### 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 usage this beam broke but the cause was initially not known.

#### Beam details, July 2015, (Zig Herzog)

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 .

#### Short comings of this simulation

A simulation is only as good at the physics represented by it. Two major points here :
• Friction in the bearings at the pivot and the hinge of the counter-weight is not included.
• The counter-weight consists in reality of a container filled with water. We assumed that this water behaves like a solid body. Don't want to even think about what it is doing in reality, in particular its influence on rotation of the counter-weight and dissipation of energy.

#### Checking the code

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.

#### The Simulation Program

The content of the dialog boxes below corresponds to the Renfrew-trebuchet with the counter-weight container filled with water excluding the space inbetween the sloped roofs. You may change the content of the dialog boxes to your hearts content. Upon clicking on the "Submit" button a new browser window will pop up containing the results of the simulation which includes a "video" of the motion of the trebuchet, a graph of normal, shear and bending stress at a set location in the long arm of the trebuchet and finally a step by print-out of various variables as function of time.

Beam
Length, a [m]
Length, b [m]
Length, c [m]
Mass of beam, Mb [kg]
Mass moment of intertia
of beam w.r.t. pivot, Ib
[kg m²]
Initial Position
Height of pivot above
shute floor, H
[m]
Initial value for angle β [m]
Initial position of sling finger, h [m]
Sling Finger
Sling finger angle, δ [deg]
Coefficient of friction ring/sling finger [-]
Counter Weight
Mass of counter weight, Mw [kg]
Length, Lw [m]
Central Mass moment of intertia of counter-weight, Iw [kg m²]
Pumpkin
Mass of pumpkin, Mp [kg]
Length, Lp [m]
Coefficient of friction in shute, μP [m]
Stress Analysis
Location, distance from tip, stressLoc [m]
Width of cross-section, stressWidth [m]
Height of cross-section, stressHeight [m]
Density of Wood, densityWood [kg/m^3]
Simulation Data
Duration of simulation [sec]
Number of time steps per second [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: 03/29/20