Linear Homogeneous Algebraic Equations Research Articles

On Paradoxes of the Formal Approach to Determining the Eigenvalues of One-Dimensional Boundary Value Problems

The conventional approach to determining the eigenvalues of a one-dimensional boundary value problem consists in writing out the solution of the differential equation in general form containing indeterminate coefficients and constructing a system of homogeneous linear algebraic equations for these coefficients on the basis of the expressions for the boundary conditions. The eigenvalue is determined from the condition that the determinant of the system thus constructed is zero.

Free vibration of anisotropic rectangular plates by general analytical method

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.

Thermal vibration analysis of thick laminated plates by the moving least squares differential quadrature method

The stresses and deflections in a laminated rectangular plate under thermal vibration are determined by using the moving least squares differential quadrature (MLSDQ) method based on the first order shear deformation theory. The weighting coefficients used in MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. By using this method, the governing differential equations are transformed into sets of linear homogeneous algebraic equations in terms of the displacement components at each discrete point. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Solving this set of algebraic equations yields the displacement components. Then substituting these displacements into the constitutive equation, we obtain the stresses. The approximate solutions for stress and deflection of laminated plate with cross layer under thermal load are obtained. Numerical results show that the MLSDQ method provides rapidly convergent and accurate solutions for calculating the stresses and deflections in a multi-layered plate of cross ply laminate subjected to thermal vibration of sinusoidal temperature including shear deformation with a few grid points.

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.

Vibration analysis of generally laminated composite plates by the moving least squares differential quadrature method

In this paper, a novel numerical solution technique, the moving least squares differential quadrature (MLSDQ) method is employed to study the free vibration problems of generally laminated composite plates based on the first order shear deformation theory. The weighting coefficients used in MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. By using this method, the governing differential equations are transformed into sets of linear homogeneous algebraic equations in terms of the displacement components at each discrete point. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Combining these algebraic equations yields a typical eigenvalue problem, which can be solved by a standard eigenvalue solver. Detailed formulations and implementations of this method are presented. Convergence and comparison studies are carried out to verify the reliability and accuracy of the numerical solutions. The applicability, efficiency, and simplicity of the present method are all demonstrated through solving several sample problems.

Differential cubature method for buckling analysis of arbitrary quadrilateral thick plates

In this paper, a novel numerical solution technique, the differential cubature method is employed to study the buckling problems of thick plates with arbitrary quadrilateral planforms and non-uniform boundary constraints based on the first order shear deformation theory. By using this method, the governing differential equations at each discrete point are transformed into sets of linear homogeneous algebraic equations. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Detailed formulation and implementation of this method are presented. The buckling parameters are calculated through solving a standard eigenvalue problem by subspace iterative method. Convergence and comparison studies are carried out to verify the reliability and accuracy of the numerical solutions. The applicability, efficiency, and simplicity of the present method are demonstrated through solving several sample plate buckling problems with various mixed boundary constraints. It is shown that the differential cubature method yields comparable numerical solutions with 2.77-times less degrees of freedom than the differential quadrature element method and 2-times less degrees of freedom than the energy method. Due to the lack of published solutions for buckling of thick rectangular plates with mixed edge conditions, the present solutions may serve as benchmark values for further studies in the future.

Asymptotic conditions for the electromagnetic form factors of hadrons represented by the VMD model

A system of linear homogeneous algebraic equations for coupling constant ratios of vector-mesons to hadrons is derived by imposing an asymptotic behaviuor upon the VMD pole parametrization of an hadron electromagnetic form factor. A similar system of equations with a simpler structure of coefficients, taken as even powers of vector-meson masses, is derived by means of integral superconvergent sum rules for the imaginary part of the considered from factor using its appropriate delta function approximation. Although both systems have been derived starting from different properties of the electromagnetic form factor and they look each in its own way, it is shown explicitly that they are fully equivalent.

Re-examination of the one-dimensional theory of a Raman free-electron laser

A new one-dimensional analysis of the collective interaction in a free-electron laser with combined helical wiggler and uniform axial magnetic fields is presented. Maxwell's curl relations and the cold-fluid equations are employed, with the appropriate form of solution for right and left circularly polarized electromagnetic waves and space-charge waves. A set of three linear homogeneous algebraic equations for the electric field amplitudes of the three propagating waves is derived. This set may be employed to obtain the general dispersion relation in the form of a tenth-degree polynomial equation. With the left circular wave assumed to be nonresonant, the dispersion relation reduces to a seventh-degree polynomial equation corresponding to four space-charge modes and three right circular modes. The results of a numerical study of the spatial growth rate and radiation frequency are presented.

Time-dependent rototranslational diffusion equation with multiple mixed boundary conditions: A formally exact analytical solution

A formally exact analytical solution of the time-dependent rototranslational diffusion equation (RTDE) with mixed boundary conditions applied over both the translational and rotational variables has been derived. The approximation of isotropic translational motion has been imposed and no further approximations have been used in the calculations. The solution has been obtained in the form of a set of homogeneous linear algebraic equations.

Solution of arbitrary cross-section waveguide using method of eigen weighted boundary integral equation

Based on the second kind of Green’s identity, a boundary integral equation for arbitrary cross-section waveguide is transformed to a system of linear homogeneous algebraic equations by means of expansion of boundary bases and by using the eigen functions of a fictitious regular boundary as weighting functions, which corresponds to less algebraic equations than BEM and simpler coefficients than the modified BEM. The numerical results for some typical metallic waveguides are given by using the method of eigen-weighted boundary integral equation, and they are accurate enough with fast convergence.

N-body scattering solution in coordinate space

The Schrodinger equation has been transformed into a set of coupled partial differential equations having hyper-variables as arguments and a procedure for embedding the boundary conditions into the N -body scattering solution by using a set of homogeneous linear algebraic equations is proposed.

Axisymmetrical Vibrations, of Tanks—Analytical

An analytical method for the computation of the axisymmetrical dynamic characteristics of partly‐filled cylindrical tanks is presented. The liquid is assumed to be inviscid and incompressible. The tank shell is assumed to be of constant thickness and its material to be linearly elastic. Under these assumptions, two coupled partial differential equations govern the vibrations of the shell. Because the tank is partly‐filled with liquid, two different solutions are obtained for the lower (wet) and upper (dry) portions of the shell. A system of linear homogeneous algebraic equations is obtained by satisfying the boundary conditions at the bottom and top of the tank and the compatibility equations at the junction of the wet and dry parts of the shell. The determinant of coefficients of this system leads to the frequency equation. The natural frequencies, mode shapes and stress distributions showed excellent agreement with those obtained from a numerical solution.

Free Vibrations of Ring-Stiffened Toroidal Shells

In this investigation, the free vibrations of ring-stiffene d toroidal shells are studied. The stiffening rings which are placed at meridional sections are treated as discrete members, each with different properties and unequally spaced. The shell is broken up into shell segments whose motion is studied on the basis of the Love-Reissner shell theory. Stiffness matrices for the shell segments are obtained by expanding the shell parameters in the meridional direction and integrating numerically in the circumferential direction. The stiffness matrices of the reinforcing rings are combined with those of the shell segments by imposing compatibility at the junctions. This results in a set of linear homogeneous algebraic equations which are solved by a fast converging iteration technique. A comparison of the frequencies of unstiffened toroidal shells shows good concurrence with available results. Plots are established showing the effect of the shell parameters on the frequencies of the unstiffened shells. The results indicate that the frequencies increase with increasing £ (thickness to radii ratio, h/R) and decreasing 77 (radii ratio, a/R). The five lowest frequencies and their corresponding mode shapes are determined for three different cases. When stiffening rings are added to the toroidal shells, the frequencies either remain the same or increase substantially. The changes being greater for shells with smaller rj. Moreover, the presence of additional modes is established as well as the coupling of circumferential harmonics.

Frequency Analysis of Thin-Walled Shear Walls

The coupled equations of motion for a thin-walled shear wall of monosymmetric cross section are presented based on thin-walled beam theory. An exact coupled solution is obtained by simultaneously solving a coupled polynomial and set of homogeneous linear algebraic equations, in order to obtain the natural frequencies and mode shapes. An approximate solution is obtained by using the uncoupled flexural and torsional mode shapes in applying the virtual work principle in order to obtain the appropriate eigenvalues. A comparison of the exact and approximate solutions for a single shear wall of E-shaped cross section indicates extremely close agreement for both frequencies and mode shapes. The approximate solution requires considerably less computer time than the exact solution and consequently provides a valuable alternative.

Critical Shear Loading of Curved Panels of Corrugated Sheets

The problem of buckling of curved shear panels of corrugated sheets, which is encountered in the design of long shells is examined theoretically. Overall panel buckling, and not local buckling, is analyzed. Thus, the corrugated sheets are treated as orthotropic curved plates. Equilibrium conditions, as well as geometric and elastic relations of a deformed infinitesimal element, are governed by two linear simultaneous coupled differential equations in deflection w and stress function F . Series presentation for w and F yielded a set of homogeneous linear algebraic equations in the unknown deflection coefficients. A nontrivial solution of these equations is possible provided that the determinant of their matrix is equal to zero. The eigenvalues of the shear load are calculated and only the lower value is considered. The few experiments available show reasonable agreement with the theoretically obtained ones. For practical use, curves are given for the calculation of critical shear load.