Four-dimensional Integral Model of Dry Friction on the Example of Wheel Movement

A four-dimensional model of dry friction in the interaction of a solid wheel and a horizontal rough surface is investigated. It is assumed that there is no separation between the wheel and the horizontal surface. The movement of the body occurs in conditions of combined dynamics, when in addition to the sliding movement, the body participates in spinning and rolling. The equation of motion of the wheel is compiled using the Appel equation. The resulting model of sliding, spinning, and rolling friction is given for the case where the contact area is a circle. The cumbersome integral expressions were replaced by fractional-linear Pade approximations. Pade approximations accurately describe the behavior of the components of the friction model. A mathematical model is proposed that describes the simultaneous sliding, spinning and rolling of a solid wheel. The dependences of the parallel and perpendicular components of the friction force and the torque of the spinning friction were ploted with respect to the parameter that characterizes the movement of the wheel. Comparisons of the integral friction model and the model based on Pade approximations are presented. The results of the comparison showed a qualitative correspondence of the models. After obtaining the equation of motion, the simulation of motion at a constant control torque of the wheel is carried out. The graphs allow you to match the logical behavior of the wheel movement.

Conventional and Evolutionary Order Reduction Techniques for Complex Systems

Order reduction of the large-scale linear dynamic systems (LSLDSs) using stability equation technique mixed with the conventional and evolutionary techniques is presented in the paper. The reduced system (RS) is obtained by mixing the advantages of the two methods. For the conventional technique, the numerator of the RS is achieved by using the Pade approximations and improved Pade approximations, whereas the denominator is obtained by the stability equation technique (SET). For the evolutionary technique, numerator of the RS is obtained by minimizing the integral square error (ISE) between transient responses of the original and the RS using the genetic algorithm (GA), and the denominator is obtained by the stability equation method. The proposed RS retains almost all the essential properties of the original system (OS). The viability of the proposed technique is proved by comparing its time, frequency responses, time domain specifications, and ISE with the new and popular methods available in the literature.

Three-dimensional integral dry friction model for the motion of a rectangular body

Introduction. A three-dimensional dry friction model in the interaction of a rectangular body and a horizontal rough surface is considered. It is assumed that there is no separation of the body from the horizontal surface. The body motion occurs under the conditions of combined dynamics when, in addition to the longitudinal movement, the body participates in twisting.Materials and Methods. Linear fractional Pade approximations are proposed, which replaced the cumbersome analytical expressions that most accurately describe the motion of bodies on rough surfaces. New mathematical models describing sliding and twisting of bodies with a rectangular base are proposed.Results. Analytical expressions of the principal vector and moment of friction for rectangular contact areas are developed and scientifically established. A friction model that takes into account the relationship between sliding and twisting speeds, which provides finding solutions for Pade dependences, is developed. After numerical solution to the equations of motion, the dependences of the sliding speed and angular velocity on time were obtained and constructed. Graphs of the dependences of the friction forces and their moment on two parameters (angular velocity and slip velocity) were constructed, which enabled to compare the integral and normalized models of friction. The comparison results showed good agreement of the integral model and the model based on Pade approximations.Discussion and Conclusions. The results obtained provide considering the dynamic coupling of components, which determines the force interaction of a rectangular body and a horizontal surface. These results can be used in mobile robotics. The analyzed motion of the body occurs through the motion control of a material point inside the body. Such mobile robots can be used when solving a wide class of problems: when creating autonomous robots for the exploration of outer space and planets; in the diagnosis and treatment in case of passing through complex structures of veins and arteries; in research under water, in places of large differential temperature; in underground operations.

Some Padé approximations and inequalities for the complete elliptic integrals of the first kind

In this paper, we present Padé approximations of some functions involving complete elliptic integrals of the first kind $K(x)$ K ( x ) , and motivated by these approximations we also present the following double inequality: $$ \frac{1-x^{2}}{1-x^{2}+\frac{x^{4}}{62}}< \frac{2 e^{\frac{2}{\pi }K(x)-1}}{ (1+\frac{1}{\sqrt{1-x^{2}}} )}< \frac{1-\frac{96}{100}x^{2}}{1-\frac{96}{100}x^{2}+\frac{x^{4}}{64}},\quad x\in ( 0,1 ). $$ 1 − x 2 1 − x 2 + x 4 62 < 2 e 2 π K ( x ) − 1 ( 1 + 1 1 − x 2 ) < 1 − 96 100 x 2 1 − 96 100 x 2 + x 4 64 , x ∈ ( 0 , 1 ) . Our results have superiority over some new recent results.

Padé Approximation to Solve the Problems of Aerodynamics and Heat Transfer in the Boundary Layer

In this chapter, we describe the applications of asymptotic methods to the problems of mathematical physics and mechanics, primarily, to the solution of nonlinear singular perturbed problems. We also discuss the applications of Padé approximations for the transformation of asymptotic expansions to rational or quasi-fractional functions. The applications of the method of matching of internal and external asymptotics in the problem of boundary layer of viscous gas by means of Padé approximation are considered.

Padé and Post-Padé Approximations for Critical Phenomena

We discuss and apply various direct extrapolation methods for calculation of the critical points and indices from the perturbative expansions my means of Padé-techniques and their various post-Padé extensions by means of root and factor approximants. Factor approximants are applied to finding critical points. Roots are employed within the context of finding critical index. Additive self-similar approximants are discussed and DLog additive recursive approximants are introduced as their generalization. They are applied to the problem of interpolation. Several examples of interpolation are considered.