Ballistic missile flight altitude. Intercontinental ballistic missiles - TOP10

In which there is no thrust or control force and moment, it is called a ballistic trajectory. If the mechanism that powers the object remains operational throughout the entire period of movement, it belongs to the category of aviation or dynamic. The trajectory of the aircraft during flight with the engines turned off high altitude can also be called ballistic.

An object that moves along given coordinates is affected only by the mechanism that drives the body, the forces of resistance and gravity. A set of such factors excludes the possibility of rectilinear movement. This rule works even in space.

The body describes a trajectory that is similar to an ellipse, hyperbola, parabola or circle. The last two options are achieved with the second and first cosmic speeds. Calculations for parabolic or circular motion are performed to determine the trajectory of a ballistic missile.

Taking into account all the parameters during launch and flight (weight, speed, temperature, etc.), the following trajectory features are distinguished:

• In order to launch the rocket as far as possible, you need to choose the right angle. The best is sharp, about 45º.
• The object has the same initial and final speed.
• The body lands at the same angle as it launches.
• The time it takes for an object to move from the start to the middle, as well as from the middle to the finishing point, is the same.

Trajectory properties and practical implications

The movement of a body after the influence of the driving force on it ceases is studied by external ballistics. This science provides calculations, tables, scales, sights and develops optimal options for shooting. The ballistic trajectory of a bullet is the curved line described by the center of gravity of an object in flight.

Since the body is affected by gravity and resistance, the path that the bullet (projectile) describes forms the shape of a curved line. Under the influence of these forces, the speed and height of the object gradually decreases. There are several trajectories: flat, mounted and conjugate.

The first is achieved by using an elevation angle that is less than the angle of greatest range. If the flight range remains the same for different trajectories, such a trajectory can be called conjugate. In the case where the elevation angle is greater than the angle of greatest range, the path becomes called a suspended path.

The trajectory of the ballistic movement of an object (bullet, projectile) consists of points and sections:

• Departure(for example, the muzzle of a barrel) - this point is the beginning of the path, and, accordingly, the reference.
• Weapons horizon- this section passes through the departure point. The trajectory crosses it twice: during release and during fall.
• Elevation area- this is a line that is a continuation of the horizon and forms a vertical plane. This area is called the firing plane.
• Trajectory vertices- this is the point that is located in the middle between the starting and ending points (shot and fall), has the highest angle along the entire path.
• Tips- the target or sighting location and the beginning of the object’s movement form the aiming line. An aiming angle is formed between the horizon of the weapon and the final target.

Rockets: features of launch and movement

There are managed and unmanaged ballistic missiles. The formation of the trajectory is also influenced by external and external factors (resistance forces, friction, weight, temperature, required flight range, etc.).

The general path of a launched body can be described by the following stages:

• Launch. In this case, the rocket enters the first stage and begins its movement. From this moment, the measurement of the height of the ballistic missile’s flight path begins.
• After about a minute, the second engine starts.
• 60 seconds after the second stage, the third engine starts.
• Then the body enters the atmosphere.
• Lastly, the warheads explode.

Launching a rocket and forming a movement curve

The rocket's travel curve consists of three parts: the launch period, free flight and re-entry into the earth's atmosphere.

Combat projectiles are launched from a fixed point on portable installations, as well as vehicles (ships, submarines). The flight initiation lasts from tenths of a thousandths of a second to several minutes. Free fall constitutes the largest portion of a ballistic missile's flight path.

The advantages of running such a device are:

• Long free flight time. Thanks to this property, fuel consumption is significantly reduced in comparison with other rockets. For prototype flight ( cruise missiles) more efficient engines are used (for example, jet engines).
• At the speed at which the intercontinental weapon moves (approximately 5 thousand m/s), interception is very difficult.
• The ballistic missile is capable of hitting a target at a distance of up to 10 thousand km.

In theory, the path of movement of a projectile is a phenomenon from the general theory of physics, the branch of the dynamics of solid bodies in motion. With respect to these objects, the movement of the center of mass and the movement around it are considered. The first relates to the characteristics of the object in flight, the second to stability and control.

Since the body has programmed trajectories for flight, the calculation of the ballistic trajectory of the missile is determined by physical and dynamic calculations.

Modern developments in ballistics

Because the combat missiles of any kind are dangerous to life, main task defense is the improvement of points for launching striking systems. The latter must ensure the complete neutralization of intercontinental and ballistic weapons at any point in the movement. A multi-tier system is proposed for consideration:

• This invention consists of separate tiers, each of which has its own purpose: the first two will be equipped with laser-type weapons (homing missiles, electromagnetic guns).
• The next two sections are equipped with the same weapons, but designed to destroy the head parts of enemy weapons.

Developments in defense missile technology do not stand still. Scientists are modernizing a quasi-ballistic missile. The latter is presented as an object that has a low path in the atmosphere, but at the same time sharply changes direction and range.

The ballistic trajectory of such a missile does not affect its speed: even at an extremely low altitude, the object moves faster than a normal one. For example, the Russian-developed Iskander flies at supersonic speeds - from 2100 to 2600 m/s with a mass of 4 kg 615 g; missile cruises move a warhead weighing up to 800 kg. During flight, it maneuvers and evades missile defenses.

Intercontinental weapons: control theory and components

Multistage ballistic missiles are called intercontinental missiles. This name appeared for a reason: due to the long flight range, it becomes possible to transfer cargo to the other end of the Earth. The main combat substance (charge) is mainly an atomic or thermonuclear substance. The latter is located in the front of the projectile.

Next, a control system, engines and fuel tanks are installed in the design. Dimensions and weight depend on the required flight range: the greater the distance, the higher the launch weight and dimensions of the structure.

The ballistic flight trajectory of an ICBM is distinguished from the trajectory of other missiles by altitude. The multi-stage rocket goes through the launch process, then moves upward at a right angle for several seconds. The control system ensures that the gun is directed towards the target. The first stage of the rocket drive separates independently after complete burnout, and at the same moment the next one is launched. Upon reaching a given speed and flight altitude, the rocket begins to rapidly move down towards the target. The flight speed to the destination reaches 25 thousand km/h.

World developments of special-purpose missiles

About 20 years ago, during the modernization of one of the missile systems medium range The anti-ship ballistic missile project was adopted. This design is placed on an autonomous launch platform. The weight of the projectile is 15 tons, and the launch range is almost 1.5 km.

The trajectory of a ballistic missile for destroying ships is not amenable to quick calculations, so it is impossible to predict enemy actions and eliminate this weapon.

This development has the following advantages:

• Launch range. This value is 2-3 times greater than that of the prototypes.
• Flight speed and altitude make military weapon invulnerable to missile defense.

World experts are confident that weapons of mass destruction can still be detected and neutralized. For such purposes, special out-of-orbit reconnaissance stations, aviation, submarines, ships, etc. are used. The most important “countermeasure” is space reconnaissance, which is presented in the form of radar stations.

The ballistic trajectory is determined by the reconnaissance system. The received data is transmitted to its destination. The main problem is the rapid obsolescence of information - in a short period of time, the data loses its relevance and can diverge from the actual location of the weapon at a distance of up to 50 km.

Characteristics of combat systems of the domestic defense industry

Most powerful weapon Currently, an intercontinental ballistic missile is considered to be stationary. The domestic missile system "R-36M2" is one of the best. It houses the heavy-duty 15A18M combat weapon, which is capable of carrying up to 36 individual precision-guided nuclear projectiles.

The ballistic flight path of such a weapon is almost impossible to predict; accordingly, neutralizing a missile also poses difficulties. The combat power of the projectile is 20 Mt. If this ammunition explodes at a low altitude, the communication, control, and missile defense systems will fail.

Modifications given rocket launcher can also be used for peaceful purposes.

Among solid fuel missiles, the RT-23 UTTH is considered especially powerful. Such a device is based autonomously (mobile). In the stationary prototype station (“15Zh60”), the starting thrust is 0.3 higher compared to the mobile version.

Missile launches carried out directly from stations are difficult to neutralize, because the number of projectiles can reach 92 units.

Missile systems and installations of the foreign defense industry

Height of the missile's ballistic trajectory American complex Minuteman 3 is not particularly different from the flight characteristics of domestic inventions.

The complex, which was developed in the USA, is the only “defender” of North America among weapons of this type up to today. Despite the age of the invention, the gun’s stability indicators are quite good even today, because the complex’s missiles could withstand missile defense and also hit a target with a high level of protection. The active part of the flight is short and lasts 160 seconds.

Another American invention is the Peakkeeper. It could also ensure an accurate hit on the target thanks to the most favorable trajectory of ballistic movement. Experts say that combat capabilities the given complex is almost 8 times higher than that of the Minuteman. The Peacekeeper's combat duty was 30 seconds.

Projectile flight and movement in the atmosphere

From the dynamics section we know the influence of air density on the speed of movement of any body in various layers of the atmosphere. The function of the last parameter takes into account the dependence of density directly on flight altitude and is expressed as a function of:

N (y) = 20000-y/20000+y;

where y is the height of the projectile (m).

The parameters and trajectory of an intercontinental ballistic missile can be calculated using special programs on a computer. The latter will provide statements, as well as data on flight altitude, speed and acceleration, and the duration of each stage.

The experimental part confirms the calculated characteristics and proves that the speed is influenced by the shape of the projectile (the better the streamlining, the higher the speed).

Guided weapons of mass destruction of the last century

All weapons of this type can be divided into two groups: ground and airborne. Ground-based devices are those that are launched from stationary stations (for example, mines). Aviation, accordingly, is launched from a carrier ship (aircraft).

The ground-based group includes ballistic, cruise and anti-aircraft missiles. Aviation - projectile aircraft, ADB and guided air combat missiles.

The main characteristic of calculating the ballistic trajectory is the altitude (several thousand kilometers above the atmospheric layer). At a given level above the ground, projectiles reach high speeds and create enormous difficulties for their detection and neutralization of missile defense.

Well-known ballistic missiles that are designed for medium flight range are: “Titan”, “Thor”, “Jupiter”, “Atlas”, etc.

The ballistic trajectory of a missile, which is launched from a point and hits specified coordinates, has the shape of an ellipse. The size and length of the arc depends on the initial parameters: speed, launch angle, mass. If the projectile speed is equal to the first cosmic speed (8 km/s), a military weapon, which is launched parallel to the horizon, will turn into a satellite of the planet with a circular orbit.

Despite constant improvements in the field of defense, the flight path of a military projectile remains virtually unchanged. At the moment, technology is not able to violate the laws of physics that all bodies obey. A small exception are homing missiles - they can change direction depending on the movement of the target.

The inventors of anti-missile systems are also modernizing and developing a weapon to destroy weapons. mass destruction new generation.

The ICBM is a very impressive human creation. Huge size, thermonuclear power, column of flame, roar of engines and the menacing roar of launch... However, all this exists only on the ground and in the first minutes of launch. After they expire, the rocket ceases to exist. Further into the flight and to carry out the combat mission, only what remains of the rocket after acceleration is used - its payload.

With long launch ranges, the payload of an intercontinental ballistic missile extends into space for many hundreds of kilometers. It rises into the layer of low-orbit satellites, 1000-1200 km above the Earth, and is located among them for a short time, only slightly lagging behind their general run. And then it begins to slide down along an elliptical trajectory...

What exactly is this load?

A ballistic missile consists of two main parts - the booster part and the other for the sake of which the boost is started. The accelerating part is a pair or three of large multi-ton stages, filled to capacity with fuel and with engines at the bottom. They give the necessary speed and direction to the movement of the other main part of the rocket - the head. The booster stages, replacing each other in the launch relay, accelerate this warhead in the direction of the area of ​​its future fall.

The head of a rocket is a complex load consisting of many elements. It contains a warhead (one or more), a platform on which these warheads are placed along with all other equipment (such as means of deceiving enemy radars and missile defenses), and a fairing. There is also fuel and compressed gases in the head part. The entire warhead will not fly to the target. It, like the ballistic missile itself earlier, will split into many elements and simply cease to exist as a single whole. The fairing will separate from it not far from the launch area, during the operation of the second stage, and somewhere along the way it will fall. The platform will collapse upon entering the air of the impact area. Only one type of element will reach the target through the atmosphere. Warheads. Up close, the warhead looks like an elongated cone, a meter or one and a half long, with a base as thick as a human torso. The nose of the cone is pointed or slightly blunt. This cone is a special aircraft whose task is to deliver weapons to the target. We'll come back to warheads later and take a closer look at them.

Pull or push?

In a missile, all warheads are located in the so-called breeding stage, or “bus”. Why bus? Because, having first been freed from the fairing, and then from the last booster stage, the propagation stage carries the warheads, like passengers, along given stops, along their trajectories, along which the deadly cones will disperse to their targets.

The “bus” is also called the combat stage, because its work determines the accuracy of pointing the warhead to the target point, and therefore combat effectiveness. The breeding stage and its work is one of the most big secrets in a rocket. But we will still take a slight, schematic look at this mysterious step and its difficult dance in space.

The dilution stage has different shapes. Most often, it looks like a round stump or a wide loaf of bread, on which warheads are mounted on top, points forward, each on its own spring pusher. The warheads are pre-positioned at precise separation angles (at the missile base, manually, using theodolites) and point in different directions, like a bunch of carrots, like the needles of a hedgehog. The platform, bristling with warheads, occupies a given position in flight, gyro-stabilized in space. And at the right moments, warheads are pushed out of it one by one. They are ejected immediately after completion of acceleration and separation from the last accelerating stage. Until (you never know?) they shot down this entire undiluted hive with anti-missile weapons or something on board the breeding stage failed.

The pictures show the breeding stages of the American heavy ICBM LGM0118A Peacekeeper, also known as MX. The missile was equipped with ten 300 kt multiple warheads. The missile was withdrawn from service in 2005.

But this happened before, at the dawn of multiple warheads. Now breeding presents a completely different picture. If earlier the warheads “stuck” forward, now the stage itself is in front along the course, and the warheads hang from below, with their tops back, inverted, like the bats. The “bus” itself in some rockets also lies upside down, in a special recess in the upper stage of the rocket. Now, after separation, the breeding stage does not push, but drags the warheads along with it. Moreover, it drags, resting against its four “paws” placed crosswise, deployed in front. At the ends of these metal legs are rearward-facing thrust nozzles for the expansion stage. After separation from the accelerating stage, the “bus” very accurately, precisely sets its movement in the beginning of space with the help of its own powerful guidance system. He himself occupies the exact path of the next warhead - its individual path.

Then the special inertia-free locks that held the next detachable warhead are opened. And not even separated, but simply now no longer connected with the stage, the warhead remains motionless hanging here, in complete weightlessness. The moments of her own flight began and flowed by. Like one individual berry next to a bunch of grapes with other warhead grapes not yet plucked from the stage by the breeding process.

K-551 "Vladimir Monomakh" - Russian nuclear submarine strategic purpose(project 955 "Borey"), armed with 16 solid-fuel Bulava ICBMs with ten multiple warheads.

Delicate movements

Now the task of the stage is to crawl away from the warhead as delicately as possible, without disturbing its precisely set (targeted) movement with gas jets of its nozzles. If a supersonic jet of a nozzle hits a separated warhead, it will inevitably add its own additive to the parameters of its movement. Over the subsequent flight time (which is half an hour to fifty minutes, depending on the launch range), the warhead will drift from this exhaust “slap” of the jet half a kilometer to a kilometer sideways from the target, or even further. It will drift without obstacles: there is space, they slapped it - it floated, not being held back by anything. But is a kilometer sideways really accurate today?

Submarines of Project 955 "Borey" - a series of Russian nuclear submarines class "strategic missile submarine cruiser" fourth generation. Initially, the project was created for the Bark missile, which was replaced by the Bulava.

To avoid such effects, it is precisely the four upper “legs” with engines that are spaced apart to the sides that are needed. The stage is, as it were, pulled forward on them so that the exhaust jets go to the sides and cannot catch the warhead separated by the belly of the stage. All thrust is divided between four nozzles, which reduces the power of each individual jet. There are other features too. For example, if there is a donut-shaped propulsion stage (with a void in the middle), this hole is attached to the rocket’s upper stage, like wedding ring finger) of the Trident-II D5 missile, the control system determines that the separated warhead still falls under the exhaust of one of the nozzles, then the control system turns off this nozzle. Silences the warhead.

The stage, gently, like a mother from the cradle of a sleeping child, fearing to disturb his peace, tiptoes away into space on the three remaining nozzles in low thrust mode, and the warhead remains on the aiming trajectory. Then the “donut” stage with the cross of the thrust nozzles is rotated around the axis so that the warhead comes out from under the zone of the torch of the switched off nozzle. Now the stage moves away from the remaining warhead on all four nozzles, but for now also at low throttle. When a sufficient distance is reached, the main thrust is turned on, and the stage vigorously moves into the area of ​​the target trajectory of the next warhead. There it slows down in a calculated manner and again very precisely sets the parameters of its movement, after which it separates the next warhead from itself. And so on - until it lands each warhead on its trajectory. This process is fast, much faster than you read about it. In one and a half to two minutes, the combat stage deploys a dozen warheads.

American Ohio-class submarines are the only type of missile carrier in service with the United States. Carries on board 24 ballistic missiles with MIRVed Trident-II (D5). The number of warheads (depending on power) is 8 or 16.

The abysses of mathematics

What has been said above is quite enough to understand how a warhead’s own path begins. But if you open the door a little wider and look a little deeper, you will notice that today the rotation in space of the breeding stage carrying the warheads is an area of ​​​​application of quaternion calculus, where the on-board attitude control system processes the measured parameters of its movement with a continuous construction of the on-board orientation quaternion. A quaternion is such a complex number (above the field of complex numbers lies a flat body of quaternions, as mathematicians would say in their precise language of definitions). But not with the usual two parts, real and imaginary, but with one real and three imaginary. In total, the quaternion has four parts, which, in fact, is what it says Latin root quatro.

The dilution stage does its job quite low, immediately after the boost stages are turned off. That is, at an altitude of 100−150 km. And there is also the influence of gravitational anomalies on the Earth’s surface, heterogeneities in the even gravitational field surrounding the Earth. Where are they from? From the uneven terrain, mountain systems, occurrence of rocks of different densities, oceanic depressions. Gravitational anomalies either attract the stage to themselves with additional attraction, or, conversely, slightly release it from the Earth.

In such irregularities, the complex ripples of the local gravitational field, the breeding stage must place the warheads with precision accuracy. To do this, it was necessary to create a more detailed map of the Earth's gravitational field. It is better to “explain” the features of a real field in systems of differential equations that describe precise ballistic motion. These are large, capacious (to include details) systems of several thousand differential equations, with several tens of thousands of constant numbers. And the gravitational field itself at low altitudes, in the immediate near-Earth region, is considered as a joint attraction of several hundred point masses of different “weights” located near the center of the Earth in a certain order. This achieves a more accurate simulation of the Earth's real gravitational field along the rocket's flight path. And more precise work flight control systems with it. And also... but that's enough! - Let's not look further and close the door; What has been said is enough for us.

The ICBM payload spends most of its flight in space object mode, rising to an altitude three times the height of the ISS. The trajectory of enormous length must be calculated with extreme accuracy.

Flight without warheads

The breeding stage, accelerated by the missile towards the same geographical area where the warheads should fall, continues its flight along with them. After all, she can’t fall behind, and why should she? After disengaging the warheads, the stage urgently attends to other matters. She moves away from the warheads, knowing in advance that she will fly a little differently from the warheads, and not wanting to disturb them. All yours further actions The breeding stage is also dedicated to warheads. This maternal desire to protect the flight of her “children” in every possible way continues for the rest of her short life. Short, but intense.

After the separated warheads, it is the turn of other wards. The most amusing things begin to fly away from the steps. Like a magician, she releases into space a lot of inflating balloons, some metal things that resemble open scissors, and objects of all sorts of other shapes. Durable air balloons sparkle brightly in the cosmic sun with the mercury shine of a metallized surface. They are quite large, some shaped like warheads flying nearby. Their aluminum-coated surface reflects a radar signal from a distance in much the same way as the warhead body. Enemy ground radars will perceive these inflatable warheads as well as real ones. Of course, in the very first moments of entering the atmosphere, these balls will fall behind and immediately burst. But before that, they will distract and load the computing power of ground-based radars - both long-range detection and guidance of anti-missile systems. In ballistic missile interceptor parlance, this is called “complicating the current ballistic environment.” And the entire heavenly army, inexorably moving towards the area of ​​impact, including real and false warheads, balloons, dipole and corner reflectors, this whole motley flock is called “multiple ballistic targets in a complicated ballistic environment.”

The metal scissors open up and become electric dipole reflectors - there are many of them, and they well reflect the radio signal of the long-range missile detection radar beam probing them. Instead of the ten desired fat ducks, the radar sees a huge blurry flock of small sparrows, in which it is difficult to make out anything. Devices of all shapes and sizes reflect different lengths waves

In addition to all this tinsel, the stage can theoretically itself emit radio signals that interfere with the targeting of enemy anti-missile missiles. Or distract them with yourself. In the end, you never know what she can do - after all, a whole stage is flying, large and complex, why not load it with a good solo program?

The photo shows the launch of a Trident II intercontinental missile (USA) from a submarine. Currently, Trident is the only family of ICBMs whose missiles are installed on American submarines. The maximum throwing weight is 2800 kg.

Last segment

However, from an aerodynamic point of view, the stage is not a warhead. If that one is a small and heavy narrow carrot, then the stage is an empty, vast bucket, with echoing empty fuel tanks, a large, streamlined body and a lack of orientation in the flow that is beginning to flow. With its wide body and decent windage, the stage responds much earlier to the first blows of the oncoming flow. The warheads also unfold along the flow, piercing the atmosphere with the least aerodynamic resistance. The step leans into the air with its vast sides and bottoms as necessary. It cannot fight the braking force of the flow. Its ballistic coefficient - an “alloy” of massiveness and compactness - is much worse than a warhead. Immediately and strongly it begins to slow down and lag behind the warheads. But the forces of the flow increase inexorably, and at the same time the temperature heats up the thin, unprotected metal, depriving it of its strength. The remaining fuel boils merrily in the hot tanks. Finally, the hull structure loses stability under the aerodynamic load that compresses it. Overload helps to destroy the bulkheads inside. Crack! Hurry! The crumpled body is immediately engulfed by hypersonic shock waves, tearing the stage into pieces and scattering them. After flying a little in the condensing air, the pieces again break into smaller fragments. Remaining fuel reacts instantly. Flying fragments of structural elements made of magnesium alloys are ignited by hot air and instantly burn with a blinding flash, similar to a camera flash - it’s not for nothing that magnesium was set on fire in the first photo flashes!

Everything is now on fire, everything is covered in hot plasma and shines well around orange coals from the fire. The denser parts go to decelerate forward, the lighter and sailier parts are blown into a tail stretching across the sky. All burning components produce dense smoke plumes, although at such speeds these very dense plumes cannot exist due to the monstrous dilution by the flow. But from a distance they are clearly visible. The ejected smoke particles stretch along the flight trail of this caravan of bits and pieces, filling the atmosphere with a wide white trail. Impact ionization gives rise to the nighttime greenish glow of this plume. Because of irregular shape fragments, their deceleration is rapid: everything that is not burned quickly loses speed, and with it the intoxicating effect of the air. Supersonic is the strongest brake! Having stood in the sky like a train falling apart on the tracks, and immediately cooled by the high-altitude frosty subsound, the strip of fragments becomes visually indistinguishable, loses its shape and structure and turns into a long, twenty minutes, quiet chaotic dispersion in the air. If you are in the right place, you can hear a small charred piece of duralumin clinking quietly against a birch trunk. Here you are. Goodbye breeding stage!

Introduction

Mechanics(Greek μηχανική - the art of building machines) - a branch of physics, a science that studies the movement of material bodies and the interaction between them; in this case, motion in mechanics is the change in time of the relative position of bodies or their parts in space.

“Mechanics, in the broad sense of the word, is a science devoted to solving any problems related to the study of the movement or equilibrium of certain material bodies and the interactions between bodies that occur during this process. Theoretical mechanics is the part of mechanics that studies general laws motion and interaction of material bodies, that is, those laws that, for example, are valid for the movement of the Earth around the Sun, and for the flight of a rocket or artillery shell, etc. The other part of mechanics consists of various general and special technical disciplines devoted to the design and calculation of all kinds of specific structures, engines, mechanisms and machines or their parts (parts).” 1

Special technical disciplines include the Flight Mechanics offered to you for study [of ballistic missiles (BMs), launch vehicles (LVs) and spacecraft (SCs)]. ROCKET- an aircraft moving due to the ejection of high-speed hot gases created by a jet (rocket) engine. In most cases, the energy to propel a rocket is obtained from the combustion of two or more chemical components (fuel and oxidizer, which together form rocket fuel) or from the decomposition of one high-energy chemical 2 .

The main mathematical apparatus of classical mechanics: differential and integral calculus, developed specifically for this by Newton and Leibniz. The modern mathematical apparatus of classical mechanics includes, first of all, the theory of differential equations, differential geometry, functional analysis, etc. In the classical formulation of mechanics, it is based on Newton’s three laws. The solution of many problems in mechanics is simplified if the equations of motion allow the possibility of formulating conservation laws (momentum, energy, angular momentum and other dynamic variables).

The task of studying the flight of an unmanned aircraft is in general very difficult, because for example, an aircraft with fixed (fixed) rudders, like any rigid body, has 6 degrees of freedom and its movement in space is described by 12 differential equations of the first order. The flight path of a real aircraft is described by a significantly larger number of equations.

Due to the extreme complexity of studying the flight trajectory of a real aircraft, it is usually divided into a number of stages and each stage is studied separately, moving from simple to complex.

At the first stage research, one can consider the movement of an aircraft as the movement of a material point. It is known that the motion of a rigid body in space can be divided into the translational motion of the center of mass and the rotational motion of the rigid body around its own center of mass.

To study the general pattern of aircraft flight, in some cases under certain conditions it is possible not to consider rotational motion. Then the movement of the aircraft can be considered as the movement of a material point, the mass of which is equal to the mass of the aircraft and to which the forces of thrust, gravity and aerodynamic drag are applied.

It should be noted that even with such a simplified formulation of the problem, in some cases it is necessary to take into account the moments of forces acting on the aircraft and the required deflection angles of the controls, because otherwise, it is impossible to establish an unambiguous relationship, for example, between lift and angle of attack; between lateral force and sliding angle.

At the second stage The equations of motion of an aircraft are studied, taking into account its rotation around its own center of mass.

The task is to study and study the dynamic properties of an aircraft, considered as an element of a system of equations, and are mainly interested in the reaction of the aircraft to the deviation of the controls and the influence of various external influences on the aircraft.

At the third stage(the most complex) they conduct a study of the dynamics of a closed control system, which includes, along with other elements, the aircraft itself.

One of the main tasks is to study flight accuracy. Accuracy is characterized by the magnitude and probability of deviation from the required trajectory. To study the accuracy of aircraft motion control, it is necessary to create a system of differential equations that would take into account all forces and moments. acting on the aircraft, and random disturbances. The result is a system of high-order differential equations, which can be nonlinear, with regular time-dependent parts, with random functions on the right-hand sides.

Missile classification

Missiles are usually classified by type of flight path, by location and direction of launch, by flight range, by type of engine, by type of warhead, and by type of control and guidance systems.

Depending on the type of flight path, there are:

– Cruise missiles. Cruise missiles are unmanned, controlled (until the target is hit) aircraft that are kept in the air for most of their flight by aerodynamic lift. The main goal cruise missiles is the delivery of a warhead to a target. They move through the Earth's atmosphere using jet engines.

Intercontinental ballistic cruise missiles can be classified depending on their size, speed (subsonic or supersonic), flight range and launch location: from the ground, air, surface of a ship or submarine.

Depending on the flight speed, rockets are divided into:

1) Subsonic cruise missiles

2) Supersonic cruise missiles

3) Hypersonic cruise missiles

Subsonic cruise missile moves at a speed below the speed of sound. It develops a speed corresponding to the Mach number M = 0.8 ... 0.9. A well-known subsonic missile is the American Tomahawk cruise missile. Below are diagrams of two Russian subsonic cruise missiles in service.

X-35 Uran – Russia

Supersonic cruise missile moves at a speed of about M=2...3, that is, it covers a distance of approximately 1 kilometer per second. The modular design of the rocket and its ability to be launched at different angles of inclination allow it to be launched from various carriers: warships, submarines, Various types aircraft, mobile autonomous installations and launch silos. The supersonic speed and mass of the warhead provides it with high kinetic impact energy (for example, Onyx (Russia) aka Yakhont - export version; P-1000 Vulcan; P-270 Moskit; P-700 Granit)

P-270 Moskit – Russia

P-700 Granit – Russia

Hypersonic cruise missile moves at a speed of M > 5. Many countries are working on creating hypersonic cruise missiles.

– Ballistic missiles. A ballistic missile is a missile that has a ballistic trajectory for most of its flight path.

Ballistic missiles are classified according to their flight range. The maximum flight range is measured along a curve along the surface of the earth from the launch point to the point of impact of the last element of the warhead. Ballistic missiles can be launched from sea and land-based carriers.

The launch location and launch direction determine the class of the rocket:

Surface-to-surface missiles. A surface-to-surface missile is a guided missile that can be launched from the hands, vehicle, mobile or stationary installation. It is propelled by a rocket motor or sometimes, if a stationary launcher is used, fired by a powder charge.

In Russia (and earlier in the USSR), surface-to-surface missiles are also divided by purpose into tactical, operational-tactical and strategic. In other countries, based on their intended purpose, surface-to-surface missiles are divided into tactical and strategic.

Surface-to-air missiles. A surface-to-air missile is launched from the surface of the earth. Designed to destroy air targets such as airplanes, helicopters and even ballistic missiles. These missiles are usually part of the air defense system, as they repel any type of air attack.

Surface-to-sea missiles. The surface (ground)-sea missile is designed to be launched from the ground to destroy enemy ships.

Air-to-air missiles. The air-to-air missile is launched from aircraft carriers and is designed to destroy air targets. Such rockets have speeds up to M = 4.

Air-to-surface (ground, water) missiles. The air-to-surface missile is designed to be launched from aircraft carriers to strike both ground and surface targets.

Sea-to-sea missiles. The sea-to-sea missile is designed to be launched from ships to destroy enemy ships.

Sea-to-ground (coast) missiles. Sea-to-surface missile ( coastal zone)" is designed to be launched from ships at ground targets.

Anti-tank missiles. The anti-tank missile is designed primarily to destroy heavily armored tanks and other armored vehicles. Anti-tank missiles can be launched from airplanes, helicopters, tanks, and shoulder-mounted launchers.

Based on their flight range, ballistic missiles are divided into:

short-range missiles;

medium-range missiles;

medium-range ballistic missiles;

intercontinental ballistic missiles.

International agreements since 1987 have used a different classification of missiles by range, although there is no generally accepted standard classification of missiles by range. Different states and non-governmental experts use different classifications of missile ranges. Thus, the Treaty on the Elimination of Intermediate-Range and Short-Range Missiles adopted the following classification:

ballistic missiles short range(from 500 to 1000 kilometers).

medium-range ballistic missiles (from 1000 to 5500 kilometers).

intercontinental ballistic missiles (over 5500 kilometers).

By engine type and fuel type:

solid propellant motor or solid propellant rocket motors;

liquid engine;

hybrid engine - chemical rocket engine. Uses components rocket fuel in different states of aggregation- liquid and solid. The solid state can contain both an oxidizing agent and a fuel.

ramjet engine (ramjet engine);

Ramjet with supersonic combustion;

cryogenic engine - uses cryogenic fuel (these are liquefied gases stored at very low temperatures, most often liquid hydrogen used as a fuel and liquid oxygen used as an oxidizer).

Warhead type:

Regular warhead. A conventional warhead is filled with chemical explosives, which explode when detonated. Additional damaging factor are fragments of the metal casing of the rocket.

Nuclear warhead.

Intercontinental and medium-range missiles are often used as strategic missiles and are equipped with nuclear warheads. Their advantage over airplanes is their short approach time (less than half an hour at intercontinental range) and high speed of the warhead, which makes them very difficult to intercept even with a modern missile defense system.

Guidance systems:

Fly-by-wire guidance. This system is generally similar to radio control, but is less susceptible to electronic countermeasures. Command signals are sent via wires. After the missile is launched, its connection with the command post is terminated.

Command guidance. Command guidance involves tracking the missile from the launch site or launch vehicle and transmitting commands via radio, radar or laser, or through tiny wires and optical fibers. Tracking can be accomplished by radar or optical devices from the launch site, or via radar or television images transmitted from the missile.

Guidance by ground landmarks. The correlation guidance system based on ground landmarks (or a terrain map) is used exclusively for cruise missiles. The system uses sensitive altimeters to monitor the terrain profile directly below the missile and compare it with a "map" stored in the missile's memory.

Geophysical guidance. The system constantly measures the angular position of the aircraft in relation to the stars and compares it with the programmed angle of the rocket along the intended trajectory. The guidance system provides information to the control system whenever it is necessary to make adjustments to the flight path.

Inertial guidance. The system is programmed before launch and is completely stored in the “memory” of the rocket. Three accelerometers mounted on a stand stabilized in space by gyroscopes measure acceleration along three mutually perpendicular axes. These accelerations are then integrated twice: the first integration determines the rocket's speed, and the second its position. The control system is configured to save in advance given trajectory flight. These systems are used in surface-to-surface (surface, water) missiles and cruise missiles.

Beam guidance. A ground-based or ship-based radar station is used, which follows the target with its beam. Information about the object enters the missile guidance system, which, if necessary, adjusts the guidance angle in accordance with the movement of the object in space.

Laser guidance. With laser guidance, a laser beam is focused on a target, reflected from it and scattered. The missile contains a laser homing head, which can detect even a small source of radiation. The homing head sets the direction of the reflected and scattered laser beam to the guidance system. The missile is launched towards the target, the homing head looks for the laser reflection, and the guidance system directs the missile towards the source of the laser reflection, which is the target.

Military missile weapons are usually classified according to the following parameters:

belonging to types of aircraft – ground troops, naval forces, air force;

flight range(from the place of application to the target) - intercontinental (launch range - more than 5500 km), medium range (1000–5500 km), operational-tactical range (300-1000 km), tactical range (less than 300 km);

physical environment of use– from the launch site (ground, air, surface, underwater, under the ice);

basing method– stationary, mobile (mobile);

nature of the flight– ballistic, aeroballistic (with wings), underwater;

flight environment– air, underwater, space;

type of control- controlled, uncontrolled;

target purpose– anti-tank (anti-tank missiles), anti-aircraft (anti-aircraft missile), anti-ship, anti-radar, anti-space, anti-submarine (against submarines).

Classification of launch vehicles

Unlike some horizontally launched aerospace systems (AKS), launch vehicles use a vertical type of launch and (much less often) air launch.

Number of steps.

Single-stage launch vehicles that launch payloads into space have not yet been created, although there are projects of varying degrees of development (“CORONA”, HEAT-1X and others). In some cases, a rocket that has an air carrier as the first stage or uses accelerators as such can be classified as single-stage. Among the ballistic missiles capable of reaching outer space, many are single-stage, including the first V-2 ballistic missile; however, none of them is capable of entering the orbit of an artificial Earth satellite.

Location of steps (layout). The design of launch vehicles can be as follows:

longitudinal layout (tandem), in which the stages are located one after the other and operate alternately in flight (Zenit-2, Proton, Delta-4 launch vehicles);

parallel arrangement (package), in which several blocks located in parallel and belonging to different stages operate simultaneously in flight (Soyuz LV);

• conditional package layout (the so-called one-and-a-half-stage scheme), in which common fuel tanks are used for all stages, from which the starting and propulsion engines are powered, starting and operating simultaneously; When the starting motors are finished operating, only they are reset.

combined longitudinal-transverse layout.

Engines used. The following can be used as propulsion engines:

liquid rocket engines;

solid propellant rocket engines;

different combinations at different levels.

Payload weight. Depending on the mass of the payload, launch vehicles are divided into the following classes:

super-heavy class missiles (more than 50 tons);

heavy class missiles (up to 30 tons);

medium-class missiles (up to 15 tons);

light class missiles (up to 2-4 tons);

ultra-light class missiles (up to 300-400 kg).

The specific boundaries of classes change with the development of technology and are quite arbitrary; currently, the light class is considered to be rockets that launch a payload weighing up to 5 tons into a low reference orbit, medium - from 5 to 20 tons, heavy - from 20 to 100 tons, super-heavy - over 100 t. A new class of so-called “nano-carriers” (payload up to several tens of kg) is also emerging.

Reuse. The most widespread are disposable multi-stage rockets, both in batch and longitudinal configurations. Disposable rockets are highly reliable due to the maximum simplification of all elements. It should be clarified that in order to achieve orbital speed, a single-stage rocket theoretically needs to have a final mass of no more than 7-10% of the starting mass, which, even with existing technologies, makes them difficult to implement and economically ineffective due to the low mass of the payload. In the history of world cosmonautics, single-stage launch vehicles were practically never created - only the so-called ones existed. one and a half stage modifications (for example, the American Atlas launch vehicle with resettable additional starting engines). The presence of several stages makes it possible to significantly increase the ratio of the mass of the launched payload to the initial mass of the rocket. At the same time, multistage rockets require the alienation of territories for the fall of intermediate stages.

Due to the need to use highly efficient complex technologies (primarily in the field of propulsion systems and thermal protection), completely reusable launch vehicles do not yet exist, despite the constant interest in this technology and periodically opening projects for the development of reusable launch vehicles (over the period of the 1990-2000s – such as: ROTON, Kistler K-1, AKS VentureStar, etc.). Partially reusable were the widely used American reusable transport space system (MTKS)-AKS "Space Shuttle" ("Space Shuttle") and the closed Soviet program MTKS "Energia-Buran", developed but never used in applied practice, as well as a number unrealized former (for example, "Spiral", MAKS and other AKS) and newly developed (for example, "Baikal-Angara") projects. Contrary to expectations, the Space Shuttle was unable to reduce the cost of delivering cargo into orbit; in addition, manned MTKS are characterized by a complex and lengthy stage of pre-launch preparation (due to increased requirements for reliability and safety in the presence of a crew).

Human presence. Rockets for manned flights must be more reliable (an emergency rescue system is also installed on them); permissible overloads for them are limited (usually no more than 3-4.5 units). At the same time, the launch vehicle itself is a fully automatic system that launches a device into outer space with people on board (this can be either pilots capable of directly controlling the device or so-called “space tourists”).

Intercontinental ballistic missiles (ICBMs) are the primary means of nuclear deterrence. The following countries have this type of weapon: Russia, USA, Great Britain, France, China. Israel does not deny the presence of these types of missiles, but does not officially confirm it either, but it has the capabilities and known developments to create such a missile.

Below is a list of intercontinental ballistic missiles ranked by maximum range.

1. P-36M (SS-18 Satan), Russia (USSR) - 16,000 km

• The P-36M (SS-18 Satan) is an intercontinental missile with the world's longest range - 16,000 km. Hit accuracy 1300 meters.
• Launch weight 183 tons. The maximum range is achieved with a warhead mass of up to 4 tons, with a warhead mass of 5825 kg, the missile’s flight range is 10200 kilometers. The missile can be equipped with multiple and monoblock warheads. To protect against missile defense (BMD), when approaching the affected area, the missile throws out decoy targets for the BMD. The rocket was developed at the Yuzhnoye design bureau named after. M. K. Yangelya, Dnepropetrovsk, Ukraine. The main missile base is silo-based.
• The first R-36Ms entered the USSR Strategic Missile Forces in 1978.
• The rocket is two-stage, with liquid rocket engines providing a speed of about 7.9 km/sec. Withdrawn from service in 1982, replaced by a next-generation missile based on the R-36M, but with increased accuracy and the ability to overcome missile defense systems. Currently, the rocket is used for peaceful purposes, to launch satellites into orbit. The created civilian rocket was named Dnepr.

2. DongFeng 5A (DF-5A), China - 13,000 km.

• DongFeng 5A (NATO reporting name: CSS-4) has the longest flight range among the Chinese Army ICBMs. Its flight range is 13,000 km.
• The missile was designed to be capable of hitting targets within the Continental United States (CONUS). The DF-5A missile entered service in 1983.
• The missile can carry six warheads weighing 600 kg each.
• The inertial guidance system and on-board computers ensure the desired direction of the rocket's flight. Rocket engines two-stage with liquid fuel.

3. R-29RMU2 Sineva (RSM-54, according to NATO classification SS-N-23 Skiff), Russia - 11,547 kilometers

• The R-29RMU2 Sineva, also known as the RSM-54 (NATO code name: SS-N-23 Skiff), is a third generation intercontinental ballistic missile. The main basing of missiles is submarines. Sineva showed a maximum range of 11,547 kilometers during testing.
• The missile entered service in 2007 and is expected to be in use until 2030. The missile is capable of carrying from four to ten individually targetable warheads. Used for flight control Russian system GLONASS. Targets are hit with high precision.
• The rocket is three-stage, liquid jet engines are installed.

4. UGM-133A Trident II (D5), USA - 11,300 kilometers

• The UGM-133A Trident II is an intercontinental ballistic missile designed for submarine deployment.
• Currently, missile submarines are based on the Ohio (USA) and Vanguard (UK) submarines. In the United States, this missile will be in service until 2042.
• The first launch of UGM-133A was carried out from the Cape Canaveral launch site in January 1987. The missile entered service with the US Navy in 1990. The UGM-133A can be equipped with eight warheads for various purposes.
• The missile is equipped with three solid-fuel rocket engines, providing a flight range of up to 11,300 kilometers. It is highly reliable; during testing, 156 launches were carried out and only 4 of them were unsuccessful, and 134 consecutive launches were successful.

5. DongFeng 31 (DF-31A), China - 11,200 km

• The DongFeng 31A or DF-31A (NATO reporting name: CSS-9 Mod-2) is a Chinese intercontinental ballistic missile with a range of 11,200 kilometers.
• The modification was developed on the basis of the DF-31 missile.
• The DF-31A missile has been operational since 2006. Based on the Julang-2 (JL-2) submarines. Modifications of ground-based missiles on a mobile launcher (TEL) are also being developed.
• The three-stage rocket has a launch weight of 42 tons and is equipped with solid propellant rocket engines.

6. RT-2PM2 “Topol-M”, Russia - 11,000 km

• RT-2PM2 "Topol-M", according to NATO classification - SS-27 Sickle B with a range of about 11,000 kilometers, is an improved version of the Topol ICBM. The rocket is installed on mobile launchers, and the option can also be used mine-based.
• The total mass of the rocket is 47.2 tons. It was developed at the Moscow Institute of Thermal Engineering. Produced at the Votkinsk Machine-Building Plant. This is Russia's first ICBM to be developed after the collapse of the Soviet Union.
• A rocket in flight can withstand powerful radiation, electromagnetic pulses and nuclear explosion in close proximity. There is also protection against high-energy lasers. During flight, it performs maneuvers thanks to additional engines.
• Three-stage rocket engines use solid fuel, the maximum rocket speed is 7,320 meters/sec. Testing of the missile began in 1994 and was adopted by the Strategic Missile Forces in 2000.

7. LGM-30G Minuteman III, USA - 10,000 km

• The LGM-30G Minuteman III has an estimated flight range of 6,000 kilometers to 10,000 kilometers, depending on the type of warhead. This missile entered service in 1970 and is the world's oldest missile in service. It is also the only silo-based missile in the United States.
• The first launch of the rocket took place in February 1961, modifications II and III were launched in 1964 and 1968, respectively.
• The rocket weighs about 34,473 kilograms and is equipped with three solid propellant engines. Rocket flight speed 24,140 km/h

8. M51, France - 10,000 km

• The M51 is an intercontinental range missile. Designed for basing and launching from submarines.
• Produced by EADS Astrium Space Transportation, for French navy. Designed to replace the M45 ICBM.
• The rocket entered service in 2010.
• Based on Triomphant-class submarines of the French Navy.
• Its combat range is from 8,000 km to 10,000 km. An improved version with new nuclear warheads is scheduled to enter service in 2015.
• The M51 weighs 50 tons and can carry six individually targetable warheads.
• The rocket uses a solid propellant engine.

9. UR-100N (SS-19 Stiletto), Russia - 10,000 km

• UR-100N, according to the START treaty - RS-18A, according to NATO classification - SS-19 mod.1 Stiletto. This is a fourth-generation ICBM in service with the Russian Strategic Missile Forces.
• The UR-100N entered service in 1975 and is expected to be in service until 2030.
• Can carry up to six individually targetable warheads. It uses an inertial target guidance system.
• The missile is two-stage, silo-based. Rocket engines use liquid rocket fuel.

10. RSM-56 Bulava, Russia - 10,000 km

• Bulava or RSM-56 (NATO code name: SS-NX-32) new intercontinental missile, designed for deployment on Russian Navy submarines. The missile has a flight range of up to 10,000 km and is designed for Borei class nuclear submarines.
• The Bulava missile entered service in January 2013. Each missile can carry from six to ten separate nuclear warheads. The total useful delivered weight is approximately 1,150 kg.
• The rocket uses solid fuel for the first two stages and liquid fuel for the third stage.

Ballistic missiles (in the 50s the term “ballistic projectiles” was used) are those missiles whose flight trajectory (with the exception of the initial section that the missile passes with the engine running) is the trajectory of a freely thrown body. After turning off the engine, the rocket is not controlled and moves like normal artillery shell, and its trajectory depends only on gravity and aerodynamic forces and represents the so-called “ballistic curve”.

Ballistic missiles are typically launched vertically upward or at angles close to 90 degrees, which necessitates the use of a control system to place the missile on its intended trajectory to hit the target.

In order for a ballistic missile to fly hundreds and thousands of kilometers, it must be given a very high flight speed. However, even under this condition, it would be impossible to obtain a greater range if the rocket were flying in dense layers of the atmosphere. Air resistance would quickly dampen its speed. Therefore, strategic ballistic missiles spend the main part of their trajectory at a very high altitude, where the air density is low, i.e., practically in airless space.

Vertical launch of a rocket allows you to reduce the time of its movement in dense layers of the atmosphere and thereby reduce the energy consumption to overcome the force of air resistance. After a few seconds of vertical ascent, the rocket’s trajectory bends towards the target and becomes inclined. Due to the operation of the engine, the speed of the rocket continuously increases until the fuel is completely consumed or the engine is turned off (cut). From this moment until it falls to the ground, the rocket moves along the trajectory of a freely thrown body. Thus, the trajectory of a ballistic missile has two sections: active - from the beginning of take-off until the engines stop working, and passive - from the moment the engines stop working until reaching the surface of the earth.

The active section can in turn be divided into segments. Ballistic missile long range starts vertically from the launcher and moves straight up within a few seconds. This part of the flight is called the starting part. Next, the rocket is launched onto its trajectory. The rocket deviates from the vertical and, describing an arc in the launch section, reaches the last inclined section (switch-off section), where the engines are cut off. The further trajectory of its flight is determined by the kinetic energy stored in the active section and can be accurately calculated.

Having described an elliptical arc outside the atmosphere, the ballistic missile or the separated warhead re-enters the atmosphere, having practically the same kinetic energy and the same angle of inclination of the trajectory to the horizon as when leaving it.