Trebuchet vs. Catapult System Analysis
Bright Wadja
Mark Hibbeler
Philemon Delouise
Ricky Casarez
Casey Rodgers
Jordan Bishop
SIGNITURES
By signing the signature page of the formal report, each member of the group acknowledges that they adhered to the Aggie Honor code and significantly contributed to the project.
Bright Wadjka
Mark Hibbeler
Philemon Delouise
Ricky Casaverez
Casey Rodgers
Jordan Bishop
INTRODUCTION
The trebuchet analysis project had its origins in a similar project carried out by Mark Hibbeler in his high school physics class. He had built an 8 foot tall trebuchet from a design he found on the internet. The original project was fairly simple, and was basically just construction, launching a projectile, and measuring the distance thrown. However, the high school project did determine the maximum range of the trebuchet. The project goals required the trebuchet to throw 150 feet, and the trebuchet exceeded that with an average throwing distance of 230 feet.
This semester for AERO 211, the project was resurrected primarily as a system analysis project. Analysis of the trebuchet was carried out to determine the forces present while throwing a projectile. As the project progressed, it became apparent that it would be fairly simple to expand the project to include a catapult analysis as well, and then to compare the two. With this idea in mind, the project goals were revised resulting in the Trebuchet/Catapult Analysis Project.
PROJECT OVERVIEW
Project Goal: Analyze a trebuchet and a catapult and compare the advantages and disadvantages of each design.
The analysis includes two areas:
1) Force Analysis: By analyzing the forces and moments present in both the trebuchet and the catapult during the throwing motion, the velocity of the throwing arm tip can be determined. Since this velocity is equivalent to the launch velocity of the projectile, it enables a comparison of the launch speeds of the two machines.
2) Air Drag Analysis: The effect of air drag on the projectile is also taken into account. This required an experiment to determine the drag coefficient of the projectile—a regulation size baseball. Once the drag coefficient is obtained, the air drag on the ball at any speed can be calculated and its effect on the projectile’s path determined.
PROJECT PROCEDURES
Drag Coefficient Experiment (Appendix A): Air drag was assumed to be an important enough factor in the trebuchet’s projectile motion that it could not be ignored. The equation for air drag is:
Drag = CD(.5ρv^2)A → CD = (Drag)/((.5ρv^2)A)
Where:
CD is Drag Coefficient
ρ is Air Density
v is free stream velocity
A is cross sectional area of body
ρ at sea level is a known quantity, approximately 1.2 kg/m.^3 The velocity was obtained from the truck speedometer. Area was obtained using the measured circumference of a baseball and is (insert). The drag of the projectile (a baseball) and the CD are both unknowns in the equation, so drag was determined using experimental data. By using the experiment apparatus (see picture) to determine the angle at which the air forced the baseball back, the drag could be calculated by using sum of forces (see appendix). From there, the CD is the only unknown so it can be easily calculated.
Experiment procedure:
1) Drive a truck at a predetermined speed
2) Hold the test apparatus into the free stream air using level on apparatus to maintain a 0 degree angle to the air
3) Read angle off the apparatus and record
This procedure was repeated at 30, 35, 40, 45, 50, and 60 mph to give different drag and velocity values. The CD values for each trial were then averaged to give a final value of .677
Force Analysis (Appendices B & C): To obtain the launch velocity of the projectile, it was necessary to calculate the speed at the end of the throwing arm of both the trebuchet and the catapult. This was done using relative motion, conservation of energy, free body diagrams, kinematic equations, moments of inertia, and experimental data.
Trebuchet: Using the geometry of the system, the governing equations for the position of various points on the trebuchet were found. Differentiating these equations with respect to time gives the velocity equations of these points. Conservation of energy gives the necessary unknown values. Next, relating the equations of one point to other points will give the equation for the velocity of the projectile as its being accelerated by the trebuchet. Then using the drag coefficient, the equation of motion of the projectile while in flight was obtained. This equation was evaluated using matlab to meet the final conditions of the flight (time in the air and distance traveled—obtained from experiment). From there the initial launch velocity was obtained. For detailed calculations see Appendix B.
Catapult: By summing the moments and finding the moment of inertia of the throwing arm, the angular acceleration of the arm can be found. Then integrating the angular acceleration with respect to theta and dividing by theta, we defined an average acceleration that could be used as constant between two angles. From there kinematics give the angular velocity (and thus the tangential velocity). For detailed calculations see Appendix C.
RESULTS
Trebuchet: Compared with the test value of 230 feet, the projected distance of 254 feet is slightly higher. However, since the trebuchet was assumed to be 100% efficient, it makes since that the real value would be slightly lower. Another assumption made in the calculations is a sea level air density, but since Southeast Texas has a fairly low altitude the difference is negligible. Also, the test distance of 230 feet is assuming no wind, but the test conditions were calm and the value is an average of many trials so it should not have caused any significant error.
Catapult: The test value for distance traveled is 7.55 feet, and the theoretical value was 7.39 feet. The discrepancy in these numbers is due almost entirely to error in the experiment. Because of the small scale of the catapult, taking accurate measurements of the distance was challenging. Even though the error was only a few inches, it threw off the measurements enough to give a measured value higher than the theoretical one, even with the efficiency losses. Also, since a catapult is powered by the energy stored in the tension of its cables, it must be assumed that the cables did not stretch out and lose some tension with repeated use.
CONCLUSION
Though a catapult and a trebuchet are similar in many respects, they differ from each other in two main ways. The first is the difference in the power source. Both transfer potential energy into kinetic energy to launch the projectile, but the potential energy comes from two different sources. The trebuchet’s potential energy results from its counterweight being acted on by gravity, whereas the catapult relies on the tension in its cords. The second main difference is very apparent from the force analysis—the cord the trebuchet has in addition to its throwing arm. This allows the sling to whip around as the throwing arm is being accelerated and greatly increases the angular acceleration of the sling and thus the launch velocity. It also means the projectile is launched farther above the ground than if it had been thrown by a catapult. These two factors allowed the trebuchet to launch a projectile 230 feet with only a 160 lb. counterweight. The addition of the cord gives the trebuchet a huge advantage over the catapult and clearly makes it the more advanced system.
APPENDIX A
APPENDIX B
APPENDIX C