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.

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