# How to extract the value of dydt from ode45 function

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Commented: John D'Errico on 14 Sep 2019
Hi all,
I have written a code of a damped mass-spring system, that takes as inputs the relevant mass, damping, stiffness and excitation force matrices/vectors, and outputs the displacement and the velocity of each mass.
What I would like to do is to be able to extract the acceleration as well (i.e. the value of dxdt(m:n) as written in the odefcn function at the bottom of the code) in addition to the displacement and velocity outputs.
Now I know that I can find the gradient of the velocity, or reconstruct my equation of motion using my solutions for the displacement and velocity and solve for the acceleration. However, having tried, I would like to avoid these approaches, as the actual problem that I am being faced with (which has significantly different equations of motion) has noisy velocity vectors, and the equation is quite complicated to reconstruct and solve for the acceleration (doable but I'd rather avoid it).
So her is the code:
clear
clc
% Inputs
% Degrees of Freedom
N = 3 ;
% Masses (kg)
m = repmat(100, 1, N) ;
% Spring coefficients (N/m)
k = repmat(15, 1, N+1) ;
% Damping coefficients (Ns/m)
c = repmat(5, 1, N+1) ;
% Maximum force (N)
f0 = repmat(100, 1, N) ;
% Phase angle between force (rad)
phi = linspace(0, pi, N) ;
% Frequency of force (Hz)
freq = 0.2 ;
% Time of simulation (s)
tmax = 100 ;
% Mass initial positions (m)
xi = zeros(1, N) ;
% ODE Inputs
% Degrees of Freedom
N = length(m) ;
% Mass Matrix
M = diag(m) ;
% Stiffness Matrix
k1 = diag(k(1:end-1) + k(2:end)) ;
k2 = diag( -k(2:end-1), 1) ;
k3 = diag( -k(2:end-1), -1) ;
K = k1 + k2 + k3 ;
% Damping Matrix
c1 = diag(c(1:end-1) + c(2:end)) ;
c2 = diag( -c(2:end-1), 1) ;
c3 = diag( -c(2:end-1), -1) ;
C = c1 + c2 + c3 ;
% ODE Initial Conditions
% Time
tspan = [0 tmax] ;
% Displacement and velocity
x0 = [ xi zeros(1, N) ] ;
% Solution
[t, xSol] = ode45(@(t, x) odefcn(t, x, M, K, C, f0, freq, phi, N), tspan, x0) ;
% Results
x = xSol(:, 1:N) ; % Displacement (m)
xdot = xSol(:, N+1:end) ; % Velocity (m/s)
% ODE Function
function dxdt = odefcn(t, x, M, K, C, f0, freq, phi, N)
% Indices
m = N + 1 ;
n = N * 2 ;
% Preallocate
dxdt = zeros(n, 1) ;
% ODE Equation
dxdt(1:N) = x(m:n) ;
dxdt(m:n) = - M \ K * x(1:N) - M \ C * x(m:n) + M \ f0' .* sin(2*pi*freq * t + phi') ;
end
KMT.

James Tursa on 13 Sep 2019
Edited: James Tursa on 13 Sep 2019
Why can't you just call your odefcn( ) function with your solution t and xSol vector elements as inputs (e.g., in a loop)? Doesn't this give you the accelerations you want?