Transient vibration phenomena in deep mine hoisting cables. Part 1: Mathematical model

The classical moving co-ordinate frame approach and Hamilton's principle are employed to derive a distributed-parameter mathematical model to investigate the dynamic behaviour of deep mine hoisting cables. This model describes the coupled lateral-longitudinal dynamic response of the cables in terms of non-linear partial differential equations that accommodate the non-stationary nature of the system. Subsequently, the Rayleigh-Ritz procedure is applied to formulate a discrete mathematical model. Consequently, a system of non-linear non-stationary coupled second order ordinary differential equations arises to govern the temporal behaviour of the cable system. This discrete model with quadratic and cubic non-linear terms describes the modal interactions between lateral oscillations of the catenary cable and longitudinal oscillations of the vertical rope. It is shown that the response of the catenary-vertical rope system may feature a number of resonance phenomena, including external, parametric and autoparametric resonances. The parameters of a typical deep mine winder are used to identify the depth locations of the resonance regions during the ascending cycles with various winding velocities.     

Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling

The response of a two-degree-of-freedom system with autoparametric coupling under the action of broad band random excitation is investigated. The system corresponds to the autoparametric vibration absorber and is also typical of many common structural configurations. A method based upon the Markov vector approach, together with an approximate treatment of third and higher statistical moments, is used to derive a set of fourteen coupled non-linear equations for the first and second moments of the system responses. A numerical integration procedure is used to obtain quantitative results for the system mean and mean square responses over a range of system parameters.The results show that large random motions of the coupled system may occur when the internal detuning parameter is close to the principal internal resonance, and that these motions may give rise to a suppression effect on the random motions of the main system. A feature of the results is that under conditions of internal resonance the random motions are found to be quasi-stationary, with steady oscillatory terms in the response moments. This suggests the possibility of entrainment of regular harmonic responses by the system random motions.     

Analytical and experimental investigations of an autoparametric beam structure

This paper discusses theoretical and experimental investigations of vibrations of an autoparametric system composed of two beams with rectangular cross sections. Different flexibilities in the two orthogonal directions are the specific features of the structure. Differential equations of motion and associated boundary conditions, up to third-order approximation, are derived by application of the Hamilton principle of least action. Experimental response of the system, tuned for the 1:4 internal resonance condition, are performed for random and harmonic excitations. The most important vibration modes are extracted from a real mechanical system. It is shown that certain modes in the stiff and flexible directions of both beams may interact, and, intuitively unexpected out-of-plane motion may appear. Preliminary numerical calculations, based on the mathematical model, are also presented.     

Nonlinear vibration of iced cable under wind excitation using three-degree-of-freedom model

High-voltage transmission line possesses a typical suspended cable structure that produces ice in harsh weather. Moreover, transversely galloping will be excited due to the irregular structure resulting from the alternation of lift force and drag force. In this paper, the nonlinear dynamics and internal resonance of an iced cable under wind excitation are investigated.Considering the excitation caused by pulsed wind and the movement of the support, the nonlinear governing equations of motion of the iced cable are established using a three-degree-of-freedom model based on Hamilton's principle. By the Galerkin method, the partial differential equations are then discretized into ordinary differential equations. The method of multiple scales is then used to obtain the averaged equations of the iced cable, and the principal parametric resonance-1/2 subharmonic resonance and the 2:1 internal resonance are considered. The numerical simulations are performed to investigate the dynamic response of the iced cable. It is found that there exist periodic, multi-periodic, and chaotic motions of the iced cable subjected to wind excitation.     

A novel multiple blade ultrasonic cutting device

In ultrasonic devices consisting of serially coupled tuned components, components whose tuned length dimension is large compared to other dimensions, or components with profiles designed for high gain, the response of the device during operation is often characterised by modal interactions, and especially the excitation of combination resonances. The effects on ultrasonic devices are high noise levels, component failures and poor operating performance. In this paper, energy exchanges between modes are characterised to illustrate the adverse effects of combination resonances. Design solutions are proposed to eliminate these effects which concentrate on reducing the number of modes. In particular, a novel half-wavelength three-blade cutting system tuned to the first longitudinal mode has been designed by finite element modelling. An experimental investigation demonstrates that the measured response does not show evidence of modal interactions.     

Dynamic sensitivity analysis for the pantograph of a high-speed rail vehicle

In this study, the dynamic characteristics of a catenary system using the finite element method (FEM) and dynamic modelling for developing a suitable pantograph at high speed is analyzed. First, the catenary system of a high-speed railway is assumed to be a beam model. Next, an analysis program using finite element analysis was performed. The pantograph of linear spring-mass system is assumed to be three-degrees-of-freedom model for the finite element analysis. The analyses of the catenary based on the FEM are executed to develop a pantograph that meets the necessary standards for high-speed rail vehicles. Using a simulation of the pantograph-catenary system, the static deflection of the catenary, the stiffness variation in the contact lines, the dynamic response of the catenary undergoing a constant moving load and the contact force analysis were executed. From the pantograph-catenary analysis, the design parameters of a pantograph could be optimized. Based on the design-parameter analysis, a pantograph with improved parameters was found to be suitable for a high-speed rail vehicle.     

Plasmonic extraordinary transmittance in array of metal nanorods

The optical response of an array of metal nanorods is studied in the case when the cylinders almost touch by their generatrices. As the cylinders approach each other, a series of surface plasmon resonances are excited. The first longitudinal mode is different from the higher-order lateral modes. The lateral resonances occur near the frequency where the real part of the metal permittivity changes sign. The plasmon resonances result in maxima and minima in the reflectance and transmittance. The resonances also result in a huge enhancement of the local electric field in the gap between cylinders.     

Autoparametric vibration absorber effect to reduce the first symmetric mode vibration of a curved beam/panel

This paper presents the implementation of autoparametric phenomena to reduce the symmetrical vibration of a curved beam/panel under external harmonic excitation. The internal energy transfer of a first symmetric mode into first anti-symmetric mode in a curved panel is one example of autoparametric vibration absorber effect. This is similar to the vibration energy transfer from the resonance of a primary structure to the resonance of a secondary spring–mass (tuned mass damper). The nonlinear response of a curved beam is analyzed using an equation with two modes, and a shaker test. The effect of different configurations of the curve beam/panel, including damping ratios and excitation levels, on the energy transfer of the first symmetric mode to the first anti-symmetric mode was studied.The conventional tuned mass damper (TMD) can reduce the resonance response by energy transfer using damping dissipation, whereas an autoparametric vibration absorber (AVA) can reduce the resonance response by energy transfer using parametric interaction. The results indicate that there is a non-absorption region in which vibration is amplified. For the AVA, the non-absorption region can be minimized by tuning the resonance frequency of the first anti-symmetric mode to half of the first symmetric mode resonance frequency using additional mass. No additional damping material is required for achieving sufficient vibration reduction. The AVA can maintain reliable performance in hot and corrosive environments where damping material cannot perform effectively. This paper presents the first successful experimental results of an autoparametric vibration absorption mechanism in a curved beam.     

Instabilities in the main parametric resonance area of a mechanical system with a pendulum

Vibrations of an autoparametric system, composed of a nonlinear mechanical oscillator with an attached damped pendulum, around the principal resonance region, are investigated in this paper. Approximate analytical solutions of the model are determined on the basis of the Harmonic Balance Method (HBM). Correctness of the analytical results is verified by numerical simulations and experimental tests performed on an especially prepared experimental rig. The influence of all essential parameters such as damping, excitation amplitude and frequency, nonlinear stiffness of the spring, on the localisation of the instability region and the system dynamics is presented in the work. Regions of regular system oscillations, chaotic motions, and full rotation of the pendulum are confirmed experimentally.     

Snap loads and torsional oscillations of the original Tacoma Narrows Bridge

In 1940, the original Tacoma Narrows Bridge was completed on June 10 and opened to traffic on July 1. On November 7, the deck collapsed. Before that day, significant vertical oscillations had occurred, but no torsion. The bridge as built was stable with respect to torsional motion under the winds of November 7 and previous winds with higher speeds. However, snap loads in the diagonal ties attached to the north midspan cable band helped to loosen the band, and the frictional resistance between the band and the north suspension cable passing through it was overcome. The cable began to slip through the band. For this new structural system, with longitudinal motion of the north cable, the wind speed was higher than the critical speed for torsional flutter, and torsional motion was initiated. Approximately 700 cycles of torsional oscillations occurred during the hour prior to the collapse. In the present study, the snap loads on the cable band are discussed first. Then a continuum model of the central span (deck, cables, and hangers) is formulated. The longitudinal motions of the cables are included, so that the slippage can be incorporated. Known information from the observed steady-state torsional motion is utilized with assumed forms of the vertical cable displacements, and the governing equations provide the horizontal cable displacements, the dynamic tensions in the cables, the vertical and torsional motions of the deck, and the resultant lift force and pitching moment (including damping) acting on the deck during its final hour.     

On the oscillatory motions of translating elastic cables

The equations of in-plane motion for an elastic catenary translating uniformly between its end supports are derived in what is, essentially, an Eulerian frame of reference. These equations are solved analytically, albeit approximately, for the case where the catenary is shallow and the tension dominated by the cable section modulus. Computed solutions for the natural frequencies and modes of a catenary of sag-span ratio 1:20 are presented and the modal characteristics are shown to be of an unusual form involving phase disparities from point to point on the cable as the cable oscillates at a natural frequency.     

Modified rate equation model including the photon�Cphoton resonance

We show that, when the longitudinal confinement factor in an edge-emitting laser is treated as a dynamic variable, the modulation transfer function has an extra term. This term produces a supplementary photon?Cphoton resonance peak in the modulation response at a frequency corresponding to the frequency separation between longitudinal modes, when these modes are phase-locked long enough (quasi-phase-locked). The photon?Cphoton resonance peak is strongest when two consecutive quasi-phase-locked dominant longitudinal modes have similar longitudinal envelopes and share equally the photon population.     

Parametric excitation of two internally resonant oscillators

The response of two-degree-of-freedom systems with quadratic non-linearities to a principal parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first-order uniform expansion yielding four first-order non-linear ordinary differential (averaged) equations governing the modulation of the amplitudes and the phases of the two modes. These equations are used to determine steady state responses and their stability. When the higher mode is excited by a principal parametric resonance, the non-trivial steady state value of its amplitude is a constant that is independent of the excitation amplitude, whereas the amplitude of the lower mode, which is indirectly excited through the internal resonance, increases with the amplitude of the excitation. However, in addition to Poincaré-type bifurcations, this response exhibits a Hopf bifurcation leading to amplitude- and phase-modulated motions. When the lower mode is excited by a principal parametric resonance, the averaged equations exhibit both Poincaré and Hopf bifurcations. In some intervals of the parameters, the periodic solutions of the averaged equations, in the latter case, experience period-doubling bifurcations, leading to chaos.     

船舶运动状态下风速风向测量补偿与数字仿真

为了研究船舶运动状态下的风速风向精确测量,设计了一种船舶风速风向动态测量及误差补偿的数字仿真系统。通过对船舶航行状态下的风速风向测量原理进行分析,建立了船舶平面运动的相对风速风向和真风风速风向的解算模型,并根据船舶空间运动的风速风向测量及其误差补偿算法,对船舶横摇、纵摇状态下的风速风向的动态测量和误差补偿进行了数字仿真。数字仿真结果表明,该方法能够消除船舶航行时的运动姿态对风速风向测量带来的影响,为船舶的操纵控制和航行安全提供了精确和可靠的风速风向数据信息。     

无人机垂直突风非线性动态过程建模与仿真

在无人机方案设计初期,需对气动关键参数进行数字仿真,以验证其能否适应复杂气象条件。为研究无人机在遭遇垂直突风时,其相关参数的动态响应情况,基于simulink建立了数字飞机模型及突风模型,完成了90s的飞行仿真。结果表明,无人机在地轴系Z向遭遇风速为50ms/s时长2s的垂直突风时,其飞行高度、速度等参数均在设计范围内波动,在突风消失后,飞机恢复稳定状态。为后期的系统设计和飞行试验奠定良好的基础。     

Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances

This study analyzed the nonlinear vibration of an axially moving beam subject to periodic lateral force excitations. Attention is paid to the fundamental and subharmonic resonances, since the excitation frequency is close to the first two natural frequencies of the system. The incremental harmonic balance (IHB) method was used to evaluate the nonlinear dynamic behaviour of the axially moving beam. The stability and bifurcations of the periodic solutions for given parameters were determined by the multivariable Floquet theory using Hsu’s method. The solutions obtained from the IHB method agreed very well with those obtained from numerical integration. Furthermore, numerical examples are given to illustrate the effects of the three-to-one internal resonance on the response of the system.     

Multiple internal resonances in MEMS arch resonators

Micromachined shallow arch resonant beams have attracted significant attention thanks to their rich dynamical behavior, inherent nonlinearities, and the potential to excite various internal resonances. Currently, there is a lack of comprehensive experimental studies for the various types of internal resonances in arches and particularly at the micro and nano scales. Here, we aim to investigate and identify different types of internal resonances of an initially curved beam, electrothermally actuated and electrostatically driven, by electrical characterization techniques. Upon changing the electrothermal voltage of silicon micromachined arches, the second symmetric natural frequency of an arch is adjusted to near twice, three times, and four times the fundamental natural frequency, which gives rise to 2:1, 3:1, and 4:1 autoparametric resonances between the two modes. These resonances are demonstrated experimentally. We show various frequency-response curves of the total response around the excitation frequency and highlight the contribution of each mode before, during, and after the internal resonances. Allan-deviation results are also shown indicating enhanced frequency stabilization of the arch oscillation when experiencing internal resonances. These studies motivate further research in this direction to exploit internal resonances of micromachined resonators for practical applications, such as sensors and mechanical amplifier.     

Bifurcation analysis of a parametrically excited inclined cable close to two-to-one internal resonance

This paper presents a study of how different vibration modes contribute to the dynamics of an inclined cable that is parametrically excited close to a 2:1 internal resonance. The behaviour of inclined cables is important for design and analysis of cable-stayed bridges. In this work the cable vibrations are modelled by a four-mode model. This type of model has been used previously to study the onset of cable sway motion caused by internal resonances which occur due to the nonlinear modal coupling terms. A bifurcation study is carried out with numerical continuation techniques applied to the scaled and averaged modal equations. As part of this analysis, the amplitudes of the cable vibration response to support inputs is computed. These theoretical results are compared with experimental measurements taken from a 5.4 m long inclined cable with a vertical support input at the lower end. In general this comparison shows a very high level of agreement.     

Resonance modes of voltage-modulated scanning force microscopy

Mechanical resonance modes of the scanning force microscope (SFM) cantilever in contact conditions provide contrast enhancement in the imaging of surface charges when using voltage modulation techniques tuned to such resonances. Extensions of the method were made as regards the lateral (twisting) and frontal (buckling) modes of the cantilever, as well as the enhanced second harmonic detection of voltage-modulated response at resonance and near-resonance detection in the SFM tapping mode. As an example of application, vibration spectra and images taken on a triglycine sulfate (TGS) single crystal are discussed.