The chapter is addressed at phenomenological mapping and mathematical analogies of oscillatory regimes in hybrid discrete-continuum systems of coupled deformable bodies. Systems consist of connected deformable bodies like plates, beams, belts, or membranes that are coupled through visco-elastic non-linear layer. The layer is modeled by continuously distributed elements of Kelvin–Voigt type with non-linearity of third order. Using the mathematical analogies, the similarities of structural models in systems of plates, beams, belts, or membranes are explained. The mathematical models consist by a set of two coupled non-homogenous partial non-linear differential equations. The proposed solution is divided into space and time domains by classical Bernoulli–Fourier method. In the time domains, the systems of coupled ordinary non-linear differential equations are completely analog for different systems of deformable bodies and are solved using the Krilov–Bogolyubov–Mitropolski asymptotic method. This paper presents the power of mathematical analytical calculus which is similar for physically different systems. The mathematical numerical experiments are a great and useful tool for making the final conclusions between many input and output values. The conclusions about non-linear phenomena in multi-body systems dynamics are revealed from the specific example of double plate’s system stationery and no stationary oscillatory regimes.

本章讨论耦合变形体的混合离散-连续系统中振荡机制的现象学映射和数学类比。系统由连接的可变形体组成，如通过粘弹性非线性层耦合的板、梁、带或膜。该层由具有三阶非线性的 Kelvin-Voigt 型连续分布元素建模。使用数学类比，解释了板、梁、带或膜系统中结构模型的相似性。数学模型由一组两个耦合的非齐次偏非线性微分方程组成。所提出的解决方案通过经典的伯努利 - 傅立叶方法分为空间和时间域。在时域中，耦合常非线性微分方程组对于不同的可变形体系统完全模拟，并使用 Krilov-Bogolyubov-Mitropolski 渐近方法求解。本文介绍了数学分析微积分的威力，它对于物理上不同的系统是相似的。数学数值实验是在许多输入和输出值之间得出最终结论的重要且有用的工具。多体系统动力学中非线性现象的结论是从双板系统平稳和无平稳振荡制度的具体例子中揭示出来的。本文介绍了数学分析微积分的威力，它对于物理上不同的系统是相似的。数学数值实验是在许多输入和输出值之间得出最终结论的重要且有用的工具。多体系统动力学中非线性现象的结论是从双板系统平稳和无平稳振荡制度的具体例子中揭示出来的。本文介绍了数学分析微积分的威力，它对于物理上不同的系统是相似的。数学数值实验是在许多输入和输出值之间得出最终结论的重要且有用的工具。多体系统动力学中非线性现象的结论是从双板系统平稳和无平稳振荡制度的具体例子中揭示出来的。