Approximation of partial differential equations on compact resistance spaces

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.

Modeling and analytical methods for a mounting system with double stage isolation

A calculation method for obtaining the displacements and rigid body modes of a Powertrain Mounting System (PMS) with double stage isolation is proposed in this paper. Firstly, the PMS with double stage isolation is modeled as a 12 Degree of Freedoms (DOFs) model, which includes six DOFs for the powertrain and the subframe respectively. The mounts are simplified as a three-dimensional spring along each axis of its Local Mount Coordinate System (LMCS), which takes the non-linear relation of the force versus the displacement of each spring into account. Secondly, the quasi-static equilibrium equation and the free vibration equation as well as the forced vibration equation of the proposed model are derived and the solutions of equations are presented. Then, the calculation and solution methods are validated by the simulation results. The differences of rigid body modes and displacements of the powertrain between single and double stage isolation are estimated, which demonstrates that the proposed model is more accurate, especially when powertrain mounts are stiff. Also, the effect of locations for powertrain mounts on car body is investigated, which shows that is beneficial for motion control of powertrain.

Influence of limiting the duration of the armature winding current on the operating indicators of a linear pulse electromechanical induction type converter

Introduction. Linear pulse electromechanical converters of induction type (LPECIT) are used in many branches of science and technology as shock-power devices and electromechanical accelerators. In them, due to the phase shift between the excitation current in the inductor winding and the induced current in the armature winding, in addition to the initial electrodynamic forces (EDF) of repulsion, subsequent EDF of attraction also arise. As a result, the operating indicators of LPECIT are reduced. The purpose of the article is to increase the performance of linear pulse electromechanical induction-type converters when operating as a shock-power device and an electromechanical accelerator by limiting the duration of the induced current in the armature winding until its polarity changes. Methodology. To analyze the electromechanical characteristics and indicators of LPECIT, a mathematical model was used, in which the solutions of equations describing interrelated electrical, magnetic, mechanical and thermal processes are presented in a recurrent form. Results. To eliminate the EDF of attraction between the LPIECIT windings, it is proposed to limit the duration of the induced current in the armature winding before changing its polarity by connecting a rectifier diode to it. It was found that when the converter operates as a shock-power device without limiting the armature winding current, the value of the EDF pulse after reaching the maximum value decreases by the end of the operating cycle. In the presence of a diode in the armature winding, the efficiency criterion, taking into account the EDF pulse, recoil force, current and heating temperature of the inductor winding, increases. When the converter operates as an electromechanical accelerator without limiting the armature winding current, the speed and efficiency decrease, taking into account the kinetic energy and voltage of the capacitive energy storage at the end of the operating cycle. In the presence of a diode in the armature winding, the efficiency criterion increases, the temperature rise of the armature winding decreases, the value of the maximum efficiency increases, reaching 16.16 %. Originality. It has been established that due to the limitation of the duration of the armature winding current, the power indicators of the LPECIT increase when operating as a shock-power device and the speed indicators when the LPECIT operates as an electromechanical accelerator. Practical value. It was found that with the help of a rectifier diode connected to the multi-turn winding of the armature, unipolarity of the current is ensured, which leads to the elimination of the EDF of attraction and an increase in the performance of the LPECIT.

Particular solutions of equations with multiple characteristics expressed through hypergeometric functions

In the paper, similarity solutions are constructed for a model equation with multiple characteristics of an arbitrary integer order. It is shown that the structure of similarity solutions depends on the mutual simplicity of the orders of derivatives with respect to the variable x and y, respectively. Frequent cases are considered in which they are shown as fundamental solutions of well-known equations, expressed in a linear way through the constructed similarity solutions.

Exact Solutions of Boundary Layer Equations in Polymer Solutions

The paper presents new exact solutions of equations derived earlier. Three of them describe unsteady motions of a polymer solution near the stagnation point. A class of partially invariant solutions with a wide functional arbitrariness is found. An invariant solution of the stationary problem in which the solid boundary is a logarithmic curve is constructed.

Equations of Electrodynamics

The equations of electrodynamics are presented, it is shown that plane and spherical electromagnetic waves are their solutions, while the spherical wave propagates only outward. Fields of uniformly moving charges are also solutions of equations. The question of finding a universal form of equations admitting a solution in the form of a field of an arbitrarily moving charge remains open. The question is raised about the existence of a two-parameter group of transformations of electromagnetic fields along with the well-known one-parameter group. The equation of motion of a charged particle in an electromagnetic field is considered without simplifying approximations. The principle of operation of an unconventional alternator in a constant electric field and a corresponding engine, as well as new types of generators of direct and impulse current, are described.

On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces

A nonlinear system of differential equations describing the rotational motion of a rigid body under the action of torque of potential and circular-gyroscopic forces is considered. For this torque, the system of differential equations has three classical first integrals: the energy integral, the area integral, and the geometric integral. For the analogue of the Lagrange case, when two moments of inertia coincide and the potential depends on one angle, an additional first integral is found and integration in quadratures is performed. A number of examples is considered where parametric families of exact solutions are considered. In these examples, polynomial or analytical functions were used as a potential. In particular, we construct families of periodic and almost periodic motions, as well as families of asymptotically uniaxial rotations. We also identified movements that have limit values of opposite signs for unlimited increase and decrease of time.

General Fractional Dynamics

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.