Vibratory motion - Soil dynamics.

1. Soil Dynamics
1. Vibratory Motion
Cristian Soriano Camelo1
1Federal University of Rio de Janeiro
Geotechnical Engineering
June 30th, 2017
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2. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
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3. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
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4. Introduction
Different types of
dynamic loading
Vibratory motion of
soils and structures
Problems of dynamic response
To solve these problems it is necessary to understand and describe the
dynamic events.
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5. Types of vibratory motion
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6. Periodic motion
u(t) is periodic if there exists some period Tf , then u(t + Tf ) = u(t)
For example, simple harmonic motion.
Simple harmonic motion can be characterized by three quantities:
amplitude, frequency and phase.
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7. Nonperiodic motion
Examples of non periodic motion are impulsive loads (explosions,
falling weights) or longer-duration transient loadings (earthquakes or
traffic).
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8. Simple harmonic motion - Trigonometric notation
Simple harmonic motion can be expressed in terms of a displacement
as:
u(t) = Asin(ωt + φ)
Where, A= amplitude, ω=circular frequency (angular speed), φ=phase
angle.
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9. Simple harmonic motion - Trigonometric notation
The phase angle describes the amount of time by which the peaks (and
zero points) are shifted from those of a pure sine function.
Another common measure is the number of cycles per unit of time,
known as frequency, expressed as the reciprocal of the period.
f = 1
T = ω
2π
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10. Simple harmonic motion - Trigonometric notation
Simple harmonic motion can also be described as a sum of sine and
cosine function, u(t) = acos(ωt) + bsin(ωt)
Using the rotating vector representation, the sum of the sine and cosine
function can be expressed as:
Therefore, u(t) = Asin(ωt + φ)
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11. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
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12. Simple harmonic motion - Complex notation
For many dynamic analyses trigonometric notation leads to long
equations. Complex notation can be derived from trigonometric
notation using Euler’s law.
u(t) = a−ib
2 eiωt + a+ib
2 e
−iωt
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13. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
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14. Other measures of motion
If the displacement with time is known, other parameters of interest
can be determined:
Trigonometric Complex -
u(t) = Asin(ωt) u(t) = Ae
iωt
Displacement
u′(t) = ωAcos(ωt) = ωAsin(ωt + π/2) u′(t) = iωAe
iωt
Velocity
u′′(t) = −ω2Asin(ωt) = ω2Asin(ωt + π) i2ω2Ae
iωt
Acceleration
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15. Outline
1 Introduction
Vibratory Motion
2 Complex Notation for Simple Harmonic Motion
Vibratory motion
3 Other measures of motion
Displacement, Velocity, Acceleration
4 Fourier Series
Trigonometric, Exponential, Discrete transform, Power spectrum
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16. Fourier series
In geotechnical earthquake engineering, it is possible to break down
complicated loading functions (earthquakes) into the sum of simple
harmonic functions.
Trigonometric Form: Is expressed as a sum of simple harmonic
functions.
x(t) = a0 + ∞
n=1(ancos(ωnt) + bnsin(ωnt))
Where,
a0 = 1
Tf
Tf
0 x(t)dt,
an = 2
Tf
Tf
0 x(t)cos(ωnt)dt,
bn = 2
Tf
Tf
0 x(t)sin(ωnt)dt.
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17. Fourier series
Example 1 (Mathematica Software): Given the following piecewise
function, obtain the Fourier expansion for 5 and 10 coefficients and
compare the results with the current function.
x[t ]:=Piecewise[{{Sin[t], 0 < t < Pi}, {0, −Pi < t < 0}}]
-3 -2 -1 1 2 3
Time
0.2
0.4
0.6
0.8
1.0
x
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18. Fourier series
Solving the integrals for the coefficients:
ao = 1
2π
π
−π x[t] dt= 1
π
a[n ]:= 1
π
π
−π x[t]Cos[n ∗ t]dt=−1−Cos[nπ]
(−1+n2)π
b[n ]:= 1
π
π
−π x[t]Sin[n ∗ t]dt=− Sin[nπ]
(−1+n2)π
Therefore, the 10 first Fourier coefficients a[n], b[n] are:
a[n]= 0, − 2
3π , 0, − 2
15π , 0, − 2
35π , 0, − 2
63π , 0, − 2
99π
b[n]= 1
2, 0, 0, 0, 0, 0, 0, 0, 0, 0
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19. Fourier series
Using directly the program commands, it is possible to obtain the
desired Fourier Series:
-Trigonometric form (10 terms):
x(t)= 1
π − 2Cos[2t]
3π − 2Cos[4t]
15π − 2Cos[6t]
35π − 2Cos[8t]
63π − 2Cos[10t]
99π + Sin[t]
2
-Trigonometric form (5 terms):
x(t)= 1
π − 2Cos[2t]
3π − 2Cos[4t]
15π + Sin[t]
2
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20. Fourier series
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21. Fourier series
Exponential form: using Eulers law is possible to obtain the Fourier
series in exponential form.
xt = a0 + (an+ibn
2 eiwnt + an+ibn
2 e−iwnt)
The function in Example 1 expressed in complex notation and using 5
coefficients is:
x(t)=1
4 ie−it − 1
4ieit + 1
π − e−2it
3π − e2it
3π − e−4it
15π − e4it
15π
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22. Fourier series
Discrete Fourier Transform (DFT): in many applications (e.g
earthquake engineering), loading motion parameters are be described
by a finite number of data points.
DFT is an algorithm that takes a signal and determines the frequency
content of the signal. To determine that frequency content, DFT
decomposes the signal into simpler parts.
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23. Fourier series
The DFT is given by:
X(ω) = N−1
n=0 x(n)e
−iπωn/N
Where x(n) is the n-value of the different points of the signal and ω
varies from 0 to N-1.
Breaking the previous equation into sines and cosines, it is possible to
identify two correlation calculations composed by a real and an
imaginary components of X(ω) :
X(ω) = N−1
n=0 x(n)cos 2πωn
N − i N−1
n=0 x(n)sin 2πωn
N
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24. Fourier series
Example 2 - DFT: Given the following signal samples, determine the
DFT.
20 40 60 80 100
w
-2
-1
1
2
Amplitude
Signal x(n)
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25. Fourier series
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26. Fourier series
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27. Fourier series
The final DFT presents four peaks that represent the real and
imaginary components of X(ω). For example, at ω=10 there is a peak
in the real component (signal similar to a cosine), and at ω=3 there is
a peak in the imaginary part (signal similar to a sine).
20 40 60 80 100
w
1
2
3
4
5
Magnitude
x(w)
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28. Fourier series
Correlation: Correlation is a measure of how two signals are. In
general correlation can be expressed as:
N
n=0 x(n)y(n)
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29. Fourier series
Example 3 - DFT: The following is a list of 200 elements containing
a periodic signal with random noise added.
data=Table N RandomReal[] − 1
2 + sin 30 2πn
200 , {n, 200} ;
50 100 150 200
w
-1.5
-1.0
-0.5
0.5
1.0
1.5
Amplitude
Signal x(t)
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30. Fourier series
The DFT in this case shows a peak at 30+1 and a symmetric peak at
201-30.
50 100 150 200
w
1
2
Q
4
5
6
U
Magnitude
x(w)
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31. Fourier series
Power Spectrum: The power spectrum answers the question ”How
much of the signal is at a frequency ω”?. In the frequency domain, this
is the square of the DFT’s magnitude:
P(ω) = Re(X(ω))2 + Im(X(ω))2
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32. Fourier series
Power spectra are often used to describe earthquake-induced ground
motions:
The figure indicates that most of the energy in the accelerogram is in
the range of 0.1 to 20 Hz and the largest amplotude at approximately 6
Hz.
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33. For Further Reading I
Steven L. Kramer.
Geotechnical Earthquake Engineering.
Prentice Hall, 1996.
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