The study of the non-linear system is usually avoided in the engineering literature because of the mathematical complications which appear in this situation. The author introduces the concept of neo-Hookean element, that is a non-linear elastic element for which the elastic force depends on the elongation both linear and at a negative integer power. These elements are then used in different mechanical systems from the most simple to very complicated. For each system are obtained the non-linear differential equations of motion, are studied the stabilities and are made comparisons between different cases of non-linearity and between the neo-Hookean case and the linear one. Keeping into account the non-linearity, the equilibrium position is not unique and in this book the reader can find a complete study of the equilibriums. Since the force depends on a negative integer power of the elongation the study must be separately made for positive and negative elongations. Some unusual behaviors very often met in practice are easily explained using the neo-Hookean elements.

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