2,200 research outputs found
Transcendental eigenvalue problems associated with vibration, buckling and control
The static and dynamic analysis of structures requires us to obtain solutions of their espective governing differential equations subject to appropriate boundary conditions. The dynamic analysis of non-uniform continuous structures is of primary interest, as most traditional methods take the help of discrete models to analyze them. Well established discrete model methods lead to an algebraic eigenvalue problem, the characteristic equation associated with which is a polynomial. The spectral characteristics of a continuous system nevertheless are represented by transcendental functions and cannot be approximated by polynomials efficiently. Hence finite dimensional discrete models are not capable of predicting the response of continuous systems irrespective of the model order used. In this research, a new low order analytical model is developed which approximates the dynamic behavior of the continuous system accurately. The idea here is to replace a non-uniform continuous system by a set of continuous system with piecewise constant physical properties. Such approximations lead to a transcendental eigenvalue problem, i.e. a problem with transcendental characteristic equation. A numerical method has been developed to solve such eigenvalue problems. The spectrum of non-uniform rods and beams are approximated with fair accuracy by solving the corresponding transcendental eigenvalue problem. This mathematical model is extended to reconstruct non-uniform rods and beams using a linear polynomial approximation of piecewise area. A piecewise tapered approximation of the physical parameters in non-uniform rods and beams leads to better accuracy in the solution. The ability to use higher order area functions as basic building blocks profoundly reduces the model order when using the mathematical model to analyze complex geometries. To further study the impact of this method in various problems of engineering the buckling of thin rectangular plates with stepped thickness has been analyzed and compared with the finite element solution. The transcendental eigenvalue method leads to the reduction in matrix sizes when compared with discrete model methods, thus making the solution computationally viable. Finally the transcendental eigenvalue problem associated with the active control of vibration in discrete mass-spring-damper systems has been developed and the proposed mathematical method has been applied
The transcendental eigenvalue problem and its application in system identification
An accurate mathematical model is needed to solve direct and inverse problems related to engineering analysis and design. Inverse problems of identifying the physical parameters of a non-uniform continuous system based on the spectral data are still unsolved. Traditional methods, for the system identification purpose, describe the continuous structure by a certain discrete model. In dynamic analysis, finite element or finite difference approximation methods are frequently used and they lead to an algebraic eigenvalue problem. The characteristic equation associated with the algebraic eigenvalue problem is a polynomial. Whereas, the spectral characteristic of a continuous system is represented by certain transcendental function, thus it cannot be approximated by the polynomials efficiently. Hence, finite dimensional discrete models are not capable of identifying the physical parameters accurately regardless of the model order used. In this research, a new low order analytical model is developed, which approximates the dynamic behavior of the continuous system accurately and solves the associated inverse problem. The main idea here is to replace the continuous system with variable physical parameters by another continuous system with piecewise uniform physical properties. Such approximations lead to transcendental eigenvalue problems with transcendental matrix elements. Numerical methods are developed to solve such eigenvalue problems. The spectrum of non-uniform rods and beams are approximated with fair accuracy by solving associated transcendental eigenvalue problems. This mathematical model is extended to reconstruct the physical parameters of the non-uniform rods and beams. There is no unique solution for the inverse problem associated with the continuous system. However, based on several observations a conjecture is established by which the solution, that satisfies the given data by its lowest spectrum, is considered the unique solution. Physical parameters of non-uniform rods and beams were identified using the appropriate spectral data. Modal analysis experiments are conducted to obtain the spectrum of the realistic structure. The parameter estimation technique is validated by using the experimental data of a piecewise beam. Besides the applications in system identification of rods and beams, this mathematical model can be used in other areas of engineering such as vibration control and damage detection
Null boundary controllability of a 1-dimensional heat equation with an internal point mass
We consider a linear hybrid system composed by two rods of equal length
connected by a point mass. We show that the system is null controllable with
Dirichlet and Neumann controls. The results are based on a careful spectral
spectral analysis together with the moment method.Comment: 12 pages, typos corrected, added references, matches version to be
submitted to Systems and Control Letter
Observation of vibrating systems at different time instants
In this paper, we obtain new observability inequalities for the vibrating
string. This work was motivated by a recent paper by A. Szij\'art\'o and J.
Heged\H{u}s in which the authors ask the question of determining the initial
data by only knowing the position of the string at two distinct time instants.
The choice of the observation instants is crucial and the estimations rely on
the Fourier series expansion of the solutions and results of Diophantine
approximation.Comment: 14 page
Revisiting Schrodinger's fourth-order, real-valued wave equation and its implications to energy levels
In his seminal part IV, Ann. der Phys. Vol 81, 1926 paper, Schrodinger has
developed a clear understanding about the wave equation that produces the
correct quadratic dispersion relation for matter-waves and he first presents a
real-valued wave equation that is 4th-order in space and 2nd-order in time. In
view of the mathematical difficulties associated with the eigenvalue analysis
of a 4th-order, differential equation in association with the structure of the
Hamilton-Jacobi equation, Schrodinger splits the 4th-order real operator into
the product of two, 2nd-order, conjugate complex operators and retains only one
of the two complex operators to construct his iconic 2nd-order, complex-valued
wave equation. In this paper we show that Schrodinger's original 4th-order,
real-valued wave equation is a stiffer equation that produces higher energy
levels than his 2nd-order, complex-valued wave equation that predicted with
remarkable success the visible energy levels observed in the visible atomic
line-spectra of the chemical elements. Accordingly, the 4th-order, real-valued
wave equation is too stiff to predict the emitted energy levels from the
electrons of the chemical elements; therefore, the paper concludes that Quantum
Mechanics can only be described with the less stiff, 2nd-order complex-valued
wave equation; unless in addition to the emitted visible energy there is also
dark energy emitted.Comment: 22 pages, 3 figure
Finite element formulation for free vibration of composite beams
Investigations concerning the dynamic response of composite beams with partial interaction are scarce. Derivation of the differential equations describing the interaction between composite elements normally involves with the solution of high order system of equations, which closed form solutions are difficult. This study concerns with the finite element formulation of composite beams for free vibration. The formulation involves with the establishment of the stiffness matrix and the mass matrix of the beam. The former was obtained through extremization of the total potential energy (or Hamilton principle for dynamic) whilst the latter was obtained by lumping the elements mass at nodes. Natural frequencies of the beam were obtained as eigenvalues. These were then verified by existing analytical solution
Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics
We present a mass lumping approach based on an isogeometric Petrov-Galerkin
method that preserves higher-order spatial accuracy in explicit dynamics
calculations irrespective of the polynomial degree of the spline approximation.
To discretize the test function space, our method uses an approximate dual
basis, whose functions are smooth, have local support and satisfy approximate
bi-orthogonality with respect to a trial space of B-splines. The resulting mass
matrix is ``close'' to the identity matrix. Specifically, a lumped version of
this mass matrix preserves all relevant polynomials when utilized in a Galerkin
projection. Consequently, the mass matrix can be lumped (via row-sum lumping)
without compromising spatial accuracy in explicit dynamics calculations. We
address the imposition of Dirichlet boundary conditions and the preservation of
approximate bi-orthogonality under geometric mappings. In addition, we
establish a link between the exact dual and approximate dual basis functions
via an iterative algorithm that improves the approximate dual basis towards
exact bi-orthogonality. We demonstrate the performance of our higher-order
accurate mass lumping approach via convergence studies and spectral analyses of
discretized beam, plate and shell models
Double scale analysis of periodic solutions of some non linear vibrating systems
We consider {\it small solutions} of a vibrating system with smooth
non-linearities for which we provide an approximate solution by using a double
scale analysis; a rigorous proof of convergence of a double scale expansion is
included; for the forced response, a stability result is needed in order to
prove convergence in a neighbourhood of a primary resonance.Comment: 36 page
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