ME580
Nonlinear Systems
Home work #2
Due: Sept. 23, 2010
1. The velocity v (t ) of a skydiver falling to the ground is governed by
mv mg kv2 ,
where m is the mass of the skydiver, g is the acceleration due to gravity, and k 0 is a
constant related to the
ME580
Nonlinear Systems
Home work #3
Due: October 7, 2010
1. Three different systems have an equation of motion of the form dx(t)/dt = a(t)x(t), where a(t) is of
period T > 0. Find the period propagator K, and determine the stability of the equilibrium po
ME 580
Nonlinear Systems
Home Work No. 4
Due: October 21, 2010
Q1. Sketch the phase diagrams for the following linear systems and classify the
equilibrium points.
(i) x x 5 y,
(iii) x 2 x y,
y x y ; (ii) x 4 x 2 y,
y x y
y 3x 2 y
Q2. Locate and classify t
ME 580: Nonlinear Engineering Systems
(Fall 2010)
Instructor:
Anil K Bajaj, School of Mechanical Engineering,
Office: Room ME 368 or ME 211: Phone # 4945688.
Time and Place:
Tuesday, Thursday: 12:00 pm  1:15 pm, Room ME 256
Text Book:
(1) Nayfeh, A. H.
Perturbation Techniques
In this series of lectures, we will like to be introduced to the basics of asymptotic
expansions and perturbation techniques. The most elementary application of perturbation
techniques is to algebraic equations which depend on a sm
Notes for Expansions/Series and Differential Equations
In the last discussion, we considered perturbation methods for constructing
solutions/roots of algebraic equations. Three types of problems were illustrated starting
from the simplest: regular (straig
Project suggestions/examples: Projects are intended to allow you to study in detail a system in which nonlinearity affects the dynamic behavior. The project is worth about 5 HWs and you should allocate effort accordingly. In the project report you should
ME580
Nonlinear Systems
Home work #1
Due: Sept. 7, 2010
1.
Consider the secondorder linear system presented in class:
mx cx kx F0 cos(t );
m, c, k 0.
Nondimensionalize the above system to
2
2n x n x ( F0 / m)cos(t ),
x
and find the complete solution
Reviewing: We are considering the use of Multiple Scales method on the Duffings oscillator 2 u + 0 u + u 3 = 0 < 1 u(0) = a 0 , u(0) = 0 We assumed a solution of the form: u (0) = 0 u = u 0 (T0 , T1, T2 ) + u1 + 2 u 2 Substituting in the equation, and col
1. INTRODUCTION
Applications of nonlinear dynamics



Structural dynamics (inertial nonlinearities,
material or geometric nonlinearities)
Automotive systems (dry friction, nonlinear
suspensions, engine mounts and isolators,
vibrations with impact or cl
Forced, linear oscillations of an undamped oscillator. If
we choose an initial condition x0 = 0 , the response is
x(t) = C1 cos n t +
Fo
2
m(n
2 )
cos t
B
It is a periodic response only if C 1 = 0 or when
p n = q, where p & q are integers
Consider = p n
We are focused on oscillators (undamped)
x+
V
=0
x
in first order form :
x1 = x 2
x 2 = V / x
2
Then, H = x 2 / 2 + V(x) = x 2 / 2 + V(x1)
is a first integral of motion, i.e., H = 0 for all solutions.
H = x 2 / 2 + V(x1) defines curves in the (x1,x2)
2
ph
Periodic Solutions
(Jordan and Smith, Chap. 5)
Consider equation
x + 2 sin( x) = cos .
(5.1)
For small motions, we can write as (see how introduced):
x + 2 x x3 = cos .
Note
Note that for =0,
Assuming a power series solution
equation (5.14) gives
Collecti
Nonresonant Hard Excitation
We still need to use K as O ( 0 )
2
i.e. u + 0 u + (2 u + u 3 ) = K cos t
Let the solution be
u(t; ) = u 0 (T0 ,T1 ) + u1 (T0 ,T1 ) + O( 2 )
Substituting into the equation gives
O( 0 ) :
2
2
D0 u0 + 0 u 0 = K cos T0
2
2
3
O( 1
Example: Laser threshold
pump
partially reflecting
mirror
Active
material
laser light
At low energy levels each atom oscillates acting as
a little antenna, but all atoms oscillate
independently and emit randomly phased photons.
At a threshold pumping leve
HartmanGrobman Theorem in n Dimensions
Definition: An equilibrium point x * of x = f(x) is said
to be hyperbolic if all eigenvalues of the Jacobian
matrix D f(x * ) have nonzero real parts.
x * is a hyperbolic equilibrium of x = f(x),
Theorem If
*
then