NORMAL MODE LOCALIZATION FOR A TWO DEGREES-OF-FREEDOM SYSTEM WITH QUADRATIC AND CUBIC NON-LINEARITIES

The non-similar normal modes of free oscillations of a coupled non-linear oscillator are examined. So far, the study of non-linear vibrations has been based on the assumption that the system is admissible. This requirement is satisfied when the stiffness of the springs are odd functions of their displacement. In this work, a two-degrees-of-freedom tuned system is considered with stiffness elements having linear, quadratic and cubic non-linearities. The potential energy function of this system is not symmetric with respect to the origin (equilibrium point) of the configuration space due to the presence of the quadratic non-linearity. Hence, the system considered is no longer admissible. A study of the balancing diagrams is performed to determine the “degenerate” and “global” similar modes of the system. Manevich–Mikhlin asymptotic methodology is used for solving the singular differential equation describing the non-similar modes and approximate analytical expressions are derived. For this system, with weak coupling, localized non-similar modes are detected in a small neighborhood of degenerate similar modes of the tuned system. Numerical integration is used to verify theoretically predicted non-similar normal modes. It is found that these modes pass periodically through a non-zero point in the configuration space.