Linear Homogeneous Algebraic Equations Research Articles

Solving systems of homogeneous linear algebraic equations in pricing problems of an alternative financial model

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained by the finite integral transform (FIT) method. While the previous studies focus on the mechanical behaviors of intact plates, this study establishes an FIT-based new scheme for analyzing the free vibration characteristics of rectangular thin plates with an edge crack. The plates may have any combined conditions of simply supported, clamped, and free boundaries. In specific solution scheme, an edge-cracked plate is decomposed into four subdomains, and a double cosine FIT is imposed on the fourth-order partial differential governing equations of each subdomain, which yields the relationship between the transformed modal deflections and naturally defined unknowns. By substituting the exterior boundary conditions of the plate, continuity conditions across the subdomains, and free conditions at the crack into the inverse transforms, the natural frequencies and unknowns are successively determined via the established homogeneous linear algebraic equations. The modal deflections are provided by further substituting the obtained unknowns into the inverse transforms. Comprehensive benchmark results for the edge-cracked plates under representative boundary conditions are shown. The results are well verified by other solution methods. A parametric study quantitatively reveals the effects of boundary conditions and crack length on the natural frequencies of edge-cracked plates. Due to the rigorous but easy-to-implement mathematical derivations, this study presents a novel solid way for exploring analytical solutions of vibration problems.

INVESTIGATION OF THE NORMAL VIBRATION MODES STABILITY IN SOME ESSENTIALLY NONLINEAR SYSTEMS

In the paper stability of nonlinear normal modes is analyzed by two approaches. One of them is the method of Ince algebraization, when a new independent variable associated with the unperturbed solution is introduced in the problem. In this case equations in variations are transformed to equations with singular points. The problem of determination of solutions corresponding to boundaries of the stability/ instability regions is reduced here to the problem of determination of functions that have singularity at the mentioned points. Such solutions can be obtained in the form of power series, which coefficients are satisfying a system of homogeneous linear algebraic equations. The condition ensuring the existence non-trivial solutions for such systems determines the boundaries between the stability / instability regions in the system parameter space. An advantage of the Ince algebraization is that we do not use the time-presentation of the solution when studying its stability. Other approach to investigating steady state stability is associated with the classical Lyapunov definition of stability. The analytical-numerical test proposed in the paper can be applied to a stability problem when the problem has no analytical solution. It also allows to obtain boundaries between the stability / instability regions in the system parameter space. In the present paper the first approach is used to analyze stability of normal vibration modes in the system of connected oscillators on the essentially nonlinear elastic support, and the second one is used to analyze stability of a horizontal vibration mode in the so-called stochastic absorber.

Calculation of the circular plates’ stability in stresses

The method is proposed for solving the plates stability problems by the finite element method based on piecewise constant approximations of moments. The solution was obtained on the basis of the principles of minimum additional energy and the possible displacements. To ensure the moment fields equilibrium, the equilibrium algebraic equations of grid nodes are compiled using the possible displacements principle. Such equilibrium equations are written as a system of linear homogeneous algebraic equations. Using the Lagrange multipliers method the equilibrium algebraic equations are including to the functional. The proposed method ensures the critical stress convergence to the exact value from below, which provides reserve of the plate stability.

R-function Theory for Bending Problem of Shallow Spherical Shells with Polygonal Boundary

The governing differential equations of the bending problem of simply supported shallow spherical shells on Winkler foundation are simplified to an independent equation of radial deflection. The independent equation of radial deflection is decomposed to two Laplace operators by intermediate variable. The R-function theory is applied to describe a shallow spherical shell on Winkler foundation with concave boundary, and then a quasi-Green’s function is established by using the fundamental solution and the normalized boundary equation. The quasi-Green’s function satisfies the homogeneous boundary condition of the problem. The Laplace operators of the problem are reduced to two simultaneous Fredholm integral equations of the second kind by the Green’s formula. The singularity of the kernel of the integral equation is eliminated by choosing a suitable form of the normalized boundary equation. The integral equations are discretized into the homogeneous linear algebraic equations to proceed numerical computing. The singular term in the discrete equation is eliminated by the integral method. Some numerical examples are given to verify the validity of the proposed method in calculating simple boundary conditions and polygonal boundary conditions. A comparison with the ANSYS finite element (FEM) solution shows a good agreement, and it demonstrates the feasibility and efficiency of the present method.

A nonlinear vibration model of fiber metal laminated thin plate with amplitude dependent property

In this paper, a nonlinear vibration model of fiber metal laminated thin plate with amplitude dependent property considering the material nonlinearity is proposed. Based on the Jones-Nelson nonlinear theory and the complex modulus method, the Lagrange energy function has been derived. Then, combining the orthogonal polynomial method and the energy method, the homogeneous linear algebraic equations of the proposed modal are solved. By using the Newton-Raphson iteration method, the nonlinear natural frequencies and responses are obtained. And, the nonlinear modal damping ratios are acquired according to the complex modulus method. As an example to demonstrate the feasibility of the developed model, the experimental test of a TA2/TC500 laminated thin plate is implemented. The calculated and measured results show a good consistency.

Algebraization in Stability Problem for Stationary Waves of the Klein-Gordon Equation

Nonlinear traveling waves of the Klein-Gordon equation with cubic nonlinearity are considered. These waves are described by the nonlinear ordinary differential equation of the second order having the energy integral. Linearized equation for variation obtained for such waves is transformed to the ordinary one using separation of variables. Then so-called algebraization by Ince is used. Namely, a new independent variable associated with the solution under consideration is introduced to the equation in variations. Integral of energy for the stationary waves is used in this transformation. An advantage of this approach is that an analysis of the stability problem does no need to use the specific form of the solution under consideration. As a result of the algebraization, the equation in variations with variable in time coefficients is transformed to equation with singular points. Indices of the singularities are found. Necessary conditions of the waves stability are obtained. Solutions of the variational equation, corresponding to boundaries of the stability/instability regions in the system parameter space, are constructed in power series by the new independent variable. Infinite recurrent systems of linear homogeneous algebraic equations to determine coefficients of the series can be written. Non-trivial solutions of these systems can be obtained if their determinants are equal to zero. These determinants are calculated up to the fifth order inclusively, then relations connecting the system parameters and corresponding to boundaries of the stability/ instability regions in the system parameter place are obtained. Namely, the relation between parameters of anharmonicity and energy of the waves are constructed. Analytical results are illustrated by numerical simulation by using the Runge-Kutta procedure for some chosen parameters of the system. A correspondence of the numerical and analytical results is observed.

A novel method for the dynamic modeling of Stewart parallel mechanism

This paper presents a new scenario for the dynamic modeling of Stewart parallel mechanism based on transfer matrix method for linear multibody systems (linear MSTMM). The mechanism is modeled as a flexible multibody system, where the platform is rigid while the legs are flexible. All the element transfer equations are re-written in the same inertial coordinate system, which significantly simplifies the deduction of the system overall transfer equation. The vibration characteristics are computed by solving homogeneous linear algebraic equations, and then the augmented eigenvector and body dynamics equation are adopted to derive the state space representation by combining modal superposition method. Without deducing system global dynamics equation, the proposed scenario is easy to formulate, systematic to apply, and can model the legs as flexible bodies without spatial discretization. Then, Hankel model reduction and LQR control are employed to demonstrate how to make use of the derived state space representation. A numerical study is performed to validate the presented method. Although the system considered in this work is Stewart parallel mechanism, the scenario is general and can be used for other linear time invariant multibody systems.

Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method

The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2N–n linear homogeneous algebraic equations with 2N–n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ω k , k = 1, 2, …, 2N–n, are positive roots of the (2N–n)th-order algebraic equation D(ω2) = 0, where D is the determinant of this set. It is shown that the system of 2N–n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2N–n)th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n. When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.

Stability and Dynamic Analysis of Laminated Composite Plates

Buckling and free vibration analysis of laminated rectangular plates with uniform and non uniform distributed in-plane compressive loadings along two opposite edges is performed using the Ritz method. Classical laminated plate theory is adopted. The static component of the applied in- plane loading are assumed to vary according to uniform, parabolic or linear distributions. Initially, the plate membrane problem is solved using the Ritz method; subsequently, using Hamilton’s variational principle, linear homogeneous algebraic equations in terms of unknown are generated, the set of linear algebraic equations can be solved as an Eigen-value problem. Buckling loads for laminated plates with different combinations of boundary conditions are obtained and their effect on the natural frequencies of plate are also investigated. The proposed method is verified by comparing results to data obtained by the finite element method (FEM) using ANSYS program, from experimental program and that obtained by other researchers. Analytical results are also presented to bring out the effects of aspect ratio, boundary conditions, lamination angle, and loading type on the critical buckling load and natural frequency.
 

Natural Frequencies and Mode Shapes of Mechanically Coupled Microbeam Resonators with an Application to Micromechanical Filters

We present a methodology to calculate analytically the mode shapes and corresponding frequencies of mechanically coupled microbeam resonators. To demonstrate the methodology, we analyze a mechanical filter composed of two beams coupled by a weak beam. The boundary-value problem (BVP) for the linear vibration problem of the coupled beams depends on the number of beams and the boundary conditions of the attachment points. This implies that the system of linear homogeneous algebraic equations becomes larger as the array of resonators becomes complicated. We suggest a method to reduce the large system of equations into a smaller system. We reduce the BVP composed of five equations and twenty boundary conditions to a set of three linear homogeneous algebraic equations for three constants and the frequencies. This methodology can be simply extended to accommodate any configuration of mechanically coupled arrays. To validate our methodology, we compare our analytical results to these obtained numerically using ANSYS. We found that the agreement is excellent. We note that the weak coupling beam splits the frequency of the single resonator into two close frequencies. In addition, the effect of the coupling beam location on the natural frequencies, and hence the filter behavior, is investigated.

Fourier–Bessel theory on flow acoustics in inviscid shear pipeline fluid flow

Flow acoustics in pipeline is of considerable interest in both industrial application and scientific research. While well-known analytical solutions exist for stationary and uniform mean flow, only numerical solutions exist for shear mean flow. Based on potential theory, a general mathematical formulation of flow acoustics in inviscid fluid with shear mean flow is deduced, resulting in a set of two second-order differential equations. According to Fourier–Bessel theory which is orthogonal and complete in Lebesgue Space, a solution is proposed to transform the differential equations to linear homogeneous algebraic equations. Consequently, the axial wave number is numerically calculated due to the existence condition of non-trivial solution to homogeneous linear algebraic equations, leading to the vanishment of the corresponding determinant. Based on the proposed method, wave propagation in laminar and turbulent flow is numerically analyzed.

TE AND TM MODES OF A VANE-LOADED CIRCULAR CYLINDRICAL WAVEGUIDE FOR GYRO-TWT APPLICATIONS

The dispersion equation governing the guided propagation of TE and TM fast wave modes of a circular cylindrical waveguide loaded by metal vanes positioned symmetrically around the wave-guide axis is derived from the exact solution of a homogeneous boundary value problem for Maxwell's equations. The dispersion equation takes the form of the solvability condition for an inflnite system of linear homogeneous algebraic equations. The approximate dispersion equation corresponding to a truncation of the inflnite-order coe-cient matrix of the inflnite system of equations to the coe-cient matrix of a flnite system of equations of su-ciently high order is solved numerically to obtain the cut-ofi wave numbers of the various propagating modes. Each cut-ofi wave number gives rise to a unique dispersion curve in the shape of a hyperbola in the !-fl plane.

Perfectly conducting tape-helix model for guided electromagnetic wave propagation

The homogeneous boundary value problem arising in the propagation of electromagnetic waves guided by an open tape helix modelled to be of infinitesimal tape thickness and infinite tape-material conductivity is shown to be inherently ill posed. It is demonstrated how the ill posed problem may be regularised using the mollification method. The regularised boundary value problem is then solved to yield the approximate dispersion equation which takes the form of the solvability condition for an infinite system of linear homogeneous algebraic equations viz., the determinant of the infinite-order coefficient matrix is zero. For the numerical computation of the dispersion characteristic, all the entries of the symmetrically truncated version of the coefficient matrix are estimated by summing an adequate number of the rapidly converging (after regularisation) series for them. The tape-current distribution is estimated from the null-space vector of the truncated coefficient matrix corresponding to a specified root of the dispersion equation. A comparison of the numerical results with those for the anisotropically conducting model (that neglects the component of the tape-current density perpendicular to the winding direction) of the tape helix reveals that the propagation characteristic computed on the basis of the anisotropically conducting model could be substantially in error even for moderately wide tapes.

Buckling and Postbuckling Behavior of Cross-Ply Composite Plate Subjected to Nonuniform In-Plane Loads

This paper presents a study of buckling and postbuckling behaviour of simply supported composite plates subjected to nonuniform in-plane loading. The mathematical model is based on higher order shear deformation theory incorporating von Kármán nonlinear strain displacement relations. Because the applied in-plane edge load is nonuniform, in the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using these stress distributions, the governing equations for postbuckling analysis of composite plates are obtained through the theorem of minimum potential energy. Adopting Galerkin’s approximation, the governing nonlinear partial differential equations are reduced into a set of nonlinear algebraic equations in the case of postbuckling analysis, and homogeneous linear algebraic equations in the case of buckling analysis. The critical buckling load is obtained from the solution of associated linear eigenvalue problem. Postbuckling equilibrium paths are obtained by solving nonlinear algebraic equations employing the Newton-Raphson iterative scheme. Explicit expressions for the plate in-plane stress distributions within the prebuckling range are reported for isotropic and composite plates subjected to parabolic in-plane edge loading. Buckling loads are determined for three plate aspect ratios (a/b=0.5, 1, 1.5) and three different types of in-plane load distributions. The effect of shear deformation on the buckling loads of composite plate is reported. The present buckling results are compared with previously published results wherever possible.

Flexural buckling of laced column with serpentine lattice

Unlike the technique accepted in existing design specifications, the buckling problem of a laced column with a serpentine lattice is formulated as a stability problem of a statically indeterminate structure. The problem is reduced to a two-point boundary value problem for a system of recurrence dependences relating the deformation parameters of column cross-sections passing through the lattice joints. These relations are derived by using the initial value method for solving differential equations of column chord equilibrium. For columns with any degree of static indeterminacy, the critical force is determined as the smallest eigenvalue of the fourth-order system of homogeneous linear algebraic equations. The obtained mode shapes have the form of irregular curves with many points of inflection and disprove a concept that the stability problem of a laced column can be reduced to the analogous problem for an ‘equivalent’ continuous solid column based on Engesser's assumption. Euler critical forces calculated for a column as a statically indeterminate system are compared with the critical forces from Engesser's equivalent solid column. The phenomenon which is similar to the Boobnov effect can occur for the serpentine column: it can lose stability so that panels of one of the chords are buckled as isolated simply supported bars. This type of buckling is possible when the lattice rigidity of the column exceeds a specific limit. For columns with identical chords, the critical force is a function of the number of sub-panels and the special lattice rigidity parameter. The relationships between the critical force and the lattice rigidity parameter for columns with a varied number of sub-panels can be applied in designing steel-laced columns.

Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads

In the present paper, buckling loads of rectangular composite plates having nine sets of different boundary conditions and subjected to non-uniform inplane loading are presented considering higher order shear deformation theory (HSDT). As the applied inplane load is non-uniform, the buckling load is evaluated in two steps. In the first step the plane elasticity problem is solved to evaluate the stress distribution within the prebuckling range. Using the above stress distribution the plate buckling equations are derived from the principle of minimum total potential energy. Adopting Galerkin's approximation, the governing partial differential equations are converted into a set of homogeneous linear algebraic equations. The critical buckling load is obtained from the solution of the associated linear eigenvalue problem. The present buckling loads are compared with the published results wherever available. The buckling loads obtained from the present method for plate with various boundary conditions and subjected to non-uniform inplane loading are found to be in excellent agreement with those obtained from commercial software ANSYS. Buckling mode shapes of plate for different boundary conditions with non-uniform inplane loadings are also presented.

The Free and Forced Vibrations of a Closed Elastic Spherical Shell Fixed to an Equatorial Beam—Part I: The Governing Equations and Special Solutions

The classical theories of shells and curved beams are used to develop the equations of motion of an elastically isotropic spherical shell attached to an elastic equatorial beam of rectangular cross section. The mass densities and elasticities of the shell and beam are, in general, different. Remarkably, for the natural frequencies, the final set of eight linear homogeneous algebraic equations uncouples into two sets of four. The only approximations made are of the same order of magnitude as those inherent in classical shell and beam theory. Although solutions of the shell equations involve Legendre functions (and not polynomials), the final set of algebraic equations involve only trigonometric and gamma functions. Several special exact solutions are given. In Part II, perturbation techniques are used to find the natural frequencies for beam-shell configurations ranging from nearly pure beams to nearly pure shells.

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.