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.     

Nonlinear stability theory of channel flow with heat transfer – an asymptotic approach

A fully developed laminar Poiseuille flow subject to constant heat flux across the wall is analysed with respect to its stability behavior by applying a weakly nonlinear stability theory. It is based on an expansion of the disturbance control equations with respect to a perturbation parameter ε. This parameter is the small initial amplitude of the fundamental wave. This fundamental wave which is the solution of the linear (Orr-Sommerfeld) first order equation triggers all higher order effects with respect to ε. Heat transfer is accounted for asymptotically through an expansion with respect to a small heat transfer parameter ε _T . Both perturbation parameters, ε and ε _T , are linked by the assumption ε _T =O(ε²) by which a certain distinguished limit is assumed. The results for a fluid with temperature dependent viscosity show that heat transfer effects in the nonlinear range continue to act in the same way as in the initial linear range. Received on 11 August 1997     

Order reduction of structural dynamic systems with static piecewise linear nonlinearities

A technique for order reduction of dynamic systems in structural form with static piecewise linear nonlinearities is presented. By utilizing two methods which approximate the nonlinear normal mode (NNM) frequencies and mode shapes, reduced-order models are constructed which more accurately represent the dynamics of the full model than do reduced models obtained via standard linear transformations. One method builds a reduced-order model which is dependent on the amplitude (initial conditions) while the other method results in an amplitude-independent reduced model. The two techniques are first applied to reduce two-degree-of-freedom undamped systems with clearance, deadzone, bang-bang, and saturation stiffness nonlinearities to single-mode reduced models which are compared by direct numerical simulation with the full models. It is then shown via a damped four-degree-of-freedom system with two deadzone nonlinearities that one of the proposed techniques allows for reduction to multi-mode reduced models and can accommodate multiple nonsmooth static nonlinearities with several surfaces of discontinuity. The advantages of the proposed methods include obtaining a reduced-order model which is signal-independent (doesn’t require direct integration of the full model), uses a subset of the original physical coordinates, retains the form of the nonsmooth nonlinearities, and closely tracks the actual NNMs of the full model.     

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation

On the Shear Number Effect in Stratified Shear Flow

The influence of the shear number on the turbulence evolution in a stably stratified fluid is investigated using direct numerical simulations on grids with up to 512 × 256 × 256 points. The shear number SK/ε is the ratio of a turbulence time scale K/ε to the shear time scale 1/S. Simulations are performed at two initial values of the Reynolds number Re _Λ= 44.72 and Re _Λ= 89.44. When the shear number is increased from small to moderate values, the nondimensional growth rate γ= (1/SK)dK/dt of the turbulent kinetic energy K increases since the shear forcing and its associated turbulence production is larger. However, a further increase of the shear number from moderate to large values results in a reduction of the growth rate γ and the turbulent kinetic energy K shows long-time decay for sufficiently large values of the shear number. The inhibition of turbulence growth at large shear numbers occurs for both initial values of the Reynolds number and can be explained with the predominance of linear effects over nonlinear effects when the shear number is sufficiently high. It is found that, at the higher initial value of the Reynolds number, the reduction of the growth rate occurs at a higher value of the shear number. The shear number is found to affect spectral space dynamics. Turbulent transport coefficients decrease with increasing shear number. Received 23 June 1998 and accepted 25 February 1999     

Elliptic balance solution of two-degree-of-freedom, undamped, forced systems with cubic nonlinearity

The solution of a system of two coupled, nonhomogeneous undamped, ordinary differential equations with cubic nonlinearity and sinusoidal driving force is obtained by the use of Jacobian elliptic functions and the elliptic balance method. To assess the accuracy of our proposed solution, we consider an example that arises in the study of the finite amplitude, nonlinear vibration of a simple shear suspension system. It is shown that the analytical results exhibit good agreement with the numerical integration solutions even for moderate values of the system parameters.     

Seismic base isolation by nonlinear mode localization

Summary In this paper, the performance of a nonlinear base-isolation system, comprised of a nonlinearly sprung subfoundation tuned in a 1∶1 internal resonance to a flexible mode of the linear primary structure to be isolated, is examined. The application of nonlinear localization to seismic isolation distinguishes this study from other base-isolation studies in the literature. Under the condition of third-order smooth stiffness nonlinearity, it is shown that a localized nonlinear normal mode (NNM) is induced in the system, which confines energy to the subfoundation and away from the primary or main structure. This is followed by a numerical analysis wherein the smooth nonlinearity is replaced by clearance nonlinearity, and the system is excited by ground motions representing near-field seismic events. The performance of the nonlinear system is compared with that of the corresponding linear system through simulation, and the sensitivity of the isolation system to several design parameters is analyzed. These simulations confirm the existence of the localized NNM, and show that the introduction of simple clearance nonlinearity significantly reduces the seismic energy transmitted to the main structure, resulting in significant attenuation in the response. This work was supported in part by the National Science Foundation Grant CMS 00-00060. The authors are grateful for this support.     

Nonlinear normal modes in homogeneous system with time delays

Periodic synchronous regimes of motion are investigated in symmetric homogeneous system of coupled essentially nonlinear oscillators with time delays. Such regimes are similar to nonlinear normal modes (NNMs), known for corresponding conservative system without delays, and can be found analytically. Unlikely the conservative counterpart, the system possesses “oval” modes with constant phase shift between the oscillators, in addition to symmetric/antisymmetric and localized regimes of motion. Numeric simulation demonstrates that the “oval” modes may be attractors of the phase flow. These attractors are particular case of phase-locked solutions, rather ubiquitous in the system under investigation.     

Development of an Analytical <Emphasis Type="Italic">β</Emphasis>-Function PDF Integration Algorithm for Simulation of Non-premixed Turbulent Combustion

Among many presumed-shape pdf approaches for modeling non-premixed turbulent combustion, the presumed β-function pdf is widely used in the literature. However, numerical integration of the β-function pdf may encounter singularity difficulties at mixture fraction values of Z = 0 or 1. To date, this issue has been addressed by few publications. The present study proposes the Piecewise Integration Method (PIM), an efficient, robust and accurate algorithm to overcome these numerical difficulties with the added benefit of improving computational efficiency. Comparison of this method to the existing numerical integration methods shows that the PIM exhibits better accuracy and greatly increases computational efficiency. The PIM treatment of the β-function pdf integration is first applied to the Burke–Schumann solution in conjunction with the k − ε turbulence model to simulate a CH₄/H₂ bluff-body turbulent flame. The proposed new method is then applied to the same flow using a more complex combustion model, the laminar flamelet model. Numerical predictions obtained by using the proposed β-function pdf integration method are compared to experimental values of the velocity field, temperature and species mass fractions to illustrate the efficiency and accuracy of the present method.     

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.     

On the transient response of forced nonlinear oscillators

We consider the transient response of a prototypical nonlinear oscillator modeled by the Duffing equation subjected to near resonant harmonic excitation. Of interest here is the overshoot problem that arises when the system is undergoing free motion and is suddenly subjected to harmonic excitation with a near resonant frequency, which leads to a beating type of transient response during the transition to steady state. In some design applications, it is valuable to know the peak value of this response and the manner in which it depends on system parameters, input parameters, and initial conditions. This nonlinear overshoot problem is addressed by considering the well-known averaged equations that describe the slowly varying amplitude and phase for both transient and steady state responses. For the undamped system, we show how the problem can be reduced to a single parameter χ that combines the frequency detuning, force amplitude, and strength of nonlinearity. We derive an explicit expression for the overshoot in terms of χ, describe how one can estimate corrections for light damping, and verify the results by simulations. For zero damping, the overshoot approximation is given by a root of a quartic equation that depends solely on χ, yielding a simple bound for the overshoot of lightly damped systems.     

Cavitation,Invertibility, and Convergence of Regularized Minimizers in Nonlinear Elasticity

We prove that energy minimizers for nonlinear elasticity in which cavitation is allowed only at a finite number of prescribed flaw points can be obtained, in the limit as ε→0, by introducing micro-voids of radius ε in the domain at the prescribed locations and minimizing the energy without allowing for cavitation. This extends the result by Sivaloganathan, Spector, and Tilakraj (SIAM J. Appl. Math. 66:736–757, 2006) to the case of multiple cavities, and constitutes a first step towards the numerical simulation of cavitation (in the nonradially-symmetric case).      

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.     

Nonlinear effects on edge wave development

The effect of nonlinearity on the evolution of free edge waves is investigated by solving numerically the nonlinear shallow water equations. The numerical scheme is the two-dimensional WAF method, originally developed by Toro [Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 1997]. First, a simple beach geometry [J. Fluid Mech. 381 (1999) 271] is considered and the numerical results are validated against available analytical solutions for small amplitude edge waves. Then, results are obtained for edge waves of larger amplitude. Nonlinear effects are found to be relevant even for moderate values of the nonlinearity parameter a which is the ratio between the wave amplitude and the water depth. In particular a resonance phenomenon has been observed which causes a modulation in time of the ultra-harmonic components of the wavefield. Larger values of a lead to a complex time evolution.     

A lower bound for a variational model for pattern formation in shape-memory alloys

The Kohn-Müller model for the formation of domain patterns in martensitic shape-memory alloys consists in minimizing the sum of elastic, surface and boundary energy in a simplified scalar setting, with a nonconvex constraint representing the presence of different variants. Precisely, one minimizes

Large amplitude oscillatory elongation flow

A filament stretching rheometer (FSR) was used for measuring the elongation flow with a large amplitude oscillative elongation imposed upon the flow. The large amplitude oscillation imposed upon the elongational flow as a function of the time t was defined as where ε is the Hencky strain, is a constant elongational rate for the base elongational flow, Λ the strain amplitude (Λ ≥ 0), and Ω the strain frequency. A narrow molecular mass distribution linear polystyrene with a molecular weight of 145 kg/mol was subjected to the oscillative flow. The onset of the steady periodic regime is reached at the same Hencky strain as the onset of the steady elongational viscosity ( Λ = 0). The integral molecular stress function formulation within the ‘interchain pressure’ concept agrees qualitatively with the experiments.     

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.     

Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation

In this paper, the nonlinear normal modes (NNMs) of a thin beamresting on a nonlinear spring bed subjected to an axial tension isstudied. An energy-based method is used to obtain NNMs. In conjunction with amatched asymptotic expansion, we analyze, through simple formulas, thelocal effects that a small bending stiffness has on the dynamics, alongwith the secular effects caused by a symmetric nonlinearity. Nonlinearmode shapes are computed and compared with those of the unperturbedlinear system. A double asymptotic expansion is employed to compute theboundary layers in the nonlinear mode shape due to the small bendingstiffness. Satisfactory agreement between the theoretical and numericalbackbone curves of the system in the frequency domain is observed.     

Numerical analysis of nonlinear modes of piecewise linear systems torsional vibrations

The nonlinear modes of essentially nonlinear piecewise-linear finite degrees of freedom systems are calculated by the numerical methods, which are suggested in this paper. The basis of these methods is numerical solutions of the equations of the systems motions in configuration space. The numerical method for the nonlinear modes of essentially nonlinear piecewise-linear systems forced vibrations is suggested. The basis of this approach is the combination of the Rauscher method and the calculations of the autonomous system nonlinear modes. The nonlinear modes of the diesel engine transmission torsional vibrations are analyzed numerically. The vibrations are described by essentially nonlinear piecewise-linear system with fifteen degrees of freedom. The NNMs of this system forced vibrations are observed in the resonance regions. Both NNMs and the motions, which are essentially differ from NNMs, are observed in the distance from the resonances. NNMs of the forced vibrations of the systems with dissipation are close to NNMs of the system without dissipation.     

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.