# Double Variable Second order Differential Equation

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Shabeel Samad on 6 Apr 2021
Edited: Shabeel Samad on 12 Apr 2021
I am attempting to solve a double mass-spring-damper system. I already solved for single mass using ode45. However I am unable to figure out how to use this with two variables and also solve a second order differential.
The Equation I used was
The function I used is shown below.
function dy=damped_spring_mass(F,m,D,w0,b,t,y)
x=y(1);
v=y(2);
dy=zeros(2,1);
dy(1)=v;
dy(2)=(F*sin(w0*t)-D*x-c*v)/m;
I used this function to in ode45 as shown below
y0=[0;0];
tspan=[0 30];
%% Spring constants
k1= 5;
k01 = 1;
k12 = 2;
c=0.1;
D1= k1+c*(k01+k12);
%% mass constants
m1=1;
%% Damper C0fficients
b1=3;
%% Force
F= 0.1;
w0=2;
%% Diff Eq and plot
[Td,Yd]=ode45(@(t,y) damped_spring_mass(F,m1,D1,w0,b1,t,y),tspan,y0);
plot(Td,Yd(:,1));
For a double mass system, I have two equations
The new constants are provided below
%% Spring constants
k2= 2;
k23 = 3;
k34 = 1;
c=0.1;
D2= k2+c*(k23+k34);
%% mass constants
m2=2;
%% Damper C0fficients
b2=5;

James Tursa on 9 Apr 2021
Just write a derivative function using four states instead of two. The states will be x, y, dxdt, and dydt. The derivitives of these states will be dxdt, dydt, d2xdt2, and d2ydt2. E.g.,
function dy = damped_spring_mass(t,y, other constants )
dxdt = y(3);
x = y(1);
dydt = y(4);
y = y(2);
d2xdt2 = your expression for this in terms of constants and x, y, dxdt, dydt
d2ydt2 = your expression for this in terms of constants and x, y, dxdt, dydt
dy = [dxdt;dydt;d2xdt2;d2ydt2];
end
Shabeel Samad on 12 Apr 2021
The mistake was on my end. Thank you so much MVP!!

Ayse Buyukbas on 6 Apr 2021
Shabeel Samad on 9 Apr 2021
I looked through the links, but none of them solve for both variables with a static input. Thank you for taking your time.