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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
m around
Frequency
inverse of period
your birthday comes once a year
Frequency is a measure of how frequently an event or a q
y
q
y
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 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 p
q
y
the system is given
fn =
1
2π
k
m
in Hz, cycles per second
1
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fn =
1
2π
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)
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
T=
m
1
= 2π
fn
k
in seconds 2
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An Example of Period and Frequency
Given: a simple spring‐mass system Find: the natural frequency of the system
the natural frequency of the system
Solution:
fn =
1
2π
k
m
=
1
2π
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
blades, CD drives
drives, Zip drives
drives, and so on
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
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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:
revolutions 2π radians
1 minute
)(
)(
)=
minutes 1 revolution 60 seconds
d
a s
r
5
.
7
6
1
1600(
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.
ΔS = rΔθ
and dividing both sides by Δt
Δθ
ΔS
=r
Δt
Δt
⇒
V r
=
ω
ω
r
=
V
Δθ
Δt
Then we have :
with ω =
4
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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.
V
=
r
miles
1h
5280 ft
)(
)(
)
h
3600 s 1 mile =
1 ft
(12.5 in.)(
)
12 in
in.
m
p
r
9
3
7
=
s
/
d
a
r
4
.
7
7
ω=
(55
Examples of Frequencies of Various Electrical and Electronic Systems
Engineering Fundamentals, By Saeed Moaveni, Third Edition, Copyrighted 2007
8‐10
5