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Calculation of the ballistic (elliptical) section of the trajectory. Program for changing the angle of attack and pitch

As already noted in the analysis of the first stage flight segment, the existing restrictions on the allowable normal overload, the maximum velocity head of the oncoming air flow, or the velocity head at the time of separation of the first and second stages lead to almost the only acceptable control in the first stage, which provides, as already noted, gravitational turn trajectory when the angle of attack is close to zero during the flight. Usually, the pitch angle program for the first stage is selected from the last condition, but the possibilities are closer to the gravity turn program. By choosing the initial negative angle of attack (up to M

After maintaining the condition d = 0 in the step separation section, the optimal derivation program in the general case may require a jump upward by the angle AO, due to different requirements and to the pitch programs at the first and second stages. The required jump can be realized practically by rotating the aircraft in pitch with the maximum allowable angular velocity |? max "Then control begins with a small constant angular velocity of rotation O a). The resulting linear change in the pitch angle in time is close (taking into account small angles) to the optimal control found in the model problem with a linear change in time in the tangent of the pitch angle.

Jump amount JSC affects mainly the height of the resulting orbit, and the constant angular velocity of rotation 0 0 - by the angle of inclination of the trajectory at the end of the active section.

During the launch process, the control system eliminates the emerging yaw and roll angles. Condition 0 = 0 is usually maintained when separating any stages, as well as when separating the payload.

In some control systems, the existing design restrictions do not allow changing the sign of the derivative of the pitch angle, i.e., the condition must be satisfied O 0. In this case, by selecting horizontal (0 = 0) and oblique (O

Let us consider possible launch schemes depending on the height of a given orbit, which, for definiteness, will be assumed to be circular.

The main generally accepted scheme for launching is such that each subsequent stage is switched on almost immediately after the spent one, and the stage engines operate at full thrust. This method is usually applied

Rice. 2.6.

for relatively low orbits with a height of 200 - 300 km(Fig. 2.7). Depending on the time of the active segment, each aircraft has its own optimal height of the circular orbit L“?.”, to which the maximum payload can be launched. When a lower altitude is launched into orbit, the payload decreases due to the strengthening of the braking effect of the atmosphere. In the case launching into higher orbits, the payload mass decreases sharply due to the appearance of large angles of attack in the flight segment of the upper stages and the strengthening of the braking effect of Earth's gravity with an increase in the steepness of the trajectory (Fig. 2.8).Increasing the steepness is necessary to achieve high orbits.

For launching aircraft with continuous operation of engines into orbits with a height of 500 - 1000 km the active section time should be increased. This can be achieved by throttling the sustainer engine (in permissible cases) or by turning off the sustainer engine of the last stage at some point in time and continuing the flight with the control engines working to accelerate the aircraft (Fig. 2.9). In the latter case, in addition to the presence of control motors

Rice. 2.7. Scheme of continuous launch into orbit: 1 - first stage operation area, 2 - second stage operation area, 3 - third stage operation area, 4 - circular orbit

Rice. 2.8.

it is necessary that they be fed with fuel from common tanks with a sustainer engine. The use of a flight segment with reduced thrust makes it possible to significantly increase the height of the orbit compared to the conventional launch method (Fig. 2.8).

We refer the mass of the output payload to its maximum value t, - t r 1 !, and for each value t r we find the relative height of the orbit A = L/,/A/, where A/, is the height of the circular orbit, to which the payload of mass t r / when using the flight segment with reduced thrust, and Ay is the height of the circular orbit to which the same payload is launched when

Rice. 2.9. Launch scheme with reduced thrust flight segment: 1 - first stage operation segment, 2 - second stage operational segment, 3 - reduced thrust flight segment, 4 - circular orbit

Rice. 2.10.

continuous operation of engines at full thrust. Typical addiction to = )t p), shown in fig. 2.10 is close to linear. With small payloads, the orbit height can be increased by a factor of 24-3 by using a flight segment with reduced thrust.

Note that such an induction mode is one of the possible optimal ones identified in the study of the model problem, and the continuous operation of the control engines ensures stability and controllability in the induction process.

The third launch scheme assumes the use of a passive flight segment between the penultimate and last stages or between the first and second firings of the last stage engine. In this way, the payload can be launched into orbits of almost any altitude.

Two modifications of this scheme are possible. The first is used for relatively lower orbits and differs in that there is a small positive trajectory inclination at the beginning of the passive leg. Due to this angle, the last stage reaches its apogee when the angular range of the passive section is significantly less than 180°. Near the apogee, located approximately at the height of a given orbit, the stage engine is turned on to increase the speed to a circular one (Fig. 2.11).

Rice. 2.11. Launch schemes with a passive section: 1 - the first stage operation section, 2 - the second stage operation section, 3 - the passive section, 4 - the third stage operation section, 5 - a circular orbit

The second modification of the launch scheme, which can be used for any orbits of practical interest, is distinguished by a large angular range of the passive section (the angular range is 180°). To do this, the passive segment must begin at a zero angle of inclination of the trajectory, i.e., the first active segment ends at the perigee of the transition trajectory, the apogee of which is located approximately at the height of the given orbit (Fig. 2.11). The stage must be properly oriented before starting the engine.

The launch scheme with a passive leg of various durations can be successfully used for any orbits, and not only for high ones.

for the entrance exam in the direction of the magistracy 160700.68 "Aircraft engines"

Classification of coordinate systems by the location of the origin of coordinates, by binding to the object. Examples from rocket technology.
Geocentric and starting coordinate system. Transfer from one to another. The concept of basic angles. Examples from rocket technology.
Bound and velocity coordinate systems. Transfer from one to another. Concepts of basic angles. Examples from rocket technology.
Equation I.V. Meshchersky: physical meaning, assumptions. The first and second tasks of K.E. Tsiolkovsky: physical meaning.
The main components of free fall acceleration. Under what conditions is it necessary to account for them?
Calculation of geodetic range and calculated azimuth.
The division of the atmosphere according to the chemical composition of the air. Character of change of viscosity, pressure and density on height. The nature of the change in temperature with height.
Determination of atmospheric parameters at an arbitrary point of the trajectory.
Basic projections of the aerodynamic force in the velocity and coupled coordinate systems. physical meaning.
The structure of the drag coefficient, the influence of M.
The structure of the lift coefficient, the influence of M.
Experimental determination of the drag coefficient.
Axial and lateral overload: physical meaning. The restrictions imposed n x and n y to the trajectory of the aircraft.
Influence of the destination of the aircraft on the type of the trajectory of the active site.
The main restrictions when choosing the trajectory of the active site.
Program for changing the angle of attack and pitch.
Parabolic and elliptical trajectories. Parameters at an arbitrary point.
Factors causing projectile dispersion. Systematic and random corrections: physical meaning, methods of determination.
Random scattering of projectiles: basic patterns. Scattering ellipse.
Dependence of speed on flight range: without atmosphere, with homogeneous atmosphere, with real atmosphere.
Optimal throwing angle: physical meaning. The value of the optimal throwing angle, taking into account the atmosphere and the curvature of the Earth.
Classification of missiles.
The layout of a solid-propellant single-stage rocket.
The layout of a liquid single-stage rocket.
Advantages and disadvantages of solid propellant rocket engines in comparison with rocket engines.
The main indicators and characteristics of the rocket engine.
Classification of solid rocket propellants. Give examples.
Classification of liquid propellants. Give examples.
The main methods of cooling the combustion chamber and the nozzle of the rocket engine.
The main types of combustion chambers and nozzles LRE. Give examples.
The main types of nozzles. Give examples.
Forms of the cooling ducts of the liquid-propellant rocket engine.
Requirements for the design of missile warheads. External forms and stabilization of head parts.
Tank requirements. Basic design schemes of tanks.
Rocket power set: spars, stringers and frames.
Turbopump unit. Purpose, composition, layout diagrams.
Methods for connecting aircraft compartments and methods for separating compartments.
The device and operation of the 8K14 rocket pressure reducer.
The device and operation of the 8K14 rocket thrust regulator.
The device and operation of the 8K14 rocket pressure stabilizer.
LRE schemes.
The law of conservation of mass.
Volumetric and surface forces in continuum mechanics. Stress tensor.
Laws of conservation of mass, momentum and energy for an ideal gas.
adiabatic processes. Poisson's adiabatic equation.
Braking parameters, critical parameters.
Gas dynamic functions. Their application for performance of gas-dynamic calculations.
Outflow from a reservoir into a medium with a given pressure.
One-dimensional unsteady flows of an ideal gas. Riemann invariants.
The formation of shock waves. Physical explanation of the formation of shock waves.
Relationships for changing the velocity at the shock wave.
Compaction jumps. Comparison of Hugoniot and Poisson adiabats.
Basic equations of plane and axisymmetric steady motions of an ideal gas.
Navier-Stokes equations for incompressible media.
Newton's equation relating the stress tensor to the strain rate tensor.
Basic similarity criteria. their physical meaning.
Poiseuille flow. Derivation of the formula for the drag coefficient. Calculation of pressure drop in laminar flow.
Derivation of equations for the boundary layer.
Calculation of friction stress on the surface of a flat plate.
Transition from laminar to turbulent flow. Critical Reynolds number.
What is called the internal energy of the system?
Give a brief description of the three principles of thermodynamics.
What is meant by a thermodynamic system, a working fluid? Give examples of thermodynamic systems.
What state is called equilibrium and non-equilibrium?
Give the equation of state for an ideal gas and describe each of its components.
Write the equation of the first law of thermodynamics and define the concepts of work of expansion, internal energy, enthalpy.
Consider the application of the first law of thermodynamics for some special cases when there is no heat exchange with the environment, the volume of the system does not change, or the internal energy does not change.
Write an expression for the first law of thermodynamics for an open thermodynamic system. What is the work flow?
What is the heat capacity of a substance? List and describe the types of heat capacities used in the calculations. How does heat capacity depend on temperature? What is the average heat capacity?
What thermodynamic process is called a cycle? What cycle is called forward and reverse?
What is the essence of the second law of thermodynamics. Name some of its expressions.
How does enthalpy change in reversible and irreversible processes?
The principle of operation of compression machines. How is compressor operation determined?
Give the classification and main characteristics of heat transfer processes.
Formulate the basic law of heat conduction.
How are the processes of cooling or heating of various bodies calculated?
What is the physical meaning of the criteria Re, Nu, Pr, Bi, Fo?
Formulate three similarity theorems.
What techniques can reduce the frictional resistance when flowing around bodies?
How to calculate the heat transfer between a gas and its surrounding shell?
Basic calculation cases. Safety factor. Margin of safety.
Mechanical properties of solid rocket propellants.
Insertable hollow charge loaded with pressure of combustion products.
Checking the depositary charge for collapse along the support end.
Calculation of the bonded charge loaded with the pressure of combustion products.
The concentration of stresses in the charge.
Calculation of the strength of the engine housing.
Basic loads, design cases and criteria for assessing the strength of the elements of the LRE combustion chamber.
Calculation of the strength of the bottom of the solid propellant rocket engine. Influence of a hole in the bottom on its strength.
Calculation of the LRE combustion chamber for the total carrying capacity.
What is the equilibrium constant of a chemical reaction? Give an example.
What is the rate constant of a chemical reaction? How is it defined?
What is the condition for the equilibrium of a mixture of substances in the combustion products.
The law of active masses. How to determine the rate of a chemical reaction?
What is meant by thermal dissociation reaction? Give examples of such reactions.
What is enthalpy? How is it related to the heat of formation of substances?
What is the stoichiometric fuel ratio?
What is the excess oxidant ratio and how is it determined?
Processes occurring during the combustion of liquid fuels.
Processes occurring during the combustion of solid fuels.

Head of direction 160700.68

Doctor of Physical and Mathematical Sciences, Professor A.V. Aliyev

Rocket movement program at OUT

ballistic missile launch overload

An analysis of real programs for the movement of guided ballistic missiles (UBR) and launch vehicles makes it possible to create approximate programs that are used in solving problems of ballistic design of guided missiles.

Thus, for the first steps of the RBS, the approximate program described by the relation is close to optimal:

In this case, the pitch angle can be replaced by the trajectory angle and an approximate program of the form, which is in good agreement with the real ones, can be used:

where is the trajectory angle at the end of the active section;

Sub-rocket fill factor;

Working fuel reserve of the i-th active stage;

Starting mass of the i-th active stage;

Mass second fuel consumption of the i-th active stage;

It will be most convenient to set various restrictions on the program of the rocket's movement on the OUT for some characteristic sections of the trajectory, depending on the number of stages of the rocket.

Fig.4.

1. Two-stage rocket (Fig. 4).

Calculations related to the choice of optimal programs show that for all flight stages, starting from the second, which are not subject to restrictions on the angle of attack, the optimal program is very close to a straight line. The flight program of the second stage includes the following sections:

the section of "calming" from the moment of time to, during the flight occurs with an angle of attack. The "calming" section is necessary to eliminate the disturbances that arise when the steps are separated;

pre-turn section (if necessary) from time to. In this section, while the angle of attack is determined and the expressions

flight segment with a constant pitch angle.

Note: The 3rd and subsequent stages are considered to be flying with a constant pitch angle.

Fig.5.

Calculation of the ballistic (elliptical) section of the trajectory

The position of the rocket at the beginning of the elliptical section is determined by the calculation of the active section of the trajectory, and at this stage of the calculation it can be considered given. The movement of the rocket from point to point, located at the same height or the same radius, occurs along the arc of an ellipse, symmetrical about the axis (Fig. 1).

The elliptical flight range is:

Earth constant.

The formula for determining the optimal trajectory angle at the end of the active section, at which the missile's flight range in the elliptical section will be maximum.

Comparing the value of the angle with the value obtained when solving the system of equations (5), it is necessary to refine the program for the flight of the rocket to the AUT in order to achieve the maximum range of the BR.

The flight time of the rocket on the elliptical section:

Calculation of the final (atmospheric) section of the trajectory

When studying the parameters of the warhead movement on the atmospheric part of the passive section of the trajectory, it is necessary to take into account the effect of aerodynamic drag.

The movement of the center of mass of the head part relative to the non-rotating Earth at zero angle of attack in projections on the axes of the velocity coordinate system is described by the following system of equations (Fig. 6):

where is the mass of the head.

Factors of overloads acting on the rocket in flight

When evaluating the strength of a rocket structure, it is necessary to know not only the resultant external forces acting on the rocket as a whole, but also their individual components.

When solving the system of equations (5) or (13), the tangential and normal accelerations of the rocket are known. Let's find the axial and transverse acceleration components in the bound coordinate system (Fig. 3).

Taking into account that, in addition to axial and transverse accelerations, the acceleration of the earth's gravity also acts on the mass of the rocket, after minor transformations, we obtain the coefficients of the total (static and dynamic) axial and transverse overloads acting on the rocket in flight.

The quantities and are purely trajectory parameters and are determined as a result of numerical integration of the rocket motion equations.

When Q=const, the law of mass change is given by m(t)=m0-Qt, where m0 is the initial mass.

The variables, while the expression of the forces included in the right-hand side, are defined by the formulas given above.

The 8th equation of system (2) is called a program. Usually this equation is a piecewise smooth curve. All eight variables must be given initial values at t=0.

We write system (3):

(3)

- for these variables initial conditions should be set.

The main calculation method is numerical integration. In addition, when solving equations, an analytical method (method of successive approximations (iterations)) can be used.

Program trajectory, requirements for the program, formulation of the problem of choosing the optimal program.

The flight program on the active leg is, in principle, set as one of the dependencies , or some other movement characteristics. Programming can be carried out not only in the vertical plane Ox0y0, but also in the horizontal plane Ox0z0, as well as for spatial trajectories. Usually proceed from software dependence, since the pitch angle is easy to measure with high accuracy by gyroscopic sensors. The program is set before the start and is not corrected during the movement. Of particular interest is the problem of choosing the optimal program for solving this problem; the main requirements are to obtain the greatest trajectory range with the least dispersion of the points of incidence.

14.10.05 *

The problem of choosing the program of the greatest range can be solved by analytical methods of the classical calculus of variations under fairly rough assumptions: if we assume that the thrust is constant, do not take into account the force of drag, take the gravitational field constant, parallel, and do not take into account restrictions on the angles of attack.

, - the initial value of the pitch angle

Such a program provides for the constancy of the pitch angle throughout the active section and the inclined launch of the rocket. This program cannot be practically implemented.

When choosing a program for changing the pitch angle, the requirements for ensuring a sufficient margin of safety of the structure with the least weight, requirements related to launch conditions, ensuring stability of movement, etc., should be taken into account, which was not provided for when solving the problem by methods of classical variational calculus. The choice of a program, taking into account all the requirements for a rocket, is one of the most important design stages. Let us dwell on these requirements and consider the methodology for choosing a program. We will consider the case of a single-stage BR. The type of this program equation depends on the purpose of the rocket, its structural and technical parameters and the type of launch (vertical, inclined). At the same time, with a correctly compiled program in accordance with the capabilities of the control system (limited deviations of the control bodies), the dependencies should change smoothly, i.e. not have corner points during the flight on the active leg. As a rule, BR start from the launcher vertically upwards so that the initial pitch angle and the initial vertical flight segment take place and remain the same for a certain time interval. The vertical launch of the BR makes it possible to have the simplest launchers and provide favorable conditions for control in the initial section of the trajectory. The latter circumstance is explained by the fact that engine thrust is used to control the BR, especially with solid propellant rocket motors, part of the main thrust is selected for control. If the thrust has not reached its nominal value, then the part of it used for control will also be insufficient. It takes several seconds for the engine to return to normal mode and usually determines the duration of the initial vertical section of the trajectory. In addition, vertical launch makes it possible to reduce the requirements for the rigidity of the BR body and, consequently, to reduce the weight of its structure.

UDC 623.4.027

SELECTION OF THE PROGRAM FOR CHANGING THE PITCH ANGLE OF THE BOAT ROCKET

AIR START

D. A. Klimovsky Supervisor - N. A. Smirnov

Siberian State Aerospace University named after Academician M.F. Reshetnev

Russian Federation, 660037, Krasnoyarsk, prosp. them. gas. "Krasnoyarsk worker", 31

E-mail: smirnov@sibsau.ru

The function of changing the pitch angle of the first stage of an air-launched carrier rocket is determined.

Key words: air launch, pitch angle.

SELECTION PROGRAMME PITCH ANGLE ROCKET WITH AIR LAUNCH

D. A. Klimovskiy Scientific Supervisor - N. A. Smirnov

Reshetnev Siberian State Aerospace University 31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation E-mail: smirnov@sibsau.ru

In paper defined a function changes the pitch angle of the first stage rocket with air launch.

Keywords: air launch, pitch angle.

In the process of launch vehicle design, the need for trajectory calculations arises in the following main cases:

1. At the stage of choosing the main design parameters of the launch vehicle (the number of stages, the choice of fuel components, the mass of fuel loaded into the boosters, the initial thrust-to-weight ratio, etc.);

2. When generating initial data for strength calculations, thermal calculations, calculations of the dynamics of the movement of the launch vehicle, including the dynamics of the start and the dynamics of stage separation, etc.

3. When forming the technical requirements for individual launch vehicle systems, such as the control system, propulsion system, pneumohydraulic system, telemetry system, etc.

4. For carrying out verification calculations with the parameters of individual elements of the launch vehicle specified during the design process.

The main problem is that all classical methods for calculating the launch vehicle are based on the launch program with a vertical launch, which makes it impossible to use them when calculating the direct launch of a rocket from a carrier aircraft, where the initial launch angles start from 0°. The upper limit is limited by the capabilities of the aircraft.

Usually, the following requirements are imposed on real programs for the movement of launch vehicles:

1) ensuring the final speed and height;

2) the possibility of a vertical launch;

3) limitation of overloads;

4) smooth change of parameters;

5) absence of angles of attack at transonic flight speeds;

Let's try to determine how the trajectory of an air-launched launch vehicle should look like. The first moments the rocket moves with the initial pitch angle. Then there should be a turn in the direction of increasing the pitch angle in order to more quickly pass through the dense layers of the atmosphere. Next, it is necessary to start reducing the pitch angle so that at the moment the engine of the last stage is turned off, the speed has the required angle of inclination to the local horizon. Under these conditions well

Actual problems of aviation and astronautics - 2015. Volume 1

suitable trigonometric functions "cosine" or "sine". So, the equation for the cosine function will take the following form:

b(tst) \u003d A co8 (yutst + f) + K

where 0 - current pitch angle; A, K, u, φ - parameters for determination, t - current relative mass of fuel consumed. An example of the required function is shown in fig. one.

Rice. 1. Pitch angle function

To determine the four unknown parameters, it is necessary to know four initial conditions:

1) 9(^r0) = 0o = 0mm for o^.0 + φ = n; Ct0 - relative mass of spent fuel at the beginning of the turn, 0o - initial pitch angle;

2) 0(Tsk1) = 0k1; ctk1 is the relative mass of the spent fuel of the first stage, 0k is the final pitch angle of the first stage;

3) 0 = 0max, for o^ + φ = 0; 0max - maximum pitch angle;

4) Since the cosine function is periodic, it is necessary that the solution fit into one period, for which the parameter u is responsible;

Considering these conditions, we obtain the following values of the unknown parameters:

A - max min . k - max min .

arccos I---l + n

The final equation will take the form:

b(|o,t) - A -yut2 + n) + K;

For a two-stage launch vehicle, the pitch angle program at 00 = 5°, tst0 = 0.05, 0s = 30, = 0.733 1, 0k2 = 0, tstk2 = 0.925 1 will take the form (Fig. 2).

Also, this equation can be used to calculate the launch vehicle with a vertical launch. On fig. 3, the dotted line shows the classical derivation program, the solid line - according to the obtained expression.

О 0.2 0.4 0.6 0.8 1

Rice. 2. Pitch angle program for a two-stage launch vehicle with air launch

Rice. 3. Derivation programs: classical and according to the obtained equation

1. Apazov R. F., Sytin O. G. Methods for designing the trajectories of carriers and satellites of the Earth. M.: Science. Ch. ed. Phys.-Math. lit., 1987. 440 p.

2. Varfolomeeva V. I., Kopytova M. I. Design and testing of ballistic missiles. M. : Military Publishing House, 1970. 392 p.

Calculation of the ballistic (elliptical) section of the trajectory. Program for changing the angle of attack and pitch

for the entrance exam in the direction of the magistracy 160700.68 "Aircraft engines"

Rocket movement program at OUT

Calculation of the ballistic (elliptical) section of the trajectory

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