Nonlinear friction in quantum mechanics

This article presents the development of a multiple inner-loop control strategy for improving the performance of hydrostatic actuation systems. In these actuators, the presence of non-linearities associated with pump/motor static friction and backlash, pressure drop in the piping system, and nonlinear friction at the load have a significant effect on the performance and positional accuracy of the system. The effect of nonlinear friction at the pump/motor interface has been overcome by the use of a high gain pump-speed inner-loop control strategy. In this article, the concept of inner-loop control will be extended to target specific sources of performance degradation. Velocity feedback will be incorporated in this manner to decrease the effects of pump backlash and nonlinear friction at the load. Simulation results supported by theoretical analysis indicate that a considerable improvement in performance can be achieved by the implementation of this control strategy.

The vital purpose of a vehicle suspension system is to isolate the car body and hence passengers, from roadway unevenness disturbances. Implementation of passive suspension systems has continuously improved disconnection from disturbances through available deflection constraints to provide maximum isolation. In the majority of relevant reported research studies, a quarter car is modelled as moving vertically straight for both a viscous damper and a stiffness spring. The motivation for this study, reported here, is to extend the modelling to take account of the actual configuration of a test rig system. Accordingly, a new passive suspension system model is presented, which includes nonlinear lubricant friction forces that affect the linear support body bearings. The friction model established relies on dynamic system analysis and the fact of slipping body on lubricant bearings; this model captures most of the friction behaviours that have been observed experimentally. The suspension model is composed of a car body and wheel unit, and only vertical motion (bounce mode) is addressed. In addition, an active actuator is used to generate the system inputs as a road simulator. Therefore, a nonlinear hydraulic actuator, including the dynamic of servovalve and proportional–integral controller model, is established. This study is validated by experimental work, with simulations achieving C++compiler. As a result, a good agreement is obtained between the experimental and simulation results, that is, the passive suspension system with considered nonlinear friction and the nonlinear hydraulic actuator with servovalve equation models are entirely accurate and useful. The suggested proportional–integral controller successfully derives the hydraulic actuator to validate the control scheme. The ride comfort and handling response are close to that expected for the passive suspension system with road disturbances.

In the machine tool industry and the semiconductor manufacture fields, the necessity of high precision and fast response is rapidly spreading. In general, it is well known that a positioning control system contains nonlinear friction force; it may cause a serious disadvantage on control performances such as tracking errors, limit cycles and undesired stick-slip motion at very low motion velocity. In order to achieve the high level control performance, the frictional effects have to be removed. This paper presents a novel nonlinear friction compensation approach based on the idea of P, PI/I-P+FF control with sliding mode compensator for a high precision stage using a synchronous piezoelectric device driver. P, PI/I-P+FF control consists of Proportional position control and Proportional-Integral/Integral-Proportional velocity control with velocity Feed-Forward compensator. Experimental results are given to show the effectiveness of our proposed control method compared to the conventional P, PI/I-P+FF control method. Especially, the positioning error of our proposed control was reduced by more than 2/3 compared to the case of the conventional control at a stepping motion.

It is demonstrated how quantum mechanics is generated by stochastic momentum kicks from the force carriers, transmitting the fundamental interactions between the point particles. The picture is consistent with quantum field theory and points out that the force carriers are the only quantum particles. Since the latter are waves in the coordinate space, they are responsible for the wavy character of quantum mechanics.

In the following Part II the deterministic and stochastic laws of motion arising from the processes depicted in Part I (particulary Section 2), are analyzed in detail. In section 4 we study the typical non-linear logistic model emerging as the deterministic equivalent of the diffusion processes of Sections 2.3 and 2.4. The interaction of firm size and firm number are also studied within the same Section. In Section 5 we analyze the (asymptotic) stochastic laws of motion of the system. In particular, we study the Langevin equation and the approximations of the Fokker-Planck equations equivalent of the master equations of the same stochastic processes. We see that stochastic laws of motion may not be equivalent (not even asymptotically) to deterministic ones (e.g. Due to variance effects which determine increasingly larger fluctuations).

This paper presents the new vibration model of the horizontal washing machine (HWM) using a nonlinear geographic of the suspension system with nonlinear friction force - velocity (F-V) relationship of dampers obtained from experiment. Furthermore, using the dissipated energy function of dampers (Ed) criteria, some results of the effect of the spring stiffness on the dampers’ loading level are achieved. The simulation was done in Matlab/Simulink® environment. The results were then compared with experimental validation results to determine the suitability of the model. The conclusions can provide some available evidences for the design and improvement of nonlinear dampers.

PurposeThis study aims to investigate the microtopography transformation at a low-speed heavy-load interface with the lubrication of powder particles and its nonlinear friction effect on the sliding pair in contact.Design/methodology/approachBased on the universal mechanical tester (UMT) tribometer and VK shape-measuring laser microscope, comparative friction experiments were conducted with graphite powder lubrication. The friction coefficient with nonlinear fluctuations and the three-dimensional morphology of the boundary layer at the interface were observed and analyzed under different operating conditions. The effects on lubrication mechanisms and frictional nonlinearity at the sliding pair were focused on under different surface roughness and powder layer thickness conditions.FindingsAt a certain external load and sliding speed, the initial specimen surface with an appropriate initial roughness and powder thickness can store and bond the powder lubricant to form a boundary film readily. The relatively flat and firm boundary layer of powder at the microscopic interface can reduce the coefficient of friction and suppress its nonlinear fluctuation effectively. Therefore, proper surface roughness and powder layer thickness are beneficial to the graphite lubrication and stability maintenance of a friction pair.Originality/valueThis research is conducive to developing a deep understanding of the microtopography transformation with frictional nonlinearity at a low-speed heavy-load interface with graphite powder lubrication.

Steady State and Transient Gain Characteristics of the Stimulated Brillouin Scattered Mode in Quantum Semiconductor Plasmas

This paper proposes a method to solve practical problems of positioning control using a ball screw system. The practical problems contain two difficulties. First difficulty is that friction affects the performance of it. In this paper, the friction is divided into three components, nonlinear term including static friction and coulomb friction, Stribeck effect and linear term including viscous friction. Nonlinear friction and Stribeck effect are considered as disturbance. These are compensated by designing controller including integrator and loop shaping. In contrast, linear term friction can be considered in framework of linear control synthesis. Second difficulty is that the change in mass of load affects the system response. Moreover, the experiment shows viscous friction coefficient varies. The controller is designed to guarantee the robust stability for uncertain parameters, which are the mass of load and the viscous friction coefficient. It is designed by using descriptor representation, polytopic representation and linear matrix inequality. The effectiveness of the proposed method is verified by simulations and experiments.

In this study, an extension of linear-quadratic-Gaussian (LQG) control theory is used to determine the optimal state feedback controller for a nonlinear energy harvesting system that is driven by a stochastic disturbance. Specifically, the energy harvester is a base-excited single-degree-of-freedom (SDOF) resonant oscillator with an electromagnetic transducer embedded between the ground and moving mass. The electromagnetic transducer used to harvest energy from the SDOF oscillator introduces a nonlinear Coulomb friction force into the system, which must be accounted for in the design of the controller. As such, the development of the optimal controller for this system is based on statistical linearization, whereby the Coulomb friction force is replaced by an equivalent linear viscous damping term, which is calculated from the stationary covariance of the closed-loop system. It is shown that the covariance matrix and optimal feedback gain matrix can be computed by implementing an iterative algorithm involving linear matrix inequalities (LMIs). Simulation results are presented for the SDOF energy harvester in which the performance of the optimal state feedback control law is compared to the performance of the optimal static admittance over a range of disturbance bandwidths.

The Helmhotz equation of light beam propagating through a medium of complex refractive index is reduced to the axial-coordinate-dependent Schrodinger equation of complex potential. The new bra vector, the new expectation value of a dynamical variable and the extended Heisenberg picture are defined by the inverse of the evolution operator instead of its Hermitian adjoint, and the complex beam propagation parameters defined in terms of the new expectation value, the complex ABCD law and the ABCD formulation of the Huygens’ integral are discussed in terms of quantum mechanics. It is shown that the evolution equations of the complex beam propagation parameters are the same as those of the beam propagation parameters of beam propagating through a medium of real refractive index. The research on an optical system of the conservative complex beam quality factor shows that the complex ABCD law holds, the evolution of its coordinate operator and the momentum operator is linear, and the Huygens’ integral is of the ABCD formulation.

A preliminary conceptual design of a new free energy nano reactor is presented. The design is based on the following: A basically topological interpretation of the Casimir effect is given as a natural intrinsic property of the geometrical topological structure of the quantum-Cantorian micro spacetime. This new interpretation compliments the earlier conventional interpretation as vacuum fluctuation or as a Schwinger source and links the Casimir energy to the so called missing dark energy density of the cosmos. We start with a general outline of the theoretical principle and basic design concepts of a proposed Casimir dark energy nano reactor. In a nutshell the theory and consequently the actual design depends crucially upon the equivalence between the dark energy density of the cosmos and the faint local Casimir effect produced by two sides boundary condition quantum waves. This Casimir effect is then colossally amplified as a one sided quantum wave pushing from the inside with nothing balancing it from the non-existent outside. In view of the present theory, this is essentially what leads to the observed accelerated expansion of the cosmos. As in any reactor, the basic principle in the present design is to produce a gradient so that the excess energy on one side flows to the other side. Thus in principle we will restructure the local topology of space using material nanoscience technology to create an artificial local high dimensionality with a Dvoretzky theorem like volume measure concentration. Without going into the intricate nonlinear dynamics and technological detail, it is fair to say that this would lead us to pure, clean, free energy obtained directly from the topology of spacetime. Needless to say the entire design is based completely on the theory of quantum wave dark energy proposed by the present author for the first time in 2011 in a conference held in the Bibliotheca Alexandrina, Egypt and a little later in Shanghai, Republic of China. The quintessence of the present theory is easily explained as the o^3 intrinsic Casimir topological energy where o=(√(5-1))/2 produced from the zero set o of the quantum particle when we extract the empty set quantum wave o^2 from it and find 〖o-o〗^2=o^3 by restructuring space via conducting but uncharged plates similar to that of the classical Casimir experiments but with some modification. Our proposed preliminary design of the reactor follows in a natural way from the above.

In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.

Oscillations caused by solid friction are treated experimentally in a rotational system. A ring-like body is connected to a spring and can oscillate coaxially with a rotating shaft under the application of spring force and nonlinear friction force between sliding contact surfaces. Pre-driving of the rotating shaft just after the cleaning of sliding surfaces has a great influence on the characteristic curve betwen kinetic friction moment and peripheral velocity of the shaft. Pre-driving produces a change in configuration of the characteristic curve, therefore the corresponding amplitude-velocity curve and the period-velocity curve obtained experimentally show different appearances from those without pre-driving. These curves are qualitatively similar closely to theoretical ones derived from Lienard's construction.

The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox’s H-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177–190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.