Research on the Flutter of Micro-Scale Cantilever Pipes——A Finite-Dimensional Analysis

GUO Yong; XIE Jianhua

doi:10.21656/1000-0887.370400

Volume 39 Issue 2

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Article Navigation > Applied Mathematics and Mechanics > 2018 > 39(2): 199-214

GUO Yong, XIE Jianhua. Research on the Flutter of Micro-Scale Cantilever Pipes——A Finite-Dimensional Analysis[J]. Applied Mathematics and Mechanics, 2018, 39(2): 199-214. doi: 10.21656/1000-0887.370400

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Research on the Flutter of Micro-Scale Cantilever Pipes——A Finite-Dimensional Analysis

doi: 10.21656/1000-0887.370400

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, P.R.China

Funds: The National Natural Science Foundation of China（11572263）

Received Date: 2016-12-30
Rev Recd Date: 2017-03-06
Publish Date: 2018-02-15

Abstract

Abstract

Based on the modified couple stress theory, the integro-differential equations of motion for micro-scale cantilever pipes were derived by means of Hamilton’s principle. The geometric nonlinearity, arising from the Lagrangian strain tensor, was taken into account. The integro-differential equations were transformed into ordinary differential equations with the Galerkin method. With different numbers of modes in the Galerkin discretization, the diagrams of critical flow velocity vs. mass ratio were given. The difference between the Galerkin approximation results and the exact solutions to the 2-point boundary problem was investigated and the effect of the internal material length scale parameter on the graphs of critical flow velocity vs. mass ratio was studied. For different numbers of modes, the first Lyapunov’s coefficient was calculated and the critical eigenvalue with respect to the flow velocity was derived with the projection method based on the center manifold theory and the normal form method, therefrom, the bifurcation model was analyzed and the effect of the number of modes on the dynamical behaviors was examined. The dynamics of hysteresis and intersection points of the curves of critical flow velocity vs. mass ratio was also investigated and then bifurcation diagrams in different directions were found. Finally, the 6-mode ordinary differential equations of the Galerkin discretization were employed to construct the bifurcation diagrams and verify the relevant results obtained, and the natural frequencies of flutter were calculated through the theoretical analysis and with the numerical method, respectively.
- micro-scale cantilever pipe,
- couple stress theory,
- critical flow velocity curve,
- hysteresis,
- Galerkin method,
- projection method,
- frequency

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References

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