This paper presents the effect of friction-induced 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.

A statistical mechanics canonical spherical energy-enstrophy theory of the superrotation phenomenon in a quasi-2D 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.) solid-body flows, (B) only solid-body 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.

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 self-organized critical state. In this case, the capture is determined by different rates of input of energy into the elements in near-boundary 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.

This paper addresses problems of mathematical modeling of heat exchange processes in the pre-nozzle volume of a solid propellant rocket engine with a charge with starlike cross-section and a recessed hinged nozzle. Methods of mathematical modeling are used to solve the quasi-stationary 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 quasi-stationary spatial conjugate problem of heat exchange. Results of validation of RANS turbulence models are presented using well-known experimental data. A comparison of numerical and experimental distributions of the heat-transfer 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 heat-transfer coefficient distributions that are particularly close to the experimental data.

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 billiard-ball 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 collision-based 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 collision-based method to compute the number $\pi$, while an identity demonstrated by Gauss revealed a new method to compute the arithmetic-geometric 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 collision-based computation with solitons.

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 second-order ordinary differential equations belonging to the classification scheme developed by Painlevé and Gambier.

Dynamical bifurcations occur in one-parameter 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 right-hand sides. We study the dynamical saddle-node 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.

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.