動態減振器於非線性簡支樑之減振研究

Vibration Mitigation of a dynamic vibration absorber on a nonlinear simple beam

呂學奇(Hsueh-Chi Lu)

指導教授 ： 王怡仁

淡江大學/工學院/航空太空工程學系碩士班/碩士(2014年)

https://doi.org/10.6846/TKU.2014.00485

摘要

本研究以一非線性Bernoulli-Euler Beam 為主體，將此非線性彈性樑以鋼纜懸掛之，且鋼纜將以非線性彈簧及線性阻尼的組成來模擬其運動。此懸吊系統可模擬一般吊橋之振動方式，若將此懸吊之非線性彈簧與阻尼倒掛置於彈性樑之下，又可模擬此彈性樑置於 Winkler Type 彈性基底的振動型態，因此本研究極具應用價值。本研究中，非線性彈性樑的兩端皆為鉸接，而樑主體上掛載一動態減振器(Dynamic Vibration Absorber (DVA))。本文使用多尺度法(Method of Multiple Scales (MOMS))分析系統於穩態固定點 (Fixed Points) 各模態之頻率響應，且藉由振幅及振動模態觀察其內共振現象。本研究將分析Mass-Spring DVA (MSDVA)的質量、彈性係數及置放於樑上的位置對於此非線性樑的減振效益。再利用Fixed Point plots(Frequency Resp.)並輔之以3D振幅投影圖(3DMACP)全面性的求出此非線性系統之最佳MSDVA的減振組合。

關鍵字

內共振、減振、動態減振器

並列摘要

Beam vibration has always been a concern for researchers and engineers and vibration within nonlinear systems is particularly problematic. This study considered a slender hinged-hinged nonlinear elastic beam with suspension cables simulated using nonlinear cubic springs and linear dampers to allow greater amplitude in the transverse direction. The model in this study could be applied to the engineering of structures with nonlinear suspension systems. In addition, inverting the system, we could simulate the beam placing on a Winkler-type elastic foundation. Therefore, there is a wide range of applications for this system. The primary objective of this study was to add a mass spring dynamic vibration absorber (MSDVA) on the beam to avoid internal resonance within this beam and achieve effective vibration damping. The internal resonance condition based on the ratio of the elastic foundation frequency to the beam frequency of the main structure was obtained. The influence of stretching effect and the location of the mass-spring were also taken into account. We employed the method of multiple scales (MOMS) to analyze this nonlinear problem. The Fixed point plots (steady state frequency response) were obtained. MSDVA with various locations and spring constants were considered and the optimal mass range for the MSDVA to reduce vibration in the main structure was also proposed by using the novel concept of 3-dimensional maximum amplitude contour plots (3D-MACP). The results of this study were verified using numerical simulation, which, in addition to confirming the accuracy by through comparison, established the applicability in this study.

並列關鍵字

Nonlinear beam vibration, Dynamic Vibration Absorber (DVA), Internal resonance, Vibration reduction.

參考文獻

[15] Y. R. Wang, and M. H. Chang, “On The Vibration Reduction of a Nonlinear Support Base with Dual-shock-absorbers,” Journal of Aeronautics, Astronautics and Aviation, Series A, Vol.42, No.3, 2010, pp.179-190.

[2] Z. Ji and J. W. Zu, Method of multiple scales for vibration analysis of rotor-shaft systems with non-linear bearing pedestal model, Journal of Sound and Vibration, 218 ,1998, pp.293-305.

[4] S. W. Shaw, and C. Pierre, “Normal modes for non-linear vibratory systems,” Journal of Sound and vibration, Vol. 164, pp.85-124, 1993.

[5] S. W. Shaw, and C. Pierre, “ Normal modes of vibration for non-linear continuous systems,” Journal of Sound and vibration, Vol. 169, 1994,pp.319-347.

[6] E. Pesheck, and C. Pierre, “A new galerkin-based approach for accurate non-linear normal modes through invariant manifolds,” Journal of Sound and vibration, Vol. 249, No.5, 2002, pp. 971-993.