Double Variable Second order Differential Equation

Question

Shabeel Samad 2021-4-6

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/794567-double-variable-second-order-differential-equation

评论： Sajawal Feroze 2022-4-20

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;

Any help would be greatly appreciated. If you need more information please let me know!

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Answer 1

James Tursa 2021-4-9

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在 MATLAB Online 中打开

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

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James Tursa 2021-4-12

在 MATLAB Online 中打开

The dy should be the last line in your derivative function code. E.g.,

function dy=damped_spring_mass(F,m,n,K,k,J,w0,d,b,t,y)
dxdt = y(3);
x = y(1);
dydt = y(4);
y = y(2);
d2xdt2 =(F*sin(w0*t)-d*dxdt+d*dydt-K*x+K*y)/m;
d2ydt2 =((d-b)*dydt-d*dxdt+k*x-J*y)/n;
dy = [dxdt;dydt;d2xdt2;d2ydt2];
end