3:1 Internal resonance of a string-beam coupled system with cubic nonlinearities

The dynamic behavior and chaotic motion of a string-beam coupled system subjected to parametric excitation are investigated. The case of three-to-one internal resonance between the modes of the beam and the string, in the presence of subharmonic resonance for the beam is considered and examined. The method of multiple scales is applied to study the steady-state response and the stability of the string-beam coupled system at resonance conditions. Numerical simulations illustrated that multiple-valued solutions, jump phenomenon, hardening and softening nonlinearities occur in the resonant frequency response curves. The effects of different parameters on system behavior have been studied applying frequency response function. Results are compared to previously published work.     

用集结坐标法对细杆中呼吸子状态的分析

研究了计入Peierls-Nabarro(P-N)力和材料粘性效应的一维无限长金属杆在简谐外力扰动下的动力响应,导出了类sine-Gordon 型的运动方程．在集结坐标(collective coordinate）下原控制方程可以用常微分动力系统描述,研究系统中呼吸子的运动．根据非线性动力学方法分析,P-N力的幅值和频率的变化将改变双曲鞍点的位置,并改变系统次谐分叉的阈值,但不改变由奇阶次谐分叉通向混沌的路径．通过实例给出了P-N力幅值和P-N力频率对细杆动力响应的详细影响过程,可见混沌发生的区域是一个半无限区域,并随着P-N力的增大而增大．P-N力的频率对系统有类似的影响．     

Nonlinear dynamics of tapping mode atomic force microscopy in the bistable phase

Nonlinear dynamics of amplitude modulation atomic force microscopy (AFM) is studied employing a reduced-order model based on a differential quadrature method (DQM). The AFM microcantilever is assumed to be operating in the dynamic contact or tapping mode while the microcantilever tip being initially located in the bistable region. We have found that the DQM is capable of precise prediction of the static bifurcation diagram and natural frequencies of the microcantilever. We have used the DQM to discretize the partial-differential equation governing the microcantilever motion and a finite difference method (FDM) to calculate limit-cycle responses of the AFM tip. It is shown that a combination of the DQM and FDM applied, respectively, to discretize the spatial and temporal derivatives provides an efficient, accurate procedure to address the complicated dynamic behavior exhibited by the AFM probe. The procedure was, therefore, utilized to study the response of the microcantilever to a base harmonic excitation through several numerical examples. We found that the dynamics of the AFM probe in the bistable region is totally different from those in the monostable region.     

Torsional oscillations in structures subject to ground motion

A multiple-scale technique has been employed to study nonlinear torsional oscillations in single-storey structures with cubic softening stiffness members subject to a single frequency ground excitation. The structures are studied under free oscillation conditions, primary resonance, and combination resonance. Time history analyses are employed to quantify the dynamic behaviour of a single-storey structure. The instabilities are related to the jump in response which exists in single degree-of-freedom softening oscillators subjected to harmonic excitation. It is shown that this jump can be quantified analytically, and appears as a cusp catastrophe in the bending mode. Time history analyses show that the bending mode can exhibit a limit cycle behaviour of constant amplitude.     

On the Nonlinear Vibration Behavior of Elastohydrodynamic Contacts

The nonlinear dynamic behavior of a simple system with an elastohydrodynamic line contact is investigated. The elastohydrodynamic problem is solved with finite differences including streamline upwind techniques. The results are stored in a lookup table for an effective calculation of the frequency response. It is found that the system exhibits rich nonlinear dynamic behavior including jump phenomena as well as sub- and superharmonic bifurcations. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison between elastohydrodynamic and dry hertzian line contacts with regard to their nonlinear vibration behaviour

The nonlinear dynamic behaviour of a simple system employing two different types of contact is compared: elastohydrodynamic line contact and Hertzian line contact with appropriate damping. The elastohydrodynamic problem is solved with finite elements. For the Hertzian line contact problem an analytical solution is known. It is found that for highly loaded cases Hertzian and elastohydrodynamic contacts lead to very similar frequency response curves. This means that instead of solving the computational expensive elastohydrodynamic problem the Hertzian contact can be used. However, considering weakly loaded contacts the two types of contact show very different behaviour. In particular the natural frequencies are different and the elastohydrodynamic contact exhibits rich nonlinear dynamic behaviour including jump phenomena as well as sub- and superharmonic bifurcations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling and nanoscale force spectroscopy of frequency modulation atomic force microscopy

In this paper, a simulation model for frequency modulation atomic force microscopy (FM-AFM) operating in constant amplitude dynamic mode is presented. The model is based on the slow time varying function theory. The mathematical principles to derive the dynamical equations for the amplitude and phase of the FM-AFM cantilever-tip motion are explained and the stability and performance of its closed-loop controller to keep the amplitude at constant value and phase at 90° is analysed. Then, the performance of the theoretical model is supported by comparison of numerical simulations and experiments. Furthermore, the transient behaviour of amplitude, phase and frequency shift of FM-AFM is investigated and the effect of controller gains on the transient motion is analysed. Finally, the derived FM-AFM model is used to simulate the single molecule/nanoscale force spectroscopy and study the effect of sample viscosity, stiffness and Hamaker constant on the response of FM-AFM.     

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.     

Active control of secondary resonances piezoelectric sandwich beams

Secondary resonances of piezoelectric/elastic/piezoelectric sandwich beams submitted to active control are studied in this paper. The proportional and derivative nonlinear potential feedback controls via piezoelectric sensor and actuator layers are used. The dynamics of the beam is modelled by a highly nonlinear ordinary-differential equation. The method of multiple scales is applied and approximate solutions are obtained for hard excitations. Analytical frequency and phase-amplitude relationships as well as the time response are explicitly given for various super- and subharmonic resonances. Static and dynamic stability criteria are elaborated and critical displacement and excitation amplitudes associated to the resulting unstable zones are analytically given. The feedback parameters effects on the subharmonic and superharmonic resonances and on their stability are investigated.     

非线性转子-密封系统1∶2亚谐共振分岔研究

研究了转子-密封系统在气流激振力作用下的低频振动——1∶2亚谐共振现象．利用流体计算动力学（CFD）方法对转子-密封系统进行了流场模拟计算,辨识出适用于气流流场的Muszynska模型参数,并建立了转子-密封系统动力学方程．采用多尺度方法将系统进行3次截断,并得到系统响应．采用奇异性理论研究了系统的1∶2亚谐共振,进一步得到系统亚谐共振的分岔方程和转迁集,根据转迁集给出了在不同奇异性参数空间内的分岔图．同时,由分岔方程得到了亚谐共振非零解存在的条件．其分析结果对抑制转子-密封系统的亚谐振动有重要的工程意义．     

Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects

Analytical solution for the steady-state response of an Euler–Bernoulli nanobeam subjected to moving concentrated load and resting on a viscoelastic foundation with surface effects consideration in a thermal environment is investigated in this article. At first, based on the Eringen's nonlocal theory, the governing equations of motion are derived using the Hamilton's principle. Then, in order to solve the equation, Galerkin method is applied to discretize the governing nonlinear partial differential equation to a nonlinear ordinary differential equation; solution is obtained employing the perturbation technique (multiple scales method). Results indicate that by increasing of various parameters such as foundation damping, linear stiffness, residual surface stress and the temperature change, the jump phenomenon is postponed and with increasing the amplitude of the moving force and the nonlocal parameter, the jump phenomenon occurs earlier and its frequency and the peak value of amplitude of vibration increases. In addition, it is seen that the non-linear stiffness and the critical velocity of the moving load are important factors in studying nanobeams subjected to moving concentrated load. Presence of the non-linear stiffness of Winkler foundation resulting nanobeam tends to instability and so, the jump phenomenon occurs. But, presence of the linear stiffness will lead to stability of the nanobeam. In the next sections of the paper, frequency responses of the nanobeam made of temperature-dependent material properties under multi-frequency excitations are investigated.     

频域有限元计算的扩展面向目标误差估计

研究了针对频域有限元直接动态分析的面向目标误差估计以及误差范围估计计算方法．面向目标的误差估计方法就是专门针对如何准确和经济地估算特定值误差的一种方法,利用原问题的共轭偶问题进行计算．频域有限元的直接动态分析是模拟频域扫描实验的一种计算方法,专门针对谐振激励的线性动态响应问题,利用将原自由度分解为实部和虚部描述频率的变化,从而计算变形体的动态响应．利用扩展针对有限元的面向目标误差估计的自由度,将该方法应用到直接动态分析中进行误差估计．通过建立同时包含实部和虚部自由度的能量弱形式及偶问题,并将其数值实现,估算频域直接动态分析有限元解的误差及误差范围,并通过悬臂梁的激振算例进行了验证．     

The vibrational behaviour of bladed disks with multiple coupling devices

In turbomachinery applications turbine blades are subjected to high static and dynamic loads. Static loads are due to centrifugal stresses and thermal strains. Especially the dynamic excitation caused by fluctuating gas forces results in high vibration amplitudes which can lead to high cycle fatigue failures (HCF). Therefore, in practical applications, coupling devices like underplatform dampers, lacing wires and tip shrouds are installed to the structure. In case of blade vibrations the relative displacements between these coupling devices and the blades generate friction forces. The resulting energy dissipation provides additional damping to the structure. Furthermore, coupling devices, in particular tip shrouds, snubbers and lacing wires, increase the stiffness of the structure. Hence, they lead to a shift of the resonance frequencies. So far, only effects of single coupling devices and the influencing properties have been examined. Within this paper the effect of multiple couplings is determined and compared with single couplings. The forced response of turbine bladings with multiple couplings is calculated under consideration of geometrical and mechanical parameters of the blading and contacts, respectively. The results are compared with the single coupled blading. Furthermore, a multiple coupled device with under-platform damper and connecting pin is compared with respect to his effectiveness. Especially the influence on the resonance frequency and the achievable damping is analysed. The results of the simulation are verified by measurements at a two-blade non-rotating test rig with an underplatform damper and connecting pin. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study on numerical simulation methods of the railway vehicle-track dynamic interaction

A very efficient numerical simulation method of the railway vehicle–track dynamic interaction is described. When a vehicle runs at high speed on the railway track, contact forces between a wheel and a rail vary dynamically due to the profile irregularities existing on the surface of the rail. A large variation of contact forces causes undesired deteriorations of a track and its substructures. Therefore these dynamic contact forces are of main concern of the railway engineers. However it is very difficult to measure such dynamic contact forces directly. So it is important to develop an appropriate numerical simulation model and identify structural factors having a large influence on the variation of contact forces. When a contact force is expressed by the linearized Hertzian contact spring model, the equation of motions of the system is expressed as a second–order linear time–variant differential equation which has a time–dependent stiffness coefficient. Applying a well–known Newmark direct integration method, a numerical simulation is reduced to solving iteratively a time–variant, large–scale sparse, symmetric positive–definite linear system. In this study, by defining a special vector named a contact point one, it is shown that this time–variant stiffness coefficient can be expressed simply as a product of the contact point vector and its transpose and so the Sherman–Morrison–Woodbury formula applied for updating the inverse of the coefficient matrix. As a result, the execution of numerical simulation can be carried out very efficiently. A comparison of the computational time is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of fast harmonic excitation on frequency-locking in a van der Pol–Mathieu–Duffing oscillator

We study the effect of high-frequency harmonic excitation on the entrainment area of the main resonance in a van der Pol–Mathieu–Duffing oscillator. An averaging technique is used to derive a self- and parametrically driven equation governing the slow dynamic of the oscillator. The multiple scales method is then performed on the slow dynamic near the main resonance to obtain a reduced autonomous slow flow equations governing the modulation of amplitude and phase of the slow dynamic. These equations are used to determine the steady state response, bifurcation and frequency–response curves. A second multiple scales expansion is used for each of the dependent variables of the slow flow to obtain slow slow flow modulation equations. Analysis of non-trivial equilibrium of this slow slow flow provides approximation of the slow flow limit cycle corresponding to quasi-periodic motion of the slow dynamic of the original system. Results show that fast harmonic excitation can change the nonlinear characteristic spring behavior and affect significantly the entrainment region. Numerical simulations are used to confirm the analytical results.     

Vibration of submerged floating tunnels due to moving loads

The concept of submerged floating tunnel (SFT) has become an increasingly attractive idea to cross the straits. The structural solution in this scheme includes buoyancy force on tunnel body plus tension in mooring tethers. This paper investigates the effect of submergence on the dynamic response of SFT due to moving load. The inertial effect of the fluid is accounted for by evaluating the added mass of tunnel using two and three dimensional models. It is found that fluid–structure interaction increases dynamic amplification of the tunnel deflection (in some cases very significantly). The results show that although the 3D model predicts lesser inertial contribution for surrounding fluid, it is not always possible to associate the larger response with the 2D or 3D models. The discrepancy between the results of the two models decreases as the tether stiffness increases. This indicates that the adoption of Morison’s equation for evaluating the fluid loading on the tunnel is a reasonable assumption when the tether stiffness is high. It is also found that by increasing the tether stiffness, it is possible to introduce a major reduction in the dynamic amplification of the response and by this way control the dynamic response of the SFT.     

Dynamic analysis and passive control of viscoelastically damped nonlinear dynamic contacts

This paper presents an analytical and finite element study on nonlinear contact dynamics and controls. Nonlinear dynamic contacts between eccentrically supported masses and simply supported beams are studied. Passive control of the dynamic contacts using viscoelastic dampers is also proposed and evaluated. A nonlinear contact finite element is modeled by a set of nonlinear stiffness and damping polynomial functions; and a nonlinear viscoelastic finite element is modeled by a Standard Linear Model with frequency-dependent nonlinear stiffness and damping functions. Analyses show that the dynamic contact force increases as the initial gap increases. Application of viscoelastic dampers can effectively reduce contact loads and prevent dynamic contacts. A simple design equation is also proposed.     

A modified scaled boundary finite element method for dynamic response of a discontinuous layered half-space

In this paper, a modified scaled boundary finite element method is proposed to deal with the dynamic analysis of a discontinuous layered half-space. In order to describe the geometry of discontinuous layered half-space exactly, splicing lines, rather than a point, are chosen as the scaling center. Based on the modified scaled boundary transformation of the geometry, the Galerkin's weighted residual technique is applied to obtain the corresponding scaled boundary finite element equations in displacement. Then a modified version of dimensionless frequency is defined, and the governing first-order partial differential equations in dynamic stiffness with respect to the excitation frequency are obtained. The global stiffness is obtained by adding the dynamic stiffness of the interior domain calculated by a standard finite element method, and the dynamic stiffness of far field is calculated by the proposed method. The comparison of two existing solutions for a horizontal layered half-space confirms the accuracy and efficiency of the proposed approach. Finally, the dynamic response of a discontinuous layered half-space due to vertical uniform strip loadings is investigated.