The coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features

The purpose of this paper is to investigate the coupling vibration characteristics of a flexible shaft-disk-blades system with mistuned features. There are some new phenomena due to the coupling effects of shaft-bending, shaft-torsion, disk-transverse and blade-bending. In this investigation, this paper mainly focuses on the influence of mistuned features of the blade's length and the stagger angle. It is found that there are four types of coupling modes: the coupling mode of shaft bending, disk transverse and blade bending (SDB), the coupling mode of shaft torsion disk transverse-blade bending (TDB), the coupling mode of disk transverse and blade bending (DB), the repeated mode of blade bending-blade bending (BB). With the effect of mistuned features, the natural frequencies and the coupling mode type will change correspondingly. With the mistuning value of blade length employed in this study, the TDB mode in the tuned system will disappear and shift into TSDB mode instead, and one of the repeated SDB modes will be replaced by STDB modes. Due to this mistuned features, the blades and disk experience a certain degree of vibration localization phenomenon. Different from the length feature, the influence of mistuning values of blade's stagger angle mainly take effect on the coupling modes. At last, by inspection on the Campbell diagrams, the influence of rotational speed on the transformation of natural frequencies is illustrated on the tuned/mistuned flexible shaft-disk-blades coupling structure.     

On some regularities in dynamic response of cyclic periodic structures

The paper deals with cyclic periodic structures modelling bladed disk assemblies of blades with friction elements for vibration damping. These elements placed between adjacent blades reduce the vibration amplitudes by means of dry friction resulting from centrifugal forces acting on the elements and relative displacements of the blades. However, the application of these friction elements results in an additional dynamical coupling which together with mistuning of some system parameters (e.g., blade eigenfrequency or contact parameters) may cause localization of vibration. In the present paper a linear approximation of such a system is investigated. The structure composed of cyclic periodic cells modelled each as a clamped-free beam interacting with each other by means of viscoelastic elements of complex stiffness is applied for dynamic system analysis. In case of free vibrations as well as in case of steady-state dynamic response to a harmonic pressure field, a perfect periodic structure and the structures with periodically mistuned parameters (blade eigenfrequencies and contact parameters) are studied. Some regularities in the dynamic response of the systems with mistuning have been noticed. Despite the fact that only a linear approximation has been used, the results and conclusions can be applied for models which describe the blade interaction in a nonlinear way.     

Resonant vibrations in harmonically excited weakly coupled mechanical systems with cyclic symmetry

The resonant vibrations in weakly coupled nonlinear cyclic symmetric structures are studied. These structures consist of weakly coupled identical nonlinear oscillators. A careful bifurcation analysis of the amplitude equations is performed in the fundamental resonance case for an illustrative example consisting of a three particle system. In case of a uniformly distributed excitation, a localized response is identified in which one of the particles exhibits large amplitude motions compared to those of the other particles. In case of single-particle excitation, it is found that for very small coupling strength and large external mistuning, a large stable localized periodic response coexists with an extended small response. With an increase in the coupling strength, multiple extended solutions arise near the exact external resonance via saddle-node bifurcations. Further increase in coupling strength and a decrease in damping results in isolated asymmetric solution branches, which bifurcate from the symmetric solutions via symmetry-breaking bifurcations. The role of coupling strength in creating/destroying localized solutions is discussed.     

Analysis of flow characteristics of granular material unloaded on nonlinear vibration inclined platform

In this study, the repeated discontinuous friction between granular material and contact platform and structural nonlinearity of inclined vibration platform giving rise to the vibration flow-aiding unloading is a complicated process, which has significant effects on the dynamic behaviors and flow characteristics of granular material. A simplified mathematical model of the inclined vibration platform and granular material is deduced by mechanical properties. Based on the equations of motion and a good degree of accuracy and applicability of the process with calculated data reported in the literature, the approximate analytical solution and flow properties are investigated by using the modified incremental harmonic balance method and numerical integration method. Moreover, the influences of friction coefficient, excitation amplitude, nonlinear stiffness and inclined angle on the complicated dynamic behaviors are explored and discussed. It is shown that the different motion paths of granular material on inclined vibration platform are observed depending on the different parameters. The increasing friction coefficient has complicated effects on the nonlinear dynamic behaviors of the granular material. The excitation amplitude and nonlinear stiffness can effectively control the flow characteristics of granular material at low excitation frequency but the inclined angle presents opposite property. The research may contribute to improve unloading efficiency, predict the motion state of granule and provide a theoretic foundation for further design the unloading system.     

Size dependent forced vibration of nanoplates with consideration of surface effects

In this article, an analytical method is presented to study the size dependent forced vibration of rectangular nanoplates under general external loading using a generalized form of Kirchhoff plate model. The effects of surface properties including surface elasticity, surface residual stresses and surface mass density are considered which are bases for size dependent behaviors due to increase in surface to volume ratios at smaller scales. At first, a complete discussion is given for size dependent natural frequencies which are then used in forced vibration analyses. It is shown that the surface properties compact the frequency spectrums of the nanoplates. Saving generality and using the superposition principle, closed form solution is derived for time response of nanoplates under general harmonic loads. As a result, some elliptic curves are obtained for which the surface properties will not change the time response of nanoplates when a point load is applied on any point of these curves. It is observed that, for actuations inside these ellipses, the surface effects reduce the vibration amplitude while increase it for actuations outside the curves. Sensitivity of the problem to the excitation frequencies is also studied and various examples are given to illustrate the trend of size dependencies.     

Transient amplification of maximum vibration amplitudes

Many low damped structures as turbine blades or drill strings are exposed to high dynamical loads causing high vibration amplitudes. These applications comprise sub-critical eigenfrequencies. Hereby, the lower eigenfrequencies have to be passed before reaching the operating point. Most investigations of vibration amplitudes caused by a resonance passage deal with the computation of single degree of freedom systems. Thereby, it has been shown that the stationary vibration response provides the highest possible amplitude. Further it can be stated that the maximum vibration response of the resonance passage decreases with an increasing sweep velocity [3]. Isolated modes of linear systems can be represented by single degree of freedom systems. Subsequently a mode shape can be described by the multiplication of the amplification function of the mode and the belonging eigenvector. There are only some recent works that deal with resonance passages of vicinal modes, e. g. [1]. In this paper the resonance passage of a three dimensional system with nearby modes is studied. To calculate the transient vibration response an analytical approach is used. It is shown that the maximum amplitude of the stationary vibration response is not the upper limit for the maximum amplitude of the resonance passage. Thus, the maximum amplitude may rise while the sweep velocity increases. Hence, regarding a multi degree of freedom system the maximum amplitude of the resonance passage can exceed the maximum amplitude of the stationary vibration response. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Curve veering and mode localization in a buckling problem

Summary It is shown on a simple example that small disorder, or mistuning, may alter drastically the eigensolution of buckling problems in nearly periodic structures with weak internal coupling. Specifically, the phenomena of eigenvalue loci veering and mode shape localization, which are known to occur for free vibration problems, are evidenced in the case of structural buckling as well.     

Mechanical model of a mistuned 2DOF-structure coupled by viscous damping

In this paper the vibration amplitude reduction of a mechanical system is investigated when applying a damper between different DOF of that structure. Thereby, an elementary mechanical model consisting of two mass-spring elements with slightly differing eigenfrequencies and which are connected via a viscous dashpot damper is regarded. To investigate the systems behavior, an analysis of the transfer function and its poles and zeros in the L APLACE domain is performed when varying the damping constant. This reveals that the achievable amplitude reduction depends on the mistuning of the system and that even ideal viscous damping leads to a coupling which results in a shift of eigenfrequencies. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of Artificial Neural Networks for Periodic Structures Analysis

The paper presents the nondeterministic, based on artificial neural network application approach analysis of periodic structures. We can distinguish several examples where the problem may be observed: conventional and magnetic railways, high building constructions that consist of repeatable blocks, ship and aeroplane bodies, space-shuttle periodic designs, long-beam antenna structures or mistuned blade disks with friction damping elements. The scope of research is to examine possibilities of use the neural networks for mistuning parameters definition and also to denominate its possible causes. The results obtained via neural network simulator training process are compared with the calculations based on mathematical model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A pendulum with an elliptic-type parametric excitation: Stability charts for a damped and undamped system

In this paper, a pendulum parametrically excited by the excitation which has the form of the Jacobi cn elliptic function is considered. Three cases related to the value of the elliptic parameter are distinguished: the case when it is smaller than zero, when it ranges between zero and unity, and when it is higher than unity. First, interpretations of the excitation with such elliptic parameter are given in terms of its period, higher harmonic content and the amplitude. These interpretations enable one to consider the elliptic-type excitation as a type of multi-cosine excitation whose frequency and amplitude are related mutually in a particular way. Stability charts are determined for damped and undamped systems. When the elliptic parameter is equal to zero, the governing equations considered transform to the well-known Mathieu equation. In all other cases, the governing equations considered can be seen as a new generalisation of the Mathieu equation. The influence of an arbitrary real elliptic parameter on the location and shape of the transition curves and instability tongues is investigated, illustrated and discussed in all three cases, which represent new and so far unknown results.     

Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity

Nonlinearly parametric resonances of axially accelerating moving viscoelastic sandwich beams with time-dependent tension are investigated in this paper. Based on the Kelvin differential constitutive equation, the controlling equation of the transverse vibration of a beam with large deflection is established. The system has been subjected to a time varying velocity and a harmonic axial tension. Here the governing equation of motion contains linear parametric terms and two frequencies, one is the frequency of axially moving velocity and the other one is the frequency of varying tension. The method of multiple scales is applied directly to the governing equation to obtain the complex eigenfunctions and natural frequencies of the system. The elimination of secular terms leads to the steady-state response and amplitude of vibrations. The influence of various parameters such as initial tension on natural frequencies and the amplitude of axial fluctuation, the phase angle between the two frequencies on response curves has been investigated for two different resonance conditions. With the help of numerical results, it has been shown that by using suitable initial tension, the amplitude of axial fluctuation, the phase angle, the vibration of the sandwich beam can be significantly controlled.     

动态环板元的动刚度矩阵

本文对于环形薄板单元取包含贝塞尔函数的谐振变形作为形状函数,解决了关于特殊函数的复杂积分问题.从而精确推导了环形单元的动刚度矩阵,并用直接刚度法进行了校核.接着,又着重于将封闭形式的动刚度矩阵,按频率平方的升幂式展开,得到了简洁完备的结果,以此作为结构动力特性分析和响应计算的基础.     

Non-stationary analysis of a rotating shaft with geometrical nonlinearity during passage through critical speeds

In this paper, analysis of a rotating shaft with stretching nonlinearity during passage through critical speeds is investigated. In the model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. First, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with non-constant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the non-stationary vibration of the rotating shaft, the asymptotic method is applied to the equations expressed in normal coordinates. Two analytical expressions, as function of system parameters that describe the amplitude and phase of motion during passage through critical speeds are derived. The effects of angular acceleration, stretching nonlinearity, eccentricity and external damping on maximum amplitude of the shaft are investigated. It is shown that the nonlinearity has important effect on maximum amplitude when the rotating shaft passing through critical speeds, especially in low angular acceleration. To validate the results of the perturbation method, numerical simulation is applied.     

Bi-harmonically excited pendulum: shifted resonances and quenching the low frequency excitation

The effect of the harmonic high-frequency vibration of the suspension point on a mathematical pendulum is well known [1]. It can be described as stiffening/softening or stabilization/destabilization of the up-pointing and down-pointing equilibriums for pure vertical or pure horizontal excitation. If the excitation is tilted the effect of biasing appears in addition [2]. The effects of bi-harmonic excitation are more complex. Two of them are discussed in the recent paper. The first one occurs if one of the excitation frequencies is high and another one is low. Then the so called shifted or conjugated resonances can be observed [3]. The second one takes place if the two excitation frequencies are high but their difference is small (slowly modulated high-frequency excitation according to [4]). It is shown that in that case it is possible to choose the bi-harmonic excitation in order to quench the resonance of the pendulum without changing its basic frequency. The analysis is fulfilled both for the external and parametric excitations. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy harvesting by coupled oscillators forced by excitations with harmonic and random components

We examine an energy harvesting system of two magnetopiezoelastic oscillators coupled by electric circuit and driven by a combination of harmonic and stochastic ambient disturbances. The effects of synchronization and escape from a single potential well are discussed. In the system with relative mistuning in the stiffness of the harvesting oscillators, we show the dependence of the voltage output for different noise levels and excitation frequency. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic analysis and suppressing chaotic impacts of a two-degree-of-freedom oscillator with a clearance

A two-degree-of-freedom impact oscillator is considered. The maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Impacts between the mass and the stops are described by an instantaneous coefficient of restitution. Dynamics of the system is studied with special attention to periodic-impact motions and bifurcations. Period-one double-impact symmetrical motion and transcendental impact Poincaré map of the system is derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motions are analyzed by using the impact Poincaré map. The Lyapunov exponents in the vibratory system with impacts are calculated by using the transcendental impact map. The influence of the clearance and excitation frequency on symmetrical double-impact periodic motion and bifurcations is analyzed. A series of other periodic-impact motions are found and the corresponding bifurcations are analyzed. For smaller values of clearance, period-one double-impact symmetrical motion usually undergoes pitchfork bifurcation with decrease in the forcing frequency. For large values of the clearance, period-one double-impact symmetrical motion undergoes Neimark–Sacker bifurcation with decrease in the forcing frequency. The chattering-impact vibration and the sticking phenomena are found to occur in the region of low forcing frequency, which enlarges the adverse effects such as high noise levels, wear and tear and so on. These imply that the dynamic behavior of this system is very rich and complex, varying from different types of periodic motions to chaos, even chattering-impacting vibration and sticking. Chaotic-impact motions are suppressed to minimize the adverse effects by using external driving force, delay feedback and feedback-based method of period pulse.     

基于非线性能量阱的双频激励非线性系统减振

针对某型民用航空发动机双频带激励特点,建立了单自由度线性振子耦合非线性能量阱（nonlinear energy sink,NES）的动力学模型.根据典型双转子发动机在巡航状态下低、高特征频率比（1∶4.74）,为系统设定双频带简谐外激励.利用四阶Runge-Kutta算法,研究了耦合NES振子时系统的振动抑制特征,并从外激励频率对系统主振子动能、系统总体能量的影响等方面,与未耦合NES系统、耦合线性动力吸振器两种情况下的数值计算结果进行对比分析.研究结果表明NES对双频带外激励具有更好的振动抑制效果,用NES降低航空发动机振动有可行性.     

相对于Wiener-Hosoya指标的最小树

Wiener-Hosoya指标是由Randic在文[1]中引入的一个指标,旨在揭示分子结构与其化学性质的更进一步的关系.任意给定点数及直径,本文确定了相对于该指标的最小树.进一步地,具有任意给定点数的最小的16个树也得到确定.     

Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects

The main objective of this study is to predict both the subharmonic and superharmonic resonances of the nonlinear oscillation of nanobeams in the presence of surface free energy effects. To this purpose, Gurtin–Murdoch elasticity theory is adopted to the classical beam theory in order to consider the surface Lame constants, surface mass density, and residual surface stress within the differential equations of motion. The Galerkin method together with the method of multiple scales is utilized to investigate the size-dependent response of nanobeams under hard excitations corresponding to various boundary conditions. A parametric analysis is carried out to indicate the influence of the surface elastic parameters on the frequency-response as well as amplitude-response of the nonlinear secondary resonance including multiple vibration modes and interactions between them. It is seen that for the superharmonic excitation, except for the clamped–free boundary condition, the jump phenomenon is along the hardening direction, while in the clamped–free end supports, it is along the softening direction. Moreover, it is revealed that for the subharmonic excitation, within a specific range of the excitation amplitude, the nanobeam is excited, and this range shifts to lower external force by incorporating the surface free energy effects. It is found that in the case of superharmonic excitation, the value of the excitation frequency associated with the bifurcation point at the peak of the frequency-response curve increases by taking the surface free energy effect into consideration.     

Torsional Vibration Damping Through Frictional Torsion Damper with Structural Friction and Slide Taken into Consideration

The paper is concerned with a non-linear discrete stationary mechanical system containing a frictional torsion damper. Proper effect of vibration damping in a two-degree-of freedom system can be reached by the right selection of geometrical parameters for given loads, as pre-determined by a mathematical model. Structural friction was considered, as well as small relative sliding of damper's discs cooperating with a plunger. The system vibrates under harmonic excitation. The problem was considered on the assumption of uniform unit pressure distribution between the contacting surfaces of friction discs and the plunger. When the discs are sliding, the friction coefficient varies, depending on relative angular velocity. Friction characteristics were assumed on the basis of the author's own research and experimental testing by other authors. Properties of the material were assumed to be in accordance with classical theory of elasticity. The author analysed the influence of parameters of the dynamic system upon amplitude and frequency characteristics as well as on phase and frequency characteristics. The equation of motion was solved by means of the slowly-varying-parameters method and, in order to compare the results, by means of numerical simulation. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)