UBC Undergraduate Research

A8-3 Model Rocket Impulse Measurement Haw, Magnus 2009-05-11

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A8-3 Model Rocket Impulse MeasurementMagnus HawScience One 2008-09May 10, 2009AbstractAn experiment was performed to measure the impulse of the A8-3 Estes modelrocket engine. The measurements were obtained by mounting an engine vertically ontoa frame and measuring the changing weight of the assembly during the  ring periodfor six trials. Thrust was calculated by the di erence in measured weight from therest weight while accounting for the mass loss rate. The impulse of each of the sixthrust curves was averaged to  nd the  nal impulse measurement: 2.36   .16 N-s.Estimates of the average thrust and exhaust velocity were also calculated from theimpulse measurement.1 IntroductionThe purpose of this experiment is to measure the impulse of an A8-3 rocket engine. Theimpulse for a rocket engine, Ir, is de ned as:Ir =Z tf0T(t)dt =  p (1)where T(t) is the thrust of the rocket, tf is the thrust duration, and  p is the change inmomentum. The impulse is important because it gives a normalized value for the launchcapacity of the rocket and is used to classify rockets into di erent ranks (A, B, C class etc.with A class being the least powerful). Since the change in momentum cannot be measureddirectly, this experiment found the impulse by measuring the thrust of the engine as a functionof time.The thrust is the force (generated by mass out ux from the engine) that the engine exertson the rocket. The equation for thrust is derived in Equations 2-5. Given m & v are therocket mass and velocity and m' & v' are the exhaust mass and exhaust velocity:dpdt =ddt(mv) +ddt(m0v0) = 0 (2)0 = mdvdt + vdmdt + m0 dv0dt + v0 dm0dt : (3)Since the exhaust experiences no acceleration after being vented, dv0dt = 0 and dm0dt =  dmdtbecause the total mass of the rocket/exhaust system is conserved:1mdvdt = dmdt (v0  v) (4)T(t) =  dmdt u (5)where u is the velocity of the exhaust relative to the rocket [3]. This experiment seeks tomeasure thrust by measuring the changing weight during the thrusting period. To  nd T(t),the rest weight is subtracted from the measured weight (Equation 6).T(ti) = W(ti)  M(ti)   g: (6)The mass loss rate must also be modeled because it changes the rest mass of the rocket;however, since the burn time is so short (0.8 s) and the mass loss rate is quite small (4 g/s),this experiment uses a linear approximation for M(t) (Equation 7).M(t) = M0    Mmaxtf  t: (7)After  nding T(ti), the general expression for the impulse Ir at a sample rate 1 t is:Ir  tfX0[W(ti)  M(ti)   g]  tFigure 1: Vertical and Horizontal Frames: These photos show the vertical and horizontalframe structures, respectively. The image on the left shows the vertical frame (20x20x30 cm)and the pre-ignition setup (Trial 6). The photo on the right shows the horizontal frame(10x20x20cm) during ignition (Trial 5).22 MethodsThis experiment measured the thrust of the A8-3 model rocket engine by securing rockets toa frame and measuring the weight of the rocket-frame system during ignition. The supportframe was constructed out of half-inch pvc pipe with angle braces mounted on the top tosecure the engine (see Figure 1).For each trial, the engine would be mounted onto the frame with screws via the anglebraces and adjusted until the nozzle was pointed vertically upward. Then the system wouldbe placed on the digital scale and the rocket would be  red. A digital camera was set up torecord the scale display values.This procedure was repeated for seven vertical trials and two horizontal trials. Thehorizontal trials used a di erent frame con guration (see Figure 1) with the rocket set uphorizontally and were designed to measure the mass loss rate of the rocket. However, due tothe induced torques on the frame, this setup was unable to measure the mass loss rate. The rst vertical trial also failed because the camera was incorrectly set up.Additionally, the frame rate of the digital camera was tested against WWV time (standardtime) to determine whether the factory value (15 hz) was accurate. This was done by usingthe camera to record WWV time over 15 minutes and comparing the two elapsed times.3 Results and Discussion3.1 Thrust as a function of timeThe data extracted from the six usable vertical trials (2, 3, 6, 7, 8, & 9) was reduced accordingto Equation 6 to calculate the thrust over time (Figure 2).The general shape of the thrust curve is a smoothed peak skewed right. The maximumthrust recorded was 5.92 N at a time of 0.4 s and the thrust duration, tf, is 1.00 s. Thesecurves di er from previously measured A8-3 thrust curves in that they do not have a regionof constant thrust after the initial peak [1]. Additionally, the peak thrust is signi cantly less(6 N versus 10 N) and occurs at a di erent time (0.4 s versus 0.2 s). Furthermore, the thrustduration (tf) is longer than previous measurements (1.0 s versus 0.7 s) [1]. This di erencemay be caused by the response time of the scale's digital display and the sampling rate.Several distinct sources of error a ect these measurements: scale error, non-vertical com-ponent error, timing error and extraction error. The extraction error is the main source oferror and is the uncertainty associated with reading the values on the scale display. Thedisplay changed so quickly that the previous values would not have fully faded by the nextvideo frame. This problem created multiple number overlaps for the ones digit making itdi cult or impossible to identify the actual value. Since the ones digit was highly uncertainthe extraction error added an uncertainty of   9 grams. The timing error analysis comparingthe camera time and WWV found that the camera takes 14.98991 frames/s compared to theproduct value of 15.0 frames/s; over the thrust period this di erence is negligible (< .1%error). The other errors: scale error (<1% error) and the non-vertical component relativeerror(<0.1%) were also negligible in comparison to the extraction error. Summing theseerrors in quadrature gives a total relative error of 3.3% for each data point (Figure 2).3Figure 2: Thrust Measurement: All curves except Trial 9 follow the same trend. Theerror bars represent 3.3% error. The uncertainty in timing is insigni cant (< 10 4 s/frame)and is not shown.3.2 Impulse MeasurementThe impulse of each trial was calculated using  nite di erence approximation (midpoint rule).The individual impulses and the 95% con dence interval are shown in the following table.Impulse DataImpulse (N-s)Trial 2 2.29Trial 3 2.48Trial 6 2.33Trial 7 2.61Trial 8 2.46Trial 9 2.01Average Impulse 2.3695% error bars   0.16The con dence interval for this measurement encompasses both the factory value (2.5N-s) and the value from T.Dooling (2.32 N-s) [1,2]. This demonstrates that both the meanimpulse and the error are not unreasonable measurements because they agree with previousvalues. Given this agreement and the small relative error (7%), it can be concluded thatthe method used in this experiment is adequate for measuring the impulse of a model rocketengine.43.3 Average Thrust and Exhaust VelocityThe average thrust of the engine is simply the impulse divided by the thrust duration:Tavg = 2:36Ns1:00s = 2:36N. Additionally, an order of magnitude approximation for the exhaustvelocity can be obtained from this experiment by combining Equations 1 and 4. Assumingthat the relative exhaust velocity, u, is constant and using the average propellant mass, m = 3:6 g: u  Z tf0dmdt dt = Ir (8)juj = I m   650m=s (9)The fact that the exhaust velocity of a hobby rocket is almost twice the speed of sound givesone an appreciation of the physics involved in a rocket engine.4 ConclusionThis measurement (2.36   .16 N-s) is a good estimate of the impulse of an A8-3 Estesrocket. If higher accuracy is needed, the most e cient way to reduce experimental error is toincorporate a data logger into the scale. This would increase the sample rate and remove theextraction error. This improvement might allow a conclusion to be drawn on which impulsevalue, Estes (2.5 N-s) or T. Dooling (2.32 N-s), is more accurate [1,2]. However, since theengines have a signi cant population variation in impulse (  .25 N-s), as measured by Estes,this could prove an unnecessary expenditure; additionally, measurement of higher poweredengines would require signi cantly better equipment [2].5 References[1] Dooling, T.A. An Eight-Parameter Function for Simulating Model Rocket Engine ThrustCurves. Phys. Teach. 45, 280- 283 (2007).[2] Estes Industries. Model Rocket Engine and Igniter Instructions. (2009).[3] Nelson, R.A., Wilson, M.E. Mathematical Analysis of a Model Rocket Trajectory Part I:The powered phase. Phys. Teach. 14, 150- 161 (1976).5


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