拱形樑於端點受到諧和激振之非線性振動實驗及理論

Abstract

本文以實驗方法與理論研究淺拱樑於端點聯結電磁激振器之非線性振動。在實驗方面，透過控制流經激振器之電流產生施加於激振器中央線圈的簡諧磁力，而此時端點受到簡諧力之拱形樑其端點運動卻不一定為簡諧運動，尤其是拱形樑之側向運動的振幅很大時。當激振頻率靠近第n個自然頻率時，本文中發現拱形樑外形之幾何瑕疵為能否激發出第n個模態的關鍵，並採用多尺度法分析以求得穩態響應振幅之解析解與幾何瑕疵之間的關係。此外，當激振頻率靠近第n個自然頻率之兩倍時，可同時存在兩個不同振幅之穩定穩態響應，且在此頻率附近將振動頻率由低頻調向高頻時，可觀察到由於穩定穩態解的不連續所造成之振幅躍增現象，對此案例而言，幾何瑕疵的影響非常小，而多尺度分析不僅僅預測出穩定與非穩定穩態解之振幅與相角，且預測出該振幅躍增現象的發生。

In this paper we study, both theoretically and experimentally, the nonlinear vibration of a shallow arch with one end attached to an electro-mechanical shaker. In the experiment we generate harmonic magnetic force on the central core of the shaker by controlling the electric current flowing into the shaker. The end motion of the arch is in general not harmonic, especially when the amplitude of lateral vibration is large. In the case when the excitation frequency is close to the n-th natural frequency of the arch, we found that geometrical imperfection is the key for the n-th mode to be excited. Analytical formula relating the amplitude of the steady state response and the geometrical imperfection can be derived via a multiple scale analysis. In the case when the excitation frequency is close to two times of the n-th natural frequency two stable steady state responses can exist simultaneously. As a consequence jump phenomenon is observed when the excitation frequency sweeps upward. The effect of geometrical imperfection on the steady state response is minimal in this case. The multiple scale analysis not only predicts the amplitudes and phases of both the stable and unstable solutions, but also predicts analytically the frequency at which jump phenomenon occurs.