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In this paper the stability of a rectangular plate in a supersonic flow in the presence of a temperature field, inhomogeneous across the thickness, is investigated. The temperature field inhomogene...

The chapter discusses the use of the spectral form of mathematical description, or the spectral method, for the statistical analysis of stochastic dynamical systems: diffusions and jump diffusions, i.e., for solving Fokker–Planck–Kolmogorov equation and Kolmogorov–Feller equation for the probability density of the state vector for these dynamical systems. The spectral form of mathematical description allows to transform linear partial differential equations or partial integro-differential equations into a system of linear algebraic equations, which determines coefficients according to orthogonal series expansions for the probability density with respect to an arbitrary orthonormal system of functions. As an example for testing, the Dryden wind turbulence model and its modification, allowing to take into account not only continuous random effects but also impulse ones, are considered.

The problem of nonlinear oscillations of isotropic rectangular plate in supersonic gas flow is examined. The study was conducted taking into account both types of non-linearity: wind (quadratic and cubic) and geometric (cubic). Due to the aerodynamic nonlinearity (especially its non-symmetric quadratic part) it is established that dependence amplitude-speed is two-valued at the certain intervals of the speed. This fact is illustrated on the figures given in the text in the form of two branches, the lower branches of which, in all probability, are unstable. The unstable branches are separated via the gravitational field of two adjacent sustainable solutions. Thus the perturbation magnitude can be easily found, which is required in order to transfer the system from one stable branch to another. Existence of specific areas of the speed is shown in which undamped flutter type oscillations cannot be excited in both pre-critical speeds and in post-critical stage.

The work is devoted to the investigation of flutter oscillations and stability of closed cylindrical shell in supersonic gas flow and placed in an inhomogeneous temperature field. It is assumed that supersonic gas flows on the outside of the shell with an unperturbed velocity U, directed parallel to the cylinder generatrix. Under the action of an inhomogeneous temperature field, the shell bulges out; this deformed state is accepted as unperturbed, and the stability of this state is studied. The main nonlinear equations and relations describing the behavior of the examined system are derived. The formulated boundary value problem is solved using Galerkin method. The joint influence of the flow and the temperature field on the relation between the amplitude and frequency of nonlinear oscillations of a cylindrical shell is studied. The critical velocity values are calculated from the corresponding linear system and are given in tables. The numerical results show that: a) the surrounding flow significantly affects the nature of the investigated relation; b) it is impossible to excite steady-state flutter oscillations (the silence zone) up to a certain value of the oscillations frequency; c) the dependence of the amplitude on the oscillations frequency can be either multi-valued or single-valued.

In this section the main equations, boundary conditions and relations of electrodynamics of moving media are brought to formulate and solve the problems of the theory of electro-magneto-elasticity describing interconnected mechanical and electromagnetic processes in deformable media in presence of external electromagnetic fields and being under the influence of several forces. Relations of mechanical-mathematical model are brought in the general form which gives a possibility to choose an appropriate model of electrodynamics. The main laws of electrodynamics are axiomatically represented in a view of Maxwell equations. The main equations of the theory of elasticity are brought in account of forces of electromagnetic origin (ponderomotive forces). Boundary conditions and state equations for the medium and field, as well as the conditions at infinity are formulated, taking into account the interaction between the medium and the field. The cases of conductive isotropic and anisotropic (as well as perfectly conductive), piezoelectric, piezomagnetic, magnetostrictive, ferromagnetic media are considered. The assumptions that limit the range of considered problems, as well as assumptions that will be used in the future are discussed. In the future for brevity, sometimes the term magnetoelastic system will be used, meaning the set of interacting elastic medium and electromagnetic field.

Wave processes in an ideally conducting half-space in a homogeneous magnetic field are investigated. The propagation of magnetoelastic waves in an anisotropic half-space under normal pressure applied to the surface of the half-space is considered in the presence of an external magnetic field, whose intensity vector is parallel to the half-space boundary. In particular, the problem of an orthotropic half-space is considered, when the principal directions of elasticity form an angle with the vector of the strength of the external magnetic field. The magnetoelastic Rayleigh problem is considered in the presence of longitudinal and transverse magnetic fields for the case of an isotropic medium with initial stresses caused by magnetic pressure. The magnetoelastic Lamb problem is investigated for the case of acting concentrated impulsive load.

In this chapter, let us study the problems of wave dynamics in the following directions of the theory of propagation of surface waves.

This chapter is devoted to the study of propagation of magnetoelastic waves in isotropic and anisotropic electro-conductive media. The basic linearized equations, boundary conditions and relations of perfectly conducting media, describing magnetoelastic wave processes, are derived. Magnetoelastic plane waves in infinite media are investigated. The characteristic equation of bulk magnetoelastic waves is derived. The classification of fast and slow, as well as of quasi-longitudinal and quasi-transverse, magnetoelastic waves is given. It is shown in which cases, depending on the orientation of the external magnetic field, the medium is under plane-strain conditions. The condition of complete hyperbolicity, which ensures the possibility of propagation of magnetoelastic waves in any direction, is established. Investigation of the roots of the characteristic equation reveals the nature of propagation of fast and slow waves, depending on the physical and mechanical properties of the medium and the magnitude of the external magnetic field intensity. The changes in wave’s phase velocities depending on the direction of propagation characteristics of the elastic medium and magnetic field.

The behavior of nonlinear oscillations of isotropic rectangular plate fluttering in a supersonic gas flow is examined. The study was conducted taking into account both types of nonlinearity...

This paper provides insights into the stability of flexible orthotropic rectangular plates in supersonic flow by revealing the amplitude-speed dependency behavior for such panels in precritical and postcritical flight conditions. Flutter and postflutter behaviors are discussed to highlight the important effect of the structural and aerodynamic nonlinearities inherently present for panels in high-speed flows. The influence of transversal shear on stability is also illustrated. Depending on the plate’s geometrical parameters, soft- and hard-type flutter behaviors are possible, and a graphical interpretation is provided along with pertinent conclusions.

This article provides new insights into the aerothermoelastic stability of thin plates. Particularly, the issue of loss of stability of an isotropic plate-strip of constant thickness immersed in a supersonic flow field and subjected to a variable temperature field through the thickness is examined. Using the basic principles of the theory of aerothermoelasticity of isotropic bodies, the theories of flexible panels, and the linear law of temperature field through the thickness of the panel, the stability equations and associated boundary conditions are obtained. As expected, the coefficients of the aerothermoelastic governing equations depend on the thermal load, and consequently the panel-flutter critical speed depends on temperature. The model takes into account quadratic and cubic aerodynamic non-linearities as well as cubic geometric non-linearities. Due to the inhomogeneity of the temperature field distribution across the thickness plate buckling instability occurs. This instability accounts for the d...

In the first part of this chapter, the possibility of controlling (i.e., using a constant magnetic field) the dynamic instability of conducting plates and cylindrical shells caused by the harmonic-in-time mechanical force is investigated.

In this chapter, based on the basic principles of the classical theory of thin plates, and the results of the previous chapter, on the basis of the hypothesis of nondeformable normals, the equations and appropriate conditions of vibrations and the stability of superconducting thin bodies (plates, cylindrical, and spherical shells) under the action of a given magnetic field are presented. Based on the solution of specific problems, the qualitative and quantitative effects of the interaction of a superconducting thin body with a magnetic field are identified. Here are some of the most significant new effects conditioned by such magnetoelastic interaction: (1) the presence of magnetic field can lead to a significant increase of the frequency of lower vibration modes, and this influence is much enhanced with the decreased relative thickness of the plate; (2) at the magnetic induction of the order 0.1Tl, plastic deformations appear; (3) in the case of magnetoelastic systems with parallel plates, it is shown that forced vibrations of the first plate, on which the disturbing force acts, can be transmitted in a noncontact way to the second plate, which is free from any external mechanical loads.

The hypotheses of magnetoelasticity of thin bodies is addressed in this section. On its basis, using the results from the previous section, two-dimensional equations and appropriate conditions are obtained that describe the behavior of conducting plates and shells in a magnetic field.

This chapter sets out the main principles and provide basic equations and relations describing interconnected mechanical and electromagnetic processes in deformable electroconducting bodies placed in an external inhomogeneous magnetic field and under the influence of various types of force interactions. Basic equations and relations are addressed in a nonlinear formulation. This allows us, by way of linearization, to obtain linear equations and surface conditions in unperturbed, as well as perturbed, states of a magnetoelastic system, which provides the basis of the theory of magnetoelasticity of thin elastic plates and shells.

In this chapter, based on the basic assumptions of the classical theory of thin cylindrical shells, the equations and appropriate conditions for the stability of superconducting thin cylindrical shells in an inhomogeneous magnetic field are obtained. On this basis, the behavior of the shell under the action of stationary and nonstationary magnetic fields is investigated. The possibility of loss of both static and dynamic stability of the unperturbed state is established.

By way of the analysis of solutions of formulated problems of natural magnetoelastic vibrations of conducting thin bodies, the following phenomena have been established conditioned by the interaction of mechanical and electromagnetic processes in deformable conducting thin bodies as follows.

The chapter is devoted to the extension of results from the previous chapter on the natural magnetoelastic vibrations of conducting cylindrical and spherical shells placed in a stationary magnetic field. The effects of the intensity of magnetic field, the conductivity of shell’s material, and the geometrical parameters on the complex frequency of magnetoelastic vibrations are studied.

Magnetoelastic forced vibrations of a conducting plate caused by either nonstationary external forces of nonelectromagnetic origin, or with the help of a nonstationary harmonic magnetic field, are examined. The following results are conditioned by the interaction of mechanical and electromagnetic processes in oscillating thin bodies. It is shown that with the help of the chosen external constant magnetic field, the possibility of dangerous resonant vibrations of the conventional type, or violent vibrations, can be eliminated in such cases when, in the absence of magnetic field, the system is out of the resonant area. It is established that the presence of a longitudinal constant magnetic field substantially increases the amplitude of forced vibrations if the frequency of the acting force is greater than the natural frequency of the plate in the absence of a magnetic field. If there is an inverse relationship between the natural frequency and the frequency of the acting mechanical force, then the amplitude of vibrations can be decreased by several hundred times (especially in the case of resonance without a magnetic field) with the help of a magnetic field.

The stability analysis of plates and shells in high speed flow deals with the determination of the flutter instability boundary. A linear analysis is made using the basic principles of the theory of aero-elasticity of isotropic bodies, the theories of flexible plates, the stability equations and associated boundary conditions obtained through a linear formulation. Herein, the nonlinear stability of flexible plate immersed in a high speed gas flow is considered. The model takes into account quadratic and cubic aerodynamic nonlinearities as well as cubic geometric nonlinearities. It is shown that the inclusion of quadratic aerodynamic nonlinear components can lead to the appearance of "amplitudefrequency" phenomena in both the pre-critical as well as in the post-critical flow speed regimes. The influence of the free stream flow speed on the "amplitude-frequency" dependence phenomena is also presented.

Solution of dynamical problems of the theory of magnetoelasticity is connected with certain mathematical di culties as it is known from [1]–[4]. In the present work the analytical formulae for elastic displacements and stresses of the induced electromagnetic field for the Lamb magnetoelastic problem are obtained with the use of Laplace’s and Fourier’s transformation technique and consequently by applying the Canyard kernel.The noted problem is discussed in the works [5], [6] for certain conditions.

On the basis of Kirchhoff hypothesis the problem of vibrations of conductive plate in a longitudinal magnetic field is brought to the solution of the singular integral–differential equation with ordinary boundary conditions. The formulated boundary-value problem solved and the influence of magnetic field on the characteristics of vibration process of the examined magnetoelastic system is investigated. Via the analysis of obtained solutions it is shown that the presence of magnetic field can: (a) increase essentially the frequency of free magnetoelastic vibrations of the plate; (b) decrease essentially the amplitude of forced vibrations if r ⩽ 1 , where r = θ / ω , θ – is the frequency of acting force, ω – is the frequency of own vibrations of the plate magnetic field is being absent; (c) increase essentially the amplitude of forced vibrations if r > 1 ; (d) decrease essentially the width of main areas of dynamic instability. It is shown that: (1) in the case of perfectly conductive plates magnetic field constricts essentially the width of main area of dynamic instability; (2) if plate’s material has the finite electroconductivity, then the certain value of the intensity of external magnetic field exists, exceeding of which excludes the possibility of appearance of parametric type resonance. It is shown also that in dependence on the character of initial excitements the plate can vibrate either across the initial non-deformable state, or across the initial bent state.