Q5.3-23E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Nonlinear Spring. The Duffing equation $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{+}{\mathbf{ry}}^{\mathbf{3}}\mathbf{=}\mathbf{0}$ where r is a constant is a model for the vibrations of amass attached to a nonlinear spring. For this model, does the period of vibration vary as the parameter r is varied?

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

with initial conditions $\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{a}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$ for a = 1, 2, and 3.]

Step-by-Step Solution

Answer

The result can get by the Runge-Kutta method.

Transform the equation

Here, the equation $\mathrm{y}\text{'}\text{'}+\mathrm{y}+{\mathrm{ry}}^3=0.$

 

The system can be written as;

 

$$\begin{gathered} {\mathrm{x}}_1=\mathrm{y}\left(\mathrm{t}\right) \\ {\mathrm{x}}_2=\mathrm{y}\text{'}=\mathrm{x}{\text{'}}_1 \end{gathered}$$

 

The transform equation is:

 

$$\begin{gathered} \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{1}}\mathbf{=}{\mathbf{x}}_{\mathbf{2}} \\ \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{2}}\mathbf{=}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{r}{{\mathbf{x}}^{\mathbf{3}}}_{\mathbf{1}} \end{gathered}$$

 

The initial conditions are:

 $$\begin{gathered} {\mathrm{x}}_1\left(0\right)=\mathrm{y}\left(0\right)=\mathrm{a}=1,2,3 \\ {\mathrm{x}}_2\left(0\right)=\mathrm{y}\text{'}\left(0\right)=0 \end{gathered}$$

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