Soil dyn __2

Soil Dynamics
2. Dynamics of Discrete Systems
Cristian Soriano Camelo1
1Federal University of Rio de Janeiro
Geotechnical Engineering
July 06th, 2017
Cristian Soriano Camelo (UFRJ) Soil Dynamics July 06th, 2017 1 / 35

Outline
1 Introduction
Dynamics of Discrete Systems
2 Vibrating systems
Categories, Degrees of freedom
3 Single Degree of Freedom Systems (SDOF)
Deﬁnition, equations of motion, response
4 Damping
Viscous damping, Other measures of energy dissipation
5 Multiple Degree of Freedom Systems
Equations of motion

Outline
1 Introduction
2 Vibrating systems
4 Damping
Equations of motion

Introduction
Vibrating systems
Discrete elements:
spring and masses
Dynamic response
of discrete systems
Dynamics of discrete systems: simple systems, damping, base motion
and nonlinearity.

Outline
1 Introduction
2 Vibrating systems
4 Damping
Equations of motion

Vibrating systems
Rigid systems: No strains occur → all points move in the same phase
(i.e rigid body).
Compliant systems: diﬀerent points within the system may move
diﬀerently (and out of phase).
Soils and structures are studied in terms of compliant systems:
structural dynamics and earthquake engineering.

Vibrating systems
Compliant systems can me characterized by the distribution of their
mass:
Discrete systems:The mass concentrated in a ﬁnite number of
locations.
Continuous systems: the mass is distributed throughout the system.
Those concepts results in the dynamic degrees of freedom that describe
the position of all the signiﬁcant masses of the system.

Outline
1 Introduction
2 Vibrating systems
4 Damping
Equations of motion

Single Degree of Freedom Systems
SDOF:Discrete systems whose position can be described by a single
variable. The degrees of freedom can be a translational displacement or
a rotational displacement.

Equation of motion - SDOF
External load Q(t), inertial force fi, viscous damping force fD and
elastic spring force fS.
- In terms of equilibrium:fi(t) + fD(t) + fS(t) = Q(t)
- In terms of motion of the mass: md2u(t)
dt2 + cdu
dt + ku(t) = Q(t)

Equation of motion - Base shaking
In earthquake engineering problems, dynamic loading often results
from vibration of the supports of a system rather than from dynamic
external loads.
Equation of motion: mu′′(t) + cu′(t) + ku = 0
Substituting: u′′
t (t) = u′′
b (t) + u′′(t) →mu′′ + cu′ + ku = −mu′′
b

Response of linear SDOF systems
Forced vibration: when the mass is subjected to loading.
Free vibration:absence of external loading or base shaking.
Undamped free vibrations c=0 Q(t)=0
Damped free vibrations c>0 Q(t)=0
Undamped forced vibrations c=0 Q(t)=0
Damped forced vibrations c>0 Q(t)=0

Undamped free vibrations
No external load is considered, c=0.
u =
u′
0
ω0
sin(ω0t) + u0cos(ω0t)
Where ω0 = k
m is the undamped natural circular frequency.

Undamped free vibrations
Example 1: An undamped oscillator has a mas m (5 and 2)kg,
k=45N/m. At time t=0, the oscillator is given an initial position
x0=2m and an initial velocity v0=4m/s.
equat(m , b , k , x0 , v0 ):= {bx′(t) + kx(t) + mx′′(t) = 0, x(0) = x0, x′(0) = v0}
response(m , b , k , x0 , v0 ):=x(t)/. Flatten[DSolve[equat(m, b, k, x0, v0), x(t), t]]
x2(t ) = response(m, 0, 45, 2, 4) = 2
15
2
√
5
√
m sin 3
√
5t√
m
+ 15 cos 3
√
5t√
m
1 2 3 4 5
t(s)
-2
-1
1
2
x1(meters)
Undamped Vibration Example
1 2 3 4 5
t(s)
-2
-1
1
2
x1(meters)
Undamped Vibration Example
m=5 kg m=2kg

Damped free vibrations
In real systems, energy may be lost as a result of friction, heat
generation, air resistance or other physical mechanisms. Hence the free
vibration response of a SDOF system will diminish with time. For
damped free vibrations:
mu′′ + cu′ + ku = 0
Where ξ is the damping ratio.

Example 2: The structure shown in the ﬁgure is released from an
initial displacement of 1 cm with an initial velocity of -5cm/s.
Compute the time history of response of the mass.

x3(t ) = response(1000, 1000, 20000, 0.01, −0.05)
0.01e− t
2 1. cos
√
79t
2 − 1.01258 sin
√
79t
2
2 4 6 8
Time(s)
-0.010
-0.005
0.005
0.010
Diplacement(m)
Displacement response

Damped forced vibrations
The equation of motion for a damped SDOF system subjected to
harmonic loading of the form Q(t) = Q0sin(ωt) is:
mu′ + cu′ + ku = Q0sin(ωt)
The solution of the equation indicates that the response has two
components. One component occurs in response occurs at the natural
frequency of the system ω0 and other at the natural frequency of the
loading ω.
u = Q0
k
1
1−β2 (sin(ωt) + βsin(ω0t))
Where β = ω
ω0
, tuning ratio.

Undamped forced vibrations
The relationship 1
1−β2 can be thought as the amount by which the
static displacement amplitude is magniﬁed by the harmonic load.
Note that for loading frequencies lower than
√
2ω0 the displacement
amplitude is greater than the static displacement.

Example 3: Simulation of an undamped SDOF system subjected to
diﬀerent excitation frequencies: twice lower than the natural frequency,
twice higher than the natural frequency and at natural frequency.
Natural angular frequency ω0 4.5 Hz
Damping ratio 0
Driving amplitude 100 kN
Driving angular frequency 2.25 Hz, 9 Hz, 4.5 Hz
Mass (m) 6500 kg
k = (2πω2
0)m 5200000 N/m

Case 1: ω = ω0/2
Case 2: ω = 2ω0

Case 3: ω = ω0
It can be seen that the response grows without bound indicating resonance of the system,
the corresponding frequency is called resonance frequency.

The most general case is a damped system subjected to forced
harmonic loading. The general equation of a damped SDOF system
subjected to harmonic loading of the form Q(t) = Q0sin(ωt) is:
mu′′ + cu′ + ku = Q0sin(ωt)
Example 4: Simulation of a resonant damped SDOF system
subjected to an excitation frequency twice lower than the natural
frequency and diﬀerent damping ratios ξ = 0.1, 0.3.

Case 1: ξ = 0.1
Case 2: ξ = 0.3

Outline
1 Introduction
2 Vibrating systems
4 Damping
Equations of motion

Damping
Viscous Damping:The most commonly used mechanism for
representing energy dissipation. For a viscous SDOF system subjected
to a harmonic displacement u(t) = u0sin(ωt):
F(t) = ku(t) + cu′(t) = ku0sin(ωt) + cωu0cos(ωt)

Damping
Evaluating these functions from a time t0 to time t0 + 2π
ω , yields the
force-displacement values for one cycle of a hysteresis loop.
When viscous damping c = 0 there is a linear elastic-stress relationship.
For nonzero damoing, the hysteresis loop becomes elliptical.

Damping
The energy dissipated in one cycle of oscillation is:
WD =
t0+ 2π
ω
t0
F du
dt dt = πcωu2
0
At maximum displacement, the velocity is zero and the energy stored
in the system is:
Ws = 1
2 ku2
0
And the damping ratio is given by: ξ = WD
4πWS
This expression is used for graphical determination of the damping
ratio.

Response spectra
The response spectrum describes the maximum response of a SDOF
system to a particular input motion as a function of the natural
frequency (or natural period) and damping ratio. The response may be
expressed in terms of acceleration, velocity or displacement.

Response spectra

Outline
1 Introduction
2 Vibrating systems
4 Damping
Equations of motion

Multiple Degree of Freedom Systems
In systems in which the motion cannot
be described by a single variable
(buildings, bridges, other structures).
Equations of motion:
[m]u”+[c]u’+[k]u=q(t)
Example 5: Determine the Equations of Motion of the two degrees of
freedom system shown in the ﬁgure.
m1=2 m2=1 c1=0.1 c2=0.3
k1=6 k2=3 p1=2sin(3t) p2=5cos(2t)

Multiple Degree of Freedom Systems
2 0
0 1
u1”
u2”
+
0.4 −0.3
−0.3 0.3
u1’
u2’
+
9 −3
−3 3
u1
u2
=
2sin3t
5cos2t
m=
2 0
0 1
; c=
0.4 −0.3
−0.3 0.3
; k=
9 −3
−3 3
u1
u2
; q=
2sin3t
5cos2t

For Further Reading I
Steven L. Kramer.
Geotechnical Earthquake Engineering.
Prentice Hall, 1996.
http://www.strongmotioncenter.org/

Soil dyn __2

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