# trebuchet paper

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Trebuch Project

Blake Johnson

Abstract— The goal of this paper is to obtain a completeunderstanding of the concepts required in the design of atrebuchet style catapult. The general concept and thinkingbehind the project, as well as the analysis of the reason behindthe success of the project is the essential purpose of thisproject. Statics and kinematics were the main mathematicalbase to the entirety of the project, as well as a small knowledgeof materials and deformation. The knowledge of Matlab wasessential in the design process of this project. The idea of tensionvs compression was made apparent during class and directlyapplied to the design. The ability to realize how each part wasgoing to react with one another was the most essential abilityto obtain during the process of the design.

I. INTRODUCTION

For the second project in our mechanical design class, wewere required to design a catapult, specifically a trebuchet(see fig. 1), entirely out of MDF (medium density fibre-board), that could launch a water filled ping pong ball 12m;and also had to be able to launch on its own. This was doneby programming an arduino board to control a servo to pull apin that locked the catapulting mechanism in place. Prior tothe design process it was required that we find the maximumreaction force acting on the main trebuchet pin, which was a1/4 inch acrylic rod. This was found by modifying the Matlabscripts that were given to us at the beginning of the project.I then had to plot a launch window as a function of twoparameters of my choosing. This was followed by havingto plot the maximum reaction force against the maximumlaunch window to determine the maximum reaction forcethat the pin was going to encounter. During this process, thedimensions of the trebuchet were decided and was the heartof the entire design process.

II. CONCEPT GENERATION AND EVALUATION

To start, we had to look up the most efficient conceptsfor a trebuchet. On multiple websites, we found that youwanted a 100:1 counter weight mass to launch object mass.We were given a 2 kg counterweight, and the water filledping-pong ball will weigh about 36 grams. This is about a55.6:1 ratio. Then, through research it was said that the mostefficient ratio for the throwing arm is about 3.75:1 for thelength of the arm on the launch side to the length of thearm on the counter weight side. But this was for a 100:1mass ratio. I then applied our mass ratio to the 3.75:1 andgot a 2.083:1 ratio. Then in order to find the longest possiblearm I had to use Pythagorean theorem on the MDF 24 inchx 18 inch boards and found that the maximum arm lengthwas 30 inches (or about 760 mm). Since the MDF is notan extremely strong material, we knew that we were goingto have to allow some extra material to surround where the

Fig. 1. Fully assembled trebuchet

counterweight was going to hang from. With the informationfound we plugged the numbers found into the Matlab codethat was given to us, and found a launch that exceeded whatwe were looking for. From here I slightly adjusted some ofthe numbers to find a better dimensional design. Yet whenthis was done, there was a negligible effect done to the launchdistance. From here the main dimensions never changed,and the design that was originally drawn was set. All thatwas added were supports to the outside of the trebuchet, togive it horizontal support to prevent torque from breaking ortwisting the design. And then supports under the rod wereadded to prevent the acrylic rod from breaking.

We had to modify the code we were given, to be ableto plot a launch window as a function of two parameters.The two parameters we chose to vary were the length fromthe pivot point to the sling and the distance length fromthe pivot point to the hinge of the counterweight. By usingall possible combinations of the two parameters, with eachover fifteen intervals we were able to compute all possiblelaunch window results for each combination. This allowedus to plot all three of them on a three dimensional plot. Theplot helped us visualize which parameter combination causedthe best launch window value. When the launch windowis large there is less chance of errors during the launchingprocess of the trebuchet, which means that the ping-pongball is more likely to travel a greater distance. We werealso able to modify the code in order to determine whatparameters caused the largest reaction force. We did thisby plotting the maximum reaction force acting on the maintrebuchet pin as a function of two parameters, each overfifteen intervals. The two parameters we chose to vary werethe length from the counterweight to the pivot point and thelength of the rope attaching the projectile to the arm. Weused all possible combinations of the fifteen intervals foreach parameter, and computed the maximum reaction forcefor each corresponding combination. We used this to plota three dimensional graph of the maximum reaction forceas a function of two parameters. The plot helped us findwhich parameters resulted in the largest reaction force. Forthe trebuchet we wanted the smallest reaction force, sincewe did not want the acrylic rod to break. With this we foundthat the maximum reaction force was going to be 22.1 N.This is the reason we decided to add supports under the rod.With these results we had to find the maximum deflection ofthe rod. This was found by first finding the second momentof inertia I by using:

I = (πr4)/4 (1)

and then using what we found for the second momentof inertia in the deflection equation (δ). Where F is themaximum reaction force, L is the length of the acrylic rod,E is young’s modulus, which was given to us by professorSpenko. With all of this, we used it in:

δacrylicrod = (Freaction ∗ LAcrylicRod)/(48 ∗ E ∗ I) (2)

. It was found that there was going to be 4.9mm of deflectionin the acrylic rod without any support under the rod. Sothe supports were added in order to decrease the amount ofacrylic rod that would be unsupported. When (5) was usedagain, it was found that there was going to be a .08mmdeflection, which is well under the recommended deflection.

III. ANALYSIS

After the design of the trebuchet, we had to design areleasing mechanism. When loaded, it was known that thecounterweight would be at rest on an angle with respectto the trowing arm. Only the force perpendicular wouldcause a moment about the pivot of the throwing arm (see

Fig. 2. Free Body Diagram of Throwing arm at rest

Fig. 3. Free Body Diagram of pin, while entire system is at rest

Fig. 4. Free Body Diagram of servo just as the servo begins to turn

fig. 2). On top of this, the arm is much shorter on thecounterweight side, meaning it would take less force to keepequilibrium than the force which is caused by the counterweight. We decided to design the release mechanism by usingan acrylic rod, with a rope wrapped perpendicularly aroundthe throwing arm in order to use the least amount of forceto keep equilibrium (see fig. 3). We used the sum of themoments equation, where Fmg is the weight of the throwingarm, d is the distance from the fulcrum, θ is the anglebetween the throwing arm and the force of the counterweighton the system, and Frope is what we are looking for, whichis the force exerted on the rope in order to keep equilibriumin the system.∑

M = −((Fmg) ∗ d ∗ cos(θ)) + ((Frope) ∗ d) (3)

was used to find the amount of force that the rope had tohold to keep the trebuchet in equilibrium. It was found byinspection that the rope would be in tension. And throughthe sum of the moments equation, it was found that the ropewould hold 4.38 N. The tension would then be translatedaround to the pin that will be lodged into the main supportof the trebuchet. It was given that the coefficient of frictionfor the acrylic rod against the MDF was 0.8. The force ofthe rope will be translated perpendicularly to the acrylic rod,and therefore the force that will pull the acrylic rod out ofthe MDF will be found with Frope which is the force foundin (3) and multiplied by µfr,s, which is the coefficient ofstatic friction.

Ffr = Frope ∗ µfr,s. (4)

Then to determine the amount of torque against the servo(τ ), (see fig. 4), we had to use the radius of the servo horn(rServoHorn).

τservo = Ffr ∗ rservohorn (5)

Through these equations, it was confirmed that the servo,which has a torque limit of 350Nmm, will be able to pullthe pin out to release the throwing arm from rest.

It would have been preferred that we had a larger coun-terweight in order for the weight ratio to have been around100:1 for the counterweight to ping-pong ball ratio. Wethen could have had a 3.75:1 ratio for our throwing armside to counterweight side ratio. This would have increasedthe distance that we could have thrown. This would havehad no effect on the locking mechanism, because we usedthe ratio of the counterweight to the ping-pong ball weightfor the lengths of each side of the throwing arm, then themoment would have remained the same. What would havechanged though, is the amount of force that the acrylic rodwould have encountered. This may have caused too greatof a deformation and may have been catastrophic. If thishad been the case, we would have had to add more supportunderneath and around the entire rod. If this would not beenough to prevent the rod from breaking, then we wouldhave had to use a larger rod, or a different material.

IV. EXPERIMENTAL RESULTS

The first trebuchet that we had cut out and tested allowedus to find design flaws that we had in our project. It helpedus find that the MDF was not going to be as reliable aspreviously thought. On top of this, it was found that thewood glue was not going to be able to withstand multiplethrows without further support. We decided to glue theentire original standing structure to a full piece of MDF inorder to prevent the structure from twisting. We decided toredesign a few components of the original design in orderto accommodate the apparent design flaws that had beenfound. These components included greater support of themain rod, and also making the throwing arm much thicker.The original arm was 1 inch thick, but when we pickedup the arm it was way to flimsy and it was apparent thatany stress would have broken the arm. The final model hadus confident that the structure was going to hold up underthe stress of multiple attempts. But in the first attempt, werealized that the bar that went through the counterweightwas going to continuously hit the sides of our structure.This forced us to be intuitive and innovative with the limitedamount of materials available to us. We ended up turningthe counterweight sideways, and mounting the counterweightwith two bolts in order to prevent it from falling from sideto side. Other than these limitations that we encountered, therest of the design was nearly flawless. The entire system didexactly what was expected of it, and minus the design flawof the width between the structures supports, we did not haveany other issues that could be attributed to the final design

V. DISCUSSION

When the counterweight was attached to the originaldesign, there was an immediate issue with what we hadoriginally planned. The counterweight bar, that runs throughthe middle of the counterweight, had only about 6.5mmclearance on both sides. But since the arm was not sturdyenough, the arm swung through and swayed from side toside, which caused it to come into contact with the supportsof the trebuchet. The impulse caused by the impact endedup being catastrophic for the acrylic rod. We then solvedthis issue by turning the counterweight 90 degrees and usedbolts to hold the counterweight in place when it was in theloading position. This allowed for plenty of clearance andallowed us to continue on with the assembly.

Another issue that was encountered was the sling. Initiallywe used tooth floss as the rope that held the sides of thesling to the throwing arm. This material ended up shreddingand tangling too easily; so we decided to use the 2mm ropesupplied to us. This solved most of the previous issues thatwe had prior to the change.

Then when testing the launch, the ball would not releaseat the right time no matter how we positioned the finger onthe end of the throwing rod. The finger is what held the loopof the one side in order to release the ball at its’ peak. Wetested the launch over fifty times before we decided to changethe sling. Instead of tying the sides of the sling, we decidedto tie the top and bottom of the sling, in which we had a hugesuccess. We went from not knowing the direction, nor theangle at which the ball would be launched, to being able toadjust the throw and each throw being extremely predictable.

The final problem encountered happened during our firstofficial launch. When the trebuchet was fully loaded, we gavepower to the arduino board to pull the locking mechanismout of place. When this started, our throwing arm snappedbefore the mechanism was ever released. We decided toreinforce the arm with two separate pieces of MDF, and drillthe broken throwing arm together in-between the two newpieces of MDF. This had negligible effects on the statics ofour structure, yet may have had some negative effects on thedistance thrown.

If we had enough time, and enough materials to re-dosome of the design of the trebuchet, we would have been ableto perfect the idea after the issues that we had encountered.This project helped me realize the amount of extra thoughtprocess that had to go into these projects. There were somany moving parts to this project, that everything that couldgo wrong had to be inspected. Instead we had to solvethe problems on the fly. Yet, in the end, it was relativelysuccessful when all was said and done.

VI. CONCLUSIONS

Through the use of statics, computer programming, andbasic knowledge of materials, we were able to design ourtrebuchet entirely. Although the design threw 9m, which was3m short of the goal, we ended up having a great learningexperience due to this project. This is entirely applicable towhat we will be doing after we receive our degrees. This

project helped me realize the amount of time that is requiredto design something that is not static. When things move,there has to be a more in-depth thought process of everysingle piece that is to be designed. Errors happen to be partof the process, and I am glad we encountered these issueson this project because of the learning experience.