Sum of torque

[EOW]

In my opinion you will find the error in analyzing the way you are calculating the integrals numerically, in that small errors introduced when summing numbers containing large differences in numerator and denominator give rise to an error in the totals.

I had this problem with the F3x and F3y forces because it is a double  integral. Even Wolfram is not perfect for the double integral I wrote a program with a high accuracy, and like that I found the sum of forces at 0 with the number of digit I want (Wolfram not, my program yes). But I don't need F3x and F3y in my use of the device because  the red axis is fixed. And I have the indefinite integrals for F2x, F2y, F1x and for the torques.

Do you understand the device ?

For Noether, don't forget there is an hypothesis: the potential energy don't change with time.

I corrected my integrals for the F3x and F3y, I divided by 4 but the true integral take x inside. The results are the same but the integrals are easier to calculate

[EOW]

The indefinite integral for the force F2. I don't think this is a problem of accuracy.

And the difference of energy is more than 20 % for an angle of 1rd, with the force I use, the difference of energy is 0.25-0.203=0.046 J it is near 20%. If the force is 1000 times higher (F2=247 N) the difference is 1000*0.046=46 J. It's easy to increase the force because the springs like the balls are always in the same relative position inside the half disk. If the device turns at 10 tr/s, the power is 460 W.

@LibreEnergia: Do you understand how the device works ?

[EOW]

With an extreme accuracy and with a very small angle it's possible to have directly the result without an integral, there is always a difference:

#include <stdio.h>

#include <gmp.h>
#include <mpfr.h>

#define DG 2000

int main()
{
mpfr_t   x,l,angle,c,s,r,r2,sum,dl,fx,fy,f,temp1,temp2,temp3,temp4,temp5,temp6,temp7,temp8,temp9;
mpfr_init2 (x, DG);
mpfr_init2 (l, DG);
mpfr_init2 (angle, DG);
mpfr_init2 (c, DG);
mpfr_init2 (s, DG);
mpfr_init2 (r, DG);
mpfr_init2 (r2, DG);
mpfr_init2 (dl, DG);
mpfr_init2 (sum, DG);
mpfr_init2 (fx, DG);
mpfr_init2 (fy, DG);
mpfr_init2 (f, DG);
mpfr_init2 (temp1, DG);
mpfr_init2 (temp2, DG);
mpfr_init2 (temp3, DG);
mpfr_init2 (temp4, DG);
mpfr_init2 (temp5, DG);
mpfr_init2 (temp6, DG);
mpfr_init2 (temp7, DG);
mpfr_init2 (temp8, DG);
mpfr_init2 (temp9, DG);

mpfr_set_d(x, 1e-20, MPFR_RNDD);
mpfr_set_d(l, 0.0, MPFR_RNDD);
mpfr_set_d(fx, 1.0, MPFR_RNDD);
mpfr_div_d(fx, fx, 6.0, MPFR_RNDD);
mpfr_set_d(fy, 0.182355, MPFR_RNDD);
mpfr_div(angle, fx, fy, MPFR_RNDD);
mpfr_atan(angle, angle, MPFR_RNDD);

mpfr_mul(temp2, fx, fx, MPFR_RNDD);
mpfr_mul(temp3, fy, fy, MPFR_RNDD);
mpfr_add(temp2, temp2, temp3, MPFR_RNDD);
mpfr_sqrt(f, temp2, MPFR_RNDD);

mpfr_mul_d(temp2, x, 2.0, MPFR_RNDD);
mpfr_sin(temp1, temp2, MPFR_RNDD);
mpfr_mul_d(temp1, temp1, 4.5, MPFR_RNDD);
mpfr_add_d(temp1, temp1, 4.5, MPFR_RNDD);
mpfr_sqrt(l, temp1, MPFR_RNDD);

mpfr_set_d(temp2,3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196,MPFR_RNDD);
mpfr_div_d(temp2, temp2, 4.0, MPFR_RNDD);
mpfr_add(temp2, temp2, angle, MPFR_RNDD);
mpfr_sub(angle, temp2, x, MPFR_RNDD);

mpfr_sin(s, angle, MPFR_RNDD);


mpfr_mul(temp5, l, s, MPFR_RNDD);
mpfr_mul(r2, temp5, x, MPFR_RNDD);


mpfr_sqrt_ui(temp7, 2.0, MPFR_RNDD);
mpfr_mul_d(temp7, temp7, 1.5, MPFR_RNDD);
mpfr_sub(dl, temp7, l, MPFR_RNDD);
mpfr_cos(c, angle, MPFR_RNDD);
mpfr_mul(r, dl, c, MPFR_RNDD);

mpfr_add(sum, r, r2, MPFR_RNDD);
mpfr_mul(sum, sum, f, MPFR_RNDD);



printf("\n");
printf("\n");
mpfr_out_str (stdout, 10, 0, l, MPFR_RNDD);
printf("\n");
printf("\n");
mpfr_out_str (stdout, 10, 0, angle, MPFR_RNDD);
printf("\n");
printf("\n");
mpfr_out_str (stdout, 10, 0, s, MPFR_RNDD);
printf("\n");
printf("\n");
mpfr_out_str (stdout, 10, 0, c, MPFR_RNDD);
printf("\n");
printf("\n");
mpfr_out_str (stdout, 10, 0, r, MPFR_RNDD);
printf("\n");
printf("\n");
mpfr_out_str (stdout, 10, 0, r2, MPFR_RNDD);
printf("\n");
printf("\n");
mpfr_out_str (stdout, 10, 0, sum, MPFR_RNDD);
printf("\n");
printf("\n");

//mpfr_clear (s1);



return 0;
}

[Gabriele]

Hello. Why don't you model your idea with WorkingModel2D... should be simple...

[EOW]

Thanks for the name of the software but it's not possible to simulate N balls I tried with Comsol.

I found my error it was in the integral about the length of the arm.

Another idea: take 3 semicircles. One like before, radius of 0.5 and 2 smaller with a radius of 0.25. There is no straight wall like before. I do the same with balls and springs with the law 1/d^2 (or 1/d^3). I place the device on an object in rotation.  Each semicircle can turn with its arm (color).

The torque from the 3 semicircles device is 0 on the green dot but the object turns and it can receive a difference.

I calculated the 3 forces (radial), I have:

Tangential forces :
F1=1/28 (black)
F2=1/6 (red)
F3=15/100 (green)

The value are exact it is indefinite integrals.

The torque at 1.5 is 1/28*1.75-1/6*1.5+15/100*1.25 = 0
But at 10 it is 1/28*10.75-1/6*10.5+15/100*10.25 = 0.1904

Don't remember the semicircles don't receive any torque around themselves from pressure of balls because the pressure is perpendicular to the surface (center of the circle). I want the device keeps its shape because walls are 3 bodies, so I need to apply a torque on the center, for example I give a torque on the red center to compensate the 1/6 torque. I need an energy but I can recover more from 2 others.

Note the main object will receive a torque on it.