Nonlinear stochastic analysis of subharmonic response of a shallow cable

The paper deals with the subharmonic response of a shallow cable due to time variations of the chord length of the equilibrium suspension, caused by time varying support point motions. Initially, the capability of a simple nonlinear two-degree-of-freedom model for the prediction of chaotic and stochastic subharmonic response is demonstrated upon comparison with a more involved model based on a spatial finite difference discretization of the full nonlinear partial differential equations of the cable. Since the stochastic response quantities are obtained by Monte Carlo simulation, which is extremely time-consuming for the finite difference model, most of the results are next based on the reduced model. Under harmonical varying support point motions the stable subharmonic motion consists of a harmonically varying component in the equilibrium plane and a large subharmonic out-of-plane component, producing a trajectory at the mid-point of shape as an infinity sign. However, when the harmonical variation of the chordwise elongation is replaced by a narrow-banded Gaussian excitation with the same standard deviation and a centre frequency equal to the circular frequency of the harmonic excitation, the slowly varying phase of the excitation implies that the phase difference between the in-plane and out-of-plane displacement components is not locked at a fixed value. In turn this implies that the trajectory of the displacement components is slowly rotating around the chord line. Hence, a large subharmonic response component is also present in the static equilibrium plane. Further, the time variation of the envelope process of the narrow-banded chordwise elongation process tends to enhance chaotic behaviour of the subharmonic response, which is detectable via extreme sensitivity on the initial conditions, or via the sign of a numerical calculated Lyapunov exponent. These effects have been further investigated based on periodic varying chord elongations with the same frequency and standard deviation as the harmonic excitation, for which the amplitude varies in a well-defined way between two levels within each period. Depending on the relative magnitude of the high and low amplitude phase and their relative duration the onset of chaotic vibrations has been verified.     

考虑弯曲刚度斜拉索面内面外内共振分析

本文研究桥梁工程中含弯曲刚度斜拉索的面内面外内共振问题．描述了工程中斜拉索变形的三种状态,考虑弯曲刚度、大变形及垂度等因素,忽略斜拉索纵向惯性力的影响,运用Hamilton变分原理建立了含弯曲刚度的斜拉索面内面外耦合偏微分控制方程,采用Galerkin方法对偏微分方程离散,并运用多尺度摄动方法进行了求解,获得了斜拉索可能存在的内共振模式,以工程中一根斜拉索为例,运用有限元法对其进行动力特性分析,列出了斜拉索前10阶面内面外振动频率,找出面内面外可能产生内共振的模态,分别研究了主共振条件下斜拉索面内和面外1:1、2:1内共振情形,获得了有意义的结论．     

Planar and non-planar non-linear dynamics of suspended cables under random in-plane loading-I. Single internal resonance

The non-linear interaction of the in-plane and out-of-plane motions of a suspended cable in the neighbourhood of 2:1 internal resonance under random loading is studied. The random loading acts externally on the in-plane mode, while the out-of-plane mode is non-linearly coupled with the in-plane mode. Any non-trivial motion of the out-of-plane mode is mainly due to this non-linear coupling, which becomes significant in the neighbourhood of internal resonance. The response statistics are estimated by employing the Fokker-Planck equation together with Gaussian and non-Gaussian closures. Monte-Carlo simulation is also used for numerical verification. Away from the internal resonance condition, the response is governed by the inplane motion, and the non-Gaussian closure solution is found to be in good agreement with numerical simulation results. The stochastic bifurcation of the out-of-plane mode is predicted by Gaussian and non-Gaussian closures, and by Monte-Carlo simulation. The non-Gaussian closure can only predict bounded solutions within a limited region. The on-off intermittency of the second mode is observed in the Monte-Carlo simulation over a small range of excitation level. The influence on response statistics of excitation level and cable parameters, such as internal detuning, damping ratios, and sag-to-span ratio, is reported.     

Nonlinear Oscillations of a Nonresonant Cable under In-Plane Excitation with a Longitudinal Control

The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.     

Damping for large-amplitude vibrations of plates and curved panels,Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.     

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.     

Dynamical modeling and non-planar coupled behavior of inclined CFRP cables under simultaneous internal and external resonances

A dynamic model for an inclined carbon ?ber reinforced polymer(CFRP)cable is established, and the linear and nonlinear dynamic behaviors are investigated in detail. The partial differential equations for both the in-plane and out-of-plane dynamics of the inclined CFRP cable are obtained by Hamilton's principle. The linear eigenvalues are explored theoretically. Then, the ordinary differential equations for analyzing the dynamic behaviors are obtained by the Galerkin integral and dimensionless treatments.The steady-state solutions of the nonlinear equations are obtained by the multiple scale method(MSM) and the Newton-Raphson method. The frequency-and force-response curves are used to investigate the dynamic behaviors of the inclined CFRP cable under simultaneous internal(between the lowest in-plane and out-of-plane modes) and external resonances, i.e., the primary resonances induced by the excitations of the in-plane mode,the out-of-plane mode, and both the in-plane mode and the out-of-plane mode, respectively. The effects of the key parameters, e.g., Young's modulus, the excitation amplitude,and the frequency on the dynamic behaviors, are discussed in detail. Some interesting phenomena and results are observed and concluded.     

Three-dimensional oscillations of suspended cables involving simultaneous internal resonances

The near resonant response of suspended, elastic cables driven by planar excitation is investigated using a three degree-of-freedom model. The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1:1 and 2:1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions. The perturbation solutions are compared with results obtained by numerically integrating the equations of motion. Furthermore, numerical simulations demonstrate the existence of quasiperiodic responses.A portion of this work was presented at the 1992 ASME Winter Annual Meeting, Anaheim, CA.     

Nonlinear vibration of viscoelastic sandwich plates under narrow-band random excitations

The effect of the narrow-band random excitation on the non-linear response of sandwich plates with an incompressible viscoelastic core is investigated. To model the core, both the transverse shear strains and rotations are assumed to be moderate and the displacement field in the thickness direction is assumed to be linear for the in-plane components and quadratic for the out-of-plane components. In connection to the moderate shear strains considered for the core, a non-linear single-integral viscoelastic model is also used for constitutive modeling of the core. The fifth-order perturbation method is used together with the Galerkin method to transform the nine partial differential equations to a single ordinary integro-differential equation. Converting the lower-order viscoelastic integral term to the differential form, the fifth-order method of multiple scale is applied together with the method of reconstitution to obtain the stochastic phase-amplitude equations. The Fokker–Planck–Kolmogorov equation corresponding to these equations is then solved by the finite difference method, to determine the probability density of the response. The variation of root mean square and marginal probability density of the response amplitude with excitation deterministic frequency and magnitudes are investigated and the bimodal distribution is recognized in narrow ranges of excitation frequency and magnitude.     

Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation

In this study, the forced vibration of a curved pipe conveying fluid resting on a nonlinear elastic foundation is considered. The governing equations for the pipe system are formed with the consideration of viscoelastic material, nonlinearity of foundation, external excitation, and extensibility of centre line. Equations governing the in-plane vibration are solved first by the Galerkin method to obtain the static in-plane equilibrium configuration. The out-of-plane vibration is simplified into a constant coefficient gyroscopic system. Subsequently, the method of multiple scales (MMS) is developed to investigate external first and second primary resonances of the out-of-plane vibration in the presence of three-to-one internal resonance between the first two modes. Modulation equations are formed to obtain the steady state solutions. A parametric study is carried out for the first primary resonance. The effects of damping, nonlinear stiffness of the foundation, internal resonance detuning parameter, and the magnitude of the external excitation are investigated through frequency response curves and force response curves. The characteristics of the single mode response and the relationship between single and two mode steady state solutions are revealed for the second primary resonance. The stability analysis is carried out for these plots. Finally, the approximately analytical results are confirmed by the numerical integrations.     

部分相干光云纹干涉法测三维位移场

本文提出了一种用部分相干光可同时获得面内位移和离面位移场的云纹干涉法。该法具有装置简单,无需防震,可用于观场测量等特点。文中论述了此法的基本原理,应用波前干涉理论和付里叶光学理论对变形前后波前的干涉、记录和再观进行了分析,导出了计算公式。实现了透射物体和反射物体表面的面内位移分量(U_0,V_(45),U_(90))和离面位移(w)的全场同时测量,典型实验证明,实验值与计算结果符合较好。     

考虑阻尼的无限长小垂度缆索弹性波的传播

缆索结构被广泛应用于电气、土木、海洋和航空工程等领域,随着缆索在工程中的应用长度越来越长,高阶振动越来越明显,研究时应该考虑扰动沿着缆索的传播.现有对缆索弹性波传播的研究中,通常不考虑阻尼项,然而阻尼对于波的传播有着重要影响.文章考虑阻尼的影响,发展了包含阻尼项的三维弹性缆索运动方程.通过求解上述含阻尼项的运动方程,分别考察了面内面外弹性波的频率关系、相速度和群速度等自由传播特性,进而通过计算无限长弹性缆索在初始余弦型脉冲作用下的位移响应,分析扰动沿着该缆索的传播规律,考察波的色散现象以及阻尼对于缆索弹性波传播的影响.结果表明,考虑阻尼后,面内波和面外波均为色散波,面内波在曲率的作用下,为高度色散波.此外,在阻尼的影响下,波的峰值在传播过程不断减小,且波的后缘端点响应总是高于前缘端点响应.     

Multiple Internal Resonance in Suspended Cables under Random In-Plane Loading

The random excitation of a suspended cable with simultaneous internal resonances is considered. The internal resonances can take place among the first in-plane and the first two out-of-plane modes. The external loading is represented by a wide-band random process. The response statistics are estimated using the Fokker-Planck-Kolmogorov (FPK) equation, together with Gaussian and non-Gaussian closures. Monte Carlo simulation is also used for numerical verification. The unimodal in-plane motion exists in regions away from the internal resonance condition. The mixed mode interaction is manifested within a limited range of internal detuning parameters, depending on the excitation power spectrum density and damping ratios. The Gaussian closure scheme failed to predict bounded solutions of mixed mode interaction. The non-Gaussian closure results are in good agreement with the Monte Carlo simulation. The on-off intermittency of the autoparametrically excited modes is observed in the Monte Carlo simulation over a small range of excitation levels. The influence of the cable parameters, such as damping ratios, sag-to-span ratio, internal detuning parameters, and excitation level on the autoparametric interaction, is studied. It is found that the internal detuning and excitation level are the two main parameters which affect the autoparametric interaction among the three modes. Due to the system's nonlinearity, the response of the three modes is strongly non-Gaussian and the coupled modes experience irregular modulation.     

A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation

A non-linear one-dimensional model of inextensional, shear undeformable, thin-walled beam with an open cross-section is developed. Non-linear in-plane and out-of-plane warping and torsional elongation effects are included in the model. By using the Vlasov kinematical hypotheses, together with the assumption that the cross-section is undeformable in its own plane, the non-linear warping is described in terms of the flexural and torsional curvatures. Due to the internal constraints, the displacement field depends on three components only, two transversal translations of the shear center and the torsional rotation. Three non-linear differential equations of motion up to the third order are derived using the Hamilton principle. Taking into account the order of magnitude of the various terms, the equations are simplified and the importance of each contribution is discussed. The effect of symmetry properties is also outlined. Finally, a discrete form of the equations is given, which is used in Part II to study dynamic coupling phenomena in conditions of internal resonance.     

Blind-hole residual stress determination using optical interferometry

The in-plane method and the out-of-plane method are used to analyze blind-hole residual stress as measured by optical interferometry. The in-plane method, which constructs a relation between the in-plane displacement field and the residual stress released from blind-hole drilling, is applicable when the sensitivity vector of the interferometer used in the measuring system is parallel to the object surface. Three in-plane displacements obtained from one interference pattern are sufficient to determine the residual stress. The out-of-plane method, which establishes a new relation between the out-of-plane displacement field and the released residual stress, is suggested when the sensitivity vector is perpendicular to the object surface. Two relative out-of-plane displacements extracted from one interference pattern are sufficient to determine the residual stress. With the adoption of these two methods, interpolating calculation is not needed to determine the fringe order of each data point, since the selections of the required data points are flexible using these two methods. Two experiments, one for the in-plane method and the other for the out-of-plane method, were carried out to illustrate the applicability and usefulness of these two methods.     

Nonlinear oscillations of suspended cables containing a two-to-one internal resonance

The near-resonant response of suspended, elastic cables driven by planar excitation is investigated using a two degree-of-fredom model. The model captures the interaction of a symmetric in-plane mode and an out-of-plane mode with near commensurable natural frequencies in a 2:1 ratio. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. The existence and stability of periodic solutions are investigated using a second order perturbation analysis. The first order analysis shows that suspended cables may exhibit saturation and jump phenomena. The second order analysis, however, reveals that the cubic nonlinearities and higher order corrections disrupt saturation. The stable, steady state solutions for the second order analysis compare favorably with results obtained by numerically integrating the equations of motion.     

Non-linear vibration of rectangular reticulated shallow shell structures

ＮＯＮ－ＬＩＮＥＡＲＶＩＢＲＡＴＩＯＮＯＦＲＥＣＴＡＮＧＵＬＡＲＲＥＴＩＣＵＬＡＴＥＤＳＨＡＬＬＯＷＳＨＥＬＬＳＴＲＵＣＴＵＲＥＳＮｉｅＧｕｏ－ｈｕａ（聂国华）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉｎｅｅｒｉｎｇＭｅｃｈａｎｉｃｓ,ＴｏｎｇｊｉＵｎｉｖｅ...     

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。     

Dynamic instability of inclined cables under combined wind flow and support motion

In this paper an inclined nearly taut stay, belonging to a cable-stayed bridge, is considered. It is subject to a prescribed motion at one end, caused by traveling vehicles, and embedded in a wind flow blowing simultaneously with rain. The cable is modeled as a non-planar, nonlinear, one-dimensional continuum, possessing torsional and flexural stiffness. The lower end of the cable is assumed to undergo a vertical sinusoidal motion of given amplitude and frequency. The wind flow is assumed uniform in space and constant in time, acting on the cable along which flows a rain rivulet. The imposed motion is responsible for both external and parametric excitations, while the wind flow produces aeroelastic instability. The relevant equations of motion are discretized via the Galerkin method, by taking one in-plane and one out-of-plane symmetric modes as trial functions. The two resulting second-order, non-homogeneous, time-periodic, ordinary differential equations are coupled and contain quadratic and cubic nonlinearities, both in the displacements and velocities. They are tackled by the Multiple Scale perturbation method, which leads to first-order amplitude-phase modulation equations, governing the slow dynamics of the cable. The wind speed, the amplitude of the support motion and the internal and external frequency detunings are set as control parameters. Numerical path-following techniques provide bifurcation diagrams as functions of the control parameters, able to highlight the interactions between in-plane and out-of-plane motions, as well as the effects of the simultaneous presence of the three sources of excitation.