Beams involving intermediate axially and transversely eccentric masses have many industrial applications such as plane wings with an intermediately positioned heavy gas turbine. In most of these applications, the beams undergo large deflections which give rise to geometric nonlinearities. So, the objective of the current research is to provide a nonlinear dynamic model for extensional-flexural vibration of such beams with variable cross sections. Hamilton's principle is employed for finding the normalized coupled extensional-flexural equations of motion and the corresponded boundary conditions. Then a new finite difference scheme is developed for finding the natural frequencies of the system and the corresponded mode shapes. The solution of the Eigen value-Eigen function problem is then verified by comparing the numerical findings with the exact solutions which are available for the simple case of constant cross section beams. A parametric study is also carried out to characterize the effect of the intermediate mass as well as its eccentricity parameters on the natural frequencies of the system. It is observed, that except for fundamental frequency, other natural frequencies are noticeably reduced with increasing the weight of the mass. Moreover, it is observed that with increasing the axial and transverse eccentric parameters, the third natural frequency is decreased, while the first, second and fourth ones are not appreciably changed. The derived modes are utilized in Lagrange equations along with a single mode approximation to derive the nonlinear coupled temporal equations of motion in time domain. These nonlinear equations are then solved analytically using the multiple time scales perturbation technique, and closed form expressions are suggested for the extensional and flexural responses of the system. The analytical findings are closely verified via numerical simulations. Finally, the frequency response of the system in the primary resonance case is derived analytically and the dependence of the amplitude of the response on the system and excitation parameters is studied in detail. The results reveal that the system exhibit hardening behavior and with increasing the excitation frequency or amplitude of the excitation, a discontinuous increase in the vibration amplitude may occur. The modeling approach suggested in this paper can be effectively used for studying the dynamic behavior of more complex systems involving single or multiple eccentric intermediate masses.

涉及中间轴向和横向偏心质量的梁具有许多工业应用，例如带有中间放置的重型燃气轮机的平面机翼。在大多数这些应用中，光束会发生大的偏转，从而导致几何非线性。因此，本研究的目的是为具有可变截面的这种梁的伸缩弯曲振动提供一个非线性动力学模型。汉密尔顿原理被用于寻找运动的归一化耦合伸缩弯曲方程和相应的边界条件。然后，开发了一种新的有限差分方案，用于查找系统的固有频率和相应的模式形状。然后，通过将数值结果与可用于恒定截面梁的简单情况的精确解进行比较，来验证特征值-特征函数问题的解。还进行了参数研究，以表征中间质量及其偏心率参数对系统固有频率的影响。可以看出，除了基频，其他自然频率会随着质量的增加而明显降低。而且，观察到随着轴向和横向偏心参数的增加，第三固有频率减小，而第一，第二和第四固有频率没有明显变化。在Lagrange方程中，将导出的模式与单模式近似一起使用，以导出时域中的非线性耦合运动时间方程。然后，使用多个时标摄动技术对这些非线性方程进行解析求解，并针对系统的拉伸和挠曲响应提出了闭式表达式。分析结果已通过数值模拟进行了严格验证。最后，通过分析得出系统在主共振情况下的频率响应，并详细研究了响应幅度对系统和激励参数的依赖性。结果表明，该系统表现出硬化行为，并且随着激励频率或激励幅度的增加，振动幅度可能会出现不连续的增加。本文提出的建模方法可以有效地用于研究涉及单个或多个偏心中间质量的更复杂系统的动力学行为。