How to find the solution of a system of equations with two known partial derivatives ?

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Cédric 2024-9-23

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编辑： Cédric 2024-11-26

Hello everyone,

I'm trying to find the solution to a system of equations with two known partial derivatives. As I am in a study with very small displacements, I consider the net variations equal to the partial derivatives...

The code I use is not correct. The equations equ_8 and equ_9 are false because when compiled it gives 0==169/10000 and 0==29/10000. Furthermore, I think that forcing the study in alpha_0 by supervising alpha as I do in the "range" is really not the solution!

Can you help me please?

% * equ_1 to equ_7 represent a macanical system
% * I can only measure alpha (absolute) and l1 & L2 (relative) therefore the
%   variations of l1 and l2
% * By measurement I know the variation of alpha compared to l1 at alpha_0
%   and the variation of alpha compared to l2 at alpha_0
% * ==> I try to know the values of l1 and l2 at alpha_0
a1=150;
a2=150;
d=2910;
P=1;
alpha_0=0.0196;
syms alpha x y l1 l2 X1 X2 Y1 Y2
equ_1=l1^2==(x-a1*cos(alpha)+d)^2+(y-a1*sin(alpha))^2;
equ_2=l2^2==(x+a2*cos(alpha)-d)^2+(y+a2*sin(alpha))^2;
equ_3=X1+X2==0;
equ_4=Y1+Y2-P==0;
equ_5=a1*(X1*sin(alpha)-Y1*cos(alpha)) == a2*(X2*sin(alpha)-Y2*cos(alpha));
equ_6=Y1/X1==(y-a1*sin(alpha))/(x-a1*cos(alpha)+d);
equ_7=Y2/X2==(y+a2*sin(alpha))/(x+a2*cos(alpha)-d);
equ_8=diff(alpha,l1)==0.0169;
equ_9=diff(alpha,l2)==0.0029;
equations = [equ_1 equ_2 equ_3 equ_4 equ_5 equ_6 equ_7 equ_8 equ_9];
vars = [alpha x y l1 l2 X1 X2 Y1 Y2];
range = [alpha_0-0.1 alpha_0+0.1 ; -2900 2900 ; 0 -2400 ; 0 6000 ; 0 6000 ; -10 0 ; 0 10 ; 0 1 ; 0 1];
S=vpasolve(equations,vars,range);
l1_0 = S.l1;
l2_0 = S.l2;

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John D'Errico 2024-9-23

This is making little sense. alpha is apparently a (unknown) number. You define it as that, and use it as such. And you apparently want to solve for alpha as a number, not as a function.

But then you want to define alpha as a function of two other variables, that is, l1 and l2. You cannot differentiate it unless alpha is a function of those variables.

And you are clearly not trying to solve a differential equation, so this all makes little sense. You probably need to explain what you want to do better, more clearly.

Cédric 2024-9-23

编辑：Cédric 2024-11-26

The first seven equations express the balance of my system. x and y are the coordinates of it's gravity center, alpha its inclination, Xi and Yi the forces according to x_axis and y_axis on it, l_i two distance from my system to two fixe points and finally P the weight.

With these seven equations, by fixing x and y to choose a desired position, if I solve the system (7 equations and 7 unknowns) I have alpha and the forces (which are experimentally correct)

In the new experience, I still know alpha by sensor mesurement. However, when the system is commissioned (moment zero), I do not know the distances l_i but I know alpha_0. I am trying to know these distances (l1_0 and l2_0) by measuring the variation of alpha when I vary the distances l_i separately (what I call delta_alpha/delta_l1 and delta_alpha/delta_l2)

I would therefore like to inject these two measurements into the system of equations (knowing that x and y are now unknown) to know l1 and l2.

I hope my question is better expressed (knowing that basically, I don't speak English!)

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Answer 1

Arnav 2024-11-25

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Hi @Cédric,

Since alpha is not a function of l1 and l2, it cannot be differentiated with respect to them. Equations 1 and 2 relate alpha, l1 and l2 in an implicit manner. From this, we can derive an implicit form of the partial derivatives using the chain rule as shown:

F1 = (x-a1*cos(alpha)+d)^2+(y-a1*sin(alpha))^2 - l1^2;
F2 = (x+a2*cos(alpha)-d)^2+(y+a2*sin(alpha))^2 - l2^2;
equ_8 = -diff(F1, l1) / diff(F1, alpha) == 0.0169;
equ_9 = -diff(F2, l2) / diff(F2, alpha) == 0.0029;

You may refer to the follow link for more information about symbolic differentiation: https://www.mathworks.com/help/symbolic/sym.diff.html

See if it helps you.

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Cédric 2024-11-26

Hi Arnav,

Thanks for your help, I'll reread the chain rules.

In this case, x and y are "functions" of α. In fact x, y, α,

,

and

all change together. I try to numericaly find

when, for an unknow equilibrium position, I only know α,

and

.

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How to find the solution of a system of equations with two known partial derivatives ?

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How to find the solution of a system of equations with two known partial derivatives ?

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