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CR MATH 55 - Modern Siege Weapons: Mechanics of the Trebuchet

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HistoryIntroductionLagrangian Mechanics and the Euler-Lagrange EquationThe See-Saw ModelPositions of the MassesThe Kinetic and Potential Energy of the SystemThe Lagrangian and the Euler-Lagrange EquationUsing a Numerical SolverParameters and Initial ConditionsVisualizing the SolutionRange of the Projectile and the Optimal Release AngleRange EfficiencyHinged Counterweight ModelPositions of the MassesThe Kinetic and Potential Energy of the SystemThe Lagrangian and the Euler-Lagrange EquationParameters and Initial ConditionsVisualizing the SolutionRange of the Projectile and the Optimal Release AngleRange EfficiencyTrebuchet With A Hinged Counterweight and SlingPositions of the MassesKinetic and Potential Energy of the SystemThe Lagrangian and the Euler-Lagrange EquationParameters and Initial ConditionsVisualizing the SolutionMaximum Range and the Optimal Release AngleRange EfficiencyConclusionsAppendix: Maple CodeSeesaw TrebuchetTrebuchet with a Hinged CounterweightTrebuchet with a Hinged Counterweight and SlingHistoryIntroductionLagrangian . . .The See-Saw ModelHinged . . .Trebuchet With A . . .ConclusionsAppendix: Maple CodeHome PageTitle PageJJ IIJ IPage 1 of 47Go BackFull ScreenCloseQuitModern Siege Weapons: Mechanics of the TrebuchetShawn Rutan and Becky WieczorekMay 18, 2005AbstractThe purpose of this project is to describe and analyze the motion of a trebuchetusing differential equations. We will begin with the original model used historicallyand gradually make the model more realistic in accordance with the various mod-ifications added to the trebuchet. Lagrangian mechanics will be used to determinethe equations of motion and to find the release angle from vertical that maximizesthe range of the projectile. By comparing the maximum range of each model withthe maximum possible range for an idealized launcher, we hope to show how eachaddition to the trebuchet makes it more efficient.1. HistoryThe first trebuchets on record appeared in Asia in the midst of the 7th century. Thesecrude siege weapons, today known as ”traction trebuchets” used human power to launchprojectiles hundreds of meters. The earliest trebuchets re sembled catapults, simplestructures with a projectile on one end and a counterweight or a series of ropes (as inthe traction-type) on the other. As trebuchets were used generation after generation towreak havoc upon villages and fortresses alike, they evolved, much as modern weaponryHistoryIntroductionLagrangian . . .The See-Saw ModelHinged . . .Trebuchet With A . . .ConclusionsAppendix: Maple CodeHome PageTitle PageJJ IIJ IPage 2 of 47Go BackFull ScreenCloseQuitevolved from the revolver to the gatling gun. The next addition to the ”seesaw”-typetrebuchets was a hinged counterweight, w hich allowed more of the potential energy ofthe system to be utilized in the projectile. As trebuchets advanced through the ages,the most important addition was the use of a sling to launch the projectile. Engineersof the day discovered that they could improve the efficiency of their weapons many-foldby extending the throwing arm of the weapon using a rope and a sling. Similar to theway a baseball player will extend his bat at the last possible instant, a sling allowsthe projectile to reach a greater velocity before it leaves the machine by convertingthe kinetic energy of the throwing arm into the kinetic energy of the projectile itself.History shows the trebuchet to be one of the most efficient siege weapons of any era,and the ingenuity of thousands of archaic engineers led to its development.2. IntroductionToday, we can mathematically model the trebuchet’s motion. This motion is extremelycomplicated but can be broken down into several manageable steps. For this project,we have assumed that every beam is massless, which simplifies calculations, althoughit would not be difficult to add the weight of the beam to the differential equations.We have also assumed that no friction exists to damp the motion of the trebuchet andthat the main beam, the hinged counter-weight, and the sling are perfectly rigid. Forthe first model, the so called ”seesaw” model, Newtonian mechanics can be used tofind the equations of motion. However, due to the complications introduced in thehinged counter-weight and sling models, it will be beneficial to model every systemusing Lagrangian mechanics.HistoryIntroductionLagrangian . . .The See-Saw ModelHinged . . .Trebuchet With A . . .ConclusionsAppendix: Maple CodeHome PageTitle PageJJ IIJ IPage 3 of 47Go BackFull ScreenCloseQuit3. Lagrangian Mechanics and the Euler-Lagrange EquationWith even a slightly complicated mechanical system, it often becomes tedious, and attimes impossible, to model the system using Newtonian mechanics. One alternative tothe classical method is to analyze the system using Lagrangian mechanics. Developedby Joseph Louis Lagrange in 1788, Lagrangian mechanics utilizes a new unit, knownas the Lagrangian, which for mechanics is given as L = T − V , where T is the kineticenergy and V is the potential energy of the system. Action is the integral over time ofthe Lagrangian. By minimizing the action, one can find which of an infinite choice ofpaths a system or particle is likely to take. A ball is thrown into the air; why does itfollow the curve it does? One answer, the Lagrangian answer, is that the path is simplythe path of smallest action.Derived from minimizing the action, the Euler-Lagrange equation gives us an easyway to solve for the equations of motion of a system. Depending on the degrees offreedom of the system , that is, in how many ways can it move, the general Euler-Lagrange equation will appear as a function of the generalized coordinates of the system,qi.∂L∂qi−ddt∂L∂ ˙qi= 0 (1)Thus, there will be an equation for each degree of freedom. In our systems, eachdegree of freedom will be an angle measure, and the x and y components are functionsof these angles. Hence our final model, which will be described as a function of θ, φ,and ψ, will require three equations.HistoryIntroductionLagrangian . . .The See-Saw ModelHinged . . .Trebuchet With A . . .ConclusionsAppendix: Maple CodeHome PageTitle PageJJ IIJ IPage 4 of 47Go BackFull ScreenCloseQuitθl1l2(x2, y2)(x1, y1)(0, 0)m2m1Figure 1: Seesaw model of a trebuchet4. The See-Saw ModelThe simplest case to consider is that of the seesaw model. In this model, both thecounterweight and the projectile are fixed to the rotating beam as depicted in Figure


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CR MATH 55 - Modern Siege Weapons: Mechanics of the Trebuchet

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