SOIL DYNAMICS - THEORY OF VIBRATIONS

GOVERNMENT COLLEGE OF
ENGINEERING, AMRAVATI
DEPARTMENT OF CIVIL ENGINEERING
SOIL DYNAMICS
THEORY OF VIBRATIONS
1

THEORY OF VIBRATION
Satisfactory design of foundations for vibrating
equipments is mostly based on displacement
considerations. Displacement due to vibratory loading can
be classified under two major divisions:
1. Cyclic displacement due to the elastic response of the
soil-foundation system to the vibrating loading
2. Permanent displacement due to compaction of soil
below the foundation
In order to estimate the displacement due to the
first loading condition, it is essential to know the nature of
the unbalanced forces (usually supplied by the
manufacturer of the machine) in a foundation.
2

THEORY OF VIBRATION
Foundation can vibrate in any or all six possible
modes. For ease of analysis, each mode is considered
separately and design is carried out by considering the
displacement due to each mode separately. Approximate
mathematical models for computing the displacement of
foundations under dynamic loads can be developed by
treating soil as a viscoelastic material. This can be
explained with the aid of Figure which shows a foundation
subjected to a vibratory loading in the vertical direction.
4

THEORY OF VIBRATION
The parameters for the vibration of the foundation can be evaluated by treating the
soil as equivalent to a spring and a dashpot which supports the foundation as
shown in Figure. This is usually referred to as a lumped parameter vibrating system.
5

THEORY OF VIBRATION
Fundamentals of Vibration:
1. Free Vibration: Vibration of a system under the action of forces inherent in
the system itself and in the absence of externally applied forces. The
response of a system is called free vibration when it is disturbed and then
left free to vibrate about some mean position.
2. Forced Vibration: Vibration of a system caused by an external force.
Vibrations that result from regular (rotating or pulsating machinery) and
irregular exciting agencies are also called as forced vibrations.
3. Degree of Freedom: The number of independent coordinates required to
describe the solution of a vibrating system.
4. Natural frequency: Number of free oscillations made by the system in unit
time.
5. Amplitude of motion: The maximum displacement of a vibrating body
from the mean position.
6. Time period: Time taken by the system to complete one cycle of vibration.
6

THEORY OF VIBRATION
Fundamentals of Vibration:
For example, the position of
the mass ‘m’ in Figure 1 can be
described by a single coordinate ‘z’,
so it is a single degree of freedom
system. In Figure 2 two coordinates
(z1 and z2) are necessary to describe
the motion of the system; hence this
system has two degree of freedom. A
rigid body has total six degrees of
freedom: three rotational and three
translational.
7

THEORY OF VIBRATION
Fundamentals of Vibration:Mathematical models
A. Free Vibration of a Spring-Mass System
8

THEORY OF VIBRATION
Figure shows a foundation resting on a spring. Let the spring
represent the elastic properties of the soil. The load W represents the weight
of the foundation plus that which comes from the machinery supported by
the foundation. If the area of the foundation is equal to A, the intensity of
load transmitted to the subgrade, q = W/A
Due to the load W, a static deflection ‘zs ‘will develop. By definition,
k = W / zs
where k = spring constant for the elastic support.
The coefficient of subgrade reaction ks can be given by
ks = q / zs
The spring supplies a restoring force equal to spring constant times the
displacement i.e.
F = k.z
9

THEORY OF VIBRATION
If the foundation is disturbed from its static equilibrium position,
the system will vibrate. If the foundation moves down, the restoring force
tends to bring it back to its equilibrium position. By the time it reaches its
equilibrium position, the mass is in motion with an acceleration ‘a’ and tends
to remain in motion and therefore overshoots the equilibrium position. At it
does so, the restoring force begins to generate again and the mass comes to
rest on the higher (upper) side of equilibrium position, when the restoring
force is equal to inertia force (m.a). As the body comes to rest, it again tend to
come back to its equilibrium position. This process repeats and body
continues to vibrate indefinitely, as kinetic energy gets converted in to
potential energy and vice-a- versa. The object (foundation) thus undergoes
simple harmonic motion.
The equation of motion can be written as
Restoring force = Inertia force
-k.z = m.a
m.a + kz = 0
10

THEORY OF VIBRATION
The equation of motion of the foundation when it has been
disturbed through a distance ‘z’ can be written from Newton’s second law of
motion
or
………………………..(1)
where,
g = acceleration due to gravity
acceleration,
t = time
m = mass = W/g
11

THEORY OF VIBRATION
In order to solve Eq. 1, let
………………….(2)
where, A1 and A2 = constants
wn = undamped natural circular frequency
Substitution of Eq. (2) into Eq. (1)
or [radians per second (rad/s)]
Hence, ….(3)
12

THEORY OF VIBRATION
In order to determine the values of A1 and A2, one must substitute
the proper boundary conditions.
At time t = 0, let Displacement z = z0 , and Velocity =
Substituting the first boundary condition in Eq.(3) (4)
Also, from Eq(3), Velocity,
……(5)
Substituting the second boundary condition in Eq. (5)
or
13

THEORY OF VIBRATION
Combination of Eqs. (3), (4), and (5) gives
…(6)
Let ---(7)
and ---(8)
Substitution of Eqs. (7) and (8) into Eq. (6)
14

THEORY OF VIBRATION
The relation for the displacement of the foundation can be
represented graphically as shown in Figure
15

THEORY OF VIBRATION
From Figure, it can be seen that the nature of displacement of
the foundation is sinusoidal. The magnitude of maximum
displacement is equal to Z. This is usually referred to as the ‘single
amplitude’. The peak-to-peak displacement amplitude is equal to 2Z,
which is sometimes referred to as the ‘double amplitude’.
The time required for the motion to repeat itself is called the
‘period of the vibration’. In Figure, the motion is repeating itself at
points A, B, and C.
The period T of this motion is given by
The ‘frequency of oscillation’ f is defined as the number of
cycles in unit time, or
16

THEORY OF VIBRATION
Now,
hence,
The term fn is generally referred to as the ‘undamped natural
frequency’.
Since
Eq. can also be expressed as
17

THEORY OF VIBRATION
Table gives values of fn for various values of zs
18

THEORY OF VIBRATION
From Eq. of ‘Z’, the expressions for the velocity and the
acceleration can be obtained as
The variation of the velocity and acceleration of the motion with
time can also be represented graphically as shown in fig.
19

THEORY OF VIBRATION
A. Free Vibration of a Spring-Mass System: Problems
1. A mass is supported by a spring. The static deflection of the spring
due to the mass is 0.381mm. Find the natural frequency vibration.
Solution :
20

THEORY OF VIBRATION
2. For a machine foundation, given weight of the foundation = 45 kN
and spring constant = 104 kN/m, determine
a) natural frequency of vibration, and
b) period of oscillation .
Solution :
21

THEORY OF VIBRATION
3. For a machine foundation, given weight of the foundation = 45 kN
and spring constant = 104 kN/m, determine
a) natural frequency of vibration, and
b) period of oscillation .
Solution :
22

THEORY OF VIBRATION
B. Free Vibration of a Spring-Mass System with viscous damping
23

THEORY OF VIBRATION
In the case of undamped free vibration , vibration would
continue once the system has been set in motion. However, in
practical cases, all vibrations undergo a gradual decrease of amplitude
with time. This characteristic of vibration is referred to as damping.
Fig. shows a foundation supported by a spring and a dashpot. The
dashpot represents the damping characteristic of the soil. The dashpot
coefficient is equal to ‘c’. [kg/(m/sec) ]. Thus dashpot exerts a force which
acts to oppose the motion of mass and has magnitude c.z
The differential equation of motion can be given by
………….(1)
Let z = A.ert be a solution to Eq., where ‘A’ and ‘r’ are arbitrary
constants. Substitution of this into Eq. (1) yields
or ………….(2)
24

THEORY OF VIBRATION
The solutions to Eq. (2) can be given as
i.e. r = 1/(2 m) [ -c + √ c2 – 4.k.m ] ………….(3)
There are three general conditions that may be developed from Eq. (3):
1. If c2 > 4.k.m, both roots of Eq. are real and negative and solution to the
equation is
z = C1 e r1t + C2 er2t
25
Since r1 and r2 are both
negative, z will decrease
exponentially without change
in sign. This is referred to as an
overdamped case. The system
which is overdamped will not
oscillate at all.

THEORY OF VIBRATION
2. If c 2 = 4.k.m
In this case, roots are equal and negative. The solution is given by
z = (C1 + C2.t) e [-c/(2m) .t]
In this case also, there is no vibratory motion. It is similar to
overdamped case, except that it is possible for the sign to change once.
Thus, for this case,
………..(4)
This is called the critical damping case and cc is known as ‘critical
damping constant’. The ratio of actual damping constant to critical
damping constant is known as ‘Damping Ratio’ (D)
D = c/cc = c/[2 √k.m ] 26

THEORY OF VIBRATION
3. If c 2 < 4.k.m, the roots of Eq. are complex conjugate and the complex
solution is given is:
z = e [-ωn D t] [ C3 sin ωn √1 – D2 + C4 cos ωn √1 – D2 ]
where, C3 & C4 are arbitrary constants. This eq. indicates that motion will be
oscillatory and the decay in amplitude with time will be proportional to e [-ωn D t] as
shown by dotted curve
This is referred to as a case of underdamping
27

THEORY OF VIBRATION
Frequency of free vibration in this case is less than undamped
natural circular frequency. The natural circular frequency for damped
oscillation in terms of undamped natural circular frequency is given by
natural logarithmic ratio of two successive peak amplitude is called
‘Logarithmic decrement’.
If the damping ratio D is small, then
If D > 1, then system is overdamped
If D = 1 , then system is critically damped
Id D < 1 then system is underdamped 28

THEORY OF VIBRATION
B. Free Vibration of a Spring-Mass System with viscous damping: Problems
1. For a machine foundation, given weight = 60 kN, spring constant = 11,000 kN/m,
and c = 200 kN-s/m, determine
(a) whether the system is overdamped, underdamped, or critically damped,
(b) the logarithmic decrement, and
(c) the ratio of two successive amplitudes.
Solution :
1.
Hence, the system is underdamped
2.
3.
29

THEORY OF VIBRATION
B. Free Vibration of a Spring-Mass System with viscous damping: Problems
2. For the previous example, determine the damped natural frequency.
Solution :
1.
2.
30

THEORY OF VIBRATION
C. Forced Vibration of a Spring-Mass System (undamped)
31

THEORY OF VIBRATION
Fig. shows foundation subjected to an alternating force
This type of problem is generally encountered with
foundations supporting reciprocating engines. The equation of
motion for this problem can be given by
…..(1)
Since the applied force is harmonic, the internal spring force
and inertia force will also be harmonic. Thus, the motion of the
system will be of the form
(A1 = const).
32

THEORY OF VIBRATION
From the equilibrium requirement
Hence the particular solution to Eq. is of the form
The general solution of Eq. is
33

THEORY OF VIBRATION
For a real system, the last two terms in Eq. will vanish due to
damping, leaving the only term for steady-state solution. If the forcing
function is in phase with the vibratory system (i.e., ß = 0), then
or
34

THEORY OF VIBRATION
However Q0/k = zs = static deflection.
If one lets = M [Magnification factor or A1/(Q0/ k)],
then
The dynamic magnification factor is defined as the ratio of
dynamic amplitude to the static deflection.
35

THEORY OF VIBRATION
The nature of variation of the magnification factor M with ω/ωn is
shown in Figure
Note that the magnification factor goes to infinity when ω/ωn = 1.
This is called the ‘resonance’ condition.
36

THEORY OF VIBRATION
At resonance,
37

THEORY OF VIBRATION
C. Forced Vibration of a Spring-Mass System (undamped):
Maximum force on foundation subgrade
The maximum and minimum force on the foundation subgrade will
occur at the time when the amplitude is maximum, i.e., when velocity is
equal to zero. It is given by
Hence, the total force on the subgrade will very between the limits
38

THEORY OF VIBRATION
C. Forced Vibration of a Spring-Mass System (undamped): Problems
1. A machine foundation can be idealized as a mass-spring system. This
foundation can be subjected to a force that can be given as
Q (kN) = 35.6 sin ωt .
Given f = 13.33 Hz
Weight of the machine + foundation = 178 kN
Spring constant = 70,000 kN/m
Determine the maximum and minimum force transmitted to the
subgrade.
Solution:
i.
39

THEORY OF VIBRATION
C. Forced Vibration of a Spring-Mass System (undamped): Problems
ii
Iii
iv.
40

THEORY OF VIBRATION
D. Steady-State Forced Vibration with Viscous Damping
The differential equation of motion for this system can be given be
41

THEORY OF VIBRATION
considering the particular solution for Eq. for the steady-state motion, let
where A1 and A2 are two constants.
Solution is obtained as
where, α =
and Z =
where ωn = k/m is the undamped natural frequency and D is the damping
ratio
42

THEORY OF VIBRATION
Equation can be plotted in a nondimensional form as Z (Q0 /
k) against ω/ωn . Maximum values of Z /(Q0 / k) do not occur at ω =
ωn, as occurs in the case of forced vibration of a spring-mass system .
43

THEORY OF VIBRATION
For maximum value of Z /(Q0 / k)
Hence,
where fm is the frequency at maximum amplitude (the resonant
frequency for vibration with damping) and fn is the natural frequency
= (1/2π) √k /m .
44

THEORY OF VIBRATION
Hence, the amplitude of vibration at resonance can be
obtained as
45

THEORY OF VIBRATION
Maximum Dynamic Force Transmitted to the Subgrade
It can be given by summing the spring force and the damping
force caused by relative motion between mass and dashpot, i.e.,
Magnitude of maximum dynamic force in this case
46

THEORY OF VIBRATION
D. Steady-State Forced Vibration with Viscous Damping : Problems
1. A machine and its foundation weigh 140 kN. The spring constant
and the damping ratio of the soil supporting the soil may be taken
as 12 × 104 kN/m and 0.2, respectively. Forced vibration of the
foundation is caused by a force that can be expressed as
Q (kN) = Q0 sin ωt
Q0 = 46 kN, ω = 157 rad/s
Determine
(a) the undamped natural frequency of the foundation,
(b) amplitude of motion, and
(c) maximum dynamic force transmitted to the subgrade.
47

THEORY OF VIBRATION
Solution
i.
ii.
iii.
48

THEORY OF VIBRATION
Solution
iv.
v. Dynamic force transmitted to the subgrade
Thus
49

THEORY OF VIBRATION
E. Rotating-Mass-Type Excitation
In many cases of foundation equipment, vertical vibration of
foundation is produced by counter-rotating masses as shown in Figure
Since horizontal forces on the foundation at any instant cancel, the
net vibrating force on the foundation can be determined to be equal to
(where me = mass of each counter-rotating element, e =
eccentricity, and ω = angular frequency of the masses). In such cases,
the equation of motion with viscous damping can be modified to the
form
50

THEORY OF VIBRATION
m is the mass of the foundation, including 2me
The solution for displacement may be given as
where &
51

THEORY OF VIBRATION
52
Angular resonant frequency for
rotating-mass-type excitation can
be obtained as
damped resonant frequency
The amplitude at damped resonant
frequency

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