Maximum Static Friction Force Research Articles

Minimum‐time control of systems with Coulomb friction: near global optima via mixed integer linear programming

AbstractThis work presents a method of finding near global optima to minimum‐time trajectory generation problems for systems that would be linear if it were not for the presence of Coulomb friction. The required final state of the system is assumed to be maintainable by the system, and the input bounds are assumed to be large enough so that the role of maintaining zero acceleration during finite time intervals of zero velocity (the role of static friction) can always be assumed by the input. Other than the previous work for generating minimum‐time trajectories for robotic manipulators for which the path in joint space is already specified, this work represents, to the best of our knowledge, the first approach for generating near global optima for minimum‐time problems involving a non‐linear class of dynamic systems. The reason the optima generated are near global optima instead of exactly global optima is due to a discrete‐time approximation of the system (which is usually used anyway to simulate such a system numerically). The method closely resembles previous methods for generating minimum‐time trajectories for linear systems, where the core operation is the solution of a Phase I linear programming problem. For the non‐linear systems considered herein, the core operation is instead the solution of a mixed integer linear programming problem. Copyright © 2001 John Wiley & Sons, Ltd.

Self-excited and Forced Oscillations with Solid Friction. 2nd Report. Occurrence of Limit Cycles and Evaluation of Solutions in Case of Combined Friction-Velocity Characteristics.

Forced frictional oscillations in a self-excited system occurred when the maximum static friction force did not become an isolation point and the kinetic friction force was given as a combined function of the relative sliding velocity composed of linear and hyperbolic functions. This process of oscillation was reviewed by the averaging method in continuation of the previous report. The properties of the limit cycle occurring where there was only one unstable first approximation solution of the harmonic oscillation with the values of the oscillation force circular frequency and the moving surface velocity are explained in this report. Various characteristics of the resultant stable, non-harmonic oscillations are also reviewed. By comparing results of the previous report with highly accurate, piecewise linear approximation solution, it was determined that this averaging method workes quite well and therefore is an effective method for judging the stability of the solution. It was learned that the highly accurate solution of the sliding motion could be estimated quantitatively by the stable first approximation solution using the averaging method. The unstable region in the first approximation solution corresponded with the occurrence region of the non-harmonic oscillations in the highly accurate solution. The solution by the averaging method showed the qualitative change of the amplitude.

Self-excited and Forced Oscillations with Solid Friction. 1st Report. Solutions and Stabilities by Averaging Method in Case of Combined Friction-Velocity Characteristics of Linear and Hyperbolic Curves.

In the present paper, the frictional oscillations in a mechanical driving system are studied theoretically on the self-excited and forced oscillations, when the friction-velocity characteristic curves are given as the combined curves of linear and hyperbolic functions and maximum static friction force is not an isolated point. Bodies acted upon by sinusoidal external force while sliding along a surface at a fixed velocity exhibit regular oscillations. We have analyzed these motions using an averaging method, without distinguishing between slipping and sticking. We have also examined the amplitude characteristic of motions found in the obtainable first approximations of the form of harmonic vibrations. Finally, we made a detailed investigation of the stability of these first approximations. Also, we have assembled the results of a comparison of the precise solutions for motions over a wide range of velocities where the kinetic velocity of the body exceeding the surface motion velocity, and we propose a usable theoretical analysis method, aimed at understanding the important aspects of regular harmonic solutions.

Studies on Contact Geometry and Limiting Resistant Torque Characteristics of the Variable Torque Slipping Clutch with Skewed Rollers.

A slipping clutch capable of delivering a constant resistant torque is described in this paper. Its principle is based on the resistance to slipping offered by the askew disposed cylindrical rollers rolling and slipping between the two specially shaped races. There are three types of this slipping clutch, namely the Right, Left and Cross Skewing respectively, which are distinguished by the directions of the roller askew arrangement and the rotation of the driving race. As a newly developed engineering surface, the surface of the races is in the spatial line contact with the skewed cylindrical rollers. After the equations of the race surface is established, the contact geometry of the tangency between the roller and races is investigated. This is essential for understanding the clutch properties and furthering the contact stress and EHL analyses. In addition, the analyzing procedure used here is also helpful for the study of other sophisticated line contact problems. Based on the assumptions of the maximum static friction force and the uniform distribution of the applied axial force along the contact line, the limiting torque characteristics are discussed in detail. The analysis of the maximum resistant torque clarifies the existence of self-locking problems under certain design parameters. On the other hand, the analysis of the minimum resistant torque shows how the resistant torque approaches the minimum values with changes of the geometrical parameters. These analyses provide the design principals for this slipping clutch in order to optimize the torque characteristics. The preliminary experiments are conducted with two types out of the three and the results prove those concluded from the theoretical analyses very well.

Self-excited and Forced Oscillations with Solid Friction. 3rd Report. Piecewise Linear Approximate Solutions in Case of Combined Friction-Velocity Characteristics.

Frictional oscillations occurring in a forced self-excited system were treated when the kinetic friction force was shown as a combined characteristic curve of linear and hyperbolic functions of the relative sliding velocity. The authors studied the characteristics and stabilities of the oscillations in the first approximate solution for harmonic oscillation form (among the steady-state oscillation solutions which can be obtained by the averaging method). The study results of various properties of the solution for non-harmonic oscillation solution accompanying the occurrence of limit cycles have been reported previously. Highly accurate, steady-state oscillation solutions (with consideration given to the two motions of slipping and sticking) are obtained with a computer by approximating the characteristic curve of the frictional force and slipping velocity with n broken lines and connecting the successive solution curves on the phase plane in the velocity boundaries of each broken line. The piecewise linear approximation method for obtaining a highly accurate solution which is very similar to the exact solution is explained first. The occurrence forms and the distribution conditions of various steady-state oscillation solutions unobtainable by the averaging method, including the maximum static friction force becoming an isolated point, are described. The resonance characteristics of the system and the effect of the velocity of the moving surface on the characteristics of the steady-state oscillation solutions are reported.

Numerical and theoretical study of friction between atomically clean surfaces and effects of impurities

Sliding friction between atomically clean surfaces with and without impurities is studied numerically and theoretically. We show that the maximum static frictional force shows complicated behavior as a function of the elastic constant of the material, which results from various discommensurate structures. We also show that impurities can diminish the maximum static frictional force. Velocity dependences of the kinetic frictional force in the absence and presence of impurities are calculated. Transverse pinning, which yields static frictional force along the transverse direction to the sliding direction, shows peculiar behavior. It is also shown that the direction of the sliding motion can deviate from that of the external force.

Theoretical and numerical study of sliding friction based on the extended-Frenkel-Kontorova model

We study sliding friction based on the extended Frenkel-Kontorova model in I and 2 dimensions. Aubry transition, that is, the transition between commensurate and discommensurate structures, is discussed. It turns out that the phonon gap and the maximum static frictional force varies in a fractal way as a function of the elastic constants of the system due to the complicated change of discommensurate structures. Transverse pinning in the case of 2 dimension is also studied.

Anomalous pinning behavior in an incommensurate two-chain model of friction

Pinning phenomena in an incommensurate two-chain model of friction are studied numerically. The pinning effect due to the breaking of analyticity exists in the present model. The pinning behavior is, however, quite different from that for the breaking of the analyticity state of the Frenkel-Kontorova model. When the elasticity of chains or the strength of interchain interaction is changed, pinning force and maximum static frictional force show anomalously complicated behavior accompanied by a successive phase transition and they vanish completely under certain conditions.

Friction Force, Contact Resistance, and Lubricant Shear Behavior at the Magnetic Head-Disk Interface During Starting

During the starting operation of magnetic rigid disks, a stiction phenomenon characterized by a high friction force may be encountered due to the smoothness of the contacting surfaces and the small thickness of the lubricant film. Since friction measurement using a force transducer yields a signal proportional to the slider displacement, the real friction force at the head-disk interface cannot be measured directly. In the present study, a dynamic data analysis scheme is developed to obtain the real friction force as a function of time based on the measured apparent friction force. Electric contact resistance measurements demonstrate that the transition from static to kinetic friction occurs before the apparent friction force reaches a maximum value. Assuming a constant acceleration of the disk, the relative slip velocity at the contact interface is obtained as a function of time. The relationship between the shear stress and shear strain rate for a relatively thick lubricant film is found to be approximately linear up to a critical value of the shear stress. Due to the extremely high shear strain rates, the maximum real friction force can be significantly greater than the maximum static friction force.

Theoretical study of friction: One-dimensional clean surfaces.

Another method has been proposed to evaluate the frictional force in the stationary state. This method is applied to the one-dimensional model of clean surfaces. The kinetic frictional force is seen to depend on velocity in general, but the dependence becomes weaker as the maximum static frictional force increases and in the limiting case the kinetic friction becomes only weakly dependent on velocity as described by one of the laws of friction. It is also shown that there is a phase transition between a state with vanishing maximum static frictional force and a state with finite static frictional force. The role of randomness at the interface and the relation to the impurity pinning of the sliding charge-density wave are discussed.