2. What is a trebuchet?
We will consider the trebuchet design which
exhibits a hinged counterweight and projectile
sling.
PE of counterweight KE of projectile
A Bit of Background
3. Project Objective
A paper titled “Trebuchet Mechanics” by
Donald Siano will serve as frame or reference
for this project.
Plan to test following assertions from the
paper:
Max “range efficiency” of a 2-D, 3 DOF model is
83%
Including component mass in analysis reduces
efficiency significantly
4. “Efficiency is the Word” –James Brown
Range Efficiency: The ratio of range
achieved to the range of an idealized
trebuchet where the total potential
energy of the counterweight is
converted to the kinetic energy of the
projectile and the projectile is released
at the optimum angle of 45 degrees.
Range Efficiency =
𝑚2 𝑣0
2(𝑠𝑖𝑛𝛼)(𝑐𝑜𝑠𝛼
𝑚1 𝑔 ℎ
Where v0 is the projectile release
velocity, α is the angle of v0 measured
with respect to the horizontal ground, g
is the acceleration due to gravity, and h
is the vertical travel of the
5. Design parameters:
Lengths L1-L4
Masses m1 (counterweight) and m2 (projectile)
2 Dimensional 3 DOF Model
7. Matlab® Optimization
Transform EOM’s to state space model
Apply Gaussian reduction since each equation
may only contain one acceleration term
Create function to solve EOM’s with Runge-
Kutta
Must pass design parameters to ODE45 function
Function returns max attainable range efficiency
Apply built in function fmincon to optimize
objective (range efficiency) for the decision
variables (design parameters)
9. System Motion for Optimum
Basis
Admittedly difficult to interpret without animation
Motion of the beam is completely arrested very near the optimum point of
10. Justification for New Optimum
Basis
“Trebuchet Mechanics”:
Optimized one parameter at a time
Applied brute force optimization, i.e. tested
objective function for many random parameter
values
Current Model
Optimizes all parameters at once
Applies gradient based approach
11. ANSYS Optimization Model
Includes additional considerations:
Component dimensions (3D)
Materials
Density, Young’s modulus, Poisson’s ratio, yield
strength, etc.
Additional considerations provide for:
Incorporation of component inertia
Stress/strain calculation
Factor of safety
12. Beam Dimensions
Most critical
component to system
efficiency
Final dimensions and
minimum FOS below
13. Simulation Animations
Beam too short - 63% range efficiency Beam too long - 66% range efficiency
Sling too long - 68% range efficiency Optimized - 83% Range Efficiency
Optimized design w/ stress - 83% range efficiency
14. Conclusions
Applying an improved optimization method to
analytical model has been demonstrated to
increase the theoretical range efficiency of a
trebuchet beyond the previously asserted
value of 83% to 92%.
Inclusion of component dimensions and
stresses encountered has been shown to be of
limited impact to the theoretical efficiency
15. References
1. D. Siano, “Trebuchet Mechanics,” Accessed online:
http://www.aemma.org/training/trebuchet/trebmath35.pdf. 2001.
2. E. Mahieu, “Optimizing the Counterweight Trebuchet,” Accessed online:
http://demonstrations.wolfram.com/OptimizingTheCounterweightTrebuch
et/. 2012
3. M. Senese, “Tuning a Trebuchet for Maximum Distance,” Accessed
online: http://www.mikesenese.com/DOIT/2010/12/tuning-a-
trebuchet/2012. 2010.