Nonlinear free undamped vibrations are investigated for ultra-precision manufacturing (UPM) machines with quadratic stiffness. The modes of the system are linearly coupled. The non-resonant case and the bounded internal resonance case are considered. The results of the non-resonant case indicate that the behavior of the system is the same as the linear behavior. However, for the internal resonance case, the results show that the amplitudes are coupled. The results also indicate that the nonlinear frequencies and amplitudes depend not only on the initial conditions, but also on the location of the isolators with respect to the center of gravity of the UPM.

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