Swivel pendulum. Fundamental doslejennya Review of the methods of keruvannya about the type of swinging pendulum
There is still one, unprecedented idea to the description of girskolizhnogo technology, also not tying with rukhs in a hinged system, but informing the parts of a liznik. Winning the basis for the model of an inverted pendulum, also called the "revolving pendulum", or the "Vitni pendulum".
This is a trickier object of theoretical mechanics, which was formulated as follows: it is possible that a material pendulum is turned upside down, and it collapses straightforwardly, albeit not evenly. It is necessary to know the position of the pendulum, also, that the fault does not fall into the winds, since the back of the house is the amount of fluidity every hour, if there is no interruption in the 2nd century.
The head of Vіtnі dosі tsіkavit mathematicians, albeit more importantly more vorotne zvorotne zavdannya: dynamical control of the movement is possible, also, when the pendulum is taken out of the way, or to beat more. The price is very important for robotics, navigation, automation of virobniting, organization of space devices, it can also be realized in case of extravagant walking.
Alle the zavdannya it is possible to go back: on the pendulum with two steps of freedom, the support that collapses in the same old, curved trajectory, in the wintry shvidkistyu, altogether for the mind of the uninterruptedness of 2 older ones. The simplest butt of a zagalny revolving pendulum: can be put on a long haircut, and it can be easily tackled in an unstable position, and can be pulled down with a hand according to a proper trajectory.
As soon as the pendulum is thrown away, then the pendulum can be struck with a wintry twist: at the same time, the frequency changes, it becomes more and more foldable. The model of a non-stable mechanical system, for example, people on a rope, is still a huge model. Along with this, it is possible to set up an error: to prevent the swing of the pendulum, in case of uneven Russia, support according to the given curved trajectory, with the addition of an active snake and the swing of the pendulum. Mi bachimo: at such a staging, I will increase the visibility of a girskolizhnik along the road!
Z'yasuvalosya: until 1973, Polish mathematician, Janusz Moravskiy, describing the mechanics of a girskolizhnik behind the aid of a revolving pendulum, the robotic ball was turned into 40 rockets.
The model of Y. Moravsky Bul is not completely complete: it’s not a good idea to lick the pendulum support, as it was necessary for the girly technical cob of the 1970s. But for the modern athletes of the highest level, the technology is not tied to the competition, and the model is more accurate in responding to reality.
The news of the revolving pendulum was born from the revision of the high-tech, practical design: to make it easier for the experiments to be carried out in the development of girskolizhnoy technology. Zazvychay, vivchennya ruhіv gіrskolizhnikіv, it is necessary to have a perfect fit of the camp, and powerless strength, so the licks, and the licker himself, it is necessary to quickly set up and test preparation of experiments.
In 2013 rotsi, Matias Gilgin, vidomy fahivets from lizhnoy mechanics, dov: if the descent traktorіya to the center of the mass, then for the model of the zagalny vortex pendulum it is possible to count the tractors, such as As a result, all the foldable possession can be replaced by an extraordinary GPS-navigator!
The experiment was carried out with a geodetic navigator, which was done using the method of differential navigation, with the accuracy of coordinates: 1 cm at the horizontal area and 2 cm vertically. Vikoristovuvalas also made a report on the 3D model of the construction, rendered behind the aid of a geodetic scanner. Infectious, for people in the United States and Europe, in the obvious access є analogous for the accuracy of the companion 3D-pictures, the zone is very popular.
With urahuvannyam micro-rel'fu, which without interruption to change on the shilia, the accuracy of the height to become 10-20 cm, i.e.. an order of magnitude lower than the accuracy of navigation. The antenna of the navigator was located on the lizhnik, the position of the CM was calculated on the basis of the early results of Robert Reid, which said: the athletes from the old CM do not see far out straight, but go through the middle of the road, in the middle of the country. And the lizhn, when turning, trims its head vertically, the middle of the shoulder is located approximately under the antenna. The appearance of the "surface-CM" should be approximately 0.45-0.5 if the "surface-head" appears; pardons for the second rivnovagi.
If a lizhnik is described by a model of a gazelle revolving pendulum, from a changeable age, then from a trajectory, and from a quickness of the CM from the surface, one can count the cutthroughs of a person who does not look like a vertical position. It is also possible to trim the trajectory of the support: points in the middle of the roadway between the walls of the lizh. And from the position of the CM, it is possible to support the center of the lizard in a later direction, and to the center of the turn, if it is not possible to calculate the position of the parts of the body and the most common lick.
Simultaneously with the GPS update, a very special possession was installed on the control device, as it was used during the preliminaries of the hydraulic equipment by the MOCAP methods, based on the model of the hinge system, from the calculations of the dynamics of the conversion of parts for a long time. The selection of the tales about the collapse of the CM became somewhat more vivid: the stench appeared to be closer, strong spirits, only on the doors with the turns, on what the pendulum was like the swing of the pendulum quickly changed when it was opened.
Ale the zavdannya was not built to induce a new model of the CM, an independent form of the lizhnik camp: no one needs a price! Practical meta: on the basis of the revolving pendulum model, correct the ringing forces, so that the reaction of the surface, the opir of the snig, the aerodynamic opir. Doctor M. Gilgian and his sportsmen took away the level of all their forces and made them equal to the values, which were counted for the dynamism of the part of the body. On the graph of the reaction of the surface, taken for the butt: the blue curve will show the force, calculated for the model of the revolving pendulum, the red for the model of the articulated system, which is the same.
Swiss exercise, Rolf Adelsberger, conducted an analogous experiment, and also changed the deformation of the licks during the descent, for additional sensors glued to the licks. The results were given to the forces, which were also calculated on the basis of GPS data, according to the method of Gilgin, in order to bring the correctness of the method.
Slovenian mathematician, Boyan Nimets, also implanted a model of a revolving pendulum with athletes from Zbirnaya Slovenia, altogether placing an antenna on the neck of the lizhn for a short approach to the CM. Winning the ryvnyannya of a spacious kuta nahila: when fallen asleep, drive down the pendulum.
Mi bachimo: rіvnyannya nabagato skladnіshe, nіzh simple formulas of kutіv, which are constantly discussed on the gіrskolizhnyh sites! It is based on experimental data, and more precisely reflects the real processes that take place before the descent hour. Bula is also a correction for the exact value of the position of the CM, ale it was gone: it’s not that big, and it’s just right on the surface, like having put M. Gilgin earlier.
Professor B.Nimets also emphasized the strong breakdowns in the distribution boxes and letting it go: the pardon is tied by the linear law of the change of the pendulum. As soon as you introduce late springiness, then the increase in the amount of time will not change, and the pardons will change drastically. Along with the pendulum, the new steps of freedom are rejected: more and more the harmony of the hands, the need for more reprocessing of the model, B.Namets plan of destruction in the offensive robots. The main problem: the introduction of the coefficient of springiness, because the frequency of the later colony is not permanent, but the value of the coefficient is also not constant.
Whenever there is a possibility of rejecting a new effect: if the support of the pendulum is twisted in a vertical straight line, at a high frequency that small amplitude, then it is the same force, if the pendulum starts at the vertical level, the frequency of the amplitude is ... In response to a single blow on the spring surfaces, the swinging pendulum is extinguished, the swinging pendulum, the setting on the spring supports, also transferring to the spring, or even for a short hour after the blow: until the swinging is extinguished. An analogous phenomenon is possible when there is a sharp change in mounting on a lick, ale, later, springiness can be found in the size of a viginu, and it grows faster.
Ale of enumeration of forces also did not boil down to the point of death: Dr. M. Gilgin, having trimmed it out on the knee of a liznik, which can lead to injuries of the slopes. The yogo method allows you to correct the assessment of the track, with a glance of safety, depriving on the basis of the given GPS for an hour of control trips.
In a very straightforward way, as the head, the establishment of the tool for the trainers, so that the dynamics of the lizher is shown without interruption, as it is prikhovannaya as a direct warning: the minds of the quickened, accelerate the forces, go on. The whole method is not a collapsible, expensive possession, but rather an expensive GPS receiver at the same time cheap from the MOCAP systems, or the energy-efficient sensors, and the simpler from the victorian.
Mi bachimo: an old idea, describing the girskolizhnu technology without the sound of the ruffs of a liznik, still not forgotten, unaffected by the appearance of new technologies. Mozhlivo, we said goodbye early with miles of spherical horses.
Good luck and good luck!
Material from Wikipedia
The inverting pendulum is a pendulum, which is the center of the mass of its own point of support, fastening on the end of a hard haircut. Often the point of support is fixed to the side, which can be shifted horizontally. At that hour, like a normal pendulum hangs down steadily, the swinging pendulum for its nature is unsteady and is guilty of constantly balancing itself out of the vertical position, due to the additional stagnation of the moment, how to twist, to the pivot point, or when the ringing point is reversed along the horizontal system language. The simplest demonstration butt can be used to balance the olivtsa at the end of the finger.
Look around
The inversion of the pendulum is a classic problem of dynamics and control theory and is widely used as a standard for testing control algorithms (PID-regulators, neural fences, fuzzy control, etc.).
The problem of the revolving pendulum is tied to the guided missiles, splinters of the rocket's propeller and rocket dislocations lower to the center of gravity, and instability becomes unstable. Qi the problem of virishena, for example, at the segway, is the self-balancing of the transport attachment.
The first way to stabilize the revolving pendulum is to keep the base at the vertical area. And here you can find a ringing sound. As soon as the swing is strong enough (in the sense of the magnitude of the accelerated amplitude), the swinging pendulum can be stabilized. If the point is that it collapses, collapses like simple harmonies, then the collapse of the pendulum will be described by the function of Mother.
Rivnyannya ruhu
With an unruffled point of support
Rivnyannya rukh is analogous to a straight pendulum, with the blame for the fact that the sign of the cobweb position changes from the vertical position of the non-stupid rivnovagi:
When transferring, vin matime the same sign of the kutovogo accelerated:
In such a rank, the swinging pendulum will rise from the vertical unstable rivnovaga into the prototype side, and the accelerated pendulum will be wrapped proportionally up to the age. The high pendulum falls more heavily, less short.
Pendulum on vizka
Rivnyannya Rukha can be recognized with the help of Rivnyannya Lagrange. Go about the hovering of the little ones, de Zavdovzka pendulum kut in relation to the vertical and dignified power or gravitation and call forces straight ... Visually significant camp viska. Lagrangian systems:
L = \ frac (1) (2) M v_1 ^ 2 + \ frac (1) (2) m v_2 ^ 2 - m g \ ell \ cos \ theta de є shvidk_styu viska, and - the speed of the material point . і can be turned through і I will write down shvidkosty yak pershoi pohidnoy camp.
v_1 ^ 2 = \ dot x ^ 2
v_2 ^ 2 = \ left ((\ frac (d) (dt)) (\ left (x- \ ell \ sin \ theta \ right)) \ right) ^ 2 + \ left ((\ frac (d) (dt ))) (\ left (\ ell \ cos \ theta \ right)) \ right) ^ 2 Forgiven virazu produce up to:
v_2 ^ 2 = \ dot x ^ 2 -2 \ ell \ dot x \ dot \ theta \ cos \ theta + \ ell ^ 2 \ dot \ theta ^ 2
Lagrangian now starts with this formula:
L = \ frac (1) (2) \ left (M + m \ right) \ dot x ^ 2 -m \ ell \ dot x \ dot \ theta \ cos \ theta + \ frac (1) (2) m \ ell ^ 2 \ dot \ theta ^ 2-mg \ ell \ cos \ theta that rivnyannya ruhu:
\ frac (\ mathrm (d)) (\ mathrm (d) t) (\ partial (L) \ over \ partial (\ dot x)) - (\ partial (L) \ over \ partial x) = F
\ frac (\ mathrm (d)) (\ mathrm (d) t) (\ partial (L) \ over \ partial (\ dot \ theta)) - (\ partial (L) \ over \ partial \ theta) = 0 Installation in the case of the fall of the pendulum, it is necessary to bring it to the level of the prying pendulum:
\ left (M + m \ right) \ ddot x - m \ ell \ ddot \ theta \ cos \ theta + m \ ell \ dot \ theta ^ 2 \ sin \ theta = F
\ ell \ ddot \ theta - g \ sin \ theta = \ ddot x \ cos \ theta Rivnyannya nonlinear, ale oskilki meta control systems - take the pendulum vertically, then the equalization can be linearized, accepting .
A pendulum with a base to swing around.
Rivnyannya rukh for such a pendulum is tied with a massless base, which is oscillated, and is trimmed in the same way, like for a pendulum to bend. The position of the material point starts with the following formula:
This flexibility is known through the first lost position:
\ ddot \ theta - (g \ over \ ell) \ sin \ theta = - (A \ over \ ell) \ omega ^ 2 \ sin \ omega t \ sin \ theta.
The price is not an elementary solution in a closed viglyade, but it can be done in a straight line. It’s close to the rivnyannya Mother, for example, if the amplitude of the kolivan is small. The analysis will show how the pendulum will get lost in the vertical position when it is swift. The first graph is shown , the pendulum falls quickly when the vertical position is low.
Yaksho swing quickly, then the pendulum can be stable for the vertical position. Another graph shows that the pendulum is now starting to swing around the vertical position when the pendulum is in a stable vertical position ( ). There is little loss from the vertical position, and the pendulum does not fall.
Gagging
Butt - balancing people and objects, for example, in acrobatics or riding a one-wheel bike. And also a segway is an electric scooter, self-balancing, with two wheels. The inverted pendulum is the central component of the early seismographs.
Div. also
Posilannya
- D. Liberzon Switching in Systems and Control(2003 Springer) pp. 89ff
Further reading
- Franklin; that in. (2005). Feedback control of dynamic systems, 5, Prentice Hall. ISBN 0-13-149930-0
Write an overview about the article "Zvorotny pendulum"
Posilannya
Urivok, which characterizes the Zvorotny pendulum
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Natasha happily marveled at the knowledge of exposing P'yra, the blatant of a pea, as she called him Peronska, and she knew that P'er їkh, and especially її, whispering at NATO. P'єr obіtsyav їy bootie at the ball and uyaviti їy cavaliers.
Ale, not dіyshovshi up to them, Bezuhiy zupinivsya bea low, archery brunet in a white uniform, like, standing bіlya vіkna, growing up with a tall head in the eyes of those strіchtsі. Natalya at once recognized a lowly young man in a white uniform: tse buv Bolkonsky, who would be even younger, happier and prettier.
- Axis shche znayomiy, Bolkonsky, bachite, mamo? - Natalya said, they were involved in Prince Andriy. - Pam'yataєte, I have spent the night with Vidradny.
- Do you know vi yogo? - said Peronska. - I can’t bear it. Il fait present la pluie et le beau temps. [The weather is good now to lay the board. (French prisliv'ya, what a meaning, what a success.)] I pride is like that, between no one! On tatka pishov. I called out to Speranskim, as a project to write. Wonder how to be happy with women! It’s like there’s him, but he’s coming in, ”she said, vkazuyuchi on nyogo. - I would have punished him, as if I had punished it like that, as if it had been done by the ladies.
With a rapt, all started talking, having started talking, poking, poking around, once again, when two ranks, who parted, at the sound of music, she sang, the sovereign woke up. The gentleman and the gentleman followed him. Sovereign ishov shvidko, bowing right-handed and lіvoruch, nіbi namagayuchis shvidshis shame on the tsієї of the first hilinny zustrіchі. The music was played by Polskoyu, vіdomy todі for the words, vigadanimi na nyogo. The words were resigned: "Oleksandra, Єlizaveta, zhooplyuєt you us ..." some of them with virases, which changed, hurriedly went through there and back. I have seen again from the doors of the virtual, de appearing the sovereign, roaming with the lord. Yaky young cholovik, stepping on the ladies with a twisted viglyad, asking them to come out. The deyaky ladies with the persons who turned to the utmost the forgetfulness of all the minds of the light, poured their toilets, pushed forward. The choloviks started to walk to the ladies and to be at Polsky's bet.
Everything fell apart, and the sovereign, laughing and not tactfully leading the master's hand, the vyshov from the doors of the virtual. Behind him came the gentlemen from M.A. More than half of the women of Mali were cavaliers and had gone to Poland. Natalya saw that she lost her mother and Sonya in the number of the few ladies who had never been taken from Poland. Vona stood with her slender arms lowered, with the worldly podnimayuyu, triple-singing breasts, streaming down, shining, overglowing eyes, marveled in front of her, with a vigor of readiness for great joy and great sorrow. Neither the sovereign, nor all the important individuals, on which Peronska ordered - she had only one thought: , і not to bang me, but to marvel at me, then marvel at such a viraz, not seem to me: Ah! it’s not won, so I don’t have to wonder. Hi, you can't be! - thought won. - "Smell blame the nobility, as I want to dance, as I dance wonderfully, and as it is fun to dance with me."
The sounds of Polsky, it was trivial to end up with dovgo, were already repaired to sound summarily - in a spogad at Natasha's vuhah. Oh, I wanted posters. Peronska came from them. Count buv at the іnshomu kіntsі zali, Countess, Sonya and vona stood alone at the fox in the whole foreign NATO, in some unpopular and unused. Prince Andriy Proyshov will be with them as a lady, obviously they do not know. Anatoly Krassen, laughing, saying something to the ladies, as if they were in viv, and glancing at Natalka to reveal them with a glance, as if they were amazed at the wall. Boris dvіchі proishov from them and schorazu came out. Berg from the squad, they didn't dance, they went to them.
Nataltsi hid in an imaginative way of being close here, at the ball, there is not a lot of time for family roses, surrounded by yak at the ball. Vona didn’t hear and didn’t marvel at Vira when she was talking about her green cloth.
Nareshty sovereign zupinivsya bilya of his own dame (vin dancesuvav iz trioma), music deputy; Sturbing ad'utants nabig on Rostov, begging them to move aside, if the stench was standing there, and the sound of waltz was sounded from the chorus. Pan with a smile glanced at the hall. Minula khvili - nichto still not fixing. The adjutant promoter of the children to the Countess Bezukhova and asking for it. Vona, laughing, lifted her hand and bowed, do not marvel at the new one, on the shoulder of the ad'yutant. Adjutant razporyadnik, meister of your right, singing, not vapidly and peacefully, minglingly embracing his lady, launching a glance with her, along the edge of the stake, they poured a drink on the cuff, turning my hand, turning the sound , chuti bouly tilki mіrnі. clacking of spurs of quick and spry nig ad'yutant, and through the skin three tactics on the turn of nibi sleptahuvala develops oxamite cloth yogo dami. Natalka was amazed at them and she was ready to play a poster, but she won’t dance the first round of waltz.
Prince Andriy at his colonel, bilom (by cavalry) uniform, panchokhs and chereviks, lively and cheerful, standing at the first rows of the stake, not far from Rostov. Baron Firgof, speaking to him about tomorrow, will pass over the future for the sake of power. Prince Andriy, as a Lyudin is close to Speransky and take the fate of the robots of the Legislative Committee, at the time of the date of the last day of tomorrow, there was a little bit about yak. Alevin did not hear what Firhof was saying to you, and marveled first at the sovereign, then at the cavaliers, who danced, did not bother to enter the colo.
DOI: 10.14529 / mmph170306
STABILIZATION OF THE GATE PENDULUM ON THE VOKOLISNY TRANSPORT STATION
V.I. Ryazkikh1, M.Є. Semenov2, A.G. Rukavitsin3, O.I. Kanishcheva4, A.A. Demchuk4, P.A. Meleshenko3
1 Voronez State Technical University, Voronezh, Russian Federation
2 Voronez State University of Architecture and Education, Voronezh, Russian Federation
3 Voronez State University, Voronezh metro station, Russian Federation
4 Vyskovy nauchno-scientific center of the Vyyskovo-spiritual forces of the Vyyskovo-miracle academy of the name of professor N.N. Zhukovsky and Yu.A. Gagarina ", metro Voronezh, Russian Federation
Email: [email protected]
A mechanical system is looked at, it is stored from a two-wheel drive, a swinging pendulum is mounted on the axis. The zavdannya polyagaє at the formulation of such a fierce flow, which is formed according to the principle of a ringing sound, yak, from one side, would not be able to assign the law to the fall of a mechanic, and from the other, it would stabilize if the pendulum was not in position.
Key words: mechanical system; two-wheeled transport system; revolving pendulum; backlash; stabilization; keruvannya.
Entry
The ability to manage unstable technical systems has been theoretically looked at for a long time, but the practical significance of such management has clearly appeared for the rest of the hour. It was found that there are no control objects with a general control, they can be low in "brown" qualities. The attachments of such objects can be a spaceship at the stage of production, a thermonuclear reactor that is rich in quality. At the same hour, when the automatic control system goes out, an unstable object can be a real threat, not safe for people, and for a new community. As a catastrophic butt, the result of turning on the automatic control can lead to an accident on the Chornobil AES. In the world, as the management system becomes more and more superior, all the wider number of technical unsteady without management is stuck in practice. One of the simplest butts of unstable objects is the classic swinging pendulum. On one side, the zavdannya about yo stabilization is somewhat simpler and on the other hand, you can know the practical stasis at the opening of models of two-legged estots, as well as anthropomorphic attachments (robots, cyberworms, and changes) In the end of the rock, robots have appeared, assigned to the problems of stabilization of the vortex pendulum, tied with a two-wheeled transport vehicle, which will collapse. The number of possible prospects of storage at bagatokh galuzyah, such as transport and development, at the connection with a compact design, quick operation, high maneuverability and low vitrate of such annexes. Protect, zavdannya, scho gaze, is still far from the residual solution. Seemingly, there are many traditional technical annexes, but they are not stylish and robotic. A characteristic butt - a segway, an electric scooter by Din Kamen, self-balancing, with two wheels, roztashanimi on the sides of the water. Two wheels of the scooter roztashovany spivvisno. Segway automatically balances for changes in the position of the water body; for the whole business, the system of indicator stabilization is used: the signals from the gyroscopic and independent sensors go up to the microprocessors, as they vibrate the electrical signals, so that they are injected onto the engine and the engine. The skin wheel of the Segway is driven at the wrapping with its own electric motor, which responds to the changes of the machine. When the Segway goes forward, the Segway starts to go forward, when the Kuta gets bigger, the Segway’s shifts. When the body is pulled back, self-
kat upovilnyuє ruh, zupinyaєatsya or kitten in reverse. Keruvannya in the first model is seen for an additional rotary handle, in new models - the column left-right. The management of colival mechanical systems may be of significant theoretical interest that is of great practical importance.
Seemingly, in the process of functioning of mechanical systems, there is inevitably a lack of backlash in the process of functioning of mechanical systems; Mathematical models of such non-linearities are usually built up to the classics before the operators, who are looking like transformations on the most common functional spaces. The dynamism of such transformations is described by the names of "vhid-stan" and "stan-vihid".
Problem statement
The robots have a mechanical system that can be stored from a two-wheel drive, on the axis of a swing pendulum. The zavdannya polyagaє at the form of such a keruyuchey pouring, yak, from one side, would neglect the assignments of the law of the mechanical collapse, and from the other, it stabilized the unstable position of the pendulum. At the same time, there is a hysteresis of power in the system contour, so that you can live. Below is a graphical representation of the elements of the mechanical system, which can be used as a two-wheeled transport system, which is fastened to the new one by a swinging pendulum.
Small. 1. The main structural elements of the mechanical annex, where you can see
here / 1 / I feili / Fr I
"1" \ 1 \ 1 i R J
Hr! / / / / /1 / / /
Small. 2.Live and right of the wheel of the mechanical attachment with a high torque
Parameters and parameters, which describe the system that can be looked at: j - corner of the vehicle; D - stand between two wheels and axles; R - radius colic; Jj - moment of inertia; Tw - difference in torque moment of left and right wheel; v -
later the speed of transport; в - the cut of the swing of the pendulum from the vertical position; m is the mass of the inverted pendulum; l - become a center of gravity
the wheels; Ти - sum of torque moments of left and right wheels; x - a change of transport means for a direct post-production; M - masa shasi; M * - masa col_s; I - clearance to backlash.
Dynamika system
The dynamics of the system can be described as follows:
n = - + - Tn, W á WR n
= - - ml C0S in Tn,
de T * = Tb - TY; Tn = Tb + TY; Mx = M + t + 2 (M * + ^ *); 1b = t / 2 + 1C; 0. = Мх1в-т2 / 2 cob2;
<Р* = Рл С)Л = ^ С № = ^ О. (4)
A model that describes the dynamics of the change in the parameters of the system is possible in the view of two independent subsystems. The first pidsystem is stored from one ryvnyannya - r-pidsystem,
of the initial carriage of the transport vehicle:
Rivnyannya (5) can be rewritten from the viglyadi system from two rivnyans:
de e1 = P-Pd, e2 = (P- (Pa).
Another pidsystem, which describes the radio transmissions of the transport system, as well as the pendulum mounted on them, is stored in two pods - (y, v) pid systems:
U = - [Jqml в2 sin в- m2l2 g sin у cos в] + Jq Tu W в S J WR u
в = - - ml С ° * Tv W WR
System (7) manually in the first order viglyadі systems and іvnyans:
?
¿6 = ~ ^ - ^^^ + c)
de W0 = MxJq-P121 2cos2 (qd + e5), e3 = X - Xd, 4 = v - vd, 5 = q-qd, 6 = q-qd
The system is visible (6), control the yoke according to the principle of a ringing ring. For the sake of this, a new change is introduced, and the surface of the intermixing in the phase space of the system is significant.
5 = in! + c1e1, (9)
de s is a positive parameter. No mid-range visibilityє:
■ I = e + c1 e1 -cp + c1 e1. (ten)
For the stabilization of the overturned ruch, the moment is significant, which controls, the next rank:
T # P - ^ b1 - -MgP (51) - k2 (11)
de - positively set parameters.
Similarly, being controlled by another subsystem (8), to control it, we will also follow the principle of the ringing sound. For the sake of introducing a new change, it is significant that there is a jumbled surface in the phase space of the system, where ■ 2 = 0.
■ 2 = vz + s2vz, (12)
de c2 - positive parameter, toi
1 . 2 2 2
■ 2 = e3 + c2 e3 = (b + b6) ^ 5 + ve) - m 1 g ^ 5 + bc1) C08 (e5 + ba)] +
7 ^ T - + c2 ez
For the stabilization of the radial movement, the moment is important when it controls:
t "2/2 ^ up to T = -Km / (bj + eb) r ^ m (eb + bj) + nn ^ + bj) e08 (e5 + bj) - 0- \ c ez - + ^ n ^) + kA ^], (14)
de k3, k4 - positively given parameters.
In order to make sure that one hour keruvati about the system's subsystems, it is introduced additional inflow, which controls:
= § Hapv - [va + c3 (v-vij) - k588n (^ 3) - kb 53], (15)
de § - accelerated
padinnya; c3, k5, kb - positive parameters; 53 - the surface of the mixing, which should be used for the relationship:
53 = e6 + c3e5.
I will formulate the main results of the robot, which are based on the principles of the flexibility of stabilization of both subsystems, in the malleable ones, which can be used to control, in the vicinity of the zero position of the equalizer.
Theorem 1. System (6) with a heavy inflow (11) is absolutely asymptotically stable:
Нш || е11 | ® 0,
Нш || е2 || ® 0.t® ¥ u 2
Proof: it is significant to the Lyapunov function yak
de a = Dj 2 RJp.
Obviously, the function V> 0 is
V = Ш1 Si = Si. (eighteen)
Submitting (14) at V,
V = - (£ Sgn (S1) + k2 (S1)) S1. (19)
Obviously, V1 Theorem 2. It is possible to understand the subsystem (8) with a kerosene inlet (14). The system is absolutely asymptotically posed in the broken downs, so that for any cob-like minds, there is a difference: lim || e3 || ® 0, t® ¥ (20) lim 11 е41 | ® о. Proof: it is significant for the Lyapunov function for system (8) for additional performance de b = Wo R! Je. Obviously, the function V2> 0, V2 = M S2 = S2, which is why the zone of insensitivity is recognized by the amount of money to be infused. Probably a short description of a hysteretic re-transformation, which is victorious in the fake - backlash, background on the operator's treatment. The output of the retype - the backlash on the monotonous inputs is described by the relationship: x (t0) for quiet t, for x (t0) - h< u(t) < x(t0), x(t) = \u(t) при тех t, при которых u(t) >x (t0), (24) u (t) + h for quiet t, for u (t)< x(t0) - h, yake іlustruє fig. 3. For the additional help of a group, the operator's activity can be extended to all shmatkovo-monotonous inputs: Г x (t) = Г [Г x (t1), h] x (t) (25) that for the additional special boundary design on all without interruption. Since the operator's input does not differentiate, it is necessary to approximate the backlash to the Bowk-Week model. Tsia vіdoma napіvfіzichna model widely used for the phenomenological description of hysteresis effects. The popularity of the Bowk-Vienna shoe to be famous for the її building of hysteretic cycles. The formal description of the model is made up to the system of offensive ryvnyans: Fbw (x, ^ = ax () + (1-a) Dkz (t), = D "1 (AX-p \ x \\ z \ n-1 z-vusi | z | n). (26) Fbw (x, t) is passed as the output of the hysteresis re-conversion, and x (t) is at the input. Here n> 1, D> 0 k> 0 ma 0<а< 1. Small. 3. Dynamics of input-output responses to backlash It is easy to understand the systems (6) and (8), for those with a high flow it goes to the input of a hysteretic revert, and the output is a high flow to the system: Fbw (x, t) = akx (t) + (1 - a) Dkz (t), z = D_1 (Ax-bx || z \ n-1 z - gx | z \ n). ¿4 = W-J mlQd + eb) 2 sin (e5 + q) - m2l2g sin (e5 + ed) cos (e5 + 0d)] b = W -Fbw (x, t) = akx (t) + (1 - a) Dkz (t), ^ z = D_1 (A x-b \ x \ z \ n-1 z-gx \ z \ n). As well as earlier in the system, the main supply was stabilization, that is, the asymptotic behavior of phase changes. The graphs are induced lower at the quiet of the physical parameters of the system and with backlash and without backlash. The system was followed by numerical experiments. Tse ze vdannya bulo virіsheno middle of the program Wolfram Mathematica. The values of the constants are shown below: m = 3; M = 5; Mw = 1; D = 1.5; R = 0.25; l = 0.2; Jw = 1.5; Jc = 5; Jv = 1.5; j (0) = 0; x (0) = 0; Q (0) = 0.2; y (0) = [j (0) x (0) Q (0) f =)