A numerical bifurcation study of friction effects in a slip-controlled torque converter clutch

Friction plays a key role in the efficiency and stability of the slip-controlled torque converter clutches. The effects of friction on the dynamics and stability of a slip-controlled torque converter clutch system using a bifurcation-analysis-based approach is presented in this paper. A three degree-of-freedom nonlinear driveline model with integral feedback action to control the clutch slip speed has been utilized for this study. The clutch interface friction is dependent on the slip speed and is a function of the static friction constant, μ ₀, the low velocity friction constant μ ₁, and the low velocity exponential rate, γ. Using one-parameter numerical continuation, local Hopf bifurcations of the subcritical type are observed as the friction parameters μ ₁ and γ were varied at low slip speeds. The continuation results are verified using simulations of the full nonlinear model. Stick-slip and undesirable oscillations of the model inertia elements are observed for certain parameter values. As the slip speed is increased, the bifurcation instability occurs at an increasingly higher value of μ ₁ signifying an improved tolerance of negative friction gradient at higher slip speeds. Smaller exponential rates γ are tolerated at higher slip speeds before the bifurcation instability occurs. For the range of parameter values considered, no bifurcations occur for a slip speeds higher than 3.4 and 4.5 rad/s with μ ₁ and γ as the continuation parameters, respectively. These values of slip speeds are much lower than the system’s first mode of torsional vibration of 16 Hz (≈100 rad/s).     

Milling Bifurcations from Structural Asymmetry and Nonlinear Regeneration

This paper investigates multiple modeling choices for analyzing the rich and complex dynamics of high-speed milling processes. Various models are introduced to capture the effects of asymmetric structural modes and the influence of nonlinear regeneration in a discontinuous cutting force model. Stability is determined from the development of a dynamic map for the resulting variational system. The general case of asymmetric structural elements is investigated with a fixed frame and rotating frame model to show differences in the predicted unstable regions due to parametric excitation. Analytical and numerical investigations are confirmed through a series of experimental cutting tests. The principal results are additional unstable regions, hysteresis in the bifurcation diagrams, and the presence of coexisting periodic and quasiperiodic attractors which is confirmed through experimentation.     

考虑压电材料非线性特性时旋转式超声电机定子的主共振响应

考虑压电材料非线性本构关系，建立了旋转式超声电机定子的非线性动力学模型，利用解析与数值方法研究超声电机定子的主共振响应，以揭示压电材料非线性本构关系对定子振动特性的影响，为深入研究旋转行波超声电机的动力学机理奠定基础．     

Free Vibration Analysis of Rotating Nonlinearly Elastic Structures with Symmetry: An Efficient Group-Equivariance Approach

A study is made of the dynamics of oscillating systems with a slowly varying parameter. A slowly varying forcing periodically crosses a critical value corresponding to a pitchfork bifurcation. The instantaneous phase portrait exhibits a centre when the forcing does not exceed the critical value, and a saddle and two centres with an associated double homoclinic loop separatrix beyond this value. The aim of this study is to construct a Poincaré map in order to describe the dynamics of the system as it repeatedly crosses the bifurcation point. For that purpose averaging methods and asymptotic matching techniques connecting local solutions are applied. Given the initial state and the values of the parameters the properties of the Poincaré map can be studied. Both sensitive dependence on initial conditions and (quasi) periodicity are observed. Moreover, Lyapunov exponents are computed. The asymptotic expressions for the Poincaré map are compared with numerical solutions of the full system that includes small damping.     

Properties of Chaotic and Regular Boundary Crisis in Dissipative Driven Nonlinear Oscillators

The phenomenon of the chaotic boundary crisis and the related concept of the chaotic destroyer saddle has become recently a new problem in the studies of the destruction of chaotic attractors in nonlinear oscillators. As it is known, in the case of regular boundary crisis, the homoclinic bifurcation of the destroyer saddle defines the parameters of the annihilation of the chaotic attractor. In contrast, at the chaotic boundary crisis, the outset of the destroyer saddle which branches away from the chaotic attractor is tangled prior to the crisis. In our paper, the main point of interest is the problem of a relation, if any, between the homoclinic tangling of the destroyer saddle and the other properties of the system which may accompany the chaotic as well as the regular boundary crisis. In particular, the question if the phenomena of fractal basin boundary, indeterminate outcome, and a period of the destroyer saddle, are directly implied by the structure of the destroyer saddle invariant manifolds, is examined for some examples of the boundary crisis that occur in the mathematical models of the twin-well and the single-well potential nonlinear oscillators.     

1／2亚谐共振──主参数共振情况下非线性振动系统的余维2退化分叉

本文应用Normal Form理论和退化向量场的普适开折理论研究了参数激励与强迫激励联合作用下非线性振动系统的余维2退化分叉,用Melnikov方法讨论了全局分叉的存在性.     

The Secondary Bifurcation of an Aeroelastic Airfoil Motion: Effect of High Harmonics

The nonlinear dynamical response of a two-degree-of-freedom aeroelastic airfoil motion with cubic restoring forces is investigated. A secondary bifurcation after the primary Hopf (flutter) bifurcation is detected for a cubic hard spring in the pitch degree-of-freedom. Furthermore, there is a hysteresis in the secondary bifurcation: starting from different initial conditions the motion may jump from one limit cycle to another at different fluid flow velocities. A high-order harmonic balance method is employed to investigate the possible bifurcation branches. Furthermore, a numerical time simulation procedure is used to confirm the stable and unstable bifurcation branches.     

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.     

APPLICATIONS OF THE SIGNS OF MELNIKOV''''S FUNCTION

The existence and stability of peridic solutions for the two-dimensional system x′=f(x)+g(x, a), O<ε<1,a∈R whoseunperturbed system is Hamlitonan be decided by using the signs of Melnikov's function. Theresults can be applied to the comstuction of phase portralts in the bifiacation set of codimension two bifurcations of flows with double aero eignvahues.     

Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincaré sections and Poincaré maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincaré sections.