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Modal truncation method for continuum structures based on matrix norm: modal perturbation method

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Abstract

Modal analysis is a widely applied method to study the vibration phenomenon of continuum structures, but there is no clear method to solve the modal truncation problem at present. To determine the contribution of different modes to the whole system, a new mode truncation method based on perturbation theory is proposed in this paper. The modes are subjected to perturbation parameters during discretization, and using norm error analysis on the stiffness matrix in different degrees of freedom (DOFs) systems confirms the model number of the continuum structure system. The results show that the DOF identified by the modal perturbation method is related to the perturbation parameter, and the smaller the perturbation parameter is, the fewer modes need to be considered. When the perturbation parameter is large enough, the response of the system can only be accurately explained by truncation to higher-order modes. Finally, the perturbation parameter is fixed to 1, and the traditional Galerkin method is connected to the modal perturbation, making traditional discretization a unique case for the modal perturbation method. This method can significantly reduce the modal truncation error, which is of great significance to the dynamic analysis of engineering applications.

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All data generated or analysed during this study are included in this published article (and its Supplementary Information files.

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Funding

The authors wish to acknowledge the support of the National Natural Science Foundation of China (11972151, 12372006, 12302007, and 12202109), and also the Specific Research Project of Guangxi for Research Bases and Talents (AD23026051).

Author information

Authors and Affiliations

  1. School of Civil and Architectural Engineering, Guangxi University, Nanning, 530004, China

    Houjun Kang, Quan Yuan, Xiaoyang Su, Tieding Guo & Yunyue Cong

  2. Scientific Research Center of Engineering Mechanics, Guangxi University, Nanning, 530004, China

    Houjun Kang, Xiaoyang Su, Tieding Guo & Yunyue Cong

Authors
  1. Houjun Kang
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  2. Quan Yuan
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  3. Xiaoyang Su
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  5. Yunyue Cong
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Correspondence to Houjun Kang.

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Appendix

Appendix

SDOF:

$$\begin{aligned}{} & {} {{\ddot{q}}_1} + 2c{{\dot{q}}_1} + \left( \int _0^1 {{\varphi _1}{\varphi _1}{{^{\prime \prime }}^{\prime \prime }}\textrm{d}x} \right. \nonumber \\{} & {} \qquad \left. - \int _0^1 {{\varphi _1}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1} \nonumber \\{} & {} \qquad -\, \left( \frac{1}{4}\int _0^1 {{\varphi _1}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}{{^\prime }^2}} \textrm{d}x \right. \nonumber \\{} & {} \qquad \left. + \int _0^1 {{\varphi _1}{\varphi _1}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1}^2\nonumber \\{} & {} \qquad -\, \frac{1}{4}\int _0^1 {{\varphi _1}{\varphi _1}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _1}{{^\prime }^2}} \textrm{d}x \cdot {q_1}^3\nonumber \\{} & {} \quad = \int _0^1 {{\varphi _1}\textrm{d}x} \cdot F\cos \Omega t \end{aligned}$$
(35)

Two-DOF:

$$\begin{aligned}{} & {} {{\ddot{q}}_1} + 2c{{\dot{q}}_1} + \left( \int _0^1 {{\varphi _1}{\varphi _1}{{^{\prime \prime }}^{\prime \prime }}\textrm{d}x}\right. \nonumber \\{} & {} \qquad \left. - \int _0^1 {{\varphi _1}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1} \\{} & {} \qquad +\, \varepsilon \left( \int _0^1 {{\varphi _1}{\varphi _2}{{^{\prime \prime }}^{\prime \prime }}\textrm{d}x} \right. \nonumber \\{} & {} \qquad \left. - \int _0^1 {{\varphi _1}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _2}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_2} \\{} & {} \quad = \left( \frac{1}{4}\int _0^1 {{\varphi _1}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}{{^\prime }^2}} \textrm{d}x\right. \nonumber \\{} & {} \qquad \left. + \int _0^1 {{\varphi _1}{\varphi _1}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1}^2 \\{} & {} \qquad +\, \varepsilon \left( \frac{1}{2}\int _0^1 {{\varphi _1}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {\varphi _2}^\prime } \textrm{d}x \right. \nonumber \\{} & {} \qquad \left. + \int _0^1 {{\varphi _1}{\varphi _1}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _2}^\prime {w_0}^\prime } \textrm{d}x \right. \\{} & {} \qquad \left. +\, \int _0^1 {{\varphi _1}{\varphi _2}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1}{q_2} \\{} & {} \qquad +\, {\varepsilon ^2}\left( \frac{1}{4}\int _0^1 {{\varphi _1}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _2}{{^\prime }^2}\textrm{d}x} \right. \nonumber \\{} & {} \qquad \left. + \int _0^1 {{\varphi _1}{\varphi _2}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _2}^\prime {w_0}^\prime \textrm{d}x} \right) \cdot {q_2}^2 \\{} & {} \qquad +\, \frac{1}{4}\int _0^1 {{\varphi _1}{\varphi _1}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _1}{{^\prime }^2}} \textrm{d}x \cdot {q_1}^3 \\{} & {} \qquad +\, \frac{1}{4}\varepsilon \left( 2\int _0^1 {{\varphi _1}{\varphi _1}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _1}^\prime {\varphi _2}^\prime } \textrm{d}x \right. \\{} & {} \qquad \left. +\, \int _0^1 {{\varphi _1}{\varphi _2}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _2}{{^\prime }^2}} \textrm{d}x \right) \cdot {q_1}^2{q_2} \\{} & {} \qquad +\, \frac{1}{4}{\varepsilon ^2}\left( 2\int _0^1 {{\varphi _1}{\varphi _2}^{\prime \prime }dx} \cdot \int _0^1 {{\varphi _1}^\prime {\varphi _2}^\prime } \textrm{d}x\right. \\{} & {} \qquad \left. +\, \int _0^1 {{\varphi _1}{\varphi _1}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _2}{{^\prime }^2}} \textrm{d}x \right) \cdot {q_1}{q_2}^2 \\{} & {} \qquad +\, \frac{1}{4}{\varepsilon ^3}\int _0^1 {{\varphi _1}{\varphi _2}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _2}{{^\prime }^2}} \textrm{d}x \cdot {q_2}^3\\{} & {} \qquad +\, \int _0^1 {{\varphi _1}\textrm{d}x} \cdot F\cos \Omega t \end{aligned}$$
$$\begin{aligned}{} & {} \varepsilon {{\ddot{q}}_2} + 2c\varepsilon {{\dot{q}}_2} + \varepsilon \left( \int _0^1 {{\varphi _2}{\varphi _1}{{^{\prime \prime }}^{\prime \prime }}\textrm{d}x} \right. \nonumber \\{} & {} \qquad \left. -\, \int _0^1 {{\varphi _2}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1} \nonumber \\{} & {} \qquad +\, {\varepsilon ^2}\left( \int _0^1 {{\varphi _2}{\varphi _2}{{^{\prime \prime }}^{\prime \prime }}\textrm{d}x}\right. \nonumber \\{} & {} \qquad \left. - \int _0^1 {{\varphi _2}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _2}^\prime {w_0}^\prime } \textrm{d}x\right) \cdot {q_2} \nonumber \\{} & {} \quad = \varepsilon \left( \frac{1}{4}\int _0^1 {{\varphi _2}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}{{^\prime }^2}} \textrm{d}x \right. \nonumber \\{} & {} \qquad \left. + \int _0^1 {{\varphi _2}{\varphi _1}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1}^2 \nonumber \\{} & {} \qquad +\, {\varepsilon ^2}\left( \frac{1}{2}\int _0^1 {{\varphi _2}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {\varphi _2}^\prime } \textrm{d}x + \int _0^1 {{\varphi _2}{\varphi _1}^{\prime \prime }\textrm{d}x} \right. \nonumber \\{} & {} \qquad \left. \int _0^1 {{\varphi _2}^\prime {w_0}^\prime } \textrm{d}x + \int _0^1 {{\varphi _2}{\varphi _2}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _1}^\prime {w_0}^\prime } \textrm{d}x \right) \cdot {q_1}{q_2} \nonumber \\{} & {} \qquad +\, {\varepsilon ^3}\left( \frac{1}{4}\int _0^1 {{\varphi _2}{w_0}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _2}{{^\prime }^2}\textrm{d}x} \right. \nonumber \\{} & {} \qquad \left. + \int _0^1 {{\varphi _2}{\varphi _2}^{\prime \prime }\textrm{d}x} \int _0^1 {{\varphi _2}^\prime {w_0}^\prime \textrm{d}x} \right) \nonumber \\{} & {} \quad \cdot {q_2}^2 + \frac{1}{4}\varepsilon \int _0^1 {{\varphi _2}{\varphi _1}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _1}{{^\prime }^2}} \textrm{d}x \cdot {q_1}^3 \nonumber \\{} & {} \qquad +\, \frac{1}{4}{\varepsilon ^2}\left( 2\int _0^1 {{\varphi _2}{\varphi _1}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _1}^\prime {\varphi _2}^\prime } \textrm{d}x \right. \nonumber \\{} & {} \qquad \left. +\, \int _0^1 {{\varphi _2}{\varphi _2}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _2}{{^\prime }^2}} \textrm{d}x \right) \cdot {q_1}^2{q_2} \nonumber \\{} & {} \qquad +\, \frac{1}{4}{\varepsilon ^3}\left( 2\int _0^1 {{\varphi _2}{\varphi _2}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _1}^\prime {\varphi _2}^\prime } \textrm{d}x \right. \nonumber \\{} & {} \qquad \left. +\, \int _0^1 {{\varphi _2}{\varphi _1}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _2}{{^\prime }^2}} \textrm{d}x \right) \cdot {q_1}{q_2}^2 \nonumber \\{} & {} \qquad +\, \frac{1}{4}{\varepsilon ^4}\int _0^1 {{\varphi _2}{\varphi _2}^{\prime \prime }\textrm{d}x} \cdot \int _0^1 {{\varphi _2}{{^\prime }^2}} \textrm{d}x \cdot {q_2}^3 \nonumber \\{} & {} \qquad +\, \varepsilon \int _0^1 {{\varphi _2}\textrm{d}x} \cdot F\cos \Omega t \end{aligned}$$
(36)

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Kang, H., Yuan, Q., Su, X. et al. Modal truncation method for continuum structures based on matrix norm: modal perturbation method. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09628-2

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