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Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik | 1999

Random eigenvalue problems for bending vibrations of beams

S. Mehlhose; J. vom Scheidt; Ralf Wunderlich

The paper deals with the determination of statistical characteristics of eigenvalues for a class of ordinary differential operators with random coefficients. This problem arises from the computation of eigenfrequencies for the bending vibrations of beams possessing random geometry and material properties. Representations of eigenvalues are found by applying the Ritz method and perturbation results for matrix eigenvalue problems. Approximations of the probability density function and the moments of the random eigenvalues are given by means of expansions in powers of the correlation length of weakly correlated random functions which are used for modelling the random terms. The eigenvalue statistics determined analytically are compared favourably with Monte-Carlo simulations.


Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik | 2002

Random transverse vibrations of a one-sided fixed beam and model reduction

J. vom Scheidt; H.-J. Starkloff; Ralf Wunderlich

This paper is devoted to the computation of second-order moment functions of the solution of large-scale systems of linear random ODEs. Such systems result from the semi-discretization of PDEs describing continuous vibration systems with random excitation. Model reduction techniques are applied to find approximations of the desired moment functions of the large-scale system by computing them for a suitable low-dimensional system. Numerical results concerning transverse vibrations of a one-sided fixed beam subjected to a random excitation modeled by a random field are presented. Special attention is paid to e-correlated excitations characterized by a short correlation length.


Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik | 2002

Remarks on randomly excited oscillators

J. vom Scheidt; H.-J. Starkloff; Ralf Wunderlich

In the paper linear oscillators with stochastic excitation are considered. Explicit formulae for second-order characteristics of random solution processes for input processes which are wide-sense stationary or possess wide-sense stationary increments are given. Further, asymptotic expansions in the case of weakly correlated excitation processes are derived.


Zeitschrift Fur Analysis Und Ihre Anwendungen | 2000

Asymptotic Expansions of Integral Functionals of Weakly Correlated Random Processes

J. vom Scheidt; H.-J. Starkloff; Ralf Wunderlich

In the paper asymptotic expansions for second-order moments of integral functionals of a family of random processes are considered. The random processes are assumed to be wide-sense stationary and c-correlated, i.e. the values are not correlated excluding an c-neighbourhood of each point. The asymptotic expansions are derived for e 0. Using a special weak assumption there are found easier expansions as in the case of general weakly correlated random processes. Expansions are given for integral functionals of real-valued as well as of complex vector-valued processes.


Stochastic Analysis and Applications | 2001

Stationary solutions of random differential equations with polynomial nonlinearities

J. vom Scheidt; H.-J. Starkloff; Ralf Wunderlich

The paper deals with systems of ODEs containing polynomial nonlinearities and random inhomogeneous terms. Applying perturbation method pathwise solutions are found in form of power series with respect to a parameter η controlling the nonlinearities. Under the assumption that for η=0 the system is stable and that the inhomogeneous terms are bounded the radius of convergence of the perturbation series is estimated. Further, it is proved that the perturbation series form stationary solutions if the inhomogeneous terms are stationary.


Zeitschrift Fur Analysis Und Ihre Anwendungen | 1997

Nonlinear Vibration Systems with Two Parallel Random Excitations

J. vom Scheidt; Ulrich Wöhrl

Systems of nonlinear vibration differential equations are investigated where the non-linearities are given by polynomials of any degree. The random excitations are induced by two parallel processes. These random excitations of an often applied type are expressed by linear functionals of weakly correlated processes with correlation length E. The moments of the solutions and their first and second derivatives are expanded with respect to s where all terms up to order e 2 are included. Approximations of the correlation functions are given explicitely. Only the quadratic and cubic non-linearities have an influence on the correlation functions in this approximation order.


Zeitschrift Fur Analysis Und Ihre Anwendungen | 1997

Distribution Approximations for Nonlinear Functionals of Weakly Correlated Random Processes

J. vom Scheidt; S. Mehlhose; Ralf Wunderlich

In this paper nonlinear functionals of weakly correlated processes with correlation length e > 0 are investigated. Expansions of moments and distribution densities of nonlinear functionals with respect to e up to terms of order o(e) are considered. For the case of a single nonlinear functional a shorter proof than in [8] is given. The results are applied to cigenvalues of random matrices which are obtained by application of the Ritz method to random differential operators. Using the expansion formulas as to e approximations of the density functions of the matrix cigenvalues can be found. In addition to [7] not only first order approximations (exact up to terms of order 0(e)) but also second order approximations (exact up to terms of order o(e)) are investigated. These approximations are compared with estimations from Monte-Carlo simulation.


Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik | 1979

Ein Grenzverteilungssatz für stochastische Eingenwertprobleme

Walter Purkert; J. vom Scheidt


Journal of Applied Mathematics and Mechanics | 1996

Nonlinear random vibrations of vehicles

J. vom Scheidt; Ralf Wunderlich


Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik | 1989

Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus. New York, Springer‐Verlag 1988. XXIII, 470 pp., 10 figs., DM 138,–. ISBN 3–540‐96535‐1 (Graduate Texts in Mathematics 113)

J. vom Scheidt

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Ralf Wunderlich

Chemnitz University of Technology

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H.-J. Starkloff

Chemnitz University of Technology

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S. Mehlhose

Chemnitz University of Technology

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Ulrich Wöhrl

HTW Berlin - University of Applied Sciences

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