This paper presents a derivation of a hinged–hinged Euler–Bernoulli beam including cubic and quintic nonlinearities. Then, a three-mode Galerkin discretization technique has been utilized to generate a system of ordinary differential equations governing the temporal deflections of the first three modes of the studied beam. The pioneering work of Nayfeh and Mook (1995) has shown the absence of internal resonances among the modes of a hinged–hinged beam with cubic nonlinearities despite there are commensurable linear natural frequencies. In this work, extra quintic nonlinearities are involved to show the presence of internal resonances among such modes. Approximate solutions of the resulted system have been approached by the method multiple scales to get the modulations governing the amplitudes and phases of the first three modal temporal deflections. Their stability is investigated via Routh–Hurwitz criterion and a shaded region of unstable solutions has been plotted to conclude picture where the stable solutions are. Different response curves are plotted to explore the effects of beam parameters on the nonlinear dynamical behavior of such beam. Finally, a simulated response of the modal temporal deflections and the overall spatial–temporal deflection are portrayed to show the accurate behavior of the beam at different initial conditions.

本文提出了包括三次和五次非线性的铰链铰链式欧拉-伯努利梁的推导。然后，采用了三模Galerkin离散技术来生成一个常微分方程组，该系统控制所研究光束的前三个模态的时间偏转。Nayfeh和Mook（1995）的开创性工作表明，尽管存在相当大的线性固有频率，但立方非线性的铰链铰链梁的模式之间却没有内部共振。在这项工作中，涉及了额外的五次非线性，以表明这些模式之间存在内部共振。该方法的多个尺度已经接近了所得系统的近似解，从而获得了控制前三个模态时间偏转的幅度和相位的调制。通过Routh–Hurwitz准则对它们的稳定性进行了研究，并绘制了不稳定溶液的阴影区域以总结出稳定溶液所在的位置。绘制了不同的响应曲线，以探索光束参数对此类光束的非线性动力学行为的影响。最后，描绘了模态时间挠度和整个时空挠度的模拟响应，以显示光束在不同初始条件下的精确行为。通过Routh–Hurwitz准则研究了它们的稳定性，并绘制了一个不稳定解的阴影区域以总结出稳定解所在的位置。绘制了不同的响应曲线，以探索光束参数对此类光束的非线性动力学行为的影响。最后，描绘了模态时间挠度和整个时空挠度的模拟响应，以显示光束在不同初始条件下的精确行为。通过Routh–Hurwitz准则研究了它们的稳定性，并绘制了不稳定解的阴影区域以得出图片，其中包含了稳定解。绘制了不同的响应曲线，以探索光束参数对此类光束的非线性动力学行为的影响。最后，描绘了模态时间挠度和整个时空挠度的模拟响应，以显示光束在不同初始条件下的精确行为。