Block diagram reduction techniques in control systems.ppt
Bahan Perkuliahan 1 Fismat II.pdf
1. JURUSAN FISIKA UNIVERSITAS ANDALAS
FISIKA MATEMATIKA 2
Slide By: Dr. rer. nat. Muldarisnur
Presented By: Rico Adrial, M.Si
Fourier Series and Transform
2. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
Pendahuluan
❑ Banyak peristiwa fisika terkait dengan dinamika yang berulang-
ulang atau periodik.
❑ Contoh yang paling sederhana adalah gerak harmonik sederhana
oleh pegas atau pendulum.
❑ Secara umum, gejala atau struktur periodik yang diamati
̶ Memiliki bentuk tidak sesederhana fungsi sinusoidal
̶ Bahkan, seringkali tidak memiliki bentuk analitik.
❑ Untuk menangani permasalahan seperti ini, kita dapat
menggunakan uraian deret dengan fungsi sinusoidal sebagai
basisnya.
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3. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
Outline
❖ Fourier Series
✓ Introduction
✓ Periodic Function
✓ Fourier Series
✓ Complex Form of Fourier Series
✓ Fourier Series with Other Intervals
✓ Even and Odd Functions
❖ Fourier Transform
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4. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
locity
angular ve
:
)
2
2
(
, f
T
t
=
=
=
y coordinate of Q (or P): t
A
A
y
sin
sin =
=
The back and forth motion of Q
→ simple harmonic motion
Harmonic motion
‒ P moves at constant speed around a circle of radius A.
‒ Q moves up and down in such a way that its y coordinate is always
equal to that of P.
For a constant circular motion,
Periodic Functions
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5. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
t
A
y
t
A
x
sin
,
cos =
=
(cos sin )
i t
z x iy A t i t
Ae
= + = +
=
In the complex plane,
imaginary part → velocity of Q
)
sin
(cos
)
( t
i
t
Ai
e
Ai
Ae
dt
d
dt
dz t
i
t
i
+
=
=
=
Complex number
The x and y coordinates of P:
Then, it is often convenient to use the complex
notation.
(Position of Q: imaginary part of the complex z)
Velocity:
Periodic Functions
5
6. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
)
sin(
,
cos
or
sin
+
t
A
t
A
t
A
cf. phase difference or
different choice of the
origin
Functional form of the simple harmonic motion:
Periodic function
Displacement
Time
amplitude
period:
2
=
T
Periodic Functions
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7. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
Displacement
Time
amplitude
period:
2
=
T
t
B
t
A
dt
dy
t
A
y
cos
cos
sin
=
=
=
Kinetic energy:
2
2 2
1 1
cos
2 2
dy
EK m mB t
dt
= =
Total energy E = Kinetic Energy EK + Potential Energy EP
2
2
2
2
2
2
1
f
A
A
mB
Displacement:
Velocity:
Periodic Functions
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8. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
x
A
y
2
sin
=
−
=
−
= t
v
x
A
vt
x
A
y
2
2
sin
)
(
2
sin
Wavelength: λ
Arbitrary periodic function (like wave)
ngth)
(or wavele
period
:
)
(
)
(
p
x
f
p
x
f =
+
Propagating wave
distance
T
f
v
T
cf
1
,
. =
=
Periodic Functions
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9. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
❑ Fundamental sinusoidal function(first order):
❑ Higher harmonics (higher order):
❑ Fundamental + the harmonics → Complicated Periodic Function.
❑ Conversely, a complicated periodic function can be expressed as the
combination of the fundamental and the harmonics
→ Fourier Series expansion
t
t
cos
,
sin
)
cos(
),
sin( t
n
t
n
Fourier Series
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10. Fisika Matematika 2 Dr. rer. nat. Muldarisnur & Mr. Co
What a-c frequencies (harmonics) make up a given signal and in what
proportions?
→ We can answer the above question by expanding these various periodic
functions into Fourier Series.
Periodic Functions: Examples
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