optimization of flight performance of pet bottle rockets

Journal of the Institute of Industrial Applications Engineers Vol.7, No.2, pp.59–71, (2019.4.25)DOI: 10.12792/JIIAE.7.59 Online edition: ISSN 2187-8811 Print edition: ISSN 2188-1758

Paper

Optimization of Flight Performance of PET Bottle Rockets byIntegrated Analysis of Dynamics Simulation System and Parameter

Design

Eiji Toma∗† Member, Hiroshi Tanaka‡ Non-memberKazushige Kikuta† Non-member

(Received November 5, 2018, revised April 9, 2019)

Abstract: PET bottle rocket is a rocket system that obtains thrust by injecting water with compressed air at highpressure, is usually manufactured using a pressure-resistant plastic bottle. Although it is compact and easy tomanufacture, the technical elements and principles included in PET bottle rockets are many in common with theactual rocket system and it is an appropriate system as a teaching material for science education and research. Inthis research, we developed a simulator based on a numerical calculation program based on dynamic considerationabout the flight performance of PET bottle rockets, and theoretically analyzed the optimum flight conditions forobtaining the flight distance and flight stability of the rocket. Based on the analysis results, factors that influencethe flight performance of PET bottle rockets by applying the ”Parameter Design Method” were extracted, and theflight conditions to obtain the optimum maximum flight distance and flight stability were verified.

Keywords: Flying principle, Flight characteristics, Dynamics Simulation System, Parameter design, SN ratio

1. IntroductionPET bottle rocket is a rocket system that obtains thrustby injecting water with compressed air at high pressure(usually about 5atm), and is usually manufactured using apressure-resistant plastic bottle (Fig. 1). Although it is com-pact and easy to manufacture, the technical elements andprinciples included in PET bottle rockets are many in com-mon with the actual rocket system and it is an appropriatesystem as a teaching material for science education and re-search. However, as a PET bottle rocket is a flying object,attention must be paid to safety during launch. There isno problem if the water rocket flies on the specified trajec-tory and land in a safe area, but its flight path is often notstable, there is a possibility of damaging things and peopleon the ground. In the case of manufacturing and launch-ing in a short time like the event, we will make a waterrocket with a simple structure, so we need to think about afuselage structure that fly stably even if made by anyone.For that purpose, numerical analysis of flight trajectory bysimulation for evaluating flight characteristics and quanti-tative analysis on flight stability are important. However,there are many empirical factors such as basic aerodynamicdata required for orbital analysis and guidelines for aero-dynamic design / structural design, which are important inachieving stable flight. In addition, it is necessary to per-

∗ Corresponding author: [email protected]† National Institute of Technology, Tomakomai College, 443, Nishikioka,

Tomakomai, Hokkaido, Japan 059-1275‡ Aichi Institute of Technology, 1247, Yachikusa, Yakusa, Toyota, Aichi,

Japan 470-0392

Figure 1: Main structural model of PET bottle rockets

form many experiments by setting various parameter con-ditions such as adjusting the amount of water to be put inthe bottle to obtain the flight distance. In reality, the opti-mum value is mostly data obtained as a result of repeatingthe launch, and its scientific basis has not been clarified atpresent [1] [2]. Therefore, in this research, we developed asimulator based on a numerical calculation program basedon dynamic consideration about the flight performance ofPET bottle rockets, theory about the optimum flight condi-tion for obtaining the flight distance and flight stability ofPET bottle rockets are analyzed. Based on the analysis re-sults, we approached optimization on flight performance byapplying Parameter Design Method. Parameter design is adesign method based on statistical theory that investigatesthe relationship between design parameters and the qualityof an object and minimizes its effect. It can be said that it isan extremely effective methodology for flight performanceevaluation of PET bottle rockets.

Published by IIAE. 2019 59

60 E. Toma, H. Tanaka and K. Kikuta

In this research, factors that influence the flight perfor-mance of PET bottle rockets were extracted, and flight con-ditions to obtain optimum maximum flight distance andflight stability were verified [3].

2. Dynamical consideration of PET bottle rockets2.1 Rocket’s equation of motion The equation thatdetermines the motion of the rocket thrust flight is expressedas follows [4].

mdvdt= F − R − mg sin θ (1)

vdθdt= −g cos θ

dmdt= −β (2)

dxdt= v cos θ

dydt= v sin θ (3)

Here,m: Mass of the rocket (including the mass of water)F: Thrust force R: Air resistance forceβ: Water discharge per unit timeg: Gravitational accelerationθ: Orbital angle of the rocketv: Speed of the rocketx, y: Rocket coordinates

2.2 Thrust due to water jetting The thrust of a PETbottle rocket includes one caused by water jetting and theother caused by the jetting out of air that occurs after thewater runs out. This thrust is generated when the pressureP0 in the bottle is larger than the atmospheric pressure Pa,and water is accelerated by the pressure difference (Fig. 2)[4] [5]. First, consider thrust obtained by squirting water.When water (density rho) is injected from the mouth of thecross-sectional area A to the rocket at the velocity u, themass dm of water ejected during the minute time dt is,

dm = ρAu · dt (4)

dm is injected backward from the rocket at the speed of u,so that the impulse received by the rocket F · dt is,

F · dt = dm · u (5)

Figure 2: Thrust model

By substituting Eq. (4) into Eq. (5) and rearranging it, weget the expression of thrust F of the rocket.

F = ρAu2 (6)

Here, if the cross sectional area A0 in the tank, the flow ve-locity u0, and that at the injection port are A and u, the fol-lowing expression is established from the Bernoulli equa-tion and Continuous equation which is the basic equation offluid dynamics.

12· ρu2 + Pa =

12· ρu2

0 + ρgh + P0 (7)

u · A = u0 · A0 (8)

Here, Pa is the atmospheric pressure, and P0 is the pressurein the pressure tank. g is the gravitational acceleration, andh is the height from the injection port to the water surface inthe tank. From these two equations, the following equationis obtained.

u =1√

1 −(

AA0

)2

√2 (P0 − Pa)ρ

+ 2gh (9)

If this is substituted into the expression of thrust F, thrust isgiven by the following equation.

F =2A

1 −(

AA0

)2 (P0 − Pa + ρgh) (10)

The pressure P0 in the pressure tank sharply decreases withthe ejection of water, but this change can be calculated byconsidering it as adiabatic change.

2.3 Thrust due to air jetting Compressed air still re-mains in the pressure tank at the time the water jetting hasended. By ejecting this air, the rocket will be further accel-erated with thrust [6]. Consider the case where the pressureand density of air flow out from the tanks with P0 and ρ0to the outside air with pressure Pa. Assuming that the pres-sure and density of the gas at the ejection port are P1 andρ1, respectively, the outflow velocity u1 at the ejection out-let can be obtained from the equation of compressive fluidas follows.

u1 =

√√√2γγ − 1

· P0

ρ0

1 −(

P1

P0

) γ−1γ

(11)

γ is the specific heat ratio of air, and if γ = 1.4 is substitutedand s rearranged, it is expressed by the following equation.

u1 =

√√7 · P0

ρ0

1 −(

P1

P0

) 27 (12)

Here, the pressure P1 at the jet port is not always equal tothe outside air pressure Pa. When the flow velocity u1of theinjected gas does not exceed the sound velocity, it can beconsidered that the jet pressure P1 is substantially equal to

IIAE Journal, Vol.7, No.2, 2019

Optimization of Flight Performance of PET Bottle Rockets by Integrated Analysis of Dynamics Simulation System and Parameter Design 61

Table 1: Basic formula

Value Unit Symbol Description

Various constants

3.141592654 - π π = 3.141592654 : Pi101300 N/m2 Pa Pa = 1.013 × 105 : Atmospheric pressure [N/m2]

1000 kg/m3 Dw Dw : Density of water [kg/m3]1.29 kg/m3 Dair Dair : Air density at 15◦C 1atm [kg/m3]1.4 γ γ = 1.4 : Specific heat ratio of air

287.03 J/kg·K R R = 287.03 : Air gas constant [J/kg·K]　 0.34 - C C = 0.34 : Drag coefficient of air resistance

0.528281788 - Kchoke Kchoke = (2/(γ + 1))(γ/(γ-1)) : Critical pressure coefficient

Varieties of rocket

0.006361725 m2 S S = π × (4.5 × 10−2)2 : Rocket body sectional area [m2]0.165 kg Mb Mb = 0.165 : Rocket mass [kg]0.18 m L L : Guide rail length [m]

6.36173E-05 m2 a a = π × (4.5 × 10−3)2 : Cross section of inje. port [m2]0.0015 m3 PVo PVo = 1.5 × 10−3 : Volume of fuel tank [m3]

Initial condition

506500 N/m2 Pstart Tank air pressure at launch [N/m2]283.2 K Temp Tank air temperature at launch [K]

6.23101662 kg/m3 Dair start Tank air density at launch [kg/m3]0.785398164 rad KAstart Launch angle [rad]（2π × θ/360◦）

0.5 kg Mw start Tank water mass at launch [kg]（at 10◦C）

the outside air pressure Pa. However, as the pressure differ-ence between the inside and the outside becomes large, evenif the flow velocity increases, it cannot exceed the soundspeed, so the jet pressure P1 does not decrease to the out-side air pressure Pa.

P1 =

(2γ + 1

) γγ−1

· P0 ≒ 0.528P0 > Pa (13)

Such a state is called choking (flow blocking). The thrust Fa

as a reaction of the momentum given to the jet air, like thecase of water jetting ρ1 ·A·u2

1 and the following equation canbe obtained by using the “Adiabatic expansion” equation.

Fa = ρ1 · A · u21 = 7 · A · P1 ·

(

P0

P1

) 27

− 1

(14)

In addition, when the pressure of the air injection port islarger than the atmospheric pressure, thrust Fp (referred toas a pressure thrust) due to a pressure difference is addedthereto.

Fp = A (P1 − Pa) (15)

In summary, thrust F can be obtained by the following equa-tion.

F = Fa + Fp

= 7AP1 ·(

P0

P1

) 27

− 1

+ A · (P1 − Pa)(16)

Here, the following conditional expression can be substi-tuted for P1.When 0.528 × P0 ≧ Pa, P1 = 0.528 × P0,When 0.528 × P0 < Pa， P1 = Pa

3. Flight trajectory simulation programBased on the solution obtained from the above differentialequation, we developed a simulator with a numerical calcu-

lation program that applied the macro function of spread-sheet software (Excel). The developed simulator was ap-plied to extract the optimal flight conditions for gettingflight distance and stability of a PET bottle rocket [7]. Ta-ble 1 shows the conditions and basic formulas in the numer-ical calculation program. It is possible to input flight condi-tions of a PET bottle rocket on the screen and simulate theoutput data of flight distance and altitude.

4. Flight characteristics analysis

The PET bottle rocket is a clear example in which the flightmotion is established based on the law of action and reac-tion, and theoretical considerations derived from the law ofimpulse and momentum conservation [8]. In general, exper-imental results have been reported that the amount of wateris 1/3 of the bottle volume, the launch angle is 55◦, and theflight distance can be extended if the airframe is made aslight as possible. However, the theoretical basis is not soclear. In order to clarify this, we examined how each condi-tion affects the rocket’s motion.

By changing each initial condition, we estimated and an-alyzed the optimum condition to extend the flight distance.

4.1 Force acting on rocket and flight distanceFig. 3, Fig. 4 shows the relationship between launch anglewater volume and flight distance of rocket. As can be seenfrom the graph, the trajectory gradually faces downwardeven if it is launched upward at a large angle. This phe-nomenon of crouching is called Gravity turn.

Fig. 5 shows various forces acting on rocket flight. Therocket’s traveling direction is determined by the resultantforce (F7), which is lower than the direction of thrust, andits orbit gradually changes downward. Also, from the graphof Fig. 4, the optimum water rate ratio for obtaining theflight distance for the bottle volume (1.5L and 1.0L) at thelaunch angle of 45◦ and the initial pressure of 5atm is 1.5L:35% ( About 530ml) and 1.0L: 40% (about 400ml) [9].



Figure 3: Launch angle and flight distance

Figure 4: Water volume and flight distance

Figure 5: Force acting on the rocket

4.2 Time change of bottle internal pressure, injectionvelocity, thrust Fig. 6 shows the calculation results ofthe time change of the internal pressure when the initialpressure is 5atm (gauge pressure), the launching angle is45◦, and the amount of water added is changed from 200mlto 500ml using 1.5L bottle. From the internal pressure, wecan calculate the volume of compressed air, and the injec-tion velocity of water and air. Since the thrust received bythe rocket is the product of the mass flow rate of the injectedfluid and the injection speed, the time change of the thrustcan also be determined. Fig. 7 shows the injection velocity,and Fig. 8 shows the thrust calculation results. It can be seenthat the flight of the PET bottle rocket relies more heavilyon water injection than air injection. It can be seen that in

Figure 6: Time change of internal pressure

Figure 7: Time change of injection velocity

Figure 8: Time change of thrust

order to obtain a large momentum, it is better to increase theinitial pressure to increase the injection speed, or to increasethe amount of water introduced to extend the injection time[10].

4.3 Initial pressure of bottle internal air If the inter-nal pressure is expanded to the atmospheric pressure beforeall the water is injected, all water in the bottle will not be re-leased and thrust cannot be obtained. Therefore, the initialpressure must be such that the internal pressure becomesgreater than the atmospheric pressure when all the water isinjected. The higher the initial pressure, the faster the in-jection velocity, the greater the thrust. Therefore, in orderto fly away the PET bottle rocket, it is considered that theinitial pressure should be made as high as possible [11].

4.4 Amount of water in the bottle In the PET bot-tle rocket, water injection is important in flying. Actually,even if it is skipped with compressed air without putting inwater, it is not able to obtain a large thrust, and it does notgive much flying distance, so be sure to put water in the bot-tle. Therefore, it can be imagined that there is an optimumamount of water to fly far. The initial pressure changes the



Figure 9: Changes in water volume and flight distance

Figure 10: Changes in water volume and flight distance

injection speed and the injection time, and the body weightdetermines the proportion of the weight of water to the to-tal weight. From this, it is thought that the optimum valuechanges with the initial pressure and the weight of the air-craft. We calculated the change in the flight distance whenthe water volume was changed from 7% (100 ml) to 55%(800 ml) of the bottle volume (Fig. 9). The initial pressureis fixed at 5atm and the weight of the aircraft is from 100gto 500g, calculation result is shown in Fig. 10. The verticalline in the figure shows the condition that the carry distanceis the maximum. When the initial weight is increased withthe aircraft weight constant, the flight distance has increasedoverall and the optimum amount of water is increasing.

In other words, it can be estimated that the ratio of 30%to 40% of the bottle volume is the optimum amount of waterfor flying away.

4.5 Aircraft weight and flight distance The momen-tum that the rocket obtains depends on the thrust, but theflight speed after the end of injection varies with the weightof the aircraft. Fig. 11 shows the calculated flight distancewhen the water volume is fixed at 30% of the bottle volumeand the initial pressure is 3 to 7atm. Fig. 12 shows the calcu-lation results when the initial pressure is 7atm and the watervolume is 20% to 50% of the bottle volume. From the cal-culation results, it was found that the optimum body weightis in the range of 100g to 130g. Therefore, in order to flyfurther, it can be estimated that the attachment should be assimple as possible and the aircraft should be lightweight.

4.6 Launch angle and flight distance It has been re-ported that the PET bottle rocket has an increase in flightdistance at a launch angle of about 55◦, but the scientificbasis of the factor has not been clarified yet. Fig. 13 and 14shows the variation of the flight distance against the launch

Figure 11: Changes in empty rocket mass and flight distance

Figure 12: Changes in empty rocket mass and flight distance

angle when the ratio of water to bottle volume is set to30% and 40%, respectively, and the initial pressure of com-pressed air is set to 3 to 7atm.

It can be estimated that the optimal launch angle for ob-taining flight distance is about 45◦ to 55◦. The rocket isconsidered to extend the flight distance by gliding duringthe descent. In order to obtain the optimum launch angleclearly, it is necessary to examine in detail the flight move-ment after the end of water injection.

Figure 13: Changes in launching angle and flight distance

Table 2 shows the analysis results of the optimization forobtaining the maximum flight distance in the flight charac-teristics of the PET bottle rocket mentioned above.

5. Optimization of flight performance by ParameterDesign

In the previous section, we analyzed the flight characteris-tics of PET bottle rockets. In this section, we have extractedthe factors that affect flight performance by parameter de-sign, and verified the flight conditions to obtain the optimal



Figure 14: Changes in launching angle and flight distance

Table 2: Results of analysis

Parameters Optimum valueBottle volume 1.5L

Bottle diameter ϕ 90mmLaunch angle 45◦∼55◦

Water volume 30%∼40%Initial pressure 5atm∼7atm

Launch nozzle dia. ϕ 9mm

Table 3: Control Factor Levels

No. Parameter Level 1 Level 2 Level 31 BOTT.DIA ϕ 80mm ϕ80mm ϕ90mm2 LAUN.ANG. 35deg 45deg 55deg3 AIR PRES. 3atmg 5atmg 7atmg4 WAT.VOL. 300ml 500ml 700ml

maximum flight distance and flight stability using a simula-tor. [12] [13].

5.1 Overview of Parameter Design The concept ofRobust design in quality engineering is shown in Fig. 15.Robust design is an idea that improves technology to bringit closer to what it should be, and Robust means stability inquality engineering. Parameter Design is one of the centralmethods of robust design, which is a method of evaluatingthe functionality and determining the parameter value of thesystem.

Parameters are design constants and components of thesystem, and are selected as control factors in parameter de-sign experiments. Improve robustness by intentionally gen-erating variations with noise factors among combinations ofsystem parameters, and optimizing the level of strong con-

Figure 15: Robust design

trol factors that can counter the variations [14] [15].

5.2 Target function and quality characteristics Inparameter design, it is important how to grasp the targetfunction. The target function refers to the role that the sys-tem should play, and the quality characteristics are classi-fied into the following four types.

By picking up the target function, the design informationcan be made use of in other similar systems and the versa-tility of the technology widens.(a) Preferably small characteristics (nonnegative andsmaller the better)(b) Preferably large characteristics (nonnegative andlarger the better)(c) Preferably target characteristics (with target value)(d) Zero preferably target characteristic (zero is the targetvalue)

In this research, the maximum flight distance and flightstability of a PET bottle rocket is the target function, andas the quality characteristic for extracting the optimum pa-rameter condition for obtaining it, each characteristic valueis set as the horizontal reaching distance Preferably largecharacteristic and Preferably target characteristic were ap-plied [16] [17]. For the quality characterization in this re-search, the following data analysis tool was adopted JUSE.StatWorks / V5 Quality Engineering Edition (Nikka GikenCo., Ltd.)

5.3 Optimization of maximum flying distance by“Preferably large characteristics”5.3.1 Determination of level table and various factorsTable 3 and Table 4 show the level table of this experiment

Table 4: Preferably Large Characteristic

No. 1 2 3 4 Noise FactorEXP.No BOTT.DIA LAUN.ANG. AIR PRES. WAT.VOL. N1 N2

1 ϕ 80mm 35deg 3atmg 300ml 21.1 21.92 ϕ 80mm 45deg 5atmg 500ml 44.8 46.33 ϕ 80mm 55deg 7atmg 700ml 29.3 29.14 ϕ 80mm 35deg 5atmg 700ml 8.1 8.45 ϕ 80mm 45deg 7atmg 300ml 64.1 666 ϕ 80mm 55deg 3atmg 500ml 13.6 13.77 ϕ 90mm 35deg 7atmg 500ml 89.2 73.68 ϕ 90mm 45deg 3atmg 700ml 13.8 14.29 ϕ 90mm 55deg 5atmg 300ml 47.4 48.8



Figure 16: Output Data Plot

Figure 17: Effect Diagram

and the experiment plan table. From this level table, Or-thogonal table (L9) which is a statistical tool of optimiza-tion method was created to formulate an experiment plan.Orthogonal table is a table that defines the assignment suchthat combinations of levels of arbitrary factors appear thesame number of times in combinations of experimental lev-els. The number of experiments is determined by the scaleof the orthogonal table and is represented by the experi-ment number shown at the left end of the orthogonal table[18] [19].

The number of experiments this time is 9× 2 = 18 times,and the upper row in the horizontal direction represents thetype of control factor allocated. The numerical values andletters listed thereunder represent the level of each controlfactor. The merit of using an orthogonal table is the reduc-tion in the number of experiments. When the orthogonal ta-ble L 9 is not used, the number of times of the experiment is34 = 81 times. Four control factors (bottle diameter, launchangle, air pressure, water volume) were extracted. Table 5shows the noise factors extracted in this study, it is takenas the temperature of water to be put in the bottle (normal

Table 5: Combination of Noise Factor Levels

No. Parameter N1 N21 water temp. N.TEMP（10◦C） H.TEMP（90◦C）

temperature 10◦C, high temperature 90◦C).Fig. 16 shows the output data of the flight distance by

simulation. From the output data, it can be seen that un-der certain conditions the difference in water temperature isa factor affecting the distance traveled. It can be inferredthat the weight of the aircraft changes due to the change inthe density of water depending on the temperature, whichaffects the flight distance.5.3.2 Calculation of SN ratio Table 6 shows thevalue of the SN ratio from the output data, and Table 7shows the calculation process of the SN ratio calculated.

The factor effect of a harmful component represents adispersion effect from an ideal function with a componentwhose output unintentionally changes. In the parameter de-sign, the measure of the evaluation is obtained by the SNratio, and ultimately the presence or absence of the effect is

Table 6: SN ratio

No. 1 2 3 4 [db]EXP.No. BOTT.DIA LAUN.ANG. AIR PRES. WAT.VOL. SN Ratio

1 ϕ80mm 35deg 3atmg 300ml 26.642 ϕ80mm 45deg 5atmg 500ml 33.173 ϕ80mm 55deg 7atmg 700ml 29.314 ϕ80mm 35deg 5atmg 700ml 18.335 ϕ80mm 45deg 7atmg 300ml 36.266 ϕ80mm 55deg 3atmg 500ml 22.707 ϕ90mm 35deg 7atmg 500ml 38.098 ϕ90mm 45deg 3atmg 700ml 22.929 ϕ90mm 55deg 5atmg 300ml 33.64



Table 7: Calculation

Data Num. Sum.Sq. Antilog [db]EXP.No. N Σ1/yˆ2 (1/Σyˆ2)/N SN Ratio

1 2 0.00000 0.002 26.642 2 0.00000 0 33.173 2 0.00000 0.001 29.314 2 0.00000 0.015 18.335 2 0.00000 0 36.266 2 0.00000 0.005 22.707 2 0.00000 0 38.098 2 0.00000 0.005 22.929 2 0.00000 0 33.64

judged on the factor effect diagram to obtain the optimumcondition. The calculation formula of the SN ratio in thepreferably large characteristic applied to each characteristicvalue is shown below [20]. SN ratio: an evaluation scale forminimizing variation, n: number of data, y: value of eachdata,SN ratio;

η = −10 log1n

1y2

1

+1y2

2

+1y2

3

+ · · · + 1y2

n

(17)

5.3.3 Factor effect and variance analysis Fig. 17shows the SN ratio for each control factor expressed in afactorial effect diagram. The factor effect table (Table 8)represents the influence of factors or factor combinationson characteristic values, and the diagram showing this isthe factor effect diagram. The meaning of this diagramcan be judged to be effective for the control factor whoseSN ratio is widened in the vertical direction to be effec-tive for extending the flight distance of the PET bottlerocket. In Fig. 17, the optimum level is the factor in thecircled portion, BOTT.DIA:ϕ90mm, LAUN. ANG:45deg,AIR PRESS:7atm, WAT. VOL:500ml.

In addition, the obtained characteristic value varies from

the average value for each level. This variation scale wasevaluated in the variance analysis table (Table 9). From thistable, it can be estimated that the influence degree to theflight distance is a high factor because the contribution ratioof the factors with large dispersion ratio (AIR PRESS. andWAT. VOL.) is also large.

Furthermore, it was also possible to estimate that thereexist factors with a high contribution rate other than the pa-rameters extracted in this experiment. This means that it isnecessary to add another parameter level to the optimizationfor obtaining the maximum flight distance.5.3.4 Optimization of maximum distance From thefactor effect diagram in the previous section, Table 10 showsthe combination of the optimum control factor level for ob-taining the maximum flight distance in this experimental re-gion with respect to the flight performance of PET bottlerockets.

Also, based on the comparison with the benchmark con-dition, since the optimum condition is a higher estimatedvalue, it can be judged that the orthogonal table experimentapplied in this experiment is reliable.

5.4 Optimization of flight stability by “Preferably tar-get characteristic”5.4.1 Determination of level table and various factorsTable 11 shows the experimental plan for this experiment.In this section, we evaluated the “Preferably target charac-teristic” with the target function of extracting the optimumflight condition concerning the flight stability of PET bottlerockets. The preferably target characteristic is a desirablecharacteristic evaluation as it is closer to the target value.Table 12 shows the noise factors extracted in this experi-ment, and it was set to 4 types of water temperature (normaltemperature 10◦, high temperature 90◦), duration (s), maxi-mum velocity (km/h) to be put in the bottle. Fig. 18 showsthe output data of the flight distance (normal temperature

Table 8: Effect Table

SN RatioTotal A.V.G 29.007 Factor Effect

No. Parameter Variance Level 1 Level 2 Level 31 BOTT.DIA 29.129 ϕ80mm ϕ90mm

-1.272 2.5442 LAUN.ANG. 7.656 35deg 45deg 55deg

-1.319 1.776 -0.4573 AIR PRES. 83.033 3atmg 5atmg 7atmg

-4.918 -0.63 5.5474 WAT.VOL. 68.355 300ml 500ml 700ml

3.176 2.314 -5.489

Table 9: Variance Analysis Table

Parameter Sum.Sq. Freedom Variance Var. ratio ContributionBOTT.DIA 29.129 1 29.129 1.249 1.568

LAUN.ANG. 15.312 2 7.656 0.328 0AIR PRES. 166.066 2 83.033 3.561 32.231WAT.VOL. 136.709 2 68.355 2.931 24.308Discrepancy 23.319 1 23.319 41.893

Total 370.535 8 46.317



Table 10: Maximum Flight Distance

Estimate GainBOTT.DIA LAUN.ANG. AIR PRES. WAT.VOL. SN Ratio

Optimal ϕ90mm 45deg 7atmg 500ml 41.188 16.515Benchmark ϕ80mm 35deg 3atmg 300ml 24.673

Figure 18: Output data Plot

10◦, high temperature 90◦), duration and maximum veloc-ity by simulation.

It is understood from the output data that flight distanceand Duration and maximum velocity are deeply involved inflight stability.5.4.2 Calculation of SN ratio and Sensitivity Ta-ble 13 shows the SN ratio and the sensitivity value from theoutput data, and Table 14 shows the calculation process ofthe calculated SN ratio and sensitivity. The preferably tar-get characteristic is a characteristic that there is a finite tar-get value and it cannot be smaller or larger than the targetvalue. Analysis of the preferably target characteristic takestwo steps: robustness (stability) improvement and outputtuning work. The SN ratio is used for improving the robust-ness (stability), and sensitivity is used in the tuning work ofthe output. From the factor effect diagram, select the high-est factor level of the SN ratio and evaluate the stability ofthe system. Factor of SN Ratio and Sensitivity use the ef-fect diagram to extract factors that have a large influence onthe SN ratio and have as little influence on the sensitivityas possible, and finally adjust the difference from the targetvalue. The calculation formula of the SN ratio and sensi-

tivity in the preferably target characteristic applied to eachcharacteristic value is shown below [20]. n: number of data,y: value of each data,Mean variation；

S m =1n

(y1 + y2 + y3 + · · · + yn)2 (18)

Error dispersion；

Ve =1

n − 1

(y2

1 + y22 + y

23 + · · · + y2

n − S m

)(19)

SN ratio；

η = 10 log1n (S m − Ve)

Ve(20)

Sensitivity；

S = 10 logS m − Ve

n(21)

Table 11: Preferably Target Characteristic

No. 1 2 3 4 Noise FactorEXP.No BOTT.DIA LAUN.ANG. AIR PRES. WAT.VOL. N1 N2 N3 N4

1 ϕ80mm 35deg 3atmg 300ml 21.1 21.9 54.5 1.92 ϕ80mm 45deg 5atmg 500ml 44.8 46.3 79.7 3.13 ϕ80mm 55deg 7atmg 700ml 29.3 29.1 48.3 2.84 ϕ80mm 35deg 5atmg 700ml 8.1 8.4 35.9 1.15 ϕ80mm 45deg 7atmg 300ml 64.1 66 107.6 46 ϕ80mm 55deg 3atmg 500ml 13.6 13.7 35.2 1.97 ϕ90mm 35deg 7atmg 500ml 89.2 73.6 144.7 4.18 ϕ90mm 45deg 3atmg 700ml 13.8 14.2 53.9 1.59 ϕ90mm 55deg 5atmg 300ml 47.4 48.8 92.9 4.1

Table 12: Combination of Noise Factor Levels

No. Parameter N1 N2 N3 N41 Mixing N.TEMP（10◦C） H.TEMP（90◦C） MAX.SPEED DURATION



Table 13: SN Ratio / Sensitivity

No. 1 2 3 4 [db]EXP.No. BOTT.DIA LAUN.ANG. AIR PRES. WAT.VOL. SN Ratio Sensitivity

1 ϕ80mm 35deg 3atmg 300ml 0.20 24.852 ϕ80mm 45deg 5atmg 500ml 2.23 43.483 ϕ80mm 55deg 7atmg 700ml 2.77 27.384 ϕ80mm 35deg 5atmg 700ml -2.97 13.385 ϕ80mm 45deg 7atmg 300ml 2.45 60.436 ϕ80mm 55deg 3atmg 500ml 0.39 16.107 ϕ90mm 35deg 7atmg 500ml 1.93 77.908 ϕ90mm 45deg 3atmg 700ml -2.32 20.859 ϕ90mm 55deg 5atmg 300ml 1.83 48.30

Table 14: Calculation

Sensitivity Data Num. Sum.Sq. Freedom Variance Antilog [db]EXP.No. S N’ ST Sm Se fe Ve (Sm-Ve)/N’/VN’ SN Ratio

1 24.85 4 3898.68 2470.09 1428.59 3 476.197 1.047 0.1992 43.48 4 10512.43 7560.303 2952.127 3 984.042 1.671 2.2293 27.38 4 4046.03 2997.563 1048.468 3 349.489 1.894 2.7744 13.38 4 1426.19 715.563 710.628 3 236.876 0.505 -2.9655 60.42 4 20058.57 14604.722 5453.848 3 1817.949 1.758 2.4516 16.1 4 1615.3 1036.84 578.46 3 192.82 1.094 0.3917 77.9 4 34328.5 24273.64 10054.86 3 3351.62 1.561 1.9338 20.85 4 3299.54 1738.89 1560.65 3 520.217 0.586 -2.3249 48.3 4 13275.42 9331.56 3943.86 3 1314.62 1.525 1.831

5.4.3 Factor effect and variance analysis Fig. 19shows the SN ratio and sensitivity for each control factorexpressed in a factor effect diagram. The factor effect table(Table 15) shows the influence of factors or factor combina-tions on characteristic values, and the diagram showing thisis the factor effect diagram. The significance of this diagramwas determined to be a factor that has a large influence onthe SN ratio and does not affect the sensitivity and that thelevel is highly significant for the flight stability performanceof PET bottle rockets.

In Fig. 19, the optimum level is the factor BOTT.DIA : ϕ80mm, LAUN. ANG: 55deg, AIR PRESS: 7atm,WAT.VOL : 300ml . As for the diameter of the bottle, it canbe inferred that a smaller diameter is more advantageous asa condition for obtaining flight stability. In addition, the ob-tained characteristic value varies from the average value foreach level. This variation scale was evaluated in the vari-

ance analysis table (Table 16).From this table, it can be inferred that AIR PRESS. and

WAT. VOL., which are large dispersion ratio factors, havehigh contribution ratios, so that the degree of influence onthe flight stability performance is a high factor.

Furthermore, it was also possible to estimate that thereexist factors with a high contribution rate other than the pa-rameters extracted in this experiment. This means that itis necessary to add another parameter for optimization toobtain flight stability.5.4.4 Optimization of flight stability From the fac-tor effect diagram in the previous section, Table 17 showsthe optimum combinations of control factor levels for ob-taining flight stability in the experimental range with respectto the flight performance of PET bottle rockets. In addi-tion, from the comparison with the benchmark condition,the optimum condition is a higher estimated value, which is

Figure 19: Effect Diagram



Table 15: Effect Table

SN RatioTotal A.V.G 0.724 Factor Effect

No. Parameter Variance Level 1 Level 2 Level 31 BOTT.DIA 0.268 ϕ80mm ϕ90mm

0.122 -0.2442 LAUN.ANG. 2.842 35deg 45deg 55deg

-1.002 0.061 0.9413 AIR PRES. 6.88 3atmg 5atmg 7atmg

-1.302 -0.359 1.6624 WAT.VOL. 5.494 300ml 500ml 700ml

0.769 0.793 -1.563

Table 16: Variance Analysis Table

Parameter Sum.Sq. Freedom Variance Var. ratio ContributionBOTT.DIA 0.268 1 0.268 0.057 0

LAUN.ANG. 5.684 2 2.842 0.601 0AIR PRES. 13.759 2 6.88 1.456 12.162WAT.VOL. 10.989 2 5.494 1.163 4.342Discrepancy 4.725 1 4.725 83.496

Total 35.426 8 4.428

a desirable condition regarding the stability of the flight tra-jectory [20]. Therefore, it can be judged that the orthogonaltable experiment applied in this experiment is reliable.

6. Conclusion

In this study, we developed a simulator based on a numer-ical calculation program based on dynamic considerationabout flight performance of PET bottle rockets, and cal-culated time change of internal pressure, water and com-pressed air injection speed by calculation. Next, consider-ing the initial conditions that can be set in the PET bottlerocket, we examine how the flight distance changes whenchanging them. Then, by calculating the equation of mo-tion by simulation, the optimum condition for obtaining themaximum flying distance was estimated.

As a result, it has been found optimal that the initial pres-sure is 5 to 7atm, the water volume is 30 to 40% of the bottle

volume (1.5 L), the aircraft weight is 100 to 130g, and thelaunch angle is 45 to 55degrees.

In addition, in the previous section, based on the analysisresult, the extraction of factors that influence the flight per-formance of PET bottle rockets by applying the ParameterDesign method, and the extraction of the factors affectingflight performance to obtain the optimum maximum flightdistance and flight stability were verified. When compar-ing the analysis result of the flight characteristics with theevaluation result of the optimum condition based on the pa-rameter design, almost the same results were obtained (Ta-ble 18). However, since various conditions can be set in thePET bottle rocket, these conditions are not independent oneby one but are closely related to each other. Therefore, it isnecessary to set so as achieve the best balance. Furthermore,it was possible to verify that it is necessary to add anotherparameter level to the optimization for obtaining the max-

Table 17: Flight Stability

Estimate GainBOTT.DIA LAUN.ANG. AIR PRES. WAT.VOL. SN Ratio Sensitivity SN Ratio Sensitivity

Optimal ϕ80mm 55deg 7atmg 300ml 4.219 33.902 4.908 8.304Benchmark ϕ80mm 35deg 3atmg 300ml -0.689 25.597

Table 18: Summary of optimization analysis results

Data analysis of Optimization ofmaximum flying distance maximum flight distance and stability

Preferably Large Preferably TargetParameters Optimum value Characteristic Characteristic

Bottle volume 1.5L 1.5L 1.0LBottle diameter ϕ90mm ϕ90mm ϕ80mmLaunch angle 45◦∼55◦ 45◦ 55◦

Water volume 30%∼40% 30%(500ml) 30%(300ml)Initial pressure 5atm∼7atm 7atm 7atm

Launch nozzle dia. ϕ9mm ϕ9mm ϕ9mm



imum flight distance and flight stability from the analysisresult of variance in the evaluation of flight conditions byparameter design. For example, as parameters not coveredin this research, the size and shape of the tail fin and its at-tachment position, the length of the rocket body, etc. can bementioned [20].

If these are changed, it can be presumed that the lift, drag,center of gravity and aerodynamic center position changein addition to the weight of the aircraft, which will affectflight stability and flight after injection. Thus, from the sci-entific knowledge obtained on the flight characteristics ofPET bottle rockets, it turned out to be an extremely compli-cated flight system.

As future research topics, we plan to add parameter con-ditions related to flight and further analyze the error be-tween the simulation analysis result and the demonstrationexperiment result. We also acquire hydrodynamic demon-stration experiment data on flight performance by analyzingthe influence on flight stability due to vibration of the rocketbody due to the behavior of water in the bottle which is therocket propellant, and we are planning to further explore thechemistry research.

References

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Eiji Toma (Member) He was born inHokkaido, Japan, on December 17, 1959.When he was engaged as a manufacturingengineer at an automobile parts manufacturer,he acquired a “Professional Engineer (P.E.jp)”and “APEC engineer” in 2018, and is presentlya professor at National Institute of Technology,Tomakomai College. He has worked on research

of the quality engineering. He is member of IIAE.

Hiroshi Tanaka (Non-member) He is a Pro-fessor in the Department of Mechanical Engi-neering at Aichi Institute of Technology, Japan.He received his Ph.D from Nagoya Universityand worked as a process engineer in automobileparts company for 20 years. His research inter-ests include environment friendly manufacturingprocess and quality engineering.



Kazushige Kikuta (Non-member) He is a pro-fessor in the Department of Mechanical En-gineering at National Institute of Technology,Tomakomai College, Japan. He received hisPh.D from Hokkaido University and his currentresearch activity is the energy management sys-tem construction for a house and a building in thecold climate region.


optimization of flight performance of pet bottle rockets

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