내부공진을 가진 보의 흡인영역

Domains of Attraction of a Forced Beam with Internal Resonance

본 연구에서는 Nayfeh등과 이원경과 소강영은 조화가진하의 핀과 꺽쇠로 고정 된 보(hinged-clamped beam)의 강제 진동 해석을 통하여 점근적으로 안정한 정상상태 응답이 둘 이상 존재할 수 있음을 알게 되었다. 본 연구에서는 이 경우에 접근적으 로 안정한두 정상상태 응답을 구하고 이들 안정한 해로 각각 흡인되는 초기조건들의 집합인 흡인영역을 보간사상법(interpolated mapping method)과 직접 수치적분에 의해 구한 후 서로 비교하였다.

A nonlinear dissipative dynamical system can often have multiple attractors. In this case, it is important to study the global behavior of the system by determining the global domain of attraction of each attractor. In this paper we study the global behavior of a forced beam with two mode interaction. The governing equation of motion is reduced to two second-order nonlinear nonautonomous ordinary differential equations. When .omega. /=3.omega.$_{1}$ and .ohm.=.omega $_{1}$, the system can have two asymptotically stable steady-state periodic solutions, where .omega./ sub 1/, .omega.$_{2}$ and .ohm. denote natural frequencies of the first and second modes and the excitation frequency, respectively. Both solutions have the same period as the excitation period. Therefore each of them shows up as a period-1 solution in Poincare map. We show how interpolated mapping method can be used to determine the two four-dimensional domains of attraction of the two solutions in a very effective way. The results are compared with the ones obtained by direct numerical integration.