Why is there a diiference in state-space velocity and derivative of state space displacement?

I stumbled on this matlab code online from mathworks. However, I am surprised that the velocity (yvector(:,1)) is different from the velocity obtained from diff(yvector(:,2))./diff(t). Why des this occur? https://www.mathworks.com/matlabcentral/answers/1613980-how-can-i-obtain-the-derivative-of-a-vector-displacement-velocity-but-what-about-acceleration-i-am

clc;  %command window
clear;%clear workspace
close all; % close figures
global m k
m = 80;  %Defining mass of SDOF system
k = 10000;%Defining stiffness of SDOF system
dt = .02; %Defining Time Step:
t = 0:dt:4;% Defining time vector.
y0 = [0.085; 0.1]; %initial vel and disp [vel disp] %velocity. 
[tsol, yvectorl] = ode45(@testode1,t,y0); 
%%plot
figure()
plot(tsol,yvectorl(:,2)), title("Displacement") %Disp.(2)

figure()
plot(tsol,yvectorl(:,1)), title("Velocity") %Vel. (1)

% THIS NEEDS TO GO AT THE END OF THE FILE
%%function statespace
function dy = testode1(t,y)
global m k
dy=[-k*y(2)/m;y(1)];
end

Answer 1

[Mathieu NOE]

hello

where do you see a difference ?

diff or gradient based dispkacement derivatives do pretty well overlay with computed velocity

clc;  %command window
clear;%clear workspace
close all; % close figures
global m k
m = 80;  %Defining mass of SDOF system
k = 10000;%Defining stiffness of SDOF system
dt = .02; %Defining Time Step:
t = (0:dt:4)';% Defining time vector.
y0 = [0.085; 0.1]; %initial vel and disp [vel disp] %velocity. 
[tsol, yvectorl] = ode45(@testode1,t,y0); 
%%plot
figure()
plot(tsol,yvectorl(:,2)), title("Displacement") %Disp.(2)

% derivative of disp
% ddisp = [0;diff(yvectorl(:,2))./diff(t)];
ddisp = gradient(yvectorl(:,2),dt);

figure()
plot(tsol,yvectorl(:,1),tsol,ddisp,'*r'), title("Velocity") %Vel. (1)

% THIS NEEDS TO GO AT THE END OF THE FILE
%%function statespace
function dy = testode1(t,y)
global m k
dy=[-k*y(2)/m;y(1)];
end

[Joe]

The plot is same but the actual values are different if you check the values compare the elements in each vector. That is, yvector(:,1) and diff(yvector(:,2))/diff(t). Why are the values different?Thanks

%Here is what I used to run the differentiation part
x=yvectorl(:,2)
v=diff(x)./diff(tsol)
tv = (tsol(1:end-1)+tsol(2:end))/2;
plot(tv,v)

[Mathieu NOE]

you will always have some difference between the "true" velocity you get directly from the ode solver and a numerical derivation from the displacement. Depending of what order / type of derivation method you use (diff is very coarse !) you can reduce the amount of error.

[Mathieu NOE]

Here we can test 3 methods

the 3rd option is the best and gives less than 1% error

also if you further reduce dt by factor 5 you can reduce that error to less than 0.1%

the difference between true and estimated velocity is plotted in the second subplot of fig 2

clc;  %command window
clear;%clear workspace
close all; % close figures
global m k
m = 80;  %Defining mass of SDOF system
k = 10000;%Defining stiffness of SDOF system
dt = .02/5; %Defining Time Step:
t = (0:dt:4)';% Defining time vector.
y0 = [0.085; 0.1]; %initial vel and disp [vel disp] %velocity. 
[tsol, yvectorl] = ode45(@testode1,t,y0); 
%%plot
figure(1)
plot(tsol,yvectorl(:,2)), title("Displacement") %Disp.(2)

% derivative of disp
% ddisp = [0;diff(yvectorl(:,2))./diff(t)]; % worst
% ddisp = gradient(yvectorl(:,2),dt);       % better in average, except first and last point
[ddisp, ~] = firstsecondderivatives(t,yvectorl(:,2));  % best

figure(2)
subplot(2,1,1),plot(tsol,yvectorl(:,1),tsol,ddisp,'*r'), title("Velocity") %Vel. (1)
subplot(2,1,2),plot(tsol,yvectorl(:,1)-ddisp(:)) %Vel. (1)

% THIS NEEDS TO GO AT THE END OF THE FILE
%%function statespace
function dy = testode1(t,y)
global m k
dy=[-k*y(2)/m;y(1)];
end

%%%%%%%%%%%%%%%%%%
function [dy, ddy] = firstsecondderivatives(x,y)

% The function calculates the first & second derivative of a function that is given by a set
% of points. The first derivatives at the first and last points are calculated by
% the 3 point forward and 3 point backward finite difference scheme respectively.
% The first derivatives at all the other points are calculated by the 2 point
% central approach.
% The second derivatives at the first and last points are calculated by
% the 4 point forward and 4 point backward finite difference scheme respectively.
% The second derivatives at all the other points are calculated by the 3 point
% central approach.

n = length (x);
dy = zeros;
ddy = zeros;
% Input variables:
% x: vector with the x the data points.
% y: vector with the f(x) data points.
% Output variable:
% dy: Vector with first derivative at each point.
% ddy: Vector with second derivative at each point.
dy(1) = (-3*y(1) + 4*y(2) - y(3)) / (2*(x(2) - x(1))); % First derivative
ddy(1) = (2*y(1) - 5*y(2) + 4*y(3) - y(4)) / (x(2) - x(1))^2; % Second derivative

for i = 2:n-1
   
	dy(i) = (y(i+1) - y(i-1)) / (x(i+1) - x(i-1));
	ddy(i) = (y(i-1) - 2*y(i) + y(i+1)) / (x(i-1) - x(i))^2;
end

	dy(n) = (y(n-2) - 4*y(n-1) + 3*y(n)) / (2*(x(n) - x(n-1)));
	ddy(n) = (-y(n-3) + 4*y(n-2) - 5*y(n-1) + 2*y(n)) / (x(n) - x(n-1))^2;

end

[Mathieu NOE]

hello

Problem solved ?