Parametric studies on relationships between flutter derivatives of slender bridge （Ⅱ）

Based on curve fitting of coefficients of three component forces of the Messina Straits Bridge, and the previously proposed semi-analytical expressions of flutter derivatives of flexible structure, the change of flutter derivatives of slender bridge cross-section with respect to its aerodynamic center, rotational speed and angle variation is studied using a parametric method. The calculated results are compared with the measured ones, and expressions of flutter derivatives of the Messina Straits Bridge are derived. The intrinsic relationships existing in flutter derivatives are validated again. It is shown that the influence of the rotational speed on flutter derivatives is not negligible. Therefore, it provides an additional semi-analytical approach for analyzing flutter derivatives of the bridge with streamlined cross-section to get its aerodynamic information.     

Semi-analytical solution to one-dimensional consolidation in unsaturated soils

In this paper, a series of semi-analytical solutions to one-dimensional consolidation in unsaturated soils are obtained. The air governing equation by Fredlund for unsaturated soils consolidation is simplified. By applying the Laplace transform and the Cayley-Hamilton theorem to the simplified governing equations of water and air, Darcy＇s law, and Fick＇s law, the transfer function between the state vectors at top and at any depth is then constructed. Finally, by the boundary conditions, the excess pore-water pressure, the excess pore-air pressure, and the soil settlement are obtained under several kinds of boundary conditions with the large-area uniform instantaneous loading. By the Crump method, the inverse Laplace transform is performed, and the semi-analytical solutions to the excess pore-water pressure, the excess pore-air pressure, and the soils settlement are obtained in the time domain. In the case of one surface which is permeable to air and water, comparisons between the semi-analytical solutions and the analytical solutions indicate that the semi-analytical solutions are correct. In the case of one surface which is permeable to air but impermeable to water, comparisons between the semi-analytical solutions and the results of the finite difference method are made, indicating that the semi-analytical solution is also correct.     

A structural topological optimization method for multi-displacement constraints and any initial topology configuration

In density-based topological design, one expects that the final result consists of elements either black （solid material） or white （void）, without any grey areas. Moreover, one also expects that the optimal topology can be obtained by starting from any initial topology configuration. An improved structural topological optimization method for multidisplacement constraints is proposed in this paper. In the proposed method, the whole optimization process is divided into two optimization adjustment phases and a phase transferring step. Firstly, an optimization model is built to deal with the varied displacement limits, design space adjustments, and reasonable relations between the element stiffness matrix and mass and its element topology variable. Secondly, a procedure is proposed to solve the optimization problem formulated in the first optimization adjustment phase, by starting with a small design space and advancing to a larger deign space. The design space adjustments are automatic when the design domain needs expansions, in which the convergence of the proposed method will not be affected. The final topology obtained by the proposed procedure in the first optimization phase, can approach to the vicinity of the optimum topology. Then, a heuristic algorithm is given to improve the efficiency and make the designed structural topology black/white in both the phase transferring step and the second optimization adjustment phase. And the optimum topology can finally be obtained by the second phase optimization adjustments. Two examples are presented to show that the topologies obtained by the proposed method are of very good 0/1 design distribution property, and the computational efficiency is enhanced by reducing the element number of the design structural finite model during two optimization adjustment phases. And the examples also show that this method is robust and practicable.     

Analysis on nonlinear effect of unsteady percolation in the inhomogeneous shale gas reservoir

The nonlinear effects of unsteady multi-scale shale gas percolation,such as desorption,slippage,diffusion,pressure-dependent viscosity,and compressibility,are investigated by numerical simulation.A new general mathematical model of the problem is built,in which the Gaussian distribution is used to describe the inhomogeneous intrinsic permeability.Based on the Boltzmann transformation,an efficient semi-analytical method is proposed.The problem is then converted into a nonlinear equation in an integral form for the pressure field,and a related explicit iteration scheme is constructed by numerical discretization.The validation examples show that the proposed method has good convergence,and the simulation results also agree well with the results obtained from both numerical and actual data of two vertical fractured test wells in the literature.Desorption,slippage,and diffusion have significant influence on shale gas flows.The accuracy of the usual technique that the product of viscosity and compressibility is approximated as its value at the average formation pressure is examined.     

Nonreciprocity of energy transfer in a nonlinear asymmetric oscillator system with various vibration states

The nonreciprocity of energy transfer is constructed in a nonlinear asymmetric oscillator system that comprises two nonlinear oscillators with different parameters placed between two identical linear oscillators. The slow-flow equation of the system is derived by the complexification-averaging method. The semi-analytical solutions to this equation are obtained by the least squares method, which are compared with the numerical solutions obtained by the Runge-Kutta method. The distribution of the ...     

A note on the definition of fractional derivatives applied in rheology

It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives—Riemann–Liouville (R–L) definition and Caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R–L definition and Caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model (Scott–Blair model). We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R–L definition with lower terminal a →∞ and the Caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R–L derivatives with lower terminal a →∞ (or, equivalently, the Caputo derivatives with lower terminal a →∞) not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott–Blair model and give an exact solution of this equation under given initial conditions.     

INCREMENTAL HARMONIC BALANCE METHOD FOR AIRFOIL FLUTTER WITH MULTIPLE STRONG NONLINEARITIES

The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.     

Parametric study on supersonic flutter of angle-ply laminated plates using shear deformable finite element method

The influence of fiber orientation,flow yaw angle and length-to-thickness ratio on flutter characteristics of angle-ply laminated plates in supersonic flow is studied by finite element approach.The structural model is established using the Reissner-Mindlin theory in which the transverse shear deformation is considered.The aerodynamic pressure is evaluated by the quasi-steady first-order piston theory.The equations of motion are formulated based on the principle of virtual work.With the harmonic motion assumption,the flutter boundary is determined by solving a series of complex eigenvalue problems.Numerical study shows that (1) The flutter dynamic pressure and the coalescence of flutter modes depend on fiber orientation,flow yaw angle and length-to-thickness ratio;(2) The laminated plate with all fibers aligned with the flow direction gives the highest flutter dynamic pressure,but a slight yawing of the flow from the fiber orientation results in a sharp decrease of the flutter dynamic pressure;(3) The angle-ply laminated plate with fiber orientation angle equal to flow yaw angle gives high flutter dynamic pressure,but not the maximum flutter dynamic pressure;(4) With the decrease of length-to-thickness ratio,an adverse effect due to mode transition on the flutter dynamic pressure is found.     

CHAOTIC MOTIONS AND LIMIT CYCLE FLUTTER OF TWO-DIMENSIONAL WING IN SUPERSONIC FLOW

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos axe very narrow.     

Fluidization science, its development and future

By revisiting the three stage theory for the progress of science proposed by Taketani in 1942, the footmarks of fluidization research are examined. The bubbling and fast fluidization issues were emphasized so that the future offluidization research can be discussed among scientists and engineers in a wider perspective. The first cycle of fluidization research was started in the early 1940s by an initial stage of phenomenology. The second stage of structural studies was kicked off in the early 1950s with the introduction of the two phase theory. The third stage of essential studies occurred in the early 1960s in the form of bubble hydrodynamics. The second cycle, which confirmed the aforementioned three stages closed at the turn of the century, established a general understanding of suspension structures including agglomerating fluidization, bubbling, turbulent and fast fluidizations and pneumatic transport; also established powerful measurement and numerical simulation tools.After a general remark on science, technology and society issues the interactions between fluidization technology and science are revisited. Our future directions are discussed including the tasks in the third cycle, particularly in its phenomenology stage where strong motivation and intention are always necessary, in relation also to the green reforming of the present technology. A generalized definition of ＇fluidization＇ is proposed to extend fluidization principle into much wider scientific fields, which would be effective also for wider collaborations.     

Parametric studies on relationships between flutter derivatives of slender bridge (Ⅱ)

Based on curve fitting of coefficients of three component forces of the Messina Straits Bridge, and the previously proposed semi-analytical expressions of flutter deriva-tives of flexible structure, the change of flutter derivatives of slender bridge cross-section with respect to its aerodynamic center, rotational speed and angle variation is studied using a parametric method. The calculated results are compared with the measured ones, and expressions of flutter derivatives of the Messina Straits Bridge are derived. The in-trinsic relationships existing in flutter derivatives are validated again. It is shown that the influence of the rotational speed on flutter derivatives is not negligible. Therefore, it provides an additional semi-analytical approach for analyzing flutter derivatives of the bridge with streamlined cross-section to get its aerodynamic information.     

MOTION-RELATED BODY-FORCE FUNCTIONS IN TWO-DIMENSIONAL LOW-SPEED FLOW

The paper examines two-dimensional force functions useful in the wind-response analysis of bluff bodies, such as bridges, that are elongated in the across-wind direction. Airfoil-type theoretical 2-D indicial, admittance, and oscillatory force functions and their interrelationships are first recalled for their analogical value. Following this, a spectral force expression is developed for the lift on a bluff section due to a cross-wind with a vertical turbulence component. The expression proposed involves an aerodynamic admittance that is based upon measured flutter derivatives for the bluff section plus information on the coherence of vertical turbulence in the atmospheric boundary layer.     

An accurate singularity-free geometrically exact beam formulation using Euler parameters

Wind-induced nonlinear oscillations of twin-box girder bridges are very sensitive to the aerodynamic shape of the deck (i.e., slot width ratio (SWR) and wind fairing shape) due to the complicated flow characteristics around the bridge deck. This paper presents a fully integrated finite element (FE) model in time domain, involving a nonlinear aerodynamic force model and a bridge FE model, to allow the investigation of nonlinear oscillation behaviors of long-span twin-box girder bridges with various SWRs and wind fairing shapes. The parameters in integrated FE model were firstly identified by using CFD simulation, and then, the proposed model was validated by conducting wind tunnel testing using sectional models and full-bridge aeroelastic models. It demonstrates that the developed integrated model has the capability of simulating the nonlinear flutter behaviors of twin-box girder bridges with various aerodynamic shapes. Furthermore, the prediction results show that the wind fairing shape has significant impact on the degree of freedom participation in coupled oscillation and failure modes, as well as flutter performance of the bridges. In addition, there is an increase in amplitudes of the limit cycle oscillations with the increase in the SWR of the twin-box girder bridges, and the relationships between the bending-torsional coupled oscillation, failure modes, and SWR of the bridges with anti-symmetric wind fairings are opposite to those with symmetric wind fairings.     

我国大跨度缆索承重桥梁的空气动力性能研究

本文介绍了我国大跨度桥梁空气动力性能方面的研究情况。以同济大学土木工程防灾国家重点实验室所从事的桥梁空气动力性能的理论和试验研究工作中的几个主要方面为重点,介绍最新进展,其中包括非线性颤振的频域分析,颤振分析的全模态技术,基于计算流体动力学的数值风洞等研究课题。     

Nonlinear parametric modeling of suspension bridges under aeroelastic forces: torsional divergence and flutter

A fully nonlinear model of suspension bridges parameterized by one single space coordinate is proposed to describe overall three-dimensional motions. The nonlinear equations of motion are obtained via a direct total Lagrangian formulation and the kinematics, for the deck-girder and the suspension cables, feature the finite displacements of the associated base lines and the flexural and torsional rotations of the deck cross-sections assumed rigid in their own planes. The strain-displacement relationships for the generalized strain parameters, the elongations in the cables, the deck elongation, and the three curvatures, retain the full geometric nonlinearities. The proposed nonlinear model with its full extensional-flexural-torsional coupling is employed to study the torsional divergence caused by the static part of the wind-induced forces. Two suspension bridges are considered as case studies: the Runyang bridge (main span 1,490?m) and the Hu Men bridge (main span 888?m) in China. The evaluation of the onset of the static instability and the post-critical behavior takes into account the prestressed condition of the bridge subject to dead loads. The dynamic bifurcation that occurs at the onset of flutter is also studied accounting for the prestressed equilibrium state about which the equations of motion are obtained via an updated Lagrangian formulation. Such a bifurcation is investigated in the context of the parametric nonlinear model considering the model parameters of the Runyang Suspension Bridge together with its aeroelastic derivatives. The calculated critical wind speeds for the onset of the static and dynamic bifurcations are compared with the results obtained via linear analysis and the main differences are highlighted. Parametric sensitivity studies are carried out to assess the influence of the design parameters on the instabilities associated with the bridge aeroelastic response.     

基于传递函数法的带多个外挂机翼颤振分析

基于带外挂机翼结构和气动特点，使用带有半解析半数值特性的传递函数方法进行处理．首先，通过机翼的运动微分方程、二元机翼非定常气动力Therdorson 模型、结合外挂挂载处的内力平衡与位移状态条件，得到了三维的带多个外挂大展弦比机翼的颤振微分方程．进而，使用传递函数方法，先将颤振计算方程整理成为状态空间方程形式，结合求解复特征值的方法，完成了带多个外挂大展弦比机翼的动气动弹性稳定性分析．对比已有文献的计算结果以及通过有限元方法进行的仿真结果，证实了文章所提计算方法的准确性和高效性，结合传递函数方法的优势，进一步将文章方法拓展到机翼的固有频率和发散速度求解．文章结尾，分析了外挂数量、外挂质量、转动惯量及位置分布等变量对带多个外挂大展弦比机翼的动气动弹性稳定性的影响．     

两种湍流模型时域颤振计算方法研究

采用时域计算分析方法进行了机翼跨音速颤振特性研究。在结构运动网格的基础上,采用格点格式有限体积方法进行空间离散和双时间全隐式方法进行时间推进求解雷诺平均N-S方程。针对流动粘性分别应用了SST湍流模型和SSG雷诺应力模型,通过对跨音速标模算例AGARD445.6机翼的计算结果与实验值的对比分析,其中应用SST湍流模型得到的颤振速度与实验值最为接近,特别是在跨音速段平均相对误差在3%以内;并且计算结果整体上反映了跨音速颤振"凹坑"物理特性,验证了方法的有效性。     

Accuracy Analysis of the Semi-Analytical Method for Shape Sensitivity Calculation∗

ABSTRACT

The semi-analytical method of design sensitivity analysis that is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements is studied in this paper. It is shown that the method can have serious accuracy problems for shape design variables in structures modeled by beam, plate, truss, frame, and solid elements. Errors are shown to be associated with an incompatibility of the sensitivity field with the structure. An error index is developed to test the accuracy of the semi-analytical method. It characterizes the difference in errors between a general finite difference method and the semi-analytical method. A method for improving the accuracy of the semi-analytical method (when possible) is provided. Examples are presented to demonstrate the use of the error index.