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Rocket Dynamics
• Forces on the Rockets - Drag• Rocket Stability• Rocket Equation• Specific Impulse• Rocket Motors
Forces on the Rocket^ Equation of Motion:
F = Ma
Forces at through the Center of Mass
Center of Mass
(1) Gravity: F
= MgGrav
F^ Grav
(2) Thrust: F
Thrust^
=^
)( MV
d dt
F^ Thrust
Need to minimize total mass M tomaximize acceleration of the rocket The thrust force seen by the rocket isequal to the rate of change of momentumcarried away in the exhaust
Thrust:Assuming that the propellant speed is approximately constant then
CM^ EP^
FThrust
Where^
C^ is the speed of the exhaustE^ is the rate of change propellant massMP
In other words, how fast you go is dependent on^ •^ the speed of the propellant^ •^ the mass available in propellant
rd^3 Force: Drag
AC^ D
12 V^^ 2
D =• is the mass density of the air (1.2 kg/m
3 @ sea level)
-^ V^ is the rocket speed•^ A is the area of the rocket perpendicular to the rocket
flow Resistance from the air to the rocket motion^ •^ C^ is the coefficient of dragD^ F^ Thrust Center of MassF^ GravDrag
There are three forms of drag, and their relative importance ishighly dependent on the speed on the rocket relative to the soundspeed, i.e. to the Mach number = V/V
V^ ~ 1050 ft/s or 331 m/ss^ s^
Types of Drag
1.^ Skin friction
arises from the friction of the fluid against the "skin" of the object that is moving through it. Skin friction arisesfrom the interaction between the fluid and the skin of the body.
Rivets on skin surface add significantlyto total drag at supersonic speeds
(2) Pressure Drag or Form Drag^ arises because of the form of the object.The general size and shape of the bodyis the most important factor in form dragBodies with a larger apparent cross-section will have a higher drag thanthinner bodies.Sleek designs, or designs that arestreamlined and change cross-sectionalarea gradually are also critical forachieving minimum form drag.
(2) Pressure Drag or Form Drag^ Drag coefficients for different shapes.^ Elimination of vortices in thewake reduce the drag
Wave drag• is caused by the formation of shock waves around the aircraft whichradiate away a considerable amount of energy producing enhanceddrag• Although shock waves are typically associated with supersonic flow,they can form at much lower speeds at areas on the aircraft wherelocal airflow accelerates to supersonic speeds.•The effect is typically seen at transonic speeds above about Mach 0.8,but it is possible to notice the problem at any speed over that of thecritical Mach of that aircraft's wing.•The magnitude peaks at about four times the normal subsonic drag.•It is so powerful that it was thought for some time that engines wouldnot be able to provide enough power to easily overcome the effect,which led to the concept of a "sound barrier".
Rocket
Equation
Neglecting drag the equation of motion for the rocket is:
V^ E
dMdt
dVM dt
M^ f EM V V^
dMM
VdV^0
ln(^
(^0) F
MVE M
V^
OR^
0 exp(
E F^
V V
M^ M
Fraction of Payload becomes exponential small when required
V
exceeds velocity of the propellant
Percentage Payload to Orbit Using^2 (1) H^ /O
and (2) R 2
(highly refined kerosene)/Op^
2
Specific Impulse (Isp)
-^ is a way to describe the efficiency of rocket and jet engines.•^ It represents the impulse (change in momentum) per unit ofpropellant.•^ The higher the specific impulse, the less propellant is neededto gain a given amount of momentum.
V^ g
dtm dtmg
V dt
Fdtmg
Isp^
E E
^
Specific impulse of various propulsion technologies
Ion thruster
Bipropellantliquid rocket
Solid rocket
Turbofan jetengine
Energy per kgof exhaust(MJ/kg) Specificimpulse(s) "Ve" exhaustvelocity(m/s) Engine
M=^
Rocket nozzlesthe hot gas produced in the combustionchamber is permitted to escape from thecombustion chamber through an opening(the "throat"), within a high expansion-ratio 'de Laval nozzle'.Provided sufficient pressure isprovided to the nozzle (about 2.5-3x above ambient pressure) thenozzle^ chokes
and a supersonic jet is formed, dramaticallyaccelerating the gas, convertingmost of the thermal energy intokinetic energy.
Pumps needed for high performanceare expensive to design, huge thermalfluxes across combustion chamber wallcan impact reuse, failure modes includemajor explosions, a lot of plumbing isneeded. Up to ~99% efficientcombustion withexcellent mixturecontrol, throttleable, Two fluid (typically liquid)propellants areintroduced throughinjectors into combustionchamber and burnt LiquidBipropellant rocket
catalysts can be easily contaminated,monopropellants can detonate ifcontaminated or provoked,
sp^ I is perhaps 1/3 of best liquids Simple in concept,throttleable, lowtemperatures incombustion chamber Propellant such asHydrazine, HydrogenPeroxide or NitrousOxide, flows over catalystand exothermicallydecomposes Monopropellantrocket
Some oxidisers are monopropellants,can explode in own right; mechanicalfailure of solid propellant can blocknozzle (very rare with rubberisedpropellant), central hole widens overburn and negatively affects mixtureratio. Quite simple, solid fuelis essentially inertwithout oxidiser, safer;cracks do not escalate,throttleable and easy toswitch off. Separate oxidiser/fuel,typically oxidiser is liquidand kept in a tank, theother solid with centralhole Hybridrocket
Once lit, extinguishing it is difficultalthough often possible, cannot bethrottled in real time; handling issuesfrom ignitable mixture, lowerperformance than liquid rockets, if graincracks it can block nozzle withdisastrous results, cracks burn andwiden during burn. Simple, often nomoving parts,reasonably good massfraction, reasonable
sp I. A thrust schedule canbe designed into thegrain. Ignitable, self sustainingsolid fuel/oxidiser mixture("grain") with central holeand nozzle Solidrocket
Disadvantages
Advantages Description Type
Solid Rocket Motors
Single Grain Motor: Thrustincrease with time due toincreased surface area as thegrain burns away.
Use of Multiple grainsso that burning betweengrains leads to moreuniform burn
