Equivalent nonlinear beam model for the 3-D analysis of shear-type buildings: Application to aeroelastic instability

Tower buildings can be very sensitive to dynamic actions and their dynamic analysis is usually carried out numerically through sophisticated finite element models. In this paper, an equivalent nonlinear one-dimensional shear–shear–torsional beam model immersed in a three-dimensional space is introduced to reproduce, in an approximate way, the dynamic behavior of tower buildings. It represents an extension of a linear beam model recently introduced by the authors, accounting for nonlinearities generated by the stretching of the columns. The constitutive law of the beam is identified from a discrete model of a 3D-frame, via a homogenization process, which accounts for the rotation of the floors around the tower axis. The macroscopic shear strain in the equivalent beam is produced by the bending of columns, accompanied by negligible rotation of the floors. A coupled nonlinear shear–torsional mechanical model is thus obtained. The coupling between shear and torsion is related to a non-symmetric layout of the columns, while mechanical nonlinearities are proportional to the slenderness of the columns. The model can be used for the analysis of the response of tower buildings to any kind of dynamic and static excitation. A first application is here presented to investigate the effect of mechanical and aerodynamic coupling on the critical galloping conditions and on the postcritical behavior of tower buildings, based on a quasi-steady model of aerodynamic forces.     

Convergence predictions for aeroelastic calculations of tuned and mistuned bladed disks

Mistuning changes the dynamics of bladed disks significantly. Frequency domain methods for predicting the dynamics of mistuned bladed disks are typically based on iterative aeroelastic calculations. Converged aerodynamic stiffness matrices are required for accurate aeroelastic results of eigenvalue and forced response problems. The tremendous computation time needed for each aerodynamic iteration would greatly benefit from a fast method of predicting the number of iterations needed for converged results. A new hybrid technique is proposed to predict the convergence history based on several critical ratios and by approximating as linear the relation between the aerodynamic force and the complex frequencies (eigenvalues) of the system. The new technique is hybrid in that it uses a combined theoretical and stochastic/computational approach. The dynamics of an industrial bladed disk is investigated, and the predicted convergence histories are shown to match the actual results very well. Monte Carlo simulations using the new hybrid technique show that the aerodynamic ratio and the aerodynamic gradient ratio are the two most important factors affecting the convergence history.     

A discrete 3D model for bridge aerodynamics and aeroelasticity: nonlinearities and linearizations

A framework for the numerical analysis of bridges under wind excitation is outlined. It is based on structural finite element scheme and cross-sectional wind load models. Two aspects are investigated: (1) how considering the mean steady configuration in the aerodynamic stability calculation; and (2) the effects of load nonlinearities on structural response. A quasi-steady load model is adopted, which is able to deal with the considered problems by using experimental data easily available in the practice. By means of numerical examples, it is pointed out (1) that both the modifications in structural tangential stiffness and in the aerodynamic coefficients due to the mean steady deformation may affect the aeroelastic stability threshold and (2) that load linearization may produce an underestimation of the structural response.     

The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

Nonlinear Dynamics - In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the...     

The parametrized strongly implicit method for solving elliptic difference equations

A new method for solving elliptic difference equations is derived based on the strongly implicit method. This parametrized strongly implicit method has three free parameters which may be functions of the field's nodal point. The method has some resemblance to the SOR techniques, but in the present method the off-diagonal entries are also over-relaxed. The main application of this method is for transport equations such as those governing the fluid flow and heat transfer fields.     

Harmonic balance method with alternating frequency/time domain technique for nonlinear dynamical system with fractional exponential

Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.     

On the convergence and causality of a frequency domain method for dynamic structural analysis

Venanico-Filho et al. developed an elegant matrix formulation for dynamic analysis by frequency domain (FD), but the convergence, causality and extended period need further refining. In the present paper, it was argued that: (1) under reasonable assumptions (approximating the frequency response function by the discrete Fourier transform of the discretized unitary impulse response function), the matrix formulation by FD is equivalent to a circular convolution; (2) to avoid the wraparound interference, the excitation vector and impulse response must be padded with enough zeros; (3) provided that the zero padding requirement satisfied, the convergence and accuracy of direct time domain analysis, which is equivalent to that by FD, are guaranteed by the numerical integration scheme; (4) the imaginary part of the computational response approaching zero is due to the continuity of the impulse response functions. The English text was polished by Yunming Chen     

A new method for detecting non-linear damping and restoring forces in non-linear oscillation systems from transient data

The purpose of this study is to recover the functional form of both non-linear damping and non-linear restoring forces in the non-linear oscillatory motions of an autonomous system. Using two sets of measured motion response data of the system, an inverse problem is formulated for recovering (or identification): the differential equation of motion is transformed into an equivalent integral equation of motion. The identification, which is non-linear, is shown to be one-to-one. However, the inverse problem formulated herein is concerned with the Volterra-type of non-linear integral equation of the first kind. This leads to numerical instability: solutions of the inverse problem lack stability properties. In order to overcome the difficulty, a regularization method is applied to the identification process. In addition, an L-curve criterion, combined with regularization, is introduced to find an optimal choice for the regularization parameter (i.e., the number of iterations), in the presence of noisy data. The workability of the identification is investigated for simultaneously recovering the functional form of the non-linear damping and the non-linear restoring forces through a numerical experiment.     

Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations

Field time integrators with second-order-accurate numerical schemes for both the fluid and the structure are considered for unsteady Euler aeroelastic computations. We show that if these schemes are simply coupled and used straightforwardly with subcycling, then accuracy and stability properties may be lost. We present new coupling staggered procedures where momentum conservation is enforced at the interface. This is done by using a structural predictor. Continuity of structural and fluid grid displacements is not satisfied at the fluid/structure interface. However, we show on a two-degree-of-freedom aerofoil that this new type of method has many advantages, e.g. accuracy of conservation at the interface and extended stability. The supersonic flutter of a flat panel is simulated in order to numerically prove that the algorithm gives accurate results with arbitrary subcycling for the fluid in the satisfying limit of 30 time steps per period of coupled oscillation. © 1997 John Wiley & Sons, Ltd.     

The extended Padé-Laplace method for efficient discretization of linear viscoelastic spectra

The discretization of linear viscoelastic spectra is valuable as a starting point for non-linear viscoelastic modeling. However, obtaining the parameters of the generalized Maxwell model from linear viscoelastic experiments with naive least squares procedures is known to be an ill-posed problem. A novel technique, the Padé-Laplace method was recently elucidated (Fulchiron et al., 1993) for robustly extracting the parameters of the generalized Maxwell model from stress relaxation experiments, without any a priori assumption about the number of Maxwellian modes. We extend this method for obtaining the Maxwellian modes from dynamic data and discuss the relationship between continuous viscoelastic spectra and the Maxwellian modes obtained by this procedure. Furthermore, the applicability of this method with experimental data in limited time/frequency windows is clarified. Finally, a procedure for assembling the discretized spectrum with the Padé-Laplace method applied to both stress relaxation and dynamic data with typical experimental time/frequency cutoffs is developed.Dedicated to Professor H. Janeschitz-Kriegl on the occasion of his 70th birthday.     

On methods for continuous systems with quadratic and cubic nonlinearities

Methods for determining the response of continuous systems with quadratic and cubic nonlinearities are discussed. We show by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbation methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems. We describe a perturbation technique that applies either the method of multiple scales or the method of averaging to the Lagrangian of the system rather than the partial-differential equations and boundary conditions.     

A method toward the construction of Liapunov function for discrete time systems and determination of their stability

The purpose of this paper is to present a new method for constructing Liapunov function and determining the stability of discrete time systems with a computer on the basis of the similarity transformation theory by directly applying the system matrix of the system under discussion instead of solving the discrete Liapunov's matrix equation.     

The relative variance criterion for stability of delay systems

A new approach to the study of delay systems, applicable to physiological control systems and other systems where little information about the time delays is available, is examined. The method is based on the fact that stability information can be deduced from the statistical properties of the probability distribution that encodes the structure of the time delay. The main statistical variables used are the usual expectation parameter,E, and a modified variance, calledrelative variance and denotedR, that is invariant under time scale changes. Recent work of the author has shown that stability often improves asR increases whileE remains fixed. A four-parameter family of delay models is analysed in this paper, and the (E, R) pair is found to be a reliable indicator of stability over the global parameter domain of the family.     

Transient growth of energy and aeroelastic stability of ground vehiclesCroissance transitoire de l'énergie et stabilité aéroélastique des véhicules terrestres

Transient growth of energy is shown to be possible in the lateral dynamics of passenger cars. This mechanism might be generated during one vehicle overtaking another. Starting from a simple linearized quasi-steady model, which couples the lateral displacement and the yaw angle of the vehicle, the transient growth appears when an initial condition in the yawing rate is applied. To cite this article: P. Hémon, C. Noger, C. R. Mecanique 332 (2004).     

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters

The p-moment exponential robust stability for stochastic systems with distributed delays and interval parameters is studied.By constructing the LyapunovKrasovskii functional and employing the decomposition technique of interval matrix and Ito's formula,the delay-dependent criteria for the p-moment exponential robust stability are obtained.Numerical examples show the validity and practicality of the presented criteria.     

Time and frequency domain nonlinear system characterization for mechanical fault identification

Mechanical systems are often nonlinear with nonlinear components and nonlinear connections, and mechanical damage frequently causes changes in the nonlinear characteristics of mechanical systems, e.g. loosening of bolts increases Coulomb friction nonlinearity. Consequently, methods which characterize the nonlinear behavior of mechanical systems are well-suited to detect such damage. This paper presents passive time and frequency domain methods that exploit the changes in the nonlinear behavior of a mechanical system to identify damage. In the time domain, fundamental mechanics models are used to generate restoring forces, which characterize the nonlinear nature of internal forces in system components under loading. The onset of nonlinear damage results in changes to the restoring forces, which can be used as indicators of damage. Analogously, in the frequency domain, transmissibility (output-only) versions of auto-regressive exogenous input (ARX) models are used to locate and characterize the degree to which faults change the nonlinear correlations present in the response data. First, it is shown that damage causes changes in the restoring force characteristics, which can be used to detect damage. Second, it is shown that damage also alters the nonlinear correlations in the data that can be used to locate and track the progress of damage. Both restoring forces and auto-regressive transmissibility methods utilize operational response data for damage identification. Mechanical faults in ground vehicle suspension systems, e.g. loosening of bolts, are identified using experimental data.     

Improved gradient descent algorithms for time-delay rational state-space systems: intelligent search method and momentum method

Nonlinear Dynamics - This study proposes two improved gradient descent parameter estimation algorithms for rational state-space models with time-delay. These two algorithms, based on intelligent...     

Reaction-diffusion systems in nonconvex domains: Invariant manifold and reduced form

A study is made of systems of weakly coupled, semilinear, parabolic equations, namely reaction-diffusion systems, subject to the homogeneous Neumann boundary conditions in parametrized nonconvex domains inR ². It is assumed that the domain approaches a union of two disjoint domains as the parameter varies. Under some conditions the long-time behavior of bounded solutions is discussed and the existence of a finite-dimensional invariant manifold is shown, together with its attractivity. This manifold is represented by a graph of some function defined in a possibly large bounded region of the phase space, and the original system is reduced to an ODE system on it. Since an explicit form of the reduced ODE system is given, its dynamics can be studied in detail, which in turn reveals the global dynamics of the original reaction-diffusion system. One can thereby prove, among other things, the existence of asymptotically stable equilibrium solutions of the original system having large spatial inhomogeneity. The existence and stability of a spatially inhomogeneous periodic solution of large amplitude are also discussed.     

Local time-stepping procedures for the space-time conservation element and solution element method

A local time-stepping procedure for the space-time conservation element and solution element (CESE) method has been developed. This new procedure allows for variation of time-step size in both space and time, and can also be extended to become multi-dimensional solvers with structured/unstructured spatial grids. Moreover, it differs substantially in concept and methodology from the existing approaches. By taking full advantage of key concepts of the CESE method, in a simple and efficient manner it can enforce flux conservation across an interface separating grid zones of different time-step sizes. In particular, no correction pass is needed. Numerical experiments show that, for a variety of flow problems involving moving shock and flame discontinuities, accurate and robust numerical simulations can be achieved even with a reduction in time-step size on the order of 10 or higher for grids across a single interface.