Learning Nonlinear Vibration with MATLAB

1. Introduction

In the physical world, vibrations are often nonlinear, meaning that their behavior cannot be accurately described by linear equations. While it is common to simplify nonlinear vibrations to linear ones for ease of analysis, there are certain systems that exhibit significant nonlinear behavior, requiring careful consideration.

Nonlinear vibration finds applications in various areas, including mechanical engineering and biology. In mechanical engineering, nonlinear vibration usually considering:

Dynamics Behavior: Nonlinear vibration involves the study of the dynamic behavior of nonlinear systems. This includes the analysis of periodic and chaotic motions, stability, bifurcations, resonance, and other complex phenomena that arise due to the nonlinearity of the system.
Self-Excited Vibration: One important aspect of nonlinear vibration is self-excited vibration, where the system generates its own vibrations without any external excitation. This phenomenon can occur in systems with nonlinearities that lead to energy transfer between different modes of vibration.
Parametric and Autoparametric Vibration: Parametric vibration refers to the case where the system's parameters vary with time, leading to nonlinear behavior. Autoparametric vibration occurs when the system's natural frequencies are close to integer multiples of each other, resulting in complex interactions and resonance phenomena.

Nonlinear systems can also be used to describe the behavior of the brain. By applying nonlinear vibration theory, researchers can model and analyze the complex dynamics of brain activity, helping in understanding cognitive processes, neurological disorders, and brain function.

To gain a simple understanding of nonlinear vibration, it is helpful to approach the topic in three steps: classification of nonlinear vibrations, introduction to nonlineat oscillators, learning the properties with MATLAB.

1.1 General Form

In general, vibration of linear systems can be considered as a special case of vibration of non-linear systems. Thus, consider a general form,

$$ \frac{d^2 x}{dt^2}+f\left(x,\frac{dx}{dt}\right)=0. $$

An common way is to rewrite it as a system of two 1st order ODE, and this x-y plane is called phase plane.

$$ \frac{dx}{dt}=y,\quad\frac{dy}{dt}=-f(x,y).\quad\Rightarrow\quad \begin{cases} \frac{dx}{dt}=F(x,y)\\ \frac{dy}{dt}=G(x,y) \end{cases} $$

For example, consider a simple pendulum with variable damping which can be written as,

$$ \ddot{x}+\nu\dot{x}+\omega_0^2\sin x=0\quad\Rightarrow\quad \begin{cases} \frac{dx}{dt}=\dot{x}\\ \frac{dy}{dt}=-\nu\dot{x}-\omega_0^2\sin x \end{cases} $$

In MATLAB, this ODE system can be written as

sys = @(t,x) [x(2); -nu*x(2)-w0^2*sin(x(1))];

1.2 Phase Plane

Consider the case that nu = 0.1, w0 = 1, and a initial state of x0 = pi/8. We can solve this system in numerical method and compute the vector plot in phase plane:

x1=linspace(-1,1,20);
x2=linspace(-1,1,20);
[x,y]=meshgrid(x1,x2);
u=zeros(size(x));
v=zeros(size(x));
t=0;
for i = 1:numel(x)
    Yprime = sys(t,[x(i); y(i)]);
    u(i) = Yprime(1);
    v(i) = Yprime(2);
end
[t,xs]=ode45(sys,[0 100],[pi/8 0]);
plot(xs(:,1),'LineWidth',2);
quiver(x,y,u,v,'AutoScaleFactor',0.5);
hold on
plot(xs(:,1),xs(:,2),'LineWidth',2);
hold off

This shows us a small deformation react as linear system. However, if we change initial state into larger amplitude (x=0, x'=2.5):

It clearly represents the nonlinear behavior that pendulum can swing a full circle.

2. Duffing Oscillator

The Duffing oscillator, named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

$$ \ddot{x}+\nu\dot{x}+\alpha x+\beta x^3=\gamma\cos(\omega t) $$

2.1 Free Vibraion

First case is no damping, and negative linear stiffness with alpha = -1, beta = 4.

2.2 Forced Vibration

Some typical examples of the time series and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are shown in the figures below. Damping sets to 0.3 and force frequency omega is 1.2. The forcing amplitude increases from gamma = 0.08, 0.14, 0.18, 0.20, 0.325, 0.40

To indicate the bifurcation, we use Poincaré map concept to draw the point at every period. Which will shows the number of harmonic mode or the appearance of chaos.

[t,xx]=ode45(sys,(100/omega)*pi:(2/omega)*pi:(400/omega)*pi,[x0,y0]);
plot(xx(:,1),xx(:,2),'k*','MarkerSize',10);

3. Van der Pol Oscillator

The differential equation

$$ \ddot{x}+x-\epsilon(1-x^2)\dot{x}=0 $$

is called the van der Pol oscillator. It is a model of a nonconservative system in which energy is added to and subtracted from the system in an autonomous fashion, resulting in a periodic motion called a limit cycle. Here we can see that the sign of the damping term changes, depending upon whether |x| is larger or smaller than unity. Van der Pol’s equation has been used as a model for stick-slip oscillations, aero-elastic flutter, and numerous biological oscillators, to name but a few of its applications.

3.1 Stable Equilibrium

When epsilon is negtive, the damping is positive and system will reach stable equilibrium. For example, epsilon = -0.1 and initial condition is x' = 1: