Definition of a linear equivalent model for a non-linear system with impacts

Modal characteristics of non-linear system are typically studied through response to harmonic excitation and using various definitions of non-linear modes. However, few results are available for systems under broadband excitation. The end objective sought here is to generate a linear system, in some sense equivalent to the non-linear system, whose modal characteristics evolve with a level of non-linearity. The considered application is the contact non-linearity found between the tubes of heat exchangers and their support plates. Such tubes, present in nuclear plants, participate to the nuclear safety and can be significantly excited by the fluid flow, so that their dynamic behavior is critical. The turbulent nature of the flow implies broadband excitation and the small gaps between the tubes and the support plate generate very significant non-linear behavior. The proposed equivalent linear system is based on a bilateral contact law whose stiffness and damping characteristics evolve with the amplitude of excitation. A non-linear model is first validated by correlation with experiments. It is then shown that three different indicators (bandwidth of main resonance, operational modal analysis of non-linear power spectral density and correlation of operational deflection shapes) lead to similar values of contact stiffness and damping in the equivalent linear model. This model is hus shown to be a very efficient tool to analyze the impact of the amplitude dependence of the non-linear behavior in the considered system

Extensions of Representable Positive Linear Functionals to Unitized Quasi *-Algebras

It is known that, under certain conditions, *-representability and extensibility to the unitized *-algebra of a positive linear functional, defined on a *-algebra without unit, are equivalent. In this paper, a new condition for an analogous result is given for the case of a hermitian linear functional defined on a quasi *-algebra (\A,\Ao) without unit.Comment: 15 pages, presented during the "Ninth Advanced Course in Operator Theory and Complex Analysis" Sevilla, 12 June 201

Random excitation of a system with bilinear hysteresis

An analysis is made of the response of a system with bilinear hysteresis to random excitation. It is shown that for moderately large inputs, the additional damping created by the bilinear hysteresis decreases the mean squared deflection compared with that for a linear system with the same viscous damping. However, for large inputs, the decrease in the stiffness of the system due to the bilinear hysteresis causes the mean squared deflection to increase over that for the equivalent linear system

Preconditioning and convergence in the right norm

The convergence of numerical approximations to the solutions of differential equations is a key aspect of Numerical Analysis and Scientific Computing. Iterative solution methods for the systems of linear(ised) equations which often result are also underpinned by analyses of convergence. In the function space setting, it is widely appreciated that there are appropriate ways in which to assess convergence and it is well-known that different norms are not equivalent. In the finite dimensional linear algebra setting, however, all norms are equivalent and little attention is often payed to the norms used. In this paper, we highlight this consideration in the context of preconditioning for minimum residual methods (MINRES and GMRES/GCR/ORTHOMIN) and argue that even in the linear algebra setting there is a ‘right’ norm in which to consider convergence: stopping an iteration which is rapidly converging in an irrelevant or highly scaled norm at some tolerance level may still give a poor answer

An efficient numerical algorithm for the transient analysis of high-frequency non-linear circuits

The paper proposes a new approach for the discrete-time integration of non-linear differential equations that describe the behaviour of high-frequency circuits, in particular those containing complex equivalent-circuit models of microwave transistor devices. The proposed approach reformulates a conventional predictor-corrector method in terms of Pad�© approximates about each function sample. The method is especially suited to the kind of non-linear stiff differential equations that arise frequently in high-frequency analysis

Adaptive Uzawa algorithm for the Stokes equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement

Derivation of equivalent linear properties of Bouc-Wen hysteretic systems for seismic response spectrum analysis via statistical linearization

A newly proposed statistical linearization based formulation is used to derive effective linear properties (ELPs), namely damping ratio and natural frequency, for stochastically excited hysteretic oscillatorsinvolving the Bouc-Wen force-deformation phenomenological model. This is achieved by first using a frequency domain statistical linearization step to substitute a Bouc-Wen oscillator by a third order linear system. Next, this third order linear system is reduced to a second order linear oscillator characterized by a set of ELPs by enforcing equality of certain response statistics of the two linear systems. The proposed formulation is utilized in conjunction with quasi-stationary stochastic processes compatible with elastic response spectra commonly used to represent the input seismic action in earthquake resistant design of structures. Then, the derived ELPs are used to estimate the peak response of Bouc-Wen hysteretic oscillators without numerical integration of the nonlinear equation of motion; this is done in the context of linear response spectrum-based dynamic analysis. Numerical results pertaining to the elastic response spectrum of the current European aseismic code provisions (EC8) are presented to demonstrate the usefulness of the proposed approach. These results are supported by pertinent Monte Carlo simulations involving an ensemble of non-stationary EC8 spectrum compatible accelerograms. The proposed approach can hopefully be an effective tool in the preliminary aseismic design stages of yielding structures and structural members commonly represented by the Bouc-Wen hysteretic model within either a force-based or a displacement-based context