Calculation of the ballistic (elliptical) section of the trajectory. Calculation of the ballistic (elliptical) trajectory section Pitch program
When Q=const, the law of mass change is given by m(t)=m0-Qt, where m0 is the initial mass.
The variables, with the expression of the forces included in the right-hand sides, are determined by the formulas given above.
The 8th equation of system (2) is called the program. Typically this equation is a piecewise smooth curve. All eight variables must be given initial values at t=0.
Let us write system (3):
(3)
- initial conditions must be specified for these variables.
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, program requirements, formulation of the problem of choosing the optimal program.
The flight program for the active phase is, in principle, specified as one of the dependencies 
, 
or some other characteristics of movement. Programming can be carried out not only in the vertical plane Ох0у0, but also in the horizontal plane Ох0z0, as well as for spatial trajectories. Usually they come from a software dependence, since the pitch angle is easy to measure with high accuracy using gyroscopic sensors. The program is set before the start and is not adjusted 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 impact points.
14.10.05 *
The problem of choosing the longest range program can be solved by analytical methods of the classical calculus of variations under fairly rough assumptions: if we assume the thrust to be constant, do not take into account the drag force, assume the gravitational field to be constant, parallel, and do not take into account restrictions on the angles of attack.
, - initial value of the pitch angle
Such a program provides for a constant pitch angle throughout the entire active section and an 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 safety margin of the structure at the minimum weight, requirements related to launch conditions, ensuring motion stability, etc. should be taken into account, which was not provided for when solving the problem using classical calculus of variations. Selecting a program taking into account all the requirements for the rocket is one of the serious design stages. Let us dwell on these requirements and consider the methodology for selecting a program. We will consider the case of a single-stage BR. The form of this program equation depends on the purpose of the rocket, its design and technical parameters and the type of launch (vertical, inclined). Moreover, with a correctly compiled program in accordance with the capabilities of the control system (limited deviations of control bodies), the dependencies 
should change smoothly, i.e. do not have corner points during the flight in the active phase. As a rule, ballistic missiles are launched from the launcher vertically upward so that the initial pitch angle and the initial vertical portion of the flight take place and remain the same for a certain time interval. Vertical launch of ballistic missiles makes it possible to have the simplest launchers and provide favorable conditions for control in the initial part of the trajectory. The latter circumstance is explained by the fact that to control the ballistic missile, especially with a solid propellant rocket engine, engine thrust is used, 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 ballistic missile body and, consequently, reduce the weight of its structure.
As already noted when analyzing the flight section of the first stage, existing restrictions on the permissible normal overload, the maximum high-speed pressure of the oncoming air flow or the high-speed pressure at the moment of separation of the first and second stages lead to almost the only acceptable control at the first stage, which ensures, as already noted, the trajectory of a gravitational turn, when during the flight the angle of attack is close to zero. Usually, the pitch angle program for the first stage is selected from the last condition, but the possibilities are closer to the gravitational turn program. Due to the choice of the initial negative angle of attack (up to M
After maintaining the condition d = 0 in the stage separation section, the optimal launch program in the general case may require an upward jump by an angle JSC, due to different requirements for pitch programs at the first and second stages. The required jump can be realized practically by rotating the aircraft in pitch with the maximum permissible angular speed |?max" Then control begins with a low constant angular speed of rotation ABOUT A). The resulting linear change in the pitch angle in time is close (taking into account small angles) to the optimal control with a linear change in time in the tangent of the pitch angle found in the model problem.
Jump size JSC affects mainly the height of the resulting orbit, and the constant angular velocity of rotation 0 0 - to the angle of inclination of the trajectory at the end of the active section.
During the insertion 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, existing design restrictions do not allow changing the sign of the derivative of the pitch angle, i.e. the condition must be satisfied О 0. In this case, by selecting horizontal (0 = 0) and inclined (ABOUT
Let us consider possible launch schemes depending on the height of a given orbit, which, for definiteness, we will assume to be circular.
The main generally accepted launch scheme is one in which each subsequent stage is switched on almost immediately after the exhausted one, and the engines of the stages operate at full thrust. This method is usually used
Rice. 2.6.
for relatively low orbits with an altitude of 200 -g 300 km(Fig. 2.7). Depending on the time of the active phase, for each aircraft there is its own optimal height of the circular orbit L“?.", to which the maximum payload can be launched. When launched into an orbit of a lower altitude, the payload decreases due to the increased braking effect of the atmosphere. In the case When inserted into higher orbits, the mass of the payload decreases sharply due to the appearance of large angles of attack in the flight section of the upper stages and the strengthening of the braking effect of earth's gravity with increasing trajectory steepness (Fig. 2.8). An increase in steepness is necessary to achieve high orbits.
For launching aircraft with continuous engine operation into orbits at an altitude of 500 - 1000 km the active section time should be increased. This can be achieved by throttling the main engine (in permissible cases) or by turning off the last stage main engine 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 insertion into orbit: 1 - operating section of the first stage, 2 - operating section of the second stage, 3 - operating section of the third stage, 4 - circular orbit
Rice. 2.8.
it is necessary that they are fed with fuel from common tanks with the main engine. The use of a flight segment with reduced thrust makes it possible to significantly increase the orbital altitude compared to the conventional insertion method (Fig. 2.8).
Let us relate the mass of the output payload to its maximum value t, - t r 1 !, and for each value t r let's find the relative altitude of the orbit A = A/,/A/, where A/, is the height of the circular orbit into which the payload of mass is launched t r / when using a flight segment with reduced thrust, and Aug is the height of the circular orbit to which the same payload is launched when
Rice. 2.9. Injection diagram with a flight section at reduced thrust: 1 - first stage operation section, 2 - second stage operation section, 3 - reduced thrust flight section, 4 - circular orbit
Rice. 2.10.
continuous operation of engines at full thrust. Typical addiction To = )t r), shown in Fig. 2.10, close to linear. With small payloads, the orbital altitude can be increased by 2-4-3 times by using a flight segment with reduced thrust.
Note that this launch mode is among the possible optimal ones identified during the study of the model problem, and the continuous operation of the control motors ensures stability and controllability during the launch 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 burns of the last stage engine. In this way, payloads 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 is distinguished by the fact that at the beginning of the passive section there is a small positive angle of inclination of the trajectory. Thanks 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 the 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 - first stage operation section, 2 - second stage operation section, 3 - passive section, 4 - third stage operation section, 5 - 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 (angular range is 180°). To do this, the passive section must begin at zero trajectory inclination angle, i.e., the first active section 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 the engine is turned on.
An injection scheme with a passive segment of varying duration can be successfully used for any orbits, not just high ones.
UDC 623.4.027
SELECTION OF THE PROGRAM FOR CHANGING THE PITCH ANGLE OF THE LAUNCHER ROCKET
AIR LAUNCH
D. A. Klimovsky Scientific supervisor - N. A. Smirnov
Siberian State Aerospace University named after Academician M. F. Reshetnev
Russian Federation, 660037, Krasnoyarsk, ave. them. gas. "Krasnoyarsk worker", 31
Email: [email protected]
The function of changing the pitch angle of the first stage of an air-launched launch vehicle is determined.
Key words: air launch, pitch angle.
SELECTION PROGRAM 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: [email protected]
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 designing a launch vehicle, the need to carry out trajectory calculations arises in the following main cases:
1. At the stage of selecting the main design parameters of the launch vehicle (number of stages, choice of propellant components, mass of fuel loaded into boosters, initial thrust-to-weight ratio, etc.);
2. When generating the initial data for strength calculations, thermal calculations, calculations of the dynamics of the LV motion, including the dynamics of launch and the dynamics of stage separation, etc.
3. When forming technical requirements for individual launch vehicle systems, such as the control system, propulsion system, pneumohydraulic system, telemetry system, etc.
4. To carry out verification calculations with the parameters of individual launch vehicle elements specified during the design process.
The main problem is that all classical methods for calculating launch vehicles are based on a 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 aircraft's capabilities.
Typically, the following requirements are imposed on real launch vehicle propulsion programs:
1) ensuring final speed and altitude;
2) the possibility of vertical launch;
3) limitation of overloads;
4) smooth change of parameters;
5) lack of angles of attack at transonic flight speeds;
Let's try to determine what the trajectory of an air-launched launch vehicle should look like. During the first moments, the rocket moves with its initial pitch angle. Then a turn should occur 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 begin decreasing the pitch angle so that at the moment the last stage engine is turned off, the speed has the required angle of inclination to the local horizon. Good under these conditions
Current problems of aviation and astronautics - 2015. Volume 1
Trigonometric functions "cosine" or "sine" are suitable. So, the equation for the cosine function will take the following form:
b(tst) = A co8(ytst +f) + K
where 0 is the current pitch angle; A, K, ω, φ are parameters for determining, ^ is the current relative mass of consumed fuel. An example of the required function is shown in Fig. 1.
Rice. 1. Pitch angle change function
To determine four unknown parameters, it is necessary to know four initial conditions:
1) 9(^r0) = 0о = 0mm at ω^.0+ φ = n; Ts0 - relative mass of spent fuel at the beginning of the turn, 0о - initial pitch angle;
2) 0(Ttk1) = 0k1; tstk1 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, with ω^ + φ = 0; 0max - maximum pitch angle;
4) Since the cosine function is periodic, it is necessary that the solution fits into one period, for which the parameter ω is responsible;
Taking these conditions into account, 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 -yct2 +n) + K;
For a two-stage launch vehicle, the pitch angle program at 00 = 5°, ot0 = 0.05, 0ы = 30, = 0.733 1, 0k2 = 0, otk2 = 0.925 1 will take the form (Fig. 2).
This equation can also be used to calculate launch vehicles with vertical launch. In Fig. 3 the dotted line shows the classic derivation program, the solid line shows the resulting expression.
O 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. Inference programs: classical and according to the resulting equation
1. Apazov R. F., Sytin O. G. Methods for designing trajectories of carriers and satellites of the Earth. M.: Science. Ch. ed. physics and mathematics lit., 1987. 440 p.
2. Varfolomeeva V.I., Kopytova M.I. Design and testing of ballistic missiles. M.: Voenizdat, 1970. 392 p.
© Klimovsky D. A., 2015
for conducting the entrance exam in the master's degree 160700.68 “Aircraft engines”
Classification of coordinate systems by the location of the origin of coordinates, by reference to an object. Examples from rocketry.
Geocentric and starting coordinate system. Conversion from one to another. The concept of basic angles. Examples from rocketry.
Coupled and velocity coordinate systems. Conversion from one to another. Concepts of basic angles. Examples from rocketry.
Equation I.V. Meshchersky: physical meaning, assumptions. The first and second tasks of K.E. Tsiolkovsky: physical meaning.
The main components of the acceleration of free fall. Under what conditions is it necessary to take them into account?
Calculation of geodetic range and calculated azimuth.
Division of the atmosphere according to the chemical composition of the air. The nature of changes in viscosity, pressure and density with height. The nature of temperature changes with altitude.
Determination of atmospheric parameters at an arbitrary point on the trajectory.
Basic projections of aerodynamic force in the velocity and coupled coordinate systems. Physical meaning.
Structure of the drag coefficient, influence of M.
Lift coefficient structure, influence of M.
Experimental determination of the drag coefficient.
Axial and lateral overload: physical meaning. Limitations imposed n x And n y to the aircraft's trajectory.
The influence of aircraft assignment on the type of trajectory of the active section.
The main restrictions when choosing the trajectory of the active section.
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 principles. Scattering ellipse.
Dependence of speed on flight range: without taking into account the atmosphere, taking into account a homogeneous atmosphere, taking into account the 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-fuel single-stage rocket.
Layout of a liquid-fuelled single-stage rocket.
Advantages and disadvantages of solid propellant rocket engines in comparison with liquid rocket engines.
Main indicators and characteristics of a rocket engine.
Classification of solid rocket fuels. Give examples.
Classification of liquid rocket fuels. Give examples.
The main methods of cooling the combustion chamber and nozzle of a liquid-propellant rocket engine.
The main types of combustion chambers and nozzles of liquid rocket engines. Give examples.
Main types of nozzles. Give examples.
Shapes of liquid-propellant rocket engine cooling tracts.
Requirements for the design of missile warheads. External shapes and stabilization of head parts.
Requirements for tanks. Basic design diagrams 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.
Design and operation of the 8K14 rocket pressure reducer.
Design and operation of the 8K14 rocket thrust regulator.
Design and operation of the 8K14 rocket pressure stabilizer.
Liquid rocket engine schemes.
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 use for performing gas-dynamic calculations.
Outflow from a tank into a medium with a given pressure.
One-dimensional unsteady flows of an ideal gas. Riemann invariants.
Formation of shock waves. Physical explanation of the formation of shock waves.
Relations for changes in velocity at a shock wave.
Compaction shocks. 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 Current. Derivation of the formula for the resistance 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 flow to turbulent. Critical Reynolds number.
What is the internal energy of a 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 nonequilibrium?
Give the equation of state of an ideal gas and characterize 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 does the work of a flow consist of?
What is the heat capacity of a substance? List and characterize the types of heat capacities used in calculations. How does heat capacity depend on temperature? What is average heat capacity?
What thermodynamic process is called a cycle? What cycle is called forward and backward?
What is the essence of the second law of thermodynamics. Name some of its formulations.
How does enthalpy change in reversible and irreversible processes?
The operating principle of compression machines. How is compressor performance determined?
Give the classification and main characteristics of heat transfer processes.
Formulate the basic law of thermal conductivity.
How are the cooling or heating processes of various bodies calculated?
What is the physical meaning of the criteria Re, Nu, Pr, Bi, Fo?
State three similarity theorems.
What technical techniques can reduce frictional resistance when flowing around bodies?
How to calculate the heat exchange between a gas and its surrounding shell?
Basic design cases. Safety factor. Safety margin.
Mechanical properties of solid rocket fuels.
An inserted hollow charge loaded with pressure from combustion products.
Checking the insert charge for crushing along the supporting end.
Calculation of a bonded charge loaded with pressure from combustion products.
Voltage concentration in the charge.
Calculation of the strength of the engine housing.
Basic loads, design cases and criteria for assessing the strength of the combustion chamber elements of a liquid propellant rocket engine.
Calculation of the strength of the bottom of the solid propellant rocket motor housing. The influence of a hole in the bottom on its strength.
Calculation of the combustion chamber of a liquid-propellant rocket engine for the total load-bearing 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 determined?
What is the condition for the onset of equilibrium of the mixture of substances in the combustion products.
Law of mass action. 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 ratio of fuel components?
What is the excess oxidant coefficient and how to determine it?
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. Aliev
Rocket propulsion program at OUT
ballistic missile launch overload
Analysis of real motion programs for guided ballistic missiles (GBMs) 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 stages of the UBR, 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 that 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 fuel filling coefficient;
Working fuel reserve of the i-th active stage;
Launch mass of the i-th active stage;
Mass second fuel consumption of the i-th active stage;
The most convenient would be to set various restrictions on the rocket motion program at the OUT for some characteristic sections of the trajectory, depending on the number of rocket stages.
Fig.4.
1. Two-stage rocket (Fig. 4).
Calculations related to the selection of optimal programs show that for all stages of flight, starting from the second, for which no restrictions on the angle of attack are imposed, the optimal program is very close to linear. The second stage flight program includes the following sections:
the “calming” section from the moment in time to, during which the flight occurs with an angle of attack. The “calming” section is necessary to eliminate disturbances that arise when the steps are separated;
additional turn section (if necessary) from time to. In this section, the angle of attack is determined by the expression
flight segment with a constant pitch angle.
Note: The 3rd and subsequent stages are considered to be flying at a constant pitch angle.
Fig.5.
Calculation of the ballistic (elliptical) trajectory section
The position of the rocket at the beginning of the elliptical section is determined by calculating 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 an elliptical arc, symmetrical about the axis (Fig. 1).
The elliptical flight range is:
Earth constant.
Formula for determining the optimal trajectory angle at the end of the active section, at which the missile’s flight range on 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 rocket flight program at the AUT in order to achieve the maximum flight range of the ballistic missile.
Rocket flight time on an elliptical section:
Calculation of the final (atmospheric) section of the trajectory
When studying the parameters of the movement of the warhead on the atmospheric part of the passive section of the trajectory, it is necessary to take into account the influence of aerodynamic drag.
The movement of the center of mass of the head part relative to the non-rotating Earth at a zero angle of attack in projections on the axis of the velocity coordinate system is described by the following system of equations (Fig. 6):
where is the mass of the head part.
G-factors acting on a rocket in flight
When assessing 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 motion are known. Let's find the axial and transverse components of acceleration in a coupled coordinate system (Fig. 3).
Considering that the mass of the rocket, in addition to axial and transverse accelerations, is also affected by the acceleration of gravity, 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 equations of rocket motion.
