Periods and Frequencies
Period
the time that it takes for an event to repeat itself
it takes about 365 days for your birthday to
come around
Frequency
inverse of period
your birthday comes once a year
Frequency is a measure of how frequently an event or a
process occurs.
Period is the time that it takes for the event to complete
one cycle.
Periods and Frequency
Consider a simple spring-mass system
shown at the right.
If the mass is pushed down and let go
it will oscillate in a manner that
manifests itself by an up-and-down
motion.
The natural undamped frequency of
the system is given
1
fn 
2
k
m
in Hz, cycles per
second
1
fn 
2
k
m
in Hz, cycles per
second
Where
fn is the natural frequency of the system in
cycles per second, or Hertz (Hz)
k represents the stiffness of the spring or an
elastic member (N/m)
and m is the mass of the system (kg)
The period of oscillation, T, for
the given system – the time that
it takes for the mass to
complete on cycle – is given by
1
m
T
 2
fn
k
in seconds
An Example of Period and Frequency
Given: a simple spring-mass system
Find: the natural frequency of the system
Solution:
1
fn 
2
k
m
1

2
5000 N/m
 8Hz
2 kg
Angular (Rotational) Speeds
Rotational motion is quite common in engineering
applications. Examples of engineering components with
rotational motion include shafts, wheels, gears, drills,
helicopter blades, CD drives, Zip drives, and so on.
The average angular speed of a line segment located on a
rotating object is defined as the change in its angular position
(angular displacement) over the time that it took the line to go
through the angular displacement.
Ω = angular speed (rad per sec)
∆θ = angular displacement (radians)
∆t = the time interval in seconds.
8-6
It is a common practice to express the angular speed of
rotating objects in revolutions per minute (rpm) instead of
radians per second (rad/s).
Convert 1600 rpm to rad/s:
1600(
revolutions 2π radians
1 minute
rad
)(
)(
)  167.5
minutes 1 revolution 60 seconds
s
8-7
There exists a relationship between linear and
angular velocities of objects that not only rotate
but also translate as well. For example, a car
wheel, when not slipping, will not only rotate
but also translate.
S  r
and dividing both sides by t
S

r
t
t

with  
t
Then we have :
V
V = rω  ω =
r
Determine the rotational speed of a car wheel if
the car is translating along at a speed of 55
mph. The radius of the wheel is 12.5 in.
miles
1h
5280 ft
(55
)(
)(
)
V
h
3600 s 1 mile  77.4 rad/s = 739 rpm
 
1 ft
r
(12.5 in.)(
)
12 in.
Examples of Frequencies of Various
Electrical and Electronic Systems
Engineering Fundamentals, By Saeed
Moaveni, Third Edition, Copyrighted 2007
8-10