Nonlinear flutter behavior of a plate with motion constraints in subsonic flow

Abstract

This paper is aimed at presenting the nonlinear flutter peculiarities of a cantilevered plate with motion-limiting constraints in subsonic flow. A non-smooth free-play structural nonlinearity is considered to model the motion constraints. The governing nonlinear partial differential equation is discretized in space and time domains by using the Galerkin method. The equilibrium points and their stabilities are presented based on qualitative analysis and numerical studies. The system loses its stability by flutter and undergoes the limit cycle oscillations (LCOs) due to the nonlinearity. A heuristic analysis scheme based on the equivalent linearization method is applied to theoretical analysis of the LCOs. The Hopf and two-multiple semi-stable limit cycle bifurcation bifurcations are supercritical or subcritical, which is dependent on the location of the motion constraints. For some special cases the bifurcations are, interestingly, both supercritical and subcritical. The influence of varying parameters on the dynamics is discussed in detail. The results predicted by the analysis scheme are in good agreement with the numerical ones.

Appendix

The elements of each matrix in Eq. 6 are defined as follows:

M is a M × M matrix with the elements

$$ M_{ij} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {1 - \int_{0}^{1} {\kappa \left[ {\varphi_{i} (\xi )} \right]^{2} d\xi } } & {i = j} \\ \end{array} } \\ {\begin{array}{*{20}c} { - \int_{0}^{1} {\kappa \varphi_{i} (\xi )\varphi_{j} (\xi )d\xi } } & {i \ne j} \\ \end{array} } \\ \end{array} } \right. . $$

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C is a M × M matrix with the elements

$$ C_{ij} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\alpha s_{i}^{4} - \frac{1}{\pi }\left[ {2\int_{0}^{1} {\kappa \varphi_{i} (\xi )\varphi^{\prime}_{i} (\xi )d\xi - \int_{0}^{1} {\ln (1 - \xi )\varphi_{i} (\xi )\varphi_{i} (\xi_{r} )d\xi } } } \right]} & {i = j} \\ \end{array} } \\ {\begin{array}{*{20}c} { - \frac{1}{\pi }\left[ {2\int_{0}^{1} {\kappa \varphi_{i} (\xi )\varphi^{\prime}_{j} (\xi )d\xi - \int_{0}^{1} {\ln (1 - \xi )\varphi_{i} (\xi )\varphi_{j} (\xi_{r} )d\xi } } } \right]} & {i \ne j} \\ \end{array} } \\ \end{array} } \right. . $$

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K is a M × M matrix with the elements

$$ K_{ij} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {s_{i}^{4} - \frac{1}{\pi }\left[ {\int_{0}^{1} {\kappa \varphi_{i} (\xi )\varphi^{\prime\prime}_{i} (\xi )d\xi - \int_{0}^{1} {\ln (1 - \xi )\varphi_{i} (\xi )\varphi^{\prime}_{i} (\xi_{r} )d\xi } } } \right]} & {i = j} \\ \end{array} } \\ {\begin{array}{*{20}c} { - \frac{1}{\pi }\left[ {\int_{0}^{1} {\kappa \varphi_{i} (\xi )\varphi^{\prime\prime}_{j} (\xi )d\xi - \int_{0}^{1} {\ln (1 - \xi )\varphi_{i} (\xi )\varphi^{\prime}_{j} (\xi_{r} )d\xi } } } \right]} & {i \ne j} \\ \end{array} } \\ \end{array} } \right. . $$

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F is a M × 1 vector with the elements:

$$ F_{i} = \left\{ {\begin{array}{*{20}l} {k\varphi_{i} (\xi_{r} )\eta_{r} } \hfill & {s_{1} > \eta_{r} } \hfill \\ {k\varphi_{i} (\xi_{r} )s_{1} } \hfill & {s_{1} \le \eta_{r} \le s_{1} + s_{2} } \hfill \\ {k\varphi_{i} (\xi_{r} )\left( {\eta_{r} - s_{2} } \right)} \hfill & {\eta_{r} > s_{1} + s_{2} } \hfill \\ \end{array} } \right. . $$

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where η _r is the dimensionless plate displacement at ξ = ξ _r :

$$ \eta_{r} = \sum\limits_{i = 1}^{M} {\varphi_{i} (\xi_{r} )q_{i} (\tau )} . $$

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