Transcript ME440 - Dan Negrut

ME 440
Intermediate Vibrations
Tu, Feb. 3, 2009
Sections 2.1-2.2, 2.6-2.8
© Dan Negrut, 2009
ME440, UW-Madison
Before we get started…

Last Time:



Frequency Spectrum
Complex form of the Fourier Series Expansion
Went through three examples


Last one was the propeller blade example
Today:


Covering material out 2.1-2.2, potentially 2.6-2.8
HW Assigned:

2.35, 2.45, 2.69 out of the book
2
Chapter 2: What Are We Up To?

Title of Chapter captures the essence:

Free Vibration of Single Degree of Freedom Systems

“Free” means that there is no forcing term



What makes the system vibrate?
Motion is due to a set of nonzero initial conditions:
“Single Degree of Freedom”



As simple as it gets…
Dealing with a system like one in figure (a)
System in (b) already has two degrees of freedom…
3
What are we up to? (Cntd)

Understand how to derive equations of motion (EOM) for 1DOF systems
Once we’ll have the EOMs, we’ll solve them

First then, how do we derive the EOM?


Newton’ Second Law


Work and Energy Methods


Always works, our workhorse. Abbreviated N2L.
Very handy for conservative systems
Lagrange’s Equation and Hamilton’s Principle

Will get limited mileage in this course, used a lot in Physics
4
Kinetic and Potential Energy:
Conservation of energy

Key assumption for conservation of energy:

We are dealing with a conservative system

Force acting on the system derived from a potential U(x)
5
Newton’s 2nd Law - Translation

y

r

(CM)
x
z
Newton’s Second Law
mv  mr   F
 F is the sum of external forces
acting on the rigid body

m is the mass of rigid body

r is the acceleration of the
center of mass (CM) of the rigid
body relative to a fixed point in
space
mvx  mx   Fx
mvy  my   Fy
mvz  mz   Fy
6
Newton’s 2nd Law – Rotation
About a Fixed Axis




Assumes that axis of rotation is fixed
in space
M0 is the sum of external moments
acting about rotation axis
I0 is the mass moment of inertia
about the rotation axis (units kg-m2)
Notation:
 
Newton’s 2nd Law Applied to Rotation:
I0  M 0
Kinetic Energy:
1
KE  I 0 2
2
7
Mass Moment
of Inertia
Sphere
I0 
2
mR 2
5
Mass rotating about point 0
I 0  mR2
I   r 2 dm
Hollow cylinder
Parallel-Axis Theorem
1
m( R 2  r 2 )
2
1
I y  I z  m(3R 2  3r 2  L2 )
12
Ix 
Rectangular Prism
Ix 
I  Icm  md
1
m(b 2  c 2 )
12
2
8
General Planar Motion

A body moving in a plan is described
by a set of 3 DOFs




Equations of motion (EOM):



Motion in the x direction
Motion in the y direction
Rotation of angle 
G – is center of mass (important!!!)
MG – total torque about point G
A consequence of Newton’s 2nd Law
9
General Planar Motion (continued)

The previous quantities are defined as:

The vector MG is the total torque:
10
Energy Approach to Derivation of EOM

For a conservative system, conservation of energy leads to

For a nonconservative system,
11
Conservative systems

A system in which only conservative forces are active.

Conservative forces are derived from a potential energy function U(x):
dU ( x )
f ( x)  
dx


Here the x represents the position where the force is to be calculated
One consequence of Newton’s second Law for conservative systems:

Sum of kinetic and potential energy is constant (conservation of energy)
1 2
mv  U ( x)  const
2
12
Example
x
f
mvx  mx  f
m
1) What are the names of the following force
functions?
2) Which ones represent conservative forces?
a)
b)
c)
d)
e)
f = -bv
f = -kx
f = -µmg (v>0)
= µmg (v<0)
f = -kx2
f=C
13
Lagrange’s Equations


Assume system has ndof degrees of freedom; array of generalized coordinates is
The second order differential equation that captures the time evolution of each
degree of freedom qj is obtained as follows:
14
Hamilton’s Principle

Gets some mileage particularly in elastodynamics
15
Example (N2L)

Determine the EOM for solid cylinder. Assume rolling without slip.
16
Deriving the Equations of Motion (EOM)
for One-DOF Systems (340 Vintage)

EOM determined following four steps:

STEP 1: Identify the displacement variable of interest

STEP 2: Write down kinematic constraints (if present)

STEP 3: Get equivalent mass/moment of inertia


Equate kinetic energy between actual system and the simplified 1-DOF
system in terms of the displacement variable of interest
Step 4: Get equivalent force/torque

Equate virtual power between actual system and the simplified 1-DOF
system in terms of the displacement variable of interest
17
Example (revisited)

Determine the EOM for solid cylinder using the equivalent mass approach
18
Short Excursion: A Word on the Solution of
Ordinary Differential Equations

Classical analytic techniques

Laplace transforms

Numerical solution


Usually found using MATLAB, or some other software package
(Maple, EES, Sundials, etc.)
MATLAB demonstrated later in this lecture (or next…)
19