2nd Order Nonlinear Differential Equation Solving with Newton Method

Hello,

I have a question about the spring mass system below.

I have solved this problem by hand till a final equation which is 2. order nonlinear differential equation. The equation is;

Now, ı need to solve this equation by using matlab.I have assigned logical values to parameters as my code below and simplified the equation,

clear all;

syms Theta(t);

ab=diff(Theta,2);
aa=diff(Theta);

m1=1;    %kg
m2=1.5;   %kg
m3=2;     %kg
b1=0.5;    %m
b2=1;      %m
k=10;      %N/m2
g=9.81;    %m/s2
T=15;       %N.m
F=20;     %N 
dx=(F/k)*cos(90-Theta);
h=b1+b2+dx;
s=sqrt((h-(b1*cos(Theta)+b2))^2+(b1*sin(Theta)^2));
c1=0.1;
c2=0.1;

ode=(m2+m3*cos(Theta).^2)*b1^2*ab-m3*b1^2*aa.^2*sin(Theta)...
 *cos(Theta)+ (c1*cos(Theta).^2+c2*sin(Theta).^2)*b1^2*aa...
 +(m2*g*b1*sin(Theta))+(k*(b1^2*sin(Theta)+((s*b1^2*sin(Theta))/(sqrt(s^2+2*b1^2*(1-cos(Theta))))))-T+F*b1*cos(Theta));

Now I am confused about how to solve this equation by using newton's method.

Can you help me ?

[darova]

Newton method is used for roots finding. BUt you want to solve ODE. Am i missing something?

wikipedia

[Mete Altay]

I wanna solve the ODE if it is possible

Answer 1

[darova]

You can try dsolve or re-write your equations and use ode45

Express = ...

ddtheta = @(th,dth) m3*b1^2*dth^2*sin(th)*cos(th) - ...
f = @(t,u) [u(2);ddtheta(u(1),u(2))];
[t,u] = ode45(f,[0 5],[1 1]);
plot(t,u)

See more: ode45

[Mete Altay]

Hi,

I re-writed the equation and tried to solve exactly as you suggest, but ı am getting the error 'Inputs must be floats, namely single or double.' I couldnt fix it. Here is my code.

clear all
syms Theta 
m1=1;    %kg
m2=1.5;   %kg
m3=2;     %kg
b1=0.5;    %m
b2=1;      %m
k=10;      %N/m2
g=9.81;    %m/s2
T=15;       %N.m
F=20;     %N 
dx=(F/k)*cos(90-Theta);
h=b1+b2+dx;
s=sqrt((h-(b1*cos(Theta)+b2))^2+(b1*sin(Theta)^2));
c1=0.1;
c2=0.1;


aa = diff(Theta);
ab= diff(Theta,2);
ab = @(Theta,aa)m3*b1^2*aa^2*sin(Theta)...
 *cos(Theta)- (c1*cos(Theta)^2+c2*sin(Theta)^2)*b1^2*aa...
 -(m2*g*b1*sin(Theta))-(k*(b1^2*sin(Theta)+((s*b1^2*sin(Theta))/(sqrt(s^2+2*b1^2*(1-cos(Theta))))))*T-F*b1*cos(Theta))... 
 /((m2+m3*cos(Theta)^2)*b1^2)


f = @(t,u) [u(2);ab(u(1),u(2))];
[t,u] = ode45(f,[0 5],[1 1]);
plot(t,u)

[darova]

I don't see this expression the same as on the picture

ab = @(Theta,aa)m3*b1^2*aa^2*sin(Theta)...
 *cos(Theta)- (c1*cos(Theta)^2+c2*sin(Theta)^2)*b1^2*aa...
 -(m2*g*b1*sin(Theta))-(k*(b1^2*sin(Theta)+((s*b1^2*sin(Theta))/(sqrt(s^2+2*b1^2*(1-cos(Theta))))))*T-F*b1*cos(Theta))... 
 /((m2+m3*cos(Theta)^2)*b1^2)

Can you explain more? what is and ? What is s?

[Mete Altay]

Here f1 and f2. I substituted them into the main equation. They are linked with potential and kinetic energy gradients of the system

[darova]

Try this

[Mete Altay]

Thanks its worked.

[Meryem EL Abdi]

Hello Darova,

i need your help in solving an ODE of a Newton’s cradle consisting of a pair of pendulums with mass 1 and 2 as shown in Figure

I determinated those equations:

I dont know how to use the ODE.

close all
clear all
clc

%Initial Conditions
r = 0.05; % Radius beider Kugeln 
l = 0.3; % Seillänge
m_rechts = 50; % Masse der rechten Kugel 
m_links = 50; % Masse der linken Kugel 
start = [pi/3;0];

phi_rechts_initial = pi/3; 
phidot_rechts_initial =0; 

phi_links_initial = 0; 
phidot_links_initial =0; 

J_rechts = (m_rechts*l^2);
J_links = (m_links*l^2);

%ts = 0.001;
%timespan = 0:ts:40;

g = 9.81;

odefun= @(t,phi) [phi(3);...
                  phi(4);...
 -(m_rechts*g*sin(phi(1)*l))/ J_links;...
  -(m_links*g*sin(phi(2)*l))/ J_rechts];
             
tspan=linspace(0,5);

[t, phi]= ode45(odefun,tspan,start);

plot(t, phi(:,1)*180/pi,'linewidth',2)
xlabel('time(s)','FontSize',16,'Fontname', 'Arial', 'FontWeight','bold')
ylabel('phi in degrees','FontSize',16,'Fontname', 'Arial', 'FontWeight','bold')
title('Pendulum Motion','FontSize',16,'Fontname', 'Arial', 'FontWeight','bold')

[darova]

please create your own question