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Chapter 1: Q5.3-23E (page 1)

Nonlinear Spring.The Duffing equationy''+y+ry3=0 where ris a constant is a model for the vibrations of amass attached to a nonlinearspring. For this model, does the period of vibration vary as the parameter ris varied?

Does the period vary as the initial conditions are varied? [Hint:Use the vectorized Runge鈥揔utta algorithm with h= 0.1 to approximate the solutions for r= 1 and 2,

with initial conditionsy(0)=a,y'(0)=0 for a = 1, 2, and 3.]

Short Answer

Expert verified

The result can get by the Runge-Kutta method.

Step by step solution

01

Transform the equation

Here, the equationy''+y+ry3=0.

The system can be written as;

x1=y(t)x2=y'=x'1

The transform equation is:

localid="1664103100476" x'1=x2x'2=-x1-rx31

The initial conditions are:

x1(0)=y(0)=a=1,2,3x2(0)=y'(0)=0

02

Apply the Runge-Kutta method and plot some graph

Apply mat lab to find the results. And some results are:

T

For

For

For

For

For

For


0

1

2

3

1

2

3

0.1

0.9900

1.9505

2.8534

0.9850

1.911

2.727

1

0.2336

-0.704

-2.689

0.0015

-1.64

-2.356

2

-0.859

-1.283

1.90

-0.999

0.841

0.990

3

-0.6618

1.8435

-0.9174

-0.0046

0.07583

0.4898

5

0.9471

-1.7494

1.151

0.00775

1.7470

2.8941

10

0.7989

1.13069

-1.6653

-0.99979

1.13987

2.4978

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