In highly nonlinear systems, the behavior of the frequency of free oscillations as the amplitude decays can vary depending on the specific system. There are no general rules that universally apply to all nonlinear systems in this regard.
In some nonlinear systems, as the amplitude of free oscillations decays, the frequency may increase. This behavior is often observed in systems where the nonlinearity introduces energy-dependent effects or changes the effective stiffness of the system as the amplitude decreases. As the oscillations lose energy, the system experiences a change in its restoring forces or dynamics, leading to an increase in frequency.
On the other hand, in certain nonlinear systems, the frequency may decrease as the amplitude decays. This behavior can occur when the nonlinear effects introduce a softening behavior or damping mechanisms that become more pronounced as the amplitude decreases. The reduction in amplitude leads to changes in the system's response, resulting in a decrease in frequency.
It is also possible for nonlinear systems to exhibit more complex behaviors as the amplitude decays. The frequency might initially increase and then decrease, or it might display irregular variations. These behaviors can arise due to the interplay of various nonlinear effects, including energy dissipation, hysteresis, and coupling between different modes of oscillation.
Therefore, the frequency of free oscillations in highly nonlinear systems as the amplitude decays is not governed by a general rule, and it depends on the specific characteristics and dynamics of the system under consideration. Analyzing the nonlinear behavior of a particular system requires a detailed understanding of its dynamics and often involves numerical simulations or experimental observations.