An Approach to Calculating Random Vibration Integrals

1987 ◽  
Vol 54 (2) ◽  
pp. 409-413 ◽  
Author(s):  
P-T. D. Spanos
Keyword(s):  
White Noise ◽  
Random Vibration ◽  
Arbitrary Order ◽  
Random Processes ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Band Limited ◽  
Stationary Random Processes ◽  
Time Invariant

Integrals required for the determination of the response statistics of an arbitrary order linear and time-invariant dynamic system under stationary excitation are examined. These integrals are found as the solution of a set of linear algebraic equations. The application of the derived general formula is exemplified by considering as excitation models white noise, band-limited white noise, and other important stationary random processes. Besides random vibration applications, the derived formula has purely mathematical merit and can be used for the calculation of complicated integrals encountered in a variety of other technical fields.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Discrete Evolutionary Spectra of Discretized Beams and Plates Under Time-Frequency Modulated Random Excitations

1995 ◽  
Author(s):  
C. W. S. To ◽  
H. W. Hung
Keyword(s):  
White Noise ◽  
Linear Time ◽  
Response Analysis ◽  
Plate Structures ◽  
Time Frequency ◽  
Linear Time Invariant ◽  
Practical Engineering ◽  
Band Limited ◽  
Evolutionary Spectra ◽  
Time Invariant

Abstract Various methods that employed the theory of evolutionary spectral density of Priestley (1965) have been proposed for the non-stationary random response analysis of linear time-invariant multi-degree-of-freedom systems (Hammond, 1968, Fugimori and Lin, 1973, To, 1982, Shihab and Preumont, 1987, To and Hung, 1989). In this paper the method presented earlier by the authors (1989) is further applied to discrete or discretized systems under time-frequency moduated random excitations in which the white noise processes are replaced by band-limited white noise processes and Kanai-Tajimi models. Applications of the method to beam and plate structures discretized by the finite element method are made so as to illustrate its capability in dealing with practical engineering systems under intensive transient disturbances that may be modelled as such time-frequency modulated random excitations.

scholarly journals Contact interaction of a predeformed plate which lies without friction on rigid base with a parabolic indenter

2021 ◽  
Vol 102 (2) ◽  
pp. 87-95
Author(s):  
Hryhorii Habrusiev ◽  
Iryna Habrusieva
Keyword(s):  
Contact Interaction ◽  
Rigid Base ◽  
Algebraic Equations ◽  
Dual Integral Equations ◽  
Linear Algebraic Equations ◽  
Vertical Displacements ◽  
Smooth Contact ◽  
Incompressible Solids ◽  
Initial Strains

Within the framework of linearized formulation of a problem of the elasticity theory, the stress-strain state of a predeformed plate, which is modeled by a prestressed layer, is analyzed in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.

Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion

2019 ◽  
Vol 968 ◽  
pp. 413-420
Author(s):  
Vitaly Yu. Miroshnikov ◽  
Alla V. Medvedeva ◽  
Sergei V. Oleshkevich
Keyword(s):  
Stress State ◽  
Reduction Method ◽  
Elastic Inclusion ◽  
Fourier Method ◽  
Theory Of Elasticity ◽  
Geometrical Parameters ◽  
Spatial Problem ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

A spatial problem of the theory of elasticity for the layer with an infinite round cylindrical inclusion is investigated. At the boundaries of the layer, displacements are given. The cylindrical elastic inclusion is rigidly coupled with the layer and their boundary surfaces do not intersect. The solution to the spatial problem is obtained by the generalized Fourier method, with regard to the Lamé system of equations. The obtained infinite systems of linear algebraic equations are solved by a reduction method. As a result, the values ​​of displacements and stresses in the elastic body are determined. A comparative analysis of the stress state for different geometrical parameters is carried out, and a comparison is made with the stress state in the layer with a cylindrical cavity.

Hilbert Transform Generalization of a Classical Random Vibration Integral

1994 ◽  
Vol 61 (3) ◽  
pp. 575-581 ◽  
Author(s):  
P. D. Spanos ◽  
S. M. Miller
Keyword(s):  
Hilbert Transform ◽  
Applied Mathematics ◽  
Classical Problem ◽  
Random Excitation ◽  
System Response ◽  
Algebraic Equations ◽  
Spectral Moments ◽  
Linear Algebraic Equations ◽  
Even Order ◽  
Time Invariant

Integrals which represent the spectral moments of the stationary response of a linear and time-invariant system under random excitation are considered. It is shown that these integrals can be determined through the solution of linear algebraic equations. These equations are derived by considering differential equations for both the autocorrelation function of the system response and its Hilbert transform. The method can be applied to determine both even-order and odd-order spectral moments. Furthermore, it provides a potent generalization of a classical formula used in control engineering and applied mathematics. The applicability of the derived formula is demonstrated by considering random excitations with, among others, the white noise, “Gaussian,” and Kanai-Tajimi seismic spectra. The results for the classical problem of a randomly excited single-degree-of-freedom oscillator are given in a concise and readily applicable format.

A Laplace Transform Technique for Direct Determination of Density of Electronic States in Disordered Semiconductors from Transient Photocurrent Data

2000 ◽  
Vol 609 ◽  
Author(s):  
Mariana J. Gueorguieva ◽  
Charles Main ◽  
Steve Reynolds
Keyword(s):  
Electronic States ◽  
Direct Determination ◽  
Algebraic Equations ◽  
Density Of Electronic States ◽  
Transient Photocurrent ◽  
Linear Algebraic Equations ◽  
Laplace Transform Technique ◽  
A New Technique ◽  
Disordered Semiconductors

ABSTRACTA new technique for direct determination of the density of electronic states (DOS) in disordered semiconductors is described. It involves Laplace transformation of transient photocurrent data I(t) followed by the numerical solution of the system of linear algebraic equations obtained from the Fredholm integral of the first kind, for a DOS represented by a series of discrete levels. No approximations are used in the solution, and no prior assumptions as to the form of the DOS are made. The fidelity of this method is assessed and compared with existing techniques by application to computer-simulated I(t) data generated from single-level and continuous DOS profiles, and to experimental data.

Parameter Estimation of Delay Systems Via Block Pulse Functions

1980 ◽  
Vol 102 (3) ◽  
pp. 159-162 ◽  
Author(s):  
Yen-Ping Shin ◽  
Chyi Hwang ◽  
Wei-Kong Chia
Keyword(s):  
Linear Time ◽  
Delay Systems ◽  
Algebraic Equations ◽  
Unknown Parameters ◽  
Linear Time Invariant ◽  
Least Squares Estimate ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations ◽  
Delay Differential ◽  
Time Invariant

Linear time-invariant delay-differential equation systems are approximately represented by a set of linear algebraic equations with the block pulse functions. A least squares estimate is then used to determine the unknown parameters. Examples with satisfactory results are given.

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

2021 ◽  
pp. 36-44
Author(s):  
Natalia Goloskubova ◽  
Yuri Mikhlin
Keyword(s):  
Parameter Space ◽  
Normal Vibration ◽  
System Parameter ◽  
Numerical Test ◽  
Vibration Modes ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Instability Regions ◽  
The Stability

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.

scholarly journals On the determination of shifting operators along geodesics on a surface

2013 ◽  
Vol 40 (1) ◽  
pp. 17-26
Author(s):  
Zoran Draskovic
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Algebraic Equations ◽  
Geodesic Line ◽  
Euclidean Spaces ◽  
Tensor Fields ◽  
Finite Element Approximations ◽  
Linear Algebraic Equations ◽  
The Future

A procedure to obtain a closed form of the shifting operators along a known geodesic line on a surface as a solution of a system of linear algebraic equations is proposed. Its correctness is numerically demonstrated in the case of a helicoid surface and a spherical one. The future use of these operators in finite element approximations of tensor fields in non-Euclidean spaces is announced.

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