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In this paper, a simplified method is proposed for deriving equilibrium equations in continuous systems. The new method is indeed the direct applying of Newton's laws on free body diagram of point. First, by describing the concept of equilibrium equations and investigating the differences between concentrated masses and continuous systems, the physical basis of new method is introduced. It is shown that, using intensive properties simplifies the analysis of continuous systems. For verifying the new method, the governing equations in Cartesian, polar and spherical coordinates systems are derived. We have to consider nonlinear terms due to developing large slopes in system. Hence, nonlinear governing equations in Cartesian system are derived too. Finally by noting to the simplicity of new method and its independency from complicated differential and vector analysis in other methods such as Hamiltonian and classic methods, the interests of new method are emphasized. By knowing concept of physical point, a united process is accessible which is extendable to other governing equations of continuous systems.

In the present paper, an exact mathematical solution has been obtained for nonlinear free transverse vibration of beams, for the first time. The nonlinear governing partial differential equation in un-deformed coordinates system has been converted in two coupled partial differential equations in deformed coordinates system. A mathematical explanation is obtained for nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of beam. It is shown that as the th mode of transverse vibration excited, the mode 2 th of in-plane vibration will be developed. The result of present work is compared with those obtained from Galerkin method and the observed agreement confirms the exact mathematical solution. It is shown that the governing equation is linear in the time domain. As a parameter, the amplitude to length ratio   Λ/l has been proposed to show when the nonlinear terms become dominant in the behavior of structure.

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