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E91: DYNAMICS
Lab 5: Design, Analysis and Prototyping of a
Simplified Trebuchet
1 Background
The trebuchet is a mechanical device derived from the catapult used for projecting artillery pieces
going back to 300 B.C. See [1] for a historical account and a review of the many trebuchet designs
including modern reconstructions. The annual Punkin Chunkin contest in Lewes, Delaware, brings
many contestants who compete in building different pumpkin launchers [2]. Our goal is to build a
mechanical device to launch a pumpkin approximately 75 - 100 feet and accurately knock down a
designated target. The device must be strictly mechanical (pulleys, cables, pins) and activated by a
simple release mechanism (for example, releasing a locking pin). It cannot have any external source
of energy during the launch. However, you can manually add energy to initialize the trebuchet in
a desired launch configuration. We will divide the project into three phases.
I. In class, we analyzed one of the simplest possible designs, shown in Figure 1. Given the
dynamic model obtained in class, think of different design modifications to improve the range
of the device using strictly mechanical modifications. You may use the model we derived in
class or come up with a more accurate model based on your design modifications.
II. Build a physical prototype that will be able to launch a projectile with a range of approxi-
mately six feet. The better your Phase I model and the more accurate it is, the better your
ability to predict the range of your trebuchet. You may want/need to revise your Phase I
model in this phase.
III. In this phase, you will analyze the dynamics of a trebuchet with a sling (see Figure 2 and
develop a dynamic model of the system that allows you to predict the range of the trebuchet.
A computer simulation will be provided to you to allow you to simulate the system with
different design parameters.
2 Phase I
2.1 Analysis
A brief summary of the process we used in class to derive the required equations and the expression
for the range is provided in this section. A simple model for Configuration 1 with non–zero cup
angle along with the Free Body Diagram (FBD) and the Inertia Response Diagram (IRD) are
shown in Figure 3.
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Figure 1: Three different configurations of a simple trebuchet with the projectile of mass mP in a
cup. The projectile is released when the normal force in the cup becomes zero. As shown in the
bottom right panel, the cup can be fabricated so it forms an angle ψ with respect to the arm of the
trebuchet. Configurations 2 and 3 have two degrees of freedom. Do either of these modifications
improve the performance of the baseline configuration, i.e. Configuration 1?
Figure 2: A trebuchet with the projectile in a sling (sling not shown).
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Figure 3: A simple trebuchet (Configuration 1) with the projectile of mass mP in a cup. The
projectile is released when the normal force in the cup becomes zero. The cup can be fabricated so
it forms an angle ψ with respect to the arm of the trebuchet.
Symbole Description
h height of the pivot O above the base
L1 length from the pivot to the center of the projectile, OP
L2 length from the pivot to the center of the counterweight, OC
LOG length from the pivot to the center of mass of the arm OG
mP mass of the projectile
mC mass of the counterweight
ma mass of the arm
(excluding the counterweight and the projectile)
IO mass moment of inertia of the arm about O
(excluding the counterweight and the projectile)
mt the total mass of moving parts mt = mP +mC +ma
Ie the effective mass moment of inertia of all moving parts about O
Ie = mPL21 +mCL
2
2 + IO
θ0 the angle from which the trebuchet is released
θs the separation angle at which the normal force goes to zero
ψ the angle made by the cup angle with the arm (see figure)
γ a nondimensional parameter
γ = mCL2−mPL1−maLOGIe
Table 1: Symbols used in the analysis.
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Assumptions
1. There is no friction in the system. No energy is lost either due to friction in the pin O, due to
the projectile rubbing against the cup, or due to damping during the motion of the projectile.
2. Assume that the projectile does not move relative to the cup. Separation (and therefore
relative motion between the projectile and the cup) occurs only when the normal force, N ,
between the projectile and the cup base goes to zero.
Observations
1. Energy is conserved. The total kinetic plus potential energy at the beginning must be equal
to the total kinetic plus potential energy at the end.
2. Considering the FBD and the IRD for the projectile (as shown in Figure 3), we should be
able to write an expression for the normal force, N , as a function of θ, θ˙, and θ¨.
3. We can consider the system consisting of the arm, the projectile and the counterweight as a
single rigid body (before separation) and calculate the angular acceleration of the rigid body
(θ¨) at any configuration.
Model
Acceleration analysis for the projectile I first expressed the position of P with respect to O
as
rP = L1b1.
This resulted in the following expression for the acceleration of P in inertial frame A
AaP = −θ˙2L1b1 + θ¨L1b2 (1)
Force balance for the projectile From the FBD and IRD, I got an expression for the normal
force
N = mP
[
g cos(θ − ψ) + L1(θ¨ cosψ − θ˙2 sinψ)
]
(2)
Force and moment balance for the system By analyzing moments about the pivot O acting
on the system of the arm, the counterweight and the projectile, I was able to get an expression for
the angular acceleration θ¨:
θ¨ =
[mCL2 −mPL1 −maLOG] g cos θ
Ie
(3)
where Ie is the effective inertia given by:
Ie = IO +mCL22 +mPL
2
1.
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Conservation of energy Assume that the arm with the projectile is released at an initial angle
θ0 with an angular velocity 0. At any angle θ before separation, the total mechanical (potential
plus kinetic) energy is the same as the mechanical energy at release. From this, I was able to obtain
the expression:
θ˙2 =
2g(mCL2 −mPL1 −maLOG)(sin θ − sin θ0)
Ie
Using the non dimensional parameter γ, this expression can be written more compactly as:
θ˙2 =
2gγ
Ie
(sin θ − sin θ0) (4)
Calculation of separation angle Let θs be the separation angle. At that angle, the angular
acceleration is given by Equation (3), and the angular velocity is given by Equation (4). Substituting
both these expressions in Equation (2) gave me the following expression:
N = mP [g cos(θs − ψ) + g cos θs cosψ − 2γg(sin θs − sin θ0) sinψ] (5)
When separation occurs, the normal force goes to zero, which gives me the following equation:
cos(θs − ψ) + cos θs cosψ − 2γ(sin θs − sin θ0)sinψ = 0 (6)
which is linear in cos θs and sin θs for given θ0, g and γ. We know how to solve such equations.
Determining the range of the projectile Establish a coordinate system at the pivot point,
O. Given θ0, g and γ, you should be able to solve for the angle at which the projectile leaves the
cup from Equation (6). Thus, the position at which the projectile is released is given by:
xP (0) = L1 cos θs
yP (0) = L1 sin θs
Substitute θs into Equation (4) to get1. Calculate the velocity of the projectile when it is released
at the separation time ts as:
AvP (ts) = L1θ˙b2
If you know the initial position and the initial velocity you should be able to solve for the range.
Candidate design The design demonstrated in class was not optimized (see Table 1). For this
poor choice of design parameters, I obtained a range of 2.52 meters.
2.2 Design optimization
You have three sets of choices to make.
1. The angle at which you release the trebuchet, θ0. This translates to choosing the height of
the pivot O and the length L1. Why? If you use the given base, the height is fixed. Now
choosing L1 determines your choice for the angle of release θ0.
2. The cup angle, ψ.
1When you solve for θ˙, make sure you take the negative root from Equation (4).
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Parameter Significance Value
h Height of the pivot O 80 cm
mP mass of the projectile 45.69 grams
mC mass of the counterweight 1100 grams
ma mass of the rod 525 grams
(excluding the counterweight and the projectile)
IO mass moment of inertia of the arm about O 0.2660 kg. m2
(excluding the counterweight and the projectile)
L1 length (pivot to projectile) 87.63 cm
L2 length (pivot to counterweight) 49.53 cm
LOG length (pivot to center of mass of arm) 22.2 cm
θ0 angle from which the trebuchet is released −66 deg
ψ cup angle 10 deg
Table 2: Examples of parameters for a trebuchet design without any optimization.
3. The lengths L1 (if you have not already selected it) and L2 and the mass of the counterweight
mC (if you choose not to use the counterweight provided), which will also affect the position
of the center of mass, G, the length, LOG, the mass mA, and the inertia IO. In reality, these
parameters only enter into the range calculations through the parameter γ and the length L1.
You want to determine the best set of parameters to maximize your range. The simplest way to
optimize the design is to write a program that will calculate the range for different values of these
parameters, and run this program for different choices of parameters. Choose 10 to 12 values for
each of these parameters. Run your program for each choice. If you choose n parameters to vary
you will have n10 to n12 runs and results to choose the best design parameters from.
2.3 Tasks
In Phase I, you will gain some intuition to help with the design in Phase II.
I.1 Make sure you go through the derivation I presented in class using Mathematica. The file is
available on the course webpage.
I.2 Derive expressions that will allow you to calculate the range of the trebuchet. The parameters
in Table 1 are for the prototype trebuchet. Include these in your report.
I.3 Write a program (Matlab, C, Java, etc.) that will allow you to change the parameters IO,
ma, L1, L2, and ψ to determine their effect on the range of the trebuchet. Please provide a
brief description of how your program works (no more than 1 page).
I.4 Assume the sum of the mass of the arm and counterweight must be less than 1750 grams
and the length of the arm is at most 0.762 meters. The counterweight provided to you is
approximately 1000 grams. Determine the trebuchet design parameters that will maximize
your range. Include your analysis in the report.
3 Phase II
The goal of this phase of the project is to build a prototype and match the experimentally observed
range with the range predicted by the model in Phase I. Since the focus of this part of the project
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is on modeling and prediction, the evaluation will be based primarily on accuracy of prediction
and secondarily on range. Any discrepancies between the predicted and observed range must be
carefully explained. You must conduct multiple experimental trials and provide an analysis of the
variations from trial to trial.
Resources
You will be given two projectiles (a golf ball (maybe) and a small rubber ball), a 1100 gram
counterweight, a 5 inches × 32 inches sheet of acrylic for fabricating the arm, and a base for your
projectile. However, you can also substitute these with materials that you have but with some
exceptions. The total mass of your arm plus counterweight cannot exceed 1750 grams and the
length of the arm can be at most 0.762 meters.
Note your arm must be designed so that the counterweights are easily mounted and removed.
Make sure you include all your SolidWorks drawings in your report.
4 Phase III
In this Phase, you are required to analyze the dynamics of a trebuchet with a sling and develop a
dynamic model of the system that allows you to predict the range of the trebuchet. This section
provides information that will help you complete this.
4.1 Analysis
You will derive the equations of motion for a trebuchet with a sling and fixed counterweight,
pictured in Figure 2. You will make two simplifying assumptions. First, assume the projectile
is small and can be modeled as a point mass. In other words, assume that its mass moment of
inertia about the center of mass is neglible. Second, while the sling (more correctly the string) is in
tension, you can model the sling as a massless rigid link that is attached to a point-mass projectile
of mass mP . Note that the arm with the pivot point O together with the sling constitute a double
pendulum, with the arm having mass and the sling assumed to be massless except for the mass of
the projectile.
You will use force balance and moment balance equations to obtain the equations of motion.
Separate the trebuchet into two components for this analysis. The free body diagram of the arm
(component 1) is shown in Figure 4. The free body diagram of the string with the projectile at the
end (in the sling), component 2 of our system, is shown in Figure 5.
4.2 Tasks
In your report, include the following (you can use Mathematica or Maple to help you with this
portion of the lab):
R1. Draw the inertia response diagrams for each component.
R2. Derive the force and moment balance equations shown below, equations (7-10). Write the
force and moment balance equations from both sets of FBDs and IRDs to derive the following
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Figure 4: Free body diagram (FBD) of the arm (Component 1).
Figure 5: Free Body Diagram of Component 2 (the massless string with the projectile at the end).
The center of mass of Component 2 is at P .
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equations:
FyL1 cos θ − FxL1 sin θ +mCgL2 cos θ
−magLOG cos θ = (IG +maL2OG +mCL22)θ¨ (7)
−Fx = mP x¨P (8)
−Fy −mP g = mP y¨P (9)
−FxL3 sinφ+ FyL3 cosφ = 0 (10)
R3. Acceleration analysis. From the geometry of Figure 2, using point O as the origin of a
Cartesian coordinate system, derive the following equations relating the accelerations of point P
(x¨P , y¨P ) to the position (θ, φ), velocity (θ˙, φ˙) and accelerations (θ¨, φ¨) of the trebuchet:
x¨P = −L1(θ˙2 cos θ + θ¨ sin θ) − L3(φ˙2 cosφ+ φ¨ sinφ) (11)
y¨P = L1(θ¨ cos θ − θ˙2 sin θ) + L3(φ¨ cosφ− φ˙2 sinφ) (12)
You now will use equations (11–12) with equations (7–10) to derive two equations of motion. First,
substitute x¨P and y¨P into equations (2) and (3) and solve for Fx and Fy.
R4. Derive the two equations of motion below. Substitute the resulting expressions for Fx and
Fy into equations (7) and (10) to get:
(IG +mPL2OG +mCL
2
2 +mPL
2
1)θ¨ +mPL1L3(φ¨ cos(θ − φ)− φ˙2 sin(θ − φ))
− g cos θ(mCL2 −mPL1 −maLOG) = 0 (13)
mPL
2
3φ¨+mPL1L3(θ¨ cos(θ − φ) + θ˙2 sin(θ − φ)) + gmPL3 cosφ = 0 (14)
Non-dimensionalized equations of motion
Let us define the effective inertia Ie, the relative angle η, and the nondimensional inertia γ as
follows:
Ie = IG +mPL2OG +mCL
2
2 +mPL
2
1 (15)
η = θ − φ (16)
γ =
(mCL2 −mPL1 −maLOG)L1
Ie
(17)
Use the shorthand notation, cφ = cosφ, sφ = sinφ, sθ = sin θ, cθ = cos θ, sη = sin η, cη = cos η,
R5. Rewrite equations (13-14) in the form:
Ieθ¨ +mPL1L3cηφ¨−mPL1L3sηφ˙2 = gIe
L1
cθ (18)
mPL1L3cη θ¨ +mPL23φ¨+mPL1L3sη θ˙
2 = −mP gL3cφ (19)
You will now define three additional nondimensional quantities for the next part:
µ =
mPL1L3
Ie
(20)
ρ =
L3
L1
(21)
σ = ρ− µc2η (22)
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R6. Obtain the Solve for θ¨ and φ¨ to obtain the equations of motion:
θ¨ = µsη
(
cη
σ
θ˙2 + (1 +
µc2η
σ
)φ˙2
)
+
g
L1
(
γcθ +
µcη
σ
(cφ + γcηcθ)
)
(23)
φ¨ =
−1
σ
(
sη(θ˙2 + µcηφ˙2) +
g
L1
(cφ + γcηcθ
)
(24)
These are the two equations of motion that allow us to calculate the accelerations (θ¨, φ¨) for any
given position (θ, φ) and velocity (θ˙, φ˙) of the trebuchet. The tension in the trebuchet can be
calculated by observing that the force on the string at Q must act along the string.
R7. Find an expression for the tension as a function of the position (θ, φ), velocity (θ˙, φ˙) and
acceleration (θ¨, φ¨). The above equations of motion (23-24) are valid while the tension is positive.
We assume that projectile leaves the sling, i.e., release or separation occurs, when this tension goes
to zero. After this point, the range can be calculated the way it was done in Phase I.
R8. Use the simulator based on the equations above (provided to you – see website for matlab
files) to develop several candidate designs and explore the effect of varying the nondimensional
parameters, γ, µ, and ρ on the range of the trebuchet.
R9. Based on the model you designed in Phase II, obtain a design for a tebuchet with a sling
with the goal of maximizing the range of the trebuchet. Assume you are limited to the resources
available to you in Phase II. No additional materials will be provided. Again, total mass of your
arm plus counterweight cannot exceed 1750 grams and the length of the arm can be at most 0.762
meters. Your system must fit into a virtual bowl with a diameter of 2.0 meters.
5 In Class Presentation
You will give a brief 15 minute Power Point presentation summarizing your analysis and design.
This should include discussion on the following:
1. The analysis you did to come up with your design.
2. The trade-offs you had to make.
3. Features that made it into the final prototype.
4. Features that did not make it into the final prototype, why?
5. Improvements to your current design.
6. Your design for the trebuchet with a sling. What are the additions/modifications you would
make to your Phase II prototype.
6 Report
For this lab you will have to write a group lab report and an individual lab report. The group lab
report should include the following all the items listed in each of the Phases. You do not have to
write a formal lab report, however, be as concise as possible without sacrificing completeness. The
individual lab report should include the following:
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1. A list of your responsibilities.
2. A list of your team members’ responsibilities broken down by team member.
3. A list of tasks you completed either individually or with other team members.
4. A list of tasks each of your team members completed whether by themselves or with other
members.
Your indivdual lab reports should be no more than 2 pages long.
References
[1] P.E. Chevedden, L. Eigenbrod, V. Foley and W. Soedel Scientific American, July 1995.
[2] The Punkin Chunkin World Championship http://www.punkinchunkin.com/, November
2005.
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