受單向拘束系統之振動分析

本論文分成兩個部分，第一部分是自由震動，以一個自由度以及兩個自由度的系統做為範例，討論單邊接觸系統在沒有間隙情況下的自由震動，系統中的非線性是不可忽略的。在一個自由度的例子中，質量一邊連接彈簧，另一邊則是與另一個彈簧單向接觸，在這個例子中可以發現，當單向接觸彈簧的彈簧質量趨近於無窮大時，頻率會是彈簧常數為零時的兩倍。這個例子，可以應用在懸臂樑以及懸臂樑尾端受單向拘束。在兩個自由度的例子中，頻率與非線性正規模態 (nonlinear normal mode) 有關，我們透過打靶法(shooting method)找出非線性正規模態，在這個例子中，頻率與振幅之間無關。在系統的兩個模態中，兩個質量並不會同時回到原點，這點與傳統的線性或非線性系統不一樣，但是在速度的部分兩個質量會同時為零。同時，我們也架設了實驗機構量測在不同條件下的頻率。第二部份分析三個自由度的系統在受到週期性外力下的穩態響應，這個系統中包含了兩個質量、三個彈簧以及基底，在這三個彈簧中，其中一個是採單向接觸，這個單向接觸的彈簧與質量之間沒有間隙，首先，以數值方法模擬得到系統穩態之後的poincare map，透過poincare map可以知道不同的外力頻率對於系統穩態之後會有不同的響應，分別是P-1、P-2、P-3、P-4、P-5、P-7、P21、P24、準週期性以及混沌，接著再將頻率從11.14Hz到33.42Hz畫成頻率與週期性關係圖。數值模擬完之後，我們以實驗驗證結果，在實驗中，以激振器施予一個週期性外力在基底上，並且透過雷射位移計量測其中一個質量的運動軌跡，並針對P-1、P-2、P-3、P-4以及混沌進行實驗，實驗結果與我們預期的相符。

關鍵字

自由震動；單邊接觸；分離式系統；穩態響應；週期性；混沌

並列摘要

In this thesis we investigate the free vibration of a discrete system constrained by a unilateral spring without any gap. A 1-dof system and a 2-dof system are taken as examples. These systems are highly nonlinear and cannot be linearized. For a 1-dof system, the mass is supported by a suspension spring and is constrained by a unilateral spring. It is found that the natural frequency of the mass-spring oscillator constrained by a unilateral rigid stop is two times the frequency of the same mass-spring oscillator when the rigid stop is absent. This property can be verified by a cantilever beam constrained by a rigid stop at the tip. For a 2-dof system constrained by a unilateral spring, the natural frequencies and the associated nonlinear normal mode can be obtained by shooting method. It is found that the natural frequency is independent of the amplitude of the mode shape. In both modes, the two masses do not return to their equilibrium positions at the same time. In other words, the two masses do not vibrate in unison. This is contrary to the conventional oscillators with bilateral springs, linear or nonlinear. However, the velocities of the two masses do vanish at the same time. The predicted natural frequencies of a 2-dof oscillator constrained by a unilateral spring can be verified experimentally with a prototype of the 2-dof oscillator. Next we study the steady state response of a three-degree-of-freedom mechanical system under harmonic excitations. The mechanical system contains a chain of two masses and three springs, which mounted on a platform. Among the three springs one of them is in unilateral contact with one of the mass. When in equilibrium configuration there is no gap between the unilateral spring and the neighboring mass. A shaker is used to exert specified harmonic force onto the platform. A laser displacement sensor is used to measure the displacement history of one of the masses. In the theoretical predictions a bifurcation diagram with excitation frequency ranging from 11.14 to 33.42 Hz is established by Poincare sampling. Very rich steady state response can be found in the bifurcation diagram, which includes which includes P-1, P-2, P-3, P-4, P-5, P-7, P21, P24, quasi-periodic, and chaotic motions. In the experiment we present five of them, which includes P-1, P-2, P-3, P-4, and chaotic motions, respectively. These Poincare maps agree well with the theoretical predictions.

並列關鍵字

Free vibration ； unilateral contact ； discrete system ； steady state response ； periodic motion ； chaotic motion

參考文獻

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受單向拘束系統之振動分析

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