【作者】 张小广；

【导师】 胡宇达；

【作者基本信息】 燕山大学 ， 工程力学， 2010， 硕士

【摘要】 功能梯度材料是一种新型的复合材料,通过各组分材料的巧妙结合,可广泛应用于各种要求的材料领域。功能梯度板壳非线性振动问题在工程实际中往往常见,其非线性振动微分方程在数学上的求解具有很大难度,因此研究其板壳的非线性振动具有重要的理论和实际意义。本文致力于研究功能梯度矩形板在外界激励作用下的非线性强迫振动问题。重点研究功能梯度板的主共振和参数振动等问题。首先,根据薄板理论和功能梯度材料的物性参数,应用伽辽金原理导出了功能梯度矩形板在横向简谐激励和面内参数激励分别作用下的非线性振动方程。其次,针对四边固支功能梯度矩形板的主共振问题和四边简支功能梯度矩形板的主参数共振问题,应用多尺度法进行求解,分别导出了系统相应的幅频响应方程,并基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件;针对四边简支功能梯度矩形板的强非线性主共振问题,基于参数变换的思想,应用改进的多尺度法对系统的达芬方程进行了求解,得到了系统的幅频响应方程。最后,通过编程计算,对几种典型的功能梯度材料板进行了分析。分别给出了在不同参数情况下,四边固支功能梯度矩形板主共振和四边简支功能梯度矩形板主参数共振下的幅频曲线及主共振下的动相平面相轨迹图和主参数共振下的解的存在区域图。对于四边简支功能梯度矩形板强非线性主共振的分析,给出其时间历程图、相图及幅频曲线。讨论了阻尼系数、材料组分指数以及板厚等参数对系统非线性振动特性的影响。更多还原

【Abstract】 Functionally Graded Material is a new type of composite materials, which can be widely applied to material areas of various requirements through the ingenious combination of component materials. The nonlinear vibration problems of functionally graded plates and cylindrical shells is particularly common in engineering practice, the solution in mathematics of the nonlinear governing partial equation is very difficult, therefore, the study on the nonlinear vibration of functionally graded plates and cylindrical shells has important theoretical and practical significance.In this paper, nonlinear forced vibration of rectangular functionally graded plates subjected to external excitation is studied. Principal resonance and parameter vibration of FGM plates and other issues are focused discussed. Firstly, according to the classical plate theory and the physical properties parameters of functionally graded materials, the nonlinear forced vibration equations of rectangular functionally graded plates respectively in the harmonic excitation and parametric excitation were deduced by using Galerkin method.Secondly, for principal resonance of FGM rectangular plate with four clamped edges and principal parametric resonance of simply supported FGM rectangular plate, the amplitude-frequency response equations were respectively derived using multi-scale method. Based on the Lyapunov stable theory, the critical conditions of steady-state solutions’stability were gotten. For the strongly nonlinear principal resonance of simply supported FGM rectangular plate, based on parameter transformation, the Duffing equation of system was studied by using modified multi-scale method. The amplitude-frequency response equation was gotten.Finally, several typical functionally graded materials were analyzed. In principal resonance, the amplitude-frequency curves, amplitude-excitation curves of FGM rectangular plate with four clamped edges, the amplitude- frequency curves and time history curves and phase trajectory of simply supported FGM rectangular plate were derived under different situations. In principal parametric resonance, the amplitude-frequency curves and amplitude- damping response curves and the solution’s existence regional diagram of the simply supported FGM rectangular plate were derived under different situations. The influence of different parameters on nonlinear resonance properties of system was discussed.更多还原