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Stochastic stability of mechanical systems
Published in W. B. Krätzig, O. T. Bruhns, H. L. Jessberger, K. Meskouris, H.-J. Niemann, G. Schmid, F. Stangenberg, A. N. Kounadis, G. I. Schuëller, Structural Dynamics, 1991
For the case of periodic variation of system parameters the well developed Floquet theory provides a mathematical concept for evaluating the stability of the system. However, there is a considerable number of applications in mechanical and structural engineering where cases of random parametric excitations (e.g. wind, earthquake loading) become increasingly important {Shih and Lin 1982, Lin and Ariaratnam 1980, Bucher and Lin 1988a). Consequently, considerable effort has been put into the development of mathematical tools for the analysis of stochastic stability (e.g. Arnold et al 1985). Those results utilize Markovian properties of the system response under consideration, i.e. (filtered) white noise is assumed to excite the system. For the case of non-white excitations only very few numerical results are available, all of them being sufficient conditions for stability (Infante 1968, Lin et al. 1986). Even for the white noise case the stability boundaries which can be computed in a straightforward way are sufficient ones in terms of e.g. second moment stability. As for as sample stability is concerned, the mathematical tools are not that well developed as yet.
Dynamic analysis for structures supported on slide-limited friction base isolation system
Published in B.F. Spencer, Y.X. Hu, Earthquake Engineering Frontiers in the New Millennium, 2017
This paper presents a new type base isolation system, i.e., Slide-limited friction base isolation system based on synthesizing the merit and demerit of P-F and R-FBI. The motion equations of S-LF are derived. Because the equations are strong nonlinear differential equations, the FGN method is used to calculate harmonic and subharmonic responses to harmonic excitation. Stability of periodic solution is investigated by using the Floquet theory. Earthquake responses of S-LF are calculated with a high precise direct integration method. The dynamical characteristics of S-LF are the following.
Examples 10.1 to 10.20
Published in L. M. B. C. Campos, Classification and Examples of Differential Equations and their Applications, 2019
*standard CCXLVI. Floquet theorem on the existence of solutions for a linear differential equation (10.560a, c) with periodic coefficients (10.560b) in the case of single or multiple eigenvalues (subsections 9.3.1–9.3.3);
Perturbation Analysis of Nonlinear Stages in Hypersonic Transition
Published in International Journal of Computational Fluid Dynamics, 2021
S. Unnikrishnan, Datta V. Gaitonde
Since the time-averaged basic-state fails to accurately capture the instability behaviour preceding transition, we now augment it with the spatio-temporal dynamics resulting from the saturated fundamental 2D wave. This results in a time-periodic basic-state, the linear analysis of which can been developed using the Floquet theory. The most natural approach, a general formulation of the corresponding linear operator for the compressible NSE in 3D curvilinear coordinates Equation (1), is very cumbersome. We circumvent this difficulty by extending the implicit-linearisation capability of MFP to time-varying basic-states. For this purpose, we reconsider the relation shown in Equation (6). Unlike MFP, the basic-state is now a function of time, which we denote as is the time-averaged mean, and is the fluctuating component, which in this case is periodic. To ease description, consider flows distorted by the primary wave alone. Therefore, where n is an integer, and is the time-period of the fundamental, with for the case under consideration. The constraining body-force now becomes a T-periodic function, , given as: which can be effectively utilised to solve the system of equations The above equation is linear in , and has known T-periodic coefficients.
Edge finite element method for periodic structures using random meshes
Published in Waves in Random and Complex Media, 2022
Ouail Ouchetto, Brahim Essakhi, Said Jai-Andaloussi, Saad Zaamoun
The Floquet theorem allows reducing the analysis of the whole periodic structure to an elementary cell with periodic boundary conditions. The Floquet theorem has been taken into account in the most used numerical methods in the electromagnetic community. In FSS, the first applications of FDTD to periodic structures were published in [11–14], where FDTD was combined with the Floquet theorem. Before 1990, the FEM was widely used in the analysis of mechanical structures and mechanical waves. The first use of the FEM in electrical engineering was done by Navarro et al. [15] and the first works on the FEM for modeling the FSS structures were reported in [16–18]. Subsequently, several other works based on the FEM and the Floquet theorem have been published, for example, [19–25]. Currently, the FSS and metamaterial science have reached a high degree of sophistication and the current interests demand increasingly complex structures. The FEM is the most adequate analysis tool for different structures, because of its ability to take into account complex inhomogeneity, to ease the electromagnetic properties manipulation, and to incorporate different boundary conditions. Finite element handling of Floquet theorem consists in imposing the periodic conditions on the elementary cell boundary. In the classical method, the opposite faces have meshed identically. We note that all existing meshing software does not have the feature to generate a periodic mesh. Besides, with complex periodic structures, this method is rather inflexible and becomes difficult to implement. It might lead to unnecessary fine elementary cell meshes or ill-conditioned FEM matrices, especially in unstructured meshing processes [26]. Therefore, the use of unconstrained meshes results in more flexible and cheaper adaptive mesh refinement procedures.