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Engineering Analysis I: Non-linear dynamics - Duffing Equation 1

Investigation of Non-LinearDynamics -

Duffing Equation

Mandeep Singh UB# 37212672

Supriya Singh UB# 37670768

MAE-507 Engineering Analysis I

Instructor: Dr.Gary Dargush

State University of New York, Buffalo

Mechanical and Aerospace Engineering

Date: 11/11/2010


1. Introduction: The Duffing equation was originally derived as a model for describing the forced vibrations of industrial

machinery, and provides a very good approximation of a damped motion. Duffing equation was studied

by G. Duffing around 1918. It is a non-linear second order differential equation. The equation is

represented in the form as given below :

mü + rů + αu + βu3 = FocosΩt (1)

The term u3 also called as Duffing term introduces the non-linearity into the equation. The ‘m’ represents

the mass of the vibrating body and r represents the damping co-efficient. Fo represents the exciting or

driving force of the system. The involvement of r and Fo represents a more complete, realistic system.

2. Physical resemblance of the model problem : During his studies, Duffing created a non-linear oscillator and introduced the non-linear terms to describe

the hardening spring effect observed in many mechanical systems. Physically speaking , ‘kx’ in the

spring mass system is replaced by αu + βu3 where α and β are constants. If β is positive (β > 0), it

represents a hard spring and if β is negative (β < 0), it represent a soft spring. Duffing’s equation has two

regions of attraction that cause local simple harmonic motion. Schematically the system can be

represented as shown in the fig. 1 below.

Figure 1 (Duffing Oscillator)


When a pendulum is forced periodically (so that it continues to oscillate), its motion can become

periodic or chaotic, depending on the amount of forcing. The study of a swinging includes

complicated "fractal" sets.

Another method of studying the Duffing equation is through Moon beam system of Moon and

Holmes which is shown in the figure below. (fig 2)

.

Figure 2

We will now investigate the system , considering the aspects of this problem defined in Greenberg (1998), Exercise 7.6 2. From the Exercise problem we find that the for the damped case

( ) ( )22

322

43 ArAAFo Ω+⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+Ω−= βα (2)

We will now plot A (Amplitude) for the different Ω with the help of Matlab toolbox, using the values for Fo = 2 , α = 1, β = 0.4, and r = 0.3. The plot (fig 3) is shown below after substituting the given values of various parameters in equation 2.


Amplitude vs Omega for the damped system

Figure 3

3. Analysation 1: a. Now we will analyze the equation 1, that is the Duffing’s Equation at Ω = 1. We have

solved it using Matlab ODE45, which uses the computerized numerical method of

Runge-Kutta for solving 4th order solution.

For the given values Fo = 2 , α = 1, β = 0.4, and r = 0.3, the program in matlab is made in two files.


First file : % Name : Mandeep Singh & Supriya Singh % Subject MAE507 % Project 01 % Associated function file for solving the Duffing Equation using ODE45 function dz=duffing(t,z) dz=[z(2);(2*cos((1*t))-0.3*z(2)-z(1)-0.4*z(1)^3)];%Focos(omega*t)-0.3udot-u-beta*u^cube Second File : % Name : Mandeep Singh & Supriya Singh % Subject MAE507 % Project 01 % Associated Function file "duffing.m" clear all close all %duffing oscillator [t,z]=ode45('duffing',[0,50],[0,0]); hold off figure(1) plot(t,z(:,1)) %Plots the time Vs u grid on title('Duffing Equation Time V/s u Plot','fontWeight','bold','fontsize',10); xlabel('t','fontWeight','bold','fontSize',10); ylabel('u','fontWeight','bold','fontSize',10); hold on [t,z]=ode45('duffing',[0,50],[0.1,0.1]); %Plots the time Vs u for small change in u(initial Condition) plot(t,z(:,1),'r') grid on title('Duffing Equation Time V/s u Plot','fontWeight','bold','fontsize',10); xlabel('t','fontWeight','bold','fontSize',10); ylabel('u','fontWeight','bold','fontSize',10); hold off hold on figure(2) [t,z]=ode45('duffing',[0,50],[0,0]); plot(z(:,1),z(:,2))%Plots the Phase Diagram title('Duffing Equation Phase Diagram','fontWeight','bold','fontsize',10); xlabel('u','fontWeight','bold','fontSize',10); ylabel('y','fontWeight','bold','fontSize',10); grid on


hold on [t,z]=ode45('duffing',[0,50],[0.1,0.1]);%Plots the Phase Diagram for small change in u(initial condition) plot(z(:,1),z(:,2),'r') title('Duffing Equation Phase Diagram','fontWeight','bold','fontsize',10); xlabel('u','fontWeight','bold','fontSize',10); ylabel('y','fontWeight','bold','fontSize',10); grid on Solving these files, generates two Graphs viz. Time Vs u (Trajectory) and Phase space diagram. Phase space plots show the velocity as a function of the position.

Figure 4(Trajectory for Ω = 1)


Figure 5(Phase space Diagram for Ω = 1)

In the above plots we can observe that the steady state has been achieved in around 18 seconds.

Also note in the two plots that once we used the initial condition (0,0) and then we did a minor

change in initial condition i.e., (0.1,0.1) and we tried to look the response of the system due to

this variation. Now we can see in the figure 3 that the steady state amplitude comes out to be near

1.9 If we compare this amplitude to the amplitude that we obtained in the Figure 3 (which is

1.85), we can see that the amplitudes obtained are considerably similar.

b. Now for Ω = 1.71 , we get the following plots :


Figure 6(Trajectory for Ω = 1.71)


Figure 7 (Phase space plot at Ω = 1.71)

Under Ω = 1.71, we find that the steady state amplitude comes out to be 2.88. We are

getting 2.85 from the Figure 3. Hence again the amplitudes obtain are considerably

similar.

c. Now we will investigate the system under Ω = 2.

Here we observe that the system takes around 25 seconds to get to the steady state

response.


Figure 8 (Trajectory For = 2.05)

Figure 9 (Phase Space plot for Ω = 2.05)


At 2.05 we find that the amplitude to be 0.63 and from the figure 2 also we are getting

the same result of amplitude to be 0.63.

d. Now we will investigate the response of the system for Ω = 1.83 . We get the

following graphs.

Figure 10 (Ω = 1.83)


Figure 11(Ω = 1.83)

At Ω = 1.8 , we observe that we get three values of the amplitude A i.e., 0.99 , 2.25, 2.88.

as shown in the plot below(Fig. 12).


Figure 12

In all of the cases above ,we observe that the amplitude that we get for the steady state

responses are the amplitude which are smallest or the largest of the three A values.

Now we will analyze the value of Ω by plugging Ω = 1.9-0.0005t in the duffing’s

equation.

We find that the steady state response is not achieved if we run the system by varying t

slowly from 0 to 800s with the initial conditions. We find that the system doesn’t get

stabilized for whole time though the amplitude keeps on dropping almost linearly with

chaos.


Figure 13 Ω = 1.9-0.0005t

Figure 14 Ω = 1.9-0.0005t


4. Analysation 2:

Now we will try to analyze the same system(damped) with similar co-efficients but

varying the value of Fo.

Fo =

2,

Ω = 2

Fo =

10, Ω

= 2

Fo

=100,

Ω = 2


Fo

=1000

, Ω =

2

Note that this plot is generated for

shorter period for better clarity

Here we would like to introduce the Chaos behavior of the system , which is a deterministic

system, over time, and is not periodic and which is sensitive to initial conditions

By varying the Fo we can see that the chaos of the system is increasing, though the response to

the variation is less compared to the system without damping as shown below.

Fo = 2 ,

Ω = 2 , r

=0


Fo = 10,

Ω = 1.2

, r =0

Fo

=100,

Ω = 1.2

,

r = 0

Fo

=10000,

Ω = 1.2

, r=0


5. Applications:

Several systems have been developed based on the concept of Duffing’s equation.

These topics include dynamic instability of an automotive disc brake pad system,

Climatic study of earth , the dynamics of the action potentials in neurons, and

molecular vibrations etc, have been studied.

6. Conclusion: 1. Behavioral plots were shown and mapped using MATLAB to understand the nature of

the non linear doffing equation and concept of chaos was observed during the analysis

study.

2. As the forcing function Fo is increased, chaos becomes more severe.

3. Considering the system to be a point mass which seems to fluctuate randomly between

bottoms of two wells in all the phase diagrams shown but, actually, it obeys a

deterministic rule.

4. Jump phenomenon associated with the Duffing equation was observed which is

something similar to hysteresis phenomenon observed in electric and magnetic fields

applied on ferromagnets

5. Results were documented in a project report.


7. References/Bibliography:

1. Webpage: http://www.maths.usyd.edu.au/u/nigel/NLD/Duffing/Duffing.html

2. Y. Ueda, Randomly transitional phenomena in the system governed by Duffing's

equation, Journal of Statistical Physics, 20, 181‐196, 1979.

3. P. Holmes and D. Whitley, On the attracting set for Duffing's equation, II. A

geometrical model for moderate force and damping, Physica, 7D, 111‐123, 1983.

4. Webpage:

http://www.emeraldinsight.com/journals.htm?articleid=1823482&show=html

5. Introduction to Chaos and Nonlinear Dynamics Authors: Takashi Kanamaru and J. Michael T. Thompson

6. Undamped doffing equation:

http://www.me.rochester.edu/~clark/ME163Web/webexamp/duffing.pdf .

7. Duffing's equation: catastrophic jumps of amplitude and phase ,

E. Christopher Zeeman, K.B., F.R.S. / UT San Antonio, March 31, 2000

8. Chaotic Motions of a Duffing Oscillator Subjected to Combined Parametric and

Quasiperiodic Excitation , Chin An Tan and Bongsu Kang

9. M.D Greenberg (1998)

10. Notes by Dr. Gary Dargush

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