We construct analytical solutions for the problem of nonlinear supersonic flow past slender bodies of revolution due to small amplitude oscillations. The method employed is based on the splitting of the time dependent small perturbation equation to a nonlinear time independent partial differential equation (P.D.E.) concerning the steady flow, and a linear time dependent one, concerning the unsteady flow. Solutions in the form of three parameters family of surfaces for the first equation are constructed, while solutions including one arbitrary function for the second equation are extracted. As an application the evaluation of the small perturbation velocity resultants for a flow past a right circular cone is obtained making use of convenient boundary and initial conditions in accordance with the physical problem.

D. E. Panayotounakos

M.  Markakis

Nonlinear unsteady supersonic flow analysis for slender bodies of revolution: Theory

This paper deals with the robustness of the class of nonlinear piecewise deterministic systems with unknown but bounded uncertainties. Under the assumption that all the modes of the markovian jump process (disturbance) communicate, the complete access to the system state and the boundedness of the uncertainties, a sufficient condition for stochastic stability of this class of systems is given. An example is presented to validate the proposed results.

E. K. Boukas

H.  Yang

Robust stability of nonlinear piecewise deterministic systems under matching conditions

This paper deals with the class of linear time-delay systems with Markovian jumping parameters (LTDSMJP). We mainly extend the stability results of the deterministic class of linear systems with time-delay to this class of systems. A delay-independent necessary condition and sufficient conditions for checking the stochastic stability are established. A sufficient condition is also given. Some numerical examples are provided to show the usefulness of the proposed theoretical results.

K.  Benjelloun

Stochastic stability of linear time-delay system with Markovian jumping parameters

The structure of the quasi-isothermal deflagration is examined by means of an asymptotic analysis of the physical-plane boundary-value problem, with Lewis&#8211;Semenov number unity, in the limit of the activation-temperature ratio, <mml:math alttext="$\beta = T_a /T_b $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#946;</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>a</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:mrow></mml:math>, greater than order unity, for the generalized reaction-rate-model case of: (1) the heat-addition-temperature ratio, <mml:math alttext="$\alpha = \left( {T_b - T_u } \right)/T_u $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#945;</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>b</mml:mi></mml:msub><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:mrow></mml:math>, of order <mml:math alttext="$\beta ^{ - 1/2} $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi> &#946;</mml:mi><mml:mrow><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>, less than order unity 
[where <mml:math alttext="$T_a $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:mrow></mml:math>, <mml:math alttext="$T_b $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>b</mml:mi></mml:msub></mml:mrow></mml:math>, and <mml:math alttext="$T_u $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mi>u</mml:mi></mml:msub></mml:mrow></mml:math>
are the activation, adiabatic-flame (and/or burned-gas), and unburned-gas temperatures, respectively]; and (2) the exponent, <mml:math alttext="$a$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi></mml:math>, which characterizes the pre-exponential thermal dependence of the reaction-rate term, unity. The examination indicates that, as in the order-unity heat-addition case, this deflagration has a four-region structure: the upstream diffusion-convection and downstream diffusion-reaction regions, and the far-upstream (or cold-boundary) and the far-downstream (or hot-boundary) regions.

W. B. Bush

L.  Krishnamurthy

On the structure of a distinguished-limit quasi-isothermal deflagration for the generalized reaction-rate model

A point mapping analysis is employed to investigate the stability of periodic systems. The method is applied to simplified rotorcraft models. The proposed approach is based on a procedure to obtain an analytical expression for the period-to-period mapping description of system's dynamics, and its dependence on system's parameters. Analytical stability and bifurcation conditions are then determined and expressed as functional relations between important system parameters. The method is applied to investigate the parametric stability of flapping motion of a rotor and the ground resonance problem encountered in rotorcraft dynamics. It is shown that the proposed approach provides very accurate results when compared with direct numerical results which are assumed to be an “exact solution” for the purpose of this study. It is also demonstrated that the point mapping method yields more accurate results than the widely used classical perturbation analysis. The ability to perform analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems. Stability analysis of higher dimensional systems, such as the ground resonance problems, by perturbation methods is not straightforward, and is usually very cumbersome.

Henryk Flashner

Ramesh S. Guttalu

Analytical stability analysis of periodic systems by Poincaré mappings with application to rotorcraft dynamics

A large class of emerging actuation devices and materials exhibit strong hysteresis characteristics during their routine operation. For example, when piezoceramic actuators are operated under the influence of strong electric fields, it is known that the resulting input&#8211;output behavior is hysteretic. Likewise, when shape memory alloys are resistively heated to induce phase transformations, the input&#8211;output response at the structural level is also known to be strongly hysteretic. This paper investigates the mathematical issues that arise in identifying a class of hysteresis operators that have been employed for modeling both piezoceramic actuation and shape memory alloy actuation. Specifically, the identification of a class of distributed hysteresis operators that arise in the control influence operator of a class of second order evolution equations is investigated. In Part I of this paper we introduce distributed,hysteretic control influence operators derived from smoothed Preisach operators and generalized hysteresis operators derived from results of Krasnoselskii and Pokrovskii. For these classes, the identification problem in which we seek to characterize the hysteretic control influence operator can be expressed as an ouput least square minimization over probability measures defined on a compact subset of a closed half-plane. In Part II of this paper, consistent and convergent approximation methods for identification of the measure characterizing the hysteresis are derived.

H. T. Banks

A. J. Kurdila

G.  Webb

Identification of hysteretic control influence operators representing smart actuators part I: Formulation

The main purpose of this paper is to incorporate a refined hysteresis model, viz. a vector Preisach model, in 2D magnetic field computations. To this end the governing Maxwell equations are rewritten in a suitable way, which allows to take into account the proper magnetic material parameters and, moreover, to pass to a variational formulation. The variational problem is solved numerically by a FE approximation, using a quadratic mesh, followed by the time discretisation based upon a modified Cranck Nicholson algorithm. The latter includes a suitable iteration procedure to deal with the nonlinear hysteresis behaviour. Finally, the effectiveness of the presented mathematical tool has been confirmed by several numerical experiments.

Luc R. Dupré

Roger Van Keer

Jan A. A. Melkebeek

A 2D finite element procedure for magnetic field analysis taking into account a vector Preisach model

In order to analyze the relaxation effects in a turbulent boundary layer with zero and nonzero free stream turbulence, the Reynolds-averaged equations of motion and energy are solved. As the closure of the Reynolds-averaged equations, the transport equation for turbulent shear stresses is used. The proposed approach leads to calculation of the relaxation scales in the turbulent boundary layer with zero and nonzero free stream turbulence. Results for friction coefficients, velocity profiles, shear stresses, thickness of the boundary layer and so called &#8220;superlayer&#8221; in a flat-plate turbulent boundary layer are presented. The results obtained are in agreement with those available from the experimental data.

Eugen  Dyban

Ella  Fridman

Relaxation process modeling in a turbulent boundary layer with nonzero free stream turbulence

The author studies an M/G/1 queueing system with multiple vacations. The server is turned off in accordance with the K-limited discipline, and is turned on in accordance with the T-N-hybrid policy. This is to say that the server will begin a vacation from the system if either the queue is empty or <mml:math alttext="$K$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi></mml:math>
 customers were served during a busy period. The server idles until it finds at least <mml:math alttext="$N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> waiting units upon return from a vacation.Formulas for the distribution generating function and some characteristics of the queueing process are derived. An optimization problem is discussed.

Alexander V. Babitsky

M/G/1 vacation model with limited service discipline and hybrid switching-on policy

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Mathematical Problems in Engineering is a broad-based journal publishing&nbsp;results of rigorous engineering research across all disciplines, carried out using mathematical tools.

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Chief Editor, Professor Guangming Xie, is currently a full professor of dynamics and control with the College of Engineering, Peking University. His research interests include complex system dynamics and control and intelligent and biomimetic robots.

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Mathematical Problems in Engineering is a broad-based journal publishing results of rigorous engineering research across all disciplines, carried out using mathematical tools.

Mathematical Problems in Engineering