DYNAMIC ANALYSIS OF LARGE-DIAMETER SAGGED CABLES TAKING INTO ACCOUNT FLEXURAL RIGIDITY

For the purpose of developing a vibration-based tension force evaluation procedure for bridge cables using measured multimode frequencies, an investigation on accurate finite element modelling of large-diameter sagged cables taking into account flexural rigidity and sag extensibility is carried out in this paper. A three-node curved isoparametric finite element is formulated for dynamic analysis of bridge stay cables by regarding the cable as a combination of an “ideal cable element” and a fictitious curved beam element in the variational sense. With the developed finite element formulation, parametric studies are conducted to evaluate the relationship between the modal properties and cable parameters lying in a wide range covering most of the cables in existing cable-supported bridges, and the effect of cable bending stiffness and sag on the natural frequencies. A case study is eventually provided to compare the measured natural frequencies of main cables of the Tsing Ma Bridge and the computed frequencies with and without considering cable bending stiffness. The results show that ignoring bending stiffness gives rise to unacceptable errors in predicting higher order natural frequencies of the cables, and the proposed finite element formulation provides an accurate baseline model for cable tension identification from measured multimode frequencies.     

Vibration analysis of horizontal self-weighted beams and cables with bending stiffness subjected to thermal loads

This paper aims at proposing an analytical model for the vibration analysis of horizontal beams that are self-weighted and thermally stressed. Geometrical nonlinearities are taken into account on the basis of large displacement and small rotation. Natural frequencies are obtained from a linearization of equilibrium equations. Thermal force and thermal bending moment are both included in the analysis. Torsional and axial springs are considered at beam ends, allowing various boundary conditions. A dimensionless analysis is performed leading to only four parameters, respectively, related to the self-weight, thermal force, thermal bending moment and torsional spring stiffness. It is shown that the proposed model can be efficiently used for cable problems with small sag-to-span ratios (typically , as in Irvine's theory). For beam problems, the model is validated thanks to finite element solutions and a parametric study is conducted in order to highlight the combined effects of thermal loads and self-weight on natural frequencies. For cable problems, solutions are first compared with existing results in the literature obtained without thermal effects or bending stiffness. Good agreement is found. A parametric study combining the effects of sag-extensibility, thermal change and bending stiffness is finally given.     

FREE VIBRATIONS OF SELF-STRAINED ASSEMBLIES OF BEAMS

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler-Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.     

Modal stability of inclined cables subjected to vertical support excitation

In this paper the out-of-plane dynamic stability of inclined cables subjected to in-plane vertical support excitation is investigated. We compute stability boundaries for the out-of-plane modes using rescaling and averaging methods. Our study focuses on the 2:1 internal resonance phenomenon between modes that occurs when the excitation frequency is twice the first out-of-plane natural frequency of the cable. The second in-plane mode is excited directly, while the out-of-plane modes can be excited parametrically. An analytical model is developed in order to study the stability regions in parameter space. In this model we include nonlinear coupling effects with other modes, which have thus far been omitted from previous models of parametric excitation of inclined cables. Our study reflects the importance of such effects. Unstable parameter regions are defined for the selected cable configuration. The validity of the proposed stability model was tested experimentally using a small-scale cable actuator rig. A comparison between experimental and analytical results is presented in which very good agreement with model predictions was obtained.     

Stability and free vibration analyses of cantilever shear buildings with semi-rigid support conditions and multiple masses

The stability and free vibration analyses of a cantilever shear building with generalized support conditions and with multiple masses (rotational and translational) rigidly attached at both ends and along its height are presented. The proposed model includes the simultaneous effects of: (1) lateral and rotational elastic restraints at the base support; (2) a uniform distributed mass and rotary inertia plus lumped rotary and translational masses rigidly attached at both extremes and along its height; (3) linearly distributed axial load plus the concentrated vertical axial loads caused by the lumped masses; and (4) shear deformations and shear forces induced by the applied axial forces. A parametric study is carried out that shows the importance of all variables included in this work on the stability and dynamic behavior of cantilever shear buildings, particularly the effects of the attached lumped masses and the rotational and translational constraints at the base support. A comparison with results presented by other researchers in previous studies shows that the proposed method and corresponding equations can be very useful in the assessment design of cantilever shear buildings. The main objective is to present readily solutions on the static stability and free vibration of cantilever shear buildings with generalized support conditions and multiple masses rigidly attached. The proposed method and corresponding expressions for the natural frequencies and modal shapes, buckling modes and axial critical loads are extensions of those presented recently by the senior author.     

Dynamic analysis of practical blades with shear center effect

Shear center effects on the natural frequencies and mode shapes of rotating and non-rotating practical blades are considered. An 18 degrees of freedom thick beam finite element is developed. Bending and shear force displacements and slopes, and torsional displacements are taken as degrees of freedom at both ends of the element. Total blade deflection slopes are considered as composed of bending and shear force deflection slopes in calculations of blade strain and kinetic energy. This element is compared with the existing thin and thick beam finite elements, and theoretical models. Results obtained for the vibration characteristics of rotating and non-rotating non-uniform aerofoil cross-sectioned blades are compared with the available calculated and experimental values. In all cases considered the element exhibits good convergence characteristics and produces accurate results.     

Vibration and stability of elastically supported multi-span beams under conservative and non-conservative loads

This paper deals with the vibration and stability of multi-span beams elastically supported against translation and rotation at several intermediate points as well as both ends. The beam is subjected to an axial or tangential load at the ends. The problem is studied on the basis of the Timoshenko beam theory. The influence of the support stiffness on the natural frequencies and the divergence and flutter instability loads are studied in detail.     

The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge

In conventional non-linear seismic analyses of cable-stayed bridges, the non-linear characteristics of the girders, stay cables and towers are considered. The non-linearity caused by cable loosening should also be considered because a large axial force fluctuation is generated in the cables of a prestressed concrete (PC) cable-stayed bridge that is subjected to strong seismic motion. In this paper, the possibility of the cable loosening in a PC cable-stayed bridge is discussed by using a cable model that can express the cable loosening. Furthermore, the effect of the cable loosening on the responses of the cables, girder and towers is evaluated using the mean value for three seismic waves. Numerical analytic results imply that the cable loosening appears in the bottom cables of the multi-cable system and the dynamic response of the bridge is slightly increased.     

Vibration modelling of helical springs with non-uniform ends

Helical springs constitute an integral part of many mechanical systems. Usually, a helical spring is modelled as a massless, frequency independent stiffness element. For a typical suspension spring, these assumptions are only valid in the quasi-static case or at low frequencies. At higher frequencies, the influence of the internal resonances of the spring grows and thus a detailed model is required. In some cases, such as when the spring is uniform, analytical models can be developed. However, in typical springs, only the central turns are uniform; the ends are often not (for example, having a varying helix angle or cross-section). Thus, obtaining analytical models in this case can be very difficult if at all possible. In this paper, the modelling of such non-uniform springs are considered. The uniform (central) part of helical springs is modelled using the wave and finite element (WFE) method since a helical spring can be regarded as a curved waveguide. The WFE model is obtained by post-processing the finite element (FE) model of a single straight or curved beam element using periodic structure theory. This yields the wave characteristics which can be used to find the dynamic stiffness matrix of the central turns of the spring. As for the non-uniform ends, they are modelled using the standard finite element (FE) method. The dynamic stiffness matrices of the ends and the central turns can be assembled as in standard FE yielding a FE/WFE model whose size is much smaller than a full FE model of the spring. This can be used to predict the stiffness of the spring and the force transmissibility. Numerical examples are presented.     

Modal interaction activations and nonlinear dynamic response of shallow arch with both ends vertically elastically constrained for two-to-one internal resonance

This study investigates the two-to-one internal resonance of the shallow arch with both ends elastically constraining, and the primary resonance case is considered. The full-basis Galerkin method and the multi-scale method are applied to obtain the modulation equations. It is shown that the natural frequencies of the first two modes cross/avoid to each other when the stiffness of elastic supports at two ends is the same/different. Moreover, the nonlinear modal interactions between these two modes may not/may be activated. The force/frequency-response curves are employed to explore the nonlinear response of the elastically supported shallow arch. The saddle-node bifurcation points and Hopf bifurcation points are observed in these cases. Moreover, the dynamic solutions, i.e., the periodic solution, quasi-periodic solution and chaotic solution are discussed. The numerical simulations are used to illustrate the route to chaos via period-doubling bifurcation.     

Rolling resistance moment of microspheres on surfaces

In the current work, for the first time, the existence of a rolling moment of resistance of an adhesion bond between a microsphere and flat surface subjected to external dynamic force has been experimentally demonstrated. The rotational motion of spherical particles deposited on a wafer is excited in the 0–3.5?MHz range using a piezoelectric transducer. The approach is based on (i) the observation that the contribution of the rotational (rocking) motion to the axial displacement of the particle are few orders of magnitude higher than those of the purely axial motion and (ii) the existence of a relationship between the rotational natural frequency of the adhesion bond and the work of adhesion. The natural frequency of the rotational (rocking) motion is extracted from the low frequency components of the transient response of the particle in the axial direction, which is measured by a laser interferometer. The existing theoretical adhesion models for rolling resistance moment are evaluated using the experimental results. Good agreement between the theoretical predictions and experimental values is found.     

A distributed-mass approach for dynamic analysis of Bernoulli-Euler plane frames

The exact dynamic analysis of plane frames should consider the effect of mass distribution in beam elements, which can be achieved by using the dynamic stiffness method. Solving for the natural frequencies and mode shapes from the dynamic stiffness matrix is a nonlinear eigenproblem. The Wittrick-Williams algorithm is a reliable tool to identify the natural frequencies. A deflated matrix method to determine the mode shapes is presented. The dynamic stiffness matrix may create some null modes in which the joints of beam elements have null deformation. Adding an interior node at the middle of beam elements can eliminate the null modes of flexural vibration, but does not eliminate the null modes of axial vibration. A force equilibrium approach to solve for the null modes of axial vibration is presented. Orthogonal conditions of vibration modes in the Bernoulli-Euler plane frames, which are required in solving the transient response, are theoretically derived. The decoupling process for the vibration modes of the same natural frequency is also presented.     

Resonance characteristics of high-speed trains passing simply supported bridges

This paper investigates the resonant characteristics of three-dimensional bridges when high-speed trains pass them. Multi-span bridges with high piers and simply supported beams were used in the dynamic finite element analysis. The dominated train frequencies proposed in this study can be clearly seen from the finite element result. To avoid resonance, the dominated train frequencies and the bridge natural frequencies should be as different as possible, especially for the first dominated train frequency and the first bridge natural frequency in each direction. If the two first frequencies are similar, the bridge resonance can be serious. This study also indicates that a suitable axial stiffness between two simple beams can reduce vibrations at a near-resonance condition. The axial stiffness of the continuous railway and the friction of the bearing plate should be enough to obtain this axial stiffness.     

Coupled horizontal and vertical bending vibrations of a stationary shaft with two cracks

This paper investigates the coupled bending vibrations of a stationary shaft with two cracks. It is known from the literature that, when a crack exists in a shaft, the bending, torsional, and longitudinal vibrations are coupled. This study focuses on the horizontal and vertical planes of a cracked shaft, whose bending vibrations are caused by a vertical excitation, in the clamped end of the model. When the crack orientations are not symmetrical to the vertical plane, a response in the horizontal plane is observed due to the presence of the cracks. The crack orientation is defined by the rotational angle of the crack, a parameter which affects the horizontal response. When more cracks appear in a shaft, then the coupling becomes stronger or weaker depending on the relative crack orientations. It is shown that a double peak appears in the vibration spectrum of a cracked or multi-cracked shaft.Modeling the crack in the traditional manner, as a spring, yields analytical results for the horizontal response as a function of the rotational angle and the depths of the two cracks. A 2×2 compliance matrix, containing two non-diagonal terms (those responsible for the coupling) serves to model the crack. Using the Euler–Bernoulli beam theory, the equations for the natural frequencies and the coupled response of the shaft are defined. The experimental coupled response and eigenfrequency measurements for the corresponding planes are presented. The double peak was also experimentally observed.     

涡旋板弹簧的有限元分析及实验

板弹簧支承技术是线性压缩机的关键技术之一,在为活塞提供轴向回复力的同时,还用来支承压缩活塞、保证密封间隙。在优化分析的基础上,文中设计制造了一款三臂涡旋板弹簧。利用ANSYS有限元软件分析了该款板弹簧的轴向刚度、径向刚度、应力分布以及固有频率特性,并对板弹簧的轴向刚度和固有频率进行了实验研究。实验结果表明,采用有限元分析方法分析板弹簧性能是有效的。     

Natural frequencies of finned heat exchanger tubes

The heat transfer of heat exchanger tubes is often enhanced by rolling external fins on to the tubes. These fins contribute to both the mass and the stiffness of the tubes. Calculations of the bending natural frequencies of finned tubes with only the mass contribution considered lead to values which are up to 10% lower than experimental values. A simple correction is derived to account also for the stiffening effect of the fins. With this correction good agreement with experiments is obtained.     

Lightning-induced currents in buried coaxial cables: A frequency-domain approach and its validation using rocket-triggered lightning

The aim of this paper is to present an experimental validation of a frequency-domain approach to the solution of the lightning electromagnetic field-to-buried cable coupling equations. The coupling to the inner conductor is evaluated using the concept of cable transfer impedance. The theoretical model and relevant computer code are tested using experimental data on lightning-induced currents in buried cables carried out in 2002 and 2003 at the International Center for Lightning Research & Testing (ICLRT) at Camp Blanding, Florida where currents induced by triggered lightning events were measured at the ends of a buried coaxial cable, both in the shield and in the inner cable conductors. Reasonably good agreement has been found between numerical simulations and recorded waveforms. In particular, the early-time response of the cable and the peak value of the induced currents were generally well reproduced by the simulations.     

Size effect in transverse mechanical behavior of one-dimensional nanostructures

This paper develops the classical strain gradient elasticity theory to investigate the scale-dependent mechanical behavior of one-dimensional (1D) nanostructures. A governing differential equation with two scale parameters is derived, where the curvature of the deflection and the higher-order bending moment are introduced as a pair of additional geometrical constraint and natural loading. Emphasis is placed on the analysis of bending deformation, free vibration and buckling of cantilever nanowires or free-standing nanocolumns. Obtained results are compared with experimental data of carbon nanotube ropes and nanowires available in the literature and they agree well, showing that transverse mechanical properties of nanowires such as bending stiffness are scale-dependent. The model proposed also indicates that the evaluated natural frequencies and critical buckling strains exhibit noticeable size effects. Bending stiffness, natural frequency and buckling load increase as the nanowire diameter drops down. The influence of rotary inertia of cross-section is also analyzed.     

桥梁拉索损伤声发射监测研究进展

大量索承体系桥梁即将达到20–30年的拉索寿命期,开展有效的拉索损伤监测方法研究有利于保障大桥结构安全。本文简要论述了现有桥梁拉索损伤检测及监测方法的适用性,重点综述了近40年来声发射技术应用于桥梁拉索监测的研究进展,以及在腐蚀、疲劳、断丝等损伤监测方面所取得的研究成果。结合实测数据在采集、数据处理过程中的难点及参数分析盲点,探讨了现有研究成果应用于特大桥梁拉索损伤监测仍需解决的问题,并针对性地提出了研究思路。     

扬声器锥壳的全频段强迫振动解

解析和数值研究了扬声器锥壳全频段的轴对称强迫振动。给出了典型低频段、典型转点频段和典型高频段的显式位移解析解、特征频率方程和轴向导纳表达式。解析结果与数值计算和实验结果结果非常吻合。在典型低频段,振动完全是纵波型的。在典型转点频段,全域的纵波运动和转点外侧域的横波运动共存,谐振和反谐振频率方程相应呈现出无矩解和弯曲解的耦合特性。在典型高频段,全域的纵波运动和横波运动互相独立,相应出现2组独立的纵波和横波固有频率。