Rocket Performance • !e rest of orbital mechanics • !e rocket equation • Mass ratio and performance • Structural and payload mass fractions • Regression analysis • Multistaging • Optimal ΔV distribution between stages • Trade-o# ratios • Parallel staging

© 2014 David L. Akin - All rights reserved • Modular staging http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Rocket Performance MARYLAND 1 ENAE 791 - Launch and Entry Vehicle Design Rocket Equation (First Look)

1 Typical Range for Launch to ) Low Earth Orbit

initial 0.1 /M ! nal

0.01

Mass Ratio, (M Mass 0.001 0 2 4 6 8 Velocity Ratio, (ΔV/Ve) U N I V E R S I T Y O F Rocket Performance MARYLAND 2 ENAE 791 - Launch and Entry Vehicle Design Sources and Categories of Vehicle Mass

Payload Propellants Structure Propulsion Avionics Power Mechanisms Thermal Etc.

U N I V E R S I T Y O F Rocket Performance MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Sources and Categories of Vehicle Mass

Payload Propellants Inert Mass Structure Propulsion Avionics Power Mechanisms Thermal Etc.

U N I V E R S I T Y O F Rocket Performance MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Basic Vehicle Parameters

• Basic mass summary mo =initial mass mo = mpl + mpr + min mpl =payload mass • Inert mass fraction mpr =propellant mass m m min =inert mass δ ≡ in = in mo mpl + mpr + min • Payload fraction m m λ ≡ pl = pl mo mpl + mpr + min • Parametric mass ratio r = λ + δ

U N I V E R S I T Y O F Rocket Performance MARYLAND 5 ENAE 791 - Launch and Entry Vehicle Design Rocket Equation (including Inert Mass)

1 ) Typical Range for Launch to Low initial Earth Orbit Inert Mass

/M Fraction δ 0.1 0 0.05 payload 0.1 0.15 0.01 0.2

0.001

Payload Fraction, (M Fraction, Payload 0 2 4 6 8 Velocity Ratio, (ΔV/Ve) U N I V E R S I T Y O F Rocket Performance MARYLAND 6 ENAE 791 - Launch and Entry Vehicle Design Limiting Performance Based on Inert Mass

5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Asymptotic Velocity Ratio, ( Δ V/Ve) Velocity Asymptotic 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Inert Mass Fraction, (Minert/Minitial) U N I V E R S I T Y O F Rocket Performance MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Limiting Performance Based on Inert Mass

5

4.5 Typical Feasible 4 Design Range for Inert Mass Fraction 3.5 3 2.5 2 1.5 1 0.5 0 Asymptotic Velocity Ratio, ( Δ V/Ve) Velocity Asymptotic 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Inert Mass Fraction, (Minert/Minitial) U N I V E R S I T Y O F Rocket Performance MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Limiting Performance Based on Inert Mass

5

4.5 Typical Feasible 4 Design Range for Inert Mass Fraction 3.5 3 2.5 2 1.5 1 0.5 0 Asymptotic Velocity Ratio, ( Δ V/Ve) Velocity Asymptotic 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Inert Mass Fraction, (Minert/Minitial) U N I V E R S I T Y O F Rocket Performance MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Regression Analysis of Existing Vehicles

Veh/Stage prop mass gross mass Type Propellants Isp vac isp sl sigma eps delta (lbs) (lbs) (sec) (sec) Delta 6925 Stage 2 13,367 15,394 StorableN2O4-A50 319.4 0.152 0.132 0.070 Delta 7925 Stage 2 13,367 15,394 StorableN2O4-A50 319.4 0.152 0.132 0.065 Titan II Stage 2 59,000 65,000 StorableN2O4-A50 316.0 0.102 0.092 0.087 Titan III Stage 2 77,200 83,600 StorableN2O4-A50 316.0 0.083 0.077 0.055 Titan IV Stage 2 77,200 87,000 StorableN2O4-A50 316.0 0.127 0.113 0.078 Proton Stage 3 110,000 123,000 StorableN2O4-A50 315.0 0.118 0.106 0.078 Titan II Stage 1 260,000 269,000 StorableN2O4-A50 296.0 0.035 0.033 0.027 Titan III Stage 1 294,000 310,000 StorableN2O4-A50 302.0 0.054 0.052 0.038 Titan IV Stage 1 340,000 359,000 StorableN2O4-A50 302.0 0.056 0.053 0.039 Proton Stage 2 330,000 365,000 StorableN2O4-A50 316.0 0.106 0.096 0.066 Proton Stage 1 904,000 1,004,000 StorableN2O4-A50 316.0 285.0 0.111 0.100 0.065 average 312.2 285.0 0.100 0.089 0.061 standard deviation 8.1 0.039 0.033 0.019

U N I V E R S I T Y O F Rocket Performance MARYLAND 8 ENAE 791 - Launch and Entry Vehicle Design Inert Mass Fraction Data for Existing LVs

0.18 0.16 0.14

0.12 LOX/LH2 0.10 LOX/RP-1 0.08 Solid 0.06 Storable

Inert Mass Fraction Mass Inert 0.04 0.02 0.00 0 1 10 100 1,000 10,000

Gross Mass (MT)

U N I V E R S I T Y O F Rocket Performance MARYLAND 9 ENAE 791 - Launch and Entry Vehicle Design Regression Analysis

• Given a set of N data points (xi,yi) • Linear curve $t: y=Ax+B – $nd A and B to minimize sum squared error N 2 error = !(Axi + B − yi) i=1 – Analytical solutions exist, or use Solver in Excel A • Power law $t: y=BNx 2 error = ! [A log(xi)+B − log(yi)] i=1 • Polynomial, exponential, many other $ts possible U N I V E R S I T Y O F Rocket Performance MARYLAND 10 ENAE 791 - Launch and Entry Vehicle Design Solution of Least-Squares Linear Regression N ∂(error) =2!(Axi + B − yi)xi =0 ∂A i=1 N ∂(error) =2 (Axi + B − yi)=0 ∂B ! N N N i=1 2 A xi + B xi − xiyi =0 ! ! ! N N i=1 i=1 i=1 A ! xi + NB − ! yi =0 i=1 i=1

2 N xiyi − xi yi yi x − xi xiyi ! ! ! B ! ! i ! ! A = 2 2 = 2 2 − N x − xi N ! xi (! xi) ! i (! )

U N I V E R S I T Y O F Rocket Performance MARYLAND 11 ENAE 791 - Launch and Entry Vehicle Design Regression Analysis - Storables

0.10

0.08

0.06

0.04

0.02 R2 = 0.0541 0.00 Inert Mass Fraction Mass Inert 1 10 100 1,000 Gross Mass (MT)

U N I V E R S I T Y O F Rocket Performance MARYLAND 12 ENAE 791 - Launch and Entry Vehicle Design Regression Values for Design Parameters

Vacuum Ve Inert Mass Max ΔV (m/sec) Fraction (m/sec) LOX/LH2 4273 0.075 11,070

LOX/RP-1 3136 0.063 8664

Storables 3058 0.061 8575

Solids 2773 0.087 6783

U N I V E R S I T Y O F Rocket Performance MARYLAND 13 ENAE 791 - Launch and Entry Vehicle Design Economy of Scale for Stage Size 0.20# M 0.18# ✏ = inert 0.16# Minert + Mprop 0.14# 0.12# 0.10# 0.08# 0.06# 0.04# Stage#Inert#Mass#Frac6on#ε# 0.02# PMF =1 ✏ 0.00# 1# 10# 100# 1,000# 10,000# Stage#Gross#Mass,#MT#

Storable#MER# LOX/RPD1# Storables# LOX/LH2# Cryo#MER#

U N I V E R S I T Y O F Rocket Performance MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design Stage Inert Mass Fraction Estimation

0.183 ✏ =0.987 (M kg ) LOX/LH2 stageh i

0.275 ✏ =1.6062 (M kg ) storables stageh i

U N I V E R S I T Y O F Rocket Performance MARYLAND 15 ENAE 791 - Launch and Entry Vehicle Design The Rocket Equation for Multiple Stages • Assume two stages

mfinal1 ∆V1 = −Ve1 ln = −Ve1 ln(r1) !minitial1 "

mfinal2 ∆V2 = −Ve2 ln = −Ve2 ln(r2) !minitial2 "

• Assume Ve1=Ve2=Ve

∆V1 + ∆V2 = −Ve ln(r1) − Ve ln(r2)=−Ve ln(r1r2)

U N I V E R S I T Y O F Rocket Performance MARYLAND 16 ENAE 791 - Launch and Entry Vehicle Design Continued Look at Multistaging

• !ere’s a historical tendency to de$ne r0=r1r2 V + V = V ln (r r ) = V ln (r ) 1 2 e 1 2 e 0

• But it’s important to remember that it’s really m m V + V = V ln (r r ) = V ln final1 final2 1 2 e 1 2 e m m initial1 initial2 ⇥ • And that r0 has no physical signi$cance, since

mfinal2 mfinal1 = minitial2 r0 = ⇥ ⇥ minitial1

U N I V E R S I T Y O F Rocket Performance MARYLAND 17 ENAE 791 - Launch and Entry Vehicle Design Multistage Inert Mass Fraction • Total inert mass fraction

min,1 + min,2 + min,3 min,1 min,2 min,3 δ0 = = + + m0 m0 m0 m0

min,1 min,2 m0,2 min,3 m0,3 m0,2 δ0 = + + m0 m0,2 m0 m0,3 m0,2 m0 • Convert to dimensionless parameters

δ0 = δ1 + δ2λ1 + δ3λ2λ1 • General form of the equation n stages j−1 δ0 = ! [δj " λ!] j=1 !=1 U N I V E R S I T Y O F Rocket Performance MARYLAND 18 ENAE 791 - Launch and Entry Vehicle Design Multistage Payload Fraction • Total payload fraction (3 stage example)

mpl mpl m0,3 m0,2 λ0 = = m0 m0,3 m0,2 m0 • Convert to dimensionless parameters

λ0 = λ3λ2λ1 • Generic form of the equation n stages

λ0 = λj j!=1

U N I V E R S I T Y O F Rocket Performance MARYLAND 19 ENAE 791 - Launch and Entry Vehicle Design Effect of δ and ΔV/Ve on Payload

0.900 V 0.800 = 0.25 0.700 Ve 0.600 1 stage 0.500 2 stages 0.400 3 stages 4 stages 0.300

Total Payload Fraction Payload Total 0.200

0.100

0.000 0 0.2 0.4 0.6 0.8 1 Inert Mass Fraction

U N I V E R S I T Y O F Rocket Performance MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design Effect of δ and ΔV/Ve on Payload

0.9000.0900.4000.0200.2500.0600.700

0.8000.0800.018 V 0.6000.350 0.050 = 12340.0525 0.7000.0700.0160.200 0.300 Ve 0.0140.500 0.6000.0600.040 0.250 0.0120.150 1 stage 0.400 0.5000.050 2 stages 0.2000.0100.030 3 stages 0.3000.4000.040 0.0080.100 0.150 4 stages 0.3000.0300.020 0.2000.006 0.100 Total Payload Fraction Payload Total 0.0040.0500.2000.020 0.010 0.0500.100 0.0020.1000.010

0.000 0 0.2 0.4 0.60.6 0.80.8 11 Inert Mass Fraction

U N I V E R S I T Y O F Rocket Performance MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design Effect of Staging Inert Mass Fraction δ=0.15 0.25 Single Stage Three Stage Two Stage Four Stage 0.20

0.15 1 stage 2 stage 3 stage 0.10 4 stage Payload Fraction

0.05

0.00 0 1 2 3 4 5 Velocity Ratio (ΔV/Ve)

U N I V E R S I T Y O F Rocket Performance MARYLAND 21 ENAE 791 - Launch and Entry Vehicle Design Trade Space for Number of Stages

4

3.5

3 Four Stage

2.5 Three Stage 2

1.5 Two Stage

Velocity Ratio DV/Ve Ratio Velocity 1 Single Stage 0.5

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Inert Mass Fraction

U N I V E R S I T Y O F Rocket Performance MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Trade Space for Number of Stages

4

3.5 Solids 3 LOX/RP-1 Four Stage Storables 2.5 Three Stage 2 LOX/LH2

1.5 Two Stage

Velocity Ratio DV/Ve Ratio Velocity 1 Velocity Ratio DV/Ve Ratio Velocity 1 Single Stage 0.5

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Inert Mass Fraction

U N I V E R S I T Y O F Rocket Performance MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Effect of ΔV Distribution

1st Stage: Solids 2nd Stage: LOX/LH2 70

60

50

40 Total mass Stage 2 mass 30 Stage 1 mass

20

10 Normalized Mass (kg/kg of payload)

0 2000 3000 4000 5000 6000 7000 8000 9000 10000 Stage 2 ΔV (m/sec)

U N I V E R S I T Y O F Rocket Performance MARYLAND 23 ENAE 791 - Launch and Entry Vehicle Design ΔV Distribution and Design Parameters

1st Stage: Solids 2nd Stage: LOX/LH2 0.45

0.4

0.35

0.3

0.25 delta_0 lambda_0 0.2 lambda/delta 0.15

0.1

0.05

0 2000 3000 4000 5000 6000 7000 8000 9000 10000 Stage 2 ΔV (m/sec)

U N I V E R S I T Y O F Rocket Performance MARYLAND 24 ENAE 791 - Launch and Entry Vehicle Design Lagrange Multipliers • Given an objective function y = f(x) subject to constraint function z = g(x) • Create a new objective function z = f(x) + [g(x) z] • Solve simultaneous equations y ⇥y = 0 = 0 x ⇥

U N I V E R S I T Y O F Rocket Performance MARYLAND 25 ENAE 791 - Launch and Entry Vehicle Design Optimum ΔV Distribution Between Stages • Maximize payload fraction (2 stage case) λ0 = λ1λ2 =(r1 − δ1)(r2 − δ2) subject to constraint function

∆Vtotal = ∆V1 + ∆V2 • Create a new objective function

−∆V1 −∆V2 λo = e Ve,1 − δ1 e Ve,2 − δ2 + K [∆V1 + ∆V2 − ∆Vtotal] ! "! " ➥Very messy for partial derivatives!

U N I V E R S I T Y O F Rocket Performance MARYLAND 26 ENAE 791 - Launch and Entry Vehicle Design Optimum ΔV Distribution (continued) • Use substitute objective function

max (λo) ⇐⇒ max [ln (λo)] • Create a new constrained objective function

ln (λo)=ln (r1 − δ1)+ln (r2 − δ2)+K [∆V1 + ∆V2 − ∆Vtotal] • Take partials and set equal to zero

∂ [ln (λ )] ∂ [ln (λ )] ∂ [ln (λo)] o =0 o =0 =0 ∂r1 ∂r2 ∂K

U N I V E R S I T Y O F Rocket Performance MARYLAND 27 ENAE 791 - Launch and Entry Vehicle Design Optimum ΔV Special Cases • “Generic” partial of objective function ∂ [ln (λ )] 1 V o = + K e,i =0 ∂ri ri − δi ri • Special case: δ1=δ2 Ve,1=Ve,2 ∆Vtotal r1 = r2 =⇒ ∆V1 = ∆V2 = 2 • Same principle holds for n stages

r1 = r2 = ···= rn =⇒ V V V ··· V ∆ total ∆ 1 = ∆ 2 = = ∆ n = n

U N I V E R S I T Y O F Rocket Performance MARYLAND 28 ENAE 791 - Launch and Entry Vehicle Design Sensitivity to Inert Mass ΔV for multistaged rocket n stages n mo,k ∆Vtot = ! ∆Vk = ! Ve,k ln " # mf,k k=1 k=1 where

n

mo,k = mpl + mpr,k + min,k + (mpr,j + min,j) j=Xk+1 n

mf,k = mpl + min,k + (mpr,j + min,j) j=Xk+1

U N I V E R S I T Y O F Rocket Performance MARYLAND 29 ENAE 791 - Launch and Entry Vehicle Design Finding Payload Sensitivity to Inert Mass • Given the equation linking mass to ΔV, take

∂(∆Vtot) ∂(∆Vtot) dmpl + dmin,j =0 ∂mpl ∂min,j and solve to $nd k 1 1 Ve,j dm j=1 mo,j mf,j pl = dmin,k N ⇣ 1 1 ⌘ @(Vtot)=0 P Ve,` `=1 mo,` mf,` • !is equation shows theP “trade-o# ⇣ratio” - Δpayload⌘ resulting from a change in inert mass for stage k (for a vehicle with N total stages)

U N I V E R S I T Y O F Rocket Performance MARYLAND 30 ENAE 791 - Launch and Entry Vehicle Design Trade-off Ratio Example: Gemini-Titan II

Stage 1 Stage 2 Initial Mass (kg) 150,500 32,630 Final Mass (kg) 39,370 6099 Ve (m/sec) 2900 3097

dmpl/dmin,k -0.1164 -1

U N I V E R S I T Y O F Rocket Performance MARYLAND 31 ENAE 791 - Launch and Entry Vehicle Design Payload Sensitivity to Propellant Mass • In a similar manner, solve to $nd k 1 Ve,j dm j=1 mo,j pl = dmpr,k N 1 ⇣ 1⌘ @(Vtot)=0 PVe,` `=1 mo,` mf,` ⇣ ⌘ • !is equation gives the Pchange in payload mass as a function of additional propellant mass (all other parameters held constant)

U N I V E R S I T Y O F Rocket Performance MARYLAND 32 ENAE 791 - Launch and Entry Vehicle Design Trade-off Ratio Example: Gemini-Titan II

Stage 1 Stage 2 Initial Mass (kg) 150,500 32,630 Final Mass (kg) 39,370 6099 Ve (m/sec) 2900 3097

dmpl/dmin,k -0.1164 -1

dmpl/dmpr,k 0.04124 0.2443

U N I V E R S I T Y O F Rocket Performance MARYLAND 33 ENAE 791 - Launch and Entry Vehicle Design Payload Sensitivity to Exhaust Velocity • Use the same technique to $nd the change in payload resulting from additional exhaust velocity for stage k k ln mo,j dm j=1 mf,j pl = dVe,k N ⇣1 ⌘ 1 @(Vtot)=0 PVe,` `=1 mo,` mf,` ⇣ ⌘ • !is trade-o # ratio (unlikeP the ones for inert and propellant masses) has units - kg/(m/sec)

U N I V E R S I T Y O F Rocket Performance MARYLAND 34 ENAE 791 - Launch and Entry Vehicle Design Trade-off Ratio Example: Gemini-Titan II Stage 1 Stage 2 Initial Mass (kg) 150,500 32,630 Final Mass (kg) 39,370 6099 Ve (m/sec) 2900 3097

dmpl/dmin,k -0.1164 -1

dmpl/dmpr,k 0.04124 0.2443

dmpl/dVe,k 2.870 6.459 (kg/m/sec)

U N I V E R S I T Y O F Rocket Performance MARYLAND 35 ENAE 791 - Launch and Entry Vehicle Design Parallel Staging • Multiple dissimilar engines burning simultaneously • Frequently a result of upgrades to operational systems • General case requires “brute force” numerical performance analysis

U N I V E R S I T Y O F Rocket Performance MARYLAND 36 ENAE 791 - Launch and Entry Vehicle Design Parallel-Staging Rocket Equation • Momentum at time t: M = mv • Momentum at time t+Δt: (subscript “b”=boosters; “c”=core vehicle) M = (m m m )(v + v) b c +m (v V ) + m (v V ) b e,b c e,c • Assume thrust (and mass %ow rates) constant

U N I V E R S I T Y O F Rocket Performance MARYLAND 37 ENAE 791 - Launch and Entry Vehicle Design Parallel-Staging Rocket Equation • Rocket equation during booster burn

mfinal min,b+min,c+mpr,c+m0,2 V = V¯e ln = V¯e ln minitial min,b+mpr,b+min,c+mpr,c+m0,2 ⇣ ⌘ ⇣ ⌘ where χ = fraction of core propellant remaining a&er booster burnout, and where

Ve,bm˙ b+Ve,cm˙ c Ve,bmpr,b+Ve,c(1−χ)mpr,c V¯e = = m˙ b+˙mc mpr,b+(1−χ)mpr,c

U N I V E R S I T Y O F Rocket Performance MARYLAND 38 ENAE 791 - Launch and Entry Vehicle Design Analyzing Parallel-Staging Performance Parallel stages break down into pseudo-serial stages: • Stage “0” (boosters and core)

¯ min,b+min,c+χmpr,c+m0,2 ∆V0 = −Ve ln ! " min,b+mpr,b+min,c+mpr,c+m0,2 • Stage “1” (core alone)

min,c+m0,2 V1 = Ve,c ln min,c+mpr,c+m0,2 • Subsequent stages are as before ⇥

U N I V E R S I T Y O F Rocket Performance MARYLAND 39 ENAE 791 - Launch and Entry Vehicle Design Parallel Staging Example: Space Shuttle • 2 x solid rocket boosters (data below for single SRB) – Gross mass 589,670 kg – Empty mass 86,183 kg – Ve 2636 m/sec – Burn time 124 sec • External tank (space shuttle main engines) – Gross mass 750,975 kg – Empty mass 29,930 kg – Ve 4459 m/sec – Burn time 480 sec • “Payload” (orbiter + P/L) 125,000 kg U N I V E R S I T Y O F Rocket Performance MARYLAND 40 ENAE 791 - Launch and Entry Vehicle Design Shuttle Parallel Staging Example m m V = 2636 Ve,c = 4459 e,b sec sec 480 − 124 χ = =0.7417 480 2636(1, 007, 000) + 4459(721, 000)(1 − .7417) m V¯e = = 2921 1, 007, 000 + 721, 000(1 − .7417) sec

862, 000 m ∆V0 = −2921 ln = 3702 3, 062, 000 sec 154, 900 m ∆V1 = −4459 ln =6659 689, 700 sec m ∆V , tot =10 360sec U N I V E R S I T Y O F Rocket Performance MARYLAND 41 ENAE 791 - Launch and Entry Vehicle Design Modular Staging • Use identical modules to form multiple stages • Have to cluster modules on lower stages to make up for nonideal ΔV distributions • Advantageous from production and development cost standpoints

U N I V E R S I T Y O F Rocket Performance MARYLAND 42 ENAE 791 - Launch and Entry Vehicle Design Module Analysis • All modules have the same inert mass and propellant mass • Because δ varies with payload mass, not all modules have the same δ! • Use module-oriented parameters m m ε ≡ in σ ≡ in min + mpr mpr δ δ • Conversions ε = σ = 1 − λ 1 − δ − λ

U N I V E R S I T Y O F Rocket Performance MARYLAND 43 ENAE 791 - Launch and Entry Vehicle Design Rocket Equation for Modular Boosters • Assuming n modules in stage 1, m nε + o2 n(min)+mo2 mmod r1 = = m 2 n(min + mpr)+mo2 n + o mmod

• If all 3 stages use same modules, nj for stage j,

n1ε + n2 + n3 + ρpl r1 = n1 + n2 + n3 + ρpl mpl where ρpl ≡ ; mmod = min + mpr mmod U N I V E R S I T Y O F Rocket Performance MARYLAND 44 ENAE 791 - Launch and Entry Vehicle Design Example: Conestoga 1620 (EER) • Small launch vehicle (1 %ight, 1 failure) • Payload 900 kg • Module gross mass 11,400 kg • Module empty mass 1,400 kg • Exhaust velocity 2754 m/sec • Staging pattern – 1st stage - 4 modules – 2nd stage - 2 modules – 3rd stage - 1 module – 4th stage - Star 48V (gross mass 2200 kg, empty mass 140 kg, Ve 2842 m/sec) U N I V E R S I T Y O F Rocket Performance MARYLAND 45 ENAE 791 - Launch and Entry Vehicle Design Conestoga 1620 Performance • 4th stage ∆V mf4 900 + 140 m V4 = Ve4 ln = 2842 ln = 3104 mo4 900 + 2200 sec • Treat like three-stage modular vehicle; Mpl=3100 kg m 1400 = in = =0.1228 mmod 11400

mpl 3100 pl = = =0.2719 mmod 11400

n1 = 4; n2 = 2; n3 =1

U N I V E R S I T Y O F Rocket Performance MARYLAND 46 ENAE 791 - Launch and Entry Vehicle Design Constellation 1620 Performance (cont.)

n1 + n2 + n3 + ⇥pl 4 0.1228 + 2 + 1 + 0.2719 r1 = = =0.5175 n1 + n2 + n3 + ⇥pl 4+2+1+0.2719 n2 + n3 + ⇥pl 2 0.1228 + 1 + 0.2719 r2 = = =0.4638 n2 + n3 + ⇥pl 2+1+0.2719

n3 + ⇥pl 1 0.1228 + 0.2719 r3 = = =0.3103 n3 + ⇥pl 1+0.2719 m m V = 1814 ; V = 2116 1 sec 2 sec m m V = 3223 ; V = 3104 3 sec 4 sec m V = 10, 257 total sec U N I V E R S I T Y O F Rocket Performance MARYLAND 47 ENAE 791 - Launch and Entry Vehicle Design Discussion about Modular Vehicles • Modularity has several advantages – Saves money (smaller modules cost less to develop) – Saves money (larger production run = lower cost/ module) – Allows resizing launch vehicles to match payloads • Trick is to optimize number of stages, number of modules/stage to minimize total number of modules • Generally close to optimum by doubling number of modules at each lower stage • Have to worry about packing factors, complexity U N I V E R S I T Y O F Rocket Performance MARYLAND 48 ENAE 791 - Launch and Entry Vehicle Design OTRAG - 1977-1983

U N I V E R S I T Y O F Rocket Performance MARYLAND 49 ENAE 791 - Launch and Entry Vehicle Design Modular Example • Let’s build a launch vehicle out of seven Space Shuttle Solid Rocket Boosters

– Min=86,180 kg

– Mpr=503,500 kg

m m ε ≡ in =0.1461 σ ≡ in =0.1711 min + mpr mpr • Look at possible approaches to sequential $ring

U N I V E R S I T Y O F Rocket Performance MARYLAND 50 ENAE 791 - Launch and Entry Vehicle Design Modular Sequencing - SRB Example • Assume no payload

• All seven $ring at once - ΔVtot=5138 m/sec

• 3-3-1 sequence - ΔVtot=9087 m/sec

• 4-2-1 sequence - ΔVtot=9175 m/sec

• 2-2-2-1 sequence - ΔVtot=9250 m/sec

• 2-1-1-1-1-1 sequence - ΔVtot=9408 m/sec

• 1-1-1-1-1-1-1 sequence - ΔVtot=9418 m/sec • Sequence limited by need to balance thrust laterally

U N I V E R S I T Y O F Rocket Performance MARYLAND 51 ENAE 791 - Launch and Entry Vehicle Design