Chapter 4
Oscillations
4.1
Harmonic oscillations
The general form of a harmonic oscillation is:
Ψ
(
t
)=
ˆ
Ψ
e
i
(
ω
t
±
ϕ
)
≡
ˆ
Ψ
cos(
ω
t
±
ϕ
)
,
where
ˆ
Ψ
is the
amplitude
. A superposition of several harmonic oscillations
with the same frequency
results in
another harmonic oscillation:
±
i
ˆ
Ψ
i
cos(
α
i
±
ω
t
)=
ˆ
Φ
cos(
β
±
ω
t
)
with:
tan(
β
)=
∑
i
ˆ
Ψ
i
sin(
α
i
)
∑
i
ˆ
Ψ
i
cos(
α
i
)
and
ˆ
Φ
2
=
±
i
ˆ
Ψ
2
i
+2
±
j>i
±
i
ˆ
Ψ
i
ˆ
Ψ
j
cos(
α
i

α
j
)
For harmonic oscillations holds:
²
x
(
t
)
dt
=
x
(
t
)
i
ω
and
d
n
x
(
t
)
dt
n
=(
i
ω
)
n
x
(
t
)
.
4.2
Mechanic oscillations
For a construction with a spring with constant
C
parallel to a damping
k
which is connected to a mass
M
, to
which a periodic force
F
(
t
)=
ˆ
F
cos(
ω
t
)
is applied holds the equation of motion
m
¨
x
=
F
(
t
)

k
˙
x

Cx
.
With complex amplitudes, this becomes

m
ω
2
x
=
F

Cx

ik
ω
x
. With
ω
2
0
=
C/m
follows:
x
=
F
m
(
ω
2
0

ω
2
)+
ik
ω
,
and for the velocity holds:
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This note was uploaded on 11/16/2010 for the course ENGR 201 taught by Professor Elder during the Spring '10 term at Blinn College.
 Spring '10
 elder
 Superposition, Frequency

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