Instability analysis in oscillators with velocity-modulated time-varying stiffness—Applications to gears submitted to engine speed fluctuations

A one degree-of-freedom model is set up which incorporates time-varying mesh stiffness functions and the influence of unsteady input rotations due to engine speed fluctuations. The stability of the associated parametrically excited system is analysed by calculating the monodromy matrix via a Newmark scheme. A piecewise constant and a sinusoidal mesh stiffness functions are considered and it is shown that additional side-band instability zones are generated because of frequency modulations. The influence of the mesh stiffness variations and damping is discussed.     

On the forced vibrations of an oscillator with a periodically time-varying mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator with a time-varying mass. Not only solutions of the oscillator equation will be constructed, but also stability properties for the forced vibrations will be presented for various parameter values.     

Subharmonic resonances of a mechanical oscillator with periodically time-varying, piecewise-nonlinear stiffness

The subharmonic (period-η, η>1) motions of a piecewise-nonlinear (PN) mechanical oscillator having parametric and external excitations are investigated. The system is formed by a viscously damped, single-degree-of-freedom oscillator subjected to a periodically time-varying, PN stiffness defined by a clearance surrounded by continuous forms of nonlinearity. A multiterm harmonic balance formulation in conjunction with discrete Fourier transforms is used to determine steady-state period-η motions of the system near the parametric instability regions. The accuracy of analytical solutions is verified through a comparison with direct numerical integration results. A parametric study is also presented to demonstrate the combined influence of a clearance and of cubic nonlinearities on period-η motions within typical ranges of system and excitation parameters.     

Impulsive control of nonlinear systems with time-varying delays

A whole impulsive control scheme of nonlinear systems with time-varying delays, which is an extension for impulsive control of nonlinear systems without time delay, is presented in this paper. Utilizing the Lyapunov functions and the impulsive-type comparison principles, we establish a series of different conditions under which impulsively controlled nonlinear systems with time-varying delays are asymptotically stable. Then we estimate upper bounds of impulse interval and time-varying delays for asymptotically stable control. Finally a numerical example is given to illustrate the effectiveness of the method.     

An investigation of an active landing gear system to reduce aircraft vibrations caused by landing impacts and runway excitations

A mathematical model is developed to control aircraft vibrations caused by runway excitation using an active landing gear system. Equations are derived to describe the integrated aircraft-active system. The nonlinear characteristics of the system are modelled and it is actively controlled using a Proportional Integral Derivative (PID) strategy. The performance of this system and its corresponding passive system are compared using numerical simulations. It is demonstrated that the impact loads and the vertical displacement of the aircraft's centre of gravity caused by landing and runway excitations are greatly reduced using the active system, which result in improvements to the performance of the landing gear system, benefits the aircraft's fatigue life, taxiing performance, crew/passenger comfort and reduces requirements on the unevenness of runways.     

Dynamic analysis for a planetary gear with time-varying pressure angles and contact ratios

In this study, the dynamic responses of a planetary gear are analyzed when component gears have time-varying pressure angles and contact ratios caused by bearing deformations. For this purpose, this study proposes a new dynamic model of the planetary gear, in which the pressure angles and contact ratios change with time. The main difference from previous studies is that the present study regards the pressure angles and contact ratios as time-varying variables, while previous studies regarded them as constants. After nonlinear equations of motion for the planetary gear are derived, the dynamic responses are computed by applying the Newmark time integration method. The time responses for the present and previous studies are compared to show the effects of the time-varying pressure angles and contact ratios on the dynamic behaviors of a planetary gear. In addition, the effects of bearing stiffness on the pressure angles and contact ratios are also analyzed.     

Nonlinear parametrically excited vibration and active control of gear pair system with time-varying characteristic

In the present work, we investigate the nonlinear parametrically excited vibration and active control of a gear pair system involving backlash, time-varying meshing stiffness and static transmission error. Firstly, a gear pair model is established in a strongly nonlinear form, and its nonlinear vibration characteristics are systematically investigated through different approaches. Several complicated phenomena such as period doubling bifurcation, anti period doubling bifurcation and chaos can be observed under the internal parametric excitation. Then, an active compensation controller is designed to suppress the vibration, including the chaos. Finally, the effectiveness of the proposed controller is verified numerically.     

具时变刚度的相对转动非线性动力系统的周期解问题

建立了一类具有时变刚度,非线性阻尼力和强迫周期力项的相对转动非线性动力系统. 运用Mawhin重合度理论,得到了该模型的周期解存在唯一性结果,推广了已有的结果, 并且列举了具体的例子来说明本文的结果是新的. 关键词： 相对转动非线性动力系统 时变刚度 周期解 存在唯一性     

Non-linear vibrations of cables in three dimensions,part II: Out-of-plane vibrations under in-plane sinusoidally time-varying load

In this paper, out-of-plane parametric vibrations of cables under in-plane sinusoidally time-varying load are analyzed by the multiple-degrees-of-freedom approach. The coupled Hill equations are analyzed by the harmonic balance method. Unstable regions where out-of-plane vibration occur are obtained.     

Effects of shear stiffness,rotatory and axial inertia,and interface stiffness on free vibrations of a two-layer beam

The linear dynamics of a two-layer beam is considered. Each beam is modelled by the Timoshenko kinematics, and all inertia terms are considered. The interface allows normal and tangential detachments, which are linearly proportional to the transmitted stresses. The eigenvalue problem providing the natural frequencies is solved exactly.An extensive study of the natural frequencies and of the normal modes is performed with the aim of understanding how the mechanical parameters of the composite influence the vibration behaviour.The study of the convergence of the solution toward that of a simplified problem obtained by neglecting axial and rotational inertia, shear deformations and by considering interface perfect adherence in the normal direction, constitutes the second objective of the work. This permits to consciously assess when these mechanical terms can actually be neglected, contrarily to what is usually done in practice, where they are disregarded ‘a priori’, without adequate care.     

Impulsive synchronization of two nonidentical chaotic systems with time-varying delay

This Letter investigates synchronization of two nonidentical Lur'e systems with time-varying delay and parameter mismatches via impulsive control. Based on the theory of impulsive functional differential equations, sufficient conditions for impulsive synchronization with a bound on the synchronization error are derived. An illustrative example is provided to validate the proposed method.     

Diffusion of vibrations in disordered systems

We consider diffusion of vibrations in random lattices with translational invariance. Above the frequency ω_IR corresponding to the Ioffe-Regel crossover (and depending on the strength of disorder), phonons cannot propagate through the lattice and transfer energy. At the same time, most of the vibrations in this range are not localized. We show that these delocalized excitations are similar to diffusons introduced by P. B. Allen, J. L. Feldman, J. Fabian, and F. Wooten (see, e.g., Phil. Mag. B 79, 1715 (1999)) to describe heat transport in glasses. In this range the energy in the lattice is transferred by means of diffusion of vibrational excitations. We have calculated the diffusivity of the modes D(ω) using both the direct numerical solution of Newton equations and the Edwards-Thouless formula. It is nearly constant above ω_IR and goes to zero at the localization threshold.     

Random vibration analysis of stochastic time-varying systems

The analysis of the free and forced vibration of a randomly time-varying system is the subject matter of this paper. This is a complicated problem which has received relatively little discussion in the literature. Herein two methods are presented, apart from the digital simulation technique, of finding the response moments. The first one is a series technique which can be considered as a generalization of the well known Galerkin method. The second method belongs to the class of closure techniques. Upon presuming some of the joint distributions to be Gaussian, equations are derived for the first two response moments. It is shown further that the non-Gaussian output density can be approximately predicted by a simple transformation. Detailed numerical results are obtained and compared with computer simulated response statistics. It is demonstrated that the methods developed here are highly efficient. In particular it is found that the Gaussian closure approximation has a wide range of application.     

Exponential networked synchronization of master-slave chaotic systems with time-varying communication topologies

The networked synchronization problem of a class of master-slave chaotic systems with time-varying communication topologies is investigated in this paper. Based on algebraic graph theory and matrix theory, a simple linear state feedback controller is designed to synchronize the master chaotic system and the slave chaotic systems with a time- varying communication topology connection. The exponential stability of the closed-loop networked synchronization error system is guaranteed by applying Lyapunov stability theory. The derived novel criteria are in the form of linear matrix inequalities (LMIs), which are easy to examine and tremendously reduce the computation burden from the feedback matrices. This paper provides an alternative networked secure communication scheme which can be extended conveniently. An illustrative example is given to demonstrate the effectiveness of the proposed networked synchronization method.