Nonlinear Models-Modeling and Simulation-Lecture, Lecture notes of Mathematical Modeling and Simulation

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We begin with 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

Introduction to Nonlinear Dynamics

We use x as a angular displacement, v(t) as its time rate of change,

and a(t) as the second derivative of x(t).

In the case of the simple pendulum the displacement is an angle

measured in radians, v(t) is the angular velocity in radians per s , and

the acceleration a is the angular acceleration in radians/s and  is

mgLsinx with I = mL^2. Making these substitutions gives

dt

• mgLSin x mL d x 2

2 2

This equation can be simplified by dividing by mL. In addition, let us

choose the pendulum with L = 9.8 m , then the equation reduces to the

following form:

-sin x dt

d x 2

2 

Because this model has sinx rather than x on the right-hand side, it

forms a second-order non-linear differential equation.

A Simple Pendulum

Solution: STEP 1: Use Newton's Second Law to find the equation of motion, or model the abstract form of the system: a = F(t, x, v), STEP 2: Choose the initial conditions, x 0 & v 0 ; choose maximum time scale for simulation ( tmax ). STEP 3: Decide on the step size h. STEP 4: Begin a loop. Execute the loop while the time is less than tmax. STEP 5: Plot (t, x) or make a portrait by plotting (x, v). STEP 6: Calculate k 1 , k 2 , k 3 , and k 4 from the Runge-Kutta formulas, and use them to update x: x (^) i  1  xi  x STEP 7: Update v with the Runge-Kutta formulas: v (^) i  1  vi  v STEP 8: Increment the time: t = t + h. STEP 9: End of STEP4 and then Stop. You may also write a program in language of your choice to implement the algorithm. Use this for h = 0.05, x 0 = 1 , and v 0 = - 0.4 ; here the domain is [0, 25], and the range is [-2, 2]. Also try a different set of initial conditions; say

x 0 = 1 and v 0 = - 0..

Example 8.1: Algorithm for Motion of a Pendulum Outline a simulation algorithm for the equation of a pendulum in an undamped system. The initial conditions are given for the position and velocity.

Phase Space: Undamped Motion of a Simple Pendulum

Let us construct the phase curves of the simple pendulum without any damping and driving force. It will be useful to make a phase portrait using a variety of initial conditions. To do this we will start the pendulum at x = 0 , but we will give it various velocities corresponding to various kinetic energies. The results are shown in Figure below.

Fig. 8.1 The angular velocity v (radians per second) as a function of displacement x (radians) for an undamped simple pendulum.

 Let us allow velocity range to be from -3 to 3 in steps of 0.5 and domain is given by [ -3p, 3p ].  We can modify the initial conditions to get a set of concentric ellipses centered on points (2np, 0).  The point in phase space (0, 0) phase space (0, 0) and the set of points ( 2np, 0), where n is an integer are points of stable equilibrium for the simple pendulum; the pendulum is hanging down from the pivot.  The set of points [(2n - 1)p, 0] are points of unstable equilibrium; the pendulum is " standing " on its pivot.

Phase Space: Undamped Motion of a Simple Pendulum

XY Graph

-u Unary Minus

sin Trigonometric Function

(^1) s Integrator

(^1) s Integrator

Phase Space: Undamped Motion of a Simple Pendulum

X(0)= 2.

V(0)= 0

XY Graph

-u Unary Minus

sin Trigonometric Function

(^1) s Integrator

(^1) s Integrator

Phase Space: Undamped Motion of a Simple Pendulum

X(0)= 2.

V(0)= 0

XY Graph

-u Unary Minus

sin Trigonometric Function

(^1) s Integrator

(^1) s Integrator

Phase Space: Undamped Motion of a Simple Pendulum

X(0)= 4

V(0)= 0

XY Graph

-u Unary Minus

sin Trigonometric Function

(^1) s Integrator

(^1) s Integrator

Phase Space: Undamped Motion of a Simple Pendulum

X(0)= 4

V(0)= -

XY Graph

-u Unary Minus

sin Trigonometric Function

(^1) s Integrator

(^1) s Integrator

Phase Space: Undamped Motion of a Simple Pendulum

X(0)= 4

V(0)= -