Trebuchet Throwing Arm Optimization

[Brandon Preble]

Summary:

Some help would be greatly appreciated for finding the ideal taper and x/y dimensions of a trebuchet throwing arm to optimize strength and minimize the moment of inertia.​

Long Version:

I am a high school junior currently in the process of working on an ambitious project to build a 21 feet tall trebuchet. I have recently built a 6-foot-tall model which successfully threw a baseball up to 80 feet with 55 pounds of counterweight. The issue now is that the next trebuchet, which is a hinged counterweight design, will require a large grant for all the hemlock wood and steel plates/axles that it will utilize. I need to prove that it will not break with a full counterweight of 2,000 pounds plus the weight of the trebuchet itself. The throwing arm will load back 135 degrees from vertical when it's in a ready to fire position. The main problem is regarding the throwing arm which is currently designed to by 4in by 12in by 15ft, but taper down to 4in by 6in at the top with 1/4" metal plates encasing the bottom. The short arm is 3ft and the long arm is 11.5ft to the base of the release pin. The counterweight box has not been modeled yet as I want to make certain the arm is structurally sound first. Ideally, the moment of inertia should be kept as low as possible along with the center of gravity to increase the release velocity of the projectile which will be a 16-pound shot put. What I have issues with due to my limited physics experience, is finding the optimal throwing arm dimensions for hemlock pine and the ideal taper for the arm to keep the arm lightweight, yet durable enough to survive a 2,000-pound launch. If anybody could help me out, I would be very grateful as I am unaware how to make an effective throwing arm on this scale.​

Specifications:

Arm Height- 15 feet
Arm Material- Hemlock wood with 1/4" thick steel reinforcements where the axles intersect.
Short Arm- 3 feet
Long Arm- 11.5 feet
Counterweight- 2,000 pounds of hinged counterweight
Projectile- Mainly 16-pound shotputs
Misc- The trebuchet is on wheels which may relieve some strain.​

 

[Dr.D]

So, did you have a specific question? How much do you know about dynamics, kinematics, and stress analysis?

 

[Brandon Preble]

So, did you have a specific question? How much do you know about dynamics, kinematics, and stress analysis?

I basically want to know the ideal shape and size of a trebuchet arm for the given counterweight and short arm to long arm ratio. Ideal refers to a low moment of inertia, yet strong enough to survive repeated use. I'm in AP Physics C, so I'm aware of basic kinematics, but am at a loss regarding stress analysis which is vital for this problem.

 

[JBA]

First, since I have no background with these machines there are two issues I would need to understand for the determination of the arm loading stress and shape. The foremost is whether the launch restraint is at the end of the long throwing or on the load box end. If the restraint is connected to the load box then the maximum bending load on the arm will only be principally due to weight of the 16 lb shot put. The contribution by the arm will be the weight of the long section applied at the center of gravity of that section.On the other hand, if the restraint is connected to throwing end, then the applied static bending load on the arm in the latched position will be 2000 lbs plus the weight of the load box (which it seems to me will exceed all dynamic arm loading effects).

The other is the species of the hemlock wood you will be using, i.e. Eastern,Western, or Mountain, because the max shear failure stress varies considerably between the Eastern and both the Western and Mountain species. For example, assuming the wood you using is at 12% moisture, according to "Mechanical Properties of Wood" by David W. Green, Jerrold E. Winandy, and David E. Kretschmann" Chapter 4, the max shear stress parallel to the grain for those three species are respectively: 1060 psi, 1290 psi and 1540 psi.

 

[Dr.D]

The trebuchet is kinematically complicated, and varies from one design to the next. To deal correctly with the kinematics, you must specify a design that can then be analyzed. There is also the matter of the projectile release and just when the release occurs. This is all before you get to the system dynamics and analysis of forces.

What you have asked is a major design engineering problem, and not one that most of us want to take on just for fun.

 

[jim mcnamara]

JBA is spot on. My only quibble would be that I cannot easily get any hemlock species with MC = 12% or less. It does exist.

We are not trying to deter you, but get you aimed in the right direction. You cannot do your calculations until you know precisely what species and grade of lumber you are using. Your question is a good one, but to get a good answer you have to go back to your materials choices.

Just so you understand: wood is a biologically produced commodity and there are lots complexities involved. So we have some standards called lumber grades, those stamps you see on lumber.

Lumber grading depends on the number (or lack of) defects in the wood (knots, waning, etc.), MC (moisture content), and species - as JBA points out. In the US, S-DRY is a grade used for framing lumber, most hemlock is sold as the species called 'white wood' along with other species, like fir, as framing timber. S-DRY means not more than 19% MC, a lumber stamp of MC-15 means a moisture content of not more than 5%.

Construction lumber for outdoor use for most locations in the US is probably okay with S-DRY. So chances are you are getting 19% MC lumber in the US and Canada, too.

FWIW - I build furniture, doors, gates for fun. For hemlock MC <12%, my suppliers can only get #1 Common KD (select), as good as it gets. It is way more than twice as expensive as S-DRY, and I would have buy an MBF (1000 board foot lot) to get that price. Point is: grade affects wood cost a lot, as well as its expected physical properties.

 

[Dr.D]

lsn't this a cart before the horse situation? At this point, as far as I understand, the motions and forces are completely unknown, so why worry about material problems at this time?

 

[JBA]

If he selects the wrong material for his components construction then all the time resolving the dynamic loadings will be pointless.

 

[Dr.D]

If he selects the wrong material for his components construction then all the time resolving the dynamic loadings will be pointless.

I totally disagree. The loads to be supported must be known before there can be an intelligent choice of materials. To go the other way around is to say, "Let's make something out of x-material. What shall we make?" The forces are mostly determined by the motions, and they exist irrespective of the material chosen to support those forces.

 

[jrmichler]

There is a PDF available on the internet titled The Algorithmic Beauty of the Trebuchet. It does an excellent job of analyzing and describing the dynamics of the trebuchet, and how to optimize the design.

Wood Handbook, Wood as an Engineering Material is available as a PDF. It is an excellent source for wood mechanical properties, and strength of various types of fasteners.

For the strength calculations, find what textbook the nearest engineering school is using for their undergrad strength of materials course.

Or you can just use the "build and break" method. Build something that looks reasonable and test it. If it does not break, you made it too strong. Reduce weight until it breaks, then add in just enough that it does not break. But read the first two books I listed first.

 

[Tom.G]

To take this sub-discussion to its limits:
Alice: Let's lift this object.
Bob: OK. I'll get a spoon.
Alice: Uhm... that's a car we are trying to lift.