几类典型粘弹系统的随机动力学研究

Dynamics of viscoelastic systems is currently an active research field. Till now, present results are primarily obtained for deterministic systems. For stochastically simulated viscoelastic systems, there is little achievement and related theory is still in infancy. In practice, most viscoelastic systems are under random excitations; therefore, compared to deterministic systems, the investigation for stochastic dynamics is even more important. This project is aimed at applied fundamental researches for kinds of typical stochastic viscoelastic systems, which mainly contains: develop and establish stochastic Melnikov method and stochastic generalized Melnikov method; study viscoelastic systems excited by bounded noises, especially the stochastic chaos phenomenon for a kind of high-dimensional strong nonlinear viscoelastic systems; establish the FPK equation for a kind of typical stochastic viscoelastic systems, and conduct the research for approximate solution and numerical solution; develop the approximation theory for chaotic polynomial, construct numerical approaches based on analog equation method and Newmark-beta algorithm; employing stochastic multi-scale technique and stochastic spectrum method, complete analytical method for stochastic dynamics of axial-motion beam viscoelastic systems. The objective of this project is to build up a set of scientific analytical and numerical methods for the dynamics of typical viscoelastic systems subjected to random excitations, and to probe a pragmatic research approach for the dynamical analysis of stochastic viscoelastic systems.

粘弹系统动力学的研究是当今最活跃的科学前沿之一。迄今为止，绝大部分的成果都属于确定性系统的研究，对随机激励下粘弹系统的动力学研究很少，理论上也不成熟。而实际问题中，大部分粘弹系统属于随机激励的粘弹系统，与研究确定性相比研究随机系统更为重要。本项目拟围绕几类典型的随机粘弹系统展开应用基础研究，主要内容包括：发展和建立随机Melnikov方法和随机广义Melnikov方法，研究有界噪声激励下粘弹系统，特别是一类高维强非线性粘弹系统的随机混沌现象；研究一类典型随机粘弹系统FPK方程的建立，开展逼近解和数值解的研究；发展和构建随机粘弹系统的混沌多项式的逼近理论和基于模拟方程方法、Newmark-beta算法的数值分析方法；基于随机多尺度方法、随机谱方法，构造和建立轴向运动梁粘弹系统随机动力学的分析方法。目标是初步建立一套随机激励下典型粘弹系统动力学的分析理论和数值方法，推动随机动力学学科的发展。

通过四年的努力和系统研究, 按照申请书的研究思路和技术路线，顺利完成了预定的研究计划。通过对粘弹系统的动力学分析，取得了一系列成果，主要在如下几个方面取得进展：发展和改进了随机Melnikov 方法、随机广义谐波平衡方法、随机线性化方法，主要研究了几类粘弹系统包括：分数阶导数描述的粘弹系统、粘弹隔振系统以及弹粘碰撞系统的研究；研究了这些典型随机粘弹系统的响应、分岔、稳定性和控制，开展了逼近解和数值解的研究；发展和构建随机粘弹系统的逼近理论。初步建立一套随机激励下典型粘弹系统动力学的分析理论和数值方法。按计划完成项目研究内容。.主办国际会议1个，主办国内会议4个，组织力学大会专题会2个，参加各类专业学术会80人次，参加项目的人员12人，包括博士生和硕士生9人。本项目在Chaos、Nonlinear Dynamics、Physical A、International Journal of Non-Linear Mechanics、Physics Letters A、International Journal of Bifurcation and Chaos 等国际和国内著名杂志发表SCI 论文20 篇。