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Page3
23
Chapt
er9Machine-in
ducedvibrations
J.W.Smith
9.1INTRODUCTION
Many
indust
rialpro
cesses
giveri
setola
rgedyn
amicf
orces.
Theunb
alance
dmass
esofla
rgerota
tingma
chines
genera
teoscil
lating
forces
,while
forge
hamme
rsand
rockc
rusher
sapply
transien
tshock
sandim
pacts.
Rotati
ngorr
ecipro
cating
machi
nesgen
erally
operate
atfixe
dfreq
uencie
s.Itis
essent
ialtod
esignth
efound
ationso
rsuppo
rtstoa
voidre
sonanc
e.Perio
dicfor
cingfu
nction
swilla
lwaysi
nduced
ynamic
respon
sesand
theses
houldb
eevalu
atedto
ensure
thatth
eydo
notdam
agethe
fabric
ofthe
enclos
ingbui
ldings,
thema
chines
thems
elves
orothe
rsensi
tivepro
cesses
nearby
.Thev
ibration
duetot
ransien
tshock
sandim
pactss
hould
alsobe
evalua
tedfor
thesam
ereaso
n.Furth
ermore
,huma
nbein
gsarev
erysen
sitive
tovib
rations
andthe
amplit
udessh
ouldn
otcaus
edisco
mfortto
person
nelwo
rkingn
earby
orto
othero
ccupan
tsofbu
ildings
thatco
ntainm
achine
s.The
rearef
ourimp
ortants
tepsin
thesuc
cessfu
ldesig
nofm
achine
founda
tions.F
irst,th
edyn
amicf
orcesh
avetob
easses
sedacc
urately
.Thisi
stheta
skofth
emech
anical
design
erof
thema
chiner
yitsel
f,and
largely
consist
soffor
cesexe
rtedby
theine
rtiaoft
hemo
vingp
arts.
Theseg
enerall
yoccu
rathar
monic
sofma
chine
speed.
Otherf
orcesa
risefro
mcylin
derign
ition,r
ockcru
shing,
impact
fromh
amme
rs,and
certain
faultc
onditio
nssuch
asshor
tcirc
uitsine
lectric
almach
inery.
Second
ly,the
ground
conditi
onshav
etobe
assess
ed.Thi
sreq
uiresa
geotec
hnical
investi
gation
withth
eaimo
fdeterm
iningre
liablev
alueso
fthe
effectiv
eelasti
cresist
ancep
rovide
dbyth
efound
ationm
aterial
.Abal
anceh
astob
eachie
vedbet
ween
thecos
tofad
etailed
survey
andthe
value
andimp
ortance
ofthe
projec
t.Ther
ewill
inevita
blybe
consid
erable
uncerta
intyabo
utthe
numeric
alvalu
eseven
tually
adopte
dforth
edes
igncal
culatio
nsand
thedes
ignern
eedsto
becon
fidenta
boutth
euppe
randlo
werbo
unds.
Thirdly
,then
umeric
almode
lofthe
system
should
besui
tablef
orthe
purpos
e.Many
machi
nefou
ndation
sconsi
stofla
rgema
ssconc
reteblo
cks.Th
eseare
effectiv
elyrigi
d,and
reliabl
edes
igncal
culatio
nscan
bedon
eusin
gquite
simple
metho
dsofan
alysis
bytrea
tingthe
soil
founda
tionasa
nelast
icsupp
orting
mediu
m.Ho
wever
,inprin
ciplet
hegro
undisa
non-lin
earsol
idwith
infinite
bounda
ries.V
ibration
ofama
chine
block
results
inwave
Page3
24pro
pagatio
nwithi
ntheso
ilradia
tingaw
ayfro
mthe
centre
ofdistu
rbance
.Thiss
houldb
erefl
ected
inthe
analyti
calmo
delling
forma
jorpro
jects.
Advan
cedana
lysisu
singfi
niteele
menta
ndlay
eredh
alfspa
cemeth
odsma
ybere
quired
.Final
ly,the
predic
tedvib
ration
should
becom
pared
withc
riteria
chosen
toensu
rethat
person
nelare
notdis
comfite
dand
that
equipm
entper
forma
nceisn
otimp
aired.S
omed
esignru
lesexi
stbuto
nlyfor
limited
typeso
find
ustrial
machi
nery.I
nforma
tionofte
nhast
obeo
btaine
dfrom
experie
ncewit
hinthe
relevan
tindus
tryorf
romres
earch.
9.2DYNAMICLOADINGBYMACHINERY
Thedyn
amicl
oading
fromi
ndustri
almach
ineryd
erivesp
rincipa
llyfro
mthe
inertia
effects
ofmo
vingp
arts.Ev
eryma
chine
behave
sdiffer
entlya
nditis
usually
theres
ponsib
ilityoft
hema
nufact
urerto
calcul
atethe
forces
thatw
illbeim
posed
onthe
suppor
tingstru
cture.
Therota
tionspe
edsorf
requen
ciesat
which
machi
nesope
rateare
alsoimp
ortanta
ndsho
uldbe
specifi
ed.9.2.1Reciprocatingengines
Large
multi-
cylind
erdiese
lengin
esareo
ftenuse
dtopro
videth
eprim
arymo
tivepow
erfor
electri
calgen
erating
planti
nremo
teregi
onslac
kingin
digeno
usfoss
ilfuels
orwher
ethere
isno
access
toothe
rforms
ofener
gypro
duction
.There
ciproc
ationto
gether
withth
ecylind
erign
itionse
quence
giveri
setop
eriodic
forces
.At
ypical
arrange
mento
fdiese
lengin
eanda
lternat
ormoun
tedon
afound
ationb
lockis
shown
inFigu
re9.1.
Acran
k—pis
tonlink
ageiss
hownin
Figure
9.2wh
erethe
masse
softhe
crank,
connec
tingrod
andpis
tonarem
1,m2,a
ndm
3respe
ctively
.Engin
espeed
saren
ormally
quoted
inrevo
lutions
permin
ute(r.p
.m.)in
which
caseth
ecrank
shaftr
otation
frequen
cy�is
given
by:
(9.1)
whereN
isthe
engine
speed.
Itisevi
dentth
atther
ewillb
eoscil
latoryi
nertia
forces
duetot
hemo
vingm
assesm
1,m2an
dm3.The
firsttw
owillh
avever
ticalan
dhoriz
ontalc
ompon
ents
whilem
3willh
aveav
ertical
compon
entonl
y.The
magni
tudeo
fthese
forces
mayb
eeva
luated
ifthev
ariousm
assesa
ndlen
gthsar
eknow
n.Fore
xample
,itcan
besho
wntha
tthe
inertia
forced
uetot
hepis
tonwil
lbe:
(9.2)
Thisis
notasi
mpleh
armoni
cexcita
tionbec
auseo
fthese
condte
rm.The
refore,
inaddi
tionto
theprim
aryeng
inefor
cesapp
liedate
ngine
speed,
therew
illbeh
igherh
armoni
csappl
iedat
integra
lmulti
plesof
theeng
inespe
ed.The
mecha
nicald
esigner
ofthe
engine
should
evalua
tethe
seforc
es.Inm
ulti-cy
lindere
ngines
itispos
siblet
obala
ncemo
stofth
einerti
aforce
s,depe
nding
onthe
number
andarra
ngeme
ntofth
ecylind
ers.H
oweve
r,there
will
Page3
25
Figure9.1
Diesel
engine
andalte
rnator.
Figure9.2
Crank-
piston
linkage
.alw
aysbe
somere
sidual
unbala
ncedfo
rcesdu
etotole
rances
onwe
ightsa
ndgeo
metry.
Furthe
rmore,
ignitio
nofa
cylind
erwillc
reatea
dynam
icmom
entabo
utthe
centre
ofgrav
ityoft
heeng
ine,as
mayb
eseen
in Figu
re9.1.
Inafou
r-strok
eengin
ethisw
illresu
ltina
pitchin
gmom
entwh
osefreq
uency
willbe
:
(9.3)
Page3
26
Figure9.3
Rotati
ngunb
alanced
mass.
wherenist
henum
berofc
ylinder
s.The
firing
orderi
sselec
tedtom
inimize
theeffe
ctofth
ismo
ment.
9.2.2Highspeedrotatingmachines
Turbin
es,cen
trifuga
lpump
s,fans
andele
ctrical
genera
torsare
examp
lesofh
ighspe
edrota
tingma
chiner
y.Even
though
consid
erable
careis
taken
tobala
ncethe
rotating
parts,
residu
alimb
alance
swilla
lwayse
xist.A
ninerti
aforce
isgene
ratedb
ytheec
centric
ityeofa
nunb
alance
dmassm
aboutt
hecen
treofr
otation
asindi
cated
inFigu
re9.3.
Strictl
yspeak
ing,
thisfor
ceacts
radiall
yand
rotates
withth
eshaft.
Butit
willha
veosc
illatory
vertica
land
horizo
ntalco
mpone
ntstha
tmay
excite
thecor
respon
dingm
odesof
vibrati
on.Thi
shappe
nsin
everyd
ayexp
erience
witha
domesti
cspind
ryer.I
tshoul
dalso
benot
edtha
tthesp
eedsof
turbine
sarem
anytim
esgrea
tertha
nthose
ofreci
procat
ingeng
inesof
similar
powera
ndthe
reforet
heout
ofbala
ncefor
cewillb
esigni
ficantly
amplif
iedbec
auseo
fthefr
equenc
ybei
ngsqu
ared(m
�2 e). 9.2.3Transienttorquesinelectricalmachines
Therea
retwo
importa
ntcase
softr
ansien
tdynam
icload
ingtha
toccur
withd
rivene
lectric
algen
erators
.Thefi
rstisk
nowna
sshortcircuittorque.Co
nsider
anele
ctrical
alterna
torbei
ngdriv
enby
aneng
ineasi
nFigu
re9.1.
Ifafau
ltoccu
rred,w
hichh
adthe
effecto
fcreat
inga
shortc
ircuitin
theout
putoft
healte
rnator,
avery
largec
urrent
would
bedem
anded
(fora
fraction
ofasec
ond,pe
rhaps)
.Thisw
ouldb
eexper
ienced
asasud
denlya
pplied
loado
rbrake
onthe
system
.Asa
result
theeng
inewo
uldapp
lyatorq
ueabo
utthe
axisof
thedriv
eshaft.
Thesec
ondcas
eiskno
wnasfaultysynchronizingtorque.
Thisoc
cursw
henan
engine
genera
torsys
temstar
tsupa
ndfee
dspow
erinto
thenat
ionalg
rid.Ift
heout
putoft
hegen
erator
isnots
ynchro
nized
withth
ea.c.w
avefor
mofth
enatio
nalgrid
abrak
ingeffe
ct,or
torque,
isex-
Page3
27per
ienced
bythe
genera
torunt
ilsuch
timeas
itisinp
hase.T
hiswil
lhave
asimil
areffec
ttosho
rtcircu
ittorqu
e,inth
atasud
dentorq
ueisa
pplied
tothe
alterna
torwh
ichhas
tobere
acted
bythe
machi
nesup
ports.
Thedyn
amice
ffectof
asudde
nlyapp
liedloa
distwi
cethe
static
effect.
Thesam
eapplie
stoasu
ddenly
applied
torque.
Thema
chine
manuf
acture
rshoul
dpro
videth
emagn
itudean
ddirec
tionoft
ransien
ttorqu
esthat
mayo
ccurdu
ringope
ration
ofama
chine.
9.2.4Gyratoryrockcrushers
Inthe
quarry
ingind
ustryt
hereis
aneed
tocrus
hexca
vated
rockin
toston
esofva
ryingsi
zefor
differe
ntend
uses.T
heraw
excava
tedroc
kisofte
ninver
ylarge
pieces
(perha
psine
xcess
ofone
metre
across
andwe
ighing
acoupl
eofto
ns),w
hereas
theend
produc
tmay
bereq
uired
forhig
hway
chippi
ngsof2
0mms
izeorl
ess.G
yratory
crushe
rsareu
sually
usedin
moder
nqua
rriesto
proces
sthero
ck.The
reares
everal
design
sinexi
stenceb
utone
ofthe
mostc
ommo
nisthe
basesu
pported
conecr
usher.
Theprin
cipleo
fopera
tionisi
llustrat
edinF
igure9
.4.The
fixedp
artsoft
hema
chine
consist
ofadru
m,hop
perand
bowl(in
verted
).Uncr
ushed
rockis
loaded
intothe
hopper
andont
othefe
edpla
teofth
ecrush
inghea
d,whic
hensu
restha
tthero
ckfall
sintot
hespa
cebetw
eenthe
bowlan
dthem
antleo
fthecr
ushing
head.T
hecru
shing
headis
driven
inagyr
atorym
otionb
y
Figure9.4Crosssectionofagyratoryconecrusher.
Page3
28
Figure9.5
Schem
aticdia
gramo
ftrans
ientfo
rcedue
toston
ecrush
ing:(a
)forcin
gfunct
ion;(b
)dire
ctiono
fdynam
icforc
e.the
mains
haftw
hichis
seated
inthe
eccent
ricsha
ftatan
angled
esigned
toachi
evethe
desired
sizeofs
toneaf
tercru
shing.
Notet
hatthe
gapbet
ween
thecru
shing
mantle
andthe
bowlva
riesaro
undthe
perime
teroft
hebow
l.The
eccent
ricsha
ftisdriv
enthro
ughag
earme
chanis
mbya
nelec
trically
powere
dcoun
tersha
ft.Crus
hingis
achiev
edwh
enlum
psof
rocka
retrap
pedbet
ween
thebow
landm
antlea
ndthe
ncrus
hedwh
enthe
gapdim
inishes
during
arevol
utiono
ftheec
centric
shaft.
Itwillb
eappre
ciated
thata
number
ofdyna
micfor
ceswil
loccur
duetor
otation
ofthe
movin
gparts
(Szcze
paniketal.199
0).The
largest
willbe
caused
bythe
gyrato
rymotio
nof
thema
ntleand
mains
haftdu
etothe
irecce
ntricity
.There
willal
sobeth
eunba
lanced
counte
rsha
ftforc
e,whic
hisusu
allyatt
wiceth
efrequ
encyo
fthem
ainsha
ft,and
vibrati
onoft
hespr
ingsho
ldingd
ownth
ehopp
erand
bowl.
Howe
ver,th
emost
importa
ntdyna
micfor
cesare
transien
tshock
sarisin
gfrom
thecru
shing
action
itself.S
mith(1
993)sh
owed
thatth
edyna
micloa
dwhen
astone
iscrus
hedcon
sistsof
animp
ulseas
thepow
erofth
edrive
motor
compre
ssesth
erock
betwe
enthe
bowlan
dmant
le,foll
owed
byasu
ddenre
leasea
sthest
onefrac
tures.T
hepea
kvalu
eofth
eforce
couldb
easmu
chast
woort
hreeto
ns,acc
ording
tothe
sizeoft
hecru
sher,a
ndiss
hownsc
hematic
allyin
Figure
9.5a.T
hereis
aneed
forexp
eriment
aldata
onthe
magni
tudeo
fthety
picalc
rushin
gfor
ce,but
initsa
bsence
anesti
matec
anbe
made
fromk
nowled
geoft
hemo
tortorq
ueand
theme
chanic
aladva
ntage
availab
lebetw
eenmo
torand
mantle
.Thed
uration
ofthe
impuls
eisbas
edon
theass
umptio
nthatt
ypical
lyfour
stones
arecru
shedp
errevo
lution.
Thefor
ceis
applied
inadire
ctionn
ormalt
othefa
ceofth
emant
le,wh
ichisu
sually
about4
5°,but
could
occuri
nany
directio
ninthe
horizo
ntalpl
anebec
auseo
fthero
tation
ofthe
pinch
pointo
fthe
crushe
r(see
Figure
9.5b).
Thesef
orcesa
retran
sientan
drando
m,asr
ocksar
efedin
tothe
machi
ne,but
mayo
ccurse
veralt
imespe
rrevol
utiono
ftheec
centric
shaft.
Page3
29Ith
asbeen
found
(Szcze
paniketal.199
0)that
vibrati
onoft
hecru
shing
machi
neon
itsfou
ndation
leadst
odyna
micstre
ssesin
thearm
softh
esuppo
rtingfr
ameth
athold
sthe
eccent
ricsha
ftand
headin
positio
n.Ultim
ately,
fatigue
cracks
mayo
ccurin
thearm
s,caus
ingbre
akdow
nofth
emach
ine.It
isimp
ortantt
hatthe
suppor
tingfou
ndation
issuff
iciently
stiffto
minimi
zethis
vibrati
on.9.2.5Hammers
Therea
remany
indust
rialpro
cesses
,typic
allyimp
actfor
ging,w
hichre
quires
ingleo
rrep
eated
blows
witha
hamme
r.Kine
ticene
rgyisg
ivento
theham
merhe
adeith
erbyso
meext
ernals
ourceo
fpowe
rsuch
asstea
m,com
presse
dair,o
rmore
usually
bygra
vity.V
elocity
isimp
artedto
theanv
iland
workp
ieceb
ytrans
ferofm
oment
um.A
schem
aticarra
ngeme
ntfor
adrop
hamme
rissho
wninF
igure9
.6.The
masso
ftheh
amme
rhead,
ortup
asitis
called
,isd
enoted
bym
0thean
vilbym
2andth
efound
ationb
ym1.Som
ekind
ofelast
iclaye
r,often
hardw
ood,is
inter-p
osedb
etween
theanv
iland
thefou
ndation
block.
Theblo
ckise
ither
suppor
tedela
stically
bythe
founda
tionma
terialo
rbyspe
cially
design
edspr
ingsto
minimi
zethe
transm
ission
ofvibra
tionton
earby
buildin
gs.Thu
sthesy
stemhas
twodeg
reeso
ffreed
om.
Figure
9.6Sch
ematic
arrange
mentf
oradro
pham
mer.
Page3
30The
veloci
tyofth
etupb
eforei
mpact
isgive
nby:
(9.4)
Inthe
caseo
fpneum
aticors
teamp
owere
ddrop
hamme
rsBark
an(19
62)fou
ndtha
t,inpra
ctice,t
hissho
uldbe
reduce
dbya
nemp
iricalf
actoro
f0.65
toallow
forfric
tionand
theres
istance
ofexha
ustair
orstea
m.Fol
lowing
theme
thodo
fBark
an(19
62),co
nserva
tionofm
oment
umcan
beexp
ressed
by: (9.5)
where
(–v')ist
hereb
oundv
elocity
ofthe
hamme
rhead
andv 0
isthe
veloci
tyimp
artedto
theanv
il.The
relative
veloci
tyafter
impact
depend
sonth
eelasti
cchara
cterist
icsoft
hecol
liding
bodies
andiso
btaine
dfrom
theexp
ressio
n:
(9.6)
whereCrist
hecoe
fficien
tofres
titution
.Thisc
onstan
tvaries
betwe
en0(f
ullypla
stic)to
1(pe
rfectly
elastic
).Thus
thevel
ocityo
fthean
vilafte
rimpac
tmay
beobt
ained
frome
qns(9.5
)and
(9.6)an
disgiv
enby:
(9.7)
whereµ0=m
2/m0.T
hisvel
ocitym
aybe
useda
saninp
uttoth
eequat
ionsof
motion
ofatwo
degree
offree
domsys
tem(se
e Sect
ion2.3
).Itis
possib
lethat
theham
merm
ightst
rikethe
anvile
ccentri
cally,
thusim
parting
arotati
onalco
mpone
nt.Thi
scondi
tionsho
uldalso
becon
sidere
d.No
vakand
ElHifna
wy(19
83)hav
everif
iedtha
ttheab
ovepro
cedure
issatis
factory
provid
edtha
tthed
uration
ofthe
impact
ismuch
shorter
thanth
enatu
ralper
iodoft
hefou
ndation
.Thism
aynot
besoi
fthefo
undatio
nsupp
ortisv
erystif
f(e.g.
piles).
Itwoul
dthen
benec
essary
totake
accoun
tofthe
force-
timefu
nction
ofthe
impuls
e.9.2.6Vibratingscreens
Thesea
reused
extens
ivelyi
nthem
ininga
ndqua
rrying
indust
riesfor
washi
ng,sep
arating
andgra
dingp
rocess
es.Ro
ckisp
assed
overa
horizo
ntalsc
reenth
atissu
pported
atitse
nds,as
shown
inFigu
re9.7.
Thescr
eenisv
ibrated
vertica
llyand
horizo
ntally
bymo
tordriv
enecc
entric
cranks
orcam
s.Ston
esthat
aresm
alleno
ughpas
sthrou
ghthe
screen
andare
collec
ted,w
hilelarg
erston
espass
overth
eslopi
ngupp
ersurf
aceand
continu
etothe
nextsi
zeofs
creen.
Sinuso
idalin
ertiafo
rcesar
ethusa
pplied
tothe
suppor
tingfram
estruc
tureand
tothe
founda
tions.I
ntheq
uarryin
gindus
tryroc
kisgra
dedinto
differe
ntsize
sbyp
assing
overa
sequen
ceofvi
brating
screen
s.Thes
eareu
sually
housed
inasin
glelarg
ebuild
ing,w
hichis
therefo
resubj
ected
tocont
inuous
dynam
icload
ingwh
ilein
Page3
31
Figure9.7
Schem
aticdia
gramo
favib
rating
screen
.ope
ration.
Thefram
eofth
ebuild
ingsho
uldbe
stiffen
oughto
preven
tunacc
eptabl
evibra
tionam
plitude
.9.2.7Rollingmills
Inproc
essess
uchasr
olling
ofstee
lsectio
ns,the
shapin
gisach
ieved
bypas
singa
hotblo
omorb
illetth
rough
thesha
pingro
llersw
hicha
redrive
nbya
d.c.m
otor.A
sabill
etente
rsthe
rollers
,thein
itialre
sistanc
eactsl
ikeasu
ddenly
applied
torque
tothe
shafto
fthed
riving
motor.
Theres
ulting
dynam
iccoup
leappl
iedtot
hefou
ndation
block
isanal
ogoust
othe
‘short
circuit’
torque
inelec
tricalg
enerato
rs.The
torque
canbe
estima
tedby
knowin
gthe
speed
andpow
erofth
edrivi
ngmo
tor.
9.3DESIGNOFSTRUCTURESTOMINIMIZEMACHINE-INDUCED
VIBRATION
9.3.1Dynamicresponseofsupportingstructureandfoundation
Mostt
ypesof
heavy
indust
rialma
chines
arepro
vided
witho
neoft
hefive
typeso
fsuppo
rting
founda
tionsho
wninF
igure9
.8.The
searem
asscon
creteb
locks,
box-typ
efound
ations,
wall
founda
tions,r
einfor
cedcon
cretef
ramesa
ndtab
letop
founda
tions.T
hecho
iceoff
oundat
ionisi
nfluenc
edby
thetyp
eofm
achine
,them
agnitud
eofth
edyna
micfor
cesand
theacc
essreq
uired
around
thema
chine.
Powerg
enerati
ngsets
,comp
rising
larged
ieselen
ginesa
ndalte
rnators
,areo
ftenmo
unted
onma
ssconc
retefou
ndation
blocks
which
distrib
utethe
loado
verthe
basear
eatoth
esup
porting
soilor
rock.P
iledfou
ndation
swoul
dbep
referred
where
thegro
undcon
ditions
arepoo
r.The
block
mayb
eanyth
ingfro
mtwic
eto
Page3
32
Figure9.8
Founda
tionsys
temsfo
rindus
trialm
achina
ry.
Page3
33
Figure
9.9Sim
plified
model
ofrigid
founda
tionblo
ckres
tingon
elastic
suppor
ts.five
timest
hema
ssofth
emach
inery.
Theper
forma
nceofb
lockfo
undatio
nswasd
iscusse
dfull
ybyS
mith(1
989).T
heide
alized
structu
ralsys
temcon
sistsof
alarge
rigidm
assres
tingon
asemi-
infinite
elastic
mediu
mwith
dynam
icforc
esand
mome
ntsapp
liedtot
hema
ss.Thi
sisan
examp
leofgr
oundst
ructure
interac
tionfor
which
thema
thema
ticalan
alysis
isrela
tively
advanc
edand
inconv
enient
forgen
eralde
signp
ractice
(Arnol
detal.195
5).Va
riousat
tempts
haveb
eenma
detod
erivesi
mplifi
edfor
mulae
includ
ingHsi
eh(19
62)wh
oprop
oseda
nequ
ivalen
tmass
restrai
nedby
elastic
eleme
nts,th
esystem
having
anapp
ropriat
eamoun
tofdam
pingd
erived
froms
emi-in
finitee
lastict
heory.
Thethe
oryofg
round
structu
reinte
raction
isdiscu
ssedb
rieflyi
n Sect
ion9.3
.2.The
metho
dofan
alysis
inmost
widesp
readu
seatth
epres
enttim
eorigi
natesf
romthe
work
ofBark
an(19
62).H
issimp
lified
system
isshow
ninFig
ure9.9
where
thesoi
lstiffn
essis
repres
ented
byver
ticalan
dhoriz
ontals
prings.
Provid
edtha
tthece
ntreofg
ravity
ofthe
mass
coinci
deswit
hthece
ntreofs
tiffnes
softh
esoil,v
ertical
vibrati
oncan
betrea
tedasa
single
degree
offree
domma
ss—spr
ingsys
tem.In
thecas
eofho
rizonta
lvibra
tion,sl
idingm
otion
willbe
accom
panied
byroc
king,r
esultin
ginatw
o-degr
eeoffr
eedom
system
.Othe
rmotio
nstha
tshoul
dbeco
nsider
edinc
ludep
itching
andhor
izonta
lmotio
ninap
lanep
erpend
icular
tothe
onesho
wn,an
dalso
yawing
motion
abouta
vertica
laxis.
Barkan
(1962)
believ
edtha
tthe
partici
pating
soilm
assdoe
snotm
akeasu
fficien
tdiffer
enceto
thecal
culatio
nofth
enatu
ralfreq
uency
andcou
ldbesa
felyneg
lected
.Furthe
rmore,
provid
edtha
tthem
achine
frequen
cyis
suffici
entlyd
ifferen
tfrom
thenat
uralfr
equenc
yofth
esystem
,amplit
udesof
forced
vibrati
onma
ybeca
lculate
dwith
reason
ableac
curacy
byign
oringd
ampin
g.Full
details
ofthe
analyti
calpro
cedure
arepro
vided
inthe
Code
ofPrac
ticefor
Founda
tionsfo
rMach
inery(
BSI,
1974).
Inform
ationa
ndme
thodsf
orcalc
ulating
thestif
fnesso
fthefo
undatio
nmate
rialwh
etheri
tbesoi
l,rock
orpiles
isprov
idedb
ySkip
p(1966
).Onth
ebasis
of
Page3
34exp
erience
ofthe
perfor
mance
ofaver
ylarge
machi
ne,Sm
ith(19
89)has
comme
ntedth
atthe
BSI(1
974)m
ethod
ofcalc
ulation
(which
isbase
donB
arkan,
1962)i
sgener
allysati
sfacto
ryfor
machi
neswit
hvery
stiffba
ses.H
oweve
r,indus
trialpr
actice
ismovi
ngtow
ardsli
ghtera
ndmo
reflex
iblebas
esforw
hichth
eflexi
ngoft
hebas
eshoul
dbeco
nsider
ed.Ina
neffo
rttored
ucethe
cost,a
ndalso
toprov
ideacc
esstoe
quipm
ent,bo
x-type
founda
tionsm
aybe
emplo
yed.Th
eBSI(
1974)m
ethod
mayb
eadopt
edfor
analys
isbut
consid
eration
should
begiv
entot
hepos
siblef
lexing
ofthe
structu
re,esp
ecially
when
design
ingfor
largem
achine
s.Wall
founda
tionsar
eparti
cularly
useful
forma
chines
which
require
equipm
entsuc
hasco
nveyor
sbenea
ththem
.Them
achine
mayb
efixed
directly
tothe
topsof
thewa
llsors
upporte
donst
eelbea
rerstha
tspan
betwe
enthe
walls.
Thiski
ndof
founda
tionisp
otentia
llyver
yflexi
bleand
particu
larcar
eshoul
dbeta
kenint
hedes
ign.Fo
rexa
mple,
itmay
notbe
wiseto
assum
ethatt
hefou
ndation
slabisr
igidsin
cetran
sverse
vibrati
onoft
hewa
llsma
ybeac
compan
iedby
flexing
ofthe
slab.
Turbin
esand
otherh
ighspe
edrota
tingma
chines
areofte
nmoun
tedon
reinfor
cedcon
crete
framesa
ndslab
s(Srina
vasulu
andVa
idyana
than,1
976).I
nthisc
asethe
suppor
tingstru
cture
cannot
betrea
tedasa
rigidb
locka
ndana
lysiso
fitsdyn
amicc
haract
eristics
mayre
quire
numeric
alcom
putatio
n.The
‘tablet
op’arra
ngeme
ntissi
milart
othefr
amety
peofs
upporti
ngstru
cture,
asshow
ninFig
ure9.1
0inwh
ichatu
rbinean
dalter
natora
remoun
tedon
aslab,
which
inturn
issupp
ortedb
ycolu
mns.I
norde
rtocon
finethe
vibrati
oncau
sedby
thema
chiner
ytothe
slab,th
eslab
itselfis
mount
edon
isolati
ngspr
ingsan
ddam
persat
thetop
soft
hecol
umns.
Thestif
fnessa
nddam
pingo
fthese
spring
—dam
peruni
tsmust
becho
sento
minimi
zetran
smissi
onoft
hevib
ration
tothe
colum
ns.The
slab/m
achine
system
hastob
eche
ckedfo
rallpo
ssiblem
odesof
vibrati
oninc
luding
flexura
lmotio
nofth
eslab.
9.3.2Ground-structureinteraction
Theabo
veme
thodsa
rehigh
lysimp
lified
theore
tically,
andtak
enoa
ccount
ofthe
infinite
orsem
i-infini
teexte
ntofth
egrou
ndtha
tissup
porting
amach
inefou
ndation
.These
simple
metho
dshave
beenu
sedext
ensive
lyform
osttyp
esoffo
undatio
nsforg
eneral
indust
rialma
chiner
y,incl
uding
quitel
argeg
enerato
rsets.
Howe
ver,in
thecas
eofve
rylarg
e,imp
ortant,
expens
iveors
afetyc
ritical
projec
tsitm
aybe
necess
arytoc
arryout
morer
igorou
sanaly
sisusi
ngadv
anced
theory
.Forex
ample
,nucle
arinst
allation
sinseis
miczon
esshou
ldbean
alysed
taking
fullacc
ountof
ground
—struc
tureinte
raction
.Thep
rinciple
sared
iscusse
dbrief
lybel
ow.
Anide
alized
model
ofafou
ndation
block
suppor
tedon
soilor
rockis
shown
inFigu
re9.11
.The
massc
onsists
ofthe
concre
tebloc
ktoget
herwit
hthem
achine
ry.The
ground
isasem
i-infi
nitereg
ionofl
ayered
elastic
media
.Thela
yersre
presen
tdiffer
entsoi
lorroc
kstrata
andwo
uldhav
ediffe
rentde
nsitya
ndela
sticcha
racteri
stics.D
ynamic
forces
areapp
liedtot
heblo
ckasa
result
ofthe
operati
onoft
he
Page3
35
Figure9.10Tab
letop
suppor
tingstru
cturef
orturb
ineand
alterna
torsys
tem.
machi
ne.Ge
nerally
thesew
ouldb
eperio
dicfor
cesaris
ingfro
mthe
outofb
alance
masse
s.Alt
hough
thegro
undpro
videse
lasticr
esistan
ce,wh
ichma
ybeev
aluate
dbyth
etheor
yof
elastic
ity,the
lacko
fbound
ariesm
eansth
atvibra
tionina
steady
statem
odesha
pedoe
snot
occur.
Instea
d,vibra
tionoft
heblo
ckres
ultsind
isplac
ement
ofthe
ground
inthe
formo
fwa
vestha
tpropa
gateaw
ayfro
mthe
source
ofdistu
rbance
.Them
ostimp
ortantw
aveso
ccur
atthe
freesur
facean
darea
nalogo
ustori
ppleso
napon
dradia
tingint
hefor
mofco
ncentri
cring
sincre
asingin
diame
terwit
htime
.Theen
ergyo
fmotio
natth
ebloc
kiscon
finedto
asm
allvol
umeo
fsoilw
hereas
itisspr
eadove
ramu
chlarg
ervolu
meafte
rsome
timew
henthe
rippleh
aswide
ned.Th
us,ene
rgyofv
ibration
isradi
ateda
wayfr
omthe
source
of
Page3
36
Figure9.11Ide
alized
model
ofafou
ndation
block
suppor
tedon
soilor
rock.
Figure9.12Dy
namicp
ointlo
adon
surfac
eofan
elastic
halfsp
ace.
disturb
ance.E
veniff
riction
aldam
pingin
thesoi
lisign
ored,t
hislos
sofene
rgydue
tothe
propag
atingw
avesis
always
presen
tandis
calledradiationdamping.Th
issurf
acewa
veeffe
ctisof
greati
mporta
ncewh
eneva
luating
thevib
ration
ofstruc
turessu
pported
byunb
ounded
media
.Ap
relimin
arystep
inthe
analys
isisto
consid
eradyn
amicp
ointlo
adon
thesur
faceo
fanela
stichal
fspace
asshow
ninFig
ure9.1
2.The
mostb
asicloa
dingfu
nction
isastea
dystat
eper
iodicf
orceg
ivenb
y:
(9.8)
noting
thatei
wt =cos
��+i
sin��.
Page3
37The
displa
cement
respon
seofth
esurfac
ey(t)
isofa
similar
form
(9.9)
Evenif
therew
asnom
aterial
dampin
ginthe
soil,th
edisp
lacem
entam
plitude
couldn
evergo
toinfin
ityatr
esonan
ce,asi
ntheca
seofst
ructure
swith
bounda
ries(se
eeqn2
.42and
Figure
2.14).
Thisis
becaus
ethesu
rfacew
avestr
avellin
gawa
yfrom
thesou
rceofd
isturba
nceabs
orbene
rgyand
giveri
setoa
pparen
tdamp
ingorr
adiatio
ndam
ping.A
furthe
rfeatu
reof
thebeh
aviour
ofwave
motion
inelast
ichalf-
spaces
isthat
therei
sacha
racteri
sticfreq
uency
below
which
wave
motion
willno
toccur
.Ife
qns(9.8
)and(9
.9)are
substit
utedin
toane
quation
ofmotio
nsuch
aseqn
(2.27)g
ivenin
Chapt
er2,it
canbe
shown
that:
(9.10)
whereyist
hedis
placem
entand
�ist
heeffe
ctivecr
iticald
ampin
gratio
(seeeq
ns2.33
,2.34
and2.3
5).The
term[k(
1+2�0
)–�2 M
]isrefe
rredtoa
sthed
ynamic
stiffne
ssand
isafun
ction
offreq
uency.
Thisre
lations
hipalso
holdsf
ormulti
-degre
eoffr
eedom
system
sforw
hichth
eequ
ationo
fmotio
nisgiv
enby:
(9.11)
where{u}ist
hevec
torofd
isplac
ement
sofall
thedeg
reeso
ffreed
omand
{P}ist
heloa
dvec
tor.[S]ist
hedyn
amics
tiffnes
smatri
xand
isgive
nby:
(9.12)
Itisgen
erally
easier
toobta
insolu
tionsfo
rperiod
icload
ingusi
ngana
lysisi
nthefr
equenc
ydom
ainasa
bove.N
otetha
tthed
ynamic
stiffne
ssmatri
xisfreq
uency
depend
ent.A
nalysis
ofgen
eralfo
rcingfu
nction
scana
lsoby
synthe
sizedfr
omfreq
uency
domain
solutio
nsusin
gFou
riertran
sforms
.Wo
lf(198
5)hasp
rovide
dadet
ailedtr
eatme
ntofgr
ound-s
tructure
interac
tionint
hefreq
uency
domain
forear
thquak
eanaly
sisofl
argest
ructure
s.Hep
ropose
dasys
temusi
ngsub
structu
res,su
itablef
orfinit
eelem
entana
lysis.
Thisis
illustra
tedin F
igure9
.13wh
erethe
structu
re—soi
lsystem
isredu
cedtot
wosub
structu
res,on
ebein
gthem
ainstru
cturea
ndthe
otherb
eingth
esurro
unding
soilof
infinite
extent
.Thed
ynamic
stiffne
ssmatri
xofth
emain
structu
re,[S],has
anord
erequa
ltothe
number
ofdegr
eesoff
reedom
inthe
finitee
lement
model
.Itma
ybeev
aluate
dbyc
onvent
ionalf
initeel
ement
metho
ds.The
displa
cement
vector
mayb
edeco
mpose
dintos
ubvec
tors{us}and
{ub}.T
hesub
scriptbden
otesal
lthen
odesat
theinte
rfacew
hilesden
otesth
erema
iningn
odeso
fthest
ructure
,assho
wninF
igure9
.13.
Similar
ly,the
stiffne
ssmatri
xmay
bedec
ompos
edinto
thesub
matric
es[S ss],[Ssb]and
[Sbb].
Hence
,theeq
uation
ofmotio
nofth
estruc
turema
ybeex
presse
das:
(9.13)
Page3
38
Figure9.13Fin
iteele
mentm
odelof
structu
reemb
edded
inanu
nbound
edreg
ionofs
oil.
Figure9.14Ref
erence
subsys
temsfo
rthest
ructure
andexc
avated
ground
.wh
erePsare
theloa
dsappl
iedton
odesof
thestru
ctureo
therth
anatt
heinte
rfacew
hereit
isass
umed
therea
renoe
xterna
lloads
.Ino
rderto
obtain
thedyn
amics
tiffnes
smatri
xofth
esoils
tructure
system
,itisn
ecessa
ryto
addthe
dynam
icstiff
nessm
atrixo
ftheex
cavate
dgrou
nd.Thi
sissho
wninF
igure9
.14and
theequ
ationo
fmotio
nofth
esystem
willbe
come:
(9.14)
where
thesup
erscrip
tssand
gden
otethe
stiffne
sssub
matric
esbelo
nging
tothe
Page3
39stru
cturea
ndthe
ground
,respe
ctively
.Itsho
uldbe
noted
thatth
estiffn
essma
trixoft
heexc
avated
ground
inprinc
iplerep
resent
sthed
ynamic
stiffne
ssofge
neraliz
edspr
ingsan
ddas
hpots
joining
thenod
esbtoa
djacen
tfixed
virtual
nodes.
Theyta
keacc
ountof
theden
sityand
stiffne
ssofth
egrou
ndma
terialt
ogethe
rwith
radiati
ondam
pingd
uetot
hewa
vemo
tion.
Thisst
iffness
matrix
isnote
asytoo
btainb
ecause
itisase
mi-infi
nitereg
ionofi
rregula
rsha
pe.Ho
wever
,itcan
beobt
ained
bysub
tracting
thestif
fnessm
atrixo
ftheex
cavate
dsoil
fromt
hatoft
heinte
rfacen
odesin
thefree
field
asfollo
ws:
(9.15)
Hence
,theeq
uation
ofmotio
nbeco
mes:
(9.16)
mayb
einterp
reteda
sthed
ynamic
stiffne
ssmatri
xofth
estruc
ture,w
iththe
stiffne
ssmatri
xofth
eexcav
atedso
ilsubt
racted
.Thela
tterma
ybeev
aluate
dbyc
onvent
ional
finitee
lement
metho
ds.The
freefiel
ddyna
micstif
fnessm
atrix
mustb
eevalu
atedu
singth
etheor
yofse
mi-infi
nitelay
eredm
edia.T
hisreq
uiresad
vanced
mathe
matica
ltreatm
enttha
tisbey
ondthe
scope
ofthis
book.W
olf(19
85)has
derive
danum
berofu
sefuls
pecial
casesf
ortwo
-dim
ension
aland
axi-sy
mmetri
cfound
ationsu
singG
reen’s
functio
ns.Mo
reover
,them
ethod
hasbee
ndeve
loped
furthe
rforan
alysis
inthe
timed
omain
(Wolf,
1988).
Wolfa
ndPar
amess
o(19
92)sol
vedap
ractica
lexam
pleofa
hamme
rfound
ationw
ithupl
iftofth
eanvil.
Thisw
asanon
-linear
proble
minw
hichth
eyuse
dalum
pedpar
amete
rmode
lfora
rigidc
ylindric
alfou
ndation
embed
dedina
soilla
yer. 9.3.3Designcriteria
Theprim
aryreq
uireme
ntofth
emach
ine/fou
ndation
system
isthat
resona
nceisa
voided
atthe
operati
ngma
chine
frequen
cies.I
tissom
etimesr
ecomm
ended
thatfo
rimpor
tantin
stallati
onsany
natura
lfrequ
encyo
fthesy
stemsho
ulddiff
erfrom
signifi
cantop
erating
frequen
ciesb
yafac
torof2
.0(BS
I,1974
).The
factor
isredu
cedto1
.5fori
nstalla
tionsof
lesseri
mporta
nce.If
thisdeg
reeofs
eparati
onofn
atural
frequen
ciesan
ddrivi
ngfreq
uencie
scanb
eachie
ved,no
furthe
ranaly
sisisg
enerall
yrequi
red.In
practic
ethisu
sually
require
sacom
promis
ebeca
usethe
dynam
icsyst
emwil
lhave
numero
usnatu
ralfreq
uencie
sofvi
bration
while
thema
chiner
yma
ygene
ratehar
monic
sinadd
itionto
theprin
cipalo
peratin
gfrequ
ency.M
orerec
entspe
cificat
ionsal
lowthe
frequen
ciesto
beeve
nclos
er(DIN
4024,1
988).
Inthe
caseo
fanins
tallatio
nwher
ethefo
rcinga
ndnat
uralfr
equenc
iesare
close,
thema
ximum
amplit
udesof
thefou
ndation
/struct
uresys
temsho
uldbe
evalua
ted
Page3
40und
erthe
princip
alforc
ingfun
ctions.
Thesew
ouldin
clude:
period
icload
ingfro
munba
lanced
rotating
masse
s;tran
sientl
oadssu
chass
hortci
rcuitto
rque;a
ndsho
ckori
mpuls
eloads
from
indust
rialpro
cesses
.The
rearet
hreep
rincipa
lconce
rnswh
encon
siderin
gwhet
herthe
predic
tedvib
rations
areacc
eptabl
eorno
t.First
,thev
ibration
sinthe
vicinit
yofth
emach
inerys
houldn
otbela
rgeeno
ughtod
isturbp
ersonn
elwork
ingnea
rby(e.g
.main
tenanc
eorco
ntrolro
omstaf
f).Hu
man
beings
aresur
prising
lysens
itiveto
vibrati
on.Av
ibration
ofsma
llamp
litude
canbe
disturb
ingora
nnoyin
gespe
cially
ifitisc
ontinu
ous.It
canimp
aircon
centrat
ion,ca
usefati
gueand
other
physica
lsymp
toms,i
ncludi
nghea
daches
andsick
nessin
extrem
ecases
.Hum
anres
ponset
ovib
ration
wasdi
scusse
dinCh
apter7
.Secon
dly,ex
cessiv
evibra
tionoft
hesup
portso
fama
chine
mayre
sultin
overst
ressof
compon
entso
fthem
achine
itself.A
nexam
pleoft
hisis
thesup
porting
framew
orkfor
theecc
entric
shafto
fagyr
atoryc
onecru
sher(s
eeFig
ure9.4
).If
thisstee
lfram
ework
issubj
ected
tocont
inuous
vibrati
onind
ucedst
resses
itmay
besus
ceptibl
etofati
guedam
age(Sz
czepan
iketal.199
0).The
founda
tionsho
uldbe
stiffen
ough
toprev
entthe
machi
nevib
rating
excess
ivelyo
nitsm
ounting
s.Bear
ingsof
turbine
sand
engine
smay
beadv
ersely
affecte
dbye
xcessiv
evibra
tion.Th
irdly,t
ransm
ission
ofmach
ine-
induce
dvibra
tiontot
hestru
ctureo
ftheb
uilding
inwhic
hthem
achine
ryisho
usedm
aybe
undesir
able.T
hevib
ration
mayb
edistu
rbingto
person
nelwo
rkingin
thesam
eorad
jacent
buildin
gs.Ele
ctronic
control
equipm
entiso
ftenhou
sedinb
oxesor
panels
andfixe
dtothe
floors,
colum
nsorw
allsofe
nclosi
ngbui
ldings.
Possib
ledam
agetos
uchequ
ipment
bycon
tinuous
vibrati
onsho
uldbe
checke
d.Agoo
dgene
ralprin
ciplei
stokee
pthefo
undatio
nsoft
hema
chines
andthe
buildin
gsepar
ate.
Therei
sasca
rcityo
fcode
provis
ionsde
alingw
ithma
chine-
induce
dvibra
tions.P
lant
manuf
acture
rsoften
workt
otheir
ownst
andard
sandb
ydefa
ultset
thestan
dardsf
orthe
relevan
tindus
try.Sup
plierso
fsensi
tiveequ
ipment
maysp
ecifyl
imitst
otheac
ceptab
leenv
ironme
ntinw
hichth
eirequ
ipment
operate
ssatisf
actoril
y.Inf
ormatio
nonli
mitsto
thevib
ration
environ
mento
frotati
ngma
chines
mayb
efound
inMo
ore(19
85).Th
eaccep
tableo
peratin
gamp
litude
decrea
seswit
hfrequ
encya
ndali
mitis
oftensp
ecified
interm
sofvel
ocityo
fvibra
tionatt
hebea
ringsof
thema
chine.
Avibra
tionvel
ocityo
flesst
han2.0
mm/sec
would
beexp
ected
toprov
idesm
oothru
nning
conditi
onswh
ereaso
ver16
mm/sec
theope
ration
ofthe
machi
newo
uldpro
bablyb
every
rough.
TheBri
tishSta
ndardf
orrota
tingele
ctrical
machi
nes(BS
I,1987
)recom
mends
alimit
ofabou
t2.5
mm/sec
r.m.s.,
althoug
hthisi
sinten
dedfor
relative
lysma
llmach
ines.I
ntheq
uarryin
gind
ustryi
ntheU
Kalim
itof0.
36mm
/secisg
enerall
yrecom
mende
dforth
efound
ationso
fcru
shers.
Thevib
ration
limits
recom
mende
dinthe
British
codeo
fpract
icefor
founda
tionsfo
rma
chiner
y(BSI,
1974)a
reactu
allyhum
antole
rancec
riteria.
Theya
rewide
lyused
forapp
lication
sother
thanfo
undatio
nsforr
ecipro
cating
engine
s.Furth
erinfo
rmation
onhum
antole
rancec
riteria
isprov
idedin
Chapt
er7.In
theabs
ence
