RU2779278C1 - Methods for simulating static friction and apparatuses for the implementation thereof (variants) - Google Patents

Abstract

FIELD: testing.

SUBSTANCE: proposed technical solutions relate to the field of controlled reversible or intermittent systems, wherein one of the disturbing forces acting on said systems is friction, and simulation modelling thereof is performed in the process of control synthesis. Claimed is a method for simulating static friction, wherein in order to obtain the driving force signal, a friction force signal is added to the signal of the projection of the sum of active forces on the direction of movement (expected direction of movement); the result is divided by the value of the mass of the moving parts, resulting in the acceleration value; the velocity value is obtained by integrating the acceleration signal; if the value of modelled velocity reaches zero, an indicator of the presence of a state of rest is formed, mutually exclusive with the indicator of the presence of a state of motion, as well as a logical command to set the modelled velocity to zero by resetting the integrator to zero when transitioning from the state of motion to the state of rest; wherein the value of the motion friction force is used as a friction force signal in the state of motion, and if the indicator of a state of rest is present, the value of static friction force is used, the absolute value whereof is formed as the smaller in modulus value from the value of initiation friction force and the value of the projection of the sum of active forces in the corresponding direction. The attribute of a state of rest in the proposed technical solutions is therein developed in advance of the time of actual zero setting of the velocity. Proposed are three basic options for receiving commands to reset the integrator and forming an indicator of a state of rest, including those accounting for the mutual direction of the driving force and velocity and the possibility of integrating using an accumulating adder. A corresponding apparatus is proposed in order to implement each of the basic variants of simulation.

EFFECT: increase in the accuracy of matching the behaviour of the model to a real physical process in a dynamic system without increasing the order of the differential equations used by the model, and a possibility of solving the problem using low-efficiency computing equipment in real time, as well as analogue computers.

17 cl, 5 dwg

Description

The proposed technical solutions relate to the field of controlled systems, one of the disturbing forces acting on which is friction. In particular, these are systems with a model (Krasnova S.A., Utkin V.A. “Cascade synthesis of observers of the state of dynamic systems”, M., “Nauka”, 2006 - [1, Fig. 1.1]) and systems with friction compensation, for example Malafeeva A.A., "Device for controlling an electric drive with friction load", RF patent for invention No. 2079961, IPC H02R 5/06, 1992 - [2]. In these systems, in particular, computational modeling (simulation) of the friction value in the working bodies is carried out. Systems of this kind can, for example, be used to control position, speed, developed force (torque), be digital or analog, or analog-to-digital.

A feature of the friction force (hereinafter, when it comes to force, mass, speed, acceleration of translational motion, it is believed that these concepts are equivalent to the concepts of moment of force, moment of inertia, angular velocity, angular acceleration for rotational motion) is that it is unable to independently initiate movement (is not an active force), but only prevents it (active forces are understood to be all forces acting on the system, except for friction forces and inertia forces). In the general case, the force (torque for rotational motion) of friction contains three components (“Open Physics 2.6”, Part I, Chapter 1 “Mechanics”, paragraph 1.13 “Friction Force”. http://www.physics.ru/courses/op25part1 /content/chapterl/section/paragraph13/theory.html#.V-umt02a21h - [3]):

a) static friction P _TP , which, in accordance with Newton's first law, is equal and oppositely directed to the resultant of external active forces P _AC , trying to cause movement until it exceeds in absolute value, respectively, the lower Rtr1 or upper Rtr2 threshold value (R _TP \u003d -R _AC at v=0 and - Рtr1≤Р _AC ≤Рtr2, Рtr1 and Рtr2>0);

b) "dry" (Coulomb) friction Рtrs is constant in absolute value (at least in a sufficiently small neighborhood) and directed opposite to the speed v of motion (Рtrs=Рtr3 at v<0, Рtrs=0 at v=0, Рtrs=-Рtr4 with v>0, Ptr3 and Ptr4>0);

c) "viscous" and non-Coulomb friction Рtrv, the value of which depends on the speed of movement (Рtrv=f(v), and f(v)=0 at v=0).

At the same time, with an increase in the absolute value of the velocity of the relative motion of bodies, the absolute value of the friction force first decreases (Striebeck effect) and, with a further increase in the modulus of velocity, increases, its sign is preserved until the sign of the velocity changes. The function f(v) can be negative in the initial section, but its sum with the Coulomb component is always directed against the velocity vector. The Coulomb component of motion friction may be less than the maximum (modulo) value of static friction (Rtr3≤Rtr1, Rtr4≤Rtr2) for the corresponding velocity sign (Monastyrshin G.I. “Mathematical modeling of dry friction” // “Automation and Telemechanics”, 1958 ., volume 19, issue 12, pp. 1091-1106 - [4]). In the general case, the values of the friction parameters may also depend on the position of the contact patch on each of the contacting surfaces. The beginning of the movement will occur in the direction of the resultant sum of active forces and reactions to them obtained (without taking into account the force of inertia). After the start of movement, in order to obtain the correct balance of forces, one should find the projection of the sum of active forces on the direction of the velocity vector, since it is in this direction that the friction force acts. (Further in the text, both for the state of rest and for the state of motion, the term "projection of the sum of active forces" is used).

In Simulink MatLab, it is proposed to use the "Block of dry and viscous friction (Coulomb and Viscous Friction)" (Studopedia, "Block of dry and viscous friction Coulomb and Viscous Friction", https://studopedia.ru/20_27818_blok-suhogo-i- vyazkogo-treniya-Coulomb-and-Viscous-Friction.html - [5]). When it is implemented, the static friction at the moment of stopping is not determined. The travel speed is given as an external input. When moving, friction is represented as an odd function with a discontinuity at zero. Therefore, in the model in the areas of reversal and starting off, friction behaves as an active force that initiates movement. In [4], it is indicated that with this version of friction simulation in the model of a dynamic system, sliding modes arise, which are not typical for a real process.

To exclude such oscillations, it is possible [4] to approximate the discontinuous velocity function with a bounded continuous one, which has a steep slope near the zero velocity value (in a particular case, a linear function with restrictions from above and below). However, the presence of an inclined section of the characteristic generates parasitic deviations in the simulation process and excludes the possibility of simulating a stop. In the article [6, p. 125] (Serebrenny V.V., Boshlyakov A.A., Ogorodnik A.I. “Mathematical model of the executive modules of gripping devices of robots” // Bulletin of BSTU named after V.G. Shukhov. 2019 ., No. 6, pp. 123-135) it is also indicated that algebraic loops arise during modeling. They can lead to looping and stopping the solution of the problem. In the same place [6, pp. 125, 126], it is said that algebraic loops can be eliminated by applying a dynamic friction model, in which the stop is simulated as a microdeformation in the zone of the contact patch. In this case, the coefficients in the differential equations are functions of the velocity, which must be calculated taking into account the values of the friction force obtained from these equations (see, for example, [6, formula (4)]).

In described in [7, paragraph 9.5] (Voronin S.G. "Electric drive of aircraft: Educational and methodological complex." - Offline version 1.0. - Chelyabinsk, 1995-2011. Chapter 9, Tracking electric drives, Paragraph 9.5. http:/ /model.exponenta.ru/epivod/glv_090.htm) method of friction modeling, it is considered that friction has a constant level, changes only when the sign of the velocity changes and is opposite to it in direction. In this case, in contrast to [5], the speed is considered here as the time integral of the acceleration obtained as a quotient from dividing the sum of all forces (moments), including the friction force (moment), by the mass (moment of inertia). That is, this method is characterized by the fact that the sign of the signal corresponding to the speed of relative movement is determined, and this sign is assigned to a positive constant, the value of which determines the level of the force (moment) of friction, and the resulting value is inverted. The disadvantages of the method are the same as for [5].

In [1, pp. 136-165], a method for controlling a robot arm is described, in which friction is considered as an external disturbance acting on a mechanical system. Since friction is not a directly measurable quantity, to form its current value, a comparison of the estimates of the observer of the state and the observer of disturbances is used. In this case, perturbation estimates are sought in the class of functions with bounded derivatives. As mentioned, friction can have a discontinuity in the transition from the area corresponding to the friction of rest, to the area corresponding to the friction of motion. In addition, sliding modes are used to obtain estimates, which leads to additional averaging of the generated value. Therefore, estimates of the current friction value obtained with this method of modeling are largely averaged.

In [2], it was proposed to model friction as a piecewise linear function of velocity with a discontinuity at zero. This method makes it possible to more accurately implement friction in the movement section, but the processes of stopping, starting and reversing, similarly to those indicated for [5], are modeled incorrectly. In addition, the friction of motion can be specified analytically (see, for example, [6, Table 1]), which simplifies experimental work on determining the corresponding function.

The method given in [8] (Shenderovich P.B., “Device for modeling non-linearity of the “dry friction” type”, AS No. 334575 for the invention of the USSR, class G06g 7/48, 1970), implements friction as a function of two arguments - the sum of active forces and velocity. In this case, for a non-zero value of the velocity, Coulomb friction is realized (in this case, a signal of the difference between the active forces and the friction force is formed), and for cases of zeroing of the velocity, static friction is realized (the difference between the active forces and the friction force is equal to zero). However, the disadvantage of the described friction modeling method is that when modeling the reverse, in the case when the sum of active forces is less in absolute value than the static friction force, the speed sign can change without switching to the rest state, and the generation of a forced command to fix the zero speed does not provided. That is, the stability of the solution and the correctness of the results are not ensured.

Although the method implemented in [9] (“Device for Simulating Dry Friction”, AS No. 1091186 for the invention of the USSR, class G06G 7/25, 1982), is supplemented compared to [8] with operations, providing for the formation of a non-linear parabolic characteristic in the movement section, it also does not provide for the generation of a command for forced zeroing of the speed.

Known "System of numerical and mathematical modeling MatLab Stateflow - Simulink. Demonstration of sliding friction in StateFlow, http://bourabai.ru/cm/friction.htm” [10], in which both the current velocity value and the sum of active forces are used to analyze the current state (motion or rest). The signal generated during the transition from the state of motion to the state of rest is used not only to switch the branches of the calculation of the friction force, but also to force the zeroing of the speed. At the same time, in the absence of a state of rest, the Coulomb component of the friction force with a level unequal to the starting friction force is realized by using a three-position (with a fixed zero) signature function of the velocity and the operation of multiplying the resulting signature by a fixed value, and in the presence of a state of rest, the friction force is also calculated using the signature function, but already from the sum of active forces, and the operation of multiplying by the smaller of the two quantities: the modulus of the sum of active forces and the level of the starting friction force.

The disadvantages of this method is that for the transition from the state of motion to the state of rest, the sign of the equality of the speed to zero is used. Since [6, p. 126] the basic Coulomb model of friction and all its variations, which have a singular point in the region of zero velocities, lead to errors in the numerical simulation of reversible systems, since the variable will be described by a real floating-point number and, except in a random way, will not may become equal to 0. The narrowing of the near-zero zone leads to an increase in the frequency of solving the problem (Mathworks, Help Center, Signal threshold, https://www.mathworks.com/help/simulink/gui/signal-threshold.html - [11]). The expansion of the zone adjacent to zero will lead to a qualitative change in the simulated motion.

The closest to the proposed technical solution is the method of digital simulation of friction, (Lushnikov B.V. “Computer simulation of the dynamics of a dry non-Coulomb friction element” // “Proceedings of the Samara Scientific Center of the Russian Academy of Sciences”, vol. 12, No. 1 (2), 2010 , pp. 439-444 - [12, Fig. 1]). The model organizes two mutually exclusive states - movement and rest. In this case, it is considered that if the speed is not zero, then the state of motion takes place. The appearance of a signal (rising edge) about the zero value of the speed in the previous cycle (the state output of the Simulink integration block) triggers the transition of the model to a state of rest and setting the current speed to zero by resetting the integrator with setting zero initial conditions. If the projection of the sum of active forces does not exceed the pull-off friction force in absolute value, then the static friction force is equal in magnitude and opposite in direction to the projection of the sum of active forces. The driving force (determined by algebraically adding the projection of the sum of active forces with the friction force) becomes equal to zero (it should be noted that the zero driving force is a necessary condition for the transition to a state of rest after zeroing the speed, but it cannot be used explicitly due to the algebraic loop and the ability to move at a constant speed). Accordingly, the acceleration obtained after dividing the driving force by the mass reduced to the point for which the speed of movement is calculated also becomes zero. Until this condition is violated, the speed at the output of the integrator will remain zero, and the state of rest in the model will be preserved. When the projection of the sum of active forces exceeds in absolute value the friction force of the breakaway in the corresponding direction, the signal at the input of the integrator becomes non-zero (the force of static friction in absolute value does not exceed the force of breakaway in the corresponding direction) / Obviously, after integration, the speed also becomes non-zero, and the prerequisite to form a sign of the state of motion in the next cycle of calculations. In the state of motion, when the friction force is formed, instead of the friction force of breaking away, the sum of the Coulomb and non-Coulomb components of friction is used, depending, respectively, the first on the sign, and the second on the magnitude of the speed of movement. While in the prototype function of friction at rest is odd with respect to the sum of active forces, and the function of friction during movement is odd with respect to the speed of movement.

The disadvantages of this method are the use to generate a command for forced zeroing of the signal from the output of the integrator of the sign of equality to zero of the speed value at the previous calculation cycle, when the acceleration for the current cycle has already been calculated. By itself, the shift of the moment of time at which the speed is checked, although it breaks the algebraic loop, but, as mentioned above, the probability that at a particular moment of time there will be a strictly zero signal is extremely small. The acceptable value of the tolerance for the velocity zeroing zone and its influence on the stability and correctness of the solution of the problem is not mentioned in [12].

The task of the proposed group of inventions is to provide a physically correct stable simulation of static friction in a dynamic system that operates in reverse or with stops.

The technical result of applying the proposed methods for simulating static friction is to increase the accuracy of the correspondence between the behavior of the model and the real physical process in a dynamic system without increasing the order of the differential equations used by the model.

As an additional technical result, it becomes possible to solve the problem using real-time analog computers and digital computing tools of the same order of performance, which provides a solution to the problem of static friction simulation using known methods.

The purpose of the proposed technical solution is achieved, as an option, due to the fact that:

- use as input signals models that depend on the speed of the Coulomb and non-Coulomb friction forces of motion and the projection of the sum of active forces on the direction of the already ongoing motion or assumed from a state of rest;

- form mutually exclusive signs of the presence of a state of rest or movement and a logical command to reset the simulated speed during the transition from the state of motion to the state of rest;

- at rest:

- form the magnitude of the static friction force as the smaller in absolute value of the values of the friction force of the breakaway and the sum of the projection of active forces in the corresponding direction (the sign of the static friction force is opposite to the sign of the projection of the sum of active forces on the intended direction of movement);

- to determine the driving force, taking into account their opposite direction, the static friction force is added to the projection of the sum of active forces on the expected direction of movement;

- use the sign of exceeding the value of the projection module of the sum of the active forces of the absolute value of the friction force of breakaway for the transition to the state of motion;

- after the input signal on the integrator becomes non-zero, the integration is carried out from zero initial conditions, and the transition of the speed from zero to a non-zero value cancels the generation of the sign of the state of rest, that is, it corresponds to the generation of the sign of the state of motion

- in a state of motion:

- to determine the driving force, taking into account their opposite direction, the force of friction of movement is added to the projection of the sum of active forces on the direction of movement;

- find the acceleration by dividing the driving force by the mass (multiplying by the reciprocal of the mass) reduced to the corresponding point of the model;

- a sign of transition to a state of rest is developed ahead of the moment when the decreasing (in absolute value) speed should reach zero, and at the same time it is taken into account that the driving force (or acceleration) is directed at this moment of time against the speed, and the module is the projection of the sum of active forces on the direction of movement does not exceed the module of the breaking force in the corresponding direction;

- when a logical command is generated to reset the simulated speed, the integrator is reset to zero.

The claimed technical result is also achieved if, in order to move from a state of motion to a state of rest, a logical command is first generated to reset the simulated speed, for which purpose, instead of the sign of equality of the expected speed to zero, the sign of the mismatch of the signs of the expected at the current moment of time and the previous speed of movement is used, and at the same time, this logical command is also used to force the zero value of the speed signal used to generate a sign of the presence of a state of rest until the next cycle of calculations.

The claimed technical result is also achieved when:

- to integrate acceleration, an accumulative adder is used (see, for example, Accumulative adder, https://life-prog.ru/view_automati.php?id=16-[13]);

- to move from a state of motion to a state of rest, first, on the basis of a mismatch in the signs of the expected at the current moment of time and the previous speed, a logical command is generated to reset the simulated speed and this logical command is used to force the assignment of the output signal of the adder with the accumulation of a zero value, and the reset signal of the speed is used to develop a sign of a state of rest.

All newly introduced operations are known individually or can be composed of more elementary known operations, as described in the presented materials.

The implementation of the proposed method is as follows (the first option).

The main reason for the occurrence of parasitic oscillations when modeling a process that includes the Coulomb component of friction is the change in the sign of the velocity of relative motion without fixing the moment when the velocity starts to zero, which leads to bypassing the branch of the model corresponding to the rest stage. As a result, in the branch of the model corresponding to the stage of motion, the friction abruptly changes sign, and if the friction exceeds the projection of the sum of active forces on the direction of motion, an acceleration occurs that is directed against the obtained velocity, which leads to the premise of another change in the velocity sign, and so on. In digital models, this is exacerbated by the presence of quantization. If at the i-th simulation step the speed is v[i], the acceleration is ε[i], and the time step is ΔT, then at the next step v[i+1]=v[i]+ε[i]×ΔT. Therefore, when the sign of ε[i] is opposite to the sign of v[i] and ε[i]×ΔT is greater than v[i] in absolute value, the sign of v[i+1] will be opposite of the sign of v[i].

Therefore, in models of systems that include a friction component containing a discontinuity at zero velocity, this leads to the need to reset the calculated velocity and force the model to go to rest. For what the corresponding logical signal should be developed. Conditions for generating this signal can be:

- the value of the speed enters the neighborhood adjacent to zero within the accuracy of the solution of the problem of determining the speed;

- the opposite of the signs of the current (or expected) and previous speed values, provided that the model is in the motion stage;

- decrease in the expected value of the time interval from the current moment to the change in the sign of the speed to a value less than the time step of solving the problem (or the expected change in the sign of the speed at the current calculation step compared to its value at the beginning of the current calculation cycle).

The first of these criteria is used in the prototype methods. In this case, the transition is carried out only in the case of a very accurate (within the assigned accuracy of calculations) hit of the speed calculated at the current step to zero, and if in the next cycle the zero speed is not determined as an integral of zero acceleration under zero initial conditions, an oscillatory process may begin. . In the prototype, this is aggravated by the fact that the speed value from the previous cycle is analyzed. The second criterion is triggered ahead of time by the moment the speed passes through zero. The time to change the sign of speed with opposite signs of speed and acceleration can be defined as (Korotkov O.V., Zhemerov V.I., Shcheglova L.A. "Method of controlling an electric drive and a device for its implementation (options)", a patent for an invention RF No. 2628757, IPC H02R 23/00, H02R 25/00, 2016 - [14, p. 7, lines 19-22]):

If the received positive value of the quotient from the division is less than the specified value, then this can serve as the basis for generating the corresponding logical signal. On the rising edge (the fact of changing the level of the logical signal from "0" to "1") of this logical signal, a command is generated to reset the simulated speed, and obtaining a zero value of the speed transfers the model to the branch corresponding to the state of rest. Since in this case the speed is reset to zero before its value with the opposite sign can be formed, then the sign of the opposite signs of the current and previous speed values can no longer be formed.

In the prototype, when modeling friction, centrally symmetric with respect to zero (odd) polynomial functions are used, while in some cases friction can be a function of a general form. Such a function, in particular, when moving along a straight line, can be modeled by two separate sets of operations for different directions of movement and the projection of the sum of the active forces that initiate it. For modeling, not only polynomials, but also other functions (for example, trigonometric, power, logarithmic, piecewise linear approximations, splines, etc.) can be used. It should be noted that the friction parameters can change not only depending on the direction of movement, but also on the position of the contact patch on each of the surfaces in contact during movement. If necessary, in the case of obtaining from the external system, along with the values of active forces and friction components of motion, also sets of coordinates (generalized phase state), the corresponding functions can also be implemented. The friction force vector will be directed against the direction of movement at each point of the trajectory when movement takes place. In this case, in the general case, the sum of active forces may not coincide with the direction of motion (for example, if there is a component pressing in the plane perpendicular to the contact spot and perpendicular to the direction of motion in the plane of the contact spot). That is, the vectors of the friction force and active forces in the process of movement can be non-parallel to each other. Therefore, for correct simulation modeling, it is necessary to use the projection of the sum of active forces on the direction of motion for summation with the friction force.

At rest, the friction force vector is directed in the plane of the contact spot (perpendicular to the pressing force at the contact point) opposite to the sum of the projections of active forces (resultant) onto the plane of the contact spot. Therefore, at the beginning of the movement, the velocity vector will be directed in the same way as the projection onto the plane of the contact spot of the vector of the sum of the active forces that initiated it.

In this regard, when moving, the calculation of friction values is carried out in the coordinate system associated with the direction of speed, and at rest - its expected direction. In this case, if necessary, the possible non-isotropy of friction from the direction of motion and the coordinates of the contacting surfaces can be taken into account.

Assume that the initial state is at rest. The speed is zero. The sign of being at rest has a level of "true" (logical "1"). The projection of the sum of active forces obtained from the outside, indicating its direction, in the simplest case of the sign (in this technical solution, the method for calculating the projection of the sum of active forces is not considered as insignificant for the problem being solved and depending on external factors, for example, the shape of the surface on which the movement occurs), compared with the pulling force in the direction of the initiated movement. If the magnitude of the projection of the sums of active forces is less than the pulling force, then the static friction force is oppositely directed and equal in magnitude to the projection of the sums of active forces. The difference in the magnitude of the projection of the sums of active forces and static friction is equal to zero. The acceleration is zero and the speed remains zero. The sign of being at rest retains its value. If the magnitude of the projection of the sums of active forces is greater than the force of breakaway in the corresponding direction, then the force of static friction at the moment of the beginning of the movement is equal in magnitude to the force of breakaway. The difference in the projection of the sums of active forces and static friction turns out to be unequal to zero. The acceleration also becomes non-zero, and the speed at the next moment of time ceases to be zero. The sign of being at rest is inverted (for example, from the level of logical "1" goes to the level of logical "0"), and a sign of the state of motion with the level "true" is formed. The value of the friction value is formed and comes from outside, depending on the speed, direction of movement and coordinates of the contacting surfaces. The resulting value of friction is algebraically summed with the projection of the sum of active forces on the direction of the velocity vector. Since the motion friction near zero speed does not exceed the breakaway friction, the increase in speed will continue until the equilibrium of the motion friction force and the projection of the sum of active forces is reached. (The presence of components of the sum of active forces that are lateral with respect to the velocity vector can, if necessary, be taken into account in an external system to correct the direction of the movement speed). If the value of the projection of the sum of active forces becomes (taking into account the direction relative to the speed of movement) less than the current value of the friction force, braking will begin. It will continue until a new equilibrium speed is reached.

When the magnitude of the projection of the sum of active forces becomes directed opposite to the movement or sufficiently small in absolute value (for example, for a dependence of the form [2, Fig. 3], it is less than the Coulomb friction force, when the speed is higher than the corresponding minimum module for motion friction, or not more than the friction modulus start when the speed is between zero and this value), the speed will tend to zero. If the allowable error limits for calculating the speed are known, then, based on the friction force of the breakaway (or with a margin - the Coulomb component of the friction of motion) and the mass, the maximum value of the time interval can be calculated for which it is permissible to set the value of the velocity to a zero value a priori before integrating the acceleration will change its sign. This time can be estimated, for example, as:

where ΔV is the permissible error of speed modeling;

ε is the value of the acceleration modeled at the current time;

Δε is the maximum possible error in calculating the acceleration.

Depending on which of the values will be less - the step of solving the problem or ΔT _V , when determining the fulfillment of the condition for the transition to a state of rest, this value can be used as a threshold Tthr.

The condition for falling into the interval can be written as:

The minus sign is due to the fact that during braking, the acceleration is opposite to the speed, if the quotient is not inverted, the interval of acceptable values from non-negative will become non-positive.

At a constant mass, acceleration can be replaced by the difference between the projection of the sum of active forces and the friction of motion, and the comparison threshold is changed in proportion to the numerical value of the mass in the corresponding system of units. A case is possible when, in the deceleration section, the projection of the sum of active forces directed against the motion exceeds in absolute value the friction force of the breakaway. Then, for a real physical process, the stop will have practically zero length in time (but with a constant value of the projection of the sum of active forces, the acceleration acceleration modulo will be less than the deceleration acceleration, since the friction will change sign). Therefore, when modeling, it is not necessary to artificially keep the speed at zero during the calculation cycle, although in the calculation cycle corresponding to the change in the sign of the speed, an overestimated (in absolute value) result for the speed will be obtained. The joint fulfillment of the condition of the expected achievement of zero speed (or already established) and the non-exceeding of the sum of the active friction forces of the breakaway by the projection determines the development of a sign of the state of rest. The transition of the sign of the state of rest from the level "false" (logical "0") to the level "true" is the condition for generating a logical command to reset the simulated speed and set zero initial conditions for integrating acceleration. Until the projection of the sum of active forces exceeds the value of the breakaway friction, the static friction is equal in magnitude and inverted in direction (sign) relative to the projection of the sum of active forces, so the driving force is zero, and the speed remains zero. This holds back the development of the sign of the state of rest (see above). The extreme condition for maintaining the state of rest is the achievement by the module of the projection of active forces of the magnitude of the friction force of the breakaway in the corresponding direction. As soon as the projection module of the active forces exceeds the value of the breakaway friction force, the driving force will cease to be zero. Integrating the acceleration will give a non-zero velocity value. The conditions of equality to zero of the speed of movement and non-exceeding by the module of the projection of the sum of active forces of the value of the module of the friction force of the breakaway, each of which is necessary to maintain the state of rest, will be violated. The model will enter the state of motion.

With this method of modeling rest friction, parasitic oscillations do not occur immediately after stopping, since at the moment of stopping, the projection of the sum of active forces according to the condition of transition to the state of rest does not exceed the pull-off friction force.

In another embodiment of the method for simulating static friction to form a state of rest, the integral of acceleration is also forced to zero, but this is done on the basis of a change in the sign of the currently expected speed value and the sign of the previous speed. The expected speed can be, in particular, determined as suggested in [14, p. 8, formula (3a)]:

or

where V is the simulated speed (V(t+μΔt) - expected, V(t) - previous, V(t-Δt) - necessary to evaluate the change in value);

t is the point in time under consideration;

Δt - delay (preemption) interval;

μ is a coefficient that takes into account the possibility of a variable step of solving the problem.

This relationship is a version of linear extrapolation. Accordingly, if it is necessary to improve the accuracy, options for extrapolation of higher orders can be proposed.

The discrepancy between the signs of the expected and the previous speed can be determined by various signs, for example:

a) separately determine the signs of the expected and previous speeds, find the product of signatures, the sign is the negative value of the product;

b) find the product of the expected and previous speeds, determine its sign, the sign is the negative value of the sign;

c) find the quotient of the velocity division (it is preferable to use a larger value as a divisor), determine the sign of the quotient, the sign is the negative value of the sign;

d) if you use the signature function, the value of which is not defined when the argument is zero, after determining the signs of the expected and previous speeds separately, the signature values are compared, the sign is their mismatch.

The generated logical command to reset the simulated speed is used, firstly, to reset the integrator and set zero initial conditions on it, and secondly, to force the speed signal used to generate a sign of transition to a rest state to zero. At the same time, at the moment of transition from the state of motion to the state of rest, the algebraic loop is broken, since the speed from the output of the integrator is not used directly either to form a logical command to reset the simulated speed, or to form a sign of the state of rest. By analogy with the above, forcibly set to zero value of the speed signal, analyzed to form a sign of the state of rest, forms a sign of the state of rest with a logic level of "1". In this state, the level of static friction is assigned the smaller of the modules of the values of the breakaway friction in the corresponding direction and the projection of the sum of active forces, and the sign is taken opposite to the direction of the projection of the sum of active forces. Accordingly, as long as the projection of the sum of active forces on the intended direction of motion does not exceed the force of friction in the same direction, the driving force will remain zero, and the integral of acceleration will also remain zero, therefore, regardless of the presence or absence of a logical command to zero the speed, the sign of the state of rest " 1" continues to be developed. If the indicated excess takes place immediately at the moment of generating a sign of the state of rest, a change in the logical command to reset the simulated speed can only occur in the next cycle of calculations using new values of the expected and previous speeds.

Another variant of the method for simulating static friction is also possible, simulating the use of an accumulating adder as an integrating element. In this case, the integral is obtained as the sum of the previous value from the output of the adder with the acceleration signal multiplied by the time step of solving the problem. Therefore, to reset to zero, it is sufficient to replace the previous value from the adder output to zero at the next cycle of calculations. The generation of a logical command to reset the speed is formed similarly to the previous version on the basis of a mismatch between the signs of the expected at the current time and the previous speeds. On the leading edge of this sign, zero is set instead of the retained speed value from the output of the adder. This zeroed signal value is used to generate a sign of transition to the resting state. As well as for the previous options, if the value of the projection of the sum of active forces does not exceed the value of the friction force of the breakaway, the static friction is equal to the projection of the sum of active forces with the opposite sign. At the output of the accumulating adder, a zero value is maintained, and regardless of the presence or absence of a logical command to reset the speed, the sign of the state of rest continues to be generated. If the projection of the sum of active forces exceeds the friction force of the breakaway, acceleration will appear, and the speed will appear at the output of the adder. The logical command to reset the speed will not be generated immediately. The state of rest will be removed. The model will enter the state of motion. If it turns out that the projection of the sum of active forces exceeds the breakaway friction immediately after the transition to a state of rest, then the zero value of the velocity will be fixed until the next cycle, after which the movement will resume. The static friction moment for this point will not be formed (the driving force is not zero).

From what has been described, it can be seen that the use of the described methods makes it possible to break algebraic loops and provide a stable and corresponding to its physical nature modeling of static friction as part of a dynamic system. That is, the desired technical result is achieved. At the same time, the number of additional operations is small and no additional reduction in the time step of solving the problem is required, since the expected speeds are calculated using the same or larger step size as for integrating the acceleration (and/or previous to the current step). The order of differential equations remains first (only acceleration is integrated). The growth of computing power and memory is practically not required.

On the basis of the described methods, devices that implement them can be created - friction simulators.

As a prototype, a subsystem for modeling dry friction is considered [12, fig. 1] (dry friction simulator). It contains the following groups of elements. The acceleration determination unit (BOA) is made as an algebraic adder and a scaling element installed in series (with a variable reduced mass, a division or multiplication unit can be used). The output of the scaling element is the output of the BOU. The first input of the algebraic adder is the first input of the friction simulator (the input of the projection of active forces), the second input is the input of the friction, and the output is the output of the driving force. The BOU output is connected to the first input of the integrator, the first output of which is the first output of the dry friction simulator. The input of the null organ (NO) is connected to the second (by state) output of the integrator, and the output is connected to the second input of the integrator (by reset) and the control input of the switch. Friction of rest is applied to one of the inputs of the switch, and friction of movement is fed to the other. The switch output is connected to the second output of the dry friction simulator and the second input of the adder. To simulate the static friction, a static friction shaper (FTF) is used, the input of which is connected to the first input of the adder. As already mentioned, the static friction enters the commutator. Internally, the FTF consists of a breakaway friction generator (BTS) and a holding force generator (FUS) connected in series. The second input of the FUS is connected to the input of the FTP, and the output is connected to the output of the FTP. The FUS contains a signature block (BSg) and a serially connected absolute value block (BAV), a comparison unit (BS) and a multiplier, the output of which serves as the output of the FUS. The BSG and BAV inputs are connected to each other and connected to the second FUS input. The BSg output is connected to the second input of the multiplier, and the second input of the BS is connected to the first input of the FUS. It is assumed that the breakaway friction generator forms its absolute value. Therefore, in the comparison block, it is determined which of the two non-negative values (the friction modulus of breakaway or the module of the projection of the sum of active forces) is less (in case of equality, for example, preference is given to one of the inputs or the one whose signal was found to be smaller in the previous step). It is transmitted to the output of the comparison block. Since this is a positive value, it is assigned the sign of the projection of the sum of active forces. This operation in the prototype is performed by a multiplier.

When the value of the signal from the second output of the integrator becomes close to zero, BUT generates a signal with a logic level of "1". The commutator connects the static friction force signal to the second input of the adder. If the projection of the sum of active forces modulo does not exceed the friction force of the breakaway, then the driving force will be zero. Since the integrator was reset to zero by the same logic signal, integrating the acceleration will also give zero speed. But since the integrator state signal is issued with a delay in relation to the signal from the first output, it may turn out to be non-zero. This may interrupt the state of rest, when it should still continue. On the other hand, when at rest the projection of the sum of active forces in absolute value exceeds the pull-off friction force, the acceleration will become non-zero, but a zero signal will continue to be received at the NO input, and the commutator will leave the rest friction signal connected to the second input of the adder. Such temporal inconsistencies distort the physics modeled by the friction simulator and can cause spurious vibrations.

The task is to provide a stable simulation of static friction, ensuring that the simulation results correspond to the real physical picture.

The technical result is to provide the possibility of accurately determining the static friction in systems that use a friction model in their work, including to compensate for its influence on the results of the system, in particular accuracy, without the need to increase the performance of its calculator and the required memory.

The specified technical result is achieved due to the fact that in the static friction simulator, containing a null body and a static friction generator connected in series, a switch, an acceleration determination unit, the second input of which is the first input of the static friction simulator, and an integrator made with the possibility of reset, the first the output of which is the first output of the static friction simulator, and the input of the static friction shaper is connected to the second input of the acceleration determination unit, the second input of the switch is the control input, and its third input is the second input of the static friction simulator, additionally connected in series a division unit, a hit sign shaper into an interval (FPPI), an "OR" block and an "AND" block. The static friction shaper is made with two outputs. The first of them gives as the static friction the smaller value of the breakaway friction modulus and the projection modulus of the sum of active forces with the sign of the projection of the sum of active forces. From the second output of the FTP, a logical signal is issued that the module of the friction force of the breakaway is not less than the module of the projection of the sum of active forces. At the same time, the output of the "AND" block is connected to the control input of the switch, the first input of the division block is connected to the output of the integrator and the input of the null organ, the second input of the division block is connected to the output of the block for determining acceleration, the second input of the integrator is connected to the output of the shaper of the sign of falling into the interval, the second input of the "OR" block is connected to the output of the null-organ, and the second input of the "AND" block is connected to the second output of the static friction shaper. The first output of the FTP is used as the second output of the static friction simulator. (OBD and FPPI are used to generate a predictive signal about zeroing the speed).

The technical result is also achieved when the static friction simulator, containing a null organ and a static friction generator connected in series, the first switch, the acceleration determination unit, the second input of which is the first input of the static friction simulator, the acceleration determination unit and the integrator, made with the possibility reset, the input of the static friction generator is connected to the second input of the block for determining acceleration, the second input of the first switch is the control input and is connected to the output of the null-organ, and its third input is the second input of the static friction simulator, the second switch is additionally introduced and the first one connected in series ( BZ1) and the second (BZ2) delay blocks, the first and second adders and the block for determining the mismatch of signs (BONZ). In this case, the output of the second switch is the first output of the static friction simulator and is connected to the input of the null element, and its input is connected to the output of the integrator and the input of the first delay block. The output of the first delay block is also connected to the second inputs of the first and second adders and the sign mismatch detection unit, and the BONZ output is connected to the second input of the integrator and the control input of the second switch. (BZ1, BZ2, the first and second adders, BONZ are used to generate a warning signal about zeroing the speed).

The technical result can also be achieved in the case when the static friction simulator, containing a null body and a static friction generator connected in series, the first switch, the acceleration determination unit, the second input of which is the first input of the static friction simulator, and the input of the static friction generator is connected with the second input of the block for determining acceleration, the second input of the first switch, is the control input and is connected to the output of the null-organ, and its third input is the second input of the static friction simulator, additionally introduced are the first delay block connected in series, accumulating adder, the second switch, the second block delays and a character mismatch detection unit. In this case, the output of the second switch is the first output of the static friction simulator and is connected to the input of the null organ and the second input of the block for determining the sign mismatch, the output of which is connected to the second input of the second switch. The output of the second delay block is also connected to the second input of the accumulating adder. (BZ2 and BONZ are used to generate a warning signal about zeroing the speed).

All components used in the proposed variants of static friction simulators are individually known or can be obtained by combining known elements in a known manner.

ZTS, FTP, BOU can be performed as in the prototype. Where necessary, the text indicates how they can be modified. Adders, integrators, commutators, logic elements, comparison blocks, delay blocks, division and multiplication blocks are well known. The null organ can be implemented either in the prototype or in [14]. The text also provides explanations about the possible implementation of the shaper of the sign of hitting the interval and BONZ.

To illustrate the proposed technical solutions in Fig. Figures 1-3 show functional diagrams of variants of execution of static friction simulators (RTS), in Fig. 4 - examples of implementations for FTP, in Fig. 5 - variants of implementations of the FUS. Shown in FIG. Elements 1-5 have the following reference designations: 1 - static friction shaper, 2 - "And" block, 3 - commutator (first), 4 - zero-organ, 5 - acceleration determination unit, 6 - integrator, 7 - division unit, 8 - the shaper of the sign of falling into the interval, 9 - the "OR" block, 10 - the adder (second), 11 - the block for determining the mismatch of characters, 12 - the switch (second), 13 - the delay block (first), 14 - the delay block (second ), 15 - adder (first), 16 - accumulating adder, 17 - holding force generator, 18 - breakout friction generator, 19 - comparator, 20 - commutator (third), 21 - multiplier, 22 - signature block, 23 - absolute values, 24 - adjustable limiter.

The functional diagram of the first version of the STP is shown in Fig. 1. It shows a null organ (4), a static friction generator (1), a commutator (3), a block (5) for determining acceleration, an integrator (6), a division block (7), a shaper (8) for a hit sign, connected in series into the interval, block (9) "OR" and block (2) "AND". The second output of the FTP (1) is connected to the second input of the block (2) "AND". The second input of the DU (7) is connected to the output of the BOU (5). The second input of the block (9) "OR" is connected to the output of the null-organ (4), the input of which is connected to the output of the integrator (6). The output of block (2) "AND" is connected to the control input of the switch (3). The output of the FPPI (8) is connected to the second input of the integrator (6). The FTP input (1) is connected to the second BOU input (5). At the same time, the division block (7) connected in series and the shaper (8) of the sign of falling into the interval form a lead generator, and the first input of the division block (7) is the first input of the lead generator, its second input is the second input of the lead generator, and the output of the generator (8 ) sign of falling into the interval is the output of the lead shaper.

In FIG. 2 (functional diagram of the second version of the STP) shows serially connected FTP (1), the first switch (3), the block (5) for determining acceleration, the integrator (6), the first block (13) of the delay (BZ1), the second block (14) of the delay (BZ2), the first adder (15), the second adder (10), the block (11) for determining the mismatch of signs, the second switch (12) and the null organ (4) connected in series. NO output (4) is connected to the control input of the first switch (5). The FTP input (1) is connected to the second BOU input (5). The output of the integrator (6) is connected to the signal input of the second switch (12). Additional inputs of the first and second adders and BONZ (11) are connected to the output of BZ1 (13). The first input of the second switch (12) is connected to the output of the integrator (6). The BONZ output (11) is connected to the reset input of the integrator (6) and the control input of the second switch (12). In this case, the first and second delay blocks ((13) and (14)) connected in series, the first and second adders ((15) and (10)) and the block (11) for determining the sign mismatch form a lead generator, and the input of the first block (13 ) delay is connected to the input of the feedforward shaper, and its output is also connected to the second inputs of the first and second adders ((15) and (10)) and block (11) for determining the mismatch of signs, the output of which is connected to the output of the feedforward shaper.

The functional diagram of the third version of the STP (Fig. 3) contains a null organ (4) and a static friction generator (1) connected in series, the first switch (3), the BOU (5), the first delay unit (13), and the accumulating adder (16) , the second switch (12), the second delay block (14) and the block (11) for determining the sign mismatch. In this case, the output of NO (4) is connected to the control input of the first switch (3), and its input and the second input of the BOZ (11) are connected to the output of the second switch (12). The second input of the accumulating adder (NS) is connected to the output of the BZ2 (14). The BONZ output (11) is connected to the control input of the second switch (12). The FTP input (1) is connected to the second BOU input (5). In this case, the second delay block (14) connected in series and the sign mismatch detection block (11) form a lead generator, and the input of the second delay block (14) is connected to the first input of the lead generator and the second input of the block (11) to determine the sign mismatch, the output of the second block (14) of the delay is connected to the first output of the lead shaper, and the output of the block (11) for determining the sign mismatch is connected to the second output of the lead shaper.

Two possible versions of the FTP (1) are given as an example. In FIG. a) 4 shows the breakout friction generator (18) and the holding force generator (17) connected in series. In FIG. b) 4 additionally shows the connection from the input of the FTP (1) to the input of the 3TS (18).

In this case, the execution of the FUS (17) can also be different. In FIG. a) 5 shows the signature block (22) and the serially connected block (23) for determining the absolute value, the comparator (19), the third switch (20), and the multiplier (21). In this case, the input of the block (22) of the signature is connected to the input of the block (23), and the output is connected to the second input of the multiplier (21). The second input of the comparator is connected to the second input of the third switch, and the third input of the third switch is connected to the first input of the comparator. In FIG. b) 5 shows an adjustable limiter (24) (RO), the control input of which is the first input of the FUS (17), the second input of which serves as the second input of the FUS, and the output is its output.

The operation of static friction simulators is carried out as follows (the state of motion is considered as the initial state).

In all proposed options for the implementation of the STP, the first input of the unit (5) for determining acceleration comes from the output of the switch (3) with a signal about the value of the friction force P _TR . The second input of the BOU (5) receives a signal from the outside about the value of the projection of the sum of the active forces Р _АС taken into account through the coefficient of friction). This input is considered as the first input of the STP and is also associated with the input of the FTP (1). The commutator (3) provides the function specified in the description of the methods to ensure the mutually exclusive use of static friction and motion friction in the corresponding states. In the state of motion, the output of the switch (3) is connected to its second input, which receives a signal from the outside about the value of the friction force of motion R _TD . This input is considered as the second input of the STP. (Issues of the formation of the projection of the sum of active forces and friction of motion are not considered in the present description and are known, see for example [2], [6], [8], [9]). In BOU (5), the signals from the first and second inputs are algebraically summed and divided by the mass reduced to the corresponding point. The corresponding operations are also given in the description of the methods. The division quotient is a signal proportional to the acceleration ε. The mass input in FIG. 1-3 is not shown due to obviousness.

In the first version of the STP (Fig. 1) from the output of the block (5) for determining the acceleration, the signal 8 is fed to the first input of the integrator (6). The integrator (6) is configured to reset the output signal and continue integration with new initial conditions, in particular zero. The need to reset the speed during the transition during the simulation from the state of motion to the state of rest was also mentioned in the description of the methods. Since the first input of the integrator receives a signal proportional to acceleration, its output will be a signal v proportional to speed. The operation of integrating a signal proportional to acceleration was also discussed in the description of the methods. In the described embodiment, the output of the integrator (6) is the first output of the STP. It is also connected to the inputs of the null-organ (4) and division block (7). While at the input of the null-organ (4) the speed differs from zero (within the sensitivity zone), it generates a signal corresponding to the logical "0". That is, on this branch, the prerequisites for the transition to a state of rest are not created. The division unit (7) connected in series and the shaper (8) of the sign of falling into the interval form a kind of lead shaper [14], while the first and second inputs of the DB (7) can be considered, respectively, as the first and second inputs of the lead shaper (FU). The signal at the output of the FPPI (8), corresponding to the logical "1", means that the expected speed in the next cycle of calculations will become equal to zero or change its sign. The desirability of generating a logical signal to reset the signal proportional to the speed ahead of its actual reset was indicated above in the description of the methods. As already mentioned, a signal proportional to the speed arrives at the first input of the DB (7). Its second input receives a signal proportional to the acceleration from the BOU output (5). Since braking occurs only when the signs of speed and acceleration are opposite, the time ΔT to stop can be estimated as the inverted quotient of dividing the signal from the first input of the DU (7) by the signal from its second input. To avoid uncertainty when dividing by zero, since at zero acceleration it is impossible to change the speed, in the case of zero acceleration, the output value of DB (7) is assigned a value that goes beyond the limits for which the FPPI (8) forms a logical "1" (or at zero value acceleration, the output signal from the FPPI (8) is immediately assigned the value of logical "0"). FPPI (8) generally checks that the calculated time to the expected stop is non-negative and less than the given (or calculated in the previous calculation cycle) value. This value does not exceed the current step of solving the problem. From the output of the FPPI (8), the logical signal U _SZ is fed to the first input of the block (9) "OR" (taking into account the specified in the description of the methods, the logical signs of the actual and predicted zeroing of the signal proportional to the speed are combined) and the second input of the integrator (6). For the integrator (6), this signal is a reset command, according to which the output speed value is set to zero, and the initial values for integration are also set to zero. Thus, a prerequisite is created for the transfer of STP to a state of rest. However, zeroing in the process of modeling the current or expected speed is not a sufficient condition for the first variant of the STP for the transition to a state of rest, since if the signal about the projection of the sum of active forces in absolute value exceeds the signal about the friction force of the breakaway, the movement does not stop. The formation of a sign that the module of the signal about the projection of the sum of active forces does not exceed the absolute value of the signal about the friction force of the breakaway (before assigning the sign Р _С is considered as a positive value) is carried out in FTP(1). Checking the ratio of the modules of these signals was also provided for in the description of the corresponding variant of the method.

In contrast to the prototype in the shaper (17) of the holding force, the comparator in Fig. a) 5 is shown deployed as a serially connected comparator (19) and a third switch (20), while the first input of the third switch (20) is the control input. The first input of the comparator (19) receives a signal R _C about the absolute value of the breakaway friction force from the first input of the holding force shaper (17), which is also fed to the second input of the third switch (20). The second input of the comparator (19) receives the _{RAS signal} about the module of the projection of the sum of active forces from the output of the block (23) of the absolute value, which also comes to the third input of the third switch (20). The comparator (19) compares the signals of the breakaway friction force modules and the projection of the sum of active forces. If the value of the signal at its first input is greater than at the second, the comparator generates a signal U _PAS with a logical level "1", according to which the third switch (20) passes the signal from its third input to the output. Otherwise, the output receives a signal from its second input. Thus, the output signal P _TPA of the third switch (20) is, as at the output of the comparison unit of the prototype, the smaller of the values of the signals at its second and third inputs. At the same time, the control signal from the output of the comparator (19) is transmitted to the second output of the FTP (1) as a sign that the signal about the module of the projection of the sum of active forces does not exceed the signal about the module of the breakaway friction force. The inputs of the block (22) of the signature and BAS (23) receives the signal R _AS from the second input of the FUS (17) proportional to the value of the projection of the sum of the active forces. Therefore, at the output of BSg (22), the value corresponds to the sign of the signal about the projection of the sum of active forces. Thus, the product of the quantities coming to the first input of the multiplier (21) from the output of the third switch (20) and to the second input - from the output of the signature block (22) gives the value Р _TP proportional to the static friction with the sign of the projection of the sum of active forces.

As soon as the first and second inputs of the block (2) "AND" simultaneously receive signals with a logical level "1", at its output the signal U _PP will also take on the value of logical "1". The simultaneous presence of these signals was provided for in the description of the corresponding variant of the method. When it is received at its control input, the switch (3) will switch its output from the second input to the first. A state of rest will be formed in the STP. At rest, the signal emitted to the outside through the second output of the STP from the first output of the FTP (1) determines the value of the static friction. The need to issue an additional signal to the external system from the output of block (2) "AND" or some other combination of signals in the present description is not considered as going beyond the subject of the described technical solutions. At rest, the first input of the BOU (5) receives a signal about static friction, and its second input receives a signal about the magnitude of the projection of the sum of active forces. Since in this state the signal about the static friction force is directed oppositely to the signal of the projection of the sum of active forces, in algebraic summation the signal about the friction force is taken inverted, so the signal about the driving force will be equal to zero. Accordingly, the signal proportional to the acceleration at the output of the BOU (5) will also be equal to zero. Therefore, the velocity signal at the output of integrator (6) will remain zero (as described above, the integrator was reset to zero before going to rest). In this regard, a signal with a logical level "1" will be generated at the output of the null-organ (4), and until the module of the projection signal of the sum of active forces exceeds the module of the signal of the breakaway force, at the second output of the shaper (1) the static friction there will also be a signal with a logical "1" level. As a result, the output of block (2) "AND" will also have a logical "1", and the signal about the static friction force will continue to pass through the switch (3). The state of rest in the STP will be maintained.

As soon as at rest the module of the signal about the projection of the sum of active forces exceeds the module of the signal about the friction force of the breakaway, the signal at the second output of the FTP (1) will take the value of logical "0" and, accordingly, the signal at the output of block (2) "AND" will take on the value logical "O". The switch key (3) will transmit a signal about the friction force of the movement to the switch output. The state of rest in the STP will be removed. The algebraic sum of the signals of the projection of the sum of active forces and the friction force of motion at the first moment of time will be (modulo) no less than the algebraic sum of the signals of the friction forces of breakaway and the projection of the sum of active forces that caused the exit from the state of rest. In this regard, at the first moment of time after leaving the state of rest, the acceleration signal at the output of the BOU (5) will be non-zero, the integrator (6) will generate a signal about a non-zero speed, and the null organ (4) will generate a signal with a logical level "O" . Since at the first (after leaving the state of rest) acceleration occurs (the signs of the acceleration and speed signals are the same), a signal with a logical “0” level will also be generated at the output of the FPPI (8). There will be no prerequisites for returning to a state of rest.

Thus, in the first embodiment of the STP, a stable transition from the state of motion and vice versa is provided, as well as a stable retention of the STP in a state of rest. At the same time, the operations performed by the STP correspond to those described for the first independent variant of the method, i.e. the unity of invention is ensured. Therefore, according to the Applicant, the specified processes for performing actions in this version of the method are carried out on material objects using material means.

In the second proposed implementation of the STP (Fig. 2) in the state of motion, the signal to the first output of the STP is fed from the output of the integrator (6) through the second switch (12). In this case, while the speed signal remains non-zero, a signal with a logical level “O” is generated from the output of the null-organ (4) to the control input of the switch (3). Accordingly, a signal about the friction of motion is transmitted to the output of the switch (3). The signal proportional to the acceleration is calculated similarly to that described for the first version of the STP in the BOU (5). The shaper (1) of the static friction can be made as in the prototype, while the shaper (17) of the holding force can also be made as shown in Fig. b) 5. The lead shaper for this option is connected, as described above, the first delay block (13), the second delay block (14), the first adder (15), the second adder (10) and the block (11) for determining discrepancies of signs, taking into account the totality of connections between them. The input BZ1 (13), which is the input of the lead shaper, is connected to the output of the integrator (6). The BONZ output (11), which is the output of the lead shaper, is connected to the second input of the integrator (6) and the control input of the second switch (12). In the state of motion, the signal from the output of the integrator (6) is fed to the input of the first block (13) of the delay. The delayed value of the signal about the simulated speed, corresponding to some previous value, is transmitted to the input of the second block (14) of the delay and the second inputs of the first and second adders and BONZ. Delayed by a certain amount, not necessarily equal to the delay of BZ1 (13), the speed value from the output of BZ2 (14) is fed to the input of the first adder (15), where it is subtracted from the speed obtained from BZ1 (13) to the first input of the first adder (15). At the output of the first adder, a change in speed is obtained during the delay in BZ2 (14). This value is fed to the first input of the second adder (10). There it is summed up with the speed value received at the second input of the second adder (10) from the output of the first delay block (13). If the time delays in BR1 (13) and BR2 (14) are equal, the output of the second adder (10) will give an estimate of the expected speed at the current time. If the time delays are not equal to each other (for example, the problem is solved with a variable step), this can be taken into account by the scaling factor at the second input of the second adder (10). Thus, the signal about the expected speed is sent to the first input of the block (11) for determining the sign mismatch, and the signal about the previous speed value is sent to its second input. BONZ (11) checks whether the signs of these speeds are opposite (for example, that the product of the speeds or their signatures is negative) and outputs a signal with a logical level "1" at its output when this condition is met.

If the condition for changing the sign of the speed is generated and a signal U _SZ with a logical level "1" appears at the output of the lead shaper, the integrator (6) is reset to zero on its leading edge and the key of the second switch (12) is transferred to the zero (grounded) position. Zero-organ (4) generates a signal with a logic level "1" at its output. The first switch (3) connects to its output a signal corresponding to static friction. The operations described above correspond to those indicated for the second independent variant of the method “to generate a sign of the state of rest, first, on the basis of a mismatch in the signs of the currently expected and previous speeds, a logical command is generated to reset the simulated speed and this logical command is also used to force hold at zero until the next cycle calculation of the value used to develop a sign of the presence of a state of rest of the speed.

If the signal proportional to the breakaway friction modulus is greater than the signal proportional to the projection modulus of the sum of active forces, the signal proportional to the static friction force will be equal to the signal proportional to the projection of the sum of active forces for the accepted calculation method (counterdirection is taken into account in BOU). In this case, the signal proportional to the driving force in the block (5) to determine the acceleration will become zero. The signal proportional to acceleration will be reset to zero. The integral of the signal proportional to acceleration under zero initial conditions will remain zero. The state of rest in the STP will be formed. In a state of rest, after a time equal to the delay of BR1 (13), the signal at the output of the lead shaper will take on the value of logical "O" (at zero speed at the output of BR1 (10), the sign of sign mismatch is not formed). The key of the second switch (12) will transmit to its output the signal from its first input, which receives the signal from the output of the integrator (6). But since the signal proportional to the speed at the output of the integrator (6) remains zero, the null element (4) will continue to generate an output signal with a logical "1" level. When the signal proportional to the module of the projection of the sum of active forces exceeds the signal proportional to the module of the friction force of the breakaway, in the BOU (5) the signal proportional to the driving force will become non-zero, the first input of the integrator (6) will receive a signal proportional to non-zero acceleration, the signal proportional to the speed at the output of the second switch (12 ) will become non-zero, and the output signal NO (4) will take the value of logical "0". The output of the first switch (3) will switch to its second input, which receives a signal proportional to the friction of the movement. Since the lead shaper will not be able to immediately develop a sign of a mismatch of signs, the transition of the STP to the state of motion will occur.

Thus, the second variant of the static friction simulator also makes it possible to ensure a stable transition from one FSW state to another and vice versa, as well as to maintain a stable state of rest. This embodiment can also be most easily implemented using an analog element base, since all elements, including delay blocks, have analog versions. In this case, the delay times in BZ1 and BZ2 do not require coordination with the step of solving the problem, and the estimated lead time can be formed taking into account dependence (2). The operations implemented in the described version of the device are similar to those described in the second independent version of the method, which corresponds to the unity of the invention. Therefore, according to the Applicant, the said processes of performing actions in this variant of the method are performed on material objects using material means.

In the third variant of the STP in the state of motion, the signal from the output of the block (5) for determining the acceleration is fed to the input of the first block (13) of the delay. Delayed by a cycle, this signal arrives at the first input of the accumulating adder (16) (“to simulate the integration of acceleration, summation with accumulation is used”). The second input of the NS (16) receives the value of the signal proportional to the speed delayed by a cycle in the second block (14) of the delay. Taking into account the fact that the coefficient on the first input of the NN (16) is proportional to the step of solving the problem, the signal at the output of the NN (16) will correspond to the current simulated speed. From the output of the accumulating adder (16) through the first input of the second switch (12), a signal proportional to the current speed arrives at the inputs of the null-organ (4), BZ2 (14) and the second input of the block (11) for determining the sign mismatch. In the state of motion, unless special measures are taken, the simulated speed almost always has a non-zero value [6]. Therefore, at the output NO (4), the signal will have a logic level of "0". Entering the control input (third) of the first switch (3), it will ensure the passage of the signal from the second input of the first switch (3) to its output. As already mentioned, a signal proportional to the friction of the movement comes to the second input from the outside. Therefore, the block (5) for determining acceleration receives signals proportional to the friction force of motion and the projection of the sum of active forces, which in the general case are not necessarily equal to each other. From the output of the second delay block (14), the delayed (previous) value of the signal proportional to the speed arrives at the first input of the BSS (11). Similarly to how it was described for the second variant of the STP, BONZ (11) forms a sign of a change in the sign of the speed of movement. In this version of the STP, the lead shaper is formed by the second block (14) of the delay and the block (11) for determining the sign mismatch (the input of BZ2 (14) and its output can be considered as the input and the first output of the FU, and the output of the BONZ (11) as the second output of the FU , while the second input of BOZ (11) is considered to be connected to the input of BZ2 (14)). If the signs of the signals proportional to the current and previous speeds are the same, the output of the BONZ (11) will be a signal corresponding to the logical "0", in which the second switch (12) remains in the position corresponding to the state of motion.

When the signs of the signals proportional to the current and previous speeds become opposite, that is, between the cycles, the signal proportional to the speed had to pass a zero value, the block (11) for determining the sign mismatch generates a signal with a logical level "1". This leads to the fact that the output of the second switch (12) is set to zero. In this case, the output of the accumulating adder (16) is disconnected from the first output of the STP. Since there is a zero value at the input of the null-organ (4), the value of the signal at its output takes the level of logical "1" (as well as if at the time of the poll a zero value was formed at the output of the accumulating adder (16)). The first switch (3) switches to the mode of transmitting a signal about static friction to its output. Since NS (16) in this cycle of calculations is already disconnected from the output of the STP, a change in the signal at its output cannot affect the change in the state of the STP. For the third independent variant of the method, the corresponding operations were described as “first, a logical command is generated to reset the simulated speed on the basis of a mismatch in the signs of the expected current and previous speeds and use this logical command to force the accumulated speed value to be replaced by a zero value, and to form a sign of the state of rest the resulting zeroed speed signal is used.

Let us consider, for example, the operation of the FTF (1) at rest, in which the output signal of the breakaway friction generator (18) depends at least on the direction of movement, and the holding force generator (17) is made in the form shown in Fig. b) 5. Suppose that in the next cycle, after the transition to the calculation branch corresponding to the state of rest, a signal about the projection of the sum of active forces will come to the input of the FTP (1). In the simplest case, this can be a signed number, in more complex cases, a multidimensional value that takes into account, for example, in addition to the strength itself, the coordinates of the contact patch and the direction of movement in its plane. This signal from the input of the shaper (1) of static friction comes to the input of the 3TS (18) and the second input of the FUS (17). In 3TS (18), for the range of possible values of the incoming signal, there is an array or an analytical dependence for calculating the corresponding responses to the direction of the projection of the sum of active forces. Therefore, the numerical value of the module of the breaking force is given to the output of the 3TS. It enters the first input of the shaper (17) of the holding force. From the second input of the FUS (17), a signed number (in a more general case, the vector components) is selected from the array of incoming data, which characterizes the projection of the sum of active forces on the intended direction of movement. This value is fed to the signal input of the adjustable limiter (24). The control input RO (24) receives a signal about the level of limitation from the first input of the FUS (17). If necessary, this value can be assigned the sign of the signal applied to the signal input, or it can be inverted (organized symmetrical limitation). Thus, at the output of RO (24) there will be a signal with the sign of the projection of the sum of active forces, and its level will be equal to the smaller of the signal modules at its inputs. The consequence of this will be that while the state of rest is maintained, the signal proportional to the acceleration at the output of the BOU (5) will be zero. Since after the forced transfer of the output signal of the second switch (12) to zero in the next cycle of calculations, the output signal from the output of the second delay block (14) will also become zero. The consequence of this, firstly, will be the zeroing of the signal at the output of the NS (16), since by this time the output of the first block (13) of the delay will also have a zero signal, and secondly, at the output of the BONZ (11) the signal will take on the value of the logical "O". Therefore, despite the fact that the output of the second switch (12) will again be connected to the output of the accumulator (16), its value will remain zero, the NO output (4) will retain the value of logical "1". The state of rest will be maintained until the signal proportional to the module of the projection of the sum of the active forces exceeds the signal proportional to the module of the breakaway force.

When the signal proportional to the module of the projection of the sum of active forces exceeds the signal proportional to the module of the friction force of the breakaway, the equality of the signals at the inputs of the control unit (5) will be violated. The signal proportional to acceleration will become non-zero. Due to the fact that the signal proportional to the speed will be different from zero, the signal at the output of the null-organ will take the value of logical "0". The first switch (3) will again pass to its output a signal corresponding to the friction force of the movement. The state of traffic in the STP will be restored.

As can be seen, in the third variant of the implementation of the STP, a stable transition from the state of motion to the state of rest and vice versa is provided, as well as a stable retention of zero speed in the state of rest. Parasitic oscillations due to a change in the direction of the friction force do not occur. This implementation is preferably used when computing devices are used that work with a fixed step of solving the problem. The operations implemented in the described version of the device are similar to those described in the third independent version of the method, which corresponds to the unity of the invention. Therefore, according to the Applicant, the specified processes for performing actions in this version of the method are also carried out on material objects using material means.

The above shows that the proposed options for methods and devices are technically feasible and new. During the simulation, the signal proportional to the friction force does not behave as an active force, while the simulation of the signal proportional to the rest friction is performed with the implementation of a stop and without parasitic oscillations. That is, the essence of the physical process is reflected. Implementation on digital and analog element base is possible. It is possible to use technical solutions in real-time systems. Since the static principle of friction force simulation is taken as a basis, there is no need to increase (at least significantly) the computing capabilities and the required amount of memory of computing devices.

SOURCES OF INFORMATION

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http://www.physics.ru/courses/op25part1/content/chapterl/section/paragraph13/the ory.html#.V-umt02a21h

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5. “Block of dry and viscous friction (Coulomb and Viscous Friction)”, Studopedia, https://studopedia.ru/20_27818_blok-suhogo-i-vyazkogo-treniya-Coulomb-and-Viscous-Friction.html

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Claims (17)

1. A static friction simulator containing a null organ, a static friction generator connected in series, a commutator, an acceleration detection unit, the second input of which is connected to the first input of the static friction simulator, and a resettable integrator, the first output of which is connected to the first output of the friction simulator rest, and the input of the static friction shaper is connected to the second input of the block for determining the acceleration, the second input of the switch is connected to the second input of the static friction simulator, characterized in that it contains serially connected lead shaper, the "OR" block and the "AND" block, the output of which connected to the third input of the switch, while the first input of the feedforward shaper is connected to the output of the integrator and the input of the null organ, the second input of the feedforward shaper is connected to the output of the acceleration determination unit, the second input of the integrator is connected to the output of the feedforward shaper, the second input of the "OR" block is connected to the output of the null organ, and the second input The “I” eye is connected to the second output of the static friction shaper, the first output of which is connected to the second output of the static friction simulator. 2. A static friction simulator, containing a null organ and a static friction generator connected in series, a first commutator, an acceleration determination unit, the second input of which is connected to the first input of the static friction simulator, and an integrator made with the possibility of reset, the input of the static friction generator being connected to the second the input of the block for determining the acceleration, the second input of the first switch is connected to the second input of the static friction simulator, and its third input is connected to the output of the null-organ, characterized in that it contains the second switch and the lead shaper, the first input of the second switch is connected to the output of the integrator and the input of the feedforward shaper, the output of which is connected to the second inputs of the integrator and the second switch, while the output of the second switch is connected to the first output of the static friction simulator and the input of the null organ, and the output of the static friction shaper is connected to the second output of the static friction simulator. 3. A static friction simulator, containing a null organ and a static friction generator connected in series, the first switch, an acceleration determination unit, the second input of which is connected to the first input of the static friction simulator, and the input of the static friction generator is connected to the second input of the acceleration determination unit, the second input of the first switch is connected to the second input of the static friction simulator, and its third input is connected to the output of the null-organ, characterized in that it contains the first delay unit connected in series, the accumulating adder, the second switch and the lead shaper, and the output of the second switch is connected to the first the output of the static friction simulator and the input of the null body, while the second input of the accumulating adder is connected to the first output of the lead shaper, the input of the first delay block is connected to the output of the acceleration determination unit, the output of the static friction shaper is connected to the second output of the static friction simulator, and the second output of the shaper i preemption is connected to the second input of the second switch. 4. The static friction simulator according to claim 1, characterized in that the static friction generator contains a breakaway friction generator and a holding force generator connected in series, the second input of which is connected to the input of the static friction generator, the first output is connected to the first output of the static friction generator, and the second output with the second output of the static friction shaper. 5. The static friction simulator according to claim 1, characterized in that the static friction generator contains a breakaway friction generator and a holding force generator connected in series, the second input of which is connected to the input of the static friction generator, the first output is connected to the first output of the static friction generator, and the second the output is with the second output of the static friction shaper, while the holding force shaper contains a signature block and a block of absolute value, a comparator, a commutator and a multiplier connected in series, the second input of the multiplier is connected to the output of the signature block, and the output is connected to the first output of the holding force shaper, the second the comparator input is connected to the second input of the commutator and the first input of the holding force generator, the input of the absolute value block is connected to the input of the signature block and the second input of the holding force generator, the output of the absolute value block is also connected to the third input of the comparator, whose output is also connected to second output of the holding force generator. 6. The static friction simulator, according to claim 1, characterized in that the lead shaper is made in the form of a division block and a shaper of a sign of hitting the interval connected in series, while the first input of the division block is connected to the first input of the lead shaper, its second input is connected to the second the input of the lead shaper, and the output of the shaper of the sign of hitting the interval is connected to the output of the lead shaper. 7. The static friction simulator according to claim 2, characterized in that the lead generator contains the first and second delay blocks connected in series, the first and second adders and the block for determining the mismatch of signs, while the input of the first delay block is connected to the input of the lead generator, and its output is also connected to the second inputs of the first and second adders and the block for determining the mismatch of signs, the output of which is connected to the output of the lead shaper. 8. The static friction simulator according to claim 3, characterized in that the lead generator contains a second delay block and a sign mismatch detection block connected in series, the input of the second delay block is connected to the first input of the lead generator and the second input of the sign mismatch detection block, the output of the second delay block connected to the first output of the lead shaper, and the output of the character mismatch determination unit is connected to the second output of the lead shaper. 9. Simulator of static friction according to any one of paragraphs. 1-3, characterized in that the static friction generator contains a breakaway friction generator and a holding force generator connected in series, the second input of which is connected to the input of the static friction generator, the first output is connected to the first output of the static friction generator. 10. The static friction simulator according to claim 2 or 3, characterized in that the static friction generator contains a breakaway friction generator and a holding force generator connected in series, the second input of which is connected to the input of the static friction generator, the first output is connected to the first output of the static friction generator, in this case, the holding force generator is an adjustable limiter, the first input of which is connected to the first input of the holding force generator, the second input - to its second input, and the output - to the output of the holding force generator. 11. A method for simulating static friction, implemented by means of the device according to claim 1, in which, in order to obtain a driving force signal, a friction force signal is added to the signal of the projection of the sum of active forces on the direction of movement (expected direction of movement), the result is divided by the value of the mass of the moving parts, obtaining acceleration value, by integrating the acceleration signal, the speed value is obtained, in the event that the simulated speed reaches zero, a sign of the presence of a state of rest is formed, mutually exclusive with the sign of the presence of a state of motion, and a logical command to zero the simulated speed by resetting the integrator to zero when moving from a state of motion to a state of rest , and in the state of motion, the value of the friction force of motion is used as a signal of the friction force, and in the presence of a sign of the state of rest, the value of the friction force of rest is used for this, the absolute value of which is formed as the smaller in absolute value of the values of the friction force of the breakaway and the projection of the sum of active forces in the corresponding direction, characterized in that the sign of the state of rest is also generated ahead of the actual zeroing of the speed signal, using the prediction of its zeroing, and at the same time, it is taken into account that the driving force (or acceleration) is directed at this moment of time against the speed, and the module of the projection of the sum of active forces on the direction of movement does not exceed the module of the pulling force in the corresponding direction. 12. A method for simulating static friction, implemented by means of the device according to claim 2, in which, in order to obtain a driving force signal, a friction force signal is added to the signal of the projection of the sum of active forces on the direction of movement (expected direction of movement), the result is divided by the value of the mass of the moving parts, obtaining acceleration value, by integrating the acceleration signal, a speed value is obtained, a sign of the presence of a state of rest is formed, mutually exclusive with a sign of the presence of a state of motion, and a logical command to reset the simulated speed by resetting the integrator to zero when moving from a state of motion to a state of rest, and in the state of motion they use as a signal of the friction force, the value of the friction force of movement, and in the presence of a sign of the state of rest, the value of the friction force of rest is used for this, the absolute value of which is formed as the smaller in absolute value from the values of the friction force of the breakaway and the projection of the sum of active forces in the corresponding direction, which differs by the fact that, in order to generate a sign of a state of rest, first, on the basis of a sign of a mismatch between the signs of the expected at the current time and the previous speeds, a logical command is generated to reset the simulated speed and this logical command is also used to forcibly hold at zero until the next cycle of computing the value used to generate the sign of the presence of the state rest speed. 13. A method for simulating static friction, implemented by means of the device according to claim 3, in which, in order to obtain a driving force signal, a friction force signal is added to the signal of the projection of the sum of active forces on the direction of movement (expected direction of movement), the result is divided by the mass value of the moving parts, obtaining the value of acceleration, by integrating the acceleration signal, the speed value is obtained, a sign of the presence of a state of rest is formed, mutually exclusive with the sign of the presence of a state of motion, and a logical command to reset the simulated speed during the transition from the state of motion to the state of rest, and in the state of motion, friction forces are used as a signal the value of the friction force of motion, and if there is a sign of the state of rest, the value of the friction force of rest is used for this, the absolute value of which is formed as the smaller modulus of the values of the friction force of the breakaway and the projection of the sum of the active forces in the corresponding direction, characterized in that for the simulation of the integriro accelerations use summation with accumulation, at the same time, at first, a logical command is generated to reset the simulated speed on the basis of a mismatch between the signs of the expected at the current time and the previous speeds, and this logical command is used to force the accumulated value of the speed to be replaced by a zero value, and to form the sign of the state of rest, the resulting speed signal to zero. 14. The method for simulating static friction according to claim 11, characterized in that, in order to predict a possible zeroing of the speed signal, zero acceleration values are excluded from consideration, the speed signal is divided by the acceleration signal, and it is checked that the quotient of the division falls into the specified range of values, the quotient falls into this interval is used as a logical indicator of the expected zeroing of the simulated speed. 15. The method for simulating static friction according to claim 12, characterized in that in order to form a sign of a mismatch between the signs of the expected at the current time and the previous speeds, a change in the speed signal is determined at the previous movement interval and an extrapolated value of the speed signal for the end of the next movement interval is found, while the extrapolated speed value is used as the expected speed value. 16. The method of simulating static friction according to claim 13, characterized in that the acceleration signal is issued for accumulation with a delay. 17. The method for simulating static friction according to claim 13, characterized in that the forced replacement of the accumulated velocity value with a zero value is performed by assigning a zero value to the signal before performing the next accumulation operation.

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