First Catapults Were Developed

Published Date: 02 Nov 2017

Dough Ball Catapult Proposal Report

Professor

W. Brent Lievers

Group 8

André Moraes

Antonio Henrique de Oliveira Fonseca

Daniel Ferraz de Campos Filho

Rebecca Hauta

Jan 31, 2013

Contents

1) Introduction

When working with design of a model, it is desirable to illustrate the methodologies and principles that support and turn the base of knowledge stronger before starting building a prototype of a project. Such a search would naturally include instruction on design selection and analysis of expectable results. Ideally, the project would also be exposed to other fundamental aspects such as problem definition, factor identification, and selection of factor levels. Additionally, the project would be hands-on, allowing a performance of tests, measurement of responses, analyze of data, and confirmation of results [1]. Considering these concepts and knowing that the aim of the project is to develop a catapult model capable to launch a projectile, it is necessary understand more about this machine and methods of launch projectiles.

The first catapults were developed by the Greeks of Syracuse (eastern Sicily, Italy). During the Middle Age the word "catapulta" was used for a machine that throws balls, while the word ballista was used for a machine that throws darts. The Roman Empire was conquered and held by the army and also by your technology. This technology and modernity were not only in the field of the Law and Literature but also in the field of Engineering and Science. Catapults and ballistae represented the light artillery pieces of the Roman Army; after the fall of the Roman Empire these weapons were widely used by several armies up to the XV century. The last major improvement in the catapult design was achieved during the Roman Empire when the most stressed components of these machines were made by metal allowing reduction in size, increase of the permissible stress levels and greater freedom of travel for the bow arms [2-4].

Although the types of catapults vary, all of them have the same objective: to hurl a projectile in the air. There are three primary methods that fall within the definition of "catapult".

• The catapult: Winched-down bucket, commonly associated with the word "catapult";

• The ballista: similar to crossbow but some parts made with steel and larger;

• A trebuchet is a weighted beam that swings a sling carrying the projectile.

Both catapults and ballistae work by storing tension either in twisted ropes or in a flexed section of wood. A trebuchet has a tendency to be easier to build because it consists of a pivoting beam and a weight that rotates the beam through an arc. The winch created by the gears allows putting a great energy into the catapult over a period of time. Then the whole energy releases at once, throwing the projectile.

2) Theory and Physics

Projectile Physics

When we are talking about projectiles being launched at an angle to the Earth’s surface, there is velocity in the horizontal and vertical directions simultaneously. In this case, the shape of the trajectory looks like a parabola (as the projectile moves up and down).

The diagram below shows the entire path of a projectile launched at an angle.

Figure - Illustration of projectile motion launched

The horizontal () and vertical () components of the initial velocity () can be determined by trigonometric identities:

(1)

(2)

Notice that the horizontal component remains equal at any point along the trajectory, because we are ignoring air resistance and by this way, there is no external force in the horizontal direction. Thus, the equations that describe the horizontal motion are the same used to describe constant horizontal speed:

(3)

(4)

(5)

The vertical component of the velocity changes along the trajectory due to the gravity force. So, as the projectile move upwards, the vertical velocity decreases at the rate of 9.81m/s2 (gravitational acceleration). At the top of the trajectory (maximum height), the vertical velocity is zero and the velocity () is equal the horizontal component of the velocity.

As the projectile moves downward, the vertical component of velocity increases in the downward direction at a rate of 9.81m/s2. Therefore, below there are the equations which describe the vertical motion:

(6)

(7)

(8)

(9)

All of them are well known and come from the basic equations of accelerated motion (equation 7). Now, putting all those equations seen for horizontal and vertical motion together, it’s possible get more 3 equations not really essentials, but resultants from the geometric analysis of the problem:

(10)

(11)

(12)

Maximum Range

When the objective is to achieve the maximum range, it's important to know what is the best angle to launch the projectile. When neglecting air resistance, the range of a projectile will be:

d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right)

We will not show the entire deduction here, but it is available at the appendix section of the report.

To determine the maximum range, we must finding the derivative of the range with respect to the elevation angle and to set the derivative equal to zero to find the extreme, thus:

\frac { dR } { d\theta} = \frac {v^2} {g} \left[ \cos \theta \left( \cos \theta + \frac {\sin \theta \, \cos \theta} {\sqrt {\sin^2 \theta + C} } \right) - \sin \theta \left( \sin \theta + \sqrt { \sin^2 \theta + C }\right)\right]

By this way we will find the below expression that gives us the angle to achieve the maximum range for uneven ground, ignoring air resistance:

\theta = \arcsin \left( \frac {1} {\sqrt {2 \left(1 + \frac{g y_0}{v^2} \right) }} \right)

As we can easily verify, for an initial position on the ground (), the angle is exactly 45Ëš. For a different position, for example and , the angle is 41.1Ëš. So, the optimum angle is generally around 45Ëš [14].

The maximum range achieved at the optimum angle is found to be

R_{\rm max} = \frac{v}{g} \, \sqrt{v^2 + 2 g y_0}

Catapult Physics

When we are talking about catapults, the physics utilized must be the physics of rotational motion.

First of all, we can think on a general arm in motion. As we can see in the diagram below, the arm rotates about an axis due to a force applied on it resulting in a torque and throwing the projectile. This torque causes the arm to accelerate and gain angular velocity and momentum. Some of this angular velocity and momentum is transferred to the projectile, which when it is thrown, becomes linear velocity and momentum with a ballistic trajectory. By considering how the physics applies, a more efficient catapult can be designed [13].

torque

Figure - Illustration of the arm showing the referenced forces

The force applied to the arm of the catapult is typically not in the direction of torque. Because of this, some of the force is wasted and composed by two perpendicular components: the component in the direction of torque and the component of wasted force that is perpendicular to it. So, minimizing the wasted force we increase the efficiency of the catapult [13].

net force

Figure - Illustration of force applied, effective and wasted on the arm

Energy

Generally, the energy involved in a process with elastics is the Elastic Potential Energy (EPE) and can be expressed by:

wherek is a property of the elastic and express how well it stores the energy; and x is de deformation applied on the material.

When an object is twisted and can still return to its original shape, torsional energy is stored. This is referred to as a "Torsion Spring" and can be used in a variety of siege devices. The energy stored in a torsional spring is represented by the following equation provided the object is not twisted beyond its elastic limit:

Where k is a constant which is large for favorable materials to use as torsional springs.

3) Methods of Launching Projectiles

The following tables describe the different methods of launching projectiles, the pros and cons, and which method each group member chose and a brief description of how it works.

Mechanisms to launch a projectile

Kinetic Energy Weapon - A weapon which launches a projectile (such as a bullet or arrow) without the use of explosives or chemicals. This type of weapon is the best option for construction of prototypes given the constraints in this course.

Table 1 - Traditional Mechanical Projectile Weapons

Name and Sketch

Description

Pros

Cons

Arbalest

European steel crossbow. The strongest windlass-pulled arbalests could have up to 22 kN of force and be accurate up to 300 m[1]

Simple

Long, can be heavy

Ballista

A weapon that stores torsional energy. The part that stores torsional energy (the prod) must scale with the size of the projectile that will be fired. [2]

simple, easy to load

Can lose torsion over multiple uses

Bow

A simple bow uses two elastic limbs connected by a string. When the bow string is pulled back, potential energy is stored in the bow limbs and the string is under tension. When the string is released, this energy is converted into kinetic energy and is transferred to the projectile (usually an arrow) [3]

Simple

No locking mechanism

Catapult

The term catapult generally refers to medieval weaponry used to hurl projectiles using different methods of energy storage such as a trebuchet (gravity) or the mangonel (torsion and/or tension).[4]

Relatively simple, powerful

Tends to shoot forward instead of upward

Circular Wheel

"Pitching Machine"

A ball is fed through spinning wheels and increases the velocity of the ball. It is usually powered by a motor.

Can ideally shoot ball sized objects with accuracy

Requires a specific ball size, dough can get stuck in it, complex

Crossbow

I basic crossbow is a bow mounted on a stock with a mechanism that can hold the bow string and can release the string using a trigger.

Simple

Requires specific materials for maximum effectiveness

Mangonel

A type of Catapult which uses torsion or tension device and a counterweight directly connected to the rotating launching arm. Due to its design, it is ideal for low angle, high velocity projectile acceleration. [4]

Can effectively shoot projectiles relative to its size when constructed and tuned properly

Complicated, may use more than one energy storage mechanism. Tends to shoot forward instead of on an angle. Needs to be calibrated.

Onager

This device uses a torsional bundle (usually several twisted ropes or torsion springs) to store energy which is released. The torsional bundle is twisted when the firing arm is in firing position.

Effective for projectile launching over great distance

Torsional bundle can lose effectiveness over time. Can be complicated to fabricate

Sling

A sling has a rope which is spun at high velocity with the projectile at the end. The projectile can have a releasing system or the rope itself can be released. Various early projectile weapons use the same principles as the sling [5]

Effective personal improvised weapon in primitive warfare

No way to lock it in place, however a sling is used as a part of other launching mechanisms.

Slingshot

A slingshot is a simple catapult that works in a similar way to a bow in principle, but generally uses a fixed y-shaped structure and the energy is stored in an elastic rope. [6]

Small, simple, effective

Ball can get stuck in holding mechanism, often does not have a locking mechanism

Spring Engines

Spring Engines can work in a variety of ways. Instead of using elastics or bow components to store energy to fire a projectile, springs can be used to store and release instead. In some jurisdictions, these ballistic springs are regulated.

Small, effective

May be illegal or regulated in some areas (like Airsoft guns). Spring can wear out.

Trebuchet

A trebuchet is a type of catapult that works by using the energy of a raised counterweight to throw the projectile.[4]

Effective for projectile launching over great distance

The structure required to get an effective lunch is complicated to build.

Other ways of firing a projectile

There are many other types of weapons that generally use chemical energy (directly or indirectly) to launch a projectile. Since these types of weapons need to be reloaded and are usually explosive, most of these weapons are not recommended for use in the CAD course.

Table 2 - Modern Mechanical Projectile Weapons (using explosive materials)

Name

Image

Primary Mechanism

Chaingun

Uses a powered mechanism to constantly feed and shoot.

Canon

Uses an explosive to launch a projectile

Compressed air guns

Uses a compressed air canister to fire a projectile such a paintball marker.

Gatling Gun

Uses cartridge fed bullets that are chemically fired.

Machine Gun

Fully automatic portable magazine or ammunition belt fed gun.

Missile

Chemically self-propelled weapon with guidance system.

Modern Guns

Fires rounds that are chemically propelled. Varies in size, shape and loading mechanisms.

Pneumatic

Use compressed air or compressed inert gasses to hit or fire a projectile.

Rocket Launcher

Rocket propelled projectile, usually no guidance system, can often be launched by an individual.

Torpedo

Warhead propelled underwater, proximity detonation

4) Concept Proposals

The following section is what type of launcher each team member will be creating or what mechanism they are considering using as a prototype for the CAD course that.

Slingshot

Use slingshot elastics attached to a support to pull back and "lock" the projectile. Pull the trigger to fire. Concerns with the slingshot are mainly how to load the dough ball without it falling out and ensuring that it fires properly. The energy is stored in the elastic.

Arbalest

The arbalest is a large weapon very similar to the crossbow with the arbalest being much larger, much more powerful, and with a greater range. Since an arbalest was much larger than earlier crossbows, and because of the greater tensile strength of steel prod, it had a greater force [5-9]. Whereas the crossbow was considered to be accurate up to 200 yards the Arbalestwas accurate up to 500 yards [5]. Arbalests were sometimes considered unfair weapons, since an inexperienced arbalester could use one to kill a knight who had a lifetime of training. So, by adapting this method, the purpose of project is build a arbalest with a steel prod (greater tensile strenght) that is easy to manipulate and capable of launching a ball instead of a bolt.

Figure - Illustration of an arbalest

Ballista

The Ballista design is similar to a giant crossbow and works by using tension (the projectiles are launched powered by twisted skeins of rope, hair, or sinew). This machine can throw heavy bolts, darts and spears along a flat trajectory. The force of the missiles launched from the Ballista is designed to have great penetration and are capable of skewering several of the enemy at one time [10].

The ballista was designed as a giant catapult. One type of ballista was called a springald (it closely resembled a crossbow in function with a vertical springboard fixed at its lower end to a timber frame). The springboard moved like a lever and the missiles thrown from the Ballista catapults were deadly. The Ballista catapults were highly accurate and could launch missiles across hundreds of yards[10,6].

Figure - Illustration of a Ballista

Mangonel

Another kind of middle aged siege weapon is the Mangonel. This medieval siege attack weapon is similar to a catapult which used torsion or counterpoise as propulsion power. Mangonels are equipped with a bowl-shaped bucket at the end of its arm which is loaded with the projectile. The rope attached to the bottom end of the arm was twisted which propelled the arm by torsion. The rope linked to the top end of the arm was responsible for arming the weapon and also served as the trigger for launching. Wheel could be added to the model to ensure mobility. [11]

Figure - Mangonel Sketch. [12]

Even though Mangonels were not as accurate as the Ballista, they could throw missiles further than a Trebuchet. The Mangonel was not as accurate as the Ballista but it was able to throw missiles further than a Trebuchet. In this type of siege weapon projectiles were thrown in an overhead arc as opposed to the straight trajectory of the dart throwing Ballista.[11]

The Mangonel's compact design makes it a good choice for the project. Even though it may be complicated due to possibility to use more than one energy storage system, it can be adapted to a simpler model. Unfortunately, this kind of structure is a little bit hard to change the launching angle after the structure is done, so the SolidWorks® approach will be very useful when trying to build the model in the optimum launch angle. Also, this model can throw objects really fast making possible to win the longest launch.

5) References

[1] Luner, J. J. "Achieving continuous improvement with the dual response approach: A demonstration of the roman catapult." Quality Engineering, 6 - 1994

[2] Pattinson, J. andSnyman, J.A. ''Mathematical modelling and optimal design of a conceptually new catapult'' South African Journal of Science 102, 2006.

[3] Cesare Rossi, FlavioRusso "Reconstruction of the Greek–Roman repeating catapult" Mechanism and Machine Theory, 45, 2010, Pages 36–45.

[4] Marco Ceccarelli (Ed.) "History of Mechanism and Machine Science 1, Distinguished Figures in Mechanism and Machine Science, Their Contributions and Legacies, Part 1." Dordrecht: Springer, 2007, ISBN 9781402063664.

[5] "The arbalest" All Thing Medieval.

Link: http://medieval.stormthecastle.com/armorypages/arbalest.htm

Accessed: 26/01/2013

[6] "Medieval Weapons Glossary"

Link: http://www.swordsknivesanddaggers.com/medieval-weapons-glossary.html

Accessed: 26/01/2013

[7] "Medieval Weapons List"

Link: http://www.buzzle.com/articles/medieval-weapons-list.html

Accessed: 26/01/2013

[8] "Crossbow Terminology"

http://windlegends.org/crossbow.htm

Accessed: 27/01/2013

[9] "The Straight Bow" New World Arbalest

Link: http://www.crossbows.net/1_straight.php

Accessed: 27/01/2013

[10] "Ballista" Middle Ages

Link: http://www.middle-ages.org.uk/ballista.htm

Accessed: 27/01/2013

[11] "Mangonel" Middle Ages

Link: http://www.middle-ages.org.uk/mangonel.htm

Accessed: 27/01/2013

[12] "A Timeline of Development of the Catapult"

Link: http://www.stormthecastle.com/catapult/development-timeline-of-the-catapult.htm

Accessed: 27/01/2013

[13] "Catapults"

Link: http://zircon.mcli.dist.maricopa.edu/mlx/warehouse/01301-01400/01331/

Accessed: 25/02/2013

[14] "Range of a projectile"

Link: http://en.wikipedia.org/wiki/Range_of_a_projectile

Accessed: 25/02/2013

6) Figures

[1] "Projectiles Motion"

Link: http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Vectors/ProjectilesMotion.html

Accessed: 26/01/2013

[2,3] "Catapults" , pg. 4

Link: http://zircon.mcli.dist.maricopa.edu/mlx/warehouse/01301-01400/01331/

Accessed: 25/02/2013