Vol. 18, no. 1
Vol. 18, no. 1, 2022
Chekirou F., Brahimi K., Bournine H., Hamouda K., Haddad M., Benkajouh T., Le Bot A.
Abstract
This paper presents the effect of frictioninduced vibration between two beams in relative
motion according to Timoshenko’s beam theory. This system is composed of two cantilever
beams screwed together, allowing friction force to occur in the contact interface. The nonlinear
behavior can be divided into two phases: stick and slip. The differential equations of motion in
the two phases are developed, with the precision of the transition condition between each phase.
A number of experiments are carried out in order to validate the theoretical model, the main
contribution of which is to test these specimens in modes greater than one. The experiments
demonstrate the influence of changing the clamping force on the stiffness of the structure and
thus on its frequency and damping ratio. The comparison between theory and experiments
reveals a good agreement. In addition, the tests show an increase in the modal damping ratio
when the frequencies are increased. This leads to a considerable increase in energy dissipation
by the structure, making it a good choice as a friction damper.

Lim C. C.
Abstract
A statistical mechanics canonical spherical energyenstrophy theory of the superrotation phenomenon in a quasi2D barotropic fluid coupled by inviscid topographic torque to a rotating solid body is solved in closed form in Fourier space, with inputs on the value of the energy to enstrophy quotient of the fluid, and two planetary parameters — the radius of the planet and its rate and the axis of spin. This allows calculations that predict the following physical consequences: (A) two critical points associated with the condensation of high and low energy (resp.) states in the form of distinct superrotating and subrotating (resp.) solidbody flows, (B) only solidbody flows having wavenumbers $l=1$, $m=0$ — tiltless rotations — are excited in the ordered phases, (C) the asymmetry between the superrotating and subrotating ordered phases where the subrotation phase transition also requires that the planetary spin is sufficiently large, and thus, less commonly observed than the superrotating phase, (D) nonexcitation of spherical modes with wavenumber $l>1$ in barotropic fluids. Comparisons with other canonical, microcanonical and dynamical theories suggest that this theory complements and completes older theories by predicting the above specific outcomes.

Cherepantsev A. S.
Abstract
This paper is concerned with the study of the patterns of the behavior of coupling elements
in the OFC model, which describes the statistical regularities of the seismic regime. It is shown
that there are two different modes of synchronous drop formation, simulating an earthquake.
Both mechanisms are determined by the capture of a neighboring element and the subsequent
synchronization of the drops. This process forms a stable drop of a larger size. The first mechanism
is typical for the initial stage of the system’s evolution toward a steady selforganized
critical state. In this case, the capture is determined by different rates of input of energy into the
elements in nearboundary regions of the lattice. The second mechanism is based on an increase
in the number of cluster boundary elements and, accordingly, an increase in the probability of
capture and synchronization of neighboring external elements. The theoretical values of the parameter
of the cluster size growth rate presented in this work are in good agreement with the
calculated values.

Chernova A. A.
Abstract
This paper addresses problems of mathematical modeling of heat exchange processes in the prenozzle volume of a solid propellant rocket engine with a charge with starlike crosssection and a recessed hinged nozzle. Methods of mathematical modeling are used to solve the quasistationary spatial conjugate problem of heat exchange. An analysis is made of the influence of RANS turbulence models on the flow structure in the flow channels of the engine and on the computed heat flow distributions over the surface of the recessed nozzle. Methods of mathematical modeling are used to solve the quasistationary spatial conjugate problem of heat exchange. Results of validation of RANS turbulence models are presented using wellknown experimental data. A comparison of numerical and experimental distributions of the heattransfer coefficient over the inlet surface of the recessed nozzle for the engine with a cylindrical channel charge is made for a primary choice of turbulence models providing a qualitative agreement between calculated and experimental data. By analyzing the results of numerical modeling of the conjugate problem of heat exchange in the combustion chamber of the solid propellant engine with a starlike channel, it is shown that the SST $k  \omega$ turbulence model provides local heattransfer coefficient distributions that are particularly close to the experimental data.

Melnik I. C.
Abstract
In this work we will show how any elliptic integral can be computed by analyzing the asymptotic behavior of idealized mechanical models. Specifically, our results reveal how a set of circular billiard systems computes the canonical set of three elliptic integrals defined by Legendre. We will treat these Newtonian systems as a particular application of the billiardball model, a ballistic computer idealized by Eduard Fredkin and Tommaso Toffoli. Initially, we showed how to define the initial conditions in order to encode the computation of a set of integral functions. We then combined our first conclusions with results established in the 18th and 19th centuries mostly by Euler, Lagrange, Legendre and Gauss in developing the theory of integral functions. In this way, we derived collisionbased methods to compute elementary functions, integrals functions and mathematical constants. In particular, from the Legendre identity for elliptic integrals, we were able to define a new collisionbased method to compute the number $\pi$, while an identity demonstrated by Gauss revealed a new method to compute the arithmeticgeometric mean. In order to explore the computational potential of the model, we admitted a hypothetical device that measures the total number of collisions between the balls and the boundary. There is even the possibility that the methods we are about to describe could one day be experimentally applied using optical phenomena, as recent studies indicate that it is possible to implement collisionbased computation with solitons.

Guha P., Choudhury A. G., Khanra B., Leach P.
Abstract
We describe a method to generate nonlocal constants of motion for a special class of nonlinear
ODEs. We employ the method of the generalized Sundman transformation to obtain certain new
nonlocal first integrals of autonomous secondorder ordinary differential equations belonging to
the classification scheme developed by Painlevé and Gambier.

Kalyakin L. A.
Abstract
Dynamical bifurcations occur in oneparameter families of dynamical systems, when the
parameter is slow time. In this paper we consider a system of two nonlinear differential equations
with slowly varying righthand sides. We study the dynamical saddlenode bifurcations that occur
at a critical instant. In a neighborhood of this instant the solution has a narrow transition layer,
which looks like a smooth jump from one equilibrium to another. The main result is asymptotics
for a solution with respect to the small parameter in the transition layer. The asymptotics is
constructed by the matching method with three time scales. The matching of the asymptotics
allows us to find the delay of the loss of stability near the critical instant.

Atepor L., Akoto R. N.
Abstract
Autoparametric vibration absorber is a machine invented to suppress vibration and has been
widely employed in many fields of engineering. Previous works reported by various researchers
have shown that dangerous motions, like the full rotation of the pendulum subsystem or chaotic
motion, can emerge due to small perturbations of initial conditions or system parameters. To
tackle this problem, a new model of the autoparametric vibration absorber with an attached
piezoelectric actuator exciter is proposed in this paper. Under the effects of parametric excitation
forces produced by the exciter, the vibration absorber will absorb more vibration energy. The
dynamic response of the new system is studied analytically using the method of multiple scales
and the results validated numerically using the continuation method and detailed bifurcation
analysis. The results show that the vibration amplitudes of the subsystems are reduced, the
region over which the absorption takes place gets widened and chaotic regions are removed with
the introduction of parametric excitation forces in contrast to that of the original model of the
autoparametric vibration absorber.
