By Ronald W. Leigh, Ph.D.
June 15, 2009
Copyright © 2007 Ronald W. Leigh
Catapults go back at least to 750 B.C., for there is a biblical reference to king Uzziah (Azariah) who provided his army with
engines, invented by skilful men, to be on the towers and upon the battlements, wherewith to shoot arrows and great stones (2 Chronicles 16:15, ASV)
These early catapults, which were used for at least 2000 years, probably employed torsion to propel the projectile. Around A.D. 1250 the trebuchet was introduced. It is a type of catapult, but operates on a different principle than the early catapults.
The trebuchet uses a "main-weight" (often referred to as a counterweight or counterpoise). As this weight is pulled down by gravity, it raises a long throwing arm to which a sling is attached. When the sling is "whipped" up to a particular angle in relation to the arm, one side of the sling is released which in turn releases the projectile.
For purposes of this discussion we will distinguish between three common types of trebuchet.
Type 1 — Fixed platform with fixed main-weight.
Type 2 — Movable platform (on wheels) with fixed main-weight
Type 3 — Fixed platform with free-swinging main-weight
Type 1 forces the main-weight to fall in a circular path. Types 2 and 3 allow it to fall in a straight vertical line, or nearly so. The calculations explained below apply to types 2 and 3 and assume a straight line descent of the weight.
The overall length of the throwing arm and the location of the fulcrum are both crucial. The ratio of the length of the main weight arm to the length of the projectile arm is typically between 1/2 and 1/5.
The following calculations approximate a trebuchet's projectile velocity at the point of release.
Calculate the distance the weight falls. (Assume that the weight falls in a straight line.)
Distance = Main weight arm • sin(A) + Main weight arm • sin(B)
We do not yet calculate the velocity of the weight at the bottom of its fall because this velocity is determined by an adjusted acceleration (due to the resistance caused by the weight of the projectile.)
Calculate the torque on both arms. For this calculation the projectile is treated as though it is suspended straight below the end of the Projectile arm. (Early in the projectile's travel the resisting torque is smaller than it would be if the projectile were suspended straight below the end of the arm, but later in its travel the resisting torque is greater since the projectile is forced to change direction. To simplify these calculations, it is assumed that this smaller torque and greater torque nearly cancel each other out.) The fact that the projectile travels through a curved path is considered in a later step.
Torque on main weight arm = Length of main weight arm • (Main weight + ½ weight of main arm)
Torque on projectile arm = Length of projectile arm • (Projectile weight + ½ weight of projectile arm)
Use the ratio of Projectile arm torque to Main weight arm torque to reduce free acceleration of the Main weight to actual acceleration. The actual acceleration is somewhere between these these two extremes: (1) If torque on both sides of the fulcrum were the same, the arms would balance. In other words, zero-acceleration. (2) If there were no projectile and thus no Projectile arm torque, the Main weight would fall at "full" acceleration, or 32.16 ft/sec².
|Main weight arm
|extreme (2)||0 ft-lb||100,000 ft-lb||0.00||32.16 ft/sec²|
|typical case||4,000 ft-lb||50,000 ft-lb||0.08||29.59 ft/sec²|
|extreme (1)||10,000 ft-lb||10,000 ft-lb||1.00||0 ft/sec²|
Actual acceleration = Free acceleration • (1 – ( Projectile arm torque / Main weight arm torque ) )
Once actual acceleration is determined, and the distance the Main weight falls is also determined, velocity of the Main weight can be calculated from the formula:
Actual velocity of the Main weight = √2 • actual acceleration • distance
Then velocity of the projectile can be calculated by determining the ratio the Projectile travel to the Main weight travel. The distance the projectile travels is best measured by using AutoCAD or SolidWorks to construct a spline representing the path of the projectile in a side view of the trebuchet.
Projectile velocity = Main weight velocity • (Projectile travel / Main weight travel )
Estimate reduction factors. Things which reduce velocity of the projectile at its point of release might include:
The final estimate of the projectile velocity (from the previous step) can be converted to throwing distance using the following formula. This formula was developed in the paper Projectile Range. Here full acceleration due to gravity is used since the projectile is now free.
D = V² / 32.16
A spreadsheet can be created which takes all of the above factors into account. This spreadsheet should be used to see which changes in the design of the trebuchet contribute most to an increased range. (Remember that changing the various factors automatically changes the ratio of the Projectile travel to the Main weight travel.)