Which angle is called zero positive and negative. trigonometric circle

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Let's call the rotation of the movable radius-vector in the counterclockwise direction positive, and in the opposite direction (in the clockwise direction) negative. The angle described by the negative rotation of the movable radius vector is called the negative angle.

Rule. An angle is measured as a positive number if it is positive and a negative number if it is negative.

Example 1. In fig. 80 shows two angles with a common initial side OA and a common end side OD: one is +270°, the other is -90°.

The sum of two angles. On the Oxy coordinate plane, consider a circle of unit radius centered at the origin (Fig. 81).

Let an arbitrary angle a (positive in the drawing) be obtained as a result of the rotation of some moving radius vector from its initial position OA, coinciding with the positive direction of the Ox axis, to its final position .

Let us now take the position of the radius-vector OE as the initial one and set aside from it an arbitrary angle (positive in the drawing), which will be obtained as a result of the rotation of some moving radius-vector from its initial position OE to its final position OS. As a result of these actions, we get an angle, which we will call the sum of the angles a and . (The initial position of the moving radius-vector OA, the final position of the radius-vector OS.)

Difference of two angles.

Under the difference of two angles a and , which we denote, we will understand such a third angle y, which, together with the angle, gives the angle a, i.e., if the Difference of two angles can be interpreted as the sum of the angles a and . Indeed, in general, for any angles their sum is measured by the algebraic sum of the real numbers that measure these angles.

Example 2. then .

Example 3. Angle , and angle . The sum of them.

In formula (95.1) it was assumed that is any non-negative integer. If we assume that - any integer (positive, negative or zero), then using the formula

where it will be possible to write down any angle, both positive and negative.

Example 4. An angle equal to -1370° can be written as follows:

Note that all angles written using formula (96.1), for different values, but the same a, have common initial (OA) and final (OE) sides (Fig. 79). Therefore, the construction of any angle is reduced to the construction of the corresponding non-negative angle less than 360°. On fig. 79 angles do not differ from each other, they differ only in the process of rotation of the radius vector, which led to their formation.

It characterizes the maximum angle at which the wheel of the car will turn with the steering wheel completely turned out. And the smaller this angle, the greater the accuracy and smoothness of control. After all, to turn even a small angle, only a small movement of the steering wheel is required.

But do not forget that the smaller the maximum turning angle, the smaller the turning radius of the car. Those. it will be very difficult to deploy in a limited space. So manufacturers have to look for some "golden mean", maneuvering between a large turning radius and control accuracy.

Changing the values ​​​​of the angles of installation of the wheels and their adjustment

The Piri Reis map has been compared to a modern map projection. Thus, he concluded that a mysterious map was taking over the world, as seen from a satellite hovering high above Cairo. In other words, over the Great Pyramid. It is surprising that Egyptologists are constantly defending these spaces, although there was a recent review of one recently opened corridor that has not yet brought any breakthroughs.

It is also worth noting that unusual psychotronic effects have been found in the pyramid, which, among other things, can affect human health. We are talking about spatial psychotronics, which creates both energy and geomagnetic "anomalous zones", which are further investigated.

Run-in shoulder - the shortest distance between the middle of the tire and the axis of rotation of the wheel. If the axis of rotation of the wheel and the middle of the wheel coincide, then the value is considered zero. With a negative value - the axis of rotation will move outward of the wheel, and with a positive value - inward.

When the wheel is turned, the tire is deformed under the action of lateral forces. And to keep the maximum contact patch with the road, the car's wheel also leans in the direction of the turn. But everywhere you need to know the measure, because with a very large caster, the wheel of the car will tilt a lot, and then lose traction.

Responsible for the weight stabilization of the steered wheels. The bottom line is that at the moment the wheel deviates from "neutral", the front end begins to rise. And since it weighs a lot, when the steering wheel is released under the influence of gravity, the system tends to take its original position, corresponding to movement in a straight line. True, in order for this stabilization to work, it is necessary to maintain a (albeit small, but undesirable) positive run-in shoulder.

Initially, the transverse angle of inclination of the axis of rotation was used by engineers to eliminate the shortcomings of the car's suspension. He got rid of such "ailments" of the car as a positive camber and a positive run-in shoulder.

During archaeological excavations, strange offerings of funerals in the form of birds with outstretched wings were also found. Later aerodynamic studies of these subjects revealed what were most likely ancient glider models. One of them was found with the inscription "Gift of Amon." The god Amun in Egypt was worshiped as a wind god so the association with flight is obvious.

But how did the members of this ancient civilization come to this knowledge without a preliminary stage of development? The answer is only in this case. This knowledge came from the governments of those times, which the Egyptians called their gods. It is quite possible for members of a technologically advanced civilization that has more than 000 years ago to have disappeared without a trace.

Many vehicles use MacPherson suspension. It makes it possible to obtain a negative or zero run-in shoulder. After all, the axis of rotation of the wheel consists of a support of a single lever, which can easily be placed inside the wheel. But this suspension is not perfect either, because due to its design it is almost impossible to make the angle of inclination of the axis of rotation small. In a turn, it leans the outside wheel at an unfavorable angle (like positive camber), while the inside wheel leans in the opposite direction at the same time.

But such facilities are still lacking. They decay, they can be destroyed, but it can also be well hidden in temples, pyramids and other iconic buildings that can lie still, properly secured against "treasure hunters".

The Great Pyramid's size and design precision has never been equaled. The pyramid weighs approximately six million tons. In its position as the Eiffel Tower, the Great Pyramid was the tallest building in the world. More than two million stones were used for its construction. Not a single stone weighs less than a ton.

As a result, the contact patch at the outer wheel is greatly reduced. And since the main load is on the outer wheel in a turn, the entire axle loses a lot of grip. This, of course, can be partially offset by caster and camber. Then the grip of the outer wheel will be good, while the inner one will practically disappear.

Car wheel alignment

There are two types of vehicle toe: positive and negative. Determining the type of convergence is very simple: you need to draw two straight lines along the wheels of the car. If these lines intersect in front of the car, then the convergence is positive, and if behind - negative. If there is a positive convergence of the front wheels, then the car will be easier to enter the turn, and will also acquire additional steering.

On the rear axle, with positive toe-in, the car will be more stable during straight-line movement, and if there is a negative toe-in, the car will behave inappropriately and scour from side to side.

And some of the more than seventy tons. Inside the chambers are connected by corridors. Today, a rough stone pyramid, but once it has been processed to a mirror-like masonry finish. It is believed that the peak of the Great Pyramid was adorned with pure gold. The sun's rays blinded hundreds of kilometers. For centuries, experts have speculated about the purpose of the pyramids. The traditional theory holds that the pyramids were a symbolic gateway to the underworld. Others believe that the pyramid was an astronomical observatory. Someone says that help is in the geographical dimension.

But it should be remembered that an excessive deviation of the toe of the car from zero will increase the rolling resistance in a straight line, in turns it will be less noticeable.

Camber

Camber, like toe, can be either negative or positive.

If you look at the front of the car, and the wheels will tilt inward, then this is negative camber, and if they deviate outward of the car, then this is already positive camber. The camber is necessary to maintain the adhesion of the wheel to the roadway.

One bizarre theory claims that the Great Pyramid was on granaries. However, experts today generally agree that the pyramids were much more than just a giant tomb. Scientists argue that the massive pyramid technology may not have been available to humans at this point in human history when these buildings were built. For example, the height of the pyramid corresponds to the distance from the Earth to the Sun. The pyramid was precisely oriented towards the four worlds with a precision that had never been achieved.

And surprisingly, the Great Pyramid lies in the exact center of the earth. Whoever built the Great Pyramid could accurately determine latitude and longitude. This is surprising because the technology for determining longitude was discovered in modern times in the sixteenth century. The pyramids were built in the exact center of the earth. Also, the height of the pyramid - seen from a great height, can be seen from the moon. Moreover, the shape of the pyramid is one of the best for reflecting radar. These reasons lead some researchers to believe that the Egyptian pyramids were built outside of their other purposes and for navigation by would-be foreign explorers.

Camber change affects the behavior of the car on a straight line, because the wheels are not perpendicular to the road, which means they do not have maximum grip. But this only affects rear-wheel drive cars when starting off with slippage.

› All about wheel alignment part 1.

For those who want to understand what Wheel Alignment (camber / toe) means and thoroughly understand the issue, this article has all the answers.

The Pyramid of Cheops is located just over eight kilometers west of Cairo. It is built on an artificially created apartment with an area of ​​1.6 square kilometers. Its base extends up to 900 square meters and is almost a millimeter wide when horizontal. Two and three quarters of a million stone blocks were used for the construction, with the heaviest weighing up to 70 tons. They fit in so that this fact is a mystery. However, the technical side of creating the pyramid remains a mystery, as it would be a major challenge for today's cutting-edge technology.

An excursion into history shows that intricate wheel alignment was used on various vehicles long before the advent of the automobile. Here are some more or less well-known examples.
It is no secret that the wheels of some carriages and other horse-drawn carriages designed for “dynamic” driving were installed with a large positive camber that was clearly visible to the eye. This was done so that the dirt flying from the wheels did not fall into the carriage and important riders, but was scattered around. In utilitarian carts for unhurried movement, everything was exactly the opposite. So, pre-revolutionary manuals on how to build a good cart recommended installing wheels with negative camber. In this case, with the loss of the dowel that locks the wheel, it did not immediately jump off the axle. The driver had time to notice the damage to the "chassis", fraught with especially great trouble if there were several tens of pounds of flour in the cart and there was no jack. In the design of gun carriages (again, vice versa), positive camber was sometimes used. It is clear that not in order to protect the gun from dirt. So it was convenient for the servants to roll the gun over the wheels with their hands from the side, without fear of crushing their legs. But at the arba, its huge wheels, which helped to easily get over the ditches, were tilted in the other direction - towards the wagon. The resulting increase in gauge contributed to an increase in the stability of the Central Asian "mobile", which was distinguished by a high center of gravity. What do these historical facts have to do with the installation of wheels on modern cars? Yes, in general, none. Nevertheless, they allow us to draw a useful conclusion. It can be seen that the installation of wheels (in particular, their collapse) is not subject to any single pattern.

Therefore, there are no hypotheses that magical powers were used in the construction of the pyramid - magical formulas written on papyrus made it possible to move heavy pieces of stone and put them on top of each other with amazing accuracy. Edgar Cayce said that these pyramids were built ten thousand years ago, while others believe that the pyramids were built by the inhabitants of Atlantis, who, before the cataclysm that destroyed their continent, mainly sought refuge in Egypt. He creates science centers, they also created a pyramidal shelter where great secrets could be hidden.

When choosing this parameter, the "manufacturer" in each case was guided by different considerations, which he considered to be a priority. So, what do car suspension designers strive for when choosing UUK? Of course, to the ideal. The ideal for a car that moves in a straight line is the position of the wheels when the planes of their rotation (rolling plane) are perpendicular to the road surface, parallel to each other, the axis of symmetry of the body and coincide with the trajectory of movement. In this case, the loss of power due to friction and wear of the tire tread is minimal, and the grip of the wheels with the road, on the contrary, is maximum. Naturally, the question arises: what makes you deliberately deviate from the ideal? Looking ahead, there are several considerations. First, we judge wheel alignment based on a static picture when the car is stationary. Who said that in motion, when accelerating, braking and maneuvering a car, it does not change? Second, reducing waste and extending tire life is not always a priority. Before talking about what factors suspension designers take into account, let's agree that out of a large number of parameters that describe the geometry of a car's suspension, we will limit ourselves to only those that are included in the primary or main group. They are so called because they determine the setting and properties of the suspension, are always monitored during its diagnosis and adjusted, if such a possibility is provided. These are the well-known convergence, camber and angles of inclination of the axis of rotation of the steered wheels. When considering these important parameters, we will have to think about other characteristics of the suspension.

The pyramid consists of 203 layers of stone blocks weighing from 2.5 to 15 tons. Some blocks at the bottom of the pyramid at the base weigh up to 50 tons. Originally, the entire pyramid was covered with a fine white and polished limestone shell, but stone was used for construction, especially after frequent earthquakes in the area.

The weight of the pyramid is proportional to the weight of the Earth 1:10. The pyramid is a maximum of 280 Egyptian cubits and the base area is 440 Egyptian cubits. If the basic scheme is divided by twice the height of the pyramid, we get the Ludolph number - 3. The deviation from the Ludolph figure is only 0.05%. The base of the base is equal to the circumference of a circle with a radius equal to the height of the pyramid.

Toe (TOE) characterizes the orientation of the wheels relative to the longitudinal axis of the vehicle. The position of each wheel can be determined separately from the others, and then one speaks of an individual convergence. It represents the angle between the plane of rotation of the wheel and the axis of the vehicle when viewed from above. The total convergence (or simply convergence) of the wheels of one axle. as the name suggests, is the sum of the individual angles. If the planes of rotation of the wheels intersect in front of the car, the convergence is positive (toe-in), if behind - negative (toe-out). In the latter case, we can talk about the divergence of the wheels.
In the adjustment data, sometimes the convergence is given not only in the form of an angular, but also a linear value. It's related to that. that the convergence of the wheels is also judged by the difference in the distances between the flanges of the rims, measured at the level of their centers behind and in front of the axle.

Whatever the truth may be, archaeologists will certainly recognize the skill of the ancient builders, for example. Flinders Petrie concluded that the errors in measurement were so small that he lined his finger. The walls connecting the corridors, falling 107 m into the center of the pyramid, showed a deviation of only 0.5 cm from ideal accuracy. Can we explain the mystery of the pharaoh's pyramid to the pedantry of the architects and builders, or unknown Egyptian magic, or the simple need to keep the dimensions as close as possible in order to achieve the maximum benefit of the pyramid?

In various sources, including serious technical literature, the version is often given that wheel alignment is necessary to compensate for the side effects of camber. Like, due to the deformation of the tire in the contact patch, the “collapsed” wheel can be represented as the base of the cone. If the wheels are installed with a positive camber angle (why - it doesn’t matter yet), they tend to “roll out” in different directions. To counteract this, the planes of rotation of the wheels are reduced. (Fig. 20)

Is it just a coincidence that this number expresses the distance from the Sun, which is reported in millions of miles? An Egyptian cubit is exactly one ten millimeter radius of the earth. The Great Pyramid expresses the ratio of 2p between the circumference and the radius of the Earth. Circle The square area of ​​a circle is 023 feet.

He also discusses the similarities between figures in the Nazca, the Great Pyramid, and Egyptian hieroglyphic texts. Bowles notes that the Great Pyramid and Nazca will be at the equator when the North Pole is located in southeast Alaska. Using coordinates and spherical trigonometry, the book demonstrates a remarkable connection between three points - ancient sites.

The version, it must be said, is not devoid of elegance, but does not stand up to criticism. If only because it suggests an unambiguous relationship between collapse and convergence. Following the proposed logic, wheels with a negative camber angle must be installed with a discrepancy, and if the camber angle is zero, then there should be no convergence. In reality, this is not at all the case.

Of course, this connection also exists between the Great Pyramid, the Nazca platform and the axis of the "ancient line", regardless of where the North Pole is located. This relationship can be used to determine the distances between three points and a plane. In the royal chamber, the diagonal is 309 from the eastern wall, the distance from the chamber is 412, the middle diagonal is 515.

The distances between Ollantaytambo, the Great Pyramid and the Axis Point on the "Ancient Line" express the same geometric relationship. 3-4 The distance of the Great Pyramid from Ollantaytambo is exactly 30% of the Earth's periphery. The distance from the Great Pyramid to Machu Picchu and the Axis Point in Alaska is 25% of the earth's perimeter. Stretching this isosceles triangle in height, we get two right-angled triangles with sides from 15% to 20% - 25%.

Reality, as usual, obeys more complex and ambiguous laws. When an inclined wheel rolls, there is indeed a lateral force in the contact patch, which is often called that - camber thrust. It arises as a result of the elastic deformation of the tire in the transverse direction and acts in the direction of the slope. The greater the angle of inclination of the wheel, the greater the camber thrust. It is she who is used by drivers of two-wheeled vehicles - motorcycles and bicycles - when cornering. It is enough for them to tilt their steed to make it “prescribe” a curvilinear trajectory, which can only be corrected by steering. The camber thrust plays an important role in the maneuvering of cars, as will be discussed later. So it is hardly worth deliberately compensating for convergence. Yes, and the very message that, due to the positive camber angle, the wheels tend to turn outward, i.e. in the direction of divergence, is incorrect. On the contrary, the design of the suspension of the steered wheels in most cases is such that, with positive camber, its thrust tends to increase convergence. So “compensation for the side effect of the camber” has nothing to do with it. There are several factors that determine the need for wheel alignment. The first is that the effect of longitudinal forces acting on the wheel when the car is moving is compensated by the previously set convergence. The nature and depth (and hence the result) of influence depend on many circumstances: the drive wheel or free-rolling, controlled, or not, finally, on the kinematics and elasticity of the suspension. Thus, a rolling resistance force acts on a freely rolling wheel of a car in the longitudinal direction. It creates a bending moment that tends to turn the wheel relative to the suspension mounts in the direction of divergence. If the car's suspension is rigid (for example, not a split or torsion beam), then the effect will not be very significant. Nevertheless, it will certainly be, since "absolute rigidity" is a term and a purely theoretical phenomenon. In addition, the movement of the wheel is determined not only by the elastic deformation of the suspension elements, but also by the compensation of structural gaps in their joints, wheel bearings, etc.
In the case of a suspension with high compliance (which is typical, for example, for lever structures with elastic bushings), the result will increase many times over. If the wheel is not only free-rolling, but also steerable, the situation becomes more complicated. Due to the appearance of an additional degree of freedom at the wheel, the same resistance force has a double effect. The moment that bends the front suspension is complemented by a moment that tends to turn the wheel around the axis of rotation. The turning moment, the value of which depends on the location of the axis of rotation, affects the details of the steering mechanism and, due to their compliance, also contributes significantly to changing the wheel toe in motion. Depending on the run-in shoulder, the contribution of the turning moment can be with a “plus” or “minus” sign. That is, it can either increase the divergence of the wheels, or counteract this. If you do not take all this into account and initially install wheels with zero toe-in, they will take a diverging position in motion. From this, the consequences that are typical for cases of violation of the toe adjustment will “follow”: increased fuel consumption, sawtooth tread wear and handling problems, which will be discussed later.
The force of resistance to movement depends on the speed of the car. Therefore, the ideal solution would be a variable toe, providing an equally ideal wheel alignment at any speed. Since this is difficult to do, the wheel is preliminarily “flattened” in such a way as to achieve minimal tire wear at cruising speed. The wheel located on the drive axle is subjected to traction force most of the time. It exceeds the forces of resistance to movement, so the resultant forces will be directed in the direction of movement. Applying the same logic, we get that in this case the wheels in static need to be installed with a discrepancy. A similar conclusion can be drawn with regard to the steerable drive wheels.
The best criterion of truth is practice. If, with this in mind, look at the adjustment data for modern cars, you can be disappointed not to find a big difference in the toe-in of the steered wheels of the rear- and front-wheel drive models. In most cases, for both, this parameter will be positive. Unless among front-wheel drive cars there are more cases of “neutral” toe adjustment. The reason is not that the above logic is not correct. It's just that when choosing the amount of convergence, along with the compensation of longitudinal forces, other considerations are taken into account that amend the final result. One of the most important is ensuring optimal vehicle handling. With the growth of speeds and dynamism of vehicles, this factor is becoming increasingly important.
Handling is a multifaceted concept, so it is worth clarifying that wheel alignment most significantly affects the stabilization of the straight trajectory of the car and its behavior at the entrance to the turn. This effect can be clearly illustrated by the example of steered wheels.

Let's suppose that while moving in a straight line, one of them is subjected to a random perturbing effect from a road roughness. The increased drag force turns the wheel in the direction of decreasing toe-in. Through the steering mechanism, the impact is transmitted to the second wheel, the convergence of which, on the contrary, increases. If initially the wheels have a positive convergence, the resistance force on the first one decreases, and on the second one it increases, which counteracts the perturbation. When the convergence is equal to zero, there is no counteracting effect, and when it is negative, a destabilizing moment appears, which contributes to the development of the perturbation. A car with such a toe adjustment will scour the road, it will have to be constantly caught by steering, which is unacceptable for a normal road car.
This "coin" has a reverse, positive side - a negative convergence allows you to get the fastest response from the steering. The slightest action of the driver immediately provokes a sharp change in the trajectory - the car willingly maneuvers, easily "agrees" to turn. Such a toe adjustment is very often used in motorsport.

Those who watch TV shows about the WRC championship, probably paid attention to how actively you have to work with the wheel of the same Loeb or Grönholm, even on relatively straight sections of the track. The toe-in of the rear axle has a similar effect on the behavior of the car - reducing the toe-in down to a slight difference increases the “mobility” of the axle. This effect is often used to compensate for understeer in vehicles such as front wheel drive models with an overloaded front axle.
Thus, the static toe parameters that are given in the adjustment data represent a kind of superposition, and sometimes a compromise, between the desire to save on fuel and rubber and achieve optimal handling characteristics for the car. Moreover, it is noticeable that in recent years the latter prevails.

Camber is a parameter that is responsible for the orientation of the wheel relative to the road surface. We remember that ideally they should be perpendicular to each other, i.e. collapse should not be. However, most road cars have it. What's the point?

Reference.
Camber reflects the orientation of the wheel relative to the vertical and is defined as the angle between the vertical and the plane of rotation of the wheel. If the wheel is actually "fallen apart", i.e. its apex is tilted outward, the camber is considered positive. If the wheel is tilted towards the body, the camber is negative.

Until recently, there was a tendency to break the wheels, i.e. give positive values ​​to the camber angles. Many, for sure, remember the textbooks on the theory of the car, in which the installation of camber wheels was explained by the desire to redistribute the load between the outer and inner wheel bearings. Like, with a positive camber angle, most of it falls on the inner bearing, which is easier to make more massive and durable. As a result, the durability of the bearing unit is improved. The thesis is not very convincing, if only because, if it is true, it is only for an ideal situation - a rectilinear movement of a car on an absolutely flat road. It is known that during maneuvers and the passage of irregularities, even the smallest ones, the bearing assembly experiences dynamic loads, which are an order of magnitude higher than static forces. Yes, and they are not distributed exactly as "dictated" by the positive camber.

Sometimes they try to interpret positive camber as an additional measure aimed at reducing the break-in shoulder. When we get to know this important parameter of the steering wheel suspension, it will become clear that this method of influence is far from the most successful. It is associated with a simultaneous change in the track width and the included angle of inclination of the axis of rotation of the wheel, which is fraught with undesirable consequences. There are more direct and less painful options for changing the break-in shoulder. In addition, its minimization is not always the goal of suspension designers.

More convincing is the version that positive camber compensates for wheel displacement that occurs with an increase in axle load (as a result of an increase in vehicle loading or a dynamic redistribution of its mass during acceleration and braking). The elasto-kinematic properties of most types of modern suspensions are such that as the weight on the wheel increases, the camber angle decreases. In order to ensure maximum grip of the wheels with the road, it is logical to “break them up” a little beforehand. Moreover, in moderate doses, camber has little effect on rolling resistance and tire wear.

It is reliably known that the choice of the camber value is also influenced by the generally accepted profiling of the roadway. In civilized countries, where there are roads, not directions, their cross section has a convex profile. In order for the wheel to remain perpendicular to the ground in this case, it needs to be given a slight positive camber angle.
Looking through the specifications for UUK, one can notice that in recent years the opposite “disintegration trend” has prevailed. The wheels of most production cars are statically installed with negative camber. The fact is that, as already mentioned, the task of ensuring their best stability and controllability comes to the fore. Camber is a parameter that has a decisive influence on the so-called lateral reaction of the wheels. It is she who counteracts the centrifugal forcesacting on the car in a turn, and helps to keep it on a curved path. From general considerations, it follows that the grip of the wheel with the road (lateral reaction) will be maximum at the largest area of ​​the contact patch, i.e. with the wheel in a vertical position. In fact, with a standard design wheel, it peaks at small negative lean angles, which is due to the contribution of the camber thrust mentioned. This means that in order to make the wheels of the car extremely tenacious in a turn, you need not to fall apart, but, on the contrary, “dump”. This effect has been known for a long time and has been used in motorsport for just as long. If you objectively look at the "formula" car, it is clearly visible that its front wheels are installed with a large negative camber.

What is good for racing cars is not so good for stock cars. Excessive negative camber causes increased wear on the inner tread area. With an increase in the inclination of the wheel, the area of ​​​​the contact patch is reduced. The grip of the wheels during rectilinear movement decreases, in turn, the efficiency of acceleration and braking decreases. Excessive negative camber affects the car's ability to maintain a straight line in the same way as insufficient toe-in, the car becomes unnecessarily nervous. The same craving for collapse is to blame for this. In an ideal situation, the camber-induced side forces act on both wheels of the axle and balance each other. But as soon as one of the wheels loses traction, the camber thrust of the other turns out to be uncompensated and causes the car to deviate from a straight path. By the way, if we recall that the amount of thrust depends on the inclination of the wheel, it is not difficult to explain the side slip of the car at different camber angles of the right and left wheels. In a word, when choosing the size of the collapse, you also have to look for the "golden mean".

To provide the car with good stability, it is not enough to make the camber angles negative in statics. Suspension designers must ensure that the wheels maintain optimal (or close to it) orientation in all modes of motion. This is not easy to do, because during maneuvers any changes in the position of the body, accompanied by a displacement of the suspension elements (dives, side rolls, etc.), lead to a significant change in the camber of the wheels. Oddly enough, this problem is solved more easily on sports cars with their "furious" suspensions, characterized by high angular rigidity and short travel. Here, the static values ​​​​of the collapse (and convergence) are the least different from how they look in dynamics.

The greater the range of suspension travel, the greater the change in camber in motion. Therefore, developers of ordinary road cars with the most elastic (for the best comfort) suspensions have the hardest time. They have to rack their brains over how to "combine the incompatible" - comfort and stability. Usually a compromise can be found by "conjuring" over the kinematics of the suspension.

There are solutions to minimize camber changes and to give these changes a desirable "trend". For example, it is desirable that in the turn the most loaded outer wheel would remain in the very optimal position - with a slight negative camber. To do this, when the body rolls, the wheel must “fall over” even more on it, which is achieved by optimizing the geometry of the suspension guide elements. In addition, they try to reduce body roll themselves by using anti-roll bars.
In fairness, it should be said that the elasticity of the suspension is not always the enemy of stability and handling. In "good hands", elasticity, on the contrary, contributes to them. For example, with the skillful use of the effect of "self-steering" of the wheels of the rear axle. Returning to the topic of conversation, we can summarize that the camber angles that are indicated in the specifications for cars will be significantly different from what they turn out to be.

Completing the "dismantling" with convergence and collapse, we can mention another interesting aspect of practical importance. In the adjustment data on the UUK, not the absolute values ​​\u200b\u200bof the camber and convergence angles are given, but the ranges of permissible values. Toe-in tolerances are tighter and usually do not exceed ±10", camber tolerances are several times looser (±30" on average). This means that the master who adjusts the UUK can tune the suspension without going beyond the factory specifications. It would seem that a few tens of arc minutes is nonsense. I drove the parameters into the "green corridor" - and order. But let's see what the result might be. For example, the specifications for the BMW 5 Series in the E39 body indicate: toe-in 0 ° 5 "± 10", camber -0 ° 13 "± 30". This means that, while remaining in the "green corridor", the toe can take on a value from -0°5" to 5", and the camber from -43" to 7". That is, both convergence and collapse can be negative, neutral or positive. Having an idea of ​​how toe and camber affect the behavior of a car, you can deliberately “sham” these parameters in such a way as to get the desired result. The effect will not be dramatic, but it will certainly be.

The camber and toe considered by us are the parameters that are determined for all four wheels of the car. Next, we will talk about the angular characteristics, which are related only to the steered wheels and determine the spatial orientation of the axis of their rotation.

It is known that the position of the axis of rotation of the steered wheel of a car is determined by two angles: longitudinal and transverse. And why not make the axis of rotation strictly vertical? Unlike cases with collapse and convergence, the answer to this question is more unambiguous. There is almost unanimity here, at least in relation to the longitudinal angle of inclination - caster.

It is rightly noted that the main function of the caster is high-speed (or dynamic) stabilization of the steered wheels of the car. Stabilization in this case is the ability of the steered wheels to resist deviation from the neutral (corresponding to rectilinear motion) position and automatically return to it after the termination of the external forces that caused the deviation. Disturbing forces constantly act on a moving automobile wheel, tending to bring it out of a neutral position. They may be the result of road roughness, unbalanced wheels, etc. Since the magnitude and direction of perturbations are constantly changing, their impact is of a random oscillatory nature. If there were no stabilization mechanism, the driver would have to parry the vibrations, which would turn the car into a torment and probably increase tire wear. With proper stabilization, the car moves steadily in a straight line with minimal driver intervention and even with the steering wheel released.

Steering wheel deflection can be caused by the driver's intentional actions associated with a change in direction of travel. In this case, the stabilizing effect assists the driver on corner exit by automatically returning the wheels to neutral. But at the entrance to the turn and in its apex, the "driver", on the contrary, has to overcome the "resistance" of the wheels, applying a certain force to the steering wheel. The reactive force generated on the steering wheel creates what is called steering feel or steering information and to which car designers and automotive journalists pay a lot of attention.

Alpha denotes a real number. The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. If we take an infinite set of natural numbers as an example, then the considered examples can be represented as follows:

To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an "infinite hotel"? An infinity inn is an inn that always has any number of vacancies, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, on the other hand, are not able to move away from banal everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers exist. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on a shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. What if you really want to? No problem. We can take a unit from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:

I have written down the operations in algebraic notation and in set theory notation, listing the elements of the set in detail. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one is subtracted from it and the same one is added.

Option two. We have many different infinite sets of natural numbers on the shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If another infinite set is added to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler for measurements. Now imagine that you have added one centimeter to the ruler. This will already be a different line, not equal to the original.

You can accept or not accept my reasoning - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add mental abilities to us (or vice versa, they deprive us of free thinking).

Sunday, August 4, 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... the rich theoretical basis of Babylonian mathematics did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."

Wow! How smart we are and how well we can see the shortcomings of others. Is it weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:

The rich theoretical basis of modern mathematics does not have a holistic character and is reduced to a set of disparate sections, devoid of a common system and evidence base.

I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole cycle of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.

May we have many BUT consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter a, the subscript with a number will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set BUT on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.

After multiplication, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question, how correctly applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.

As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.

As you can see, units of measurement and common math make set theory a thing of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.

Finally, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.

Wednesday, July 4, 2018

I already told you that, with the help of which shamans try to sort "" realities. How do they do it? How does the formation of the set actually take place?

Let's take a closer look at the definition of a set: "a collection of different elements, conceived as a single whole." Now feel the difference between the two phrases: "thinkable as a whole" and "thinkable as a whole." The first phrase is the end result, the multitude. The second phrase is a preliminary preparation for the formation of the set. At this stage, reality is divided into separate elements ("whole") from which a multitude ("single whole") will then be formed. At the same time, the factor that allows you to combine the "whole" into a "single whole" is carefully monitored, otherwise the shamans will not succeed. After all, shamans know in advance exactly what set they want to demonstrate to us.

I will show the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, and there are without a bow. After that, we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.

Now let's do a little trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received sets "with a bow" and "red" the same set or two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.

This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We formed a set of "red solid pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.

The letter "a" with different indices denotes different units of measurement. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to which the set is formed, is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it with “obviousness”, because units of measurement are not included in their “scientific” arsenal.

With the help of units of measurement, it is very easy to break one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Saturday, June 30, 2018

If mathematicians cannot reduce a concept to other concepts, then they do not understand anything in mathematics. I answer: how do the elements of one set differ from the elements of another set? The answer is very simple: numbers and units of measure.

It is today that everything that we do not take belongs to some set (as mathematicians assure us). By the way, did you see in the mirror on your forehead a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of sets to which this thing belongs. Sets are all inventions of shamans. How do they do it? Let's look a little deeper into history and see how the elements of the set looked before the mathematicians-shamans pulled them apart into their sets.

A long time ago, when no one had heard of mathematics yet, and only trees and Saturn had rings, huge herds of wild elements of sets roamed the physical fields (after all, shamans had not yet invented mathematical fields). They looked like this.

Yes, do not be surprised, from the point of view of mathematics, all elements of the sets are most similar to sea urchins - from one point, like needles, units of measurement stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any quantity can be represented as a bundle of segments sticking out in different directions from one point. This point is the zero point. I will not draw this work of geometric art (no inspiration), but you can easily imagine it.

What units of measurement form an element of the set? Any that describe this element from different points of view. These are the ancient units of measurement used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are units of measurement unknown to us, which our descendants will come up with and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. And what about physics? Units of measurement - this is the direct connection between mathematics and physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. I personally can’t imagine a real science of mathematics without units of measurement. That is why, at the very beginning of the story about set theory, I spoke of it as the Stone Age.

But let's move on to the most interesting - to the algebra of elements of sets. Algebraically, any element of the set is a product (the result of multiplication) of different quantities. It looks like this.

I deliberately did not use the conventions adopted in set theory, since we are considering an element of a set in a natural habitat before the advent of set theory. Each pair of letters in brackets denotes a separate value, consisting of the number indicated by the letter " n" and units of measurement, indicated by the letter " a". Indexes near the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of values ​​\u200b\u200b(as long as we and our descendants have enough imagination). Each bracket is geometrically represented by a separate segment. In the example with a sea urchin one bracket is one needle.

How do shamans form sets from different elements? In fact, by units of measurement or by numbers. Understanding nothing in mathematics, they take different sea urchins and carefully examine them in search of that single needle by which they form a set. If there is such a needle, then this element belongs to the set; if there is no such needle, this element is not from this set. Shamans tell us fables about mental processes and a single whole.

As you may have guessed, the same element can belong to a variety of sets. Next, I will show you how sets, subsets and other shamanic nonsense are formed. As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...

And now I have the most interesting question: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.

Look here. We select football stadiums with the same field area. The area of ​​the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

If you are already familiar with trigonometric circle , and you just want to refresh individual elements in your memory, or you are completely impatient, then here it is, :

Here we will analyze everything in detail step by step.

The trigonometric circle is not a luxury, but a necessity

Trigonometry many are associated with an impassable thicket. Suddenly, so many values ​​​​of trigonometric functions pile up, so many formulas ... But it’s, after all, it didn’t work out at first, and ... off and on ... sheer misunderstanding ...

It is very important not to wave your hand at values ​​of trigonometric functions, - they say, you can always look at the spur with a table of values.

If you constantly look at the table with the values ​​​​of trigonometric formulas, let's get rid of this habit!

Will save us! You will work with it several times, and then it will pop up in your head on its own. Why is it better than a table? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!

For example, say, looking at standard table of values ​​of trigonometric formulas , which is the sine of, say, 300 degrees, or -45.

No way? .. you can, of course, connect reduction formulas... And looking at the trigonometric circle, you can easily answer such questions. And you will soon know how!

And when solving trigonometric equations and inequalities without a trigonometric circle - nowhere at all.

Introduction to the trigonometric circle

Let's go in order.

First, write down the following series of numbers:

And now this:

And finally this one:

Of course, it is clear that, in fact, in the first place is, in the second place is, and in the last -. That is, we will be more interested in the chain .

But how beautiful it turned out! In which case, we will restore this “wonderful ladder”.

And why do we need it?

This chain is the main values ​​​​of sine and cosine in the first quarter.

Let's draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius along the length, and declare its length to be unit).

From the “0-Start” beam, we set aside in the direction of the arrow (see Fig.) corners.

We get the corresponding points on the circle. So, if we project the points onto each of the axes, then we will get exactly the values ​​from the above chain.

Why is that, you ask?

Let's not take everything apart. Consider principle, which will allow you to cope with other, similar situations.

Triangle AOB is a right triangle with . And we know that opposite the angle at lies a leg twice as small as the hypotenuse (our hypotenuse = the radius of the circle, that is, 1).

Hence, AB= (and hence OM=). And by the Pythagorean theorem

I hope something is clear now.

So point B will correspond to the value, and point M will correspond to the value

Similarly with the rest of the values ​​of the first quarter.

As you understand, the axis familiar to us (ox) will be cosine axis, and the axis (oy) - sinus axis . later.

To the left of zero on the cosine axis (below zero on the sine axis) there will, of course, be negative values.

So, here it is, the ALL-POWERFUL, without which nowhere in trigonometry.

But how to use the trigonometric circle, we'll talk in.