Soil Dynamics

Soil Dynamics
1. Vibratory Motion
Cristian Soriano Camelo1
1Federal University of Rio de Janeiro
Geotechnical Engineering
June 30th, 2017
Cristian Soriano Camelo (UFRJ) Soil Dynamics June 30th, 2017 1 / 33

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

Outline
1 Introduction
Vibratory Motion
Vibratory motion
4 Fourier Series

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.

Types of vibratory motion

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.

Nonperiodic motion
Examples of non periodic motion are impulsive loads (explosions,
falling weights) or longer-duration transient loadings (earthquakes or
traﬃc).

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.

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

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)

Outline
1 Introduction
Vibratory Motion
Vibratory motion
4 Fourier Series

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

Outline
1 Introduction
Vibratory Motion
Vibratory motion
4 Fourier Series

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

Outline
1 Introduction
Vibratory Motion
Vibratory motion
4 Fourier Series

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.

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

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

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

Fourier series

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

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.

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

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)

Fourier series

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)

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)

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)

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)

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

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.

For Further Reading I
Steven L. Kramer.
Geotechnical Earthquake Engineering.
Prentice Hall, 1996.

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