使用Matlab计算弓弦的力量

Question

I'm trying to calculate the force in a bow string while using MATLAB 's ode45 , I have this differential equation

M(x)=d^2y/dx^2 * (1+(dy/dx)^2)^(-3/2)=-q * y where M(x) is the average torque in the bow, q=11.4716, and y(0)=0.3, y'(0)=0 and 0≤ x ≤0.3668

I have calculated the differential equation with ode45 but I have tried to calculate the force with this code;

X=0.3668
tolerance=odeset('Abstol', 1e-9, 'Reltol', 1e-11);
[x,M]=diffekv45(tolerance,X)    

force=zeros(length(x),1);
for i=1:length(x)
    if x(i)==0
        force(i)=0;
    else
        force(i)=(0.3-M(i,1))./x(i); %%% 0.3-M(i,1) is because it is equilibrium in the         %%center of the bow
    end
end
totalforce=sum(force);

Line 6 is calculating the torque in all the readings. I suppose that the first element in M is the average torque.

I'm using these functions

function [x,M] = diffekv45(tolerance,X)
    u0=[0.3;0];
    [x,u]=ode45('M',[0,X],u0,tolerance);
end

function M=M(~,u)
global q 
    M=[u(2)
        -q*u(1)*(1+u(2)^2)^(3/2)];
end

When I calculate the force with a higher or lower tolerans I get a significant higher or lower value of the force. That's why I'm wrong. So how can I calculate the force in the bow string?

Answer 1

You have a 2nd order equation in terms of the displacement, which needs to be split up into two first order equations for use in orde45() .

Use a state vector h=[y; diff(y,x)] h=[y; diff(y,x)] and create the following differential function for use in ode45()

function hp = deriv(x,h)
  y = h(1);       %First element of h is the deflection
  v = h(2);       %Last element of h is the slope
  hp = [v; -q*y/(1+v^2)^(-3/2)];
end

and then call

h0 = [0.3; 0];
ode45(deriv, [0,X], h0, tol);