Simplified Determination of Stability Bounds in Nonlinear Vibration Systems

Even slight nonlinearities in vibrating systems introduce instability frequency bands and unstable amplitudes. In many vibration problems it is desirable to know precisely the bounds of the instability frequencies and the associated amplitude ranges. Using a general nonlinear single degree of freedom system, based on a Coulomb friction augmented Duffing model, simplified graphical/analytical modus operandi was developed for computing singular amplitudes in the frequency domain. The method may be used in lieu of the usual phase plane approach wherein the physical meaning of the different vibration modes is obscured. In previous publications approximate estimation of instability frequency bands of the Duffing system was achieved by assuming variations about the steady state solution. The new method presented herein allows accurate determination of instability frequency ranges in a more general class of vibration systems, while quantifying “Horizontal Tangents Amplitudes,” in addition to the usual “Resonance Amplitudes” and “Vertical Tangents Amplitudes.”