Very interesting! I will be continuing to make a complete model in Mathematica including the 3 phases of the projectile,s flight: 1. The constrained sliding, 2. The slinging on the trebuchet and 3. The free flight under gravity. As mentioned before: the sliding is essential to gain projectile range and will improve substantialy your trebuchet,s efficiency. Good luck and keep on experimenting!

@Frederick Wu these are some wonderful images and event looks awesome. So did you guys end up using Mathematica for building or using those trebuchets? Did it help to win?

Also I see that your PDF file is a Mathematica notebook. Could you please also attach the actual notebook in .NB format, please?

Indeed, this would avoid the constrained phase when the projectile is sliding on the floor. But this is not how the real trebuchets worked. One can prove, as Siano did, that the range and efficiency is much higher when you have this initial sliding phase. By the way yr idea of a pit to avoid the constrained sliding is the intermediary solution mentioned in Siano's paper (page 12)

@Erik, Thanks for share the new information from Mr. Siano.

However, I am not sure if this drawing are made by da Vinci. But it looks not same style of da Vinci's drawing. I guess, it's probably an entertain idea made by others, but in da Vinci's name.

I happend to have a few image below from a book, "The Invention's of Leonardo da Vinci", Edited by Charles Gibbs-Simith and Gareth Rees, Printed by Charles Scibner's Sons, in 1978.

Leonardo improved catapult at his time, instead of trebuchet.

@Erik Mahieu,

If I understand you right. You mean the 1st phase of Trebuchet, can be seperated two (1.1 and 1.2 ) in detail. http://demonstrations.wolfram.com/TrebuchetBallistics/

phase 1:

1.0 Start from all stationary state

1.1 Only two masses ( counterweight and main beam ) involved in physical with 2 DOF rotation. Becasue, sling or rope at projectile side is not straight and no force. projectile is still on ground in this phase.

1.2 Three masses ( counterweight, projectile and main beam ) involved in physical with 3 DOF rotation, sling is straight with a force to pull projectile until it released.

phase 2: the projectile is free throw with ballistic trajectory in air, with air drag. ( Air drag is clear.)

I think below factors in phase 1 might need to consider for efficiency.

1. mass of main beam as a rigid bar;

2. damping or friction on three axes;

3. collision at relearing or after releasing (or the remaining energy of the system, where is it ? ) ;

I think 3rd part of energy is consumed by collision when the beam hit something as end-stop. Usually we can use the position of end-stop to control the velocity direction of projectile. See Siano's page: http://algobeautytreb.com/

It appears Siano uses a bar as end-stop to control the velocity direction of projectile. The moment, when main bean hit on a bar, should be the moment the projectile releasing. Maybe you have conside the 3 part in side equations ?

SystemModeler is indeed a better choice for modelling this type of "machinery"

But a real trebuchet is more complicated than is depicted in all the above (including my own demonstrations!) If you look at the paper "Trebuchet Mechanics" by Donald B. Siano and at this design from the paper:

you can see that there is a third link in the kinematic chain that was neg;acted in the above: the part indicated l3 (between {x2,y2} and {x3,y3} laying horizontally on the ground where the projectile is attached. Adding this link complicates the analysis (Siano does it at the end!) but it is essential for the trebuchet deadly efficiency since it will throw the load much further and with a higher speed! One day, when more time available, I will attempt to add tis feature but it would be a nice Mathematica/SystemModeler combined exercise.

Any volunteers?

Perhaps you could make a use of System Modeler too? Take a look at Model Your Own Medieval Catapult Design.

Hello Erik,

Excellent job for developing two trebuchet demos ! I have two questions.

1. Inside you notebook (Trebuchet revisited-9.nb), It state Lagrangian 3 equations, the first equation is like below. However, in demo code, you use anther equation, Which one is correctlly? Is code equations better or solvable? How you get three equations from Lagrangian?

2. I make some approximate calucation based on your demo, see attachment notebook. Input energy ( potential energy from countermass) = mgh ; Output energy ( kinematic energy of projectile weight with released velocity) = 1/2 m v^2 ; I found, Input energy is almost same as output energy. That means the effiency of device is always 100%. Usually, a practical trebuchet effiency is about 10% - 85% (maximum) based on Donald B. Siano's paper. Is that means this demo is a perfect theoretical simulation so far, without consider energy loss (like conflict and friction)? Am I understanding right ?

If I want to add a coefficient for effiencency as engineering way, where I should put a coefficient inside the 3 Lagrangian equations? What do you suggest ?

BTW, One of your fans found you changed your logo. ^_^

Attachments:

This is a better version with some loose ends fixed.

One can experiment and see that the most critical parameter to obtain a long shooting range is the moment the projectile is released from the poach. In this Manipulate, this is set by the angle between the vertical and the sling {rlsAngle) at the moment of release. This was the most critical job of the trebuchet operator: pull a string to open the poach at the right moment to obtain the desired range. Good luck!

Attachments:

This is exactly what I suggested David. here is an attempt to "glue" the two together using the trebuchet NDSolve output as the initial conditions of the projectile equations. I attach a preliminary notebook based on my previous demonstration. Still needs some refining!

Attachments:

Hi Varun. I took a look at the notebook. I don't follow the physical logic. I only see NDSolve in use to solve the trebuchet equations. To elaborate on my response of 1 day ago, the solution for ballistic flight of the projectile would go like this:

1) Solve the trebuchet just as done by Erik Mahieu in the demonstration. This solution terminates at a final time tMax, which is when the projectile leaves the pouch to begin ballistic flight. At that time tMax we have numerical values for the 3 generalized coordinates and their derivatives. These are the arm angles defining the configuration of the trebuchet as it goes through its motion.

2) Erik also gives the equations for the x, y location x[t], y[t] and the velocity components of the projectile in the pouch as derived from the general coordinate angles. So at the point where the projectile leaves the pouch it is at x[tMax], y[tMax] and has velocity components x'[tMax] and y'[tMax].

3) The ballistic flight of the projectile after leaving the pouch is a completely separate problem from the trebuchet problem. The trebuchet equations have nothing to do with it. The equations for ballistic flight without drag are the usual: xx''[t]==0, yy''[t]==-g. These are solved using NDSolve, and the connection to the trebuchet problem is in the initial conditions xx[tMax]==x[tMax], xx'[tMax]==x'[tmax], yy[tmax]==y[tmax], yy'[tmax]==y'[tMax]. (Notice that x and y refer to the values derived from the trebuchet solution while xx and yy are the coordinates for the ballistic flight.) The ballistic flight is solved with NDSolve with a "StopIntegration" when yy[t]==0, which is when the projectile hits the ground. So the ballistic flight is solved for {t, tMax, Infinity} with Event Location or WhenEvent used to stop the simulation on impact.

4) So you see there are two separate solves for two different differential equation systems. First for the trebuchet, and then for the ballistic flight with initial conditions provided by the end point of the trebuchet solution. These will provide separate interpolating functions for the solutions: the first set for coordinates of the trebuchet, and the derived values for the location of the projectile in the pouch; and the second set for the coordinates of the projectile in ballistic flight after leaving the pouch. Both are required for plotting the full path of the projectile.

If you don't mind, I would like to make a few suggestions about using Mathematica. First, your notebook puts all the calculations in a single cell. It is easier to find your way through a problem if you put each calculation in its own cell, and you evaluate and look at the result as you go along. Second, if you are going to put something in Manipulate, first solve it without. For example, in the method I outline above, do each of these things in separate cells, one at a time, and look at the result by plotting the NDSolve solutions using Plot and ParametricPlot, so you can see that you are getting what you expect. Do the same for the following graphics. Once you have everything going as you expect, then you can delete the output cells, merge the separate cells into one with semicolons as separators, and then encapsulate in animations or Manipulations.

Best regards, David

David said >

It is easier to find your way through a problem if you put each calculation in its own cell, and you evaluate and look at the result as you go along.

I had thought Mathematica 11.0 has a feature that debugs functions step by step, it has breakpoints i cannot get it to work yet. Documentation is very thin - despite setting breakpoints it never stops :)

I made the below for an earlier Mathematica (4.0). I'm unsure if it works "for complicated functions" in Mathematica 11.0 or not. It is in jutilspkg part of fNBookForm2 (orig. part of PeriodicTable` pkg). It is supposed to do what you said without all the ; editing. This allows being able to freeform work a cell at a time, then have a function made auto. It works but needs tender care, and has a help page.

Attachments:

I would not combine the equations but use the output (speed and direction) as the input of the projectile equations. Two separate computations.

Hi, Thanks

I was able to generate the equations of motions for the Trebuchet and calculate the Lagrangian.

I tried using the equations of Projectile motion as well. But I am not quite sure as to how to combine the equations from Projectile with Air Drag into the Trebuchet equations to animate the projectile to actually leave the Trebuchet and make an impact against something like a wall at a distance say x='constant'.

Is there any specific way to do that?

The equations I got were:

cn = {L1, L2, L3, L4, m1, m2, mb, rg, rc};


(*Coordinates of the short end of the beam*)     (*Defined 'a'*)
x1[?_] := L1 *Sin[?];
y1[?_] := -L1 *Cos[?];
(*Coordinates of the long end of the beam:*)    (*Defined 'b'*)
x2[?_] := -L2* Sin[?];
y2[?_] := L2 *Cos[?];
(*Coordinates of the center of mass of the Counterweight*)     \
(*Defined 'c'*)
x4[?_, ?_] := 
  L1* Sin[?] - L4 *Sin[? + ?];
y4[?_, ?_] := -L1 *Cos[?] + 
   L4* Cos[? + ?];
(*Coordinates of the Projectile end of the Sling *)    (*Defined 'd'*)


x3[?_, ?_] := -(L3 *Sin[? - ?] + 
     L2* Sin[?]);
y3[?_, ?_] := -(L3 *Cos[? - ?] - 
     L2* Cos[?]);


(* potential energy of the three parts:*)
PE[?_, ?_, ?_] := 
  m1 *g* y4[?, ?] + m2 *g* y3[?, ?] - 
   mb* g* rc *Cos[?];

(* kinetic energy of the three parts:*)
KE[?_, ?_, ?_] := 
  0.5*m1*((Dt[x4[?, ?], t, Constants -> cn])^2 + (Dt[
         y4[?, ?], t, Constants -> cn])^2
     ) + 0.5*
    m2 *((Dt[x3[?, ?], t, Constants -> cn])^2 + (Dt[
         y3[?, ?, Constants -> cn], t])^2) + 
   (mb/2) *(rg^2)* Dt[?, t, Constants -> cn]^2;
(* the lagrangian*)
Lag = KE - PE;
L[?_, ?_, ?_] := 
  KE[?, ?, ?] - PE[?, ?, ?];


(* define what is constant, and the derivatives of the angles, \
denoted by the "d" suffix:*)
EL = L[?, ?, ?] /. {Dt[?, t, 
      Constants -> {L1, L2, L3, L4, m1, m2, mb, rg, rc}] -> ?d,
        Dt[?, t, 
      Constants -> {L1, L2, L3, L4, m1, m2, mb, rg, rc}] -> ?d,
           Dt[?, t, 
      Constants -> {L1, L2, L3, L4, m1, m2, mb, rg, rc}] -> ?d, 
    Dt[?, t] -> ?d, Dt[?, t] -> ?d, 
    Dt[?, t] -> ?d};



eq? = 
  Simplify[ 
   Dt[D[EL, ?d], t] - D[EL, ?] == 0 /. {Dt[L1, t] -> 0, 
     Dt[L2, t] -> 0, Dt[L3, t] -> 0, Dt[L4, t] -> 0, Dt[mb, t] -> 0, 
     Dt[m1, t] -> 0, Dt[m2, t] -> 0, Dt[g, t] -> 0, Dt[rg, t] -> 0, 
     Dt[rc, t] -> 0,
          Dt[?, t] -> ?d, Dt[?, t] -> ?d, 
     Dt[?, t] -> ?d, Dt[?d, t] -> ?dd, 
     Dt[?d, t] -> ?dd, Dt[?d, t] -> ?dd}];

eq? = 
  Simplify[Dt[D[EL, ?d], t] - D[EL, ?] == 
     0 /. {Dt[L1, t] -> 0, Dt[L2, t] -> 0, Dt[L3, t] -> 0, 
     Dt[L4, t] -> 0, Dt[mb, t] -> 0, Dt[m1, t] -> 0, Dt[m2, t] -> 0, 
     Dt[g, t] -> 0, Dt[rg, t] -> 0, Dt[rc, t] -> 0,
          Dt[?, t] -> ?d, Dt[?, t] -> ?d, 
     Dt[?, t] -> ?d, Dt[?d, t] -> ?dd, 
     Dt[?d, t] -> ?dd, Dt[?d, t] -> ?dd}];

eq? = 
  Simplify[Dt[D[EL, ?d], t] - D[EL, ?] == 
     0 /. {Dt[L1, t] -> 0, Dt[L2, t] -> 0, Dt[L3, t] -> 0, 
     Dt[L4, t] -> 0, Dt[mb, t] -> 0, Dt[m1, t] -> 0, Dt[m2, t] -> 0, 
     Dt[g, t] -> 0, Dt[rg, t] -> 0, Dt[rc, t] -> 0,
          Dt[?, t] -> ?d, Dt[?, t] -> ?d, 
     Dt[?, t] -> ?d, Dt[?d, t] -> ?dd, 
     Dt[?d, t] -> ?dd, Dt[?, t] -> ?dd}];


sol = Solve[{eq?, eq?, eq?}, {?dd, ?dd, ?dd}];

Thanks