A qualitative analysis of the brachistochrone problem with dry friction and maximizing the horizontal range

The problem of maximizing the horizontal coordinate of a point moving in a vertical plane under the action of gravity and dry friction and the corresponding brachistochrone problem are considered. The optimal control problem is reduced to a boundary value problem for a system of two nonlinear differential equations. A qualitative analysis of the trajectories of this system is carried out, their typical features are found and illustrated by numerical solving of the boundary value problem. It is shown that the normal component of the support reaction should be positive when moving along the optimal curve. The optimality of the found extremals is discussed.     

Brachistochrone for a rolling cylinder

The motion of a heavy homogeneous cylinder is considered as a no-slip rolling along the desired curve. We obtain a functional in the form of the total time of the cylinder rolling and solve the corresponding variational problem of minimizing this functional. We obtain an algebraic equation for the directional line of steepest descent, brachistochrone, in parametric form. We use the equation of motion of the cylinder with constraint reaction to determine the conditions of implementation of its pure rolling without separation and slip with respect to the brachistochrone.     

Cycloidal pendulum with a rolling cylinder

Free vibrations of a heavy homogeneous cylinder rolling in a cylindrical cavity whose directing curve is a brachistochrone are considered. The equation of motion of the cylinder is derived and the circular frequency of free vibrations of the cylinder center of mass is determined. An analogy between the cycloidal pendulum with a rolling cylinder and the classical cycloidal pendulum in the form of a material point is obtained.     

Evolutionary solutions to the brachistochrone problem with Coulomb friction

The classical brachistochrone problem was originally proposed by Johann Bernoulli more than 300 years ago, and is still one of the most elegant problems in modern mathematics. This paper examines a generalisation of the original brachistochrone problem by the inclusion of a non-conservative force in the form of Coulomb friction. Solutions to this problem are found using an evolutionary computational technique based on Darwinian natural selection and survival of the fittest. These evolutionary solutions tend to indicate that as friction is introduced the optimum curves develop from cycloidal (the analytical solution for the conservative problem) toward straight lines. The flexibility of the evolutionary method to optimise for other objectives is then discussed.     

Conditions for pure rolling of a heavy cylinder along a brachistochrone

Algebraic equations for the line of steepest descent of a cylinder are derived in parametric form. Conditions for rolling without slipping and separation of the cylinder along a brachistochrone are established based on the equations of motion with constraint reaction. The important conclusion is drawn that the center of mass of a cylinder moving along a brachistochrone describes a cycloid     

Brachistochrone with Coulomb friction

The classical brachistochrone is considered with the inclusion of a resistant force, which is due to Coulomb friction, in addition to the uniform gravitational force that is present. The solution to this problem is expressed in terms of standard functions, and it is developed in two separate ways by means of constrained variational calculus methods. These ways involve formulations of the problem in terms of temporal and spatial independent variables, respectively. The equations of motion that result in both cases are non-linear and coupled. The utilization of path variables is a central feature of the developments provided.     

The Okhotsimskii-Pontryagin method in control theory and analytical mechanics. Part II

The Okhotsimskii method for the differentiation of functionals in control theory is discussed on the basis of the Pontryagin formalism. The relation of this method to other approaches to solving control problems and to other methods of analytical mechanics is studied. Some typical cases of solving optimal control problems are considered. The brachistochrone problem is solved with consideration of viscous friction.     

A STUDY OF THE CATASTROPHE AND THE CAVITATION FOR A SPHERICAL CAVITY IN HOOKE’S MATERIAL WITH 1/2 POISSON’S RATIO

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具有黏滞阻力时的最速降线

针对实际应用中存在黏滞阻力的最速降线的问题, 首先推导出适于此类问题的解除约束的 广义变分原理, 它适用于具有摩擦阻尼和多自由度系统优化的问题. 得到描述有黏滞阻力情 况下最速降线相关函数的微分方程, 它在黏滞阻力为零时即退化为滚轮线. 利用MATLAB数值 计算给出了最速降线受黏滞阻力的影响: 在黏滞阻力系数较小时最速降线趋于变凹, 当阻力 系数增大到一定值之后最速降线趋于平缓, 当阻力系数很大时最速降线趋于直线.     

Brachistochronic motion of nonholonomic nonlinear mechanical system of the Chaplygin type

In this paper, the differential equations of brachistochronic motion of nonholonomic nonlinear system of the Chaplygin type were determined. The additional forces R_α realizing the corresponding motion of the system together with the given forces were also determined. Lastly, the brachistochrone was solved for a concrete model using mathematical numerical methods.     

Dynamics of vibration isolation system with a quasi-isochronous roller shock absorber

The low-frequency vibrations of a vibration isolation system of rigid bodies (roller shock absorber and carrying body) under external harmonic loading are considered. The working surface of the absorber has the form of a brachistochrone. The equations describing the slip-free motion of the absorber over the hinged roller and the motion of the carrying body are derived. A graphical method for optimizing the parameters of the roller absorber as a component of the vibration isolation system is proposed     

Efficiency of a vibroprotection system with an isochronous roller damper

Low-frequency vibrations of a vibroprotection “roller damper-movable bearing body” system of rigid bodies under the action of an external harmonic excitation are considered. The working surface of the damper working body is formed by a brachistochrone. The dynamic equations of common no-slip motion of the damper working body on a hinged roller and of the bearing body are formulated. The roller damper tuning parameters are determined.     

On Constructing the Unique Solution for the Necking in a Hyper-Elastic Rod

In this paper, a novel approach which considers gradient effects and uses non-deforming boundary conditions is adopted to construct the unique solution for necking in a hyper-elastic rod. We study the problem of the large axially symmetric deformations of a rod composed of an incompressible Ogden’s hyper-elastic material subject to a tensile stress (or a given displacement) when its two ends are fixed to rigid bodies. The attention is on the class of energy functions for which the stress–strain curve in the case of the uniaxial tension has a peak and valley combination. A phase-plane analysis is introduced to study the qualitative behaviour of the solutions. Then, by using the non-deforming conditions at two ends, the solutions corresponding to trajectories in different phase planes are obtained. It turns out that the non-deforming conditions play an important role in selecting the solutions. Further, by converting the problem into a displacement-controlled problem, the unique solution is obtained. The engineering strain and engineering stress curve plotted from our solution exhibits two interesting phenomena: (i) After the stress reaches the peak value there is a sudden stress drop; (ii) Afterwards it is followed by a stress plateau. Some mathematical explanations on these two phenomena are then given.     

Trajectory planning and tracking control for autonomous bicycle robot

Control of the autonomous bicycle robot offers considerable challenges to the field of robotics due to its nonholonomic, underactuated, and nonminimum-phase properties. Furthermore, instability and complex dynamic coupling make the trajectory planning of the bicycle robot even more challenging. In this paper, we consider both trajectory planning and tracking control of the autonomous bicycle robot. The desired motion trajectory of the contact point of the bicycle’s rear wheel is constructed using the parameterized polynomial curve that can connect two given endpoints with associated tangent angles. The parameters of the polynomial curve are determined by minimizing the maximum of the desired roll angle’s equilibrium of the bicycle, and this optimization problem is solved by the particle swarm optimization algorithm. Then, a control scheme that can achieve full-state trajectory tracking while maintaining the bicycle’s balance is proposed by combining a planar trajectory tracking controller with a roll angle balance controller. Simulation results are presented to demonstrate the effectiveness of the proposed method.     

Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model

The problem of limit cycles for the Kolmogorov model is interesting and significant both in theory and applications. In this paper, we investigate the center-focus problems and limit cycles bifurcations for a class of cubic Kolmogorov model with three positive equilibrium points. The sufficient and necessary condition that each positive equilibrium point becomes a center is given. At the same time, we show that each one of point (1,2) and point (2,1) can bifurcate 1 small limit cycles under a certain condition, and 3 limit cycle can occur near (1,1) at the same step. Among the above limit cycles, 4 limit cycles can be stable. The limit cycles bifurcations problem for Kolmogorov model with several positive equilibrium points are hardly seen in published references. Our result is new and interesting.     

Problem of equilibrium of the timoshenko plate containing a crack on the boundary of an elastic inclusion with an infinite shear rigidity

A problem of equilibrium of a composite plate consisting of a matrix and an elastic inclusion with a through crack along the boundary of this inclusion is studied. The matrix deformation is described by the Timoshenko model, and the elastic inclusion deformation is described by the Kirchhoff-Love model. Conditions of mutual non-penetration of the crack edges are imposed on the curve that describes the crack. Unique solvability of the variational problem is proved. A system of boundary conditions on the curve bounding (in the mid-plane) the elastic inclusion is obtained. A differential formulation of the problem equivalent to the initial variational formulation is given.     

考虑剪切效应时压杆临界载荷的新数值方法

熟知,计入剪切效应时压杆临界载荷的求解归为某些常微分方程的特征值问题,本文提出一种求解该种问题的新数值方法.此种方法的思想是基于将特征值问题转化为同一方程的初值问题,要点为:(1)首先在积分始端取一个补充的初值,(2)又取一个暂设的特征值,(3)数值求解此微分方程的一个初值问题,(4)多次试凑和改变暂设特征值,让终端条件得到满足.数值算例已在文中给出.     

线性力下质点运动轨迹的几何分析

在质点所受的作用力为线性变化的情形下,当初始时刻作用力、末端时刻作用力与时间间隔上总位移三者不在同一平面时,质点运动轨迹必为空间挠曲线和正则的简单曲线,否则,必为平面曲线,且曲率和挠率处处不等于零。当质点运动轨迹为平面曲线时,给出了质点的运动轨迹曲线,有尖点、拐点和自交点的条件,并讨论了相关的性质,这些性质在几何上是直观的。     

Inverse problem of elastica of a variable-arc-length beam subjected to a concentrated load

An inverse problem of elastica of a variable-arc-length beam subjected to a concentrated load is investigated. The beam is fixed at one end, and can slide freely over a hinge support at the other end. The inverse problem is to determine the value of the load when the deflection of the action point of the load is given. Based on the elasitca equations and the elliptic integrals, a set of nonlinear equations for the inverse problem are derived, and an analytical solution by means of iterations and Quasi-Newton method is presented. From the results, the relationship between the loads and deflections of the loading point is obtained. The project supported by the National Natural Science Foundation of China(10272011) The English text was polished by Keren Wang     

Reservoir fluid flow through vertical fracture of thick stratum with underlying fluid interface

An exact solution is obtained for the problem of steady-state filtration of a heavy dense incompressible fluid in a thin, infinitely deep, inclined reservoir having a crack of given depth along the reservoir rise. The region of filtration of the lighter liquid (oil) has an impermeable upper boundary in the form of a horizontal fault line. Below the filtration region there is a free boundary, below which lies the region of stationary fluid (bottom water). The interface of the fluids, the fissure profile, and the reservoir fluid flow rate are determined from the solution of the problem on the basis of the given parameters (permeability of the reservoir and of the material filling the fissure, viscosity of the filtering fluid, specific weight of the upper and lower fluids, depth of the fissure, pressure differential between a point at the fissure and a point at the interface of the fluids). In the case when the thin reservoir is a vertical filtering layer, the considered flow is interpreted as the motion of the reservoir fluid through a vertical fissure of a thick reservoir (half-space) in the presence of an underlying fluid interface. The problem is solved in finite form with the aid of known analytic functions using integrals of the Cauchy type. The fundamental solution is first found of the special problem of flow with a point singularity. The fundamental solution is also of independent importance as an extension of the solution of certain known problems [1–4].