Phanikrishna Thota

TC-HAT (

This paper describes the underlying formulation and functionality of the newly developed software program

\widehat{TC}

\widehat{\text{{\sc tc}}}

): A Novel Toolbox for the Continuation of Periodic Trajectories in Hybrid Dynamical Systems

(“tc-hat”), to perform bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, often referred to as hybrid dynamical systems. Boundary-value-problem formulations corresponding to single- and two-parameter continuations of periodic trajectories and selected associated codimension-one bifurcations in such systems are presented. Finally, the capabilities of the program are illustrated by performing bifurcation analysis of a few example hybrid dynamical systems.

Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

Controlling bistability in tapping-mode atomic force microscopy using dual-frequency excitation

This letter discusses an experimental method to suppress spontaneous transitions between low- and high-amplitude oscillatory responses in tapping-mode atomic force microscopy in the absence of feedback control. Here, the cantilever is excited at two frequencies and the dynamic force curves for different excitation amplitudes are recorded. Experimental observations of the dual-frequency excitation strategy are reported for three different cantilevers. These suggest that such transitions may indeed be eliminated from a region of interest of separations between the sample surface and the average position of the cantilever support even with relatively small secondary excitation amplitudes.

Bifurcation Analysis of a Microactuator Using a New Toolbox for Continuation of Hybrid System Trajectories

This paper presents the application of a newly developed computational toolbox, TC-HAT (TCˆ), for bifurcation analysis of systems in which continuous-in-time dynamics are interrupted by discrete-in-time events, here referred to as hybrid dynamical systems. In particular, new results pertaining to the dynamic behavior of an example hybrid dynamical system, an impact microactuator, are obtained using this software program. Here, periodic trajectories of the actuator with single or multiple impacts per period and associated saddle-node, perioddoubling, and grazing bifurcation curves are documented. The analysis confirms previous analytical results regarding the presence of co-dimension-two grazing bifurcation points from which saddle-node and period-doubling bifurcation curves emanate.Copyright

Bifurcation analysis of nose landing gear shimmy with lateral and longitudinal bending

This work develops and studies a model of an aircraft nose landing gear with torsional, lateral, and longitudinal degrees of freedom. The corresponding three modes are coupled in a nonlinear fashion via the geometry of the landing gear in the presence of a nonzero rake angle, as well as via the nonlinear tire forces. Their interplay may lead to different types of shimmy oscillations as a function of the forward velocity and the vertical force on the landing gear. Methods from nonlinear dynamics, especially numerical continuation of equilibria and periodic solutions, are used to asses how the three modes contribute to different types of shimmy dynamics. In conclusion, the longitudinal mode does not actively participate in the nose-landing-gear dynamics over the entire range of forward velocity and vertical force.

Numerical Continuation Analysis of a Dual-sidestay Main Landing Gear Mechanism

A model of a three-dimensional dual-sidestay landing gear mechanism is presented and employed in an investigation of the sensitivity of the downlocking mechanism to attachment point deflections. A motivation for this study is the desire to understand the underlying nonlinear behavior, which may prevent a dual-sidestay landing gear from downlocking under certain conditions. The model formulates the mechanism as a set of steady-state constraint equations. Solutions to these equations are then continued numerically in state and parameter space, providing all state parameter dependencies within the model from a single computation. The capability of this analysis approach is demonstrated with an investigation into the effects of the aft sidestay angle on retraction actuator loads. It was found that the retraction loads are not significantly affected by the sidestay plane angle, but the landing gear’s ability to be retracted fully is impeded at certain sidestay plane angles. This result is attributed to the lan...

Influence of Tire Inflation Pressure on Nose Landing Gear Shimmy

This work investigates shimmy oscillations in the nose landing gear of a passenger aircraft and studies how they depend on changes in the tire inflation pressure. To achieve this, a mathematical model of a landing gear is considered that includes the influence of the tire pressure via different tire properties, such as cornering force and contact patch length. Experimental data obtained from two radial tires are used as a basis for modeling the influence of inflation pressure on tire properties. Bifurcation analysis of the mathematical model is then performed. It yields stability diagrams in the plane of velocity and vertical force for different values of the tire inflation pressure. Specifically, two-parameter bifurcation diagrams for five different inflation pressures are presented. This allows the conclusion that for the type of tires considered, the landing gear is less susceptible to shimmy oscillations at higher than nominal inflation pressures.

Geometric nonlinearities of aircraft systems

Nonlinearities due to geometric effects, in particular, via angular variables that are not small, are important for aircraft operation. Geometric nonlinearities have a strong effect on the dynamics of the aircraft system under consideration, and they are especially pronounced in aircraft ground operations. As a concrete example we consider here the effect of a non-zero rake angle on the dynamics of a nose landing gear. More specifically, we use tools from bifurcation theory to investigate the stability of the straight-rolling motion during a take-off run.