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.

Hilbert Transform Generalization of a Classical Random Vibration Integral

1993 ◽  
Author(s):  
Pol D. Spanos ◽  
Scott M. Miller
Keyword(s):  
Hilbert Transform ◽  
Linear Time ◽  
Applied Mathematics ◽  
Classical Problem ◽  
Random Excitation ◽  
System Response ◽  
Algebraic Equations ◽  
Spectral Moments ◽  
Linear Time Invariant ◽  
Linear Algebraic Equations

Abstract Integrals which represent the spectral moments of the stationary response of a linear, 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.

scholarly journals Approximate Stochastic Response of Hysteretic System With Fractional Element and Subjected to Combined Stochastic and Periodic Excitation

2021 ◽  
Author(s):  
Fan Kong ◽  
Renjie Han ◽  
Yuanjin Zhang
Keyword(s):  
Fractional Order ◽  
Harmonic Balance Method ◽  
System Response ◽  
Statistical Linearization ◽  
Algebraic Equations ◽  
Stochastic Response ◽  
Hysteretic System ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Linear Differential

Abstract A method based on statistical linearization is proposed, for determining response of the single-degree-of-freedom (SDOF) hysteretic system endowed with fractional derivatives and subjected to combined periodic and white/colored excitation. The method is developed by decomposing the system response into a combination of a periodic and of a zero-mean stochastic components. In this regard, first, the equation of motion is cast into two sets of coupled fractional-order non-linear differential equations with unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization for the fractional-order deterministic and stochastic subsystems are used, to obtain the Fourier coefficients of the deterministic component and the variance of the stochastic component, respectively. This yields two sets of coupled non-linear algebraic equations which can be solved by appropriate standared numerical method. Pertinent numerical examples, including both softening and hardening Bouc-Wen hysteretic system endowed with different fractional-orders, are used to demonstrate the applicability and accuracy of the proposed method.

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.

The Contact Problem of a Plate Pressed Between Two Spheres

1964 ◽  
Vol 31 (4) ◽  
pp. 659-666 ◽  
Author(s):  
Yih-O Tu ◽  
D. C. Gazis
Keyword(s):  
Contact Problem ◽  
Legendre Polynomials ◽  
Plate Thickness ◽  
Elastic Contact ◽  
Algebraic Equations ◽  
Total Load ◽  
Linear Algebraic Equations ◽  
Even Order ◽  
Axisymmetric Bodies ◽  
Maximum Contact

The elastic contact of a plate with two axisymmetric bodies, generally of dissimilar elastic properties and curvatures, is considered. The solution of the resulting two integral equations is given as a truncated series of Legendre polynomials of even order whose coefficients may be determined from a system of linear algebraic equations. The relationships between the total load, radii of curvature of the bodies, contact radii, approach, radial displacements, maximum contact stress, and the plate thickness can then be computed in terms of these coefficients. Numerical computations have been carried out for the contact of a plate with two identical spheres.

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.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Differential-algebraic systems are generically controllable and stabilizable

2021 ◽  
Author(s):  
Achim Ilchmann ◽  
Jonas Kirchhoff
Keyword(s):  
Algebraic Geometry ◽  
Linear Time ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Algebraic Systems ◽  
Linear Time Invariant ◽  
Survey Article ◽  
Link Type ◽  
Invariant Differential ◽  
Time Invariant

AbstractWe investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. 10.1007/978-3-642-34928-7_1)), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.

scholarly journals Exact statistical solution for the hopping transport of trapped charge via finite Markov jump processes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrey A. Pil’nik ◽  
Andrey A. Chernov ◽  
Damir R. Islamov
Keyword(s):  
Charge Transport ◽  
Thin Layers ◽  
Jump Processes ◽  
Dielectric Films ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Statistical Solution ◽  
Linear Algebraic Equations ◽  
Special Cases ◽  
Thin Dielectric Films

AbstractIn this study, we developed a discrete theory of the charge transport in thin dielectric films by trapped electrons or holes, that is applicable both for the case of countable and a large number of traps. It was shown that Shockley–Read–Hall-like transport equations, which describe the 1D transport through dielectric layers, might incorrectly describe the charge flow through ultra-thin layers with a countable number of traps, taking into account the injection from and extraction to electrodes (contacts). A comparison with other theoretical models shows a good agreement. The developed model can be applied to one-, two- and three-dimensional systems. The model, formulated in a system of linear algebraic equations, can be implemented in the computational code using different optimized libraries. We demonstrated that analytical solutions can be found for stationary cases for any trap distribution and for the dynamics of system evolution for special cases. These solutions can be used to test the code and for studying the charge transport properties of thin dielectric films.

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