Effect of Unsteady Wind Flow on Galloping of Tall Prismatic Structures

The galloping of tall prismatic cantilevertype structures due to unsteady wind is solvedanalytically. The unsteady wind was considered byadding a time varying wind speed component to the meanwind speed component. In reality, the time-varyingwind speed component is a random phenomenon that can bemodeled as a series of harmonic terms using thetransformation of the unsteady wind speed spectruminto the time domain. In doing this it is apparentthat the structure is subjected to multiharmonicexternal and parametric excitations due to theunsteady wind in addition to the nonlinearself-excited wind forces due to the steady wind speedcomponent. To have a clear insight into the unsteadywind effect, only one harmonic term is considered outof all the harmonic terms. The multiple-scale methodis used to study the effect of primary and secondaryresonances on the galloping response of the structure. Comparisons between the analytical results obtainedfrom the method of multiple scales and the numericalsolutions obtained from numerical integration indicate the accuracy of the analysis and thecomprehensive information obtained from the analyticalsolutions.     

Response of a non-linearly damped oscillator to combined periodic parametric and random external excitation

An analytical solution, using the Fokker-Planck-Kolmogorov equation, is obtained for the problem of response of a non-linearly damped oscillator to combined periodic parametric and random external excitation. The solution yields first-order probability densities of amplitude and phase. These expressions are employed to distinguish between oscillations excited by external and parametric periodic forces in the presence of additional broadband random external excitation. Through decoupling of fast and slow motions an approximate expression is obtained for expected value of time to phase “switch”.     

Nonlinear dynamics and small damping signal control of chaos in a model of flow-induced oscillations of cylinders

Nonlinear dynamics of flow-induced oscillations of cylinders is investigated. The approach in our paper is made to introduce an harmonic forced vibration in the coupling term of the structural equation since this may be the consequence of approximating the potential force that could act as a periodic excitation. The method of multiple scales is used to determine the steady state responses. Amplitude and phase modulation equations as well as external force-response and frequency-response curves are obtained. We show that harmonic excitation can induce resonance phenomena in the oscillation of the structure for a range of frequencies of potential force, and also lock-in phenomena appear in the structure part. Also, we find that the structure can be damaged as the amplitude of the potential excitation increases. Numerical simulations confirm the existence of chaotic vibration in the system, a small damping signal control is used to suppress it since it may cause fatigue in the system. The model developed is expected to yield better results for structure in water.     

Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations

The principal parametric resonance of a single-degree-of-freedom system with non-linear two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations that describe to first-order the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict the existence of steady state responses. Stability is investigated. The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed.     

Fast parametrically excited van der Pol oscillator with time delay state feedback

We investigate the effect of a fast vertical parametric excitation on self-excited vibrations in a delayed van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic in the vicinity of the trivial equilibrium. Then, we apply the multiple scales method on this slow dynamic to derive a second-order slow flow system describing the modulation of slow dynamic. In particular we analyze the slow flow to obtain the effect of a fast excitation on the regions in parameter space where self-excited vibrations can be eliminated. We have shown that in the case where the time delay and the feedback gains are imposed, fast vertical parametric excitation can be an alternative to suppress undesirable self-excited vibrations in a delayed van der Pol oscillator.     

Stability analysis of a nonlinear rotating asymmetrical shaft near the resonances

An asymmetrical rotating shaft with unequal mass moments of inertia and flexural rigidities in the direction of principal axes is considered. In this system, there are two excitation sources, including a harmonic excitation due to the dynamic imbalances and a parametric excitation due to shaft asymmetry. Nonlinearities are due to the in-extensionality of the shaft and large amplitude. In this study, harmonic and parametric resonances due to the mentioned effects are considered. The influences of inequality of mass moments of inertia and flexural rigidities in the direction of principal axes, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation of steady state response of the rotating asymmetrical shaft are investigated. In addition, the characteristic of stable stationary points and loci of bifurcation points as function of damping coefficient are determined. In order to analyze the resonances of the system the multiple scales method is applied to the complex form of partial differential equations of motion. The achieved results show a good agreement with those of numerical computation.     

On the dynamics of rings rotating with variable spin speed

This work investigates nonlinear dynamic response of circular rings rotating with spin speed which involves small fluctuations from a constant average value. First, Hamilton's principle is applied and the equations of motion are expressed in terms of a single time coordinate, representing the amplitude of an in-plane bending mode. For nonresonant excitation or for slowly rotating rings, a complete analysis is presented by employing phase plane methodologies. For rapidly rotating rings, periodic spin speed variations give rise to terms leading to parametric excitation. In this case, the vibrations that occur under principal parametric resonance are analyzed by applying the method of multiple scales. The resulting modulation equations possess combinations of trivial and nontrivial constant steady state solutions. The existence and stability properties of these motions are first analyzed in detail. Also, analysis of the undamped slow-flow equations provides a global picture for the possible motions of the ring. In all cases, the analytical predictions are verified and complemented by numerical results. In addition to periodic response, these results reveal the existence of unbounded as well as transient chaotic response of the rotating ring.     

Chaos control and synchronization of the Φ<Superscript>6</Superscript>-Van der Pol system driven by external and parametric excitations

The dynamical behavior of the Φ⁶-Van der Pol system subjected to both external and parametric excitation is investigated. The effect of parametric excitation amplitude on the routes to chaos is studied by numerical analysis. It is found that the probability of chaos happening increases along with the parametric excitation amplitude increases while the external excitation amplitude fixed. Based on the invariance principle of differential equations, the system is lead to desirable periodic orbit or chaotic state (synchronization) with different control techniques. Numerical simulations are provided to validate the proposed method.     

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.     

Principal resonance of a nonlinear system with two-frequency parametric and self-excitations

The principal resonance of a single-degree-of-freedom system with two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations of modulation of amplitude and phase. Qualitative analyses are employed to study the behaviour of steady state responses, limit cycle responses and 2-torus responses, including their stability and bifurcation. The effects of damping, detuning, and magnitudes of self-excitation and parametric excitations are analyzed. The theoretical analyses are verified by numerical integration results of the governing equation and the modulation equations.     

Nonlinear dynamic responses of high-speed railway vehicles under combined self-excitation and forced excitation considering the influence of unsteady aerodynamic loads

The dynamic characteristics of a railway vehicle system under unsteady aerodynamic loads are examined in this study. A dynamic analysis model of the railway vehicle considering the influences of aerodynamic loads was established. The model not only considers the forced excitation effect of unsteady aerodynamic loads but also accounts for the effect of unsteady aerodynamic loads on the change of the wheel–rail contact normal forces as well as changes of the wheelset creep coefficients and creep forces/moments. Therefore, this model also considers the influences of unsteady aerodynamic loads on the self-excited vibration characteristics of the vehicle system. The time-history curves, phase trajectory diagrams, Poincaré sections, and Lyapunov exponents of the vehicle system running on a smooth straight track under unsteady aerodynamic loads were determined. The results show that when the critical speed is exceeded, the vehicle system usually performs quasi-periodic motion under unsteady aerodynamic loads, which is significantly different from the periodic motion under steady aerodynamic loads. In different cases, the amplitude and phase of motion are significantly different. The amplitude of the motions can be increased by more than 159%, and the difference of phase can be up to 173°. (The phase is almost reversed.) The dynamic responses of the vehicle system under unsteady aerodynamic loads contain abundant frequency components, including the frequency of the self-excited vibration, the frequency of the forced excitation, and combinations of their integer multiples. The vibration forms corresponding to the main harmonic components under unsteady and steady aerodynamic loads were compared, and the self-excited vibration component of the vehicle system under unsteady aerodynamic loads was identified. The variations in the critical speed with various parameter combinations were computed. The variation range of the critical velocity can reach 73%.

     

On the quasi-steady analysis of one-degree-of-freedom galloping with combined translational and rotational effects

Galloping is the low-frequency, self-excited oscillation of an elastic structure in a wind field. Its analysis is commonly based on a quasi-steady aerodynamic analysis, in which the instantaneous wind forces are derived from force data obtained in static wind tunnel tests. For the galloping of a rigid prismatic beam the validity of the quasi-steady assumption is critically assessed for the case that rotational effects must be included in the aerodynamics. An oscillator structure with one (torsional) degree of freedom is proposed which allows a reliable modelling. Its effective motion can be considered as being composed of a translation with a coupled rotation of the cross section, and can be regarded as a natural extension of pure translational galloping. The analysis reveals that the resulting aerodynamic damping is determined by the sectional aerodynamic normal force coefficient alone. An aerodynamic damping coefficient is defined that can be expressed uniquely in terms of an aerodynamic amplitude, allowing a normalization of the galloping curve. This result can be used to analyze both purely translational and combined galloping, which are found to differ only by the way the structural amplitude (displacement) is related to the aerodynamic amplitude. An interesting result is that for large wind speeds rotational galloping displays an aerodynamic limit, in contrast to translation galloping where the limit-cycle amplitude increases linearly with wind speed. Results obtained from wind tunnel experiments confirm the major findings of the analysis.     

Local bifurcation theory of nonlinear systems with parametric excitation

This paper summarizes the authors' research on local bifurcation theory of nonlinear systems with parametric excitation since 1986. The paper is divided into three parts. The first one is the local bifurcation problem of nonlinear systems with parametric excitation in cases of fundamental harmonic, subharmonic and superharmonic resonance. The second one is the experiment investigation of local bifurcation solutions in nonlinear systems with parametric excitation. The third one is the universal unfolding study of periodic bifurcation solutions in the nonlinear Hill system, where the influence of every physical parameter on the periodic bifurcation solution is discussed in detail and all the results may be applied to engineering.     

The Subharmonic Bifurcation of a Viscoelastic Circular Cylindrical Shell

In this paper the nonlinear dynamic behavior of a viscoelastic circular cylindrical shell under a harmonic excitation applied at both ends is studied. The modified Flugge partial differential equations of motion are reduced to a system of finite degrees of freedom using the Galerkin method. The equations are solved by the Liapunov–Schmidt reduction procedure. In order to study 1/2 and 1/4 subharmonic parametric resonance of the shell, the transition sets in parameter plane and bifurcation diagrams are plotted for a number of situations. Results indicate that, for certain static loads, the shell may display jumps due to the presence of dynamic periodic load with small amplitude. Additionally, different physical situations are identified in which periodic oscillating phenomena can be observed, and where 1/4 subharmonic parametric resonance is simpler than the 1/2-one.     

Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems with time delay

The effect of time-delayed feedback and fast harmonic excitation (FHE) on stationary periodic vibration and quasi-periodic responses in a parametric and self-excited weakly nonlinear oscillator is analyzed in this paper. The method of direct partition of motion and two stages of multiple scales analysis are conducted to obtain analytical approximation for quasi-periodic oscillation envelopes and frequency-locking area near primary resonance. A parameter study shows that, in the absence or the presence of high-frequency excitation, time-delayed feedback may reduce significantly the amplitude and the envelopes of quasi-periodic oscillations leading to a quasi synchronization of the response over the whole frequency range around the resonance. The results presented for the parameters tested agree well with results obtained by numerical simulation.     

谐和与窄带随机噪声联合作用下Duffing系统的参数主共振

研究了Ｄｕｆｉｎｇ振子在谐和与窄带随机噪声联合激励下的参数主共振响应和稳定性问题．用多尺度法分离了系统的快变项，并求出了系统的最大Ｌｙａｐｕｎｏｖ指数．本文还分析了失稳及跳跃现象，及系统的阻尼项、非线性项、随机项、确定性参激强度对系统响应的影响．数值模拟表明本文提出的方法是有效的．     

Nonlinear dynamic analysis of a quarter vehicle system with external periodic excitation

In this paper, a nonlinear dynamic model of a quarter vehicle with nonlinear spring and damping is established. The dynamic characteristics of the vehicle system with external periodic excitation are theoretically investigated by the incremental harmonic balance method and Newmark method, and the accuracy of the incremental harmonic balance method is verified by comparing with the result of Newmark method. The influences of the damping coefficient, excitation amplitude and excitation frequency on the dynamic responses are analyzed. The results show that the vibration behaviors of the vehicle system can be control by adjusting appropriately system parameters with the damping coefficient, excitation amplitude and excitation frequency. The multi-valued properties, spur-harmonic response and hardening type nonlinear behavior are revealed in the presented amplitude-frequency curves. With the changing parameters, the transformation of chaotic motion, quasi-periodic motion and periodic motion is also observed. The conclusions can provide some available evidences for the design and improvement of the vehicle system.     

Effet d'une modulation magnétique sur le seuil d'instabilité convectif au sein d'une couche de liquide magnétique chauffée par le haut

The effect of a time-periodic magnetic field on the onset of convection in a horizontal magnetic fluid layer heated from above and bounded by isothermal nonmagnetic boundaries is investigated. We consider the case where the magnetic field obeys a periodic rectangular pulse. A first-order Galerkin method is performed to reduce the governing linear system to a parametric differential equation. Therefore, the Floquet theory is used to determine the convective threshold for the rigid–rigid and free–free cases. With an appropriate choice of the ratio of the magnetic and gravitational forces, we show the possibility to produce a competition between the harmonic and subharmonic modes at the onset of convection.     

Nonlinear characterization of concurrent energy harvesting from galloping and base excitations

An energy harvester is proposed to concurrently harness energy from base and galloping excitations. This harvester consists of a triangular cross-sectional tip mass attached to a multilayered piezoelectric cantilever beam and placed in an incompressible flow and subjected to a harmonic base excitation in the cross-flow direction. A coupled nonlinear-distributed-parameter model is developed representing the dynamics of the transverse degree of freedom and the generated voltage. The galloping force and moment are modeled by using a nonlinear quasi-steady approximation. Under combined loadings and when the excitation frequency is away from the global natural frequency of the harvester, the response of the harvester mainly contains these two harmonic frequencies. Thus, the harvester’s response is generally aperiodic and is either periodic with large period (i.e., period- \(n\) ), or quasi-periodic, or chaotic. To characterize the harvester’s response under a combination of vibratory base excitations and aerodynamic loading, we use modern methods of nonlinear dynamics, such as phase portraits, power spectra, and Poincaré sections. A further analysis is then performed to determine the effects of the wind speed, frequency excitation, base acceleration, and electrical load resistance on the performance of the harvester under separate loadings.     

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.