Periodic Delay Effects on Cutting Dynamics

We consider a delay equation whose delay is perturbed by a small periodic fluctuation. In particular, it is assumed that the delay equation exhibits a Hopf bifurcation when its delay is unperturbed. The periodically perturbed system exhibits more delicate bifurcations than a Hopf bifurcation. We show that these bifurcations are well explained by the Bogdanov-Takens bifurcation when the ratio between the frequencies of the periodic solution of the unperturbed system (Hopf bifurcation) and the external periodic perturbation is 1:2. Our analysis is based on center manifold reduction theory.     

On the existence and bifurcation of minimal normal modes

Minimal normal modes (MNMs) are defined as non-linear normal modes which give a true minimum to Jacobi's Principle of Least Action. It is shown that for a certain class of two degree of freedom non-linear conservative systems, MNMs generically occur in pairs. The nature of both generic and non-generic bifurcations of MNMs is derived and illustrative examples are given.     

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

The non-linear normal modes (NNMs) and their bifurcation of a complex two DOF system are investigated systematically in this paper. The coupling and ground springs have both quadratic and cubic non-linearity simultaneously. The cases of ω₁:ω₂=1:1, 1:2 and 1:3 are discussed, respectively, as well as the case of no internal resonance. Approximate solutions for NNMs are computed by applying the method of multiple scales, which ensures that NNM solutions can asymtote to linear normal modes as the non-linearity disappears. According to the procedure, NNMs can be classified into coupled and uncoupled modes. It is found that coupled NNMs exist for systems with any kind of internal resonance, but uncoupled modes may appear or not appear, depending on the type of internal resonance. For systems with 1:1 internal resonance, uncoupled NNMs exist only when coefficients of cubic non-linear terms describing the ground springs are identical. For systems with 1:2 or 1:3 internal resonance, in additional to one uncoupled NNM, there exists one more uncoupled NNM when the coefficients of quadratic or cubic non-linear terms describing the ground springs are identical. The results for the case of internal resonance are consistent with ones for no internal resonance. For the case of 1:2 internal resonance, the bifurcation of the coupled NNM is not only affected by cubic but also by quadratic non-linearity besides detuning parameter although for the cases of 1:1 and 1:3 internal resonance, only cubic non-linearity operate. As a check of the analytical results, direct numerical integrations of the equations of motion are carried out.     

Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

On the Hamiltonian Hopf Bifurcations in the 3D Hénon–Heiles Family

An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the linear behaviour is not that of a typical Hamiltonian Hopf bifurcation as the eigen-values completely vanish at the bifurcation. However, the nonlinear structure is that of a Hamiltonian Hopf bifurcation. The result is obtained by formulating geometric criteria involving the normalized Hamiltonian and the reduced phase space.     

Fundamental and subharmonic resonances in a system with a ‘1-1’ internal resonance

The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a 1–1 internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses nonlinear normal modes. For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.     

Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of-freedom system

Forced, weakly nonlinear oscillations of a two degree-of-freedom autoparametric vibration absorber system are studied for resonant excitations. The method of averaging is used to obtain first-order approximations to the response of the system. A complete bifurcation analysis of the averaged equations is undertaken in the subharmonic case of internal and external resonance. The locked pendulum mode of response is found to bifurcate to coupled-mode motion for some excitation frequencies and forcing amplitudes. The coupled-mode response can undergo Hopf bifurcation to limit cycle motions, when the two linear modes are mistuned away from the exact internal resonance condition. The software packages AUTO and KAOS are used and a numerically assisted study of the Hopf bifurcation sets, and dynamic steady solutions of the amplitude or averaged equations is presented. It is shown that both super-and sub-critical Hopf bifurcations arise and the limit cycles quickly undergo period-doubling bifurcations to chaos. These imply chaotic amplitude modulated motions for the system.     

多自由度内共振系统非线性模态的分岔特性

利用多尺度法构造了一个立方非线性1:3内共振系统的内共振非线性模态(NonlinearNormal Modes associated with internal resonance)．研究表明,内共振非线性系统除存在单模态运动外还存在耦合模态运动．耦合内共振模态具有分岔特性．利用奇异性理论对模态分岔方程进行分析发现此类系统的模态存在叉形点分岔和滞后点分岔这两种典型的分岔模式．     

轴承-转子系统Hopf分叉及其后动力学行为研究

采用长轴承解析模型研究滑动轴承支承的平衡单盘柔性转子-轴承系统的自激振动,把结合打靶法的延续算法应用于柔性平衡转子-轴承系统Hopf分叉后周期解的追踪和求解上,基于Floquet理论对周期解的稳定性加以分析.通过持续追踪周期解频率变化并与失稳固有频率进行对比,分析了自激锁相现象,研究了非线性油膜力自激源对系统的作用机理.运用Poincare映射、分叉图、及Lyapnov指数对周期解分叉、混沌及进入和脱离混沌的过程进行了分析.     

Nonlinear Normal Modes and Their Bifurcations for an Inertially Coupled Nonlinear Conservative System

This work concerns the nonlinear normal modes (NNMs) of a 2 degree-of-freedom autonomous conservative spring–mass–pendulum system, a system that exhibits inertial coupling between the two generalized coordinates and quadratic (even) nonlinearities. Several general methods introduced in the literature to calculate the NNMs of conservative systems are reviewed, and then applied to the spring–mass–pendulum system. These include the invariant manifold method, the multiple scales method, the asymptotic perturbation method and the method of harmonic balance. Then, an efficient numerical methodology is developed to calculate the exact NNMs, and this method is further used to analyze and follow the bifurcations of the NNMs as a function of linear frequency ratio p and total energy h. The bifurcations in NNMs, when near 1:2 and 1:1 resonances arise in the two linear modes, is investigated by perturbation techniques and the results are compared with those predicted by the exact numerical solutions. By using the method of multiple time scales (MTS), not only the bifurcation diagrams but also the low energy global dynamics of the system is obtained. The numerical method gives reliable results for the high-energy case. These bifurcation analyses provide a significant glimpse into the complex dynamics of the system. It is shown that when the total energy is sufficiently high, varying p, the ratio of the spring and the pendulum linear frequencies, results in the system undergoing an order–chaos–order sequence. This phenomenon is also presented and discussed.     

Bifurcation of nonlinear internal resonant normal modes of a class of multi-degree-of-freedom systems

The method of multiple scales is applied for constructing nonlinear normal modes (NNMs) of a three-degree-of-freedom system which is discretized from a two-link flexible arm connected by a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internal resonance between the second and the third modes. The approximate solution for the NNM associated with internal resonance are presented. The NNMs determined here tend to the linear modes as the nonlinearity vanishes, which is significant for one to construct NNM. Greatly different from results of those nonlinear systems without internal resonance, it is found that the NNM involved in internal resonance include coupled and uncoupled two kinds. The bifurcation analysis of the coupled NNM of the system considered is given by means of the singularity theory. The pitchfork and hysteresis bifurcation are simultaneously found. Therefore, the number of NNM arising from the internal resonance may exceed the number of linear modes, in contrast with the case of no internal resonance, where they are equal. Curves displaying variation of the coupling extent of the coupled NNM with the internal-resonance-deturing parameter are proposed for six cases.     

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities – single zero and Hopf bifurcation – have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.     

A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter , we study thedynamics of this system at the limit 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.     

Non-linear normal modes and their bifurcation of a class of systems with three double of pure imaginary roots and dual internal resonances

The method of multiple scales is used to construct non-linear normal modes (NNMs) of a class of systems with three double of pure imaginary roots and 1:2:5 dual internal resonance. It is found that the three NNMs associated with dual internal resonances include two uncoupled NNMs as well as a coupled NNM. And the bifurcation problem of the coupled NNM is in two variables, which is greatly different from the bifurcation of the NNMs of systems with single internal resonance. Because no results in singularities can be straightly applied, a practical way is proposed to do singularity analysis for bifurcation of two dimensions. It is also noted that with the variation of the bifurcation parameters, the modes may convert to each other or suddenly emerge and disappear, which give rise to the number of the NNMs more or fewer than the number of the degrees of freedom.     

Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

Donnell equations are used to simulate free nonlinear oscillations of cylindrical shells with imperfections. The expansion, which consists of two conjugate modes and axisymmetric one, is used to analyze shell oscillations. Amplitudes of the axisymmetric motions are assumed significantly smaller, than the conjugate modes amplitudes. Nonlinear normal vibrations mode, which is determined by shell imperfections, is analyzed. The stability and bifurcations of this mode are studied by the multiple scales method. It is discovered that stable quasiperiodic motions appear at the bifurcations points. The forced oscillations of circular cylindrical shells in the case of two internal resonances and the principle resonance are analyzed too. The multiple scales method is used to obtain the system of six modulation equations. The method for stability analysis of standing waves is suggested. The continuation algorithm is used to analyze fixed points of the system of the modulation equations.     

Singular Characteristics of Nonlinear Normal Modes in a Two Degrees of Freedom Asymmetric Systems with Cubic Nonlinearities

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Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

Nonlinear normal modes of a fixed-fixed buckled beam about its first post-buckling configuration are investigated. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate solutions for the nonlinear normal modes are computed by applying the method of multiple scales directly to the governing integral-partial-differential equation and associated boundary conditions. Curves displaying variation of the amplitude of one of the modes with the internal-resonance-detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess one stable uncoupled mode (high-frequency mode) and either (a) one stable coupled mode, (b) three stable coupled modes, or (c) two stable and one unstable coupled modes. For the same resonance, the beam possesses one degenerate mode (with a multiplicity of two) and two stable and one unstable coupled modes. On the other hand, for a one-to-one internal resonance between the first and second modes, the beam possesses (a) two stable uncoupled modes and two stable and two unstable coupled modes; (b) one stable and one unstable uncoupled modes and two stable and two unstable coupled modes; and (c) two stable uncoupled and two unstable coupled modes (with a multiplicity of two). For a one-to-one internal resonance between the third and fourth modes, the beam possesses (a) two stable uncoupled modes and four stable coupled modes; (b) one stable and one unstable uncoupled modes and four stable coupled modes; (c) two unstable uncoupled modes and four stable coupled modes; and (d) two stable uncoupled modes and two stable coupled modes (each with a multiplicity of two).     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Subcritical, nontypical and period-doubling bifurcations of a delta wing in a low speed wind tunnel

Limit Cycle Oscillations (LCOs) involving Delta wings are an important area of research in modern aeroelasticity. Such phenomena can be the result of geometric or aerodynamic nonlinearity. In this paper, a flexible half-span Delta wing is tested in a low speed wind tunnel in order to investigate its dynamic response. The wing is designed to be more flexible than the models used in previous research on the subject in order to expand the airspeed range in which LCOs occur. The experiments reveal that this wing features a very rich bifurcation behavior. Three types of bifurcation are observed for the first time for such an aeroelastic system: subcritical bifurcations, period-doubling/period-halving and nontypical bifurcations. They give rise to a great variety of LCOs, even at very low angles of attack. The LCOs resulting from the nontypical bifurcation display Hopf-type behavior, i.e. having fundamental frequencies equal to one of the linear modal frequencies. All of the other LCOs have fundamental frequencies that are unrelated to the underlying linear system modes.     

Effects of rotary inertia on sub- and super-critical free vibration of an axially moving beam

The most important issue in the vibration study of an engineering system is dynamics modeling. Axially moving continua is often discussed without the inertia produced by the rotation of the continua section. The main goal of this paper is to discover the effects of rotary inertia on the free vibration characteristics of an axially moving beam in the sub-critical and super-critical regime. Specifically, an integro-partial-differential nonlinear equation is modeled for the transverse vibration of the moving beam based on the generalized Hamilton principle. Then the effects of rotary inertia on the natural frequencies, the critical speed, post-buckling vibration frequencies are presented. Two kinds of boundary conditions are also compared. In super-critical speed range, the straight configuration of the axially moving beam loses its stability. The buckling configurations are derived from the corresponding nonlinear static equilibrium equation. Then the natural frequencies of the post-buckling vibration of the super-critical moving beam are calculated by using local linearization theory. By comparing the critical speed and the vibration frequencies in the sub-critical and super-critical regime, the effects of the inertia moment due to beam section rotation are investigated. Several interesting phenomena are disclosed. For examples, without rotary inertia, the study overestimates the stability of the axially moving beam. Moreover, the relative differences between the super-critical fundamental frequencies of the two theories may increase with an increasing beam length.