An investigation of the stability in the large for an autonomous second-order two-degree-of-freedom system

An extension to an algorithm due to Simpson has been developed for the analysis of a second-order two-degree-of-freedom autonomous system. The form of equations considered arises from the study of mechanical systems with a single concentrated non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. For a system possessing a stable equilibrium point and an unstable limit cycle arising from a subcritical Hopf bifurcation, the method has been applied to the problem of predicting the basin of attraction of the equilibrium point. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution. The method was applied to four weakly non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in the cases considered. However, it would be expected that the method would not give such accurate results if the non-linear effect was more significant. This was illustrated for the case of the cubic hardening non-linearity.     

Excitation symmetry analysis method (ESAM) for calculation of higher order non-linearities

A non-linear system with 3rd order non-linearity is fully characterized using symmetry analysis (SA) applied to the excitation, as it is done in 2nd order non-linear systems using the pulse inverted method. Symmetry analysis is performed using irreductible representations and the character table of C₃ rotation point group, which leads to the construction of three eigen-excitations allowing extraction of the 3rd order non-linearity parameter without the perturbation of fundamental and 2nd order terms. Validation of this concept is based on excitation symmetry analysis method (ESAM) which was tested on simulated noisy signals and compared with classical spectral analysis.     

The prediction of time-dependent non-linear stresses in viscoelastic materials: I. Selection of constitutive equation and strain measure

The theories for the prediction of time-dependent, non-linear stresses in viscoelastic materials such as polymers are reviewed, and it is noted that the commonly observed stress non-linearity may be ascribed either, as is usually done, to memory-function non-linearity or, alternatively, to strain-measure non-linearity. To investigate the latter alternative whilst retaining a general memory-function non-linearity, a single-integral constitutive equation of the Bird—Carreau type is employed but with an arbitrary strain measure I in place of the normally employed Finger tensor F. This model includes as special cases a large proportion of the constitutive equations previously employed for predictive purposes and in particular with a linear memory function it is shown to be indistinguishable, with the normally conducted shear experiments, from the successful BKZ model.In the new model the shear component I₁₂ of the strain measure can be found from experimental results obtained in the startup of steady shear flow, without specification or restriction of memory-function non-linearity. The form of I₁₂ found from experiment is quite non-linear in shear a for ¦a¦> 2, and hence differs from the F tensor for which F₁₂ = a. The same form for I₁₂ found for a variety of polymer solutions and a polymer melt and consequently a simple function describing I₁₂ is proposed as a new, material-independent, strain measure.     

Large Uniform Expansions of Periodic Solutions to Strongly Non-Linear Evolution Equations with Odd Polynomial Non-Linearity

A new method of uniform expansions of periodic solutions to ordinary differential equations with arbitrary odd polynomial non-linearity is constructed to study quasi-harmonic processes in non-linear dynamical systems, in particular when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter. In the particular case of a single-periodic solution, this small parameter appears due to descending the amplitudes of harmonics monotonically with increasing their number. Due to uniform expansion the amplitudes of higher harmonics turn out to be rational and fractional functions in the amplitude of the first harmonic and the frequency of oscillations. We show that the method of uniform expansions is an effective tool for obtaining convergent expansions of periodic solutions in explicit form all over the domain, where periodic solutions exist, independently of the magnitude of non-linearity. In each subsequent approximation, one more higher harmonic is taken into account, with all the other harmonics being corrected. We demonstrate the effectiveness of the method on the examples of the harmonically forced Duffing oscillator; free vibrations of the oscillator with fifth-power non-linearity and mathematical pendulum.     

TTS in LAOS: validation of time-temperature superposition under large amplitude oscillatory shear

The validation of time-temperature superposition of non-linear parameters obtained from large amplitude oscillatory shear is investigated for a model viscoelastic fluid. Oscillatory time sweeps were performed on a 11?wt.% solution of high molecular weight polyisobutylene in pristane as a function of temperature and frequency and for a broad range of strain amplitudes varying from the linear to the highly non-linear regime. Lissajous curves show that this reference material displays strong non-linear behaviour when the strain amplitude is exceeding a critical value. Elastic and viscous Chebyshev coefficients and alternative non-linear parameters were obtained based on the framework of Ewoldt et al. (J Rheol 52(6):1427?C1458, 2008) as a function of temperature, frequency and strain amplitude. For each strain amplitude, temperature shift factors a _T (T) were calculated for the first order elastic and viscous Chebyshev coefficients simultaneously, so that master curves at a certain reference temperature T _ref were obtained. It is shown that the expected independency of these shift factors on strain amplitude holds even in the non-linear regime. The shift factors a _T (T) can be used to also superpose the higher order elastic and viscous Chebyshev coefficients and the alternative moduli and viscosities onto master curves. It was shown that the Rutgers-Delaware rule also holds for a viscoelastic solution at large strain amplitudes.     

Time delay Duffing’s systems: chaos and chatter control

The effect of a delay feedback control (DFC), realized by displacement in the Duffing oscillator, for parameters which generate strange chaotic Ueda attractor is investigated in this paper. First, the classical Duffing system without time delay is analysed to find stable and especially unstable periodic orbits which can be stabilized by means of displacement delay feedback. The periodic orbits are found with help of the continuation method using the AUTO97 software. Next, the DFC is introduced with a time delay and a feedback gain parameters. The proper time delay and feedback gain are found in order to destroy the chaotic attractor and to stabilize the periodic orbit. Finally, chatter generated by time delay component is suppressed with help of an external excitation.     

Parametric excitation in non-linear dynamics

Consider a one-mass system with two degrees of freedom, non-linearly coupled, with parametric excitation in one direction. Assuming the internal resonance 1:2 and parametric resonance 1:2 we derive conditions for stability of the trivial solution by using both the harmonic balance method and the normal form method of averaging. If the trivial solution becomes unstable, a stable periodic solution may emerge, there are also cases where the trivial solution is stable and co-exists with a stable periodic solution; if both the trivial solution and the periodic solution(s) are unstable, we find an attracting torus with large amplitudes by a Neimark-Sacker bifurcation. The results of the harmonic balance method and averaging are compared, as well as the results on the Neimark-Sacker bifurcation obtained by the numerical software package CONTENT and by averaging. In all cases we have good agreement.     

Uniform Expansions of Periodic Solutions for the Third Superharmonic Resonance

A new method of uniform expansions of periodic solutions to ordinary differential equations has recently been proposed to study quasi-harmonic processes in non-linear dynamical systems, in particular, when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter that appears due to descending the amplitudes of harmonics monotonically with increasing their number (this is the condition that the term quasi-harmonic implies). In this paper, the method is generalized for the third superharmonic resonance (when the first and the third harmonics become of the same magnitude) in a harmonically forced oscillator with arbitrary odd polynomial non-linearity.     

Determination of the non-linear parameter (mobility factor) of the Giesekus constitutive model using LAOS procedure

The paper introduces a novel procedure to determine the non-linear parameter of the Giesekus model, in relation to the characterization of the non-linear oscillatory shear regime of viscoelastic polymer solutions based on polyacrylamide. Instead of using the shear-thinning viscosity as the representative non-linear effect, the third harmonic in the Fourier spectrum of the shear stress response signal is considered for computing the mobility factor. The fluid is subjected to large amplitude oscillatory shear (LAOS) and its response is recorded. Deviations of this signal from the sinusoidal form are specific to each material and gives both qualitative and quantitative measures of the non-linearity. By fitting the material response with the corresponding numerical solutions of the n-modes Giesekus constitutive relation, one can extract the values of the non-linear α_i-parameters that describe the fluid rheology. It is demonstrated that this procedure, which can be successfully applied to semi-concentrated polymer solutions, provides better results than the classical viscosity-fit method.     

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.     

Solar sail dynamics in the three-body problem: Homoclinic paths of points and orbits

In this paper we consider the orbital dynamics of a solar sail in the Earth-Sun circular restricted three-body problem. The equations of motion of the sail are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the sail. We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described.     

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡λV/R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De_c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De_Lc≡λV/L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.     

Non-linear dynamics of wind turbine wings

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli-Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω. Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω₂/ω₁?2, internal resonances between these modes have been studied. It is demonstrated that for ω/ω₁?0.66,1.33,1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω/ω₁?1.0 and ω/ω₂?1.0 partly caused by the additive load term.     

多自由度参激系统稳定性分析的数值解法

现有参激系统的动力稳定性问题研究主要集中在主不稳定区域上。为获得组合不稳定区域,基于Floquet方法,采用Bolotin方法在不同周期数下设解形式,结合特征值分析法得到确定多自由度参激系统动力不稳定区域的数值解法。对一个两自由度受周期轴向力的旋转轴系算例的稳定性分析,发现通过增加设解近似项数可获得高阶不稳定区域,且各阶不稳定区域边界随近似次数的增加逐渐趋于稳定,此外,增大阻尼可使各不稳定区域边界变得更加平滑。本文方法可用于一般多自由度周期参激阻尼系统,是一种简明易操作的直接数值解法。     

Stability chart of parametric vibrating systems using energy-rate method

Based on the integral of energy and numerical integration, we introduce, develop, and apply a general algorithm to predict parameters of a parametric equation to produce a periodic response. Using the new method, called energy-rate, we are able to find not only stability chart of a parametric equation which indicates the boundaries of stable and unstable regions, but also periodic responses that are embedded in stable or unstable regions.There are three main important advantages in energy-rate method. It can be applied not only to linear but also to non-linear parametric equations; most of the perturbation methods cannot. It can be applied to large values of parameters; most of the perturbation methods cannot. Depending on the accuracy of numerical integration method, it can also find the value of parameters for a periodic response more accurate than classical methods, no matter if the periodic response is on the boundary of stability and instability or it is a periodic response within the stable or unstable region.In order to introduce the energy-rate method and indicate its advantages we apply the method to the standard Mathieu's equation,

     

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 the flow over a cylinder moving harmonically in the cross-flow direction

The flow over a stationary cylinder is self-excited with a specific natural frequency f_N. When the cylinder is moved harmonically in the cross-flow direction, the response of the flow (in terms of the lift force) will contain two frequencies, namely, the natural frequency f_N and the excitation frequency f_E. When f_E is close to f_N, the natural flow response will be entrained by the excitation, and the response will be periodic with frequency f_E, and dynamicists refer to this phenomenon as lock-in or synchronization. When f_E is away from f_N, the flow will be either periodic with a period that is multiple of the excitation period (i.e., period-n) and dynamicists refer to this phenomenon as secondary synchronization or quasiperiodic consisting of two incommensurate frequencies, or chaotic. We use modern methods of non-linear dynamics to characterize these responses and show that the route to chaos is torus breakdown.     

On infinitesimal stability and instability of pendulum type oscillations

For the pendulum type of oscillations governed by the equation ? + φ(x) = 0, with φ(x) an odd function, it is shown that according to the linearized disturbance equation, stability is predicted if and only if dTd_x = 0. where T is the period and α is the amplitude of the non-linear steady-state oscillations. From this it follows that for a given non-linear function φ(x). infinitesimal stability can at most be predicted only for certain discrete values of α. It is shown analytically that for a simple pendulum, a power-law spring and a cubic hard or soft spring, the oscillations are infinitesimally unstable for all α. It is further shown, however, that particular cases of non-linear restoring forces do exist for which infinitesimal stability is predicted for certain α's, in contrast to the actual Liapunov instability in these cases.     

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.