The Application Of Complex Inner Averaging Method To Discontinuous Oscillating Multi－Degrees－of－Freedom System

Since 60's of last century, nonlinear science is interesting subject of studying nonlinear phenomenon in many scientific feild, which is said as 'third revolution' of natural science in 20 century. Nonlinear dynamics is the one of major research contents of nonlinear science, although many scholars have gotten some accomplishments about it, but for engineering project, there still are a lot of problems to be solved, such as global bifurcation, chaos control and bifurcations control ect.In the article we use double-body resonance box as an example. We studied the discontinuous nonlinear oscillating multi-degrees-of-freedom system. We obtained the bifurcation equation of it in main resonance case by complex inner averaging method. We study it nearly all by computer with Maple. We discussed the dynamic behavior of the system in the case of main resonance and inner resonance.People have not studied inner resonance of resonance box before. When we observe the resonance curve, we found that the ratio of low nature frequency and high frequency near 1:3. When we change the value of system parameter slightly, is easy to make the ratio of low nature frequency and high frequency equal 1:3. So it can appear new phenomena that not found before. With given value of system parameter, we find the amplitude of below box is several times of the comman working amplitude. So it should be avoided in the designing.To the resonance box, the solving of piecewise-linear function is question must solve first. In the case of main reonance, the vibration is made of one mode. We can solve it by Fourier series. But in inner resonance, the vibration is made of two mode. It's very hard to solve it by integral method directly and Fourier series method. So we use the curve-fit method to solve piecewise function. The result is same with the result by the method of integral method.In the computing we use the advantage of complex inner averaging method and take advantage of computer. Compare with real numer averaging method, the complex inner averaging method is simple and quick. The result can be used to the optimization of system parameter and trouble diagnosing.