analysis of trebuchet

Trebuchet AnalysisSam Higginbotham, Ray Ng, Brooke Waln

Author: Sam

December 1, 2014

Abstract

This document provides an analysis of a trebuchet from a mechanical point of view. The concept ofLagrangian Mechanics has provided enough motivation to see how well the Lagrangian and HamiltonianFormalism describes a mechanical system, such as a trebuchet, versus standard elementary techniques,such as scalar energy calculations. The goal of this project is to explore the difference in analysis ofLagrangian Mechanics to Elementary methods.

Contents

1 Introduction 2

2 Measurement 22.1 Build . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Gathering Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Data 4

4 Analysis 54.1 Elementary Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5 Discussion and Error Analysis 95.1 Elementary Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Conclusion 10

7 Appendix: Code for Mathematica Algorithm from [2] 12

1

2 MEASUREMENT

1 Introduction

The trebuchet is a complicated mechanical weapon of warfare that was common in the Middle Ages. Ithas been built in various shapes and sizes; however, every trebuchet shares a common launching mechanism.Trebuchets use a gravitational pull on a large object and a fulcrum to launch a smaller projectile. Thisproject examines the behavior of the trebuchet with two different methods of analysis. . By categorizing thetrebuchet as a double pendulum, Lagrangian mechanics can be used to analyze its complicated behavior.Shown below is a time lapse photo of a trebuchet firing from a software simulation program [4].

As one can see, unique behavior of a trebuchet that makes it more complicated than other catapults isthe behavior of the hanging masses shown above. The doubled coupled pendulum allows the user to optimizestall points [5], which allows extremely efficient energy transfer to the projectile. The ”youtube” video linkposted has an excellent simulation of a coupled pendulum and explanation of stall points. The idea here isto analyze the system and see if we can take the behavior of the double pendulum in our analysis to calculatethe speed the projectile leaves - and its horizontal range - more accurately.

2 Measurement

This section explores how the team setup the experiment and gathered data. Data was gathered bybuilding a miniature trebuchet and launching small projectiles in doors. The following sections discuss theconstruction of the trebuchet and how we gathered the data for our analysis.

2.1 Build

In order to launch projectiles, we had to construct a trebuchet. We chose a simple model that was sturdyand capable of being fired in a systematic manner. The model is found on the instructables site [3]. Belowshows the process of glueing the wooden dowel rods together. We chose to increase the size of the model by50% to increase durability.

2

2.2 Gathering Data 2 MEASUREMENT

We fixated a nail into the throwing arm and made a pouch out of an old t-shirt to hold the small projectile.

2.2 Gathering Data

To gather the data we used an empty lab room and placed the trebuchet on flat ground. The team placedthe trebuchet on a rubber mat and weighed it down to minimize precession and the effects of sliding causedby the counterweight.

After building and firing the trebuchet, the team was ready for data collection. We shot the projectilesdown range after coating them in chalk so that we could precisely measure where they landed. We made

3

3 DATA

sure to measure their weight after applying the chalk for precision. From the picture one may notice themeasuring tape on the floor, we used this tape to measure the range of the projectile after it left its markon a dark towel (sliding of the towel should be considered negligible).

3 Data

This section tabulates and explores the data for the entire experiment, from weight to range of projectiles.Each firing required precise measurement of distances as well as specific documentation of the length andangle of the trebuchet components. We fired seven small glass marbles in two trials and recorded thedistances.

Weights Because the masses that we measured were all within .1 grams of each measurement we cansimply assume that they are invariant.

Type Mass (kg)Counterweight 0.5006Projectile 0.0022

Dimensions

Type length (cm)Throwing Arm R 24.2Sling r 11.2Counter Weight Arm L 4.3Counter Weight Swing l 3.2

Trial 1

Projectile Range (meters)1 7.1762 7.1433 7.0264 6.9885 6.9126 6.8747 6.645

Average Range: 6.966m Standard Deviation: 0.166m

Trial 2

Projectile Range (meters)1 7.5222 7.3183 6.9984 6.9505 6.9026 6.9567 6.598

Average Range: 7.035m Standard Deviation: 0.278m

Overall Average Range: 7.000m Overall Standard Deviation: 0.232m

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4 ANALYSIS

4 Analysis

Because of the trebuchet?s design, the launching process can be analyzed by both elementary Newtonianmechanics and Lagrangian mechanics. Because we are firing very small projectiles at low speeds, it is rathersafe to assume that the drag coefficient for spherical-like objects in air at low speeds is negligible [1].

4.1 Elementary Mechanics

The key to the analysis is the launch angle, if we can accurately measure the launch angle then we canseparate the velocity into components and calculate the range. Luckily we obtained a picture that describesthat angle well.

We measured the angle of launch to be 42 degrees. To calculate the theoretical max range we must take theenergy perspective:

E = T + U = constant

1

2m~r2 = Mgh

where h is the height the counterweight fell, g the gravity, M the mass of the counterweight, m the massof the projectile, and ~r the velocity magnitude of the projectile. We are interested in the horizontal rangewhich is simply found by calculating ~r and separating the components using standard kinematics.

~r =

√2Mgh

m=

√2(0.5006kg)(9.8m

s2 )(0.075m)

0.0022kg= 18.289

m

s

~ry = sin 42◦18.289m

s= 12.238

m

s

time of flight = tf =2 ~ryg

= 2.498s

Range = ~rxtf = cos 42◦ ∗ 18.289m

s∗ 2.498s = 33.945m

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4.2 Lagrangian Mechanics 4 ANALYSIS

4.2 Lagrangian Mechanics

The goal of this section is to analyze the trebuchet as a coupled double pendulum. The prospect ofan analytically closed solution is not really reachable, so the hope is that we can use this method to moreaccurately characterize the trebuchet and its behavior. The following section sets up the equations of motionand given the initial conditions attempts to calculate the velocity at release. For an extremely thoroughdiscussion of all the different types of trebuchets and their analyses please consider glancing at [2], whichour analysis is based off of. Below is the mathematical setup, we chose to use the pivot point of the beamon the stands as the origin and because of the mass of our projectile being of less mass than our beam, thebeam should be considered in our system.

Calculating the LagrangianL = T − U

L =1

2m(x2m + y2m

)+

1

2M(x2M + y2M

)+

1

6mb

(L2 − LR+R2

)θ2 −mgym −MgyM +

mb

2g(L−R)

We can change the coordinates over from cartesian to the polar coordinates.

xm = −r sin (φ− ψ)−R sinψ

ym = −r cos (φ− ψ) +R cosψ

xM = −l sin (θ + ψ) + L sinψ

yM = l cos (θ + ψ) + L cosψ

As we can see the Lagrangian is rather complicated, I will choose to omit the derivation of the Euler-Lagrangeequations here as there are too many terms to warrant display in normal latex script. But just to give anidea of the complicated behavior of the system, here is the Lagrangian:

L =1

2m(r2(ψ − φ)2 cos2 (φ− ψ)− 2rR(ψ − φ)ψ cosψ cos (φ− ψ) +R2ψ2 cos2 ψ

+ r2(φ− ψ)2 sin2 (φ− ψ)− 2rR(φ− ψ)ψ sinψ sin (φ− ψ) +R2ψ2 sin2 ψ)

+1

2M(l2(ψ + θ)2 cos2 (θ + ψ)− 2lL(ψ + θ)ψ cosψ cos (ψ + θ) + L2ψ2 cos2 ψ

+(l2(ψ + θ)2 sin2 (θ + ψ)− 2lL(ψ + θ)ψ sinψ sin (ψ + θ) + L2ψ2 sin2 ψ

)−mgr(φ− ψ) sin (φ− ψ) +mgRψ sinψ

−Mgl(ψ + θ) sin (θ + ψ) +MgLψ sinψ

+1

6mb(L

2 − LR+R2)ψ2 +1

2mbg(L−R) cosψ

6


L =1

2m(r2(ψ − φ)2 − 2rR(ψ − φ)ψ cosψ cos (2φ− φ) +R2ψ2

+1

2M(l2(ψ + θ)2 − 2lL(ψ + θ)ψ cosψ cos (−θ) + L2ψ2

−mgr(φ− ψ) sin (φ− ψ) +mgRψ sinψ −Mgl(ψ + θ) sin (θ + ψ) +MgLψ sinψ

+1

6mb(L

2 − LR+R2)ψ2 +1

2mbg(L−R) cosψ

The differential equations can be obtained from the Euler-Lagrange Formalism...We notice that r = 0, R = 0,l = 0, and L = 0 so we need not worry about those equations of motion.

∂L∂θ

=d

dt

∂L∂θ

∂L∂φ

=d

dt

∂L∂φ

∂L∂ψ

=d

dt

∂L∂ψ

After fitting the algorithm to our needs - graciously found in [2] - We have the following plots for theangles of interest with respect to radians vs time.

0.1 0.2 0.3 0.4 0.5

-1.0

-0.5

0.5

1.0

1.5

2.0

psi@tD

0.1 0.2 0.3 0.4 0.5

2

4

6

8

phi@tD

7


This algorithm, setting the center of mass of the beam to be zero maintains the following maximumrange:

The maximum range is 11.164m at time 0.1328 s

We may also consider adding the mass of the beam into the equations in some way, unfortunately thenumerical methods do not appreciate the value of the mass of the beam we have of 0.057kg as it renders aboundary issue. However we multiply the projectile mass by two (which is rather reasonable as the centerof mass is located on the long arm of the beam and the mass is comparable to that of the projectile from atorque standpoint), we obtain the following results.

0.1 0.2 0.3 0.4 0.5

-1.0

-0.5

0.5

1.0

1.5

2.0

psi@tD

0.1 0.2 0.3 0.4 0.5

2

4

6

8

phi@tD

This algorithm, setting the center of mass of the beam to be zero, but multiplying the projectile’s massby two maintains the following maximum range:

The maximum range is 7.870m at time 0.1492 s

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5 DISCUSSION AND ERROR ANALYSIS

5 Discussion and Error Analysis

This section discusses the results and the error of the theoretical range to what was observed in ourexperiment. It should be mentioned that the error in our theoretical methods were omitted numerically,but that in reality there is some error involved with the theoretical techniques. The elementary analysiserror would be rather easy to use the tangential error method, however an estimation for the error in theLagrangian formalism would be an academic report in itself so the error is omitted. Again, the real goal wasto compare Elementary and Lagrangian methods in order to find which one better describes our system.

5.1 Elementary Mechanics

As we can see, the elementary analysis is quite ambitious! Thirty-four meters is a very long distance withour current setup. If we compare to the overall average range found in the experiment we are over a factorof four off! This is most likely due to the complexity of the setup. The coupled double pendulum must betuned to optimize the range and this is clearly the maximum range possible, but this analysis doesn’t takeinto account the energy dissipation in the system such as moving the beam, translating the counterweight,flinging the sling, and so forth.

5.2 Lagrangian Mechanics

The Lagrangian method proved far more mathematically taxing than the elementary analysis, but itseemed to have paid off. The Euler-Lagrange equations were analytically derived from the Lagrangian usingMathematica and the generous script found from [2]. The algorithm predicted a maximum range of 11meters without using a mass for the beam, which is about 60% off value of what was observed.

When we added the effect of the beam mass in the algorithm I was unable to calculate because ofcomputer error - we therefore added the center of mass by just doubling the projectile’s mass. The resultingmaximum range was extremely pleasing. We obtained 7.870m which is only a percent difference of 12% thusthe extra mass of the beam was really important in the analysis. The range given by the algorithm alsoassumes a 45 degree angle of launch which we were very close to obtaining, but we where still slightly off -which means that this approach was even closer than expected. In comparison with the elementary analysis,we were able to obtain results that were much closer to the experimental results.

If given more time, and perhaps a time lapse camera, we could detect the exact angle of release and tracethat to a time stamp to compute the velocity of the projectile at release to extreme precision and accuracy.The numerical method approach to the trebuchet seem to be paramount to analyzing it effectively.

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6 CONCLUSION

6 Conclusion

This project explored the mechanical nature of a trebuchet. Trebuchets use a gravitational pull on a largeobject and a fulcrum to launch a smaller projectile, and are often categorized by complicated mechanicaladditions such as slings. The type of trebuchet analyzed here was a type with a hanging counterweight anda hanging projectile. The system can be categorized mathematically as a coupled double pendulum.

The team gathered the experimental data after tuning the trebuchet by shooting chalk covered glass beadsand measuring the range of the projectiles. The data showed that the hand-built system was systematicwith an average of 7.000 meters and standard deviation of 0.232 meters.

Two different techniques were used to analyze the trebuchet and compare theoretical to experimentalresults. The first technique was simple elementary techniques that used conservation of energy and kinematicsto calculate the range of the projectile based on the launch angle. This analysis proved far too ambitious,predicting a value around 30 meters.

The second technique used the Euler-Lagrange formalism, which is complicated using the double coupledpendulum model. The prospect of a closed form analytical solution was lost and the team had to find anumerical method to generate and solve the equations of motion. The team used a Mathematica algorithmfrom [2] and found that the algorithm was sufficient in analyzing the trebuchet. Omitting the mass of thebeam that coupled the slings produced a predicted value of 11.164 meters and adding the mass of the beamby doubling the projectile’s mass produced a range of 7.870 meters which was far closer to the experimentalresults.

Both analyses of the trebuchet certainly had error involved but the team was more concerned with fittingthe data from the experiment with an accurate model. One may compute the tangential error method forthe elementary analysis and do a numerical error analysis of the Euler-Lagrange method. Again, the realgoal was to compare Elementary and Lagrangian methods in order to find which one better describes oursystem.

The Euler-Lagrange numerical method produced results that were closer to the experimental data. Thusthe coupled double pendulum is sufficient in describing the trebuchet. The inference to best explanationof the error in the elementary analysis is due to tuning parameters. Although the trebuchet had a ratherimpressive range, from conservation of energy, it should be expected that one could tune our system further toreach the theoretical limit. So the efficiency of the trebuchet was rather low. With more time and equipmentone could analyze the trebuchet and optimize parameters based on results of the numerical analysis. Theproject was successful in establishing a mathematical model and analyzing results from a trebuchet.

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REFERENCES REFERENCES

References

[1] NASA, Editor: Tom Benson, NASA Official: Tom Benson, http://www.grc.nasa.gov/WWW/k-12/airplane/shaped.html Last Updated: June 12, 2014.

[2] Donald B. Siano, ’Trebuchet Mechanics’, March 28, 2001. http://asme.usu.edu/wp-content/uploads/2013/09/trebmath35.pdf

[3] User: Acceptable Risk. ’Wooden Desktop Trebuchet’. 2006. http://www.instructables.com/id/Wooden-Desktop-Trebuchet/?ALLSTEPS

[4] Superimposed picture of trebuchet from ”Atreb Simulator” analysis software. 2014 RLT Industries.http://www.trebuchet.com/99001

[5] User: Rabbit on Da Moon. ”Trebuchet Physics Tutorial: Make Your Trebuchet Throw Farther”. Pub-lished on Mar 18, 2014. https://www.youtube.com/watch?v=8hAX72Xgf1U

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7 APPENDIX: CODE FOR MATHEMATICA ALGORITHM FROM [?]

7 Appendix: Code for Mathematica Algorithm from [2]

Below is the mathematica code from Donald B. Siano with slight modifications - it is from 1997.

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th=.;phi=.;psi=.;l1=.;l2=.;l3=.;l4=.;m1=.;m2=.;m3=.;mb=.;g=.cn={l1,l2,l3,l4,m1,m2,mb};

x1[th_]:=l1 Sin[th];y1[th_]:=-l1 Cos[th];x2[th_]:=-l2 Sin[th];y2[th_]:=l2 Cos[th];x4[th_,phi_]:=l1 Sin[th]-l4 Sin[phi+th];y4[th_,phi_]:=-l1 Cos[th]+l4 Cos[phi+th];x3[th_,psi_]:=-(l3 Sin[psi-th]+l2 Sin[th]);y3[th_,psi_]:=-(l3 Cos[psi-th]-l2 Cos[th]);

V[th_,phi_,psi_]:=m1 g y4[th,phi]+m2 g y3[th,psi]-mb g (l1 - l2)Cos[th]/2

T[th_,phi_,psi_]:=(m1/2)((Dt[x4[th,phi],t,Constants->cn])^2+ (Dt[y4[th,phi],t,Constants->cn])^2)+(m2/2) ((Dt[x3[th,psi],t,Constants->cn])^2+(Dt[y3[th,psi],t,Constants->cn])^2)+(mb/6) (l2-l1 l2+l1) Dt[th,t,Constants->cn];

Lag[th_,phi_, psi_]:=T[th,phi,psi]-V[th,phi,psi];ltrr=Lag[th,phi,psi]/.{Dt[th,t,Constants->{l1,l2,l3,l4,m1,m2,mb}]->thd,Dt[phi,t,Constants->{l1,l2,l3,l4,m1,m2,mb}]->phid,Dt[psi,t,Constants->{l1,l2,l3,l4,m1,m2,mb}]->psid};

The lagrangian: 0.-0.02156 (-0.112 Cos[psi-th]+0.242 Cos[th])-4.90588 (-0.043 Cos[th]+0.032 Cos[phi+th])+0.0011 ((-0.112 Cos[psi-th] (Dt[psi,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]-Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}])-0.242 Cos[th] Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}])2+(0.112 (Dt[psi,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]-Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]) Sin[psi-th]-0.242 Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}] Sin[th])2)+0.2503 ((0.043 Cos[th] Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]-0.032 Cos[phi+th] (Dt[phi,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]+Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]))2+(0.043 Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}] Sin[th]-0.032 (Dt[phi,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]+Dt[th,t,Constants->{0,0.0022,0.032,0.043,0.112,0.242,0.5006}]) Sin[phi+th])2)

ltrr=Lag[th,phi,psi]/.{Dt[th,t,Constants->{l1,l2,l3,l4,m1,m2,mb}]->thd,Dt[phi,t,Constants->{l1,l2,l3,l4,m1,m2,mb}]-


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>phid,Dt[psi,t,Constants->{l1,l2,l3,l4,m1,m2,mb}]->psid};

eqbig=Simplify[{Dt[D[ltrr,thd],t]-D[ltrr,th]0,Dt[D[ltrr,phid],t]-D[ltrr,phi]0,Dt[D[ltrr,psid],t]-D[ltrr,psi]0}/.{Dt[l1,t]->0,Dt[l2,t]->0,Dt[l3,t]->0,Dt[l4,t]->0,Dt[mb,t]->0,Dt[m1,t]->0,Dt[m2,t]->0,Dt[g,t]->0,Dt[th,t]->thd,Dt[phi,t]->phid,Dt[psi,t]->psid,Dt[thd,t]->thdd,Dt[phid,t]->phidd,Dt[psid,t]->psidd}];

(*input the parameters here*)m1=0.5006;m2= 0.0022;mb=0;l1= 0.043(*0.141076;*);l2= 0.242(*0.793963;*);l3= 0.112(*0.367454;*);l4= 0.032(*0.104987;*);g=9.8;l5=l4/Sqrt[2];(psis=ths-Pi/2;);ths=(3*Pi)/4;phis=-ths+Pi;(*set up the equations*)eqs=eqbig/.{th->th[t],thd->Derivative[1][th][t],thdd->Derivative[1][Derivative[1][th]][t],phi->phi[t],phid->Derivative[1][phi][t],phidd->Derivative[1][Derivative[1][phi]][t],psi->psi[t],psid->Derivative[1][psi][t],psidd->Derivative[1][Derivative[1][psi]][t]};

(*Solve the DE:*)solslcw=NDSolve[{eqs,th[0]ths,phi[0]phis,psi[0]psis,th'[0]0,phi'[0]0,psi'[0]0},{th[t],phi[t],psi[t]},{t,0.,1}];

(*get th,psi and the velocity from this solution*)thint[t_]=Chop[th[t]/.Flatten[solslcw][[1]]];psiint[t_]=Chop[psi[t]/.Flatten[solslcw][[3]]];Plot[thint[t],{t,0,.5},PlotLabel->"psi[t]"]Plot[psiint[t],{t,0,.5},PlotLabel->"phi[t]"]

(*now to get the range,we first need to get the velocity of the projectile*)vx3[t_]:=Chop[-l3 Cos[psiint[t]-thint[t]]*(psiint'[t]-thint'[t])-l2*Cos[thint[t]]*thint'[t]];vy3[t_]:=Chop[l3*(psiint'[t]-thint'[t])*Sin[psiint[t]-thint[t]]-l2*thint'[t]*Sin[thint[t]]];range[t_]=2 vx3[t]*vy3[t]/g;Plot[range[t],{t,0,.4},PlotLabel->"range[t]"]


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(*get the maximum range*)tstrt=0;tend=.4;del=.001;rangetime=Table[{range[t],t},{t,tstrt,tend,del*(tend-tstrt)}];allowedranges=Transpose[rangetime];mallowedrange=Max[First[allowedranges]];postallowedrange=First[Flatten[Position[rangetime,mallowedrange]]];tallowedrange=Last[rangetime[[postallowedrange]]];Print[" The maximum range is ",mallowedrange," at time ",tallowedrange," s"];Print["theta at time of release is ",180/Pi*thint[tallowedrange]," deg"];Print["psi at time of release is ",180/Pi*psiint[tallowedrange]," deg"];Print["thoret max range=",rth=N[2 m1 l1 (1+Sin[Pi/4])/m2]," ft"];Print["efficiency=",mallowedrange/rth]


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analysis of trebuchet

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