Nonlinear interactions in the planar dynamics of cable-stayed beam

An analytical model is proposed to study the nonlinear interactions between beam and cable dynamics in stayed-systems. The integro-differential problem, describing the in-plane motion of a simple cable-stayed beam, presents quadratic and cubic nonlinearities both in the cable equation and at the boundary conditions. Mainly studied are the effects of quadratic interactions, appearing at relatively low oscillation amplitude. To this end an analysis of the sensitivity of modal properties to parameter variations, in intervals of technical interest, has evidenced the occurrence of one-to-two and two-to-one internal resonances between global and local modes. The interactions between the resonant modes evidences two different sources of oscillation in cables, illustrated by simple 2dof discrete models.In the one-to-two global–local resonance, a novel mechanism is analyzed, by which cable undergoes large periodic and chaotic oscillations due to an energy transfer from the low-global to high-local frequencies.In two-to-one global–local resonance, the well-known parametric-induced cable oscillation in stayed-systems is correctly reinterpreted through the autoparametric resonance between a global and a local mode. Increasing the load the saturation of the global oscillations evidences the energy transfer from high-global to low-local frequencies, producing large cable oscillations. In both cases, the effects of detuning from internal and external resonance are presented.     

谐波激励下斜拉桥面内非线性振动试验研究

斜拉桥拉索的振动问题一直是桥梁工程领域的研究热点。为揭示拉索大幅振动的力学机理，课题组建立了斜拉桥的全桥精细化模型，本文测试和研究了单频激励下的斜拉桥可能的非线性振动行为。首先，通过自由振动试验测试了模型的模态参数，并与两类有限元模型（OECS模型和MECS模型）进行对比，结果吻合良好。其次，试验研究了在单个竖向简谐激励下斜拉桥模型的非线性响应。研究发现：当激励频率与斜拉桥某阶全局模态频率接近时，主梁产生主共振，并引起多根长索产生大幅的参强振动；当激励频率与某根斜拉索面内一阶频率之比为1：2或者2：1时，可以观测到索中产生超谐波和亚谐波共振现象。     

Detection and description of non-linear phenomena in experimental modal analysis via linearity plots

Ground vibration tests (GVTs) on aircraft prototypes are mainly performed to experimentally identify the structural dynamic behaviour in terms of a modal model. This assumes a linear dynamic behaviour of the structure. However, in the practice of ground vibration testing it is often observed that structures do not behave in a perfectly linear manner. Non-linearities can be determined, for example, by free play in junctions, hydraulic systems in control surfaces, or friction. This paper compiles measured, typical, non-linear phenomena from various GVTs on large aircraft. The standard procedure in GVTs nowadays is the application of the Harmonic Balance method which linearizes the dynamic behaviour on the level of excitation. The procedure requires a harmonic excitation of the structure which is usually performed during phase resonance testing. The non-linear behaviour is investigated in terms of linearity plots in which the resonance frequency of a mode is plotted as a function of the excitation level. The experimental data is then compatible with all post-processing procedures for the measured results, e.g. updating of the finite element model or flutter calculations. This paper shows measured linearity plots for some typical non-linear phenomena. In the second part of the paper analytical linearity plots for different non-linear stiffness and damping models are considered in order to investigate whether the type of non-linearity can be identified from measured linearity plots. The analytical linearity plots are discussed with respect to their application limits. The analytical linearity plots are used to interpret the experimental linearity plots stemming from various GVTs on different aircraft prototypes. Finally, the observability of non-linear stiffness and non-linear damping characteristics via linearity plots is assessed.     

Non-linear closed-form computational model of cable trusses

In this paper the non-linear closed-form static computational model of the pre-stressed suspended biconvex and biconcave cable trusses with unmovable, movable, or elastic yielding supports subjected to vertical distributed load applied over the entire span and over a part (over the half) of the span is presented. The paper is an extension of the previously published work of authors [S. Kmet, Z. Kokorudova, Non-linear analytical solution for cable trusses, Journal of Engineering Mechanics ASCE 132 (1) (2006) 119-123]. Irvine's linearized forms of the deflection and the cable equations are modified because the effects of the non-linear truss behaviour needed to be incorporated in them. The concrete forms of the system of two non-linear cubic cable equations due to the load type are derived and presented. From a solution of a non-linear vertical equilibrium equation for a loaded cable truss, the additional vertical deflection is determined. The computational analytical model serves to determine the response, i.e. horizontal components of cable forces and deflection of the geometrically non-linear biconvex or biconcave cable truss to the applied loading, considering effects of elastic deformations, temperature changes and elastic supports. The application of the derived non-linear analytical model is illustrated by numerical examples. Resulting responses of the symmetric and asymmetric cable trusses with various geometries (shallow and deep profiles) obtained by the present non-linear closed-form solution are compared with those obtained by Irvine's linear solution and those by the non-linear finite element method. The conditions for the use of the linear and non-linear approach are briefly specified.     

Analytical prediction and experimental validation for longitudinal control of cable oscillations

The paper summarizes the knowledge acquired from the analytical studies and the experimental implementation of a longitudinal non-collocated control strategy for the reduction of cable oscillations. The control is introduced by imposing a longitudinal action at one support based on the knowledge of transverse displacements and velocities of a few selected points. A spatially one-dimensional continuous model of a suspended cable has been used to describe the main features of the non-collocated longitudinal active control strategy. A discrete modal representation has permitted the introduction of suitable non-linear state-feedback controllers. The results have been used to derive an implementable strategy, based on direct output feedback, which preserves the main previous control features. A physical model of an actively controlled cable has been used to demonstrate the control effectiveness of the proposed strategy through a large campaign of experiments, conducted in various frequency ranges and amplitude levels including meaningful external resonance conditions. The responses predicted by the analytical model and the experimental results show good qualitative agreement with one another, in both the uncontrolled and controlled experienced cable dynamics.     

Non-linear stochastic response of a shallow cable

The paper considers the stochastic response of geometrical non-linear shallow cables. Large rain-wind induced cable oscillations with non-linear interactions have been observed in many large cable stayed bridges during the last decades. The response of the cable is investigated for a reduced two-degrees-of-freedom system with one modal coordinate for the in-plane displacement and one for the out-of-plane displacement. At first harmonic varying chord elongation at excitation frequencies close to the corresponding eigenfrequencies of the cable is considered in order to identify stable modes of vibration. Depending on the initial conditions the system may enter one of two states of vibration in the static equilibrium plane with the out-of-plane displacement equal to zero, or a whirling state with the out-of-plane displacement different from zero. Possible solutions are found both analytically and numerically. Next, the chord elongation is modelled as a narrow-banded Gaussian stochastic process, and it is shown that all the indicated harmonic solutions now become instable with probability one. Instead, the cable jumps randomly back and forth between the two in-plane and the whirling mode of vibration. A theory for determining the probability of occupying either of these modes at a certain time is derived based on a homogeneous, continuous time three states Markov chain model. It is shown that the transitional probability rates can be determined by first-passage crossing rates of the envelope process of the chord wise component of the support point motion relative to a safe domain determined from the harmonic analysis of the problem.     

斜拉桥中斜拉索的面内外耦合内共振分析

研究了桥面侧振引起的斜拉索非线性振动问题。基于Hamilton原理建立了拉索的非线性振动控制方程,并利用多尺度法得到了斜拉索振动方程的二阶近似解。通过具体算例分析了斜拉索面内一阶模态与面外一阶模态相互耦合发生内共振的可能性,讨论了拉索倾斜角对拉索振动的影响,比较了在零初始条件和非零初始条件下拉索振动响应的区别。研究发现:拉索内共振发生在一定的激励频率和激励幅值区域内;改变倾斜角度,会影响拉索发生内共振时激励频率区域的大小;初始条件的不同,拉索的振动形式会相差很大。     

An experimental and analytical study of autoparametric resonance in a 3DOF model of cable-stayed-beam

Autoparametric interaction and the associated phenomenon of amplitude saturation are experimentally observed in a physical model of cable-and-beam structure. In this system, the horizontal beam is fixed at one end and supported at the other end by an inclined taut cable. The longitudinal axes of beam and cable are in a vertical plane. Three natural frequencies of the system are approximately of the ratio 1:1:2. This is a combination of two conditions that are very likely to occur in relatively long-span, multi-stay-cable bridges, namely, 1:1 tuning and 1:2 superharmonic tuning. While the beam is vertically excited with sufficiently large force near a primary resonance, the cable vibrates horizontally at half of excitation frequency. The beam also vibrates horizontally at half-frequency, as well as vertically. As the vertical excitation on the bean is further increased in amplitude, the vertical vibration amplitude gets saturated instead of increasing proportionately. A 3DOF analytical model of the structure is also derived, where the finite motion of the cable introduces geometric nonlinearities in quadratic and cubic forms. The system parameters having been carefully measured from the experimental model, steady-state solutions of the coupled nonlinear equations of motion are obtained, by the perturbation method of multiple time scales. Agreement between experimental observation and analytical prediction is very good, both qualitatively and quantitatively. Very good agreement is found also in the case of horizontal excitation of the beam, where effects of linear and nonlinear interaction are apparent.     

A computational study of local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens

In this paper, the local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens are investigated by finite element analyses. Based on the experimental observations of kinked crack growth mechanisms in lap-shear specimens under cyclic loading conditions, three-dimensional and two-dimensional plane-strain finite element models are established to investigate the local stress intensity factor solutions for kinked cracks emanating from the main crack. Semi-elliptical cracks with various kink depths are assumed in the three-dimensional finite element analysis. The local stress intensity factor solutions at the critical locations or at the maximum depths of the kinked cracks are obtained. The computational local stress intensity factor solutions at the critical locations of the kinked cracks of finite depths are expressed in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The three-dimensional finite element computational results show that the critical local mode I stress intensity factor solution increases and then decreases as the kink depth increases. When the kink depth approaches to 0, the critical local mode I stress intensity factor solution appears to approach to that for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The two-dimensional plane-strain computational results indicate that the critical local mode I stress intensity factor solution increases monotonically and increases substantially more than that based on the three-dimensional computational results as the kink depth increases. The local stress intensity factor solutions of the kinked cracks of finite depths are also presented in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. Finally, the implications of the local stress intensity factor solutions for kinked cracks on fatigue life prediction are discussed.     

基于附加虚拟质量的结构损伤识别方法

针对大型土木结构的模态信息往往对局部损伤灵敏度较低的特点,提出通过在各个子结构上附加虚拟质量的方法提高局部灵敏度,实现整体结构的准确损伤识别。该方法无需在结构上布置真实质量,它首先利用虚拟变形法（VDM）可进行结构快速重分析的思想,由实测结构的激励和加速度响应,构造结构附加虚拟质量后虚拟结构的频率响应;然后结合灵敏度分析和附加质量与频率关系,确定所需附加质量和对应具有较高灵敏度的频率;最后分别在每个子结构上附加虚拟质量,联合所有虚拟结构和对应的频率即可准确快速地识别出整体结构各个子结构的损伤。本文通过两层平面框架有限元模型验证了附加质量损伤识别方法的有效性。     

Using instrumental inertia in controlled stress rheometry

We describe a new method for characterizing the non-linear behavior of complex fluids at both small and large deformations. For creep measurements, we use the coupling between the instrumental inertia and the material‘s elasticity to follow the rheological behavior of a solution of iota carrageenan both above and below the yield stress. It is shown that this coupling selectively excites one particular frequency of the relaxation spectrum. An analytical calculation is used to quantify the non-linear behavior near the yield stress. The “free“ oscillations observed during the first few seconds allow us to choose the most appropriate mechanical model. Comparison with experiment shows that even above the yield stress, a linear model can still give independently reliable information about the changes in each element of the mechanical model. A comparison of free and forced oscillations in controlled stress rheometry shows both experimentally and theoretically the conditions under which the use of free oscillations is advantageous. Received: 4 September1997 Accepted: 13 January 1998     

Nonlinear FE-based investigation of flexural damping of slacking wire cables

The flexural damping of wire cable due to the flexural hysteresis influences the dynamic behavior of slacking wire cables significantly. However, the details of the local model, accounting for the flexural hysteresis between the wire strands, are quite challenging to include in large-scale engineering applications. This paper addresses these difficulties by modeling the flexural damping of slacking wire cables using homogenized Rayleigh damping. By using the nonlinear finite element method and high-speed imaging technique, three aspects of the work were examined. First, the mechanical properties of the slacking cable were identified experimentally. Second, a sample cable was fixed at one end and allowed to vibrate freely at the other end. The shapes of the vibrating cable were captured by a high-speed digital camera and processed by photogrammetry. The cable demonstrated a high flexural damping at zero tension and its damping was measured to be as high as 37.7% of the critical damping. Third, the cable was modeled and analyzed using our newly developed nonlinear curved beam element with the Rayleigh damping. The finite element predictions of the cable motion agree well with the experimental measurement. These predictions demonstrate that the new element is capable of describing the dynamic response of the cable and that the Rayleigh damping is sufficient to model the flexural damping of slacking wire cables.     

Super-Harmonic and Internal Resonances of a Suspended Cable with Nearly Commensurable Natural Frequencies

In this paper, super-harmonic and internal resonance characteristics ofa viscously damped cable with nearly commensurable natural frequenciesare investigated by use of a novel method. The proposed frequency-domainsolution method is based on the combined use of a three-dimensionalnonlinear finite element approach and the incremental harmonic balancetechnique. It is an accurate algorithm in the sense that it accommodatesmulti-harmonic components and no mode-based model reduction is utilizedin the solution process. The alternating frequency/amplitude-controlledalgorithm enables complete solution to the frequency-response curvesincluding unstable branches, sub- and super-harmonic resonance andinternal resonance. A suspended cable paradigm under internal resonancecondition is studied using the proposed method. Nonlinear response andmodal interaction characteristics of the cable at different frequencyregions are identified from analysis of response profiles and harmoniccomponent features. The super-harmonic and internal resonance responsesare respectively characterized based on the harmonic distribution. Underan in-plane harmonic excitation, the two-to-one internal resonancebetween the in-plane and out-of-plane modes and the super-harmonicresonance around the second symmetric in-plane mode are revealed. Strongnonlinear interaction among different modes in the parameter spaceranging from primary resonance to super-harmonic resonance is observed.     

悬索在考虑1:3内共振情况下的动力学行为

研究了悬索在受到外激励作用下考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内悬索的弹性-几何参数而言,悬索的第三阶面内对称模态的固有频率接近于第一阶面内对称模态固有频率的三倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动得到主共振情况下的平均方程.接下来对平均方程的稳态解、周期解以及混沌解进行了研究.最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应.     

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.     

Super element approach to cable passing through multiple pulleys

A parametric super element model for cable passing through multiple pulleys is presented in this study for the static analysis of structures. The amounts of cable passages over pulleys are introduced as additional degrees-of-freedoms in the finite element model and the relationship between cable tensions at the two sides of each pulley is imposed based on the friction law or empirical data. The proposed finite element model is firstly verified by a simple pulley cable system and then applied to the analysis of real complex engineering structures. The verification results satisfy the static equilibrium and deformation compatibility conditions of the structural system and basic engineering principles. With the application of the proposed super element model, the global deformation and stress distribution for structures with multiple-pulley cable systems can be effectively and accurately computed. Numerical results for structural analysis show that the effect of friction of pulleys on the cable tensions is significant and the friction-free and fixed models both give unrealistic and incorrect results in cable tensions in some cases.     

Dynamical modelling of wind-induced vibrations of overhead lines

A dynamical model is presented for the wind-induced vibrations of overhead lines due to Karman vortex shedding. Because of the complexity of the excitation mechanism the model for the continuous structure is based on an appropriate non-linear model of a circular rigid cylinder that is oscillating transversally in a flowing fluid. The exciting non-linear forces acting on the cylinder are deduced from known experimental data. It is shown that the observed well-known “lock-in”-effect can be described in satisfactory manner by this model. The transition from the rigid body model to the continuous, flexible structure leads to a non-linear boundary value problem. Approximate solutions are derived by perturbation theory. The model is able to predict the observed vibration frequency and the corresponding vibration mode (depending on the wind velocity) as well as the observed vibration amplitude. An important consequence of the model consists in the self-restricting behaviour of the aerodynamic forces. On this basis, a new conception of the vortex shedding mechanism on a vibrating cable will be presented.     

Buckling driven debonding in sandwich columns

A compression loaded sandwich column that contains a debond is analyzed using a geometrically non-linear finite element model. The model includes a cohesive zone along one face sheet/core interface whereby the debond can extend by interface crack growth. Two geometrical imperfections are introduced; a global imperfection of the sandwich column axis and a local imperfection of the debonded face sheet axis. The model predicts the sandwich column to be very sensitive to the initial debond length and the local face sheet imperfection. The study shows that the sensitivity to the face sheet imperfection results from two mechanisms: (a) interaction of local debond buckling and global buckling and (b) the development of a damaged zone at the debond crack tip. Based on the pronounced imperfection sensitivity, the author predicts that an experimental measurement of the strength of sandwich structures may exhibit a large scatter caused by geometrical variations between test specimens.     

Non-linear model updating of a three-dimensional portal frame based on Wiener series

The common metrics used in linear finite element (FE) model updating using vibration test data are generally functions of relationships based on unidimensional convolution, for example, distances involving natural frequencies, frequency response or impulse response functions, modal shapes, etc. When a structure has local elements or geometry, like joints, bolts, gaps, backlash, etc., these approaches can fail once it could to induce non-linear behavior. Thus, the methods for FE model updating, when considering the existence of localized non-linear parameters, have been receiving much attention in the last years. In this sense, the present paper proposes the use of a strategy through objective functions based on multiples convolutions described by the first order and second order discrete-time Volterra kernels. These kernels are effective metrics for a model updating into large FE model with local non-linearity. In order to improve the non-linear coefficient identification, an orthogonal basis involving Kautz filter is used to expand the kernels, called by Wiener kernel. To exemplify in full details the steps of the updating procedure, an FE model of a three-dimensional portal frame with commons non-linearities is simulated with different excitation forces and used to identify the non-linear parameters. These results allow us to characterize the practical applicability and the drawbacks of the proposed method with suggestions and remarks for further use in industrial structures.