Nonresonance parametric interactions of surface waves in isotropic solid bodies

A solution in parametric approximation is given of the nonlinear interaction problem of Rayleigh surface waves propagating in a solid body with given elastic fields. Shortened equations that govern the modulation effect of the surface waves, and also expressions for the modulation index in terms of the third-order elasticity constant, are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 163–172, July–August, 1973.     

Investigation of thermocapillary and thermogravitational convection in a fluid with local heating

The results of a numerical solution of the problem of the unsteady convective motion generated in a fluid layer by the formation at the initial instant of a heated zone in the form of a thin cylindrical column, extending from the surface into the interior of the fluid, are presented. The problem is formulated with allowance for both thermocapillary and thermogravitational convection. The influence of the thermocapillary and thermogravitational effects on the fluid motion for various layer thicknesses is subjected to parametric analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 25–29, November–December, 1989.     

The application of the projection method for constructing the shape of a body with minimum drag

The problem is considered of constructing a semi-infinite axisymmetric body with minimum drag in subsonic flow of an ideal gas. This problem is formulated as the problem of finite-dimensional minimization by prescribing the shape of the body in parametric form and applying the projection method for solving a flow problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 108–113, March–April, 1985.     

Influence of nozzle profile on the characteristics of a gas-dynamic laser

The results are given of numerical profiling and analysis of the influence of nozzle shape and the gas-dynamic parameters on the characteristics of gas-dynamic lasers. Investigation of the two-dimensional nonequilbrium flow in a family of similar nozzles and nozzles with different angles of inclination of the contracting part show that it is expedient to choose a shape of the subsonic part that ensures a straight sonic line. Relationships between the geometrical parameters of the subsonic and transonic part of the nozzle are recommended which ensure separationless flow and a shape of the sonic surface that is nearly flat. A parametric investigation was made of the supersonic section of two classes of planar gas-dynamic laser nozzles constructed on the basis of uniform and symmetric characteristics at the exit. The parametric investigations of the influence of the degree of expansion, the total pressure and the temperature, and also the gas composition show that the smallest losses of useful vibrational energy in the cavity are achieved for nozzles constructed on the basis of uniform characteristics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 163–167, November–December, 1982.     

Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

Nonlinear wave effects in a fluid-saturated porous medium

Some one-dimensional nonlinear effects associated with wave propagation in weakly permeable fluid-saturated porous media are investigated. The effect of nonlinearity on the damping of monoharmonic waves is estimated and, moreover, the characteristics of the nonlinear parametric interaction of two waves excited in the medium by two monoharmonic sources of different frequencies are established.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 74–77, January–February, 1992.     

The Nonlinear Vibrations and Stability of Orthotropic Cylindrical Liquid-Carrying Shells Under Periodic Actions

The nonlinear resonance vibrations and stability of nonlinear orthotropic shells partially filled with a liquid and subjected to longitudinal and transverse periodic loading are studied. The equations of motion are derived with regard for the existence and nonlinear interaction of conjugate flexural modes of the shell. Stationary regimes for forced and parametric vibrations are found and analyzed for stability under the conditions of fundamental and subharmonic resonances. The amplitude–frequency characteristics of these regimes are constructed for various parameters of the system     

Example of parametric resonance in a rotating flow

Parametric resonance is one of the common types of instability of mechanical systems [1]. A standard example of the equations describing parametric oscillations is the Mathieu equation and its generalizations. In hydrodynamics these oscillations have been closely studied in connection with the problem of the vertical oscillations of a vessel containing an incompressible fluid in a uniform gravity field [1–5]. In this paper a new example of a flow whose stability problem reduces to the Mathieu equation is given. This is a flow of special type in a rotating cylindrical channel. The direction of the angular velocity is perpendicular to the channel axis, and its magnitude varies periodically with time. Flows with this geometry are of potential interest in technical applications [6, 7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 175–177, March–April, 1987.     

Stochastic stability of parametrically excited random systems

Multidegree-of-freedom dynamic systems subjected to parametric excitation are analyzed for stochastic stability. The variation of excitation intensity with time is described by the sum of a harmonic function and a stationary random process. The stability boundaries are determined by the stochastic averaging method. The effect of random parametric excitation on the stability of trivial solutions of systems of differential equations for the moments of phase variables is studied. It is assumed that the frequency of harmonic component falls within the region of combination resonances. Stability conditions for the first and second moments are obtained. It turns out that additional parametric excitation may have a stabilizing or destabilizing effect, depending on the values of certain parameters of random excitation. As an example, the stability of a beam in plane bending is analyzed.Published in Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 135–144, October 2004.     

Parametric excitation of internal waves in a continuously stratified liquid

A study is made of the parametric excitation of internal waves in a continuously stratified liquid in a vessel executing oscillations in the vertical direction. It is shown that vertical oscillations of the vessel will excite oscillatory modes with eigenfrequencies equal to half of the oscillation frequency of the vessel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 167–169, September–October, 1982.I thank S. V. Nesterov for interest in the work.     

Influence of anisotropy of the permeability on the convection and heat transfer in a porous annular layer

Investigations into the convective transport of heat in porous materials are of interest for many applications in connection with the problem of increasing the efficiency of thermal insulation. In [1–5], convection in Isotropic porous media was considered. However, in many cases porous materials have an essential anisotropy of their permeability. Convective heat transfer has been inadequately studied for this case. In [6], the linearized equations were used to study the convection between infinite horizontal planes with a filling of an anisotropic material; the value of the critical Rayleigh number was found, and this agreed satisfactorily with experimental data. In the present paper, we investigate numerically convection between two infinite coaxial cylinders with an anisotropic porous filling, using the equations of convection in the Darcy—Boussinesq approximation [1–3]. The permeability tensor in the annular region is constructed from its principal values, which can be found experimentally. A method of calculation is developed and a parametric study made of the structure of the flow and of the local and averaged characteristics of the heat transfer, which are of interest for the design of thermal insulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–64, January–February, 1980.     

Formation of the three-dimensional structure of the subharmonic transition regime in Blasius flow

The formation of the angles of the three-dimensional structure associated with subharmonic-type transition in a Blasius boundary layer is investigated. It is shown that the observed structure is determined by the selection of a spatially symmetric pair of Tollmien-Schlichting waves at parametric resonance of the background perturbations in the presence of induced oscillation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 77–81, May–June, 1989.     

Sphere scattering of nonlinearly interacting acoustic waves

Sphere scattering of the field of nonlinearly interacting plane acoustic waves when the sphere is located in the region of nonlinear interaction between the primary pumping waves of a parametric antenna is considered. An analytic expression for the secondary field pressure at the difference frequency is obtained. This expression describes the process of nonlinear interaction of the incident and scattered waves. The secondary-field total pressure components, which characterize the interaction between the incident plane waves and scattered spherical waves are analyzed. The numerical results and experimental data are given.Taganrog. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 4–12, March–April, 1995.     

Integral equations in two-dimensional airfoil theory

A system of integral equations is obtained for the tangential components of the velocities on the upper and lower surfaces of a profile which does not have a parametric singularity connected with the thickness of the airfoil. An example of the solution of these equations by the discrete vortex method is given. This shows that they are highly effective for solving problems of flow over an airfoil of any, including arbitrarily small, thickness.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 173–177, July–August, 1992.     

Nonlinear Vibrations of a Cylindrical Shell Containing a Flowing Fluid

The Bogolyubov-Mitropolsky method is used to find approximate periodic solutions to the system of nonlinear equations that describes the large-amplitude vibrations of cylindrical shells interacting with a fluid flow. Three quantitatively different cases are studied: (i) the shell is subject to hydrodynamic pressure and external periodical loading, (ii) the shell executes parametric vibrations due to the pulsation of the fluid velocity, and (iii) the shell experiences both forced and parametric vibrations. For each of these cases, the first-order amplitude-frequency characteristic is derived and stability criteria for stationary vibrations are established__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 75–84, April 2005.     

Axisymmetric flow of ideal fluid past a gas bubble

A generalization of the analytic functions is proposed to solve the problem of axisymmetric flow past a gas bubble. By the use of an integral transformation the problem is reduced to that of two-dimensional parallel flow, which can be investigated by the well-known methods of the theory of analytic functions. The problem of plane flow is solved by means of a total approximation for the parametric plane.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 98–103, September–October, 1993.     

Complex Parametric Vibrations of Flexible Rectangular Plates

In this paper we consider parametric oscillations of flexible plates within the model of von Kármán equations. First we propose the general iterational method to find solutions to even more general problem governed by the von Kármán–Vlasov–Mushtari equations. In the language of physics the found solutions define stress–strain state of flexible shallow shell with a bounded convex space R ² and with sufficiently smooth boundary . The new variational formulation of the problem has been proposed and his validity and application has been discussed using precise mathematical treatment. Then, using the earlier introduced theoretical results, an effective algorithm has been applied to convert problem of finding solutions to hybrid type partial differential equations of von Kármán form to that of the ordinary differential (ODEs) and algebraic (AEs) equations. Mechanisms of transition to chaos of deterministic systems with infinite number of degrees of freedom are presented. Comparison of mechanisms of transition to chaos with known ones is performed. The following cases of longitudinal loads of different sign are investigated: parametric load acting along X direction only, and parametric load acting in both directions X and Y with the same amplitude and frequency.     

Comparative analysis of methods of areal flooding of stratified reservoirs of anomalous oil

When anomalous oils in productive reservoirs are displaced by water [1], residual undisplaced oil remains [2, 3]. For calculating the volume of the unrecovered oil, most studies have been of homogeneous reservoirs [4–8]. In the present paper, the residual volume is assessed using an inhomogeneous-stratified-model. The inhomogeneity of the strata is described by the random distributions of Weibull or Pearson (gamma-distribution), which are characteristic of oilfields possessing anomalous properties (see, for example, [9]). A parametric study is made of four known schemes of areal flooding: two-point (straight line), five-, seven-, and nine-point [10]. The obtained estimates of the volume of the residual oil can be used for rational selection of the parameters for exploiting anomalous oilfields.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 121–126, May–June, 1985.     

Dispersion theory of surface waves on an inhomogeneous plasma in a strong high-frequency field

The spectrum of the characteristic surface oscillations and stability of a plasma in a strong high-frequency (hf) electric field are studied. It is shown that inhomogeneity of the plasma leads to spatial dispersion and to specific damping of stable oscillations of a plasma in an external hf field, the frequency of which greatly exceeds the plasma frequency. A systematic theory of parametric resonance at the frequency of the electronic surface oscillations is developed taking account of the inhomogeneity of the plasma.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 11–14, January–February, 1973.The authors thank V. P. Silin for interest in this work and for valuable observations.