### Modeling the Motion of a Trebuchet

This work was conducted in an attempt to understand the mechanical stresses inside the beam of an existing trebuchet (the Renfrew trebuchet). After several years of uusage this beam broke but the cause was initially not known.

#### Beam details, July 2015, Zig

The developed code (strictly javascript using the canvas-capability of HTML 5) simulates a trebuchet with a hinged counter-weight and a projectile which slides initially inside a horizontal shute with friction present. Classical F=m*a considerations lead to a set of three second order differential equations which are solved numerically using a 4th order Runge-Kutta method. Following the motion of all the parts of the trebuchet we are able to conduct a stress analysis (only for the Renfrew treb).

As a first check on our calculations we used conservation of energy.

Secondly, by setting the mass of the projectile to near zero we are able to simulate a classical double pendulum and successfully compared our results to the results of the simulator available at http://complexity.stanford.edu/blog/double-pendulum-simulation
The input for the Standford simulator is somewhat restrictive but you can enter there
Θ1=90 ω1=0 mass1=20 length=25
Θ2=90 ω2=0 mass2=15 length=18
and click on to provide our program with the corresponding input.

Finally, a javascript-based trebuchet simulator is available at http://www.virtualtrebuchet.com/. You can enter there the input values of the Renfrew-trebuchet with the following exceptions :
Projectile Diameter : 0.1 m
Pivot to Arm CG : +0.2 m
Inertia of Arm : 539.3 kg m^2 (they require central moment of intertia)
Release angle : 39.0 (what my program gets) for given sling finger angle
As a result you should get from them a flight time of 6.231 s, a distance of 132.591 m and a release velocity of 37.045 m/s.
My simulation produces a flight time of 6.26737 s, a distance of 136.2 m and a release velocity of 35.6 m/s. Note that my simulation does not include air resistance, but their's does.

 Length, a [m] Length, b [m] Length, c [m] Height of pivot aboveshute floor, H [m] Initial value for angle β [m] Initial position of sling finger, h [m] Sling finger angle, δ [deg] Coefficient of friction ring/sling finger [-] Mass of beam, Mb [kg] Mass moment of intertiaof beam w.r.t. pivot, Ib [kg m²] Mass of counter weight, Mw [kg] Length, Lw [m] Central Mass moment of intertia of counter-weight, Iw [kg m²] Mass of pumpkin, Mp [kg] Length, Lp [m] Coefficient of friction in shute, μP [m] Duration of simulation [sec] Number of time steps per characteristic time [1/sec]

 ```# Fraction Mw Lw Iw # 1.0 601.048 0.731 69.305 # 0.75 516.098 0.762 63.039 # 0.5 431.147 0.776 56.938 # 0.25 346.197 0.760 48.892 # 0.0 261.246 0.683 34.040 ```

Zig Herzog; hgnherzog@yahoo.com
Last revised: 10/10/15