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Analytic Considerations for Lifting Ascent Launch Vehicle Trajectories
Martin Bayer
Motivation
Is There Another Way?
???
Coarse analytical solution vs. Complex numerical approachCoarse analytical solution vs. Complex numerical approach
Tsiolkovsky Revisited (I)
The thrust of a rocket equals the negative propellant mass flow times the exhaust velocity:
(1) esp cmgIdtdmF 0
Note: The mass flow must be taken as negative in order to obtain a positive thrust value, since from the perspective of the vehicle it constitutes a mass loss due to propellant consumption
According to Newton’s Second Law, the vehicle acceleration equals the thrust divided by the mass:
(2) mF
dtdva
Inserting (1) in (2) yields:
(3) mc
dtdm
dtdv e
Tsiolkovsky Revisited (II)
Multiplication with dt and subsequent integration under the assumption of a constant ec leads to:
(4) e ev
v
m
me m
dmcdv0 0
Note: The integration from a larger 0m to a smaller em implies that dm is negative, as outlined above
Solving the integrals finally leads to the well known Tsiolkowsy rocket equation:
(5) e
ee mm
cvvv 00 ln
An analogous approach can be taken for the performance determination of propulsion systems with variable specific impulse
Specific Impulses (Isp) of Various Propulsion Systems
Isp as a Linear Function of Flight Velocity
Derivation of a Propulsive Equation for Variable Isp (I)
Airbreathing engines generally have specific impulses that vary considerably with the flight velocity and typically decrease for higher velocities
Often the specific impulse can at least for certain segments be approximated with reasonable accuracy as a linear function of velocity:
(6) )( '0
vIII spspsp
Analogous to the derivation of the rocket equation, the acceleration can be written as:
(7) m
vIIgdtdm
dtdv spsp )( '
0 0
Multiplication with dt , separation of the variables and subsequent integration yields:
(8)
e ev
v
m
mspsp mdm
vIIgdv
0 00)( '
0
Derivation of a Propulsive Equation for Variable Isp (II)
Solving the integrals leads to:
(9) 00
'0
'0
'0
ln)(
)(ln1
0
0
mm
vIIg
vIIg
Ige
spsp
espsp
sp
Some rearrangement yields:
(10) '
0
0
0
00'
' spIge
spsp
espsp
mm
vIIvII
The final result is:
(11)
'0
0
00'0 1
spIge
sp
spe m
mv
I
Ivvv
For a value of 'spI approaching zero, this equation transforms into the Tsiolkowsy
equation
Derivation of a Propulsive Equation for Variable Isp (III)
The maximum achievable ev (for 0m
me approaching zero) is:
(12) 'max0
sp
spe I
Iv
This boundary value is the velocity, for which the specific impulse becomes zero
Apart from fundamental evaluations, equation (11) enables quantitative analyses, if an effective specific impulse, which includes all loss terms, is known as a function of the flight velocity:
(13)
F
gmFDII speffsp
sin1
This requires however advance knowledge of the main trajectory parameters; if these are not known yet, the influence of drag and gravity losses has to be included explicitly
Typical Ascent Trajectory Constraints for HTHL Vehicles
Ascent Trajectory of Airbreathing HTHL TSTO Booster
Ascent Trajectory of Airbreathing HTHL SSTO
Ascent Trajectory of Airbreathing VTHL SSTO
Modeling of Ascent Trajectories
It is assumed, that for ascent flight segments of vehicles with horizontal takeoff a relationship between flight velocity and air density of the following form is valid:
(14) .constlvk
Some typical values for k are:
:1k Constant air mass flow (subsonic flight)
:0k Constant altitude (transonic flight)
:2k Constant dynamic pressure (supersonic flight)
:8.2k Constant total pressure/total temperature ratio aft of airbreather intake
:5.4...9.3k Airbreathing engine pressure and temperature limit (hypersonic flight)
:3.6...0.6k Constant aerothermodynamic stagnation point heat flux
Transonic dives like the SR-71 ‘dipsy doodle’ maneuver can also be described by a negative value of k
Ascent Gravity Loss Relationships
Determination of Ascent Gravity Losses (I)
The momentary gravity loss occuring during ascent can be expressed as:
(15) dtdh
vg
dtdh
dsdtg
dsdhgg sin
Equating the hydrostatic equation:
(16) dhgdp
With the differential formulation of the polytropic equation of state:
(17)
dpndp
Leads after division with dt to:
(18) dtg
dpndtdh
2
Determination of Ascent Gravity Losses (II)
From equation (14) follows:
(19) dtdv
vkl
dtd
k
1
Inserting equation (19) into equation (18) and using equation (14) leads to:
(20) dtdv
vgpkn
dtdh
Using the ideal gas equation:
(21) TRp
Finally leads to the expression:
(22) dtdv
vTRkng
2sin
Determination of Ascent Gravity Losses (III)
Using the Mach number relationship:
(23) 221Mv
TR
Yields the alternative formulation:
(24) dtdv
Mkng
2sin
The total gravity loss during a flight segment following the relationship defined in equation (14) is:
(25) dvv
TRkndtgve et
t
v
vg
0 02sin
Note: The integral gravity loss is independent of flight duration and acceleration for trajectory segments that follow the relationship defined in equation (14), since lower acceleration and associated longer flight duration lead to shallower flight path angles
Integration of Equation of Motion along Flight Path (I)
The simplified equation of motion along an ascent trajectory is:
(26) sin gmD
mF
dtdv
Using equation (22) leads to:
(27) m
DFdtdv
vTRkn
21
Inserting the relationship:
(28) mdmdt
With both dm and m once again being negative yields:
(29) mdmdv
DFTRknv
vm
2
2
Integration of Equation of Motion along Flight Path (II)
This can be rewritten as:
(30) mdmdv
DFTRknv
vgIF
sp
2
20
If all factors and variables on the left side of the equation are given for example either as constants or as linear functions of v , this equation can be integrated
Alternatively, m , F and D can also be expressed as the following functions:
(31) 0)()()( gvIvmvF sp
(32) 2
)()(
2 krefD vlAvc
vD
(33) kvlAvAA
vvm 10
0
)()()(
Equations (32) and (33) require however k to be an integer in order to lead to an integrable solution
Integration of Equation of Motion along Flight Path (III)
Solving the integrals:
(34)
ee m
m
v
v mdmdv
DFTRknv
vm
00
2
2
respectively the transformation obtained using equation (1):
(35)
ee m
m
v
v sp mdmdv
FDTRknv
vgIF
00
2
20
allows to determine the mass ratio em
m0 , and with that the propellant consumption,
which is required to achieve a given v
Integration of Equation of Motion along Flight Path (IV)
Integration of Equation of Motion along Flight Path (V)
A derivation of equation (29) can also be used to determine the theoretical optimum switching point in flight from one propulsion system to another, such as from airbreather to rocket:
(36)
DFTRknv
vmm
dvdm 2
2
As soon as the value of dvdm , which denotes the propellant mass increment necessary
for achieving a given velocity increment, for the airbreather exceeds the one for the rocket for the respective values of k , switching will lead to lowering the propellant flow required for vehicle acceleration and hence the propellant mass for the total mission
Analytically Recalculated Air Launch/HTHL Examples
Boundary Conditions and Results of Calculated Examples
Advantages and Limitations
Symbols and Abbreviations (I)
Symbols and Abbreviations (II)
Literature
Martin Bayer: Analytic performance considerations for lifting ascent trajectories of winged launch vehicles, Acta Astronautica 54, 2004, pp. 713 – 721
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