Data Processing in Bifurcation Analysis of Maps with Time-Delays in the Frequency Domain

A unified frequency-domain approach to analyze the NS (Neimark-Sacker) bifurcations and the period-doubling bifurcations of nonlinear maps with time-delays in the linear feed-forward term is presented. The technique relies on the HBA (harmonic balance approximation, a very important method in data processing ) and feedback systems theory. The expressions of the bifurcation solution and the stability are derived.

Research on Dynamic Characteristic and Experiments of Double-Disc Rotor System with Oil Film Support

Multi-DOF model of double-disc rotor-bearing system taking oil film support into account is established, and Newmark method is also applied to dynamic response of continuous system. To simplify the calculation in double-disc eccentric situation, the research aims at time domain, frequency response and bifurcation solution, simultaneously qualitative experiments are also carried out on the experiment bench. Experiments show that the numerical algorithm and calculation results are credible. The conclusions conclude: For the rotor system shown in the paper, with the other parameters constant, small eccentricity system is prone to appear quasi-periodic instability, but for big eccentricity system it is period-doubling instability, and the instability speed will increase with eccentricity enlargement; initial eccentric phase has severe effects on the dynamic characteristic of system, so it is worth studying it more in depth. This method and results in this paper provides a theoretical reference for stability analysis and vibration control in more complex relevant rotor-bearing system.

Bifurcation to a corner-like formation in a slender nonlinearly elastic cylinder: asymptotic solution and mechanism

In this paper, we study the corner-like formation in a slender nonlinearly elastic cylinder due to compression. Mathematically, this is a very challenging problem: one needs to study the bifurcation of the nonlinear field equations (complicated nonlinear partial differential equations) and show that there is a bifurcation leading to a solution with a corner-like profile. We also aim to obtain the asymptotic expression for this post-bifurcation solution. As far as we know, there is no available analytical method to obtain the post-bifurcation solution of nonlinear partial differential equation(s). Here we use a novel approach to tackle the present problem. Through a method of coupled series–asymptotic expansions, we manage to derive the normal form equation of the original nonlinear field equations, which can be written as a singular ordinary differential equation (ODE) system (the vector field has a singularity at one point). With welding end conditions, the problem is reduced to study the boundary-value problem of a singular ODE system. It seems that such a boundary-value problem of a singular ODE system was not formulated or studied before in the context of the present problem. With the help of phase planes, we manage to solve such a boundary-value problem. It turns out that there is a vertical singular line in phase planes, which causes the bifurcation to a corner-like solution. The expression for this post-bifurcation solution is also obtained. From the analytical results obtained, we reveal that the coupling effect of the material nonlinearity and geometrical size is the physical mechanism that causes the formation of a corner-like profile.

Bifurcation Analysis of Nonlinear Vibration Isolation System With Hard Stiffness by Cell Mapping

Period-doubling bifurcation is one of the major routes to chaos, but the methods widely used have some shortages in analyzing the bifurcation of the nonlinear vibration isolation system with hard stiffness. The location and property of bifurcation solution can be obtained conveniently by using numerical methods. Therefore, numerical research is important. In this paper, global bifurcation diagram is achieved by using Poincare´ mapping method. Subsequently, cell mapping as a useful numerical method is applied to analyze the static and dynamic bifurcation of nonlinear vibration isolation system with hard stiffness.

INVESTIGATION OF METAL TREATMENT PROCESS

Differential equations of self‐excited oscillations, arising in metal cutting process and metal drilling process, are presented in this paper. The causes of these oscillations are delaying forces, arising in metal treatment processes. The linear analysis of differential equations of metal cutting process is performed and an area of asymptotically stability is chosen. The non‐linear analysis is performed by the theory of bifurcation. Solution of this differential equation is compared with results of numerical experiment. Differential equations of dynamic of metal drilling process are presented.

Out-of-Plane Solutions and Bifurcations of Submersibles in Free Positive Buoyancy Ascent

The problem of motion stability of submersible vehicles in free positive buoyancy ascent is analyzed. Motion is allowed to occur in combined vertical and horizontal planes. Continuation and catastrophe theory techniques are employed to trace all possible steady-state solutions in six degrees of freedom, while local linearization reveals their stability properties. Vehicle geometric properties and control surface deflections are used as the primary bifurcation parameters. It is shown that multiple solutions may exist in the form of pitchfork bifurcation, solution separation, hysteresis, and teardrop branches. Regions in parameter spaces are identified where extreme sensitivity of solutions to geometric properties and hydrodynamic modeling is present.