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)

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:

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