Lecture 2

Marat Kulakhmetov

AS8
Did some rockets tumble?
 Did some rockets wobble?
 Did some rockets flip over?


Maybe some rockets were unstable

l6w

Stability refers to how likely an object will return to
its initial position or orientation if it is disturbed
◦ Stable – Object returns to initial position
◦ Neutrally Stable – Object does not move
◦ Unstable – Object continues moving away from its initial
position




Moment describe the object’s tendency to rotate
◦ Moment = Force * Perpendicular Distance
In the example above, the moments generated by the two
weights generate 20 N*m and -20 N*m. They are balanced
Moments are usually calculated about their center of gravity
(CG)
Unbalanced moments on a rocket will cause the rocket to
tumble.

Location where the forces will balance

CG = Moment / Total Weight

Example:
◦ Moment = 10 * (0) + 20 * 3 = 60 N * m
◦ Total Weight = 10 + 20 = 30 N
◦ CG = Moment / Total Weight = 60 / 30 = 2 m
X=0
X=2
X=3
Beer, Russell, Johnston, DeWolf Mechanics of Materials
X=0
5
7
11
13 14
20
Part
Length
(cm)
Weight
(g)
Nose Cone
5
10
Parachute
sys.
3
5
Recovery
1
1
Launch
Lug
3
2
Engine
Mount
5
15
Rocket
Engine
5
30
Fins
5
3
Rocket
Body
15
40
X=0
5
7
11
13 14
20
Part
Centroid
Formula
Distance To
Centroid
Mass
Moment
Nose Con
h/3 =1.67
5/3 = 1.67
10
16.7
Parachute
h/2 =1.5
11+1.5 =12.5
5
62.5
Recovery
h/2=0.5
13+0.5=13.5
1
13.5
Launch Lug
h/2= 1.5
7+1.5=8.5
2
17
Engine
Mount
h/2 = 2.5
14+2.5=16.5
15
247.5
X=0
Part
5
Centroid
Formula
7
11
13 14
Distance To
Centroid
From Above
20
Mass
Moment
33
357.2
Rocket
Engine
h/2 =2.5
14+2.5=16.5
30
495
Rocket Body
h/2=7.5
5+7.5
40
300
103
1152.2
Total
X=0




5
7
11
13 14
Moment = 1152.2
Mass = 103
CG = Moment / Mass
= 1152.2/103 = 11.19 cm
20
B3=2

1
3
H=5
1
1
1
2



B1=2




Break it up into a
triangle, rectangle and
triangle
Area 1 = ½ *b1 * h = 5
Area 2 = b2 * h =5
Area 3 = ½ * b3 * h=5
B2=1
Total Area = Area 1 + Area 2 + Area 3 = 15
Mass1 = Total Mass * Area 1 / Total Area = 1
Mass2 = Total Mass * Area 2 / Total Area =1
Mass3 = Total Mass * Area 3 / Total Area =1
b3


1
3
h
1
1
b1
1
2


b2


Part 1 is a triangle
Centroid 1 = b1/3 =.66
Part 2 is a rectangle
Centroid 2 = b2/2 = 0.5
Part 3 is a triangle
Centroid 3 = b3/2 =.66

Moment Fin = Mass1 * (b1 – Centroid 1)
+ Mass2 * ( b1 + Centroid 2)
+ Mass3 * ( b1 + b2 + Centroid 3)= 7.5

CG Fin = Moment Fin / Total Fin Mass =2.5

X=0



5
7
11
13 14
20
Moment with fins = 1152.2 +(2.5+14)*3
Mass = 103+3
CG = Moment / Mass =11.34 cm

If :
◦ Rocket has no fins
◦ Thrust is aligned
◦ Rocket pitched a little
X

Moment = -1*Lift * x
y

x
This rocket will keep
pitching and fly out of
control

Little Drag
Lots of Drag

If :
◦ Thrust is aligned
◦ Rocket turned a little
X

Moment = -1* Lift *x +
Fin * x1
X1
Fin Force

If Fin * x1 > Lift * x , the
rocket will right itself

Fin force =
1
2
F  Cd V A
2
◦ Larger Area = More force provided by fins
◦ Larger Velocity = More Force provided by fins

Fin Moment = Fin Force * Distance
◦ Larger Force = Larger Moment
◦ Larger Distance = Larger Moments


For stability, we want large fins as far away from
CG as possible.
If fins are too large they create more drag
X=0

5
7
11
13 14
20
Calculating aerodynamic center will require
Computational Fluid Dynamic (CFD)
analysis.

We will estimate that the aerodynamic
center is at Fin centroid

We calculated that this is at 16.5cm


Nozzles push on high
gasses and accelerate them
out the back
In return, the gasses push
on the nozzle and
accelerates it forward
High
Pressure
Low Pressure

Air wants to go from high pressure to low
pressure

Pressure Force ( P1 – P2) * A

Remember that Pressure = Force / Area



Action-Reacting
If you throw something out one way it will
push you the other way
If the rocket nozzle throws gases down, the
gasses push the rocket up




It is usually easy to study gas flows using
control volumes
Forces on the rocket could be calculated by
only looking at control surfaces
Fpressure =(Pe - Pa ) Ae
2 A
=
ρ
U
Fgas
e
e

Why did rockets filled with water go higher
than those filled with just air?
Changes
Thrust  [( Pe  Pa)  V ] Ae
2
Exit
Pressure
Constant
Ambient
Pressure
Constant
Exit
Velocity
Assumed
Constant


Rockets usually use
converging-diverging
nozzles. These could also be
called isentropic nozzles
The thrust through the C-D
nozzle depends on chamber
pressure, ambient pressure,
and nozzle shape



Upstream of the
nozzle, in the
combustion chamber,
the gas velocity is
small
All fluids (water, air,
etc.) accelerate
through a converging
section
The fastest they could
get in the converging
section is Mach 1



If the gases reached
Mach 1 in converging
section then they will
continue accelerating in
the diverging section
If the gasses did not
reach Mach 1 in the
converging section then
they will decelerate in the
diverging section
This is why our water
bottle rockets only had
converging section
Ambient Conditions:
Pa = 101,000 Pa
Area = 0.05 m^2
Mass = 0.5 kg
Exit Conditions:
Pe = 150,000 Pa
Ve = 100 m/s
Density = 1.2 kg/m3

Lets Calculate Rocket Thrust and
acceleration
2
Thrust  [( Pe  Pa)  V ] A
Thrust  [(101,000  150,000)  1.2(100) ]0.05  3050 N
2
 A = F/m = 3050 / 0.5 = 6100 m/s^2

Pressurized Air
◦ Balloon




Solid Propellant
Liquid Propellant
Nuclear
Electric
F
ISP 
mg
F  Thrust
m  massflow(kg / s )
g  gravity

ISP is used to classify how well a rocket performs

Low ISP = need a lot of fuel to achieve thrust

High ISP =do not need as much fuel to achieve same
thrust

Propellant is initially in the solid state and it
becomes a hot gas during combustion

Pros:

Cons:
◦
◦
◦
◦
Simple
Cheap
Easy to store
Can be launched quickly
◦ ISP only 150-350
◦ Cannot turn off after ignition
◦ Cannot throttle during flight

Fuel and Oxidizer are both stored
separately in liquid form

Pros:
◦ Better performance (ISP 300-460)

Cons:
◦ More complex
◦ Requires pumps or pressurized gas
tanks
◦ Heavier

Nuclear Reactor heats working gas that is
accelerated through a nozzle

Pros:

Cons:
◦ Isp 800-1000
◦ Requires shielding, can be heavy
◦ It’s a NUKE

Two types:
◦ Arcjet: Electricity is used to superheat the gases
◦ Ion Thrusters: ionized (charged) atoms are
accelerated through an electro-magnetic field

Pros:
◦ ISP 400-10,000

Cons:
◦ Thrust usually <1N

VASMIR