Piston Pressure and Massive Models

R&D Report for NARAM 47
August 2005
Bumbling Brothers Flying Circus, NAR Team 011
Robert Alway and Peter Alway

Summary

Our experience with eggloft contest flying had led us to believe that long pistons are advantageous for heavier models. To examine this idea, we tested floating-head, zero-volume piston launchers of varying lengths (control flights with no piston, 9-inch pistons, 15-inch pistons, and 30-inch pistons) on models of varying masses (66 grams, 98 grams, 125.5 grams, and 191.5 grams). We evaluated flight performance with two-station altitude tracking and digital video photography of the launches.

Using commercially available simulation software, we determined the impulse contribution of the piston by simulating flights with engines having incremental increases in impulse, and comparing those simulations with the tracking data. From the impulse contribution, we could calculate an average piston pressure. We analyzed the video data to measure launch speed and acceleration on the piston.  From these we could calculate a redundant figure for the impulse contribution of the piston, as well as pressure inside the piston. By using these two independent methods, we were able to confirm our conclusions. 

We also reviewed previous works on pistons, and analyzed some of the published data in terms of impulse and piston pressure

We saw that pistons produced a greater percentage increase in altitude with more massive models than with lighter ones. We also saw that pistons contributed more impulse to more massive models than to lighter ones. We attribute this to greater time on the piston during launches of heavier models. We also saw that longer pistons gave a greater performance enhancement for the models tested. We attribute this to the fact that massive models spend more time on the piston, allowing the model to accumulate more impulse from a given, and possibly accumulate more pressure as well.

Piston Pressure and Massive Models

R&D Report for NARAM 47
August 2005
Bumbling Brothers Flying Circus, NAR Team 011
Robert Alway and Peter Alway

Introduction

Robert H. Goddard patented one of the earliest methods of boosting the launch of a rocket by capturing the energy of exhaust gas before it reaches optimum thrust (US Patent No. 2,307,125, "Launching Apparatus for Rocket Craft," Robert H. Goddard, Filed Dec. 9, 1940, issued Jan. 5, 1945). His method used turbines to power what was essentially a rack-and-pinion driven elevator for the rocket. While this scheme looks preposterous in hindsight, the fundamental idea of capturing pre-flight exhaust had merit. In the late 1950's, Atlantic Research Corporation successfully applied the closed breech launcher to its Arcas sounding rocket, using the pressure of exhaust gas to directly propel the rocket forward.  This system saw many years of use (Rockets of the World, Peter Alway, Saturn Press, 1999, pp. 132-135).  In the early 1970’s, contest modelers introduced the zero-volume piston launcher to hobby rocketry with great success. In the 1980's Chuck Weiss and Jeff Vincent refined the zero-volume piston launcher by introducing the floating-head piston. Our team, the Bumbling Brothers Flying Circus, has had considerable success in NAR competition flying egglofters from long, floating-head pistons in a tower.

A few questions, and bits of common wisdom worth questioning, have come up in the course of idle conversations at the contest range. How much do pistons really help? Why does everyone else fly with shorter pistons, when it seems like longer ones will do more good? What is the actual pressure inside the piston? The last question got us started on our project, though the other questions have proved to be connected.

We were aware that pistons have been studied in R&D reports in the past, so we set about to search the literature as best we could, to see if our questions had been answered. Our sources were a stack of technical papers available from NARTS, old Model Rocketeer magazines, and those contest R&D reports we found indexed on the Internet. From these, we learned the following background:

Trip Barber laid out the basic physics of the piston launcher in 1974 ("Pressurization Effect Launchers," Trip Barber, Model Rocketeer, Vol. XVI, No. 6, July 1974, pp. 14-16) but his theoretical paper gave no firm numbers beyond a suggestion that 3 psi was a likely piston pressure.

Thoelen, Bauer, and Porzio ("Optimization of the Zero Volume Piston Launcher," Robert E. Tholen, Thomas J. Bauer, and Paul A Porzio, NAR Technical Review, Volume 2, Fall, 1974, pp. 3-19) wrote an extensive paper on the optimization of piston launchers. While they compiled huge amounts of data from roughly 100 flights, close scrutiny of their data shows a critical flaw—almost all flights appeared to use engines with too-short ejection delays. Re-creating the flights in Rocsim confirmed our suspicion. Few of their flights could have reached apogee, and variation in ejection time would have been a major source of error. 

Essentially their data showed unambiguously that pistons work, but their conclusion that the altitude benefit of a piston is about 25% seems tainted by their unfortunate choice of ejection delays, and their conclusion that 12-inch pistons are optimal is similarly questionable.

Two papers by Weiss and Vincent were available online. The first ("The Floating Head Piston Launcher," Research and Development report submitted at NARAM 28, August 1986, by the Odd Couple Team, T-085, Chuck Weiss and Jeff Vincent, http://members.acmenet.net/~jvincent/RandD/fhp1/fhp1.html) describes the floating head piston. By measuring velocity shortly after leaving the piston, Weiss and Vincent made a strong case for the advantages of their design. Their second paper ("Empirical Evaluation of Optimum Piston Tube Length for a Floating-Head Piston Launcher," Research and Development report submitted at NARAM-30, August 1988, by the Crunch Birds Team, T-471, Chuck Weiss and Jeff Vincent http://members.acmenet.net/~jvincent/RandD/fhp2/fhp2.html) provides detailed plots of velocity, position, and time for their piston-launched models. They showed that the effective piston acceleration occurred early in the engine burn, before any appreciable thrust that would have boosted a control model. 

The separation in time of piston effect and thrust means that it is possible to measure the piston effect in isolation—simplifying analysis of the piston’s function.

Their data also showed that for their lightweight models (16.5 grams on a 1/2A3-2T) the rocket achieved its piston-related acceleration within the first 9 inches of flight. The model did not decelerate after that point, but rather a coasted or accelerated slightly until the engine came up to full power and began to accelerate the model rapidly.

Weiss and Vincent concluded from this data that a piston longer than 9 inches would not improve performance for their models. They did suggest, however, that additional piston length might be advantageous as a safety precaution, for its effect in guiding the model. We found it notable that excess piston length, while not especially valuable for improving performance, did not seem to penalize performance. We interpret this observation to also mean that piston optimization is not so much a matter of finding an ideal length (not too short and not too long) but one of finding a minimum length that is adequate for capturing all gas-pressure benefit available.

None of the papers we located dealt with the case of heavier models, such as egglofters. We suspected that heavier models would benefit from longer pistons, because they are slower to accelerate, and would have the time to accumulate more gas, which could pressurize a larger piston fully. 

We also questioned the way results were expressed in the papers we found. We noticed that results were expressed either in altitude benefit or in launch velocity.  While altitude is the bottom line for evaluating a piston's worth, this figure only applies to a specific rocket with a specific mass, frontal area, drag coefficient, and engine. While the speed of a model off the piston is of some value, it is removed from the ultimate question of altitude, and is dependent on model mass. We feel that a better metric of piston performance would be the impulse benefit of the piston. That impulse would be independent of drag, upper stages, or thrust duration (the impulse boost would be the same for a B6-4 as a C6-7).  Knowing this bonus impulse would allow us to simulate piston-boosted flights in the future.

Objectives

In this project, we set out to answer these questions: 

•  Do the Bumbling Brothers' unusually long pistons really help heavy models, like our egglofters, perform?
•  What is the real impulse benefit of a piston launcher?
•  What is the pressure inside a piston?

Approach

Our practical question about the use of long pistons for heavy rockets suggests two independent variables for our experimental design, piston length and model weight. Our interest in impulse benefits and resulting altitude suggested measuring both altitude and motion at the pad. Altitude tracking put a practical limit on the number of flights we could make, since we would require the services of volunteers from a local NAR section for several hours. We chose to fly a matrix of three piston lengths (and one control of no piston) and four models of varying weights. Our raw data would be in the form of peak altitude tracking elevations and azimuths and video of the launches. From these we hoped to calculate the impulse contributions of pistons, launch velocities, and piston pressures

Equipment

Our models were Estes Baby Bertha kits, 1.637 inches in diameter.  We modified them by leaving out the engine hooks, building them with three fins, gluing their launch lugs to the fin roots, and by adding varying amounts of oil-based modeling clay (plasticene) into the nose cones: 

Model 0:  66.0 grams at launch (no clay in the nose)
Model 30:  98.0 grams at launch (30 grams of clay in the nose)
Model 60:  125.5 grams at launch (60 grams of clay in the nose, the mass of one egg)
Model 120:  191.5 grams at launch (120 grams of clay in the nose, the mass of two eggs) 

Figure 1 shows the components of our floating-head pistons.  The piston tubes consisted of lengths of generic Estes BT-20 (Totally Tubular white T-20 for 30-inch and 15-inch pistons, Red Arrow Hobbies BT-20 for 9-inch pistons). We glued a wound paper adapter ring in at the base of each tube as a stop. We machined the heads from epoxy for a smooth, sliding fit inside the piston. We set the head on top of a length of 1/4-inch carbon fiber tubing to hold it close to the engine at ignition, and to transmit the pressure force to the ground. We taped the engines, Estes B6-4's and B6-2's, into the models, with Ring Rocketry yellow tape, the tape also serving to give a tight fit into the top of the piston. We installed Estes Solar igniters with their standard plugs, and arranged their leads to leave the piston upward between the engine and the piston tube. We launched the control flights and the 9-inch and 15-inch piston flights from 1/8-inch rods, while we launched the 30-inch piston flights from a BMS Medalist tower. We modified the tower to accommodate a long piston by adding extended legs, and a blast deflector with a hole for the piston. The lengths and weights of the pistons were:

30-inch piston at 15 grams
15-inch piston at 8 grams
9-inch piston at 4.5 grams

Figure 1:  Piston Components

Figure 2:  Piston Operation

Figure 2 shows our version of the floating head piston in operation.  The first diagram shows the apparatus at rest, the second during piston operation when the inside of the piston is pressurized, and the third after the piston head has reached its stop.  The model has blown free, and the piston head has disengaged from the piston rod.  The piston in the illustration is much shorter than those we used. 

We measured altitudes with two theodolites, built from Trip Barber’s Triple Track plans ("Triple Track Tracker," Tracking Tech Report Collection, National Association of Rocketry Technical Service), of the sort commonly used for altitude competition. Tracking stations were 400 feet apart.  We collected and reduced the data after the fashion of NAR competition. We asked out trackers to track to apogee.

We recorded the launches with a Kodak DX4330 Digital Camera in video mode. The camera ran at 15 frames per second. Next to the launchers, we placed a vertical calibration mast consisting of  four 36-inch × 1/2-inch dowels held together with expended model rocket engines colored black. All pads were at the same distance from the camera as the vertical mast.

The three lightest models flew on Estes B6-4 engines, all from the same bulk pack. The heaviest model flew with B6-2's. Three were from the same package, the second non-piston flight used an engine from a second pack.

We reduced altitudes on the field with an HP-32 programmable calculator, and further reduced data on our home PC’s using Apogee Components Rocksim 5 simulation software and spreadsheet software. 

Facilities

We conducted our test flights at Plainwell Airport, in Plainwell Michigan, during a club launch of SMASH, NAR Section 500.

Budget

We already had the trackers, launchers, calculator, and computers. We purchased the following for this project:

4 Baby Bertha kits:  $25
16 B6-4 and B6-2 engines:  $30
12 piston tubes with paper rings: $25
Total equipment cost: $80

This does not include cost of travel between our homes (120 miles apart) and the launch field to discuss and perform the work, which may have added up to as much a $300.

Data

The altitude tracking data appears in Appendix B.

The raw video is not available online due to limited server space

Data Reduction

Calculating altitude

We reduced our elevation and azimuth data using both geodesic and vertical midpoint algorithms as recommended for NAR competition.  However, altitude data was just the beginning of calculations of the piston’s impulse contribution, and ultimately piston pressure.

Calculating piston impulse contribution from tracked altitude

We used Rocksim to find predicted altitudes for our control flights. We also simulated flights with modified B6 engines. These modified engines had extra impulse added to the beginning of their burn to simulate the impulse contribution of the piston. Our modified engines were designated B6XX-2 or B6XX-4, the XX indicating the percentage of the motor’s impulse that was added. For instance, a B610-4 had 10% more impulse (0.44 N-s) than Rocksim's standard 4.4 N-s B6-4. A B6-10-4 would have 10% (0.44 N-s) less impulse than Rocksim's standard B6-4. These predicted altitudes are presented in Appendix A.

We could then match our measured altitudes to the predicted altitudes to find the combined impulse of the engine and piston. To find the impulse in finer increments than 5% (0.22 N-s) we interpolated the results linearly. These results are presented in Appendix E.

Calculating piston impulse contribution from launch video

A second means of measuring a piston's impulse is to measure the model's momentum as it leaves the piston. Momentum (p = m × v, where p is momentum, m is mass, and v is velocity) and impulse (I = F × t, where I is impulse, F is average force, and t is elapsed time) are closely related—the momentum of a rocket is equal to the impulse applied to it since it was at rest. We would expect significant impulse from three effects—piston pressure, gravity, thrust, and drag:

m × v = p = I_piston + I_gravity + I_thrust+ I_drag

Weiss and Vincent showed that the effect of thrust is small during piston operation. Drag is low at launch, so we chose to neglect it as well:

m × v = I_piston + I_gravity

With the mass of the models known, we need only measure speed as the rocket leaves the piston.  Using flight video, we can measure the distance traveled in 1/15 second, and apply the formula v = Dx/Dt, where x is the distance traveled over time interval Dt. The impulse due to gravity is the weight of the model, mg, times time, t. This gives us a means of calculating piston impulse:

I_piston = m × Dx/Dt + mgt

This requires a pair of altitude (x) measurements at the exact moment the model clears the piston—a impossibility with a camera that shoots 15 frames per second—but we can get a pair of measurements close to that time.  If the interval is biased toward the time after separation, we would expect to measure a higher speed and higher impulse, and if the interval is biased toward the time before separation, we would expect to measure a lower speed and lower impulse. With that in mind, we can use these speeds as a check on our altitude data. The calculation is carried out in stages in our spreadsheets for each flight.

This method is much better, however, in analyzing existing data published in Weiss and Vincent's second paper, which has sufficiently fine time resolution to show a period of essentially constant speed.

Piston Pressure

We have two means at our disposal for measuring pressure inside the piston. Both build on the calculation of force from the piston, and use of the relationship F = P × A, where F is force on the rocket, P (uppercase) is pressure, and A is area at the base of the rocket. We calculate A with the formula A = pi × r², where r is the radius of the motor. To find force, we can either apply Newton’s second law directly, or use work-energy considerations.

Calculating piston pressure by means of acceleration in launch video

In the first method, we find the force using Newton’s second law, F = m × a, where m is the mass of the model, and a is acceleration. To calculate acceleration, we need two adjacent velocity measurements, so that we can measure the change in velocity over time:

a = Dv/Dt

where Dv is the change in velocity, and Dt is the time interval. This requires catching at least three images of the model in motion on the piston. Because we are interested in the physical state inside the piston, rather than the effect on the rocket’s flight, we are interested in the force on the combined mass of the piston and rocket.  Again, we neglect the forces of thrust and drag:

SF = m × a

F_piston + F_gravity = m × Dv / Dt

F_piston - mg = m × Dv / Dt

F_piston = (m × Dv / Dt) + mg

P = F_piston / A

P = ((m × Dv / Dt) + mg) / pr²

Our spreadsheet performs this calculation in stages. We carried out this calculation for three of our flights which had three successive position measurements on the piston. We also tried these calculations on the data published in Weiss and Vincent’s second paper.

Calculating pressure by means of energy considerations from tracked altitude

It is also possible to estimate piston pressure on the basis of our altitude measurements.  Neglecting thrust and drag for the short period on the piston, we can assume that the momentum of the rocket as it leaves the piston is equal to the impulse contribution of the piston plus the (negative) impulse of gravity:

p = m_r × v = I_piston + I_gravity

v = I/m_r-gt

Where p (lowercase) is the momentum of the model as it leaves the piston, v is the velocity of the model leaving the piston, m_r is the mass of the model, g is gravitational acceleration, and I is the piston’s contribution to impulse. From the velocity, we can determine the combined kinetic energy of model and piston as the model leaves the piston.  Because we are interested in analyzing the forces on the rocket before it leaves the piston, we use the total mass of the model and piston, m_t:

KE = ½ m_t × v²

KE = ½ m_t ((I /m_r) –gt) ²

Where KE is the kinetic energy of the rocket as it leaves the pad. The kinetic energy of the model as it leaves the pad is equal to the work done on the model by the piston and by gravity

W= òF_piston dx +-m_tgl = m_t ((I /m_r) –gt) ²/2

Where F_piston is the force of the piston and l is the lenght of the piston. The piston force may vary along the length of its run, but for a spatially averaged force, F:

Fl – m_tgl = m_t ((I /m_r) –gt)² /2

Fl  = (m_t ((I /m_r) –gt)² /2) + m_tgl

and

F = ((m_t ((I /m_r)–gt) ²)/ 2 + m_tgl)/l

Applying P = F / A:

P= F / A = ((m_t ((I /m_r)–gt) ²)/ 2 + m_tgl)/ l A

And finally substituting pr² for A:

P= ((m_t ((I /m_r)–gt) ²)/ 2 + m_tgl)/ lpr²

Where r is the radius of the piston.

Results

Altitude Benefit from pistons

We were able to make 16 flights.  The altitudes are shown in Table 1. The piston failed to operate on one flight, and we were unable to make flights to fill in our intended matrix of flights for equipment reasons in two other cases.  We re-flew combinations in those cases.  Re-flight altitudes are indicated with slashes in Table 1.  Raw data appears in Appendix B.  We performed one non-standard "fix" on the data.  The After every four flights, the trackers took measurements on the pad to check their settings.  We noticed one tracker consistently measured the launch tower to be 4 degrees in elevation.  Corrected altitude data appears in Appendix C.   Later analysis of video will show that two of these altitudes could not have resulted from piston effect alone—they are augmented by a "hot" B6-4 or tracking error.  Those numbers are presented in parentheses

Table 1:  Altitude

Launch Mass                 66 gm                  98 gm                  125.5 gm              191.5 gm

No piston                      118 m                  77m                     46 m                    15 m/10 m*
9-inch piston                  126 m                  (101 m)                No flight               No flight
15-inch piston                (159 m)                103 m/98 m          73 m                    32 m
30-inch piston                149 m                  No flight               87 m                    44 m

Percentage increase
with  30-inch piston        27%                     -                          88%                     196%
* 10 m altitude with engine from different batch.

A quick glance at Table 1 reveals that the heavier models benefit more in terms of percentage altitude increase from the piston than the lighter models. This data is plotted in Figure 1 below.

Figure 3:  Altitude vs. Piston Length for Various Masses

Error bars in Figure 3 represent tracking closure.  The data points for the two questionable flights are shaded, and we did not run our lines through those points.  Again, this plot shows that the increase in the piston's effectiveness in terms of percentage altitude benefit is stronger with heavier models.  The effect is large compared to the tracking error bars, and large with respect to the differences within the two pairs redundant flights (no piston on the 191.5 gram model and the 15-inch piston on the 125.5 gram model). 

Impulse contribution from pistons

Impulse contribution calculated from tracked altitude

We also wish to examine these results in terms of the piston's contribution to the impulse. Table 2 indicates how much extra impulse had to be added to a Rocksim flight simulation of the models to match the tracked altitude. Note that the no-piston control flights have non-zero values as well. These might be considered measures of error in the simulation. The control flights were simulated with B6-4 and B6-2 engines of 4.4 N-s total impulse.

 Table 2:   Impulse Contribution from Piston

Launch Mass              66 gm                  98 gm                  125.5 gm              191.5 gm

No piston                   -0.19 N-s              0.13 N-s               -0.23 N-s              -0.28/-0.71* N-s
9-inch piston               0.03 N-s               0.82 N-s               No flight               No flight
15-inch piston             0.92 N-s               0.87/0.72 N-s        1.1 N-s                 1.08 N-s
30-inch piston             0.76 N-s               No flight               1.57 N-s               1.97 N-s

* B6-2 from different batch.

We can massage this data with the following steps: First, we can remove the second non-piston flight of the 191.5-gram model, on the grounds that it used a different batch of engines, and is much farther from the Rocksim value than any other control flight. Second we can normalize the piston contribution subtracting the mean piston contribution from the flights that did not use pistons (subtracting -0.14 N-s, effectively adding 0.14 N-s), so that the controls average zero. Third, we can average the two flights of the 98-gram model on the 15-inch piston. Table 3 reflects all of these refinements.

Table 3:  Corrected Impulse Contribution from Piston

Launch Mass              66 gm                  98 gm                  125.5 gm              191.5 gm

No piston                   -0.05 N-s              0.27 N-s               -0.09 N-s              -0.14 N-s
9-inch piston               0.17 N-s               (0.96 N-s)             No flight               No flight
15-inch piston             (1.06 N-s)             0.94 N-s               1.24 N-s               1.22 N-s
30-inch piston             0.90 N-s               No flight               1.71 N-s               2.11 N-s

The standard deviation of the error based on the non-piston flights is 0.19 N-s

We believe the two values in parentheses are artifacts of "hot" engines or tracking error, as discussed in the next section.  It is clear from this table that the absolute added impulse from the piston tends to increase with model mass.

Impulse contribution determined by speed off piston in launch video

We also measured impulse from the piston by measuring the speed of the models as they left the piston, using digital video images. The velocities presented in Table 4 represent the first two frames of video after the model left the piston. The original video is on the CD as a series of .mov files. Figure 4 is a composite of two video frames used in this measurement.  In Photoshop, we marked lines at the center of the nose cone as a reference point. We measured the distances on a printout to get the numbers given.  Note the black engine casings on the mast 36 inches apart. We printed the images at a scale such that one millimeter on the image corresponded to 1 inch on the mast. The image at the left was the first showing smoke under the rocket (indicating the model had left the piston), and the image at the right followed 1/15 of a second later.  Dividing the 0.676 meters traveled by the 1/15 second between frames gave us 1.14 m/s (rounded to 1.1 m/s in Table 4, as we could not measure to better than about a half inch of the 26.5 inches traveled).

Figure 4:  Measurement of Speed of the Piston

It is clear from visible flame in Figure 4 that the engine is providing some thrust by the end of this interval.  Because we expect that there would be some acceleration due to thrust in this period, this velocity would be biased on the high side. The times in parentheses indicate the period from interval between two frames when first motion occurred and the interval of two frames when velocity was measured.

Table 4:  Upper limit on Speed off the Piston (time after first motion)

Launch Mass           66 gm                     98 gm                     125.5 gm                 191.5 gm

9-inch piston            No data                   5.5 m/s (0.13 s)       No flight                  No flight
15-inch piston          10.1 m/s (0.13 s)     8.7 m/s (0.20 s)       9.3 m/s (0.20 s)       6.1 m/s (0.13 s)
                                                            9.7 m/s (0.20 s)
30-inch piston          17.9 m/s (0.20 s)     No flight                  12.7 m/s (0.20 s)     14.5 m/s (0.27 s)

It is worth noting that, in spite of the variability in the time and altitude of the interval where speed was clocked, in every case the speed off the piston was significantly higher for the longer piston. To relate this data more directly to the altitude data, we can determine an upper limit on the piston impulse from these numbers using the equation:

I_piston = m Dx/Dt + mgt

Applying this to our data, we have a check on our altitude-based impulse data, presented in Table 5:

Table 5:  Upper limit on Impulse Contribution from the Piston

Launch Mass        66 gm                     98 gm                     125.5 gm                 191.5 gm

9-inch piston         No data                   0.67 N-s*                 No flight                  No flight
15-inch piston       0.75 N-s*                 1.04 N-s                  1.41 N-s                  1.41 N-s
                                                         1.14 N-s
30-inch piston       1.31 N-s                  No flight                  1.84 N-s                  3.28 N-s

*These numbers contradict the values calculated from altitudes.

As expected, the data in Table 5 agrees more or less with the impulse contribution found from altitude-based data found in Table 3.  Also as expected, these numbers are a few tenths of a Newton-second higher, showing the contributions of rocket thrust. There are three notable exceptions. In two cases, (66-gram model with a 15-inch piston and 98-gram model with a 9-inch piston) the altitude data in Table 3 suggest an impulse contribution greater than the highest possible impulse from the piston as measured with the velocity data in Table 5. This contradiction is well beyond our error in velocity measurement, and we must attribute it to "hot" B6-4’s or tracking error. It is interesting to notice that these flights were the only flights that bucked the trend of longer pistons giving higher flights.

The third exception to the close match is the case of the 191.5-gram model on a 30-inch piston.  Because this massive model took the longest to leave the piston, we measured its velocity later in the flight, when the engine was approaching its peak thrust. This would cause the velocity measurement to overestimate the impulse contribution more than usual. Because the altitude-backtracked value does not exceed the launch video value, this is not an actual contradiction.

Figure 4 is a plot combining the impulse values determined from altitude backtracking (Table 3) with the maximum values from the video analysis (Table 5).  The error bars show the standard deviation of the difference in the backtracked impulses of the control flights from the standard Rocsim impulse of the unassisted engines.  We have added the maximum possible piston impulses from the launch video as well.  Note the “hot” B6-4 flights, shown shaded.

Figure 5:  Impulse Contribution from Piston Determined from Altitude Data

Taking the data together, we see that the impulse contribution from a piston was greater for a more massive model in general. We see that the impulse contribution from a longer piston was greater for a given model in all cases, once the impossible data points are eliminated.

Impulse contribution from Analysis of Wiess and Vincent data

In addition to our own data, we have calculated the impulse for a 16-17 gram model flown off a 13mm piston with a 1/2A3-4T, based on Wiess and Vincent’s data.  In their first paper, they published data for both a fixed head and floating head piston. They provide mean velocity off the piston for each case, and mean model mass for each case. In their second paper, they publish representative data for a thrust-time curve, and we estimated a velocity from the "plateau" between the loss of piston acceleration and the onset of significant thrust. We have converted units to meter-kilogram-second metric units in order to find momentum in Kg-m/s, which are equivalent to Newton-seconds of impulse.  The results of our calculations appear n Table 6:

Table 6:  Impulse Contributions from Weiss & Vincent’s Pistons

Type of piston       Mean model mass            Mean speed off piston          Momentum (Impulse)

Fixed head           16.76 gm (0.01676 kg)       22.51 ft/s (6.861 m/s)           0.1150 Kg-m/s (0.1150 N-s)
Floating head        16.77 gm (0.01677 kg)       30.09 ft/s (9.171 m/s)           0.1538 Kg-m/s (0.1538 N-s)

                                                                  Estimated speed off piston

Floating head        16.5 gm (0.0165 kg)          26 ft/s (7.9 m/s)                   0.13 Kg-m/s (0.13 N-s)

Wiess and Vincent report a standard deviation of 5-10% for their speeds, suggesting an error on the order of 5-10% in the momentum/impulse values presented here. These piston impulses are around 10% of the nominal 1.25 N-s impulse of a 1/2A engine.

These models weighed about a quarter as much as our lightest models, used very different motors, and shorter pistons. The results are consistent with our results in that we would expect the piston’s contribution to be much smaller, and it is.

While it is a stretch to claim that the impulse contribution of a piston is directly proportional to model mass or piston length, it is interesting to note that speed off the piston on the order of 10 meters per second (within a factor of two) for models ranging over more than a factor of ten in mass and a factor of four in impulse.

Pressure Inside Piston

Pressure by way of acceleration from launch video

We were able to measure average pressure over one time interval for three of our 16 flights. Each of three successful flights from the 30-inch piston yielded three usable images—acquired in sequence after first motion, and before separation from the piston. Figure 6 is a composite of video frames from the launch of the 66-gram model from the 30-inch piston.  The position of the approximate center of the nose cone is marked on each frame, and we measured the altitudes from a printout of this image.  Again, notice the lines 36 inches apart centered on the engine casings that joined the 36-inch dowel sections of the mast. We used the second, third, and fourth frames of this composite to measure acceleration. Note that the model has moved between the first and second frames, and smoke is not yet visible in the fourth frame, indicating that this measurement was made entirely during the operation of the piston.

For each pair of adjacent images, we subtracted altitudes and divided by the time interval of 1/15 second, giving an average velocity. To find acceleration, we subtracted adjacent velocities, and again divided by 1/15 second. Because the rockets were not accelerating uniformly, we cannot specify exactly which 1/15-second time interval had the average acceleration, but we could be assured that our acceleration was representative of some time around the middle of the piston’s visible operation.

Figure 6:  Measuring Acceleration on a Piston from Video Images

We multiplied the combined mass of the piston and rocket times the average acceleration to find an average force. We then divided by the cross-sectional area of the piston to find pressures presented in Table 7.

Table 7: 30-inch Piston Pressure Based on Digital Video

Mass of Rocket          Pressure

66 grams                    4.1 psi
125.5 grams               5.9 psi
191.5 grams               6.8 psi

The first source of error in this measurement is the limit on our ability to measure altitude from the photos. The critical figure to consider in this regard is the change in velocity. This is proportional to the difference in distance traveled (displacement) in the two time intervals examined. We can estimate error by comparing altitude measurement error to the difference in displacement.  

In the cases of the 66-gram and 125.5 gram rockets, we are confident that we could measure altitudes within a half-inch.  The difference in displacements were 15.4 inches and 13.5 inches respectively.  We estimate that our final error is no worse than one part in 20

Because of a poor choice of camera settings on our first flight, we believe our measurements of the flight of the 191.5-gram rocket could be as far off as 2 inches. The difference in displacement was 11 inches, giving us an error on the order of one part in five.

A second error in pressure is represented by the engine coming up to thrust, which would add a reaction force in addition to the pressure inside the piston. Weiss and Vincent convincingly showed that pistons act before the engine comes up to thrust. All of the measurements used in the pressure calculation were taken within 0.200 seconds of first motion. Video of our control images confirms that none of our B6 engines attained liftoff thrust less than 0.200 seconds after first showing smoke.  Assuming that the appearance of smoke corresponds to the first motion on a piston, we believe that thrust is not a major source of error in these calculations.

Piston Pressure from Weiss and Vincent data

Weiss and Vincent, in their second paper, provided time and velocity data at 2.5-inch intervals for the flight of their 16.5-gram model’s trip along a 6-5-gram, 18-inch-long piston.  This model used a 13 mm-diameter 1/2A3-4 engine.  By dividing the difference in velocity in each interval by the time interval, we could quickly calculate force, and dividing by the cross-sectional area, we could determine pressures.

P = ((m × Dv / Dt) + mg) / pr²

The results of this calculation appear in Table 8:

Table 8:  Piston Pressure for Lightweight Weiss and Vincent model

Position            Time                 Speed               Pressure

2.5 inches         0.023 s             5.12 m/s           5.8 psi
5.0 inches         0.033 s             6.77 m/s           4.4 psi
7.5 inches         0.042 s             7.77 m/s           3.1 psi
10.0 inches       0.050 s             7.86 m/s           0.5 psi
12.5 inches       0.058 s             8.17 m/s           1.2 psi
15.0 inches       0.066 s             8.17 m/s           0.2 psi
17.5 inches       0.074 s             8.81 m/s           2.4 psi

Time-averaged over 17.5-inch interval                   3.3 psi

We note that the initial pressure of this piston is comparable to the average pressures of our pistons, while the average value is lower.  We also note that this model cleared the piston in just 0.074 seconds—roughly half to a quarter the time our heavier models took to clear their pistons. This suggests that the pistons with our heavy models either experienced higher pressures, or maintained their pressures for a longer period.

Piston pressure determined from altitude data

We applied our equation:

P= ((m_t ((I /m_r)–gt) ²)/ 2 + m_tgl)/ lpr²

using our backtracked impulse value for I, the time between the first video image showing motion and the first video image of the model free of the piston for t, and the piston length minus 1 inch for l.  We did not calculate pressures in the two cases where the altitudes measured were due to "hot" engines or tracking error (marked "poor data").   The results appear in Table 9:

Table 9:  Piston Pressure determined from Altitude

Launch Mass              66 gm                  98 gm                  125.5 gm              191.5 gm

9-inch piston               0.56 psi                Poor data             No flight               No flight
15-inch piston             Poor data             5.54 psi                7.45 psi                5.26 psi
30-inch piston             4.72 psi                No flight               8.17 psi                6.75 psi

Video values for
30-inch piston             4.1                                                  5.9                       6.8

The data in Table 9 is particularly sensitive to errors in impulse measurement—Neglecting gravity, the pressure goes with the square of impulse, so the percentage error in impulse is doubled in the pressure measurement. 

We do not detect strong patterns in the data in Table 9.  However, one pattern is conspicuously absent.  Weiss and Vincent’s data clearly show that the pressure in their piston with the lightweight model drops significantly over piston extension.  Our data does not show a decrease in pressure with longer pistons.  If anything, it suggests an increase in pressure with longer pistons.

Conclusions

•  Pistons have a greater effect on the altitude performance of heavy models than they do on lighter models

A 191-gram model's altitude was nearly tripled by the addition of a 30-inch piston. Looking at the piston's impulse contribution as a function of weight, we see a clear pattern that extends from the realm of Weiss and Vincent's small models to our most massive model on our longest piston:

Table 10:  Piston benefit vs. Model Mass

Model Mass               Impulse benefit     Altitude benefit
                                                            (longest piston tested)

16.5 g                        0.13 N-s               -
66 g                           0.90 N-s               27%
125.5 g                      1.71 N-s               88%
191.5 g                      2.11 N-s               196%

•  Longer pistons, up to the practical limit of piston length, are more effective than shorter pistons, for heavier models. 

There is an upward trend in altitude vs. piston length in our data, as seen in Figure 3, and once we eliminate two flights whose performance could only be explained by "hot" engines or tracking error, the data is monotonic—a longer piston always produced a higher flight in a given model. The plots in 4 illustrate that this holds true for impulse contribution measured by altitude tracking, and it holds true without exception for impulse contribution measured by launch video. 

•  Pistons impart more impulse to heavy models because the models spend more time on the piston.

Because pistons do their work over a fixed distance rather than over a fixed time, a slower ride on a piston with a given pressure will allow the piston impart more impulse (force × time). This translates into a considerable increase in altitude.

•  Longer pistons impart more impulse to heavy models because pressure in the piston does not decay over its run.

While the pressure in a piston carrying a lightweight model decays rapidly, pistons carrying heavy models are slower moving, allowing the engine to fill the piston with more gas as the burn begins to increase.  We see time-averaged pressures in long pistons under heave rockets that are only equaled in the first moments of a piston operating on a lightweight model.

•  Pistons may impart more impulse and energy to heavy models because more pressure builds up in the piston over time.

Our data hints that piston pressure is higher with heavier models.  Piston pressure is directly proportional to the piston’s thrust augmentation.  There is considerable error in these measurements, however. 

Possible Further Work

We have shown conclusively that pistons are especially advantageous for heavy models, and that in the case of heavy models, longer pistons are superior to shorter ones.  This can be explained by piston pressure, time on the piston, or both.  However, we could not clearly tease out the relative effects of pressure and time.  This would require higher time-resolution studies on heavy models leaving the pad.  The apparatus described by Weiss and Vincent in their second paper used infrared detectors to measure a fin passing several points with high resolution in time which could produce position and velocity curves with high time-resolution.  Such a setup, applied to heavier models, would be ideal for this work.

Acknowledgements

We would like to thank the members of  SMASH, NAR section 500, for making our day of test launches possible.  We would like to especially thank Randy Boadway, Jay Calvert, William Geresy, Jack Rose, Jill Rose, Nancy Vander Voord, and Kelo Wavio for tracking, recovering, helping with range setup and other assistance to our data collection.

References

US Patent No. 2,307,125, "Launching Apparatus for Rocket Craft," Robert H. Goddard, Filed Dec. 9, 1940, issued Jan. 5, 1945.

Rockets of the World, Peter Alway, Saturn Press, 1999, pp. 132-135.

"Pressurization Effect Launchers," Trip Barber, Model Rocketeer, Vol. XVI, No. 6, July 1974, pp. 14-16.

"Optimization of the Zero Volume Piston Launcher," Robert E. Tholen, Thomas J. Bauer, and Paul A Porzio, NAR Technical Review, Volume 2, Fall, 1974, pp. 3-19.

"The Floating Head Piston Launcher," Research and Development report submitted at NARAM 28, August 1986, by the Odd Couple Team, T-085, Chuck Weiss and Jeff Vincent, http://members.acmenet.net/~jvincent/RandD/fhp1/fhp1.html

"Empirical Evaluation of Optimum Piston Tube Length for a Floating-Head Piston Launcher," Research and Development report submitted at NARAM-30, August 1988, by the Crunch Birds Team, T-471, Chuck Weiss and Jeff Vincent http://members.acmenet.net/~jvincent/RandD/fhp2/fhp2.html

"Triple Track Tracker," Tracking Tech Report Collection, National Association of Rocketry Technical Service.

Appendices

Appendix A:  Altitude Simulations with Nominal and Modified B6 Engines

Appendix B:  Altitude Data

Appendix C:  Altitudes from Corrected Data

Appendix D was not used (its content appears as Table 9)

Appendix E:  Interpolated Impulse Calculations

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