Nonlinea Vib a ions P oduced by
Unbalanced Mo o s
Doc o al Thesis
Ja ie González Ca bajal
Supe ised by
P o . Jaime Domínguez and P o . Daniel Ga cía
Depa men o Mechanical and Manu ac u ing Enginee ing
Facul y o Enginee ing
Uni e si y o Se ille
Ma ch 2017
i
Con en s
1 In oduc ion 1
1.1 S a e o he A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Mo i a ion and Objec i es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 O ganiza ion o he Documen . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Ma hema ical Me hods 13
2.1 Fi s O de A e aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Second O de A e aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Singula Pe u ba ion Theo y . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Hop Bi u ca ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 The Poinca é-Béndixson Theo em . . . . . . . . . . . . . . . . . . . . . 25
3 The Case o La ge Slope o he Mo o Cha ac e is ic: Analy ical
App oach 27
3.1 P oblem S a emen and Assump ions . . . . . . . . . . . . . . . . . . . . 28
3.2 Al e na i e Fi s O de A e aging . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Pe u ba ion App oach: De i a ion o he Reduced Sys em . . . 36
3.4 Analysis o he Reduced Sys em . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Classi ica ion o he Hop bi u ca ions . . . . . . . . . . . . . . . . . . 55
ii
3.6 Condi ions unde which all Sys em T ajec o ies a e A ac ed
owa ds a Limi Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 The Case o La ge Slope o he Mo o Cha ac e is ic: Nume ical
Simula ions 71
4.1 Global Bi u ca ions o he Limi Cycles . . . . . . . . . . . . . . . . . 72
4.2 Nume ical Valida ion o Analy ical Resul s . . . . . . . . . . . . . . . 80
5 The Case o Small Slope o he Mo o Cha ac e is ic: Analy ical
App oach 89
5.1 Ou e Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Inne Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6 The Case o Small Slope o he Mo o Cha ac e is ic: Nume ical
Simula ions 113
7 To que-Speed Cu es o he Whole F equency Range 127
7.1 Compu a ion o he To que-Speed Cu es . . . . . . . . . . . . . . . 128
7.2 A Global Pe spec i e o he Cases o La ge and Small Slope 136
8 Modelling and Simula ion o he Vib ocompac ion P ocess 141
8.1 Some No es on he Real P ocess . . . . . . . . . . . . . . . . . . . . . . 142
8.2 Full Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.3 Simpli ied Model and To que-Speed Cu es . . . . . . . . . . . . . 156
8.4 Analy ical In es iga ion o a Quasis a ic Vib ocompac ion
P ocess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.5 Nume ical Resul s and Discussion . . . . . . . . . . . . . . . . . . . . . 171
9 Summa y and Conclusions 203
Appendix 211
iii
Acknowledgemen s
T as es os cua o años de esis, es oy en deuda con muchas pe sonas y me gus a ía
pode inclui las a odas en es e ag adecimien o. Con ío en que sepan disculpa me
las que, po al a de espacio o po mi despis e exis encial, no apa ezcan
exp esamen e nomb adas.
En p ime luga , quie o ag adece a mis di ec o es, Jaime y Dani, odos sus
buenos consejos y el clima de libe ad que han p opiciado desde el p incipio de
es e abajo. Es oy con encido de que no pod ía habe enido mejo es maes os.
G acias ambién al p o eso Emilio F ei e, po o ien a me sabiamen e cuando me
aden aba en e enos ma emá icos pan anosos.
I would like o exp ess my g a i ude o P o esso Gaë an Ke schen o my s ay a
he Uni e si y o Liège. Wo king wi h his g oup was a wonde ul esea ch
expe ience, in an en i onmen which could no ha e been iendlie . Thanks o
Peppe, who kindly con ibu ed o co ec ypos in he hesis, Lionel, Thibau ,
Chia a, Edoua d, Vincen , I án, Mi co and many mo e good iends ha I hope o
keep o many yea s and made my s ay in Liège wa m and enjoyable.
Thanks o P o esso Jan A. Sande s, who was e y willing o ad ise me abou he
a ac ion p ope ies o a e aged sys ems. His keen e-mail esponses we e uly
help ul o he de elopmen o Chap e 5 in he hesis.
G acias a Joselu, compañe o de a igas desde los mí icos iempos en que
in en amos la osca cuad ada, a Acei uno, Jo ge Julio, Se gio, Me che, Juan,
Da id y al es o de compañe os del depa amen o, po que ha sido una aleg ía
pode abaja odeado de buenos amigos. Ex iendo con gus o es e ag adecimien o
al g an Guido, pese a su obsesión posesi a con el espacio de abajo y con las
papas i as.
A Cha ly, Me chan e, Diego, O ega, Bill, Angeli o, And és, Miguel, Yoshua,
Koc el, Nacho, De ey, Vale a y muchos más, po los buenos a os y ce ezas que
una esis necesi a, aunque sigan pensando que la mía a a de o nillos.

i
A Ma a, mi equipo alpha pa icula , que ha sabido anima me como nadie y
hace lo odo muy lle ade o es os úl imos meses.
Muchas g acias a mis pad es, que son el mejo ejemplo que engo, y a Pablo y Ana,
po su apoyo y su ca iño dia ios. También al es o de mi g an amilia, desde mis
abuelas y abuelos has a Julia y Ra illa. Es e abajo hab ía sido mucho más du o si
no hubie a es ado an bien odeado.
Y p incipalmen e g acias a Dios, a quien le debo odo.
1 INTRODUCTION
The linea heo y o ib a ions is one o he mos use ul ools in he handbook o a
mechanical enginee . I u nishes a solid and well-es ablished ma hema ical
amewo k o concep s such us esonance and linea modes o ib a ion, which a e
cen al o he modal analysis o s uc u es and mechanical sys ems.
A mos appealing quali y o his linea heo y is i s ela i e simplici y. Linea
models exhibi special a ibu es which make hem pa icula ly use ul om a
p ac ical poin o iew, such us he p inciple o supe posi ion –whe eby he
esponse o a linea combina ion o exci a ions can be ob ained as he linea
combina ion o indi idual esponses– o he ac ha se e al a ac o s ne e coexis
– he s a iona y mo ion o a o ced, damped, linea sys em, is independen o he
ini ial condi ions–. I is hanks o hese p ope ies, oge he wi h he ac ha many
eal s uc u es a e well ep esen ed by linea models in hei anges o ope a ion,
2 1 In oduc ion
ha linea ib a ion analysis has become a chie ool in he design o mechanical
sys ems and s uc u es in indus y.
Howe e , linea i y is clea ly an idealiza ion. Na u e is no linea in gene al,
al hough i can be app oxima ely ep esen ed by linea models in some cases. Fo
ins ance, he linea heo y o elas ici y is known o gi e good esul s o s uc u es
unde going small displacemen s and small s ains. Howe e , as soon as
displacemen s become signi ican , nonlinea e ec s need o be conside ed (Luongo,
Rega, & Ves oni, 1986). Some o he possible sou ces o nonlinea i y in mechanical
sys ems a e (Thomsen, 2003)
- Ma e ial nonlinea i ies due o a nonlinea ela ion be ween s esses and
s ains in some ma e ials.
- Nonlinea body o ces, such as magne ic o ae odynamic in e ac ions.
- Nonlinea i ies due o he physical con igu a ion, such as hose associa ed o
discon inuous couplings, clea ances o s ops.
Nonlinea i y plays a key ole in he dynamic beha iou o nume ous eal-wo ld
applica ions. Some examples a e he mo ion o la ge wind u bines, he
c ashwo hiness o ehicles o he ib ocompac ion o g anula ma e ials. The
ae onau ical indus y, wi h g ea in e es in minimizing weigh , is inc easingly
p oducing e y ligh and slende s uc u es, wi h he subsequen ac i a ion o
geome ical nonlinea i ies. The use o ma e ials such as ca bon- ibe composi es in
ae ospace applica ions can also p oduce a signi ican de ia ion om linea i y. I is
e iden ha , in all hese si ua ions, a linea ized model would no be able o cap u e
he eal sys em dynamics. Thus, in o de o ha e eliable p edic ions, nonlinea i y
would need o be included in he model, which in u n implies en e ing he complex
ield o nonlinea dynamics.
Nonlinea dynamics is he b anch o ma hema ics which in ends o unco e he
empo al e olu ion o nonlinea dynamical sys ems. Unlike i s linea coun e pa ,
nonlinea dynamics is no a closed subjec –no e en a ma u e one–. The main
eason is ha , while all linea sys ems a e essen ially he same, each nonlinea
1 In oduc ion 3
sys em is nonlinea in i s own way. This means ha , in gene al, conclusions abou
he beha iou o a pa icula nonlinea sys em canno be gene alized o any o he .
Besides, he dynamics o nonlinea sys ems is ex emely ich, exhibi ing a wide
ange o phenomena which canno occu in linea sys ems, such as mul is abili y,
chaos o limi cycles. All his complexi y ende s i ex emely ha d o ob ain
analy ical solu ions o he nonlinea di e en ial equa ions go e ning he sys em
dynamics. In gene al, i is necessa y o eso o nume ical compu a ions in o de o
ge some insigh in o he sys em beha iou .
F om he abo e conside a ions, i is clea ha a signi ican esea ch e o needs o
be o ien ed o a be e comp ehension o he dynamic beha iou o nonlinea
mechanical sys ems. This will hope ully lead o e icien p edic ions abou he
pe o mance o sys ems whe e nonlinea i y has a signi ican e ec , and will also
mo i a e he design and de elopmen o new nonlinea componen s which a e able
o ou pe o m hei linea coun e pa s. I is he aim o his hesis o con ibu e o
his gene al objec i e, by analysing in de ail a pa icula class o nonlinea sys ems,
namely hose exci ed by unbalanced mo o s.
The mo ion o unbalanced o o s cons i u es one o he mos common ib a ion
sou ces in mechanical enginee ing (Boyaci, Lu, & Schweize , 2015; Yang e al.,
2016). Vib a ions due o unbalance may occu in any kind o o a ing sys ems, such
as u bines, lywheels, blowe s o ans (Shabana, 1996). Ac ually, in p ac ice, o o s
can ne e be comple ely balanced because o manu ac u ing e o s such as po osi y
in cas ing, non-uni o m densi y o he ma e ial, manu ac u ing ole ances, e c. (Xu
& Ma angoni, 1994). E en a subsequen balancing p ocess will ne e be pe ec
due o he ole ances o he balancing machines. Mo eo e , some amoun o
unbalance gene ally appea s du ing he ope a ion o he machine, as a consequence
o une en wea , co osion o unequal build-up o deposi s (di , lime, e c.).
Usually, o o unbalance has a ha m ul e ec on o a ing machine y, since ib a ion
may damage c i ical pa s o he machine, such as bea ings, seals, gea s and
couplings (Xu & Ma angoni, 1994). Howe e , he e a e applica ions whe e
10 1 In oduc ion
ixed poin s o bo h a e aged sys ems a e ob ained and hei s abili y is analysed.
Admi edly, some o he de elopmen s o his Chap e a e no o ally new, bu a
e o mula ion o al eady published ea men s o he p oblem. Howe e , he
s abili y o he s a iona y mo ions o he sys em nea esonance was no o ally
sol ed hi he o. By using a ac ion a gumen s, a de ailed s abili y analysis o hese
solu ions is conduc ed.
Chap e 6 p esen s nume ical esul s which alida e he esul s and conclusions o
Chap e 5.
In Chap e 7, an al e na i e app oxima e me hod is used o ob ain he s a iona y
solu ions o he sys em s udied in p e ious chap e s. This app oach is based on he
Me hod o Di ec Sepa a ion o Mo ions, p oposed by Blekhman (Blekhman, 2000)
and has he ad an age o p o iding a g aphical ep esen a ion o he s a iona y
mo ions which is highly con enien wi h a iew o he ib ocompac ion analysis o
he nex chap e . Fu he mo e, his al e na i e p ocedu e p o ides a e y clea
g aphical compa ison be ween he cases o la ge and small slope conside ed in
p eceding chap e s.
A new model is in oduced in Chap e 8, mo e complex han he one analysed in
p e ious chap e s. This sys em is a i s a emp o model he ib ocompac ion
p ocess, which has no been done be o e, o he au ho ’s knowledge. In addi ion o
he nonlinea i y in insically associa ed o nonideali y, he model includes con ac
and impac s be ween he mix u e and he pla o m suppo ing he unbalanced
mo o , and also be ween he mix u e and he mould whe e i is con ained.
Fu he mo e, a nonlinea cons i u i e law o he mix u e, which allows modelling
he compac ion i sel , is p oposed. I is shown ha , unde some condi ions, his
model can be ans o med in o he simple sys em analysed in Chap e s 3-7, which
makes he p eceding analy ical de elopmen s use ul o he ib ocompac ion
analysis. Se e al nume ical simula ions illus a e how he p oposed model can be
used o in es iga e he e ec o di e en pa ame e s on he inal le el o
compac ion achie ed.

1.3 O ganiza ion o he Documen 11
Finally, Chap e 9 summa ises he conclusions o his wo k and p oposes possible
u he in es iga ions.
2 MATHEMATICAL
METHODS
Be o e he analysis o he p oblem unde s udy, a b ie desc ip ion o he
ma hema ical me hods used wi hin he hesis is p esen ed.
2.1 Fi s O de A e aging
Pe u ba ion me hods cons i u e a b oad class o ma hema ical echniques aimed a
inding app oxima e solu ions o a p oblem, based on he solu ion o a simple
ela ed p oblem. In pa icula , a e aging p ocedu es a e among he mos widely
used pe u ba ion me hods, p esen ing wo ele an s eng hs (Sande s e al., 2007):
- They a e suppo ed by igo ously p o ed heo ems.
- They can be sys ema ically ex ended o any o de o accu acy.
In o de o unde s and he basic idea behind a e aging, conside a sys em o he
o m 󰇗 ( ), whe e is a small pa ame e and is -pe iodic in . The
14 2 Ma hema ical Me hods
dynamics o such sys ems con ains wo di e en ime scales: a as scale, associa ed
o he ac ha depends on , and a slow scale, associa ed o he ac ha is a
slow a iable ( 󰇗 ( )). Then, i can be shown ha he essen ial ea u es o he
sys em a e main ained when i is eplaced by i s co esponding a e aged sys em
󰇗 ∫ ( )
( ). The idea is o ake in o accoun he mean e ec o
he as oscilla o y dynamics h ough a e aging, so ha he second sys em e ains
he long e m beha iou o he i s one. This ans o ma ion is use ul because he
a e aged sys em is au onomous and, he e o e, conside ably easie o analyse han
he non-au onomous o iginal sys em.
Al hough he abo e desc ip ion has been gi en o a e aging o e ime, i is
some imes use ul o a e age o e as o a ing angles, which play essen ially he
same ole as ime. Once he in ui i e idea has been explained, some mo e igo ous
esul s a e now gi en, which will be used h oughou he hesis.
Then, conside an ini ial alue p oblem o he o m
{ 󰇗 ( )
󰇗 ( ) ( )} ( )
( )
(2.1)
whe e is a ec o o slow a iables, is a as o a ing phase and is a
su icien ly small, posi i e, dimensionless pa ame e , . No e ha
ep esen s he ci cum e ence. Hence o say ha me ely means ha ,
wi h unc ions and being -pe iodic in .
Le be he solu ion o
󰇗 ( ) ( )
(2.2)
whe e
2.1 Fi s O de A e aging 15
( )
∫ ( )
(2.3)
Then, i ( ) emains in ,
( ) ( ) ( ) (
⁄)
(2.4)
The p oo can be ound in (Sande s e al., 2007).
This is he s anda d e sion o he heo em o i s o de a e aging o e angles.
No e ha , hanks o his heo em, an asymp o ic app oxima ion o he solu ion ( )
can be ob ained by eplacing he o iginal sys em (2.1) wi h he app oxima e sys em
(2.2). In o he wo ds, i is possible o educe he sys em dimension, om o
, upon a e aging o e he as o a ing phase.
No e also ha , as explici ly s a ed in (2.4), he app oxima ion ob ained h ough
a e aging is only alid du ing a limi ed ime scale. This is a gene al ea u e o
pe u ba ion app oaches, which can be o e come in some si ua ions (e.g. when
a ac ion exis s, as desc ibed la e on in his sec ion). The limi a ion in he ime
scale whe e he app oxima ion is alid will p o e o be c ucial in Sec ion 5.2.
Gene aliza ion o mul i- equency sys ems
The gene aliza ion o his heo em o mul i- equency sys ems is s aigh o wa d
(Sande s e al., 2007). The only ca ea ha needs o be aken in o accoun is ha
unc ion mus be w i en as a sum o unc ions, each o one depending on only
one o he as angles. Thus, conside he mul i- equency sys em gi en by
{ 󰇗 ( )
󰇗 ( ) ( )} ( )
( )
(2.5)
whe e ep esen s he - o us. Then, assuming ha unc ion can be w i en as

16 2 Ma hema ical Me hods
( ) ∑ ( )
(2.6)
he i s o de a e aged sys em can be ob ained as
󰇗 ∑  ( )
( )
(2.7)
wi h  de ined as in (2.3). Then, he e o es ima e is gi en by
( ) ( ) ( ) (
⁄)
(2.8)
This mul i- equency e sion o he heo em will be used in Sec ion 5.1.
The concep o ‘Resonance Mani old’
A c ucial poin o his a e aging p ocess is ha , o he app oxima ion o be alid,
each equency ( ) mus be bounded away om ze o. The eason is ha , when
any ( ) app oaches ze o, a anishing denomina o appea s in he highe o de
e ms o eqs. (2.4) and (2.8). (This will be seen clea ly in equa ion (2.12) o Sec ion
2.2.)
The se s o o which one o mo e unc ions ( ) anish a e called ‘ esonance
mani olds’. The ailu e o a e aging in he icini y o esonances can be easily
unde s ood by no icing ha , nea a esonance mani old, one o mo e angles a e
no as and, consequen ly, we canno a e age o e hem.
The main consequence is he ollowing: away om esonances, i is possible o
a e age o e all angles. Howe e , in he icini y o a esonance mani old, we can
only a e age o e pa icula angles, namely hose whose equencies emain
bounded away om ze o. This dis inc ion will be necessa y in Chap e 5.
2.1 Fi s O de A e aging 17
A e aging wi h A ac ion
Asymp o ic app oxima ions ob ained by a e aging a e, in gene al, alid on a ime
scale
⁄. Howe e , his ime scale can be enla ged i he e exis s a ac ion in he
a e aged sys em (Sande s e al., 2007).
Conside again a sys em o he o m (2.1), wi h a e aged sys em (2.2).
Suppose is an asymp o ically s able ixed poin o sys em (2.2), wi h domain
o a ac ion . Then, o all , i can be shown ha
( ) ( ) ( ) , )
(2.9)
Thus, he app oxima ion is uni o mly alid –i.e. alid o all ime– o solu ions o
he a e aged sys em which a e a ac ed by an asymp o ically s able ixed poin .
Two di e en p oo s o his heo em ha e been gi en by Sánchez-Palencia
(Sanchez-Palencia, 1975) and Eckhaus (Wik o Eckhaus, 1975).
A simila esul holds o ajec o ies o he a e aged sys em which a e a ac ed by
an asymp o ically s able limi cycle. In his case, he app oxima e solu ion is alid
on , ) o all a iables excep he angula one, i.e. he a iable which
measu es he low on he closed o bi (Sande s e al., 2007). This is equi alen o
say ha he closeness o he limi cycle can be uni o mly app oxima ed, ye no he
posi ion on i . The eason is ha any small de ia ion on he equency is
accumula ed o e he cycles, gi ing ise o la ge e o s a e a su icien numbe o
pe iods.
2.2 Second O de A e aging
In some si ua ions, mo e accu a e app oxima ions han hose p o ided by a i s
o de a e aging (Sec ion 2.1) a e equi ed, as will be he case in Chap e 5 o his
18 2 Ma hema ical Me hods
hesis. Fo una ely, he e exis gene al esul s o o de a e aging which allow
inc easing he p ecision o he app oxima ion as much as needed. He e we only
show some esul s o o de 2, which is enough o he pu pose o his hesis. Hence
conside a sys em o he o m
8 󰇗 ( ) ( ) , -( )
󰇗 ( )9 ( )
( )
(2.10)
whe e, ollowing he no a ion in (Sande s e al., 2007), b acke s a e used in , - o
s ess ha his e m is a emainde o an expansion in powe s o . This is also
no ed by he ac ha , - depends on , while and do no .
Le * ( ) ( )+ be he solu ion o
8 󰇗  ( )  ( )
󰇗 ( )9 ( ) ( )
( )
(2.11)
whe e  is de ined as in (2.3) and
( )
∫[ ( )  ( )]
(2.12)
 ( )
∫[ ( ) ( ) ( )]
(2.13)
Cons an in (2.12) is chosen in such a way ha
∫ ( )
(2.14)
2.2 Second O de A e aging 19
while symbol in (2.13) ep esen s di e en ia ion wi h espec o . Then, he
e o es ima e is
( ) ( ) ( ( ) ( )) ( ) (
⁄)
(2.15)
This esul will be used in Chap e 5. The p oo can be ound in (Sande s e al.,
2007).
No e he p esence o in he denomina o o exp ession (2.12), in connec ion wi h
he concep o ‘ esonance mani old’ explained in Sec ion 2.1. A anishing
equency would make unbounded and, acco ding o (2.15), he
app oxima ion would no be alid anymo e. In o he wo ds, he a e aging
ans o ma ion becomes singula when app oaches ze o.
2.3 Singula Pe u ba ion Theo y
The Singula Pe u ba ion Theo y (SPT) explains he beha iou o a pa icula class
o as -slow sys ems (Hun e , 2004; Lesne, 2006; Ve huls & Bak i, 2006). As has
been done wi h a e aging, an in ui i e desc ip ion o he heo y is i s gi en,
ollowed by he exposi ion o some mo e igo ous esul s which will be used la e
on in he hesis.
Sys ems whe e he SPT is applicable a e hose exhibi ing a singula limi . This
means ha he sys em depends on a pa ame e in such a way ha , when he
pa ame e app oaches some limi ing alue, he gene al solu ion o he p oblem is
quali a i ely di e en o he solu ion o he limi ing p oblem. Typically, sys ems o
his ype exhibi wo sepa a e ime scales –o space scales– wi h e y di e en
beha iou s. A well-known example is he bounda y laye heo y in luid mechanics,
whe e he e ec s o iscosi y a e ele an in a e y hin laye o luid, close o a
bounding su ace, while being negligible o he es o he domain.
26 2 Ma hema ical Me hods
is o ien ed inwa ds, as depic ed in Fig. 2.3. Then, all ajec o ies which s a inside
a e es ic ed o emain inside.
Fig. 2.3 Phase po ai showing a closed, bounded egion wi h he low di ec ed
inwa ds a he bounda ies. I con ains no ixed poin s, hen i con ains a leas one
s able limi cycle.
R

3 THE CASE OF LARGE
SLOPE OF THE MOTOR
CHARACTERISTIC:
ANALYTICAL
APPROACH
This Chap e in es iga es he dynamics o a 2-DOF sys em consis ing in an
unbalanced mo o a ached o he ixed ame by a nonlinea sp ing and a linea
dampe . As commen ed in he in oduc ion, wo di e en scena ios need o be
conside ed sepa a ely, depending on he o de o magni ude o he slope o he
mo o cha ac e is ic. The Case o a La ge Slope is conside ed in his Chap e and
he ollowing, while he al e na i e si ua ion will be s udied in Chap e s 5 and 6.
28 3 The Case o La ge Slope: Analy ical App oach
3.1 P oblem S a emen and Assump ions
Conside he sys em depic ed in Fig. 3.1. I consis s in an unbalanced mo o
a ached o a ixed ame by a nonlinea sp ing –whose o ce has linea and cubic
componen s– and a linea dampe . The cubic componen o he sp ing gi es he
possibili y o model a nonlinea beha io o he s uc u e suppo ing he mo o
(Me le , 1962). The e ec o g a i y can be shown o ha e no ele ance
(Dimen be g e al., 1997) and, he e o e, i will no be included in he model.
Fig. 3.1 Model
Va iable s ands o he linea mo ion, is he angle o he o o , is he
unbalanced mass wi h eccen ici y , is he es o he ib a ing mass, is he
o o ine ia (wi hou including he unbalance), is he iscous damping coe icien
and and a e, espec i ely, he linea and cubic coe icien s o he sp ing. The
equa ions o mo ion o he coupled 2-DOF sys em a e (El-Badawy, 2007)
󰇘 󰇗 ( 󰇗 󰇘 )
󰇘 ( 󰇗) 󰇘
(3.1)
whe e , and an o e do ep esen s di e en ia ion wi h
espec o ime, .
Func ion ( 󰇗) is he d i ing o que p oduced by he mo o –gi en by i s o que-
speed cu e, also known as s a ic cha ac e is ic– minus he losses o que due o
0
I
0
m
1
m
b
x

,
k

3.1 P oblem S a emen and Assump ions 29
ic ion a he bea ings, windage, e c. We assume his ne o que o be a linea
unc ion o he o o speed:
( 󰇗) 󰇗
(3.2)
Al hough ( 󰇗) includes he damping o o a ional mo ion, we will usually e e o
i sho ly as ‘ he mo o cha ac e is ic’.
As will be seen la e , i is con enien o he pu pose o his chap e o w i e he
d i ing o que in an al e na i e way. Then, deno ing by he linea na u al
equency o he oscilla o , gi en by √
⁄, he mo o o que can be w i en
as
( 󰇗) ( 󰇗 )
(3.3)
whe e ep esen s he d i ing o que a esonance ( ( ) ). F om equa ions
(3.2) and (3.3), he ela ion be ween cons an s and can be di ec ly deduced:
(3.4)
Along he whole hesis, he mo o cha ac e is ic will be w i en as (3.2) o (3.3),
depending on he si ua ion. I should be kep in mind ha hese wo exp essions a e
o ally equi alen . The impo an poin is ha he d i ing o que is assumed o
ollow a linea ela ion wi h he o o speed. I is u he assumed ha – he
d i ing o que dec eases wi h he o o speed–, as is usual o mos kinds o mo o .
This assump ion will p o e o be o majo impo ance.
30 3 The Case o La ge Slope: Analy ical App oach
Fig. 3.2 Typical s a ic cha ac e is ic o an asynch onous mo o
Fig. 3.3 S a ic cha ac e is ic co esponding o equa ion (3.3)
As an example, he s a ic cha ac e is ic o an induc ion mo o is depic ed in Fig. 3.2.
No e ha such a mo o is usually designed o wo k on he egion 󰇗 , whe e
he cu e could be easonably app oxima ed by a s aigh line wi h nega i e slope.
The simpli ied mo o cha ac e is ic gi en a (3.3) is ep esen ed in Fig. 3.3.
m
L

m
L
n

C
1
D
peak


3.1 P oblem S a emen and Assump ions 31
In he second o equa ions (3.1), which imposes he equilib ium o he o o , he las
e m is o g ea signi icance, since i accoun s o he o que on he o o caused by
linea mo ion o he sys em. I s physical in e p e a ion can be eadily unde s ood
wi h he aid o Fig. 3.4. Due o displacemen ( ), a ho izon al ine ial o ce ac s on
he unbalanced mass and gene a es a o que wi h espec o he o o axis. This
pa icula e m o he equa ions o mo ion is wha makes he exci a ion nonideal, o
i akes in o accoun how ib a ion in luences o a ion. I his o que due o
ib a ion did no exis –o i i was negligible–, he o o equilib ium equa ion
would educe o 󰇘 ( 󰇗), and i could be sol ed o ( ) ega dless o he
linea mo ion. Then, his solu ion ( ) could be in oduced in he i s o equa ions
(3.1) as a p esc ibed exci a ion.
Fig. 3.4 To que on he o o due o ib a ion
By de ining
⁄
⁄
√
(
)
(3.5)
he equa ions o mo ion can be w i en in a mo e con enien dimensionless o m
󰇘 󰇗 ( 󰇗 󰇘 )
󰇘 ( 󰇗 ) 󰇘
(3.6)
1
mx
sin


x
1
m

32 3 The Case o La ge Slope: Analy ical App oach
whe e a do now ep esen s di e en ia ion wi h espec o dimensionless ime, .
In o de o apply pe u ba ion echniques o sys em (3.6), some assump ions on he
o de o magni ude o he sys em pa ame e s ha e o be made. Thus, we assume he
damping, he unbalance and he nonlinea i y o be small. This is exp essed by
making he co esponding coe icien s p opo ional o a su icien ly small, posi i e
and dimensionless pa ame e :
(3.7)
whe e pa ame e s wi h subsc ip a e -independen . I is also assumed ha he
o que gene a ed by he mo o a esonance ( 󰇗 ) is su icien ly small:
(3.8)
Finally, he slope o he mo o cha ac e is ic is assumed o be o he o de o uni y,
i.e. independen o :
(3.9)
This assump ion co esponds o wha we ha e called ‘la ge slope cha ac e is ic’.
The case o small slope, wi h p opo ional o , is ea ed in Chap e s 5 and 6.
Taking he p oposed scaling (3.7)-(3.9) in o accoun and d opping he subsc ip ‘ ’
o con enience, sys em (3.1) akes he o m
󰇘 [ 󰇗 ( 󰇗 󰇘 )]
󰇘 ( 󰇗 ) , 󰇘 -
(3.10)
3.2 Al e na i e Fi s O de A e aging 33
3.2 Al e na i e Fi s O de A e aging
Be o e u ning o he ea men o sys em (3.10) h ough some pe u ba ion
echniques, an al e na i e a e aging p ocedu e is de eloped in his sec ion, which
will be use ul in wha ollows. In o de o make he p ocedu e as gene al as
possible, conside a sys em o he o m
{ 󰇗 , ( )- ( )
󰇗 ( ) ( )
󰇗 ( )}
(3.11)
whe e and a e ma ices o cons an coe icien s and is a scala cons an ,
bounded away om ze o. I will be shown in he nex sec ion ha sys em (3.10) can
be w i en in he o m (3.11).
Suppose ha we y o pe o m a i s o de a e aging on sys em (3.11) o e angle
. Acco ding o he esul s explained in Sec ion 2.1, such a echnique is no
applicable in his case, because he se o a iables is no slow. In o de o use he
heo em o Sec ion 2.1 we would need o ha e 󰇗 ( ) and 󰇗 ( ), which is
no he case. This jus i ies he in oduc ion o he modi ied echnique p esen ed
below.
Fi s , he a e aged a iables a e de ined as
( ) ∫ ( )
⁄
⁄ ( ) ∫ ( )
⁄
⁄
(3.12)
whe e
⁄. As illus a ed in Fig. 3.5, he e ec o he ope a o de ined in
(3.12) is o smoo h ou he sho - e m luc ua ions o each a iable, while e aining
he long- e m beha io .
34 3 The Case o La ge Slope: Analy ical App oach
Fig. 3.5 De ini ion o he a e aged a iables
Suppose we a e in e es ed in he e olu ion o he a e aged a iables ( ) and ( ).
Then, we can a e age he i s wo equa ions in (3.11), which yields
{
󰇗 [  ∫ ( ( ) ( ))
⁄
⁄] ( )
󰇗  ∫ ( ( ) ( ))
⁄
⁄ ( )
}
(3.13)
whe e i has been used ha he a e age, as de ined in (3.12), is a linea ope a o ( he
a e age o he sum is he sum o he a e ages).
The nex s ep consis s in ans o ming he in eg als in (3.13). Since he p ocess is
exac ly he same o bo h in eg als, we only ocus on he i s o hem.
Fi s , we can w i e
6.4 6.6 6.8 7 7.2
0
0.2
0.4
0.6
0.8
1
x
x
3.2 Al e na i e Fi s O de A e aging 35
∫ ( ( ) ( ))
⁄
⁄ ∫ ( ( ) ( ))
⁄
⁄ ( )
(3.14)
whe e i has been used he p ope y ha , in one pe iod , ( ) can only change by
( ), acco ding o (3.11). Thus, we can w i e ( ) ( ) ( ). Changing he
in eg a ion a iable om o yields
∫ ( ( ) ( ))
⁄
⁄ ( )
∫ ( ( ) )
(
⁄)
(
⁄) ( )
(3.15)
whe e he las o ela ions (3.11) has been used ( ( )). The
in eg a ion limi s can also be ans o med by using again 󰇗 ( ):
∫ ( ( ) )
(
⁄)
(
⁄) ( )
∫ ( ( ) )
( )
( ) ( )
(3.16)
Finally, as unc ion is -pe iodic in , we can w i e
∫ ( ( ) )
( )
( ) ( )
∫ ( ( ) )
( )
(3.17)
By compa ing his las exp ession o de ini ion (2.3), sys em (3.13) can be ew i en
as
8 󰇗 ,  ( )- ( )
󰇗  ( ) ( )9
(3.18)
42 3 The Case o La ge Slope: Analy ical App oach
can be w i en as
( )
(3.41)
which is he exp ession o he Slow Mani old: a 2D su ace in he 3D phase
space. Thus, he i s condi ion is sa is ied.
2. Fo ixed alues o and , i is ound ha ( ) is a globally
asymp o ically s able ixed poin o he 1D sys em
󰇗
(3.42)
p o ided ha assump ion holds. The e o e, he second condi ion is
also ul illed.
Once bo h equi emen s ha e been e i ied, i can be s a ed ha sys em (3.38)
displays wo quali a i ely di e en beha io s a wo sequen ial ime scales –see
Sec ion 2.3–, which co espond o he second and hi d s ages o he o iginal sys em
(3.26). Using he esul s o Sec ion 2.3, we ha e ha , a he i s o hese s ages –
second s age o (3.26)–, he sys em can be w i en as
( )
( )
󰇗 ( )
(3.43)
whe e i has been aken in o accoun ha , a he beginning o s age 2,
( ) and ( ). Then, a his s age, he sys em is a ac ed owa ds he
Slow Mani old, wi h he slow a iables emaining nea ly cons an :
( )
(3.44)

3.3 Pe u ba ion App oach: De i a ion o he Reduced Sys em 43
Summing up, he second s age co esponds o a ime leng h ( ), jus as he
i s one. I ends once a iable has eached an ( )–dis ance o ( ).
Du ing his phase o he mo ion, and do no change signi ican ly.
Thi d s age
The hi d s age o he o iginal sys em (3.26) –which is he second s age o he
a e aged sys em (3.38)– occu s a a ime scale (
⁄). This can be easily
unde s ood by no icing ha , once he sys em is nea he slow mani old, a iable
becomes slow (in oducing (3.41) in (3.38) leads o 󰇗 ( )). The e o e, nea he
slow mani old, all a iables a e slow and, as a consequence, he sys em na u al ime
scale is (
⁄).
By in oducing he exp ession o he slow mani old in (3.38), he equa ions
co esponding o he hi d phase o he mo ion a e ob ained:
{
󰇗 ( ) ( )
󰇗 ( ( )
) ( )
( ) ( )
}
(3.45)
As usual, highe o de e ms in (3.45) can be elimina ed, gi ing ise o an ( )
app oxima ion o a ime leng h (
⁄):
{ 󰇗 ( )
󰇗 ( ( )
)
( ) }
(3.46)
I is con enien o obse e ha , al hough (3.46) con ains h ee equa ions, only wo
o hem a e di e en ial equa ions. Thus, (3.46) ep esen s a 2D au onomous
dynamical sys em. The e olu ion o and no longe depends on , once is
w i en as a unc ion o and . The las equa ion is w i en wi h he only pu pose
o acking he e olu ion o a iable .
44 3 The Case o La ge Slope: Analy ical App oach
In summa y, he hi d s age co esponds o a ime leng h (
⁄). A his phase
o he mo ion, he a e aged sys em e ol es along he slow mani old gi en by
(3.41). Va iables , and obey equa ions (3.46), wi h ( ) p ecision.
Fig. 3.6 shows a schema ic ep esen a ion o he h ee di e en s ages o he sys em
dynamics, summing up he esul s ob ained in he p esen sec ion. No e ha , in Fig.
3.6, he use o o e ba s o he a e aged a iables is eco e ed. The mos ele an
esul is ha , once he ini ial ansien co esponding o he i s wo s ages has
inished, he e olu ion o a iables and is go e ned by equa ions (3.46) –wi hin
an ( ) e o –.
F om Fig. 3.6, i is clea ha sui able ini ial condi ions o sys em (3.46) a e
* +. Recalling de ini ion (3.31), his can be w i en as * ( )
⁄+,
whe e * + is he se o ini ial condi ions o sys em (3.26)
Fig. 3.6 O e iew o he sys em dynamics, wi h * + being he solu ion o
sys em (3.46) wi h app op ia e ini ial condi ions.
S age 3
S age 2
S age 1
0
0
0
a


0
0
1
()
()
()
*
aa O
O
O

  







00
0
0
()
()
( ) ( )
*
*
*,
aa
a
O
O
O

  
   






0


0
0
1
()
*
aa O






00
0
0
()
()
()
*
**,
aa
a
O
O

  
  





()
()
( ) ( )
*,
RR
R
R
aa
a
O
O
O

  
   







(1)O


(1)O


()
1
O


3.3 Pe u ba ion App oach: De i a ion o he Reduced Sys em 45
Howe e , we may be in e es ed in a pa icula se o ini ial condi ions o sys em
(3.10), gi en as { 󰇗 󰇗 }. I is, hen, con enien , o exp ess he ini ial
condi ions o (3.46) as unc ions o he ini ial condi ions o (3.10):
√ 󰇗
( 󰇗
) 󰇗
(3.47)
as can be eadily deduced om ela ions (3.20), (3.25) and (3.31).
Recapi ula ing, we ha e been able o elimina e om he o mula ion a iable by
A e aging, and a iable by applying he Singula Pe u ba ion Theo y.
3.4 Analysis o he Reduced Sys em
This sec ion ocuses on he beha iou o sys em (3.46), once i has been shown o
cap u e, wi h ( ) p ecision, he dynamics o he o iginal sys em (3.10) du ing he
hi d s age o he mo ion.
Fi s ly, i is use ul o make a compa ison be ween he sys em unde s udy and i s
ideal coun e pa , whe e he o o speed is cons an . Clea ly, o his ideal case, he
equa ion o mo ion o he sys em shown in Fig. 3.1 is gi en by
󰇘 󰇗 󰇗
(3.48)
wi h 󰇗 ixed. Equa ion (3.48) desc ibes a Du ing oscilla o , subjec ed o ha monic
exci a ion. This is a e y well-known p oblem, which has been widely s udied in he
li e a u e (B ennan, Ko acic, Ca ella, & Wa e s, 2008; Fidlin, 2006; Nay eh &
Mook, 1995; Thomsen, 2003). Unde he assump ions o small damping, small
46 3 The Case o La ge Slope: Analy ical App oach
nonlinea i y, small unbalance and nea - esonan exci a ion ( 󰇗 ), he
A e aging Me hod can be applied o sys em (3.48), leading o
{ 󰇗 ( )
󰇗 (
)}
(3.49)
whe e all he pa ame e s and a iables a e de ined as in Sec ions 3.1 and 3.3. I is
easy o e i y ha sys em (3.49) is exac ly he same as (3.46), wi h he only
di e ence o eplacing ( ) by he cons an alue . This is a clea illus a ion
o he concep o nonideal exci a ion. In he ideal case, he o o speed appea s in
equa ions (3.49) as a cons an alue , ex e nally imposed by he mo o . Howe e ,
in he nonideal case, he o o speed en e s equa ions (3.46) as a unc ion o he
sys em ib a o y mo ion, ( ).
I is also impo an o obse e ha an ideal mo o displays a e ical s a ic
cha ac e is ic, co esponding o he limi case . The mo o is, hen, able o
gene a e any o que o he same o o speed. This sugges s he idea ha a eal
mo o wi h a s a ic cha ac e is ic o e y la ge slope (in absolu e alue) is mo e
likely o beha e in an ideal manne han ano he one wi h a smalle slope.
Fixed poin s
Going back o he objec i e o analyzing sys em (3.46), i is i s con enien o look
o i s ixed poin s, { }:
( )
( )
(3.50)
F om he i s o equa ions (3.50), we ha e
3.4 Analysis o he Reduced Sys em 47
√
(3.51)
Combining (3.41), (3.50) and (3.51) yields
√
(3.52)
Solu ions o (3.52), o bo h alues o , gi e o all he ixed poin s o (3.46).
This can be done analy ically, bu he exp essions become cumbe some and
di icul o in e p e . An al e na i e p ocedu e is p oposed, which leads o he ixed
poin s o (3.46) in a g aphical way. To his end, he las o equa ions (3.46) can be
ew i en as
(3.53)
whe e de ini ion (3.41) has been used. Now, ecall he las o equa ions (3.38),
which go e ns he e olu ion o he o o speed o he a e aged sys em:
󰇗 ( )
(3.54)
In he ligh o (3.54), (3.53) can be in e p e ed as an equilib ium be ween wo
o ques on he o o . The le hand e m in (3.53) ep esen s he d i ing o que
p oduced by he mo o , while he igh hand e m ep esen s he esis ing o que due
o ib a ion. Thus, he ac ha he a e aged sys em is on he slow mani old –which
is exp essed in equa ion (3.53)– can be unde s ood as a o que equilib ium
condi ion.
Equa ion (3.53), pa icula ized o he ixed poin { }, akes he o m

48 3 The Case o La ge Slope: Analy ical App oach
(3.55)
whe e (3.50) has been used. We now de ine he ollowing unc ions:
( )
( )
(3.56)
Clea ly, acco ding o he commen s below equa ion (3.54), ep esen s he
d i ing o que p oduced by he mo o , while co esponds o he esis ing o que
due o ib a ion. Then, (3.55) can be ew i en as
( ) ( )
(3.57)
which is he o que equilib ium condi ion, pa icula ized o he ixed poin .
In o de o sol e (3.57) in a g aphical way, i would be desi able o w i e bo h
o ques explici ly in e ms o . Howe e , his would in u n need explici ly
w i ing in e ms o , which p oduces long and complica ed exp essions.
Thus, an implici p ocedu e o he g aphical ep esen a ion is p oposed. Combining
(3.50) and (3.51) esul s in
( )
(3.58)
whe e unc ion ( ) is de ined as
( ) √
(3.59)
The p oposed ep esen a ion can be cons uc ed as ollows: i s , g aph e sus
acco ding o (3.56). Then, g aph on he same plo he pa ame ic cu e gi en by
3.4 Analysis o he Reduced Sys em 49
* ( ) ( )+, o and ( -. The ac ha is s ic ly posi i e
comes om he de ini ion o as he adius o a pola coo dina e ans o ma ion –
see (3.20)–. On he o he hand canno be g ea e han 1, acco ding o he i s o
equa ions (3.50).
The abo e p ocedu e gi es ise o a plo like ha shown in Fig. 3.7. Conside ing
equa ion (3.57), he ixed poin s can be ound as he in e sec ions o he wo o que
cu es. In he pa icula case displayed in Fig. 3.7, he e a e h ee equilib ium
poin s, ma ked wi h ci cles. No e ha he cu e associa ed o he ib a ion o que is
composed o wo b anches, which collide a he maximum o he cu e. They
co espond o he wo possible alues o pa ame e , as speci ied in Fig. 3.7.
Fig. 3.7 Fixed poin s o sys em (3.46)
We no e ha he ‘Somme ed e ec ’, which was desc ibed in he in oduc ion, can
be eadily explained by using Fig. 3.7. Fo such an explana ion, he in e es ed
eade can e e o (Blekhman, 2000; Dimen be g e al., 1997; Kononenko, 1969;
Nay eh & Mook, 1995).
-2 0 2 4 6 8 10 12
0
0.5
1
1.5
2
2.5
1z

T
T
1z
m
T
50 3 The Case o La ge Slope: Analy ical App oach
S abili y Analysis
Once he ixed poin s o he educed sys em ha e been ob ained, i is con enien o
in es iga e hei s abili y. Fo a 2D sys em, his educes o calcula ing he ace and
de e minan o he jacobian ma ix, e alua ed a he equilib ium poin o in e es :
[
(
)
]
(3.60)
whe e s ands o √
.
The condi ions o a ixed poin o be asymp o ically s able a e
( )
(3.61)
( )
(3.62)
A e some algeb a, hese condi ions can be exp essed as
(3.63)
{
}
(3.64)
whe e deno es he slope o he cu e a he conside ed equilib ium poin (see
Fig. 3.8 and Fig. 3.9), and has he exp ession
3.4 Analysis o he Reduced Sys em 51
(3.65)
as can be deduced om (3.56), (3.59).
Condi ions (3.63) and (3.64) a e now applied o e alua e s abili y egions in
di e en scena ios. The p ocedu e is as ollows. Conside pa ame e s ixed,
so ha he cu e –see Fig. 3.7– is ixed oo. Conside a pai o alues ( )
which gi es a pa icula cu e ( ). The in e sec ions be ween he wo cu es
ep esen he equilib ium poin s o he sys em. Selec one o hem –i he e a e mo e
han one– and le pa ame e s ( ) a y in such a way ha he selec ed equilib ium
poin emains an equilib ium poin . In o he wo ds, le pa ame e s ( ) a y so as
o make he cu e ( ) o a e a ound he selec ed equilib ium poin , sa is ying
es ic ion . Finally, use condi ions (3.63) and (3.64) o analyze how he
s abili y o he ixed poin is a ec ed by he slope o he mo o cha ac e is ic.
Fig. 3.8 displays he ou come o applying he abo e p ocedu e o a ixed poin
loca ed a he le b anch o he ib a ion o que cu e ( ). Two scena ios a e
conside ed, depending on he sign o slope , e alua ed a he ixed poin unde
conside a ion. I is obse ed ha a change o s abili y occu s when bo h o que
cu es become angen ( ). This can be shown o co espond o a ansc i ical
bi u ca ion. No e ha , in Fig. 3.8, he mo o cu e co esponding o has been
di ec ly labeled as , ins ead o ( ). This sho ened no a ion will be
widely used in he igu es o he documen .
Fig. 3.9 shows analogous esul s o a ixed poin loca ed a he igh b anch o he
ib a ion o que cu e ( ). The sys em beha io is iche in his case, since
s abili y may change in wo di e en ways, depending on he compa ison
whe e is de ined below.
58 3 The Case o La ge Slope: Analy ical App oach
T ans o ma ion o he eal eigenbasis o ma ix
A new change o a iables, using he eal eigenbasis o ma ix , is de ined:
0 1 0
1
(3.76)
whe e he columns o ma ix a e he eal and imagina y pa s o he complex
conjuga e eigen ec o s o , deno ed by :
0
1 0
1 0
1
(3.77)
wi h
√4
5
(3.78)
Sys em (3.73), w i en in e ms o he new a iables, akes he o m
[ 󰇗 󰇗] ([
]0
1 [ ( )
( )])
(3.79)
whe e unc ions and , con aining he nonlinea e ms o he sys em, can be
w i en as Taylo se ies:

3.5 Classi ica ion o he Hop Bi u ca ions 59
( ) ∑
( ) ∑
(3.80)
Coe icien s and a e speci ied in he Appendix.
No e ha he sys em is inally w i en in he o m (2.21). Thus, he esul explained
in Sec ion 2.4 can be di ec ly applied.
T ans o ma ion o No mal Fo m
Sys em (3.79) can be ans o med o i s No mal Fo m by a s anda d p ocedu e
(Guckenheime & Holmes, 1983; Kuzne so , 1998), as desc ibed in Sec ion 2.4:
󰇗
(3.81)
whe e pa ame e is ob ained as
{
, ( ) ( ) -}
(3.82)
In summa y, i can be said ha , a e a la ge numbe o a iable ans o ma ions,
sys em (3.46) can be w i en as (3.81), om which i is concluded ha he
bi u ca ion is supe c i ical (subc i ical) i ( ).
Despi e he ac ha coe icien s and a e o a he complica ed o m, we ind
–wi h he aid o so wa e o symbolic compu a ion (Ma lab)– ha he condi ion o
supe c i icali y o subc i icali y can be exp essed in a su p isingly simple manne :
60 3 The Case o La ge Slope: Analy ical App oach
(3.83)
Fig. 3.11 De ini ion o slope
F om (3.83), i is clea ha a nonlinea i y o he so ening ype ( ) is needed o
ha e a supe c i ical bi u ca ion.
I is also wo h no ing ha condi ions (3.83) admi a e y clea g aphical
in e p e a ion. Conside a cu e which in e sec s a he equilib ium poin
unde conside a ion and also a he peak o cu e . Le deno e he slope o his
pa icula mo o cha ac e is ic, as depic ed in Fig. 3.11.
In o de o ob ain , he coo dina es o he wo poin s de ining he s aigh line a e
de ined below. Fi s , he highes peak o cu e can be shown o co espond o
. Subs i u ing his condi ion in (3.56) and (3.59) yields
(3.84)
T
T
P
d

3.5 Classi ica ion o he Hop Bi u ca ions 61
On he o he hand, he ( ) coo dina es o he equilib ium poin unde s udy a e
di ec ly gi en in (3.56) and (3.59):
(3.85)
Then, om (3.84) and (3.85), he exp ession o can be eadily ob ained:
(3.86)
By compa ing (3.86) and (3.66), condi ions (3.83) can be exp essed as
(3.87)
This las manne o cha ac e izing he bi u ca ion is ce ainly appealing om a
g aphical poin o iew, since he basic in o ma ion abou he bi u ca ion can be
di ec ly obse ed om he o que–speed cu es, as shown in Fig. 3.12 o wo
pa icula examples.
62 3 The Case o La ge Slope: Analy ical App oach
Fig. 3.12 Examples o (a) subc i ical and (b) supe c i ical bi u ca ions.
(a)
(b)
-4 -3 -2 -1 01234
0
0.1
0.2
0.3
0.4
0.5

T
-10 -5 0 5
0
0.1
0.2
0.3
0.4
0.5

T
(a)
(b)
T
P
d
H
d
T
P
d
H
d
3.6 Condi ions o he Sys em o be Always A ac ed by a Limi Cycle 63
3.6 Condi ions unde which all Sys em T ajec o ies a e
A ac ed owa ds a Limi Cycle
In Sec ion 3.5, a simple condi ion has been ob ained o asce ain whe he he Hop
bi u ca ion unde s udy is subc i ical o supe c i ical, which in u n allows
p edic ing he kind o limi cycle gene a ed by he bi u ca ion (see Fig. 3.10).
Al hough his dis inc ion is ele an , i is based on a local analysis and,
consequen ly, i only gi es local in o ma ion abou he sys em beha iou . This is so
in wo senses: he analysis o Sec ion 3.5 p o ides insigh in o he sys em dynamics
- o alues o close enough o ( esul s a e local in he pa ame e space)
and
- o ajec o ies close enough o he in es iga ed ixed poin ( esul s a e local
in he phase plane).
In iew o he a o emen ioned limi a ions, his sec ion add esses a new global esul
ha complemen s hose o Sec ion 3.5. I will be shown ha , unde ce ain
ci cums ances, he Poinca é-Bendixson (P-B) heo em can be used o p o e ha all
ajec o ies o he sys em unde s udy a e a ac ed owa ds a limi cycle. Fo a b ie
explana ion o he heo em, see Sec ion 2.5.
Fi s , i can be easily deduced om (3.46) ha
󰇗
(3.88)
Le and ep esen pola coo dina es on he phase plane, acco ding o (3.70), and
le deno e a ci cle cen ed a he o igin o he phase plane wi h a adius sligh ly
g ea e han , say . F om (3.88), i can be said ha e e y ajec o y s a ing
ou side egion will en e and emain inside o all subsequen ime. Ob iously,
ajec o ies s a ing inside will also emain inside o e e . This kind o beha io
would p esen as a sui able candida e o he ole o egion in he P-B heo em
–see Sec ion 2.5 –, i i we e no o he p esence o ixed poin s inside .

64 3 The Case o La ge Slope: Analy ical App oach
Conside now he ollowing pa icula si ua ion:
{
}
(3.89)
whose o que cu es a e depic ed in Fig. 3.13. We suppose ha he only ixed poin
o he sys em is on he igh b anch o cu e and unde goes a Hop bi u ca ion. I
is also assumed ha he ac ual slope o he mo o cha ac e is ic is and,
he e o e, he equilib ium is uns able.
Fi s , le us p o e ha he ixed poin is a epelle . Since he equilib ium is al eady
known o be uns able, we only need o p o e ha i is no a saddle. Le be he
jacobian ma ix o sys em (3.46), e alua ed a he equilib ium poin . Taking in o
accoun ha a saddle poin has wo eal eigen alues wi h di e en signs, we
can s a e
( )
(3.90)
Wi h some simple algeb a, i can be shown ha , o , condi ion ( )
can be w i en as . Then, i is clea ha , o a ixed poin sa is ying (3.89), we
ha e ( ) . Thus, he equilib ium is a epelle .
A new egion is now de ined as minus a ci cle o in ini esimal adius a ound
he equilib ium poin . F om he abo e conside a ions –all ajec o ies en e and
he ixed poin is a epelle –, i is clea ha he low on he bounda y o is
di ec ed inwa ds, as depic ed in Fig. 3.14.
In summa y, a closed, bounded egion o he phase plane has been ob ained,
which con ains no ixed poin s and such ha all ajec o ies o he sys em en e
and emain inside o e e . Then, all condi ions o he P-B heo em a e ul illed, and
i can be assu ed ha any ajec o y o he sys em is a ac ed owa ds a closed o bi
as , i i is no a closed o bi i sel .
3.6 Condi ions o he Sys em o be Always A ac ed by a Limi Cycle 65
Fig. 3.13 Schema ic iew o he o que cu es co esponding o condi ions (3.89)
Finally, i should be no ed ha , al hough he P-B heo em does no gua an ee ha
all ajec o ies end o he same closed o bi , all he nume ical expe imen s
conduc ed wi hin his hesis show he p esence o only one s able limi cycle,
namely ha c ea ed by he Hop bi u ca ion. This sugges s ha , o a sys em
e i ying (3.89), all he sys em dynamics is a ac ed owa ds a unique limi cycle.
-8 -6 -4 -2 0 2 4
0
0.1
0.2
0.3
0.4
0.5
T
T
d
H
d

66 3 The Case o La ge Slope: Analy ical App oach
Fig. 3.14 Flow on he bounda y o egion (dashed), unde condi ions (3.89)
3.7 Discussion
Time Validi y
A c ucial poin in any pe u ba ion analysis is he ime scale o which he ob ained
app oxima e solu ion is alid. I has been shown in Sec ion 3.3 ha he solu ion
gi en by he educed sys em is alid, a leas , o a ime scale (
⁄) –see Fig.
3.6–.
Howe e , he si ua ion is e en be e han ha . As desc ibed in Sec ion 2.1
(A e aging wi h A ac ion), he asymp o ic app oxima ions a ained h ough
a e aging a e alid o all ime, whene e hey a e a ac ed by a s able ixed poin
o a s able limi cycle. In he la e case, he uni o m alidi y holds o all a iables
excep he angula one, i.e. he a iable which measu es he low on he limi cycle.
As will be seen la e , all he nume ical solu ions ob ained in Chap e 4 ul ill he
abo e condi ion o a ac ion.
Region 𝑄
Fixed
Poin
101
.
a
x
y
3.7 Discussion 67
Compa ison wi h o he au ho s’ esul s
In his subsec ion, he p esen ed app oach and esul s a e compa ed o some
p oposed by o he au ho s.
Fi s o all, as a as he au ho s know, he e has been no a emp in he li e a u e o
use he SPT o he analysis o nonideally exci ed sys ems. Thus, he analy ical
p ocedu e add essed in his Chap e appea s o be a no el app oach o he p oblem.
On he o he hand, he possibili y o a Hop bi u ca ion on he igh b anch o he
ib a ion o que cu e (Fig. 3.9b.2) has been add essed. An impo an implica ion
o his esul is ha he s abili y o he s a iona y solu ions nea esonance does no
only depends on he compa ison be ween he slopes o he wo o que cu es
( ), as commonly s a ed in he li e a u e (Blekhman, 2000; Dimen be g e al.,
1997; Kononenko, 1969; Nay eh & Mook, 1995). Le us y o explain his
di e gence in he esul s.
Kononenko’s book (Kononenko, 1969) is one o he mos ele an e e ences in he
subjec . He conside ed se e al linea and nonlinea sys ems exci ed by nonideal
mo o s. By using he a e aging me hod, he was able o analy ically in es iga e he
s a iona y mo ions o he mo o and hei s abili y. His app oach was as ollows.
Conside ing he o o speed o be in he icini y o esonance, he expanded i as
󰇗
(3.91)
Thus, he ound equa ions o mo ion o he o m
{
󰇗 ( )
󰇗 ( )
󰇗 ( )
󰇗
}
(3.92)
74 4 The Case o La ge Slope: Nume ical Simula ions
Fig. 4.2 Phase po ai s co esponding o pa ame e s (4.1). The ixed poin s a e
ma ked wi h do s. The dashed loop ep esen s he uns able limi cycle
(a) , (b)
The dynamical mechanism whe eby he limi cycle is des oyed, which u ns ou o
be a homoclinic bi u ca ion (Kuzne so , 1998), is shown in Fig. 4.2 and Fig. 4.3.
Le us ollow he e olu ion o he phase po ai . F om Fig. 4.2 (a) o Fig. 4.2 (b),
he Hop bi u ca ion akes place: he ocus becomes s able, while an uns able limi
cycle is bo n a ound i . In Fig. 4.3(a), he cycle has swelled conside ably and passes
close o saddle poin . The homoclinic bi u ca ion occu s when he cycle ouches
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
-0.8
-0.6
-0.4
-0.2
0
0.2
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
-0.8
-0.6
-0.4
-0.2
0
0.2
x
y
x
y
(a)
(b)

4.1 Global Bi u ca ions o he Limi Cycles 75
he saddle poin ( ), becoming a homoclinic o bi . In Fig. 4.3(b), we ha e
and he loop has been des oyed.
Fig. 4.3 Phase po ai s co esponding o pa ame e s (4.1). The ixed poin s a e
ma ked wi h do s. The dashed loop ep esen s he uns able limi cycle
(a) , (b)
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
-0.8
-0.6
-0.4
-0.2
0
0.2
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
-0.8
-0.6
-0.4
-0.2
0
0.2
x
y
x
y
(a)
(b)
S
76 4 The Case o La ge Slope: Nume ical Simula ions
I is wo h no ing ha , when he uns able limi cycle exis s –namely, o
–, i ac s as a on ie be ween he domains o a ac ion o he wo s able
equilib ium poin s o he sys em –see Fig. 4.2(b) and Fig. 4.3(a)–.
Many o he cases exhibi ing a subc i ical bi u ca ion, which a e no shown he e,
ha e also been nume ically sol ed. In all o hem, he uns able limi cycle has been
ound o disappea h ough a homoclinic bi u ca ion.
The Supe c i ical Case
Conside he ollowing se o dimensionless pa ame e s:
(4.5)
which migh be associa ed o dimensional pa ame e s
{
⁄
⁄
⁄
}
(4.6)
wi h . Equa ions (3.66) and (3.86) yield he alues o slopes and ,
depic ed in Fig. 4.4.
(4.7)
C i e ion (3.87) allows cha ac e izing he bi u ca ion as supe c i ical. Then, as
ep esen ed in Fig. 3.10, i can be assu ed ha a s able limi cycle enci cles he
uns able equilib ium o , wi hin a ce ain neighbo hood o . As a ma e
o ac , he esul s o Sec ion 3.6 can be used he e o in es iga e he ange o slopes
o which he limi cycle exis s.
4.1 Global Bi u ca ions o he Limi Cycles 77
Conside he cu e which in e sec s a he ixed poin unde s udy and is
angen o cu e a ano he poin . Le s and o he slope o ha pa icula
o que cu e, as displayed in Fig. 4.4. Then, i is s aigh o wa d o show ha , o
, condi ions (3.89) a e ul illed and, consequen ly, i can be assu ed
ha all sys em ajec o ies end o a pe iodic o bi . In he case unde analysis, we
ha e
(4.8)
Fig. 4.4 To que cu es co esponding o pa ame e s (4.5)
No e ha he Poinca é-Bendixson Theo em gi es su icien , bu no necessa y,
condi ions o he exis ence o a s able pe iodic o bi . Thus, i canno be deduced
om he Theo em whe he he limi cycle su i es o no when . To he end
o answe ing his ques ion, we eso again o a nume ical esolu ion o sys em
(3.46), o inc easing alues o . The esul s a e displayed in Fig. 4.5 and Fig. 4.6.
-6 -4 -2 0 2
0
0.2
0.4
0.6
0.8
1

T
T
H
d
T
d
P
d
78 4 The Case o La ge Slope: Nume ical Simula ions
Fig. 4.5 Phase po ai s co esponding o pa ame e s (4.5). The ixed poin s a e
ma ked wi h do s. The solid loop ep esen s he s able limi cycle
( ) , (b)
Le us ack he e olu ion o he phase po ai . In Fig. 4.5(a) we ha e and
all sys em ajec o ies a e a ac ed owa ds he only ixed poin o he sys em. I
may seem om Fig. 4.5(a) ha ajec o ies a e ac ually a ac ed owa ds a limi
cycle su ounding he ixed poin . The eason o his alse imp ession is ha he
a ac ion o he ixed poin is e y weak, as i is close o becoming uns able ( is
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y
x
y
(a)
(b)
4.1 Global Bi u ca ions o he Limi Cycles 79
close o ). Hence he equi ed ime o ajec o ies o app oach he equilib ium is
ex emely long.
Fig. 4.6 Phase po ai s co esponding o pa ame e s (4.5), o . The
ixed poin s a e ma ked wi h do s. The solid loop ep esen s he s able limi cycle
Fig. 4.5b co esponds o . The Hop bi u ca ion has occu ed and,
he e o e, he ocus has los i s s abili y a he same ime ha a s able limi cycle has
appea ed a ound i . No e ha , in Fig. 4.5(b), condi ions (3.89) hold. Consequen ly,
all sys em ajec o ies a e a ac ed owa ds a pe iodic o bi . Ac ually, Fig. 4.5b can
be obse ed as a pa icula example o he gene al pic u e shown in Fig. 3.14.
The nume ical esul s men ioned abo e a e only use ul o con i m he analy ical
de elopmen s o p e ious sec ions. By con as , Fig. 4.6 does p o ide new
in o ma ion abou he global dynamics o he sys em. I shows ha he s able limi
cycle is des oyed h ough a saddle-node homoclinic bi u ca ion (Kuzne so , 1998),
which occu s a . This means ha he cycle disappea s exac ly when
condi ions (3.89) a e no ul illed anymo e. The mechanism is as ollows. A
a new ixed poin , which immedia ely spli s in o a saddle and a node, is
c ea ed h ough a saddle-node bi u ca ion. This new equilib ium appea s p ecisely
on he limi cycle, ans o ming i in o a homoclinic o bi . Wha is ound a ,
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
x
y

80 4 The Case o La ge Slope: Nume ical Simula ions
as obse ed in Fig. 4.6, is ha he limi cycle has been eplaced by a couple o
he e oclinic o bi s connec ing he saddle and he node.
I has been shown ha , o he pa icula se o pa ame e s (4.5), condi ions (3.89)
a e necessa y and su icien o he exis ence o a s able limi cycle. Thus, he
pe iodic o bi ne e coexis s wi h any o he a ac o o he sys em. Ne e heless, i
should be s essed ha his is no always he case. In ac , cases ha e also been
ound whe e he s able limi cycle is des oyed h ough a homoclinic bi u ca ion,
jus like in he subc i ical case. In hese si ua ions, he global bi u ca ion occu s a
ce ain slope and, he e o e, he limi cycle coexis s wi h a s able
equilib ium o .
As an example, conside a case wi h sa is ying . Clea ly,
acco ding o (3.87), he Hop bi u ca ion is supe c i ical. Howe e , i is no possible
o he limi cycle o be des oyed h ough a saddle-node homoclinic bi u ca ion,
because he saddle and he node a e c ea ed be o e he limi cycle. In ac , in hese
cases, he closed o bi has been ound o die in he same way as he uns able limi
cycle shown in Fig. 4.3, i.e. h ough a homoclinic bi u ca ion due o he p esence o
a saddle poin .
In summa y, he simula ions ca ied ou sugges ha , while uns able limi cycles a e
des oyed by homoclinic bi u ca ions, he s able ones can disappea ei he h ough
homoclinic bi u ca ions o saddle-node homoclinic bi u ca ions.
4.2 Nume ical Valida ion o Analy ical Resul s
A Subc i ical Case
Conside again he se o pa ame e s gi en a (4.1), which gi es ise o a subc i ical
Hop bi u ca ion, as depic ed in Fig. 4.2 and Fig. 4.3. Two di e en scena ios a e
s udied, co esponding o he ollowing slopes o he mo o cha ac e is ic:
4.2 Nume ical Valida ion o Analy ical Resul s 81
(4.9)
By compa ing (4.9) wi h Fig. 4.2 and Fig. 4.3, i can be e i ied ha , o ,
he sys em has a s able ocus su ounded by an uns able limi cycle, while, a
, he ocus has become uns able h ough a Hop bi u ca ion. As poin ed ou
in Sec ion 4.1, he uns able limi cycle o is he bounda y which sepa a es
he basins o a ac ion o he wo a ac ing ixed poin s p esen in he sys em–see
Fig. 4.3(a)–.
Fo , wo se s o ini ial condi ions, I.C. (1) and I.C. (2), a e selec ed, ou side
and inside he limi cycle, espec i ely:
( ){
} ( ){
}
(4.10)
Then, by using ela ions (3.47), co esponding ini ial condi ions o he o iginal
sys em can be compu ed:
( )
{
󰇗
󰇗
}
( )
{
󰇗
󰇗
}
(4.11)
No e ha his s ep has no a unique solu ion, because di e en se s o o iginal ini ial
condi ions can p oduce he same educed ini ial condi ions.
The ob ained nume ical solu ions a e shown in Fig. 4.7, o . A good
ag eemen be ween solu ions o bo h sys ems is obse ed. Clea ly, he wo
conside ed se s o ini ial condi ions lead he sys em o di e en a ac o s.
82 4 The Case o La ge Slope: Nume ical Simula ions
Fig. 4.7 Compa ison o nume ical solu ions o he o iginal (solid line) and educed
(dashed line) sys ems o pa ame e s (4.1), and
(a) Displacemen s
(b) Ro o Speed
I is con enien o make he e an obse a ion abou he size o pa ame e . The
p ocedu e used in Chap e 3 o ans o m he o iginal sys em in o a simple educed
sys em is based on pe u ba ion me hods. These echniques a e use ul o dynamical
sys ems which con ain a small pa ame e , and hey explain how such sys ems
beha e o a su icien ly small . This means ha he smalle is, he mo e accu a e
0 0.5 1 1.5 2 2.5 3 3.5 4
x 104
-1
-0.5
0
0.5
1

u, a
0 1 2 3 4
x 104
0.998
1
1.002
1.004


I.C. (2)
I.C. (1)
I.C. (2)
I.C. (1)
(a)
(b)
4.2 Nume ical Valida ion o Analy ical Resul s 83
pe u ba ion p edic ions a e. Fig. 4.7 shows ha , o he case unde conside a ion, a
alue o gi es a ema kable acco dance be ween solu ions o he o iginal
and educed sys em. As an illus a i e example, he same nume ical compu a ion is
done, o ini ial condi ions I.C. (2) and . This la ge gi es ise o a less
accu a e p edic ion, as displayed in Fig. 4.8. The equi ed o ha e an accu a e
esul depends on he case unde s udy. Fo ins ance, in he ollowing simula ion
(Fig. 4.9), i was necessa y o ake o a good ma ching be ween solu ions
o he exac and app oxima e sys ems. Howe e , in he majo i y o simula ions
conduc ed wi hin his wo k, p o ed o be small enough.
Conside now he case whe e, acco ding o Fig. 4.2(a), he ocus is uns able
and he e is a unique a ac ing ixed poin in he sys em. Ini ial condi ions
( ){
}
(4.12)
a e selec ed o he educed sys em, om which co esponding ini ial condi ions o
he o iginal sys em can be ob ained:
( )
{
󰇗
󰇗
}
(4.13)
The o iginal and educed sys ems a e nume ically sol ed wi h and ini ial
condi ions (4.13) and (4.12), espec i ely. The esul s a e displayed in Fig. 4.9,
whe e i is clea ly obse ed how he sys em mo es away om he uns able ocus, as
he oscilla ion ampli ude inc eases, un il i is a ac ed o he s able node.
I.C. (1)
90 5 The Case o Small Slope: Analy ical App oach
o que a esonance–, and eplacing assump ion (3.9) –la ge slope o he mo o
cha ac e is ic– wi h
(5.1)
The assump ion is kep wi hin his chap e . Mo eo e , we assume ,
wi h de ined in (3.5).
Wi h hese new assump ions, he dimensionless equa ions o mo ion a e
󰇘 [ 󰇗 ( 󰇗 󰇘 )]
󰇘 [ ( 󰇗 ) 󰇘 ]
(5.2)
whe e subsc ip ‘0’ has been d opped o con enience.
I is use ul o ans o m sys em (5.2), acco ding o change o a iables
{ ( )
󰇗 ( )}
(5.3)
and de ine a new a iable o he o o speed:
󰇗
(5.4)
No ice ha he p ocedu e ollowed in Chap e 3 is being epea ed: a change o pola
coo dina es is pe o med by eplacing he pai o a iables * ( ) 󰇗( )+ wi h he
pai o ampli ude–phase a iables * ( ) ( )+. Thus, he in e media e s eps can be
skipped, since hey a e exac ly he same as in Chap e 3. The sys em, w i en in he
new a iables, akes he o m

5 The Case o Small Slope: Analy ical App oach 91
{
󰇗 ( ) ( ) ( )
󰇗 , ( ) ( )- ( )
󰇗 ( )
( ) ( )
󰇗
}
(5.5)
whe e
( ) ( )
(5.6)
Equa ions (5.5) and (5.6) a e analogous o (3.26) and (3.27).
A di ec inspec ion o sys em (5.5) e eals ha i con ains wo non-angula eal
a iables * + which a e slow – hey e ol e wi h a e ( )– and wo angula
a iables * + which a e, in p inciple, as – hey e ol e wi h a e ( ) unless
o –. Hence his is a sui able scena io o a e aging o e he as
angles. Howe e , in o de o a e age o e se e al angles, he sys em needs o be
w i en in he o m (2.5), (2.6), as explained in Sec ion 2.1. To his end, new
angula a iables a e de ined:
(5.7)
Then, by expanding he p oduc s o sines and cosines in (5.5), he sys em can be
w i en as
{
󰇗 [ ( ) ( ( ) ( ))] ( )
󰇗 0 ( ) 1 ( )
󰇗 ( )
󰇗 ( )
󰇗 ( )
}
(5.8)
92 5 The Case o Small Slope: Analy ical App oach
Now, assume a posi i e o o speed, . Then, he only esonance mani old
p esen in sys em (5.8) is gi en by condi ion
(5.9)
As explained in Sec ion 2.1, i is necessa y o dis inguish be ween wo scena ios,
depending on whe he o no he sys em is close o he esonance mani old.
5.1 Ou e Region
Suppose he o o speed is away om . Then, we can a e age sys em (5.8) o e
he h ee as angles , and . The esul ing a e aged sys em is
{ 󰇗
󰇗 ( )}
(5.10)
whe e
( ) ( )
(5.11)
Acco ding o he a e aging heo em s a ed in Sec ion 2.1, sys em (5.10) is alid on
a ime scale (
⁄), wi h ( ) p ecision.
A s aigh o wa d analysis o sys em (5.10) yields he conclusion ha i has one
only ixed poin , gi en by
{
( ) } 8
 9
(5.12)
which is globally asymp o ically s able as long as .
5.1 Ou e Region 93
No e ha , acco ding o assump ions , he equilib ium poin (5.12)
co esponds o a pos - esonan egime, . This solu ion has a e y clea
physical in e p e a ion. Fi s , no e by compa ing (5.11) o (3.3) ha unc ion ( )
is simply a dimensionless e sion o he mo o cha ac e is ic ( 󰇗):
( ) ( )
(5.13)
Clea ly,  is he only ze o o ( ), as ep esen ed in Fig. 5.1. Then, he ou e
ixed poin (5.14) co esponds o a pos - esonan mo ion whe e he oscilla ion
ampli ude is ze o and he o o speed akes he alue which makes he mo o o que
anish. No e ha his holds o he a e aged sys em (5.10), whose solu ions a e a
an ( ) dis ance o hose o he o iginal sys em (5.2). Then, ega ding he o iginal
sys em, i can be said ha he ou e ixed poin (5.14) ep esen s a pos - esonan
mo ion wi h small oscilla ion ampli udes ( ) and wi h he o o speed close o
he ze o o he mo o cha ac e is ic  ( ).
Fig. 5.1 Dimensionless mo o cha ac e is ic ( )
In physical e ms, i is consis en ha a non- esonan exci a ion p oduces a small
oscilla ion o he ib a ing sys em. No e ha , o such small oscilla ions, he o que


()
m
H
1
c
1
d
94 5 The Case o Small Slope: Analy ical App oach
on he o o due o ib a ion is e y small ( 󰇘 ( )). Then, du ing his
pos - esonan mo ion, he mo o does no ha e o p o ide any signi ican o que o
main ain he sys em ib a ion and, consequen ly, he o o speed akes such a alue
ha he d i ing o que is i ually ze o:  ( ) , ( )-
( ).
Two di e en scena ios can be conside ed:
- I ( ) , sys em (5.10) is exponen ially a ac ed owa ds equilib ium
(5.12) wi hou app oaching he esonance mani old. Then, acco ding o
Sec ion 2.1 (A e aging wi h A ac ion), he ou e a e aged sys em is alid
o all .
- I ( ) , sys em (5.10) is also exponen ially a ac ed by equilib ium
(5.12). Howe e , in i s way owa ds he equilib ium, he sys em will
necessa ily each he icini y o he esonance mani old, making sys em
(5.10) no longe alid. The e a e, in p inciple, wo op ions:
o The sys em emains close o he esonance mani old o all
subsequen ime ( esonan cap u e).
o The sys em s ays nea he esonance mani old o some ini e ime,
a e which i con inues i s e olu ion owa ds ixed poin (5.12)
(passage h ough esonance).
The nex Sec ion in es iga es he dynamics o (5.5) close o he esonance mani old.
5.2 Inne Region
In o de o s udy he sys em beha io in he icini y o he esonance mani old, he
o o speed is expanded as
√
(5.14)
5.2 Inne Region 95
No e ha he de ini ion o he de uning a iable is no he same as in he case o
la ge slope –compa e (5.14) o (3.35)–. The eason is ha , unde he assump ion o
small slope made in his chap e , he p oblem equi es a di e en pe u ba ion
app oach, which in u n equi es a di e en scaling o he o o speed. I can be
checked ha a scaling such as (3.35) would no yield any ele an esul in he
p esen case.
Replacing (5.14) in (5.5), (5.6) yields
{
󰇗 ( ) ( ) ( √ )
󰇗 √ ( )
( ) ( √ )
󰇗 √ , ( )- ( √ )
󰇗 √
}
(5.15)
wi h
( ) ( )
(5.16)
Clea ly, sys em (5.15) con ains h ee slow a iables * + and a as o a ing
phase . I is, hen, sui able o a second o de a e aging p ocedu e. Following he
p ocedu e desc ibed in Sec ion 2.2, we a i e a a e aged sys em
{
󰇗 (  )
󰇗 √  4  
 5
󰇗 √ 0  1 
󰇗 √ 
}
(5.17)
whe e, wi h an app op ia e ela ion be ween he ini ial condi ions o he o iginal
and a e aged sys ems, he e o es ima es a e

96 5 The Case o Small Slope: Analy ical App oach
{
 ( )
 ( )
 √  (  ) ( )
 (√ )
}
(
√ )
(5.18)
No e ha he (√ ) e ms in he i s wo o ela ions (5.18) u n ou o be ze o in
his pa icula case.
Despi e he ac ha he e olu ion {   } is independen o  –as is e iden , since
his is p ecisely he pu pose o a e aging–, sys em (5.17) includes  as a s a e
a iable. The eason is ha a iable ( ) is necessa y o cons uc he e o
es ima es in (5.18). Howe e , in o de o in es iga e he dynamics o he a e aged
sys em, i is con enien o ew i e i wi hou he as angle:
{
󰇗 (  )
󰇗 √  4  
 5
󰇗 √ 0  1 
}
(5.19)
A di ec analysis o (5.19) allows deducing ha , i
⁄, he e a e no ixed
poin s in he inne egion. On he o he hand, i
⁄, sys em (5.19) exhibi s wo
ixed poin s, gi en by
{ √ ( )
√ ( )
√ ( )}
(5.20)
wi h
5.2 Inne Region 97
{
√
(
)
}
(5.21)
whe e √ . The wo possible alues o co espond o he wo di e en
equilib ium poin s.
No e ha , unlike in he case o la ge slope –see (3.52)–, he analy ical exp essions
o he ixed poin s a e e y simple in he p esen scena io. Howe e , he o que-
speed plo s used in Sec ion 3.4 may also be illus a i e he e and will p o ide an
in e es ing compa ison wi h he case o la ge slope.
Thus, conside he equilib ium condi ion applied o a iable . F om he hi d o
equa ions (5.19) we ha e, a i s o de ,
(5.22)
On he o he hand, condi ion 󰇗 yields
(5.23)
which allows w i ing (5.22) as
(5.24)
Equa ion (5.24) can be clea ly in e p e ed as a o que equilib ium condi ion:
( )
(5.25)
wi h
98 5 The Case o Small Slope: Analy ical App oach
( )
(5.26)
ep esen s he d i ing o que p oduced by he mo o and co esponds o he
esis ing o que due o ib a ion. In o de o ob ain he usual o que-speed plo , we
would need o w i e and in e ms o . Ne e heless, his would in u n
equi e w i ing in e ms o , and hen subs i u e in (5.25). Since his
yields e y long and cumbe some exp essions, we eso o an al e na i e implici
p ocedu e o he g aphical ep esen a ion.
F om condi ion 󰇗 , we ha e
√
(5.27)
Then, we can w i e
( )
(5.28)
whe e unc ion ( ) is de ined as
( ) √
(5.29)
The p oposed ep esen a ion can be cons uc ed as ollows. Fi s , g aph e sus
(in his case, as , a cons an unc ion is ob ained). Then, ep esen he
pa ame ic cu e gi en by * ( ) ( )+ o and ( -. The ac ha
is s ic ly posi i e comes om i s de ini ion as he adius o a pola coo dina e
ans o ma ion –see (5.3)–, while condi ion (5.23) o bids o be g ea e han .
This p ocedu e gi es ise o a plo like ha shown in Fig. 5.2.
5.2 Inne Region 99
Fig. 5.2 Fixed poin s o sys em (5.19)
A di ec compa ison be ween Fig. 5.2 and Fig. 3.7 is qui e illus a i e as o he
di e ence be ween he wo scena ios conside ed in his hesis. In he case o la ge
(small) slope, he d i ing o que cu e exhibi s a slope which is compa able
(negligible) wi h espec o ha o he ib a ion o que cu e.
Fig. 5.2 shows he exis ence o wo ixed poin s, as long as
⁄. The s abili y
o hese equilib ia is now in es iga ed. To his end, he jacobian ma ix o sys em
(5.19) needs o be ob ained and e alua ed a he equilib ium poin o in e es :
√ ( √ )
(5.30)
wi h
[
] [
]
(5.31)
The eigen alues o ma ix a e now compu ed. A e some algeb a, we ind, o
he igh b anch o he o que cu e ( ),
-2 0 2 4 6 8 10 12
0
0.5
1
1.5
2
2.5
1z

T
T
1z
m
T
c
2

106 5 The Case o Small Slope: Analy ical App oach
whe e is a eal ma ix whose columns con ain he eal eigen ec o s o he
jacobian o (5.45), e alua ed a . In he case o complex conjuga e
eigen ec o s, he eal and imagina y pa s a e s o ed in di e en columns o .
Clea ly, change o a iables is composed o wo ans o ma ions. Fi s , he
coo dina e sys em is ansla ed so ha he o igin coincides wi h he ixed poin .
Then, a ans o ma ion o he eal eigenbasis o he jacobian is pe o med, by means
o ma ix .
Vec o con ains he s a e a iables o he sys em, exp essed wi h espec o he
basis o med by he eigen ec o s o he jacobian. Then, i is clea ha hese
eigen ec o s a e necessa ily o hogonal in he space o coo dina es .
Consequen ly, o wo solu ions s a ing in he Poinca é-Lyapuno domain, we ha e
‖ ( ) ( )‖ ‖ ( ) ( )‖
(5.53)
By conside ing a ime inc emen
⁄ in (5.53), we ha e
‖ ‖ ‖ ( ) ( )‖
(5.54)
whe e
( )
(5.55)
Thus, i has been shown ha ( ) exhibi s exponen ial con ac ion on he ime scale
⁄, e en hough his con ac ion is weak.
Now, a iable ( ) –solu ion o he o iginal sys em (5.35)– can also be ans o med
acco ding o (5.52):
(5.56)

5.2 Inne Region 107
Conside he ollowing pa i ion o ime in in e als o (
⁄):
[
] [
] [
] ( )
(5.57)
whe e cons an s a e chosen such ha
4 (
) (
)5
(5.58)
No e ha his choice o cons an s is always possible, because anishes a leas
once pe pe iod, acco ding o (5.44).
We de ine
(
) ( ) ( ( ) )
(5.59)
(
) ( ) ( ( ) )
(5.60)
( ) (
) (
)
(5.61)
Thus, ( ) ep esen s he alue o a he end o an in e al when, as an ini ial
condi ion, is imposed o be equal o a he beginning o he in e al.
Fi s , we ha e ha
‖ ( )‖
(5.62)
as is clea om (5.37),(5.49) and (5.56), using ha a he conside ed
ins an s.
108 5 The Case o Small Slope: Analy ical App oach
On he o he hand, by i ue o ela ion (5.54), we can w i e
‖ ( )‖ ‖ ‖
(5.63)
Combining (5.62) and (5.63), we ha e
‖ ‖ ‖ ‖
(5.64)
By using (5.64) ecu si ely, we a i e a
‖ ‖ ( ) ‖ ‖
(5.65)
No e ha , acco ding o (5.37), (5.49) and (5.56), we can w i e
‖ ‖
(5.66)
hanks o he ac ha ( (
⁄) (
⁄)) .
Finally, in oducing (5.66) in o (5.65) and aking he limi o yields
‖ ‖
(5.67)
whe e (5.55) has been used.
Al hough (5.67) only holds, in p inciple, o he pa icula ins an s
⁄, i can be
eadily gene alized o any . No e ha , as s a ed in (5.44), any is a ( )–
dis ance om an ins an whe e . Clea ly, could be aken as
⁄ and,
he e o e, (5.67) holds a . On he o he hand, and can only unde go ( )
a ia ions in an ( ) ime inc emen , which jus i ies he gene aliza ion o (5.67) o
any . Then, o , we can w i e
5.2 Inne Region 109
‖ ( ) ( )‖
(5.68)
Reco e ing he o iginal a iables, we ha e
‖ ( ) ( )‖ ‖ ‖‖( ( ) ( ))‖ , )
(5.69)
Finally, in oducing (5.68) in o (5.69) yields
‖ ( ) ( )‖ , )
(5.70)
whe e
‖ ‖
(5.71)
The conclusion is ha , o ini ial condi ions close enough o he conside ed
equilib ium, he solu ion o he a e aged sys em is a an ( ) dis ance om he
solu ion o he o iginal sys em, o all . Then, i he equilib ium is
asymp o ically s able in he a e aged sys em, i is asymp o ically s able as well o
he o iginal sys em.
Now, he ob ained esul is pa icula ized o he case o he mo o wi h small slope
cha ac e is ic. Equa ion (5.70) akes he o m
{  (√ )
 (√ )
 (√ )} , )
(5.72)
110 5 The Case o Small Slope: Analy ical App oach
o solu ions s a ing close enough o he ixed poin (5.20), (5.21), wi h .
By compa ing (5.70) wi h (5.18), i is clea ha he ime alidi y o he
app oxima ion has been ex ended, paying he p ice o a less accu a e solu ion.
Final Rema ks
In summa y, he sys em has been ound o exhibi wo equilib ium poin s in he
esonance egion as long as
⁄. Fig. 5.2 ep esen s bo h equilib ia on a
o que-speed plo , whe e he ixed poin on he igh b anch is uns able and he one
on he le b anch is s able. No e ha he exis ence o a s able ixed poin in he
inne egion jus i ies he possibili y o ‘ esonance cap u e’.
Recall ha he sys em eaches he esonance mani old whene e ( ) . Fo
some se s o ini ial condi ions, he ajec o y will en e he Poinca é-Lyapuno
domain o he s able ixed poin and, he e o e, i will emain nea esonance o all
subsequen ime – esonan cap u e–. Clea ly, he e may also be se s o ini ial
condi ions such ha he ajec o y does no each he Poinca é-Lyapuno domain o
he s able ixed poin . In hese cases, he sys em will p obably lea e he inne egion
and e ol e owa ds ixed poin (5.12) in he ou e egion –passing h ough
esonance–.
In p inciple, i would also be possible ha he sys em was a ac ed by a di e en
objec in he inne egion, such as a s able limi cycle o a chao ic a ac o . This
would ep esen ano he kind o esonance cap u e, no due o he p esence o he
s able ixed poin analysed in his sec ion. Howe e , he nume ical simula ions
ca ied ou ha e no e ealed he exis ence in he inne egion o any a ac o o he
han he analysed ixed poin .
No e also ha his chap e has coped wi h he inne and ou e app oxima ions
sepa a ely. We ha e no ied o cons uc a ‘composi e expansion’ by ma ching he
inne and ou e solu ions, which is a a he in ica e and complex subjec , ea ed,
o example, in (W. Eckhaus, 1979; Sande s e al., 2007).
5.2 Inne Region 111
Be o e showing nume ical esul s o con i m he analy ical de elopmen s o his
Chap e , i is con enien o commen some o he wo ks on he subjec .
Sande s, Ve huls and Mu dock conside ed he sys em s udied in his Sec ion as an
illus a i e example in Chap e s 7 and 8 o hei book (Sande s e al., 2007).
Rega ding he ou e egion o he phase space, hey conduc ed he same analysis as
in his Chap e , a e aging o e he h ee as angles in (5.8) and ob aining
equilib ium (5.12). Howe e , in he inne egion hey only ca ied ou a i s o de
a e aging in con as o he second o de a e aging add essed in his Chap e . This
p ocedu e did no allow hem o analyse he s abili y o he ixed poin s nea
esonance, since a i s o de a e aging is no accu a e enough o his pu pose.
In addi ion, Alexande Fidlin de o ed Chap e 5 o his book (Fidlin, 2006) o he
s udy o nonideal exci a ions, aking sys em (5.2) as a ele an example. He ocused
only on he esonance egion, a i ing a a sys em analogous o (5.19) a e a second
o de a e aging. Howe e , he e a e wo main di e ences be ween his esul s and
hose p esen ed in he p esen Chap e :
- Acco ding o he analysis p oposed in his hesis, he equilib ium poin o
is s able as long as . Howe e , Fidlin came o he conclusion
ha he condi ion o s abili y is . The eason o his di e ence is
a small e a um in he eigen alues compu a ion in (Fidlin, 2006), namely in
he s ep om equa ion (5.21) o (5.22) o he men ioned e e ence: whe e i
eads , i should ead .
- Fidlin add essed he sho ime scale whe e he inne a e aged sys em is
alid, ( √
⁄), as an impo an limi a ion o he analysis. He
p oposed a hie a chic a e aging p ocedu e as a way o enla ge he ime o
alidi y o he app oxima ion (Pechene , 1992). Howe e , his me hod ails
p ecisely in he icini y o he equilib ium poin o in e es , because he
equi ed a iable ans o ma ion becomes singula a ha poin . The e o e,
he hie a chic a e aging scheme canno be used o jus i y he asymp o ic
s abili y o he ixed poin .

112 5 The Case o Small Slope: Analy ical App oach
In conclusion, he chie con ibu ion o his Chap e wi h espec o p e ious
published wo ks is he igo ous jus i ica ion o he asymp o ic s abili y o one o he
s a iona y mo ions o he sys em nea esonance, which in u n gi es a solid
explana ion o he possibili y o esonan cap u e, o ‘locking in o esonance’.
6 THE CASE OF SMALL
SLOPE OF THE MOTOR
CHARACTERISTIC:
NUMERICAL
SIMULATIONS
This Chap e is in ended o e i y he esul s o Chap e 5 by means o nume ical
simula ion. Following an analogous scheme o ha in Chap e 4, pa icula alues
a e assigned o he sys em pa ame e s and bo h he o iginal and app oxima e
sys ems a e nume ically sol ed in o de o compa e he ob ained solu ions.
Thus, conside he ollowing pa ame e s
114 6 The Case o Small Slope: Nume ical Simula ions
(6.1)
which migh be associa ed o dimensional pa ame e s
{
⁄
⁄
⁄
}
(6.2)
wi h . This se o pa ame e s gi es ise o he o que-speed cu es
depic ed in Fig. 6.1. No e ha condi ion
⁄ is ul illed, which implies ha
he e exis wo ixed poin s in he inne egion o he phase space, co esponding o
he wo in e sec ions be ween and in Fig. 6.1.
Fig. 6.1 To que-speed cu es co esponding o pa ame e s (6.1).
S and U label he s able and uns able ixed poin s, espec i ely.
-6 -4 -2 0 2 4 6 8
0.5
1
1.5
2

T
m
T
T
S
U
6 The Case o Small Slope: Nume ical Simula ions 115
As discussed in Chap e 5, he equilib ium on he le b anch o cu e is s able,
while he one on he igh b anch is uns able. The ixed poin s can be eadily
compu ed by in oducing (6.1) in o (5.21):
{
}
(6.3)
{
}
(6.4)
The hi d equilib ium, in he ou e egion o he phase space, can be ob ained by
in oducing (6.1) in o (5.12):
2
3
(6.5)
The simula ions ha e been ca ied ou as ollows. A se o ini ial condi ions o he
o iginal sys em (5.2) is chosen wi h ( ) , i.e. in he p e- esonan egion o he
phase space. Then, he o iginal sys em o equa ions is nume ically sol ed o a ime
in e al [ ] which is long enough o asce ain whe he he sys em is cap u ed o
passes h ough esonance.
Suppose he sys em passes h ough esonance. Looking a he nume ical solu ion o
he o iginal equa ions, wo pa icula ins an s, and a e de ined, a which he
sys em en e s and lea es he esonance egion, espec i ely. Al hough he choice o
hese wo alues is somewha a bi a y, hey gi e an app oxima ion o he limi s
be ween he inne and ou e solu ions. Then, he ou e app oxima e sys em (5.10) is
sol ed o , - and [ ], wi h he ini ial condi ions ob ained as he
solu ion o he o iginal sys em e alua ed a and , espec i ely. The
inne app oxima e sys em (5.17) is sol ed o , -, wi h he ini ial condi ions
co esponding o he solu ion o he o iginal equa ions pa icula ized a .
Finally, he solu ions o he o iginal, ou e and inne sys ems a e ep esen ed
122 6 The Case o Small Slope: Nume ical Simula ions
app oxima e equa ions. Once he sys em has o e come he esonance egion, i
e ol es owa ds he ou e s able equilib ium gi en by (6.5).
I is also illus a i e o conside a di e en si ua ion. Suppose pa ame e is
inc eased un il he e a e no ixed poin s in he a e aged sys em nea esonance. The
condi ion o mee is
⁄ . Then, conside he ollowing se o pa ame e
alues:
(6.8)
which is exac ly he same as (6.1) excep o a la ge d i ing o que a esonance.
The o que-speed g aph o his scena io, ob ained h ough ela ions (5.26) and
(5.29), is depic ed in Fig. 6.6, exhibi ing no in e sec ions be ween he cu es.
Fig. 6.6 To que-speed cu es co esponding o pa ame e s (6.8)
Clea ly, esonance cap u e canno occu in his si ua ion, unless an a ac o o he
han a ixed poin exis ed in he inne egion. As s a ed be o e, no nume ical
e idence o such an a ac o has been ound. The conclusion is ha he sys em will
pass h ough esonance o any p e- esonan ini ial condi ion and will lead owa ds
he ou e s able equilib ium, which is now gi en by
-6 -4 -2 0 2 4 6 8
0
0.5
1
1.5
2

T
m
T
T

6 The Case o Small Slope: Nume ical Simula ions 123
2
3
(6.9)
as can be ob ained by in oducing (6.8) in o (5.12):
Despi e he esonance being no ac i e –i.e. he e a e no a ac o s in he esonance
egion–, i can be expec ed ha ajec o ies a e somehow dis o ed when passing
h ough he esonance mani old. To he end o obse ing his e ec , wo di e en
simula ions ha e been conduc ed wi h di e en ini ial condi ions:
{
󰇗
󰇗
}
(6.10)
{
󰇗
󰇗
}
(6.11)
The esul s a e displayed in Fig. 6.7-Fig. 6.9. No e ha , o ini ial condi ions (6.10),
he e olu ion o he o o speed is nea ly una ec ed by esonance, while he e is a
signi ican e ec on he ib a ion ampli ude. I is in e es ing ha exac ly he
opposi e case is encoun e ed o ini ial condi ions (6.11): whe eas he s uc u e
ib a ion is almos unal e ed by esonance, he o o speed unde goes signi ican
oscilla ions when he sys em passes h ough he esonance mani old. Thus, i is
clea ha he in luence o esonance on he sys em beha iou depends on he ini ial
condi ions. In gene al, we can s a e ha some ansien esonan e ec s can be
expec ed in he sys em, e en when he e a e no a ac o s in he esonance egion.
124 6 The Case o Small Slope: Nume ical Simula ions
Fig. 6.7 Nume ical solu ions o pa ame e s (6.1), ini ial condi ions (6.10) and
. Solid, dashed and do ed lines co espond o he o iginal, inne and ou e
sys ems, espec i ely
a) Displacemen
b) Ro o speed
0500 1000 1500 2000 2500
-0.1
-0.05
0
0.05
0.1

u, a
0500 1000 1500 2000 2500
0
1
2
3


(a)
(b)
1

2

1

2

6 The Case o Small Slope: Nume ical Simula ions 125
Fig. 6.8 Nume ical solu ions o pa ame e s (6.1), ini ial condi ions (6.11) and
. Solid, dashed and do ed lines co espond o he o iginal, inne and ou e
sys ems, espec i ely
a) Displacemen
b) Ro o speed
0500 1000 1500 2000 2500
-2
0
2

u, a
0500 1000 1500 2000 2500
0
1
2
3


1

2

1

2

(a)
(b)
126 6 The Case o Small Slope: Nume ical Simula ions
Fig. 6.9 Close-up a ound esonance o nume ical solu ions o pa ame e s (6.1),
ini ial condi ions (6.11) and . Solid, dashed and do ed lines co espond o
he o iginal, inne and ou e sys ems, espec i ely
a) Displacemen
b) Ro o speed
150 250 350 450
0.5
1
1.5


150 250 350 450
-3
-2
-1
0
1
2
3

u, a
1

2

1

2

(a)
(b)
7 TORQUE-SPEED CURVES
FOR THE WHOLE
FREQUENCY RANGE
This Chap e se es as a connec o be ween he analyses p esen ed in Chap e s 3-6
and he s udy o he ib ocompac ion p ocess expounded in Chap e 8. Recall ha
o que-speed cu es ha e al eady been success ully used o ob ain he s a iona y
mo ions o he ib a ing unbalanced mo o in Chap e s 3-6. Howe e , in hese
p e ious app oaches, he o que-speed plo is only ep esen ed o he esonance
egion, like in Fig. 3.7 and Fig. 5.2, o o he non- esonan egion, like in Fig. 5.1.
This necessa y dis inc ion be ween esonan and non- esonan egions o he phase

128 7 To que-Speed Cu es o he Whole F equency Range
space is a di ec consequence o he pe u ba ion app oaches ha ha e been u ilized
in p e ious chap e s.
The objec i e is now o gene alize he use o hese cu es, so ha he ib a ion
o que and he mo o o que can be plo ed oge he in a single g aph o he whole
equency ange, he eby ep esen ing he esonan and non- esonan s a iona y
mo ions o he mo o . This will u n ou o be e y use ul in he nex chap e , as will
be seen, and will also p o ide a clea global pe spec i e o he p oblem s udied in
Chap e s 3-6.
7.1 Compu a ion o he To que-Speed Cu es
Conside again he mechanical sys em shown in Fig. 3.1, whose equa ions o
mo ion (3.1) a e ew i en below.
󰇘 󰇗 ( 󰇗 󰇘 )
󰇘 ( 󰇗) 󰇘
(7.1)
No e ha he cubic nonlinea i y is now assumed o be ze o o simplici y. As usual,
he mo o cha ac e is ic is assumed o be a linea unc ion o he o o speed:
( 󰇗) 󰇗
(7.2)
wi h .
We no e ha he analysis shown in wha ollows is based on Blekhman’s app oach
o di ec sepa a ion o mo ions (Blekhman, 2000). In o de o app oxima ely ob ain
he s a iona y mo ions o he sys em, i is easonable o look o solu ions whe e he
o o speed has he o m
7.1 Compu a ion o he To que-Speed Cu es 129
󰇗( ) ( )
(7.3)
wi h cons an and ( ) a pe iodic unc ion o ime wi h ze o a e age. I is also
assumed ha he solu ion sa is ies
{ ( )
󰇗 ( ) }
(7.4)
Condi ions (7.4) will be e i ied a e wa ds. In oducing (7.3) in o he i s o
equa ions (7.1) yields
󰇘 󰇗 [( ) 󰇗 ]
(7.5)
By aking (7.4) in o accoun , equa ion (7.5) can be app oxima ed as
󰇘 󰇗 ( )
(7.6)
whe e ( ) has been assumed o simplici y. No e ha , o i s app oxima ion,
he small oscilla ion o he o o speed ( ) does no a ec he sys em ib a ion.
I may seem om (7.6) ha he p oblem has been ende ed linea wi h he p oposed
app oxima ion. In ac , equa ion (7.6) ep esen s a ha monically o ced linea
oscilla o . Howe e , he sys em as a whole is s ill nonlinea , due o he nonideal
in e ac ion wi h he exci e . This can be seen by no icing ha cons an in (7.6) is
no known a p io i. Hence he linea mo ion ( ) needs o be sol ed as a unc ion o
. Then, he o que p oduced by his ib a ion will be in oduced in he o o
equilib ium equa ion –second o equa ions (7.1)–, which will allow ob aining .
The e o e, he e is s ill a wo-way coupling be ween ib a ion and o a ion.
The s a iona y solu ion o (7.6) is e y well-known om linea ib a ion heo y:
130 7 To que-Speed Cu es o he Whole F equency Range
( ) ( )
(7.7)
wi h
.
/
√6.
/ 7 0
1
(
.
/
)
(7.8)
whe e √
⁄, ( )⁄ . The ib a ion ampli ude is ep esen ed
agains he a e age o o speed in Fig. 7.1, acco ding o (7.8).
Fig. 7.1 Ampli ude o he s a iona y ib a ion e sus a e aged o o speed,
co esponding o equa ion (7.8)
Once he linea mo ion has been ob ained, i can be in oduced in he o o
equilib ium equa ion in o de o compu e he s a iona y o o speed. Fi s , he
p oposed solu ion o he o o speed (7.3) is eplaced in he second o equa ions
(7.1):
0
xmax
n

7.1 Compu a ion o he To que-Speed Cu es 131
󰇗 ( ) 󰇘 ( )
(7.9)
whe e assump ion (7.4) has been used. Then, in oducing solu ion (7.7) in o (7.9)
yields
󰇗 ( ) ( ) ( )
(7.10)
I is con enien o ew i e he las e m in (7.10) as he sum o i s mean alue and an
oscilla ing componen :
󰇗 ( )
( )
(7.11)
No e ha (7.11) con ains cons an and oscilla ing e ms wi h ze o a e age. Clea ly,
i equa ion (7.11) is a e aged, only he cons an e ms emain:
( )
(7.12)
Subs ac ing (7.12) o (7.11) yields
󰇗 ( )
(7.13)
Equa ion (7.12) can be in e p e ed as an equilib ium condi ions be ween he a e age
o ques ac ing on he o o du ing he s a iona y mo ion. Ac ually, by inse ing (7.8)
in o (7.12), he ollowing ela ion is ob ained:
( ) ( )
(7.14)
138 7 To que-Speed Cu es o he Whole F equency Range
Fig. 7.4 To que-speed cu es co esponding o a case o small slope o he mo o
cha ac e is ic
(a) Resonan cap u e can occu
(b) Resonan cap u e does no occu
To que
To que
m
L
L
To que
n



n

m
L
L
(a)
(b)

7. 2 A Global Pe spec i e o he Cases o La ge and Small Slope 139
The nea - esonan s a iona y mo ions in Fig. 7.4(a) co espond o he ixed poin s
shown in Fig. 5.2. As was widely discussed in Sec ion 5.2, he i s o hese wo
ixed poin s is asymp o ically s able, whe eas he second one is uns able. The hi d
s a iona y mo ion ep esen ed in Fig. 7.4(a), which is ou side he esonance egion,
co esponds o he ixed poin o he ou e a e aged sys em, depic ed in Fig. 5.1.
This solu ion was shown o be s able in Sec ion 5.1.
Then, o he case o small slope –assuming he mo o o que a esonance o be
smalle han he esonance peak o he ib a ion o que cu e–, wo s able s a iona y
beha iou s exis , one o hem in he esonance egion, and he o he being a away
om esonance. Fo any p e- esonan ini ial s a e, he sys em can be a ac ed by
ei he he nea - esonan o he pos - esonan s able s a iona y mo ions. These wo
scena ios a e e e ed o as esonan cap u e and passage h ough esonance,
espec i ely. In he simple case ep esen ed in Fig. 7.4(b), whe e no s a iona y
mo ions close o esonance exis , he sys em always passes h ough esonance, and
e ol es owa ds i s only a ac o , away om he esonance egion.
Hope ully, i has been shown ha mos o he conclusions abou he sys em
beha iou ob ained in p e ious chap e s can be summed up and easily e ained by
using he o que-speed cu es desc ibed in his chap e . Howe e , he igo ous
pe u ba ion app oaches o Chap e s 3 and 5 a e necessa y o assess he s abili y o
he s a iona y mo ions.
8 MODELLING AND
SIMULATION OF THE
VIBROCOMPACTION
PROCESS
This Chap e ocuses on he ib ocompac ion p ocess which has mo i a ed he
whole hesis. A e desc ibing he eal indus ial p ocedu e, a 4-DOF nume ical
model o he ib ocompac ion sys em is p esen ed. Al hough his model is sui able
o nume ical in es iga ion o he p ocess, i is s ill oo complex o an analy ical
ea men which may e eal mo e gene al in o ma ion abou he sys em dynamics.
Then, a second model wi h 2 DOFs is de i ed, h ough some easonable
simpli ica ions, which u ns ou o be e y use ul in o de o analyse he p ocess and
e en une he pa ame e s o he compac ing machine. Finally, nume ical
simula ions on he i s model ( ull model) and compa ison wi h he p edic ions o

142 8 Modelling and Simula ion o he Vib ocompac ion P ocess
he second (simpli ied model) illus a e hei abili y o p edic he e ec o di e en
pa ame e s on he inal le el o compac ion achie ed.
8.1 Some No es on he Real P ocess
Qua z agglome a es, made o g anula ed qua z mixed wi h a polyes e esin, a e
widely used as an a i icial s one o coun e ops in ki chens o ba h ooms. The
manu ac u ing p ocess o a slab o his ma e ial s a s wi h he illing o a mould
wi h he mix u e o qua z and esin. Once he mould is ull, a con eyo bel ca ies
i o he ib ocompac ion zone, whe e he hickness o he slab is educed o nea ly
hal o i s ini ial alue, by elimina ing he ai ou o he ma e ial. Then, he mix u e
is cu ed in a kiln, du ing a speci ied ime in e al, a a sui able empe a u e o he
polyme iza ion o he esin. A e he esin is polyme ized, an ai s eam is used o
cool he slab be o e i en e s he mechanical inishing s age. Du ing his p ocess he
edges a e cu , p oducing a slab o p esc ibed dimensions, and he su aces a e
polished. Then, he p oduc is eady o he quali y con ol s age.
I is wo h gi ing some mo e insigh in o he ib ocompac ion s age o he p ocess,
which is he one o in e es o he pu pose o his s udy. Be o e he mix u e has
been compac ed, i is composed o h ee di e en phases: solid ( he qua z g ains),
liquid ( he esin) and gas (ai ). The ai is p esen in he ma e ial in wo di e en
ways: as bubbles wi hin he esin o as gaps be ween g ains o qua z ha he esin
has no been able o ill. The aim o he compac ion p ocess is o elimina e he ai
ou o he mix u e, since he p esence o po es a he su ace o he inal coun e op
is clea ly de imen al om a p ac ical poin o iew: he po es end o accumula e
di and a e a he di icul o clean.
The compac ion is conduc ed by means o se e al unbalanced elec ic mo o s,
moun ed on a pis on wi h he dimensions o he slab su ace. A he beginning o
he ib ocompac ion p ocess, he pis on descends on o he mix u e and exe s a
s a ic p essu e, due o i s weigh and o an ai p essu e applied on i . Then, he ai
8.1 Some No es on he Real P ocess 143
p essu e inside he mould is educed by using a acuum sys em, a e which he
mo o s a e swi ched on. The ib a ion p oduced by he unbalanced mo o s is he
main esponsible o he compac ion. Du ing he mo ion o he sys em, he e can be
sepa a ions and impac s be ween he pis on and he slab, which a e gene ally
bene icial o he compac ion, as hey p oduce e y high peaks o comp ession
o ces. In o de o educe ib a ions in he icini y o he compac ion machine,
elas ic elemen s a e placed be ween he ounda ion o he machine and he g ound,
ac ing as a ib a ion abso be and hus p o ec ing nea by equipmen . Fig. 8.1 shows
a pilo plan used o es ing pu poses, which p ese es he main ea u es o he
ac ual indus ial machine. I is in e es ing o no e ha he e a e wo mo o s moun ed
on he pis on, which o a e in opposi e di ec ions in o de o cancel he ho izon al
componen s o he cen i ugal o ces on he unbalanced masses. Hence he ne
e ec o he o a ion o bo h mo o s is an oscilla ing e ical o ce.
Fig. 8.1 Pilo plan o he analysis o he ib ocompac ion p ocess
144 8 Modelling and Simula ion o he Vib ocompac ion P ocess
F om he abo e commen s, i is clea ha he ib ocompac ion p ocess is ex emely
complex om a physical poin o iew. A la ge numbe o ac o s –some o hem
being in insically nonlinea – in luence he inal esul o he compac ion:
- The qua z g anulome y, he heological p ope ies o he esin and he
mass a io be ween qua z and esin a ec he mechanical beha iou o he
compac ing mix u e. This beha iou is necessa ily nonlinea , since he
mix u e su e s i e e sible de o ma ion du ing compac ion. Mo eo e , an
accu a e desc ip ion o his cons i u i e law would equi e modelling he
mo ion o he bubbles h ough he mix u e, he ic ion be ween qua z
pa icles, he in e ac ion be ween qua z and esin, e c. Some in es iga ions
abou he beha iou hese ypes o h ee-phase mix u es can be ound in
(Alonso, Gens, & Josa, 1990; Pie uszczak & Pande, 1996; S ickel &
Powell, 2005).
- The dynamic p ope ies o he di e en elemen s o he machine – he
pis on, he con eyo bel suppo ing he mould, he elas ome be ween he
ounda ion and he g ound, e c. – may in luence he ib ocompac ion as
well.
- The speed o he mo o s, hei a ailable powe and he amoun o unbalance
a e key pa ame e s o he p ocess.
- The inal esul o he compac ion may also depend on he du a ion o he
p ocess.
- The spa ial dis ibu ion o he acuum channels in luences he ex ac ion o
he ai ou o he mix u e, he eby a ec ing he compac ion.
8.2 Full Model
Building a eliable model o such a complex manu ac u ing p ocess, able o
accu a ely p edic he esul o he compac ion depending on he sys em pa ame e s,
is an ex emely ha d ask, which clea ly exceeds he scope o his hesis. I should
be no ed ha , as a as he au ho know, such a model is no a ailable ye .
8.2 Full Model 145
The aim o his Sec ion is o p esen an app oxima e model which, wi hou
in ending o gi e accu a e quan i a i e p edic ions, p o ides use ul quali a i e
esul s ega ding he ib ocompac ion p ocess. This may be seen as a i s s ep
owa ds he ambi ious goal o achie ing a mo e complex model which eliably
cap u es he dynamics o he eal sys em. No e ha he name ull model is used he e
only o dis inc ion om he simpli ied model p esen ed in he nex sec ion.
The simpli ica ion ca ied ou can be obse ed in Fig. 8.2 and Fig. 8.3. The o me
shows a schema ic pic u e o he eal machine, while he la e displays he
app oxima e 4-DOF model.
The qua z- esin mix u e is ep esen ed in he model by a couple o masses a ached
o each o he by a linea dampe and a nonlinea sp ing, which models he
compac ion i sel by allowing o pe manen de o ma ion when he sp ing is
comp essed. Then, he dis ance be ween bo h masses would ep esen he hickness
o he compac ing mix u e. The mould is modelled as a igid base, while he pis on
wi h he unbalanced mo o s is ep esen ed by a mass wi h a single unbalanced
mo o . The mix u e is in con ac –wi h sepa a ions and impac s allowed– wi h he
mould a he bo om and wi h he pis on a he op. The acuum sys em is no
included in he model.
I should be no ed ha he model assumes he ho izon al mo ion o he pis on o be
comple ely es ained, which makes unnecessa y o include a couple o mo o s
o a ing in opposi e di ec ions.
As ep esen ed in Fig. 8.3, he model has 4 DOFs: , , and , which
co espond, espec i ely, o posi ion o he bo om o he mix u e, posi ion o he
op o he mix u e, posi ion o he pis on and o a ion o he mo o .
The pa ame e s ep esen ed in Fig. 8.3 a e as ollows: s ands o he mass o
he mix u e, is he unbalanced mass, is he mass o he pis on and he mo o ,
is he eccen ici y o he unbalance, is he o o ine ia, is he damping