {\text{span}}\{ e_{1} = p_{m}^{2} ,\;\;e_{2} = \dot{p}_{m}^{2} \;\} \oplus {\text{span}}\{ e_{3} = p_{m} \dot{p}_{m} \} ; \hfill \\ \overline{H}_{2}^{3} = {\text{span}}\{ e_{1} = p_{m}^{3} ,\;\;e_{2} = p_{m} \dot{p}_{m}^{2} \} \oplus {\text{span}}\{ e_{3} = \dot{p}_{m}^{3} ,\;\;\;e_{4} = p_{m}^{2} \dot{p}_{m} \} \hfill \\ \end{aligned} $$

(60)

Both are denoted as direct sum of two invariant subspaces, with respect to the operator ${{{\hat{\text{d}}}^{2} } \mathord{\left/ {\vphantom {{{\hat{\text{d}}}^{2} } {{\hat{\text{d}}}t^{2} }}} \right. \kern-0pt} {{\hat{\text{d}}}t^{2} }}$. This can be directly verified by similar procedures used in Eqs. (56) or (58).

More explicitly, corresponding to quadratic nonlinearity $p_{m}^{2}$, only the invariant subspace $\hat{H}_{2}^{2} = {\text{span}}\{ e_{1} = p_{m}^{2} ,\;\;e_{2} = \dot{p}_{m}^{2} \}$ is demanded for constructing passive patterns. Similarly, corresponding to cubic nonlinearity $p_{m}^{3}$ and/or $\dot{p}_{m}^{2} p_{m}$, only the invariant subspace $\hat{H}_{2}^{3} = {\text{span}}\{ e_{1} = p_{m}^{3} ,\;\;e_{2} = \dot{p}_{m}^{2} p_{m} \}$ is required for passive patterns. This explains why subspaces $\hat{H}_{2}^{2} ,\;\;\hat{H}_{2}^{3}$ are employed in Sect. 2.3. However, for cubic nonlinearity due to damping, for example, $\dot{p}_{m}^{3}$ or $p_{m}^{2} \dot{p}_{m}$ [55], another invariant subspace ${\text{span}}\{ e_{3} = \dot{p}_{m}^{3} ,\;\;\;e_{4} = p_{m}^{2} \dot{p}_{m} \}$ should be employed to construct passive patterns. The current framework using invariant subspaces of real-valued polynomials is different from normal form analysis using complex-valued notations, built around either linear eigen-frequency [26] or nonlinear response frequency [56, 57].