Vibrations: a special class of motion
Oscillatory in time: Position ~ cos (frequency * t)
We have already seen/examined vibrations in some dynamical systems:
Vibration control in vehicles; Millenial Bridge; two-story building
Systems that vibrate (nearly

RIGID BODY DYNAMICS
(stop treating everything like a point object)
RIGID BODY
No change in shape during motion
No change in distances between points on body
(Relative positions are the same)
Can rotate with no motion of Center of Mass or some other point

Summary on Kinematics:
Motion of Rigid Body can be described by
rA / B
x(t ), y (t ) of any one point
TRANSLATIONAL MOTION
+
ROTATIONAL MOTION
(t )
of entire body
Vector description:
rA rB rAB (cos i sin ) j
VA VB x rA / B
a A aB x rA / B x ( x r

Always the Same Equation of Motion
d 2x
Mass m on a spring: m
+ kx = 0
2
dt
d 2
m 2 + ( mg / L) = 0
Pendulum:
dt
5m d 2 x
Two-mass/Pulley system:
+ kx = 0
2
4 dt
2
d
Rotating bar:
I c 2 + 12 kL2 = 0
dt
Equation of Motion always of the form:
2
d x
M 2 + Kx

Types of Forcing:
External Forcing
Base Excitation
Rotor Excitation
All of these situations are of practical interest. Some subtle but important
distinctions to consider, so we will look at each.
BUT
Strategy is simple: derive Equation of Motion and put i

Summary of Dynamics of Rigid Bodies
F1
M O IO
ma
F
F
G
M G IG
aG
2
G
O
F3
If O is fixed
M O mrG aG I G
+ Kinematics relations between points on a rigid body:
x A xB rAB cos
y A y B rAB sin
x A x B rAB sin
y y r cos
A
B
AB
x A xB rAB sin - rAB

Chapter 3
Analyzing motion of systems of particles
In this chapter, we shall discuss
1. The concept of a particle
2. Position/velocity/acceleration relations for a particle
3. Newtons laws of motion for a particle
4. How to use Newtons laws to calculate t

Damped Vibrations: Equation of Motion
dx
d 2x
kx c
=m 2
dt
dt
Equation of motion:
d 2x
dx
2
+
2
+
n
nx = 0
2
dt
dt
Only two parameters:
k
n =
m
c
=" zeta" =
2mn
Three Classes of Solutions:
>1
Overdamped:
Damping is strong
x (t ) = C1e
Critical Damping:

Summary of Dynamics of Rigid Bodies
F1
M O IO
ma
F
F
G
M G IG
aG
2
G
O
F3
If O is fixed
M O mrG aG I G
+ Kinematics relations between points on a rigid body:
x A xB rAB cos
y A y B rAB sin
x A x B rAB sin
y y r cos
A
B
AB
x A xB rAB sin - rAB

Types of Forcing:
External Forcing
Base Excitation
Rotor Excitation
All of these situations are of practical interest. Some subtle but important
distinctions to consider, so we will look at each.
BUT
Strategy is simple: derive Equation of Motion and put i

Last time: Defined Angular Velocity and Angular Acceleration Vectors:
,
And could then express the velocity of a point
a on a rotating (fixed axis) rigid body as
VA x rA
same as:
V A r sin i r cos j
and the acceleration as:
a A x rA x ( x rA )
Ta