Analysis of a Form of Trebuchet - Assignment Example

Introduction The trebuchet is well known as a weapon that was used long before the invention of the modern weaponry and can be described as being a catapult whose power source is a massive counterweight. In the recent years the trebuchet has undergone a revival of interests from various quarters. In this paper we aim at obtaining some analysis of a form of trebuchet. First we have description of the geometry of some models including the sea saw and hinged counterweight. Lagrangian Mechanics and the Euler-Lagrange Equation As mechanical systems become more complex it becomes very tedious or even an impossibility for a system to be modeled by use of Newtonian mechanics. Use of Lagrangian mechanics is an alternative to the classical method. This method there is utilization of a new unit referred to as Lagrangian given by L=T-V where T represents kinetic energy (KE) and V being potential energy. The Euler-Lagrange equation which is derived from minimization of action gives an easy way of solving the equations of motion of a system. Dependent upon the number of degree of freedom a general Euler-Lagrange equation will exhibit a function of the a generalized coordinate system  that is given by Calculate the projectile motion of a ping pong ball (with drag). The range of the ball is given by Calculate the motion of a catapult arm with a fixed (i.e. non-pivoting) mass. Here we apply the principle of see saw model where both the projectile mass and counterweight mass are attached to the beam. Physically speaking it can easily be seen that the counterweight being much heavy in comparison to the projectile mass will in charge of the motion. When the system is looked at in long term it can be seen that the system will exhibit a periodic motion. This system can be seen as being a dual mass pendulum just like in the case of a metronome. With the interest in the system being in the launching of a projectile the repeated periodic movement is not regarded as being important. With the projectile motion for equation after its release conforming to a simplified case of kinematics, the only this that is to be put into consideration is the projectile motion differential equations before release are considered and thus it calls for application of Euler-Lagrangian equation. Mass position The initial step is finding the position of each of the two masses and aiming at limiting the degree of freedom for x components and y components of the positions as a function of  that in turn is a function of time. The coordinates of m1 can be given as vectorizing the traced path by m1 the corresponding position vector, P1 which is represented by The coordinates for m2 are given by x2 = −l2 sin(_) and y2 = l2 cos(_). The position vector that gives the description of path of m2, P2, is given by : System’s Kinetic Energy (KE) and Potential Energy (PE) KE for a mass = Velocity is found from a derivative of the position function to time. Finding m1 velocity is by taking the derivative for each component of P1. For m1, this gives us a velocity ( ) given by ( ) =< l1 cos(), l1  sin() > Similarly the velocity of m2 is calculated as ( ) =< l2 cos(), l2  sin() > Having KE and PE it makes it simple to calculate Lagrangian relationship Equations of motion Application of Euler-Lagrange equation to the Lagrangian relation results to equation The solution to this equation gives equation of motion as The parameters used in calculation were as follow Range The range calculation involves use of the x and y velocity of the projectile upon release When air friction is not put into consideration, a projectile on the earth surface with obey the following kinematic equations In the equations y0, y˙0, x0, and x˙0 are initial conditions that obtained from the system at the time of releasing the mass and they are functions of  and . By employing appropriate release times of y is set at 0 this being the moment when the ball will hit the ground then the solutions of the kinematic equations are obtained. By using the value of t obtained in the equation of x the ranges of the projectiles for various release angles is obtained. The variation of range with angle of release is as shown in figure 1. From the figure it can be seen that the angle with maximum range is 38.5 with a range of 66.9m Trebuchet With a Hinged Counterweight This is a system resembling a double pendulum. In this system there is more conversion of the PE in the counterweight mass into KE of the projectile. The system depends on both  and  calling for use of two motion equations. Mass position It is only the counterweight mass position that has changed. This can be observed as being a geometric addition of the initial position vector for the case of m1 and having a new position vector to the mass from that point. Thus we have Equations of motion Application of Euler-Lagrange equation to the Lagrangian relation results to equation The solution to this equation gives equation 1 of motion as Solution of ELE with respect to  This gives as equation to as The parameters used in calculation were The variation of range with angle of release is as shown in figure 2. From the figure it can be seen that the angle with maximum range is 19 with a range of 394.5m which is about 6 times the range in the seesaw model. Location of pivot A mass ration of 100:1 was used after being picked arbitrary with a counter weight of 200g and projectile of 2g being used. With results being as shown in figure 3. Mass relationship Using a variety of counterweights for the projectile of 2g; the results were as shown in Figure which reveals the optimum results point. The calculation done by use of perfect engine model, the expectation was that the range was to have a linear relationship with respect to mass ratio. As can be seen from figure 4 that this was true for true with a small increase in counterweight mass having a large effect on the range. However, it reaches at appoint where this relationship does not hold true and at this point huge increase of counterweight are required so as to have some significant level of change in the range. Reference Siano, D. B. (2001).Trebuchet Mechanics, Read More