Linear Conservative Systems Research Articles

Theoretical analysis of linear oscillations of bubbles in a polydisperse cluster

The dynamics of a single air bubble in an unbounded liquid are significantly different from the dynamics of an individual bubble in clusters due to the hydrodynamic interaction between the bubbles. Studying the mechanism of this interaction is one of the important aspects in the study of the fundamental nature of acoustic and hydrodynamic cavitation. In this work, to analyze small oscillations of bubbles in a spherical cluster near a stable equilibrium position, the mathematical theory of a linear conservative system with several degrees of freedom is applied to explain the mechanism of interaction between bubbles of different sizes. Using this theory, in the general case, it has been proven that the number of resonance frequencies in a polydisperse cluster coincides with the number of fractions. It is shown that in the regions of the main resonance (at low frequencies) bubbles of different fractions oscillate in phase, and in the regions of secondary resonances (at high frequencies) the phases successively change to the opposite, starting with the fraction containing bubbles of the largest radius, and further changing in order of decreasing. Using the example of a two-fraction cluster, it was found that there is an inertial connection between the bubbles, but there is no force connection; when the number of bubbles of one of the fractions is small, the connection between them and the bubbles of the other fraction is weak, while the interaction between them can be strong. An analysis of the energy transfer between bubbles of different fractions showed that the change in the nature of bubbles oscillations in the fraction with a small radius, while the nature of vibration of bubbles in the other fraction does not change, is the result of dynamic damping.

A simple proof of the Greenlee’s stability theorem for gyroscopic systems

We give a short, elementary proof of the Greenlee’s stability theorem for linear conservative gyroscopic systems. The proof is based on the direct method of Lyapunov.

Моделювання коливальних процесів віброударних систем

In order to improve the efficiency of methods and means of mathematical modeling of vibro-impact systems, a generalized function of the periodic mode of movement of the executive body has been developed. It is presented in the form of the dependence of the shock impulse on the ratio of the angular velocities of the linear conservative system and its own. When obtaining this function, the Heaviside integral jump function and the periodic Green's function were used. The function of the dependence of the oscillation frequency on the impact impulse is determined from the impact conditions for the function of the system's response to a periodic sequence of impulses. The design model of a vibroimpact system is considered, both with one impact element and a motion limiter, and with a double-sided impact pair with alternate impact interactions with the limiters. In the intervals between impacts, there is a linear force interaction. When developing the mathematical model, a stereomechanical impact model was used, which is characterized by the velocity recovery coefficient after the impact. The analysis of the dependence function of the oscillation frequency on the shock impulse made it possible to obtain skeletal diagrams of resonant and quasi-resonant oscillations of vibro-impact systems with one and many degrees of freedom. Based on the obtained phase diagrams of the state of vibro-impact systems, it was determined: in a system with a gap, an increase in the impact speed increases the oscillation frequency, and the vibro-impact nonlinearity is «hard»; in a system with tension, with an increase in the value of the shock impulse, the oscillation frequency decreases (nonlinearity is «soft»). In the absence of a gap, the system is isochronous. Depending on the initial energy reserve and the location of the limiters in an asymmetric oscillatory system, with one degree of freedom, there can be vibro-impact modes with both one (closer located) and both limiters. In a linear conservative system with several degrees of freedom, a single-impact T-periodic regime is realized. If the dissipation during motion and impact is very small, then a regime close to resonant can exist in the system. In this case, periodic oscillations are supported by a weak external periodic force. The developed mathematical model makes it possible to fully describe the process of changing the relative coordinate of the movement of the working body, both in transient and in the established modes of movement of the system.

Averaging and mixing for stochastic perturbations of linear conservative systems

We study stochastic perturbations of linear systems of the form \begin{equation} dv(t)+Av(t) dt =\varepsilon P(v(t)) dt+\sqrt{\varepsilon} \mathcal{B}(v(t)) dW (t), \qquad v\in\mathbb{R}^D, \tag{*} \end{equation} where $A$ is a linear operator with non-zero imaginary spectrum. It is assumed that the vector field $P(v)$ and the matrix function $\mathcal{B}(v)$ are locally Lipschitz with at most polynomial growth at infinity, that the equation is well-posed and a few of first moments of the norms of solutions $v(t)$ are bounded uniformly in $\varepsilon$. We use Khasminski's approach to stochastic averaging to show that, as $\varepsilon\to0$, a solution $v(t)$, written in the interaction representation in terms of the operator $A$, for $0\leqslant t\leqslantConst\cdot\varepsilon^{-1}$ converges in distribution to a solution of an effective equation. The latter is obtained from $(*)$ by means of certain averaging. Assuming that equation $(*)$ and/or the effective equation are mixing, we examine this convergence further. Bibliography: 27 titles.

Averaging and mixing for stochastic perturbations of linear conservative systems

We study stochastic perturbations of linear systems of the form \begin{equation*} dv(t)+Av(t) dt=\varepsilon P(v(t)) dt+\sqrt{\varepsilon} \mathcal{B}(v(t)) dW (t), \qquad v\in\mathbb{R}^D, \tag{*} \end{equation*} where $A$ is a linear operator with non-zero imaginary spectrum. It is assumed that the vector field $P(v)$ and the matrix function $\mathcal{B}(v)$ are locally Lipschitz with at most polynomial growth at infinity, that the equation is well posed and a few of first moments of the norms of solutions $v(t)$ are bounded uniformly in $\varepsilon$. We use Khasminski's approach to stochastic averaging to show that, as $\varepsilon\to0$, a solution $v(t)$, written in the interaction representation in terms of the operator $A$, for $0\leqslant t\leqslantConst\cdot\varepsilon^{-1}$ converges in distribution to a solution of an effective equation. The latter is obtained from $(*)$ by means of certain averaging. Assuming that equation $(*)$ and/or the effective equation are mixing, we examine this convergence further. Bibliography: 27 titles.

A note on the Lyapunov stability of linear conservative gyroscopic systems

A note on the Lyapunov stability of linear conservative gyroscopic systems

Convergent term of the Galerkin truncation for dynamic response of sandwich beams on nonlinear foundations

Dynamic response of beams on elastic foundations is of great theoretical significance in the field of engineering. Galerkin truncation method is the most commonly used powerful method to study the vibration of these structures. The number of truncation terms affects the convergence of the analysis and determines the accuracy of the dynamic responses. However, there is still no basis for how to choose the appropriate number of truncation terms. In this paper, two effective schemes based on the natural frequency of the corresponding linear conservative system are proposed to determine the convergent terms of the Galerkin truncation. Since determining the natural frequencies of the corresponding linear conservative system are much easier, the convergence determination method is convenient for application. Based on the dynamic response of the sandwich beam on nonlinear foundations under a moving load, the feasibility of the two schemes for determining the convergent term is confirmed. Moreover, the effects of system parameters on the two schemes are presented. In short, this paper provides two simple and easy schemes for selecting the Galerkin truncation terms in the research of the dynamic response of elastic structures resting on elastic foundations.

System theory and orthogonal multi-wavelets

In this paper we provide a complete and unifying characterization of compactly supported univariate scalar orthogonal wavelets and vector-valued or matrix-valued orthogonal multi-wavelets. This characterization is based on classical results from system theory and basic linear algebra. In particular, we show that the corresponding wavelet and multi-wavelet masks are identified with a transfer function F(z)=A+Bz(I−Dz)−1C,z∈D={z∈C:|z|<1},of a conservative linear system. The complex matrices A,B,C,D define a block circulant unitary matrix. Our results show that there are no intrinsic differences between the elegant wavelet construction by Daubechies or any other construction of vector-valued or matrix-valued multi-wavelets. The structure of the unitary matrix defined by A,B,C,D allows us to parametrize in a systematic way all classes of possible wavelet and multi-wavelet masks together with the masks of the corresponding refinable functions.

Asynchronous modes of vibrations in linear conservative systems: an illustrative discussion of plane framed structures

The theme of asynchronous modes of vibration is recast in this paper to address the case of linear conservative discrete systems. In fact, it is known that linear non-conservative discrete systems such as those characterized by a non-symmetric stiffness matrix may undergo asynchronous free vibrations. Yet, non-conservativeness is not mandatory for the occurrence of asynchronous modes, as it is illustrated in this investigation of simple plane frames, provided the system parameters are properly chosen. Firstly, a one-storey-frame model, with consistent-mass and stiffness matrices, is studied and conditions for the occurrence of asynchronous modes are established. Next, a three-storey shear building is modelled, with lumped-mass matrix, and isolated asynchronous modes are not observed. However, non-isolated asynchronous modes are seen to exist, including a case in which the first floor is at rest. In this scenario, the upper floors play the role of vibration controllers, sparing the lower columns from oscillating. This might be welcome from the architectural viewpoint, so that pilotis would not be a priori ruled out in case of earthquake-borne vibration. Usually the participating masses of the asynchronous modes are low (and even null). To maximize the asynchronous-mode participating mass, a final model of the three-storey frame is discussed, this time taking into account the flexural stiffness of both the columns and the beams, although still using lumped-mass matrix. In this case, it is seen that much larger participating masses can be obtained for the asynchronous mode. Since asynchronous modes are associated with the vibration localization phenomenon, they can be potentially explored in the design of vibration controllers or, alternatively, energy harvesting systems.

Uniform polynomial observability of time-discrete conservative linear systems

In this paper we study time semi-discrete approximations of a class of polynomially observable infinite dimensional systems. By using a method based on the resolvent estimate, we derive uniform polynomial observability inequalities within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. We also present an application of our result to stabilization problems.

Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-Hamiltonian formulation

This paper introduces linear and nonlinear damping models, which preserve the eigenspaces of conservative linear mechanical problems. After some recalls on the finite dimensional case and on Caughey's linear dampings, an extension to a nonlinear class is introduced. These results are recast in the port-Hamiltonian framework and generalized to infinite dimensional systems. They are applied to an Euler-Bernoulli beam, excited by a distributed force. Simulations yield sounds of xylophone, glockenspiel (etc) and some interpolations for nonlinear dampings.

Transmutation techniques and observability for time-discrete approximation schemes of conservative systems

In this article, we consider abstract linear conservative systems and their time-discrete counterparts. Our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time-discrete one. As an application, we show how observability properties for the time continuous model yield uniform (with respect to the time-step) observability results for its time-discrete approximation counterparts, provided the initial data are suitably filtered. The main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time-discrete models. Besides, using a reverse representation formula, we also prove that this estimate on the time of uniform observability for the time-discrete models is sharp. We then conclude with some general comments and open problems.

A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems

For conservative mechanical systems, the so-called Caughey series are known to define the class of damping matrices that preserve eigenspaces. In particular, for finite-dimensional systems, these matrices prove to be a polynomial of one reduced matrix, which depends on the mass and stiffness matrices. Damping is ensured whatever the eigenvalues of the conservative problem if and only if the polynomial is positive for positive scalar values.This paper first recasts this result in the port-Hamiltonian framework by introducing a port variable corresponding to internal energy dissipation (resistive element). Moreover, this formalism naturally allows to cope with systems including gyroscopic effects (gyrators).Second, generalizations to the infinite-dimensional case are considered. They consist of extending the previous polynomial class to rational functions and more general functions of operators (instead of matrices), once the appropriate functional framework has been defined. In this case, the resistive element is modelled by a given static operator, such as an elliptic PDE. These results are illustrated on several PDE examples: the Webster horn equation, the Bernoulli beam equation; the damping models under consideration are fluid, structural, rational and generalized fractional Laplacian or bi-Laplacian.

A NEW EXPLICIT TIME INTEGRATION METHOD FOR STRUCTURAL DYNAMICS

A family of new explicit time-integration method is proposed herein, which inherits the numerical characteristics of any existing implicit Runge–Kutta algorithms for a linear conservative system. Based on an exact derivation of the increment of mechanical energy, the method proposed is demonstrated to be unconditionally stable. Also, the stability condition of the proposed method is derived when applied to solving a nonlinear system. The characteristics of the proposed method are investigated by observing the mechanical-energy time history of a nonlinear conservative system. The numerical results can be explained by the stability condition derived in the nonlinear regime. Finally, the computational accuracy and efficiency between the Newmark time integration method and the proposed explicit method are compared in solving the dynamic response of a couple of linear oscillators.

On Integral Transforms and Matrix Functions

The Sumudu transform of certain elementary matrix functions is obtained. These transforms are then used to solve the differential equation of a general linear conservative vibration system, a vibrating system with a special type of viscous damping.

A new measure of efficiency for model reduction: Application to a vibroimpact system

The aim of this paper is to propose a new way to measure the efficiency of the proper orthogonal decomposition (POD) to construct a reduced-order model in Structural Dynamics. It investigates the efficiency of three reduced-order models for a vibroimpact problem: (1) PODdir-basis, which is the basis constructed using the direct method of the proper orthogonal decomposition; (2) PODsnap-basis, which is the basis constructed using the snapshot method of the POD; and (3) LIN-basis, which is the basis composed by the normal modes of the associated linear (LIN) conservative system. The efficiency is measured in terms of (1) number of elements to represent the dynamics with a given precision and (2) computational cost to simulate the time response within a given precision.

Stability, resonance and Lyapunov inequalities for periodic conservative systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.

Conservative realizations of the functions associated with Schur's algorithm for the Schur class operator-valued function

Let M and N be separable Hilbert spaces and let Θ(λ) be a function from the Schur class S(M,N) of contractive functions holomorphic on the unit disk. The operator generalization of the classical Schur algorithm associates with Θ the sequence of contractions (the Schur parameters of Θ ) Γ0 =Θ(0) ∈ L(M,N), Γn ∈ L(DΓn−1 ,DΓ∗n−1 ) and the sequence of functions Θ0 =Θ , Θn ∈ S(DΓn ,DΓ∗n ) n = 1, . . . (the Schur iterates of Θ ) connected by the relations Γn = Θn(0), Θn(λ) = Γn +λDΓ∗nΘn+1(λ)(I +λΓ ∗ nΘn+1(λ)) DΓn , |λ | < 1. It is well known that the function Θ(λ) ∈ S(M,N) can be realized as the transfer function Θ(λ) = D+λC(I −λA)−1B of a linear conservative and simple discrete time-invariant system τ = {[ D C B A ]

Uniformly exponentially stable approximations for a class of damped systems

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi-discretizations to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a suitable CFL type condition on the time and space discretization parameters.

On the observability of time-discrete conservative linear systems

We consider various time discretization schemes of abstract conservative evolution equations of the form z ˙ = A z , where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair ( A , B ) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by Burq and Zworski in [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2) (2004) 443–471 (electronic)]. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations.