sd lecture02 simple vibrations

Soil Dynamics

Lecture 02

Simple Vibrations

© Luis A. Prieto-Portar, August 2006.

Some simple initial definitions:

- A free vibration is any system that vibrates under the action of forces that are part of,

or are inherent to the system itself.

- A forced vibration is any system that vibrates under the action of an external force to

the system itself.

- The degree of freedom: The figure at left below can be described by a single

coordinate z so it is a single degree of freedom system; the figure in the middle needs z1

and z2 to describe the motion of the system, so it is a two-degrees of freedom system;

the figure at right is also a two-degree of freedom system, requiring z and θ to describe

the motion.

The basic design criterion for foundations subjected to vibrations (whether seismic,

machinery or impact loadings) is to control their displacements.

These displacements are of two kinds,

1) temporary cyclic elastic displacements (that is, they return to their original

position after the loading stops, and

2) permanent plastic displacements (the foundation remains displaced

from its original position after the loading ceases).

Foundations can vibrate in all six possible modes, as shown below.

These six modes of

vibration may contribute

to unbalanced forces in a

simple foundation. These

unbalanced forces in turn

generate the vibrations.

Each mode is analyzed separately. The most common simplification is to represent the

foundation-soil system subjected to a dynamic loading Q with a spring and a dashpot

analog model system (also known as a lumped parameter vibration system).

For example, a foundation subjected to a vertical axis dynamic loading, show below at

left is represented by the figure on the right:

this is equivalent to

a lumped parameter vibration system

Q

In this lecture, we will consider the following four cases, in progressively increasing

complexity:

1) A free vibration system without damping;

2) A steady-state forced vibration system without damping;

3) A free vibration system with viscous damping; and

4) A steady-state forced vibration system with viscous damping.

(1) A free-vibration system (with only a spring-mass).

The soil subgrade reaction q is the foundation load W over an area A. In the lumped

parameter system, the displacement zs of the soil is proportional to the load W, or

s

W lbk

z inch=

3s

s s

q W lbk

z Az in= =

expressed as an equality by using the spring

constant k,

The coefficient of sub-grade reaction ks is,

When the foundation is disturbed from its

static equilibrium, the foundation-soil system

will vibrate. The resulting equation of motion

can be written from Newton’s second law of

motion,

0 0 + = + =

ɺɺ ɺɺk

mz kz or z zm

1 2

2

1 2 1

0 0

+ = + =

= +

− + +

ɺɺ ɺɺ

n n

n

n n n

W kz kz or z z

g m

with a solution z A cos t A sin t

where is the undamped natural circular frequency.

Substituting z back int o the differential equation yields,

k( A cos t A sin t ) ( A cos

m

ω ωω ωω ωω ω

ωωωω

ω ω ω ωω ω ω ωω ω ω ωω ω ω ω2

1 2 0 0

01 0 2

00

0

0

+ =

=

= = =

= =

= +

ɺ

n n

n

t A sin t )

kand obviously the parenthesis is not zero, therefore

m

Find A and A from boundary conditions at t z z and z v

vwhichyields, A z and A

k

m

vk kThus z z cos t sin t

m mk

m

ωωωω

ωωωω

00

1 2 200 0

0

−

= +

= −

= = +

n

Now we can rewrite this equation,

vk kz z cos t sin t

m mk

m

as,the extremely simple ,

z Z cos( t )

where,

vtan and Z z (m / k )v

z k / m

ω αω αω αω α

αααα

0

0

22 0

3 23 2 0

22

n

n

n

n

n

n

The displacement z of the foundation, z Z cos( t )

at different time int ervals ,

t z Z cos( ) Z cos

t z Z cos( ) Z

/t z Z cos( / )

t z Z cos( ) Z

/t z Z cos( / )

t z Z cos( ) Z etc...

Thes


α αα αα αα αααααωωωω

π απ απ απ αππππ

ωωωω


ωωωω


ωωωω


ωωωω

= −

= = − =

= = =

+= = =

+= = = −

+= = =

+= = =

e results are shown on the next slide...

use the analogy of a pendulum.

The maximum displacement Z is called the single amplitude. The peak-to-peak amplitude

is 2Z and is also referred to as the double amplitude. The time required for the

sinusoidal displacement to repeat itself is called the period T, which is given by,

2 1

2

1 1 1

2 2 2 2

= = =

= = = =

n

n

n

n

s s

T and the frequency of oscillation f is, fT

k Wg gThe undamped natural frequency f

m zW z

ωωωωππππω πω πω πω π

ωωωωπ π π ππ π π ππ π π ππ π π π

The plots for the velocity and the acceleration of the foundation can be found from the

first and second derivatives of the displacement z,

2 2

2

= −

= − − = − +

= − − = − +

ɺ

ɺɺ

n

n n n n

n n n n

z Z cos( t )

z ( Z ) sin( t ) Z cos( t / ) and

z Z cos( t ) Z cos( t )


ω ω α ω ω α πω ω α ω ω α πω ω α ω ω α πω ω α ω ω α π

ω ω α ω ω α πω ω α ω ω α πω ω α ω ω α πω ω α ω ω α π

Example 1.

Find the natural frequency of vibration of a foundation with a mass m supported by a

soil that has experienced a static deflection of 0.5 inches under that mass.

21 1 32 2 124 42

2 2 0 5

= = =

n

s

g ( . ft / sec )( inches / feet )f . cycles per sec ond

z ( . inches )π ππ ππ ππ π

Example 2.

Find the period of oscillation and the natural frequency of vibration of a foundation

with a weight of 60 kN and a soil (spring) constant of 103 kN/m.

3

2

1 1 102 04

2 2 60 9 81

1 10 5

2 04

n

n

k kN / mf . cycles per sec ond

m kN / . m / s

T . sec ondsf .

π ππ ππ ππ π

= = =

= = =

(2) A steady-state forced-vibration system without damping.

The previously studied free vibration foundation-soil system has now an additional

external alternating force Qo sin(ωt + β). This is shown below, where the spring

constant is still k. This type of problem is typical of footings supporting internal

combustion engines that have reciprocating pistons. The equation of motion is now,

0mz kz Q sin( t )ω βω βω βω β+ = +ɺɺ

A particular solution for that equation of

motion could be of the form,

( )

1

2

1 1 0

01 2

z A sin( t )

Substituting int o the differential equation,

mA sin( t ) kA sin( t ) Q sin( t )

Q / mA

k / m

ω βω βω βω β

ω ω β ω β ω βω ω β ω β ω βω ω β ω β ω βω ω β ω β ω β

ωωωω

= +

− + + + = +

∴ =−

01 2

2 3

0

n n

Thus, the particular solution is,

Q / mz A sin( t ) sin( t )

( k / m )

The complementary solution must satisfy mz kz

and the solution to this equation was found to be,

z A cos t A sin t

Therefore, the general soluti

ω β ω βω β ω βω β ω βω β ω βωωωω


= + = + −

+ =

= +

ɺɺ

1 2 3

0

0

1

0

0

n n

n

n

on will be the addition of these two,

z A sin( t ) A cos t A sin t

For the boundary conditions at t o,

z z

z v

which yields,

z A sin( t ) cos( t ) sin sin( t )cos

The sec ond term will disappear whe

ω β ω ωω β ω ωω β ω ωω β ω ω

ωωωωω β ω β ω βω β ω β ω βω β ω β ω βω β ω β ω β

ωωωω

= + + +

=

= =

= =

= + − −

ɺ

0

n the system has damping, and if the force

function i s in phase with the vibratory system, that is , thus,ββββ =

0

2 2

0

2 2

1

1

1

n

n n

s

n

s n

n

Q / kz sin t sin t

/

The ratio Q / k was seen before as the static deflection z , and the rest is

known as the magnification factor M ,

M/

Therefore,

z z M sin t sin t

ωωωωω ωω ωω ωω ω

ω ω ωω ω ωω ω ωω ω ω



ωωωω

= − −

=−

= −

The magnification factor M varies with the ratio of ω/ωn . Notice that when that ratio

approaches unity, the magnification increases without bound. This is called a resonant

condition, and is of great interest to all structural engineers.

( )

( )

( )

2 21

1

2

n

n n

s

n

s n n n

At resonance, the general solution can be solved u sin g L' Hopital' s rule ,

dsin t / sin t

dlim ( z ) zd

/d

from whence,the displacemen t and the velocity at resonance are,

z z sin t t cos t

z


ω ω ω ωω ω ω ωω ω ω ωω ω ω ωωωωω

ω ωω ωω ωω ωωωωω


→

− =

−

= −

ɺ ( )

( )

2

2

1

2

10 0

2

1

2

s n n

s n n n n

max s maxresonance

z t sin t

Notice that the displacement ismax imum when the velocity is zero,

z z t sin t or sin t which is t n

and

z n z which means that as n so does z


ω ω ω ω πω ω ω ω πω ω ω ω πω ω ω ω π

ππππ

=

= = = =

= →∞

ɺ

The plot of z versus time t for the resonant condition shows its unending increase.

The largest and smallest forces from the foundation upon the soil sub-grade will occur

when the amplitude is the greatest, in other words, when the velocity is zero.

0

2 2

0

2 2

1

01

0

1

2

n

n n

n

n

n

n

Q / kSince z sin t sin t

/

the velocity is ,

Q / kz ( cos t cos t ) for max imum deflection

/

or ( cos t cos t )

( )t m ( the other solution is"beating"and is not relevant )

or ,

t



ω ω ω ωω ω ω ωω ω ω ωω ω ω ωω ωω ωω ωω ω

ω ω ω ωω ω ω ωω ω ω ωω ω ω ω

ω ω πω ω πω ω πω ω π

= − −

= − = −

− =

+ =

ɺ

2

n

mππππω ωω ωω ωω ω

=+

0 0

0

2

1 1

1

max

n n n

max imum max

n

Substituting this value for the time int o the displacement yields,

Q / k Q / kmz z sin

/ /

From this,themax imum dynamic force on the subgrade is,

QF kz

/

Thus, themax imum a

π ωπ ωπ ωπ ωω ω ω ω ω ωω ω ω ω ω ωω ω ω ω ω ωω ω ω ω ω ω


= = = − + −

= =−

0 0

1 1n n

nd min imum forces from the foundationW are,

Q QW and W

/ /ω ω ω ωω ω ω ωω ω ω ωω ω ω ω+ −

− −

Example 3.

Determine the maximum and minimum forces placed by a vibrating machine’s

foundation upon its soil sub-grade, if: (1) the machine and its foundation weigh 40 kips,

(2) the soil’s equivalent spring constant is 400 kip/in, and (3) the machine operates at

800 rpm. Assume that the force Qo (kips) upon the foundation is 8 sinωt.

2

0

40062 2

40 32 2 12

2 2 800 1 60 83 8

823

1 1 83 8 62 2

40 23 63

n

dynamic

n

dynamic

k kip / in. radians / s

m kip / ( . ft / s )( ft / in )

f ( cycles / minute )( minute / sec onds ) . radians / s

QF kips

/ . / .

The max imum force on the subgrade W F k

ωωωω

ω π πω π πω π πω π π


= = =

= = =

= = =− −

= + = + =

40 23 17dynamic

ips

The min imum force on the subgrade W F kips= − = − =

(3) A free-vibration system with viscous damping.

We saw before that a free vibration imposed upon an undamped foundation-soil

system will continue to vibrate without end. In reality, all vibrations will gradually

decrease in amplitude with time due to damping. The figure below shows the modeling

of the damping through the use of a dashpot and a coefficient c.

The differential equation of motion is now,

2

22

2

0

0

02 4

12

rt

rt rt rt

overdam

mz cz kz

Let z Ae be a solution;therefore, backsubstituting yields,

mAr e cAre kAe

c k c c kor r r with solution r

m m m m m

There are three possible cases :

c k) If both proots are real : d

mc

me

+ + =

=

+ + =

+ + = = − ± −

>

ɺɺ ɺ

ase.

22 2

2c

critical dampinc k c

) If then r and this is the case.m m m

For this case ,

c

g

c k m

= = −

= =

2

2

32

2 4

c k) If then the roots are complex ,

m m

c k cr i This case is called .

munderdam e

mp d

m

<

= − ± −

22

2

2

12 4

c

n n

The damping ratio D is defined as,

c cD

c k m

The solution can now be written as,

c c k kr ( D D ) where

m m m mω ωω ωω ωω ω

= =

= − ± − = − ± − =

( ) ( )

2

2 20 00

2

1

1

1 11

1

n

n

D t n

n n

n

n

For the ( D ),

r ( D D ) and the displacement z may be written as,

v D z

overdamped case

critically damped conditi

z e z cosh D t sinh D tD

For the ( D ),

r and the displacement z may be writ en

o

t

n

ωωωω

ωωωω


ωωωω

ωωωω

−

>

= − ± −

= − + −

−

=

= −

( )0 0 0nt

n

as,

z z v z t eωωωωωωωω − = + +

( )

2

2

2 0 00

2

1 0 0

2

0

1

1

1

1

n

n

d

D t n

n

n

n

d

For the ( D ),

r ( D i D ) and the displacement z may be written as,

z Z cos t where,

v D zZ e z

unde

D

v D ztan

z D

where is the da

rdamped c

mped natural circular freq

s

n y

a e

ue c

ωωωω

ωωωω


ωωωω

ωωωω

ωωωωαααα

ωωωω

ωωωω

−

−

<

= − ± −

= −

+ = + −

+ = −

21n

Dωωωω= −

( )( )

( )

( )

111

1

2

1

2

2 2

1

22

1

n nnn n n

n n n

n n

d n

n

n

exp D tZexp D t t

Z exp D t

However , the factor t t is the period

log arithmic dec

T ,

TD

Defining as the ,

Z Dln D

Z D

rement

ωωωωωωωω

ωωωω

π ππ ππ ππ πωωωω ωωωω

δδδδ

ππππδ πδ πδ πδ π

+++

+

+

− = = − − −

−

= =−

= = ≈

−

Example 4. Determine (1) whether a foundation-soil system is over-damped,

critically damped or under-damped, (2) the logarithmic decrement δ, and (3) the ratio

of two successive amplitudes, if the weight of the foundation is 60 kN, the soil spring

constant is 11 MN per meter and the damping coefficient is 200 kN-s/m.

( )( )2 2

1

60 2002 2 11 000 520 039 1

981 520

2

2 0392

1 1 03

3

3

2

9

6

c

c

c

( )Thecritical dampingcoefficient c is,

c kN s / mc km , kN s / m D .

. c kN s / m

( )The logarithmicdecrement is,

.D

D .

( )The ratioof twos

underdamped

uccessiv

.

δδδδ

ππππππππδδδδ

− = = = − ∴ = = = < ∴ −

= = =− −

263

1

2 2

4

1 1 11 000 981675

2 2 60

1 1 039 6

139

62375

.n

n

n

d n

eamplitudes is,

Ze e

Z

( )Thedamped natural frequency is,

k , ( . )f . cps

m

f D f (

.

. cp. ( ) s) .

δδδδ


+

= = =

= = =

= − = − =

(4) A steady-state forced vibration system with viscous damping.

The differential equation of motion with a sinusoidal varying force Q = Q0 sinωt, is

( )

( )( )

( )

( ) ( )

0

1 2

2 221 1

0

22 2 2 2 2

1

2

1 4

n

n

n n

n

mz cz kz Q sin t

A particular solution for the steady state motion could be,

z A sin t A sin t or

z Z cos t where

/k mtan tan and

c D /

Q / kZ

/ D /

where k / m is the

ωωωω



ω ωω ωω ωω ωωωωωαααα



ωωωω

− −

+ + =

−

= +

= +

− − = =

= − +

=

ɺɺ ɺ

undamped natural frequency

( ) ( )

0

22 2 2 2 20

1

1 4n n

The amplitude can be plotted normalized as Z /(Q / k ),

Z

(Q / k )/ D /ω ω ω ωω ω ω ωω ω ω ωω ω ω ω

= − +

Note that the maximum values do

not occur at ω = ωn as occurs in

the case of a forced vibration of

the foundation-soil system.

The maximum values occur at fm,

called the maximum amplitude.

( ) ( )

( )( )

22 2 2 2 20

0

22 2

2

2 2

1

1 4

0

1 2 0 1 2

11 2 1 2

2

n n

n

n

n n n

m n

res

resamplitude of vibration at resonanc

From

Z

(Q / k )/ D /

the max imum value occurs at ,

Z / Q / k

/

D or D

kf f D D

m

Therefore, the is

Z

e Z ,

Q



ω ω ωω ω ωω ω ωω ω ωω ωω ωω ωω ω


ππππ

= − +

∂ =∂

∴ − − = = −

∴ = − = −

=

( ) ( )0 0

2 22 2 2

1 1

2 11 1 2 4 1 2

Q

k k D DD D D

=− − − + −

The maximum dynamic force that is transmitted to the soil sub-grade can be found

through the spring force and the damping force caused by the relative motion between

the foundation mass and the dashpot,

( )( )

( ) ( )

dynamic

dynamic

F cz kz

U sin g the exp ression for the displacement z ,

z Z cos t and therefore the velocity z is ,

z Z sin t and replacing z and z int o the differential equation,

F c Z sin t kZ cos t

Let kZ Acos and c Z As


ω ω αω ω αω ω αω ω α

ω ω α ω αω ω α ω αω ω α ω αω ω α ω α

φ ωφ ωφ ωφ ω

= +

= +

= − +

= − + + +

= =

ɺ

ɺ

ɺ ɺ

( ) ( ) ( )2 2 22

dynamic

in

F Acos( t )

where A Acos A sin Z k c

Notice that A is the magnitude of the max imum dynamic force.

φφφφω φ αω φ αω φ αω φ α

φ φ ωφ φ ωφ φ ωφ φ ω

∴ = + +

= + = +

Example 5.

Determine (1) the undamped natural frequency fn of the foundation, (2) the amplitude

of the motion, and (3) the maximum dynamic force transmitted to the sub-grade, for

an electrical generator driven by a diesel engine placed on a isolated footing that

weighs 140 kN. The spring constant and the damping ratio of the soil is 12x104 kN/m

and 0.2 respectively. The vibrating force Qo is 46 kN at ω = 157 rad/s.

( )

( )

( ) ( )( )

( ) ( ) ( )

( ) ( )

4

2

4

0

2 2 22 2 2 2 2 2 2 2 2

4

1 1 12 101

2 2 140 981

2 2 2 146 917

46 12 10

1 4 1 157 917 4 02 157 917

1403 2 2 02 12 10

146

01

523

7

81

8

9

n

n n

n n

k x kN / m) f

m kN / . m/ s

) f ( . ) . radians / s

/ xQ / kZ

/ D / / . . / .

. cps

. mm

)c D km . x kN s / m.


ω π πω π πω π πω π π


= = =

= = =

= = = − + − +

= = = −

( ) ( ) ( ) ( )22 22 40000187 12 2710 523 157 2A Z k c . x . kNxωωωω= + = + =

References.

1. Das, B., “Principles of Soil Dynamics”, PWS-Kent Publishing Co., Boston, 1993;

2. Richart F.E., Hall J.R., Woods R.D., “Vibrations of Soils and Foundations”,

Prentice-Hall Inc., New Jersey, 1970;

3. Humar J.L., “Dynamics of Structures”, Prentice-Hall, New Jersey, 1990;

4. Prakash S., “Soil Dynamics”, McGraw-Hill, New York, 1981;