Spectrum of a network of Euler-Bernoulli beams

A network of N flexible beams connected by n vibrating point masses is considered. The spectrum of the spatial operator involved in this evolution problem is studied. If λ² is any real number outside a discrete set of values S and if λ is an eigenvalue, then it satisfies a characteristic equation which is given. The associated eigenvectors are also characterized. If λ² lies in S and if the N beams are identical (same mechanical properties), another characteristic equation is available. It is not the case for different beams: no general result can be stated. Some numerical examples and counterexamples are given to illustrate the impossibility of such a generalization. At last the asymptotic behaviour of the eigenvalues is investigated by proving the so-called Weyl's formula.     

Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations

We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A₁, A₁⊕A₂, 3A₁, or A3,3⊕A₁. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕A₁ corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250].     

Controllability of a multichannel system

We consider the system consisting of K coupled acoustic channels with the different sound velocities $c_j$. Channels are interacting at any point via the pressure and its time derivatives. Using the moment approach and the theory of exponential families with vector coefficients we establish two controllability results: the system is exactly controllable if(i) the control $u_j$ in the jth channel acts longer than the double travel time of a wave from the start to the end of the j-th channel;(ii) all controls $u_j$ act more than or equal to the maximal double travel time.     

Discrete Ingham type inequalities and simultaneous observability of strings or beams

Harmonic analysis has been an efficient tool in control theory for a long time, see, e.g., Russell [D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev. 20 (1978) 639-739] and its numerous references. Here we establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened gap condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams.     

Numerical analysis of a contact problem between two thermoviscoelastic beams

We propose and analyze in this article a finite element approximation, based on a penalty formulation, to a quasi‐static unilateral contact problem between two thermoviscoelastic beams. An error bound is given and some numerical experiments discussed. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 644–661, 2011     

具有耗散接头的三维欧拉─伯努利杆系的特征频率

本文计算了具有耗散接头的一般非共线的欧拉─伯努利或铁木辛柯杆系结构振动特征频率的传递矩阵．将允许结构是三维的，因此，一定要同时研究各种振动模式，包括纵向和扭转的振动。这种结构一般可以看作是由许多杆件首尾相接构成一条链条。允许有各种不同的减振器，甚至有些减振器是本结构内部形成的．我们也允许在结构中采用不同材料，杆件也可以是不同宽度。本文证明了可以利用渐近估计方法来求得近似的特征频率。     

A generating function for a class of generalized Bernoulli polynomials

First we derive a generating function and a Fourier expansion for a class of generalized Bernoulli polynomials. Then we derive formulas that allow certain Dirichlet series to be evaluated in terms of these generalized Bernoulli polynomials.      

A note on rearrangement of Fourier series

We give an example of a permutation of integers which preserves all convergent Fourier series and makes some divergent Fourier series converge.     

Convergence of Double Fourier Series after a Change of Variable

In this paper, we prove that for any compact set there exists a homeomorphism of the closed interval such that for an arbitrary function f the Fourier series of the function F(x,y) = f((x),(y)) converges uniformly on simultaneously over rectangles, over spheres, and over triangles.     

Euler-Bernoulli梁的非同位控制

对一端固定,一端加剪切力反馈的Euler-Bernoulli梁,运用Legendre谱方法对一个非同位控制系统进行研究,得到了最优反馈增益系数和系统衰减率.结果表明这样的非同位控制系统可以有效的增大系统衰减率,使系统具有更好的稳定性.同时指出所研究的系统是极小相位的.     

Generalized absolute Cesaro summability of a Fourier series

In an attempt to study the scope of a theorem due to Pati, the authors have established that φ(t) logK|t∈B _u V in (0,π)⟹ΣA _n (x) is |C, 0,β| forβ>1, at the pointt = x.     

Analysis of subdivision schemes for nets of functions by proximity and controllability

In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C¹ limits with approximation order 2.     

Riesz basis property, exponential stability of variable coefficient Euler-Bernoulli beams with indefinite damping

We study damped Euler–Bernoulli beams that have nonuniformthickness or density. These nonuniformfeatures result in variablecoefficient beam equations. We prove that despite the nonuniformfeatures, the eigenfunctions of the beam form a Riesz basisand asymptotic behaviour of the beam system can be deduced withoutany restrictions on the sign of the damping. We also providean answer to the frequently asked question on damping: ‘howmuch more positive than negative should the damping be withoutdisrupting the exponential stability?’, and result ina criterion condition which ensures that the system is exponentiallystable.     

Singular Optimal Control for a Transport-Diffusion Equation

In this paper we consider a transport-diffusion equation with coefficient of diffusion ε > 0 small and coefficient of transport M(x, t). We study the asymptotic behavior of the cost of the null controllability of such a system when ε 0⁺.

If at least one trajectory associated to M(x, t) does not enter the control zone, we prove that this cost explodes exponentially as ε 0⁺. On the other hand, as long as trajectories reach the control region and the controllability time is sufficiently large, we prove that the cost is bounded as ε 0⁺, and moreover decays exponentially as ε 0⁺ as soon as all trajectories cross the boundary.     

L p bounds for a maximal dyadic sum operator

The authors prove L ^p bounds in the range 1<p< for a maximal dyadic sum operator on R ⁿ . This maximal operator provides a discrete multidimensional model of Carlesons operator. Its boundedness is obtained by a simple twist of the proof of Carlesons theorem given by Lacey and Thiele [7] adapted in higher dimensions [9]. In dimension one, the L ^p boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunts extension [4] of Carlesons theorem on almost everywhere convergence of Fourier integrals. Mathematics Subject Classification (2000):Primary 42A20, Secondary 42A24Grafakos is supported by the NSF. Tao is a Clay Prize Fellow and is supported by a grant from the Packard Foundation.     

On a decay rate for nonlinear extensible viscoelastic beams with history setting

In this paper we deal with a nonlinear extensible viscoelastic beam model whose memory term is considered in a history setting. The goal is to extend an approach on stability first provided by Guesmia and Messaoudi (J Math Anal Appl. 2014;416:212–228) to a class of viscoelastic beams/plates with nonlinear extensible and source terms. Our stability result contributes in clarifying how the constants appearing in the decay rate depend upon the nonlinearities and the size of initial data. Thus, it also complements some results dealing with this methodology.     

一类数项级数和的表示

求出函数f(x)=xk的Fourier系数并将其代人Parseval等式,继而利用第二数学归纳法可证明:数项级数∞∑n=1 1/n2k的和能够表示为π2k/dk的形式.其中对于任意确定的k值.dk以为一常数.证明过程同时给出了求解dk的方法.     

Mixed linear complementary formulation of lateral contacts among a system of parallel beams

A novel linear complementary formulation is developed in this paper to deal with multiple lateral contacts at a number of cross sections for a structure consisting of many parallel beams and having rigid body displacements. The proposed method can be used to handle a challenging situation where the number of potential contact pairs exceeds the total number of available lateral displacements at a cross section. The traditional displacement-based contact algorithms cannot be employed to solve this type of contact problem. In this paper, a mixed force–displacement scheme is developed. This scheme removes the restriction on the number of contact pairs at a cross section and is suitable for handling contacts among many parallel beams. It is proven that a unique solution to the linear complementary equations obtained using the mixed force–displacement scheme exists and can be obtained using the Lemke algorithm. Numerical results for an example application show that the scheme is indeed convergent and accurate.     

Utilization of characteristic polynomials in vibration analysis of non-uniform beams under a moving mass excitation

Vibration of non-uniform beams with different boundary conditions subjected to a moving mass is investigated. The beam is modeled using Euler–Bernoulli beam theory. Applying the method of eigenfunction expansion, equation of motion has been transformed into a number of coupled linear time-varying ordinary differential equations. In non-uniform beams, the exact vibration functions do not exist and in order to solve these equations using eigenfunction expansion method, an adequate set of functions must be selected as the assumed vibration modes. A set of polynomial functions called as beam characteristic polynomials, which is constructed by considering beam boundary conditions, have been used along with the vibration functions of the equivalent uniform beam with similar boundary conditions, as the assumed vibration functions. Orthogonal polynomials which are generated by utilizing a Gram–Schmidt process are also used, and results of their application show no advantage over the set of simple non-orthogonal polynomials. In the numerical examples, both natural frequencies and forced vibration of three different non-uniform beams with different shapes and boundary conditions are scrutinized.     

Approximation of the Singularities of a Bounded Function by the Partial Sums of Its Differentiated Fourier Series

In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those formulas.