A number of practical structures such as compliant offshore towers can be physically modeled, as a first approximation, as an inverted pendulum. The nonlinear dynamics of such model can present some complex features due to the nonlinear coupling among its degrees-of-freedom. In this paper a model of a spatial inverted pendulum restrained by three inclined extensional springs is adopted. Geometric imperfections are considered, and the solutions for both perfect and imperfect systems are presented herein. Especial attention is given to the determination of the nonlinear vibration modes. Non-similar and similar NNMs are obtained analytically by direct application of asymptotic methods and the results show important NNM features such as instability and multiplicity of modes. Poincare´ maps of the conservative system are used to identify the existence of modes that do not have a linear counterpart. Analytical approximations are derived for these nonlinear modes by the use of a two-dimensional polynomial fitting procedure obtained by the application of the Levenberg-Marquardt method. The points coordinates used in the fitting procedure were obtained through numerical integration of the governing equations of motion. Such analytical expressions can then be used in modal reduction, parametric analysis and in the derivation of important features of the system such as its frequency-amplitude relations and resonance curves.

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