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• Rahul Garg

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Project Report The analysis of Trebuchet Dynamics Instructors Prof. Harish P.M

Group Members Ankita Sharma, 12110010 Ashish Anarse, 12110014 Rahul Garg, 12110070

Department of Mechanical Engineering, Indian Institute of Technology, Gandhinagar.

Date of Submission: November 28,2014

Abstract The Lagrange equations are used to write down the equations of motion of a system, with the help of its kinetic and potential energies expressed in terms of generalized co-ordinates. It can be applied to solve various problems in dynamics, including the complicated motion of a trebuchet. The trebuchet is a machine that is used to throw payloads over a large range with great velocity, utilizing the energy from a potentially declining heavy counterweight. This work focuses on how Lagrange equations can be used to generate the equations of motion of a trebuchet, under some assumptions, which can be used to simulate its trajectory in Matlab. Further, the trebuchet can be optimized for the maximum velocity, under various given parameters.

1. Theoretical Background 1.1 Lagrange Equations The Lagrange equations are used to define the trajectory of a system, making use of independent generalised co-ordinates, which can describe the motion completely, considering its constraints and degrees of freedom. The kinetic energy of the system can be then calculated as follows:-

T = Total kinetic energy of system mi = Mass of each of the sub-component of the system ṙi = Velocity of each of the subcomponent of the system Further, we need to compute the total potential energy of the system, which is defined as :-

V = ∑ m i g hi + ∑ h = displacement from the considered datum 𝑘𝑖 = Modulus of elasticity xi = Extension in the component The Lagrange function can now be defined as :-

L =T - V The Lagrange equation can be then defined as:-

1 2

𝑘𝑖 𝑥𝑖2

𝑞𝑟 = generalized coordinate 𝑄𝑞𝑟 = These are those generalized forces that are not derived from a potential function W = Work done by each of the non-conservative force

1.2 Trebuchet A trebuchet is a battle machine used in the middle ages to throw heavy payloads at enemies. The payload could be thrown a far distance and do considerable damage. A trebuchet works by using the energy of a falling counterweight to launch a projectile (the payload), using mechanical advantage to achieve a high launch speed. For maximum launch speed the counterweight must be much heavier than the payload. If we consider the complete conversion of potential energy of counterweight to the kinetic energy of projectile on launch, it will give us an ideal maximum for the range that can be achieved.

R = range of projectile v0 = Launch velocity α = Launch angle For maximum range, α = 450 and by conservation of energy in ideal scenario, T of projectile =

1 2

𝑚 𝑣 2 = M g h = Initial V of counterweight

m= mass of projectile M = mass of counterweight h= initial height of the counterweight Maximum range Rm = 2

𝑀 𝑚

ℎ

To get the maximum range, the ratio of mass of counterweight to that of the projectile needs to be maximized, for a given height that is constrained by design. The efficiency can thus be defined as,

ε=

𝑅 𝑅𝑚𝑎𝑥

=

𝑉02 𝑚 𝑔ℎ 𝑀

2. Analysis 2.1 Problem Statement The model of trebuchet considered is quite close to a realistic one, with the projectile attached to one end of the beam through a sing and the counterweight attached to the other end through another rope. The projectile is however kept unconstrained. After taking the described model, Lagrange equations are used to obtain the trajectory equations which are solved using Matlab. The range is then calculated, and plotted against M/m, ratio of distance from fulcrum of projectile to counterweight and the efficiency. The values of trebuchet design are taken as Launch angle = 45, thetai = 5, thetadoti =0, betai = 5, betadoti =0, alphai = 85, mass of counterweight = 8, mass of projectile = 0.5kg, mass of beam = 1kg, Max height that counterweight can fall = 2m. There are, however a lot of assumptions taken. The entire system is taken to be frictionless and the resistance due to viscous drag is neglected. The trebuchet is fixed to the ground and the sling and rope are considered to be massless and inextensible. Bodies are considered to be rigid with free rotation happening about the pivot.

3. Simulation and Calculations

fig 3.1 The trajectory of projectile

Fig 3.2 Range v/s Ratio of length of Counterweight distance from pivot to projectile distance from pivot

Fig3.3 Range of projectile v/s M/m

Fig 3.4 Efficiency of trebuchet v/s mass ratio.

4. Results and Discussions  The projectile follows a complicated path before being released, with a launch velocity of V after which it follows a simple projectile motion, with range = v2 /2g

 The range can be maximised with respect to the mass ratio of M/m, that is counterweight to projectile. The range increases with increasing value of M/m, with very less change till the ratio is 20, and exponentially, above 25.

 The range, however decreases with increasing ratio of length of Counterweight distance from pivot to projectile distance, making it necessary to have more distance of fulcrum from projectile, within design constraints.

 The efficiency of the trebuchet can be then calculated v/s M/m. This first increases, but later decreases for the ratio above 40.

5. References 1. Donald B. Siano, Feb. 3, 2006, Trebuchet Mechanics 2. Donald T. Greenwood, Principles of Dynamics 3. www.realworldphysicsproblems.com 4. www.wikipedia.org