The purpose of this work is the investigation of the dynamical behavior of a mechanical network consisting of discontinuous elastically coupled system oscillators with strong irrational nonlinearities. This mechanical network is consisted of a set of N nonlinear oscillators, elastically coupled where nonlinearity in each unit cell is just due to the geometric configuration, which is the inclination of linear strings. By using the Lagrangian formulation, the set of model equations governing the dynamics of the system are established. These set of equations have strong irrational nonlinearities, with smooth or discontinuous characteristics depending just to the inclination angles of strings, and they are used to study the steady states as well as their stabilities analysis. Then the behaviors of the system when its frequency approaches it eigenfrequency are studied, the well-known resonance phenomenon, which shows the appearing of hysteresis as the frequency shift increases while the displacements of masses tend to infinite. Next, the partial differential equation governing the dynamics of continuous signals in the system is derived. This partial differential equation admits for strong amplitude the train of envelope bursting like signal as solution which is very interesting in this work, and for weak amplitude the well-known envelope like pulse or kink soliton as well as elliptic solutions, with their amplitudes just functions of the inclination angle. All these solutions are found as well as their stabilities criteria which are just functions of the inclination angle. Next by using the perturbation method, the nonlinear Schrödinger (NLS) equation governing the small amplitude modulated signal in the network is found and used to seek modulated pulse and dark solitons as approximated solutions of the network equation. Finally, the conditions (threshold amplitude) for which the input signal will propagate in the network for the input frequency belonging to the forbidden band zone is established, the well-known supratransmission criterion, which is checked by numerical investigations.

这项工作的目的是研究由具有强非理性非线性的不连续弹性耦合系统振荡器组成的机械网络的动力学行为。这个机械网络由一组 N 个非线性振荡器组成，弹性耦合，其中每个单元中的非线性只是由于几何配置，即线性弦的倾斜。通过使用拉格朗日公式，建立了控制系统动力学的模型方程组。这组方程具有很强的非有理非线性，仅取决于弦的倾角而具有平滑或不连续的特性，用于研究稳态及其稳定性分析。然后研究了系统在其频率接近其特征频率时的行为，即众所周知的共振现象，它表明随着频移的增加而出现滞后现象，而质量的位移趋于无穷大。接下来，推导出控制系统中连续信号动力学的偏微分方程。这个偏微分方程允许强振幅的包络脉冲串作为解决方案，这在这项工作中非常有趣，对于弱振幅，众所周知的包络，如脉冲或扭结孤子以及椭圆解，它们的振幅只是函数的倾角。找到了所有这些解决方案以及它们的稳定性标准，这些标准只是倾角的函数。接下来通过使用扰动方法，找到了控制网络中小幅度调制信号的非线性薛定谔 (NLS) 方程，并用于寻找调制脉冲和暗孤子作为网络方程的近似解。最后，建立输入信号将在网络中传播的条件（阈值幅度）属于禁带区域的输入频率，这是众所周知的超传输标准，通过数值研究进行检查。