Stability Of Steady-state Motions Research Articles

Dynamic analysis of atomic force microscope in tapping mode

The Atomic force microscope (AFM) is an enormously valuable tool in a wide variety of applications because of its function to depict in different mediums with the sub-nano-meter resolution, and also manipulating objects with nano-meter-scale features and measuring forces with better than pico-newton resolution. In this paper, the lumped parameter model is used to construct the mathematical model of the AFM cantilever. The cantilever tip, excited by the harmonic external force, is under the influence of the tip-sample interaction force. Since, we consider the AFM in the operation mode named dynamic contact mode, the Deryagin-Muller-Toporov (DMT) force is considered as the interaction force between the cantilever tip and the surface of the sample. DMT force causes non-linearity. To solve the equation of motion, the Van der Pol method is used to obtain the frequency response equation to investigate the non-linearity effect as well as the amplitude of the excitation on the response. The stability of steady state motion is investigated.

Stability of Steady-State Motion of an Isolated System Consisting of a Rotating Body and Two Pendulums

An isolated mechanical system consisting of a rotating body and two pendulums fit on its longitudinal axis is studied. This system models how pendulum, ball, or fluid (ring) dampers decrease or increase the nutation angle of a spin-stabilized artificial satellite. The conditions of origin, existence, and cessation of the steady-state motion of the system, depending on its parameters, and the stability conditions for the primary motion (the body rotates about the longitudinal axis and the pendulums lie on the same line) and secondary motions (the body does not rotate around the longitudinal axis) are established. The residual nutation angle is estimated

Stability of steady-state motions of a mechanical system with deformable elements

Stability of steady-state motions of a mechanical system with deformable elements

Saddle-Node Bifurcations of Cycles in a Relief Valve

We extend the asymptotic perturbation (AP) method to the studyof a linear partial differential equation with nonlinear boundaryconditions. A relief valve under the combined effects of static anddynamic loadings is considered with the following resonances between thenth linear mode and the external periodic excitation: primaryresonance, subharmonic resonance of order one-half or one-third,superharmonic resonance of order-two or order-three and combinationresonance. The AP method uses two different procedures for thesolutions: introducing an asymptotic temporal rescaling and balancing ofthe harmonic terms with a simple iteration. We obtain amplitude andphase modulation equations and determine external force-response andfrequency-response curves. Stability of steady-state motions is alsoinvestigated. Saddle-node bifurcations of cycles are observed and underappropriate conditions the performance of the relief valve may beunsatisfactory due to the presence of jumps and hysteresis effects inthe system response. Global analysis is used in order to exclude theexistence of modulated motion. The validity of the method is highlightedby comparing approximate solutions with results of the numericalintegration.

Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Asymptotic expansions for the exponential growth rate, known as the Lyapunov exponent, and rotation numbers for two coupled oscillators driven by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems. Almost-sure stability or instability of dynamical systems depends on the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable. The results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging, where approximate ItO equations in amplitudes and phases are obtained in the sense of weak convergence.

A remark on the stability of steady state motions

A remark on the stability of steady state motions

Stability of steady-state motions of a free rigid body with a rotor situated on an elastic shaft

Stability of steady-state motions of a free rigid body with a rotor situated on an elastic shaft

On the stability of steadystate motions: PMM vol. 32, no. 3, 1968, pp. 504–508

On the stability of steadystate motions: PMM vol. 32, no. 3, 1968, pp. 504–508

On the stabilization of steady-state motions of a nonlinear control system in the critical case of a pair of pure imaginary roots

On the stabilization of steady-state motions of a nonlinear control system in the critical case of a pair of pure imaginary roots