Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.     

NONLINEAR INTERNAL RESONANCE OF FUNCTIONALLY GRADED CYLINDRICAL SHELLS USING THE HAMILTONIAN DYNAMICS

Internal resonance in nonlinear vibration of functionally graded （FG） circular cylin- drical shells in thermal environment is studied using the Hamiltonian dynamics formulation. The material properties are considered to be temperature-dependent. Based on the Karman-Donnell＇s nonlinear shell theory, the kinetic and potential energy of FG cylindrical thin shells are formu- lated. The primary target is to investigate the two-mode internal resonance, which is triggered by geometric and material parameters of shells. Following a secular perturbation procedure, the underlying dynamic characteristics of the two-mode interactions in both exact and near resonance cases are fully discussed. It is revealed that the system will undergo a bifurcation in near resonance case, which induces the dynamic response at high energy level being distinct from the motion at low energy level. The effects of temperature and volume fractions of composition on the exact resonance condition and bifurcation characteristics of FG cylindrical shells are also investigated.     

Nonlinear vibrations of rotating thin circular cylindrical shell

Nonlinear vibrations of thin circular cylindrical shells are investigated in this paper. Based on Love thin shell theory, the governing partial differential equations of motion for the rotating circular cylindrical shell are formulated using Hamilton principle. Taking into account the clamped-free boundary conditions, the partial differential system is truncated by using the Galerkin method. Sequentially, the effects of temperature, geometric parameters, circumferential wave number, axial half wave number and rotating speed on the nature frequency of the rotating circular cylindrical shell are studied. The dynamic responses of the rotating circular cylindrical shell are also investigated in time domain and frequency domain. Then, the effects of nonlinearity, excitation and damping on frequency responses of steady solution are investigated.     

Nonlinear energy harvesting with dual resonant zones based on rotating system

An electromagnetic nonlinear energy harvester(NEH)based on a rotating system is proposed,of which the host system rotates at a constant speed and vibrates harmonically in the vertical direction.This kind of device exhibits several resonant phenomena due to the combinations of the rotating and the vibration frequencies of the host system as well as the cubic nonlinearity of the NEH.The governing equation of motion for the NEH is derived,and the dynamic responses and output power are investigated with the multiple scale method under the 1:1 primary and 2:1 superharmonic resonant conditions.The effects of system parameters including the nondimensional external frequency,the rotating speed,and the nonlinear stiffness on the responses of free vibration for the system are studied.The results of the primary resonance show that the responses exhibit not only the resonant characteristics but also the nonlinear dynamic characteristics such as the saddle-node(SN)bifurcation.The coexistence of multiple solutions and the varying trends of responses are verified with the direct numerical simulation.Moreover,the effects of system parameters on the average output power are investigated.The results of the analyses on the two resonant conditions indicate that the large power can be harvested in two resonant frequency bands.The effect of resonance on the output power is dominant for the 2:1 superharmonic resonance.Moreover,the results also show that introducing the nonlinearity can increase the value of the output power in large frequency bands and induce the occurence of new frequency bands to harvest the large power.The efficiency of the harvested power could be improved by the combined effects of the resonance as well as the nonlinearity of the NEH device.Suitable parameter conditions could help optimize the power harvesting in design.     

Free vibrations of rotating composite conical shells with stringer and ring stiffeners

In this paper, an analytical solution for the free vibration of rotating composite conical shells with axial stiffeners (stringers) and circumferential stiffener (rings), is presented using an energy-based approach. Ritz method is applied while stiffeners are treated as discrete elements. The conical shells are stiffened with uniform interval and it is assumed that the stiffeners have the same material and geometric properties. The study includes the effects of the coriolis and centrifugal accelerations, and the initial hoop tension. The results obtained include the relationship between frequency parameter and circumferential wave number as well as rotating speed at various angles. Influences of geometric properties on the frequency parameter are also discussed. In order to validate the present analysis, it is compared with other published works for a non-stiffened conical shell; other comparison is made in the special case where the angle of the stiffened conical shell goes to zero, i.e., stiffened cylindrical shell. Good agreement is observed and a new range of results is presented for rotating stiffened conical shells which can be used as a benchmark to approximate solutions.     

两尺度Duffing系统的动力吸振器减振研究

两尺度耦合的Duffing系统存在复杂振动, 此类振动具有振幅大、频率高的特点, 对系统的危害不容忽视. 研究了线性动力吸振器对低频参数激励下Duffing系统的振动控制问题, 通过对比耦合动力吸振器前后系统的时间历程图、相图, 发现加入动力吸振器后系统会由单一振动模式转变为混合振动模式(簇发振动), 振动幅值明显减小, 尤其对高频振动部分抑制明显. 利用快慢分析法, 当参数激励为慢变过程时得到相应的自治系统, 并发现自治系统稳定性与分岔行为对非自治系统振动响应具有明显调节作用. 研究结果表明, 虽然耦合动力吸振器前后自治系统均发生叉形分岔, 但是加入吸振器后自治系统稳定性发生变化, 稳定中心变为渐进稳定的焦点, 稳定平衡线对非自治系统轨线的吸引力增强, 使得响应振动幅值减小; 另外轨线在不同吸引子之间的跳跃次数减少, 也是导致响应振动幅值减小的另一个原因. 通过对参数激励的相关参数减振效果分析, 发现加入的动力吸振器在较大的振动幅值和频率范围内都能起到抑制系统振动的作用. 为两尺度系统耦合线性动力吸振器减振研究提供了理论依据.      

轴向运动导电板磁弹性非线性动力学及分岔特性

研究磁场环境中轴向运动导电薄板磁弹性动力学及分岔特性。考虑几何非线性因素,在给出薄板运动的动能、应变能及外力虚功的基础上,应用哈密顿变分原理,得到磁场中轴向运动薄板的非线性磁弹性振动方程,并给出洛伦兹电磁力的确定形式。针对横向磁场环境中条形板共振特性进行分析,应用多尺度法和奇异性理论,得到稳态运动下的分岔响应方程以及普适开折对应的转迁集。通过算例,分别得到以磁感应强度、轴向运动速度和激励力为分岔控制参数的分岔图、最大李雅普诺夫指数图和庞加莱映射图等计算结果,讨论不同分岔参数对系统呈现的倍周期和混沌运动的影响。结果表明,通过相应参数的改变可实现对系统复杂动力学行为的控制。     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

一类三维非线性系统的复杂簇发振荡行为及其机理

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.      

多层环形钢板式阻尼器钢板厚度对其性能影响的实验研究

多层环形钢板式隔振器是一种新型全金属隔振器。适用于重载高速、高低温及大温差等工况下的阻尼减振。本文对该隔振器的结构特点及制备方法进行了研究。主要分析了钢板厚度对隔振器性能的影响,制备出了具有不同结构形式和性能特点的样件,对样件进行了动静态实验研究;获得了钢板厚度不同时隔振器的弹性迟滞回线、隔振器系统频响特性、传递率及固有频率。研究结果表明:钢板厚度是影响隔振器性能的主要几何参数。改变钢板厚度,能够使隔振器具有不同的弹性阻尼性能和能量耗散性能,该结果对实际应用具有重要价值。     

COMPLICATED BURSTING BEHAVIORS AS WELL AS THE MECHANISM OF A THREE DIMENSIONAL NONLINEAR SYSTEM 1)

由多时间尺度耦合效应引起的簇发振荡行为是非线性动力学研究的重要课题之一.本文针对一类参数激励下的三维非线性电机系统(该系统可以描述两种自激同极发电机系统的动力学行为,两种系统在数学上等效),研究了当参数激励频率远小于系统自然频率时的各种复杂簇发振荡行为及其产生机理.通过快慢分析方法, 将参数激励作为慢变参数,得到了非自治系统对应的广义自治系统及快子系统和慢变量,并给出了快子系统的稳定性和分岔条件以及系统关于典型参数的单参数分岔图.借助转换相图与分岔图的叠加, 分析了对称式delayed subHopf/fold cycle簇发振荡的产生机理及其动力学转迁, 即delayed subHopf/fold cycle簇发振荡、焦点/焦点型对称式叉形分岔滞后簇发振荡和焦点/焦点型叉形分岔滞后簇发振荡.研究结果表明, 系统会出现两种不同的分岔滞后形式, 一种是亚临界Hopf分岔滞后,另一种是叉形分岔滞后,而且控制参数显著影响平衡点的稳定性和分岔滞后区间的宽度.同时初始点的选取则会影响系统动力学行为的对称性.本文的研究进一步加深了对由分岔滞后引起的簇发振荡的认识和理解.     

Asymptotic Nonlinear Behaviors in Transverse Vibration of an Axially Accelerating Viscoelastic String

This paper investigates longtime dynamical behaviors of an axially accelerating viscoelastic string with geometric nonlinearity. Application of Newton's second law leads to a nonlinear partial-differential equation governing transverse motion of the string. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. By use of the Poincare maps, the dynamical behaviors are presented based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for varying one of the following parameter: the mean transport speed, the amplitude and the frequency of transport speed fluctuation, the string stiffness or the string dynamic viscosity, while other parameters are fixed.     

Bifurcation and chaotic response of a cracked rotor system with viscoelastic supports

Cracks appearing in the shaft of a rotary system are one of the main causes of accidents for large rotary machine systems. This research focuses on investigating the bifurcation and chaotic behavior of a rotating system with considerations of various crack depth and rotating speed of the system’s shaft. An equivalent linear-spring model is utilized to describe the cracks on the shaft. The breathing of the cracks due to the rotation of the shaft is represented with a series truncated time-varying cosine series. The geometric nonlinearity of the shaft, the masses of the shaft and a disc mounted on the shaft, and the viscoelasticity of the supports are taken into account in modeling the nonlinear dynamic rotor system. Numerical simulations are performed to study the bifurcation and chaos of the system. Effects of the shaft’s rotational speed, various crack depths and viscosity coefficients on the nonlinear dynamic properties of the system are investigated in detail. The system shows the existence of rich bifurcation and chaos characteristics with various system parameters. The results of this research may provide guidance for rotary machine design, machining on rotary machines, and monitoring or diagnosing of rotor system cracks.     

Response analysis of a dual-disc rotor system with multi-unbalances–multi-fixed-point rubbing faults

Rotor unbalance and rub-impact are major concerns in rotating machinery. In order to study the dynamic characteristics of these machinery faults, a dual-disc rotor system capable of describing the mechanical vibration resulting from multi-unbalances and multi-fixed-point rub-impact faults is formulated using Euler beam element. The Lankarani–Nikravesh model is used to describe the nonlinear impact forces between discs and casing convex points, and the Coulomb model is applied to simulate the frictional characteristics. To predict the moment of rub-impact happening, a linear interpolation method is carried out in the numerical simulation. The coupling equations are numerically solved using a combination of the linear interpolation method and the Runge–Kutta method. Then, the dynamic behaviours of the rotor system are analysed by the bifurcation diagram, whirl orbit, Poincaré map and spectrum plot. The effects of rotating speed, phase difference of unbalances, convex point of casing and initial clearance on the responses are investigated in detail. The numerical results reveal that a variety of motion types are found, such as periodic, multi-periodic and quasi-periodic motions. Moreover, the energy transfer between the compressor disc and the turbine disc occurs in the multi-fixed-point rubbing faults. Compared with the parameters of the turbine disc, those of the compressor disc can affect the motion of the rotor system more significantly. That is, the responses exhibit simple 1T-periodic motion in the wide range of rotating speed under the conditions of sharp convex point and larger initial clearance. These forms of dynamic characteristics can be effectively used to diagnose the fixed-point rub-impact faults.     

NONLINEAR VIBRATION RESPONSE AND BIFURCATION OF CIRCULAR CYLINDRICAL SHELLS UNDER TRAVELING CONCENTRATED HARMONIC EXCITATION

The nonlinear vibration of a cantilever cylindrical shell under a concentrated harmonic excitation moving in a concentric circular path is proposed. Nonlinearities due to large-amplitude shell motion are considered, with account taken of the effect of viscous structure damping. The system is discretized by Galerkin's method. The method of averaging is developed to study the nonlinear traveling wave responses of the multi-degrees-of-freedom system. The bifurcation phenomenon of the model is investigated by means of the averaged system in detail. The results reveal the change process and nonlinear dynamic characteristics of the periodic solutions of averaged equations.     

Free vibrations of an anisotropic cylindrical fiberglass shell reinforced by annular ribs and containing fluid flow

This paper presents the results of determining the free vibration frequency of a structurally anisotropic, cylindrical fiberglass shell reinforced by annular ribs and containing flowing fluid. Boundary Navier conditions are imposed on the ends of the shell. Natural vibration frequencies are calculated as dependences of the frequency on the fiberglass winding angle and fluid flow velocity for different values of the wave formation parameters and the parameters characterizing the geometric dimensions of the shell.     

The effect of blade vibration on the nonlinear characteristics of rotor–bearing system supported by nonlinear suspension

In this paper, the complicated nonlinear dynamics of the harmonically forced quasi-zero-stiffness SD (smooth and discontinuous) oscillator is investigated via direct numerical simulations. This oscillator considered that the gravity is composed of a lumped mass connected with a vertical spring of positive stiffness and a pair of horizontally compressed springs providing negative stiffness, which can achieve the quasi-zero stiffness widely used in vibration isolation. The local and global bifurcation analyses are implemented to reveal the complex dynamic phenomena of this system. The double-parameter bifurcation diagrams are constructed to demonstrate the overall topological structures for the distribution of various responses in parameter spaces. Using the Floquet theory and parameter continuation method, the local bifurcation patterns of periodic solutions are obtained. Moreover, the global bifurcation mechanisms for the crises of chaos and metamorphoses of basin boundaries are examined by analysing the attractors and attraction basins, exploring the evolutions of invariant manifolds and constructing the basin cells. Meanwhile, additional nonlinear dynamic phenomena and characteristics closely related to the bifurcations are discussed including the resonant tongues, jump phenomena, amplitude–frequency responses, chaotic seas, transient chaos, chaotic saddles, and also their generation mechanisms are presented.     

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

     

Rubbing analysis of a nonlinear rotor system with surface coatings

Taking into account surface coatings painted on disc and casing, a nonlinear rotor system is established for simulating unbalance-rubbing coupled fault. During the modeling process of shaft, the nonlinear geometric relation between displacement and strain is considered. According to the Hamilton's principle, the restoring force between disc and shaft is described by the equivalent spring and the equivalent damper, and then the equivalent dynamic model is further set up. For contact analysis, the novel contact force model is employed to describe the impact force between disc and casing. Meanwhile, the Coulomb model is used to simulate the frictional force. The motion equations of the equivalent dynamic model are solved numerically and the corresponding dynamic behaviors are analyzed by bifurcation diagram, whirl orbit and Poincaré map. The relationship between equivalent linear/nonlinear stiffness and structural parameters is analyzed in detail. Moreover, the effects of eccentricity of disc, surface coatings and radius of shaft on the dynamic characteristic of the equivalent model are discussed.     

四边简支功能梯度圆锥壳的非线性振动分析

研究了四边简支条件下功能梯度圆锥壳的非线性自由振动。首先,通过Voigt模型和幂律分布模型描述了功能梯度材料的物理属性。然后,考虑von-Karman几何非线性建立了功能梯度圆锥壳的能量表达式,利用Hamilton原理推出圆锥壳的运动方程。在此基础上,采用Galerkin法,只考虑横向振动,功能梯度圆锥壳运动方程可简化为单自由度非线性振动微分方程。最后,通过改进的L-P法和Runge-Kutta法求解非线性振动方程,讨论功能梯度圆锥壳的非线性振动响应,分析几何参数和陶瓷体积分数指数对圆锥壳非线性频率响应的影响。结果表明,几何参数对非线性频率和响应的影响相较于陶瓷体积分数指数更明显;圆锥壳的几何参数和陶瓷体积分数指数通过改变非线性频率影响振动响应;功能梯度圆锥壳呈弹簧渐硬非线性振动特性。