This document describes parametric design optimizations of trebuchet models conducted with Matlab and ANSYS software. A 2D trebuchet model in Matlab was optimized using a Lagrangian formulation to achieve a maximum range efficiency of 92.6%, higher than the previously reported maximum of 83%. A 3D trebuchet model in ANSYS including component dimensions and stresses was also optimized, achieving a maximum range efficiency of 83% with component safety factors above 2. Both models support the initial assertions regarding theoretical trebuchet efficiency.

1. TREBUCHET
PARAMETRIC DESIGN
OPTIMIZATION
Image taken from: http://joemonster.org/art/28956
Ben Johnson
4/20/2016
ME 644 Final Project: Trebuchet
Parametric Design Optimization
Comparative trebuchet design optimizations conducted with
Matlab® (3 degree of freedom model) as well as ANSYS®
(full 3D finite element model).
PA R A M ETR I C D ES I G N O PTI M I ZA TI O N FO R R A N G E
EFFI C I EN C Y

2. ME 644 Final Project – Ben Johnson
Page 1
INTRODUCTION | ONE
Once upon a time, the trebuchet was a mighty medieval weapon used to hurl projectiles at the walls of a
castle during the terror of a siege. Although today they may be more commonly associated with launching
pumpkins for reality TV, the motion exhibited by a trebuchet is of academic interest to someone studying
nonlinear dynamics. A trebuchet operates by converting the potential energy of a large raised mass to the
kinetic energy of a projectile at the end of a sling. From an engineering standpoint we may quantify the
effectiveness of a particular trebuchet design by the efficiency in which it completes this conversion.
PROJECT OBJECTIVE | TWO
The overall objective of this project is to apply parametric design optimization techniques in an attempt to
arrive at an optimal solution for trebuchet design. Utilizing an online article “Trebuchet Mechanics” by
Donald Siano [1] as a primary basis this project will also attempt to verify a couple assertions made in the
article regarding optimum trebuchet design. The first assertion to be tested is that the maximum range
efficiency1 of an idealized analytical model of a trebuchet is 83%. The second assertion to be tested is that
including component mass in the model necessarily reduces the system efficiency below 83%.
Although the 2D model provides both a revealing analytical tool and a promising starting point for a real
design, it is a mathematical abstraction whose assumptions do not hold up in reality. Addressing the
reduction in range efficiency upon the inclusion of component mass will involve applying an ANSYS
response surface optimization to a parametric trebuchet design and comparing efficiency results achieved by
the 2D and 3D models. The ANSYS model overcomes the limitations of the analytical model providing for
the inclusion of realistic considerations such as part dimensions, material properties, and factors of safety.
BACKGROUND | THREE
Geometry of the Trebuchet
Centuries of design iterations have produced several popular trebuchet designs available for present day
analysis. The type of trebuchet to be considered for this project may be described by its two main
characteristics, a hinged counterweight and projectile sling.
1 Range efficiency defined in section three (Background).

3. ME 644 Final Project – Ben Johnson
Page 2
Figure 1.1 Trebuchet Diagram with Dimension Definitions
The variable parameters available to the designer include the illustrated lengths L1-L4 as well as the masses
of the counterweight and projectile m1 and m2 respectively.
Trebuchet Operation
A trebuchet is loaded by rotating the main beam to elevate the counterweight thus imparting to the weight
some potential energy. A firing mechanism fixes the beam rotation maintaining the weight in the elevated
position until ready to fire. A projectile is loaded to the sling whose attachment to the main beam provides
for releasing the projectile at some predetermined angle. When ready to launch, the firing mechanism is
unlocked and the weight of m1 imparts an angular acceleration to the beam. The beam motion is transferred
to the projectile through the sling whose rotation accelerates through an arc until reaching the predetermined
firing angle at which point the sling partially disengages from the beam thus releasing the projectile.
Figure 3.2 Illustration of Trebuchet Operation2
2 Image obtained from online resource at http://www.tasigh.org/ingenium/physics.html

4. ME 644 Final Project – Ben Johnson
Page 3
Trebuchet Efficiency
Although the efficiency of a trebuchet may be described in multiple ways, the means chosen for this analysis
is what Siano has designated “range efficiency” in “Trebuchet Mechanics.” The range efficiency in this
context is the ratio of a given trebuchet’s range 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. The formula for range efficiency is straightforward to obtain
and is verified in multiple online resources to be:
𝑅𝑎𝑛𝑔𝑒 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =
𝑚2 𝑣0
2(𝑠𝑖𝑛𝛼)(𝑐𝑜𝑠𝛼)
𝑚1 𝑔 ℎ
Eq 3.1
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 counterweight.
As was previously mentioned, “Trebuchet Mechanics” asserts the maximum range efficiency for the hinged
counterweight/projectile sling type of trebuchet to be 83%. A major goal of this paper is to verify this
maximum because although other sources [2] [3] make reference to this optimum basis independent
verifications of the theoretical maximum have not been forthcoming.
TWO DIMENSIONAL MODEL (MATLAB®) | FOUR
Assumptions of the 2D model
The two dimensional model assumes that the only system components to exhibit mass are the counterweight
and the projectile. All components are assumed to be rigid and all connections are assumed to be
frictionless. All aerodynamics forces are neglected as was done in “Trebuchet Mechanics” and all
simulations use a starting angle for the beam of 45 degrees from vertical i.e. the beam rotates through 135
degrees to the point of minimum potential energy upon firing.
Defining the Lagrangian
The Lagrangian of the system may be defined with respect to the potential and kinetic energies of the system
components. It is similar to that of a double pendulum but with an additional pendulum segment (the
counterweight and pivot) which points upwards contributing similar but opposite sign terms to the equation.
𝑇 (𝑖. 𝑒. 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦) =
1
2
𝜃̇1
2
(𝑚1 𝐿1
2
+ 𝑚2 𝐿4
2
+ 𝜃̇1 [2𝐿3 𝐿4 𝑚2 𝜃̇3 cos( 𝜃1 − 𝜃3 ) −
2𝐿1 𝐿2 𝑚1 𝜃̇2 cos( 𝜃1 − 𝜃2 )] + 𝐿2
2
𝑚1 𝜃̇2
2
+ 𝐿3
2
𝑚2 𝜃̇3
2
)
Eq 4.1
𝑉 (𝑖. 𝑒. 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦) = 𝑔𝑐𝑜𝑠( 𝜃1)(𝐿4 𝑚2 − 𝐿1 𝑚1 ) + 𝑔𝐿2 𝑚1cos(𝜃2 ) + 𝑔𝐿3 𝑚2cos(𝜃3 )
Eq 4.2
The Lagrangian is:

5. ME 644 Final Project – Ben Johnson
Page 4
𝐿 = 𝑇 − 𝑉 Eq 4.3
And the equations of motion are produced for each coordinate, qj, according to:
𝑑
𝑑𝑡
(
𝜕𝐿
𝜕𝑞̇ 𝑗
) − (
𝜕𝐿
𝜕𝑞 𝑗
) = 0 Eq 4.4
Evaluating Eq. 4.4 for each qj, (θ1, θ2, θ3) provides the following 3 equations of motion for the system:
𝑚1 𝐿1
2
𝜃̈1 + 𝑚2 𝐿4
2
𝜃̈1 − 𝐿1 𝐿2 𝑚1 𝜃̇2
2
sin( 𝜃1 − 𝜃2) − 𝐿1 𝐿2 𝑚1 𝜃̈2cos( 𝜃1 − 𝜃2 ) + 𝐿3 𝐿4 𝑚2 𝜃̇3
2
sin( 𝜃1 −
𝜃3) + 𝐿3 𝐿4 𝑚2 𝜃̈3cos( 𝜃1 − 𝜃3) − 𝑔𝐿1 𝑚1 sin( 𝜃1 ) + 𝑔𝐿4 𝑚2sin( 𝜃1) = 0
Eq 4.5
𝐿2 𝜃̈2 + 𝐿1 𝜃̇1
2
sin( 𝜃1 − 𝜃2) − 𝐿1 𝜃̈1cos( 𝜃1 − 𝜃2) + 𝑔 sin( 𝜃2) = 0 Eq 4.6
𝐿3 𝜃̈3 − 𝐿4 𝜃̇1
2
sin( 𝜃1 − 𝜃3) + 𝐿4 𝜃̈1cos( 𝜃1 − 𝜃2) + 𝑔 sin( 𝜃3) = 0 Eq 4.7
Formatting the Equations of Motion for Runge-Kutta
In order to simulate the system motion equations 4.5-4.7 must be formatted in a manner suitable for a Runge-
Kutta solver. By representing the equations in matrix form and utilizing Gaussian elimination, the equations3
may be rearranged so that each includes only a single acceleration term. The equations may then be
transposed to a state space representation providing six first order expressions for the state variable
derivatives which are ready to be solved with Matlab.
Optimizing System Design Parameters
Utilizing the Matlab function “fmincon” the system may be optimized for range efficiency. The output of
the optimization is provided in figure 4.1. The syntax required to pass extra arguments (L2-L4 and M2 in this
case) to the Matlab built in ODE45 function may be found in the appendix. Another special provision found
in the appendix is the events function which stops ODE45 execution once a given value of sling rotation is
achieved. Adding events was necessary to bound the objective function output since changing the system
parameters significantly impacts the operational period of the trebuchet.
3 See appendix for Matlab® .m file code.

6. ME 644 Final Project – Ben Johnson
Page 5
Figure 4.1 Optimization Results
As can be seen in the figure, after 20 iterations a new optimum basis which provides the underlined value of
92.6% range efficiency has been produced. The “final point” values of 4.846, 1.737, 4.12, and 130.024
correspond to parameters L2, L3, L4, and M1 respectively. Length L1 is given a value of 1m and projectile
mass is given a value of 1kg. A plot of the motion is displayed in figure 4.2 below.

7. ME 644 Final Project – Ben Johnson
Page 6
Figure 4.2 System Motion for Optimum Basis
The lower datatip displays the max range efficiency of 92.6% occurring at t=0.693 seconds and the upper
datatip shows the sling rotation of 2.418 radians (for t=0.693 seconds) which corresponds to an angle of 38
degrees with the ground (this is not the same as projectile velocity relative to horizontal). The physical
meaning of the chart is admittedly difficult to appreciate without a graphical rendering of the motion. At this
point it is sufficient to convey that the motion of the beam is arrested very near the optimum point of release.
This indicates that the rotational energy of the beam has been efficiently converted to sling rotation and that
the timing of this conversion is near optimal.
Justification for New Optimum Basis
The optimization presented herein differs from that of “Trebuchet Mechanics” in a couple aspects. First of
all, in “Trebuchet Mechanics” the optimization was performed on individual parameters as a ratio to a single
other parameter. In effect, only one parameter was optimized at a time. The fmincon function presented
here approaches the optimum basis by varying all parameters at once. In this way interacting terms in the
objective function can be optimized as well as the first order terms. The second difference between
optimization methods was that for “Trebuchet Mechanics” a “Monte Carlo” method of optimization was
used. Monte Carlo in this context amounts to testing the objective function for a series of random parameter
values. The method applied by the fmincon function uses knowledge of the objective function gradients
relative to decision variables to direct its iterative progression toward an optimum basis. The combination of
these two differences provides justification for the improved performance of the new trebuchet design.

8. ME 644 Final Project – Ben Johnson
Page 7
THREE DIMENSIONAL MODEL (ANSYS)| SIX
Additional Considerations of the 3D Model
For the finite element model the dimensions and material of each component are included. Material
considerations include but are not limited to density, Young’s modulus, Poisson’s ratio, tensile and
compressive yield strength. All of the prior system length and mass parameters (L1-L4 and M1-M2) are still
included in the optimization with the addition of various component dimensions critical to system function
and stresses encountered during operation.
Optimizing Part and Assembly Geometry
The ANSYS response surface optimization feature was utilized to tune the range efficiency of the 3D model.
While the geometry of the counterweight arm and the projectile sling were optimized for stresses, the impact
of their cross section on range efficiency was of insufficient significance to justify investigating any sort of
tapered designs. The beam, on the other hand, is a different story. The inclusion of several geometric
parameters was necessary to achieve maximum range efficiency. The most important aspect of the beam is
the tapered portion which becomes narrower approaching the sling connection. This design aspect reduces
the moment of inertia of the beam but must be optimized against bending stresses encountered during
operation. The geometry was created by defining a spline from the corners near the pivot to the corners near
the sling, the length of which is defined by dimensions L21+L46. The construction points that are set a
width of L22 apart control the rapidity of the taper. Length L21 sets the distance of the taper point from the
start of the spline. Length L44 at the bottom right sets the final tapered beam width. The length of the beam
extending prismatically beyond the spline is set by L26, the dimension which corresponds to L4 in the 2D
model. The second most important aspect of the geometry is the widened portion to the left of the pivot
(located at the origin) which moves the center of mass in that direction.

9. ME 644 Final Project – Ben Johnson
Page 8
Figure 6.1 Beam Geometry and Dimensioning Scheme
Optimal Solution
After several design iterations, a maximum range efficiency of 83% was achieved by the ANSYS model with
a minimum beam factor of safety around 2. Factors of safety for all other parts are >10. All parts are
specified as steel except the sling which is polyethylene. Beam dimensions for the final design may be found
in figure 6.2.
Figure 6.2 Beam Final Dimensions, Range Efficiency, and Minimum Factor of Safety
Behavior of Optimized Trebuchet

10. ME 644 Final Project – Ben Johnson
Page 9
Results of the ANSYS optimization model indicate that the inclusion of component mass and flexibility need
not lead to a dramatic reduction in trebuchet efficiency. It becomes clear upon watching the simulation4
that
for an efficiently designed trebuchet almost all rotational motion imparted to the beam by the counterweight
is passed on to the projectile sling at as close to the optimum release angle as is feasible.
CONCLUSIONS|SIX
The models presented herein have achieved their intended purpose. Applying an improved optimization
method to the analytical model has been demonstrated to increase the theoretical range efficiency of a
trebuchet beyond the previously asserted value of 83% to 92%. In addition, the inclusion of component
dimensions and stresses encountered has been shown to be of limited impact to the theoretical efficiency. If
more time was spent on the ANSYS model the efficiency could no doubt be further increased possibly
approaching the 92% provided by the Matlab simulation.
Although the improvement in range efficiency is a positive result, it seems the time savings reaped from the
technique is the most significant. In this instance, the application of modern computational tools replaced
countless design iteration by a few weeks of simulation. The progress produced by these evolving tools
should be remarkable to witness, assuming they don’t produce robots wielding optimized trebuchets.
REFERENCES| SEVEN
[1] D. Siano, “Trebuchet Mechanics,” Accessedonline: http://www.aemma.org/training/trebuchet/trebmath35.pdf. 2001.
[2] E. Mahieu, “Optimizingthe Counterweight Trebuchet,”Accessedonline: http://demonstrations.wolfram.com/OptimizingTheCounterweightTrebuchet/.
2012
[3] M. Senese, “Tuninga Trebuchet for Maximum Distance,”Accessedonline: http://www.mikesenese.com/DOIT/2010/12/tuning-a-trebuchet/2012.
2010.
APPENDIX| EIGHT
Equations of Motion: State Space .m File
function thetadot=TrebODEfcn(t,theta,L1,L2,L3,L4,m1,m2,g); %% Function outputs
state variable values
%% Variables to achieve Gaussian elimination
a=[(m2*g*L4-m1*g*L1)*sin(theta(1))-m1*L1*L2*theta(4)^2*sin(theta(1)-
theta(3))+m2*L3*L4*theta(6)^2*sin(theta(1)-theta(5))];
b=g*sin(theta(3))+L1*sin(theta(1)-theta(3))*theta(2)^2;
c=g*sin(theta(5))-L4*sin(theta(1)-theta(5))*theta(2)^2;
4 A video of the motion may be viewed online at: https://youtu.be/cinPR2tznsU

11. ME 644 Final Project – Ben Johnson
Page 10
d=(m1*L1^2+m2*L4^2)-L1*cos(theta(1)-theta(3))*m1*L1*cos(theta(1)-theta(3))-
L4*cos(theta(1)-theta(5))*m2*L4*cos(theta(1)-theta(5));
e=[a+b*m1*L1*cos(theta(1)-theta(3))-c*m2*L4*cos(theta(1)-theta(5))]/d;
thetadot(1,1)=theta(2);
thetadot(2,1)=-1*e;
thetadot(3,1)=theta(4);
thetadot(4,1)=-1*[b+e*L1*cos(theta(1)-theta(3))]/L2;
thetadot(5,1)=theta(6);
thetadot(6,1)=-1*[c-e*L4*cos(theta(1)-theta(5))]/L3;
end
Function to Apply Runge-Kutta to EOM’s and Report Range Efficiency
function [mineR,t,theta,eR,v0sqr]=runTrebODEfcn(x); %% Function called by
optimization routine fmincon which tries to minimize –1*(max efficiency ratio)
%% Establish parameter values passed to function by x()
l2=x(1);
l3=x(2);
l4=x(3);
M1=x(4);
L1=1; %% these two parameters held constant at 1
m2=1;
g=9.81;
L2=l2;
L3=l3;
L4=l4;
m1=M1;
opts=odeset('Events',@events); %% Set ode45 options (Stops integration when
rotation of sling reaches 180 deg)
[t,theta]=ode45(@(t,theta)TrebODEfcn(t,theta,L1,L2,L3,L4,m1,m2,g),[0.01:1e-
4:2],[pi/4,0,0,0,0,0],opts); %% Runs ode45
v0sqr=L4^2*(theta(:,2).^2)+L3^2*(theta(:,6).^2)+2*L4*L3*theta(:,2).*theta(:,6).*cos
(theta(:,1)-theta(:,5)); %% Calculates velocity squared vs time for efficiency
calc
h=L1*sin(pi/4)+L1; %% calculate distance counterweight falls
xvel=L4*theta(:,2).*cos(pi-theta(:,1))+L3*theta(:,6).*cos(pi-theta(:,5)); %%y
velocity of projectile

12. ME 644 Final Project – Ben Johnson
Page 11
yvel=-1*L1*theta(:,2).*sin(pi-theta(:,1))-1*L3*theta(:,6).*cos(pi-theta(:,5)); %%
x velocity of projectile
alpha=atan(yvel./xvel); %% angle of projectile velocity
eR=(m2/m1)*v0sqr.*sin(alpha).*cos(alpha)/9.81/h; %% Calculate efficiency ratio
[mineR]=min(eR); %% Find minimum of efficiency ratio
end
Function to Stop ODE45 Integration When Sling Rotation is 180 °
function[value,isterminal,direction]=events(t,theta)
value=theta(5)-pi; %detect theta3 = 180 deg
isterminal=1; %stop integration
direction=1; %positive rotation
end