Simplified Determination of Stability Bounds in Nonlinear Vibration Systems

Abstract

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.”

Existence and control of weighted pseudo almost periodic solutions for abstract nonlinear vibration systems

“…Even slight nonlinearities in vibrating dynamics can cause instability bands and unstable amplitudes. By the works of Peleg and Hing, 24 in many vibration dynamics or vibration problems, it is vital to know precisely the bounds of the instability/stability frequencies and the associated amplitude rang. In this paper, we calculated the precise Lyapunov exponents of vibrating dynamics with bounded or unbounded nonlinearities.…”

“…12 Zhang 13–15 firstly studied the pseudo almost periodicity. The authors 16–27 studied the existence, uniqueness and/or stability of the solutions to the abstract differential equations.…”

Shock Characteristics of the Opposed Disc Springs (ODS) Shock Isolator with Pretightening under Boundary Friction Condition

“…Multiple disc springs can be combined in different stacking ways to form a variety of nonlinear elastic elements with different stiffness. Elements of this type have the damping characteristics of Coulomb and viscous damping due to the deformation of disc springs [14]. erefore, this study presents an ODS shock isolator with pretightening, which ensures a good shock isolation effect and stability at the static equilibrium position and can be applied in strong shock environment with large displacement response.…”