Unwinding characteristics of thin cables for inner and outer dispensers

When thin cables of materials such as yarn and tether are unwound from a spool dispenser, various balloon shapes are generated depending on the initial tensile force at the guide-eyelet point. The shapes are dependent on the properties of the cable because the point of unwinding depends on the thickness of the cable. In this paper, the unwinding characteristics are analyzed for inner and outer dispensers by using three types of thin cable. The dimensionless steady-state equation of motion is first derived from Hamilton’s principle for an open system and the perturbation scheme. The second-order differential equation is then solved by means of shooting method because the boundary conditions at the lift-off point are not fully sufficient. Several parameters such as cable diameter and initial tensile force are checked to study their effects on the balloon shapes. For a given material, the larger is the diameter of the cable, the higher is the air-drag coefficient on the cable, and the effect of the centrifugal force is weaker than that of the fluid resistance. Moreover, the maximum radius of the balloon and the total length of the control volume are small when the cable is thick.     

碳纤维复合材料拉索的非线性力学性能

采用解析法分析碳纤维复合材料(CFRP)拉索的非线性力学性能,通过与钢拉索对比分析,得出了CFRP拉索的受力特点.由于密度低,CFRP拉索自重应力、垂度约为钢拉索的1/5,承载效率也比钢拉索高许多,其承载极限长度为钢拉索的7倍,且随跨径增大,钢拉索的等效弹性模量下降非常快,而CFRP索仍保持较高值.由于CFRP线胀系数比钢材的线胀系数要小得多,约为其1/14,在温差与约束相同时,无垂度CFRP索温度应力仅为无垂度钢索的1/23.当有垂度时,钢索的温度应力降低,CFRP索的温度应力变化很小.     

Revisited modeling and nonlinear oscillation behaviors of multi-segment damaged suspended cables in thermal environments

Meccanica - This paper revisits the mathematical modeling of suspended cables with multi-segment damaged and thermal effects simultaneously. Damage effects are more complex than the thermal ones,...     

具有形位误差的液压滑阀阀芯动态建模与仿真

制造和装配的误差使滑阀副不可避免的存在几何形状和同轴度误差,径向缝隙流动会对滑阀副换向过程中的力学特性产生影响。在分析径向缝隙流动及经过阀口的粘性流动的基础上,建立了阀芯换向过程的运动微分方程,得出了缝隙间的库埃特流动会使阀芯运动更平稳而泊肃叶流动对阀芯移动有正反两方面的影响。最后仿真分析并试验验证了偏心及缝隙越大阀芯运动越不平稳的结论。     

Effect of gravity and tangential air resistance on unwinding cable

In the field of unwinding dynamics, most of the researches so far have only considered the normal component of air resistance. In this research, the transient-state equation of motion that accurately contains all the boundary conditions at the guide-eyelet and lift-off points is derived. The transient-state equation of motion is derived from Hamilton??s principle for an open system, because the total mass of an unwinding cable varies continuously. The virtual work in Hamilton??s principle includes gravity, normal air resistance, and tangential air resistance. The air resistances are assumed to be proportional to the square of the normal and tangential velocities, and the effects of gravity and tangential air resistance on the unwinding cable are verified on the basis of the maximum balloon radius and the associated error. The results show that the effect of gravity and tangential air resistance on the maximum balloon radius is within 4?%, which is negligible.     

Static and dynamic response of elastic suspended cables with damage

In cable-stayed structures cables are subjected to potential damage, mainly due to fatigue and galvanic corrosion. The paper presents an analysis of damage effects on the statics and dynamics of suspended cables. An elastic continuous monodimensional model for damaged cables, including geometric nonlinearities, is formulated for the purpose. The damage is described as a diffused reduction of the cable axial stiffness, and defined through its intensity, extent and position. Exact solutions of the equations governing the cable static equilibrium under self-weight are achieved, and the significance of the tension loss and sag augmentation resulting from damage are investigated under variation of practically significant parameters. The system spectral properties characterizing the free undamped dynamics are obtained in a closed-form fashion for shallow cables within the low frequency range. The sensitivity of the frequencies to the intensity and extent of damage is discussed, outlining two damage effects, which alternatively stiffen or soften the cable modes, whose respective static and geometric origin is recognized. Finally, the symmetry-breaking induced by damage on the static profile is verified to destroy the crossing phenomenon (crossover) characterizing the frequency loci of undamaged cables, which degenerates into a narrow frequency veering phenomenon.     

损伤拉索的等效弹性模量及其参数分析

损伤拉索会出现线形松弛、应力水平降低的情况,必然会影响拉索的等效弹性模量。本文首先引入损伤程度、位置及范围3个参数,用以描述拉索损伤形态的特征,建立损伤拉索索力和线形计算公式,采用数值方法计算了损伤拉索弦向等效弹性模量精确数值,并和经典的等效弹性模量公式的计算结果进行了比较分析,分析了考虑损伤时两种不同计算方法结果的误差。计算表明,对于500m弦向长度以内的损伤拉索,拉索的弦向长度Lc越大,倾角越小,等效弹性模量的损失越大,并且应用割线模量公式计算的误差也越大,当Lc=500m时,损伤拉索相对误差值在2.5%~4.5%之间。弦向应变越小,等效弹性模量损失越大,弦向应变在[0.001,0.004]内,应用割线模量公式计算的相对误差小于3.5%。损伤程度及损伤范围对引用等效弹性模量公式的误差影响较大,倾角对等效弹模公式相对误差的影响也不容忽视。弦向长度、弦向应变、倾角和损伤程度参数都是通过改变拉索的松弛程度进而影响等效弹性模量的数值以及公式的误差。     

Modelling and transient planar dynamics of suspended cables with moving mass

In this study, the 3D nonlinear equations of motion of the suspended cable with moving mass are obtained via the Hamilton principle, and its transient linear planar dynamics is investigated. Considering the quasi-static assumption, the condensed planar model accounting for the effect of the moving mass is derived, and it is then discretized by choosing the static deflection and sine series as shape functions. It is shown that this expansion shows good convergence features. The Newmark method is used to investigate the transient response. The effects of the inertia force, mass, sag and velocity of the moving mass on the transient dynamics of the suspended cable are systematically investigated. Finally, the horizontal tension of the suspended cable and the case of sequentially moving masses are examined.     

Static and dynamic response of elastic suspended cables with thermal effects

Cable structures are often subjected to severe and variable environmental conditions, and their mechanical behavior is known to be particularly sensitive to different ambient factors. The paper analyzes temperature effects on the static and dynamic response of suspended inclined cables through a continuous monodimensional model including geometric nonlinearities. Uniform temperature changes are introduced through a non-homogeneous constitutive law for the material linear elasticity. Exact and approximate solutions of the equations governing the cable static equilibrium under self-weight are achieved, and the significance of the temperature-dependent variation of tension and sag are parametrically investigated. The spectral properties characterizing the free dynamics are obtained in a closed-form fashion for shallow parabolic cables within the low frequency vibration range. The sensitivity of the linear frequencies to temperature changes is discussed, outlining two thermal effects, which are distinguished by their different origins, geometric or static. For a generic temperature change, the geometric effect produces a systematic increment or reduction of all the frequencies, for both symmetric and anti-symmetric modes. The static effect stiffens or softens only the symmetric modes, and may prevail over the competing geometric effect, depending on the cable Irvine parameter. Finally, the thermal effects on the frequency veering and modal hybridization phenomena, which characterize quasi-resonant shallow cubic cables, are analyzed.     

Prediction of thermally induced fracture from notch and crack front

The strain energy density criterion is applied to predict fracture trajectories emanating from existing notch and crack front in nonisothermal environments. When temperature gradients are raised sufficiently high across a notch or crack, the resulting fracture trajectories are non-self-similar and curved in shape. Influence of mechanical loading is also considered in addition to stresses induced by thermal changes. Increase in the applied mechanical load tends to shift or restore the fracture trajectories toward the plane of notch or crack symmetry. The notch sharpness can be varied by adjusting the ration of the minor to major axes of an elliptical cavity. Narrowing the notch primarily increases the local intensity of the strain energy density function dW/dV that is inversely proportional to the radial distance measured from the focal point of the ellipse. This singular character of dW/dV prevails, in general, for all materials and loadings. Numerical results are obtained and displayed graphically for several examples involving fracture trajectory shapes that are not intuitively obvious.     

A numerical and experimental hybrid approach for the investigation of aerodynamic forces on stay cables suffering from rain-wind induced vibration

The aerodynamic forces on a stay cable under a rain-wind induced vibration (RWIV) are difficult to measure directly in a wind tunnel test. This paper presents a hybrid approach that combines an experiment with computational fluid dynamics (CFD) for the investigation on aerodynamic forces of a stay cable under a RWIV. The stay cable and flow field were considered as two substructures of the system. The oscillation of the stay cable was first measured by using a wind tunnel test of a RWIV under an artificial rainfall condition. The oscillation of the cable was treated as a previously known moving boundary condition and applied to the flow field. Only the flow field with the known moving cable boundary was then numerically simulated by using a CFD method (such as Fluent 6.3). The transient aerodynamic forces of the stay cable with a predetermined cable oscillation were obtained from numerical calculations. The characteristics of the aerodynamic forces in the time domain and frequency domain were then analysed for various cases. To verify the feasibility and accuracy of the proposed hybrid approach, the transient aerodynamic forces were applied to a single-degree-of-freedom model (SDOF) of the stay cable to calculate the RWIV of the cable. A comparison was performed between the oscillation responses of the stay cable obtained from the calculated (SDOF model) and experimental results, and the results indicate that the hybrid approach accurately simulates the transient aerodynamic forces of the stay cable. The equivalent damping ratios induced by the aerodynamic forces were obtained for various wind speeds. Furthermore, a nonlinear model of the aerodynamic force is proposed based on the calculation results, and the coefficients in the model were identified by a nonlinear least-squares technique.     

A fast and stable first-order method for simulation of flexible beams and cables

Nonlinear Dynamics - Gradient deficient beam elements (commonly known as cable elements in the literature) based on the absolute nodal coordinate formulation (ANCF) have great potential for...     

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.     

Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances

Nonlinear planar oscillations of suspended cables subjected to external excitations with three-to-one internal resonances are investigated. At first, the Galerkin method is used to discretize the governing nonlinear integral–partial-differential equation. Then, the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance. The equilibrium solutions, the periodic solutions and chaotic solutions of the modulation equations are also investigated. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency/force–response curves. The supercritical Hopf bifurcations are found in these curves. Choosing these bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. Numerical simulations are used to illustrate the cascades of period-doubling bifurcations leading to chaos. At last, the nonlinear responses of the two-degree-of-freedom model are investigated.     

Static and dynamic responses of suspended arch bridges due to failure of cables

Archive of Applied Mechanics - A mathematical model is proposed to investigate the behavior of a suspended arch bridge, subjected to sudden failure of cables. The main aim of this study is to...     

Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables

The global bifurcations in mode of a nonlinear forced dynamics of suspended cables are investigated with the case of the 1:1 internal resonance. After determining the equations of motion in a suitable form, the energy phase method proposed by Haller and Wiggins is employed to show the existence of the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the two cases of Hamiltonian and dissipative perturbation. Furthermore, some complex chaos behaviors are revealed for this class of systems.     

Prediction of turbulent wall shear flows directly from wall

The fully elliptic Reynolds-averaged Navier–Stokes equations have been used together with Lam and Bremhorst's low-Reynolds-number model, Chen and Patel's two-layer model and a two-point wall function method incorporated into the standard k-? model to predict channel flows and a backward-facig step flow. These flows enable the evaluation of the performance of different near-wall treatments in flows involving streamwise and normal pressure gradients, flows with separation and flows with non-equilibrium turbulence characteristics. Direct numerical simulation (DNS) of a channel flow with Re =3200 further provides the detailed budgets of each modelling term of the k and ?-transport equations. Comparison of model results with DNS data to evaluate the performance of each modelling term is also made in the present study. It is concluded that the low-Reynolds-number model has wider applicability and performs better than the two-layer model and wall function approaches. Comparison with DNS data further shows that large discrepancies exist between the DNS budgets and the modelled production and destruction terms of the ? equation. However, for simple channel flow the discrepancies are similar in magnitude but opposite in sign, so they are cancelled by each other. This may explain why, even when employing such an inaccurately modelled ?-equation, one can still predict satisfactorily some simple turbulent flows.     

Prediction of the creep behaviour of polyethylene and molybdenum from stress relaxation experiments

In many applications it is useful to be able to convert observed creep data of a material to corresponding stress relaxation data or vice versa. If the material exhibits non-linear viscoelasticity such a conversion can be rather difficult. In this paper two semi-empirical flow equations, the power law and the exponential law, are used to convert stress relaxation data into corresponding creep behaviour data. These two flow equations are often used to describe non-linear viscoelastic behaviour. The procedure adopted here is based on the assumption that the creep data during the experiment decrease due to an increase in the internal stress level, thus decreasing the effective stress for flow. The conversion method is applied to high density polyethylene and polycrystalline molybdenum at room temperature. In general predictions using the power law are in better agreement with the experimental results than predictions using the exponential formula. The concepts of secondary and ceasing creep are discussed in terms of build-up of internal stress during the creep process.     

Forced nonlinear oscillations of semi-infinite cables and beams resting on a unilateral elastic substrate

In this work, we study the nonlinear oscillations of mechanical systems resting on a (unilateral) elastic substrate reacting in compression only. We consider both semi-infinite cables and semi-infinite beams, subject to a constant distributed load and to a harmonic displacement applied to the finite boundary. Due to the nonlinearity of the substrate, the problem falls in the realm of free-boundary problems, because the position of the points where the system detaches from the substrate, called Touch Down Points (TDP), is not known in advance. By an appropriate change of variables, the problem is transformed into a fixed-boundary problem, which is successively approached by a perturbative expansion method. In order to detect the main mechanical phenomenon, terms up to the second order have to be considered. Two different regimes have been identified in the behaviour of the system, one below (called subcritical) and one above (called supercritical) a certain critical excitation frequency. In the latter, energy is lost by radiation at infinity, while in the former this phenomenon does not occur and various resonances are observed instead; their number depends on the statical configuration around which the system performs nonlinear oscillations.