Duffing Pendulum-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

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Nonlinear Second Order Models

Duffing Pendulum

Mathematical Modeling

& Simulation

Consider a model of two-dimensional dynamical system that obeys

the following ordinary differential equations (ODEs):

f ( x , y ) dt

dy  g(x,y) dt

dx 

It is an autonomous system as it does not explicitly involve the

independent variable t. If such a system involves a nonlinear term such

as x^2 (t), x(t)y(t), sinx(t), exp(ay) etc., then the system is said to be

nonlinear one.

Recall the chain rule for differentials: (dy/dx)(dx/dt) is equal to dy/dt.

It means that

g(x,y)

f(x,y)

dx / dt

dy / dt

dx

dy  

Non-Linear Models

MATLAB program: Duffingx.m

% main program for duffing pendulum: % Equation : acc = -cv + ax - bx^3 + dcos(wt) % this equation is used in a separate m-file for function: duff.m % we get chaos and a strange attractor if the driving force d =1. % Poincare section is a complicated curve namely a fractal ....... close all clear clc global c a b d w c=0.1; a=1; b=0.25; w =2; % constant values % the driving force d =1.5 and a Poincar section is developed d=1.5; % amplitude of force time_period = 2pi/w del = time_period/100; h = del; tmax = 150000.0; nx = tmax ; nmax = tmax/del; t = 0:del:nx; % time array is formed.... %%%%% for solver; [t,y] = solver(odefun,tspan,y0,options) [t x]=ode45(@duf,t,[0 1]); % the model eq. is solved .... %%%%%% now plot graphs

MATLAB program: Duffingx.m

%%%%%% now plot graphs figure(1) plot(t(1:nx),x(1:nx,1),'r') axis tight title('time series') figure(2) plot(x(1:nx,2),x(1:nx,1),'b') axis tight title('phase space') figure(3) for i=200:100:nmax kk=(i-100)/100; x1(kk)=x(i,2); x2(kk)=x(i,1); end plot(x1(:),x2(:),'k.') axis tight title('Poincare section') %

% function program: duf.m % Forced Duffing Oscillator % function xdot=duf(t,x) global c a b d w xdot(1)=-cx(1) + a^2x(2)- bx(2)^3 + dcos(w*t); xdot(2)=x(1); xdot=xdot'; end

Separate file for function.

First Case Study:

Variable time step

Sim. Time = 5000

x(0) = 1,

v(0) = -1,

The x(t) versus t. The system stabilizes after some transient behavior. The trajectory encircles a point (2, 0); It evolves to a period motion with fix amplitude and fixed period.

c = 0.1; a = 1; b = 0.25; w = 1.4; d = 0.1;

First Case Study:

Variable time step

Sim. Time = 5000

x(0) = 1,

v(0) = -1,

The v(t) versus x(t). The system stabilizes after some transient behavior. The trajectory encircles a point (2, 0); It evolves to an orbit called a limit cycle.

c = 0.1; a = 1; b = 0.25; w = 1.4; d = 0.1;

Second Case Study:

Variable time step

Sim. Time = 5000

x(0) = 1,

v(0) = -1,

c = 0.1; a = 1; b = 0.25; w = 1.4; d = 0.1;

The system stabilizes after some transient behavior. The trajectory encircles a point (2, 0); It evolves to an orbit called a limit cycle.

We have increased force amplitude = d from 0.05 to 1.

Third Case Study:

Variable time step

Sim. Time = 5000

x(0) = 1,

v(0) = -1,

c = 0.1; a = 1; b = 0.25; w = 1.4; d = 0.2;

The system again stabilizes after some transient behavior. However, trajectory encircles a new point (-2, 0); It evolves to an orbit called a limit cycle.

Here We have increased force amplitude = d from 0.1 to 2.

Fifth Study Case

Variable time step

Sim. Time = 5000

x(0) = 1,

v(0) = -1,

c = 0.1; a = 1; b = 0.25; w = 1.4; d = 0.45;

The system again stabilizes after some transient behavior. However, trajectory encircles a new point (2, 0); It evolves to an orbit called a limit cycle.

Here force amplitude is increased to 0.45.

Sixth Study Case

Variable time step

Sim. Time = 5000

x(0) = 1,

v(0) = -1,

c = 0.1; a = 1; b = 0.25; w = 1.4; d = 0.5;

The system again stabilizes after some transient behavior. However, trajectory encircles a new point (2, 0) with three different periods;

Here force amplitude is increased to 0.5.

Seventh Study Case

Variable time step

Sim. Time = 50000

x(0) = 1,

v(0) = -1,

Here force amplitude is increased to 0.7. The system never stabilizes …. There is now chaos … The Poincare section is a beautiful picture.

c = 0.1; a = 1; b = 0.25; w = 1.4; d = 0.7;

Seventh Case

Variable time step

Sim. Time = 100000

x(0) = 1,

v(0) = -1,

c = 0.1; a = 1; b =0.25; w =1.4;

d = 0.7;

Here force amplitude is increased to 0.7. The system never stabilizes ….

There is now chaos …

The Poincare section is a beautiful picture.

Now w has been increased from 1.4 to 2.0 ;

The Poincare section is again very fine picture.

This repeats itself and has further details in it. We show magnified section here and its shows the pattern repeating…

c=0.1; a=1; b=0.25; w = 2.0; d = 1.

Variable time step

New Study Case Sim. Time = 100000.0;

Now w has been increased from 1.4 to 2.0 ;

The Poincare section is again very fine picture.

This repeats itself and has further details in it. We show magnified section here and its shows the pattern repeating…

c=0.1; a=1; b=0.25; w = 2.0; d = 1.

Variable time step

New Study Case Sim. Time = 100000.0;