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VIBRATION MECHANICS OF A SEMI-CYLINDRICAL
CUSHION WITH INERTIAL PROPERTIES
R. Akba li1, P. Ga aye 2, D. Imamalie 3
Depa men o Mechanics, Aze baijan Uni e si y o A chi ec u e and Cons uc ion, Baku,
Aze baijan1,2
Depa men o Ci il Enginee ing, Tashken s a e anspo uni e si y, Tashken , Uzbekis an3
h ps://doi.o g/10.5281/zenodo.17702781
Abs ac . In he p esen s udy, he ee and o ced ib a ions o an ine ial semi-cylind ical
cushion a e in es iga ed. A dynamic model is conside ed, in which he mo ion o he cushion is
desc ibed by he Lamé equa ions in e ms o displacemen s. To sol e he posed p oblem, he
na u al equencies and mode shapes a e de e mined, as well as he no mal displacemen s o
poin s on he semi-cylind ical cushion unde a ious bounda y condi ions. Pa icula a en ion is
paid o he in luence o ine ial p ope ies and ma e ial pa ame e s, such as densi y, elas ic
modulus, and Poisson’s a io, on he equency cha ac e is ics o he sys em. The ob ained esul s
allow he iden i ica ion o pa e ns in he a ia ion o ib a ion ampli udes and equencies
depending on he geome ic and ma e ial pa ame e s o he cushion. Cha ac e is ic dependencies
a e cons uc ed o illus a e he e ec o ine ial pa ame e s on he dynamic beha io o he semi-
cylind ical s uc u e. A compa a i e analysis o heo e ical esul s and nume ical calcula ions is
ca ied ou , con i ming he alidi y o he p oposed model. The indings can be applied in he
design and op imiza ion o s uc u es ope a ing unde ib a ional loads, as well as in he
de elopmen o ib a ion isola ion and damping sys em componen s. This s udy con ibu es o he
achie emen o he Sus ainable De elopmen Goal 9 (Indus y, Inno a ion and In as uc u e) by
p o iding insigh s o sa e , mo e du able, and op imized enginee ing s uc u es.
Keywo ds: dynamic beha io , ine ial p ope ies, na u al equencies, semi-cylind ical
cushion, ib a ions.
In oduc ion
Vib a ions o b idge s uc u es ep esen an impo an aspec o enginee ing design, as hey
can signi ican ly a ec he du abili y and sa e y o cons uc ions. To educe oscilla o y p ocesses,
cushions o a ious shapes a e employed o p o ide damping and load dis ibu ion. The s udy o
he dynamic cha ac e is ics o such cushions allows o he op imiza ion o b idge design and
imp o emen o hei ope a ional eliabili y. In [1], [8], he ee ib a ions o a h ee-laye ci cula
pla e on an elas ic ine ial ounda ion unde he mal e ec s a e conside ed. The kinema ics o he
hickness-asymme ic package a e desc ibed by he b oken-line no mal hypo hesis, while he
ounda ion eac ion is modeled using he Winkle scheme. Analy ical solu ions o he ini ial-
bounda y alue p oblems a e ob ained, and a nume ical compa a i e analysis is pe o med. I is
shown ha an inc ease in empe a u e educes he na u al equencies o he s uc u e, while he
displacemen ampli udes inc ease due o he educ ion in ma e ial s i ness. An inc ease in he
ine ia o he ounda ion also leads o highe ib a ion ampli udes and a sho e ib a ion pe iod,
wi h a nonlinea cha ac e o in luence. In [2], [9], axisymme ic ee ib a ions o a h ee-laye
elas ic pla e on a Winkle - ype ounda ion a e in es iga ed, while in [3], [10], a simila p oblem is
examined unde ec angula loading condi ions, whe e he ounda ion eac ion is desc ibed by he
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Pas e nak model. In all cases, he b oken-line no mal hypo hesis is applied, and he co e is assumed
o be ligh weigh . Fu he s udies on ela ed opics can be ound in [11–15], which explo e aspec s
such as wa e oo p in educ ion in i iga ion sys ems [11], assessmen o comp essi e s eng h o
eco-conc e e using machine lea ning [12], de elopmen o acous ic abso be s om wood ibe
composi es [13], ibological p ope ies o high-speed s eel unde high empe a u es [14], and soil
bed de o ma ions caused by ain loads [15]. In addi ion o de i ing he analy ical model, his s udy
includes a nume ical alida ion s ep. A h ee-dimensional ini e elemen model (FEM) was
de eloped in ABAQUS o e i y he analy ical esul s. By compa ing he na u al equencies and
displacemen ields ob ained om bo h app oaches, he accu acy and applicabili y o he p oposed
model a e con i med.
Table 1. Ma e ial and ounda ion pa ame e s used in he s udy
Pa ame e
Symbol
Value
Uni
Desc ip ion
Densi y o
cushion
ma e ial
ρₛ
2500
kg/m³
Mass densi y o he semi-
cylind ical cushion
Elas ic
modulus
E
3.0 × 10⁷
Pa
Young’s modulus o he
cushion ma e ial
Poisson’s
a io
ν
0.25
-
T ans e se con ac ion
a io
Lamé
pa ame e (λ)
λₛ
compu ed
Pa
De i ed om E and ν
Shea
modulus (μ)
μₛ
compu ed
Pa
De i ed om E and ν
Founda ion
s i ness
kθ
4.0 × 10⁶
N/m²
Winkle s i ness
coe icien
Founda ion
ine ia
m_d
3000
kg/m²
Mass pe uni a ea o
suppo ing ounda ion
Cushion
leng h
L
0.3–1.2
m
Range used in pa ame ic
s udy
Cushion
adius
R
0.1–0.2
m
Geome ic cha ac e is ic
o he semi-cylinde
P oblem S a emen . In he p esen s udy, he ib a ions o an ine ial semi-cylind ical
cushion loca ed be ween he elemen s o a b idge s uc u e — he span and he suppo pie (Fig.
1) — a e in es iga ed. The aim o he wo k is o de e mine he dynamic cha ac e is ics o he
cushion, o assess he in luence o i s ine ial and elas ic p ope ies on he ib a ion beha io , and
o es ablish he dependence o ib a ion ampli udes and equencies on he s uc u al pa ame e s.
Fig. 1. Ine ial semi-cylind ical cushion
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p — load applied o he cushion om he b idge side
𝑞𝑟— load applied o he cushion om he suppo side
To sol e he o mula ed p oblem, he a ia ional Hamil on–Os og adsky p inciple [4,5] is
applied:
𝛿 𝑊=0 (1)
He e, 𝑊=∫Jd
𝑡′′
𝑡′ is he ac ion unc ional, and, 𝑡′and 𝑡′′ a e a bi a ily chosen momen s
o ime.
𝐽=𝑉𝑘𝑠−𝑉𝑝𝑠−𝐴𝑝−𝐴𝑞 (2)
He e, 𝑉𝑝𝑠,𝑉𝑘𝑠 a e he po en ial and kine ic ene gies o he semi-cylind ical cushion,
espec i ely; 𝐴𝑞 is he wo k o he o ce ac ing on he cushion om he b idge side du ing i s
e ical displacemen , and 𝐴𝑝 is he wo k o he o ce ac ing om he suppo side. These
quan i ies a e calcula ed as ollows:
𝑉𝑝𝑠 =∭ [𝜆𝑠+2𝜇𝑠
2(𝑒11
2+𝑒22
2+𝑒33
2)+(𝜆𝑠+2𝜇𝑠)(𝑒11𝑒22+𝑒22𝑒33+𝑒11𝑒33)] (3)
222
2
s x
ks V
s s s
V dxdyd
(4)
0
q
S
A q s dxdy
; (4)
, /2
( )s
p
R
A p x dx
(5)
The quan i ies 𝑠𝑥,𝑠𝜃,𝑠𝑟 deno e he displacemen s o he cushion poin s along he
co esponding coo dina e di ec ions, and ep esen s ime. The s ain componen s 𝑒11, 𝑒22,𝑒33 a e
exp essed in e ms o hese displacemen s 𝑠𝑥,𝑠𝜃,𝑠𝑟 as ollows:
𝑒11 =𝜕𝑠𝑟
𝜕𝑟 ,𝑒22 =𝑅(𝜕𝑠𝜃
𝜕𝜃 +𝑠𝑟),𝑒33 =𝜕𝑠𝑥
𝜕𝑥 (6)
I is assumed ha he o ce 𝑞𝑟 , ac ing om he suppo side on he semi-cylind ical cushion
du ing i s e ical displacemen 𝑠𝑟, ollows he Winkle - ype law:
2
2
d
s
q k s m
(7)
whe e
k
is he s i ness coe icien , and
d
m
is he speci ic weigh o he suppo ma e ial.
The load om he b idge side ac ing on he cushion is de ined by he unc ion:
P(x)=𝑝0𝑐𝑜𝑠𝑛𝜋
2𝑠𝑖𝑛𝑘𝑥𝑠𝑖𝑛𝜔𝑡 (8)
The displacemen s o he ine ial semi-cylind ical cushion 𝑠𝑥,𝑠𝜃,𝑠𝑟 a e desc ibed by he
Lamé equa ions in ec o o m [6,7]:
𝑎𝑙2𝑔𝑟𝑎𝑑𝑑𝑖𝑣𝑢
−𝑎𝑡2𝑟𝑜𝑡𝑟𝑜𝑡𝑢
+𝜌𝑠𝜕2𝑢
𝜕𝑡2=0(9)
whe e 𝑎𝑡=√𝜆𝑠+2𝜇𝑠
𝜌𝑠,𝑎𝑒=√𝜇𝑠
𝜌𝑠a e he longi udinal and ans e se wa e p opaga ion
eloci ies, 𝜌𝑠 is he densi y, λs and μs a e he Lamé coe icien s.
Solu ion
The displacemen s o he ine ial semi-cylind ical cushion 𝑠𝑥,𝑠𝜃,𝑠𝑟 a e gi en in he
ollowing o m [6]:
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2cos cos sin
s
x s n e n
C
s A kI I n kx
%
%
sin sin sin
n
s s s
n e n
I
A n C nk B
s I I n kx
n
%% %
(10)
cos sin sin
n e n
ss
s n
I I
C k B n
s A I n kx
%%
%
whe e 𝐴
~𝑠,𝐶
~𝑠,𝐵
~𝑠 a e unknown cons an coe icien s, and k, n,
e
,
– a e he wa e
numbe s and
e e
k k
2 2 2 2 2 2
,
By subs i u ing exp ession (10) in o o mulas (3)–(6), he exp essions o he po en ial and
kine ic ene gies o he cushion a e ob ained:
𝑉𝑝𝑠 =𝜋𝑙(𝜆𝑠+2𝜇𝑠)
16 {𝐴
~𝑠
2∫𝑅
0
+𝛽1(𝑅2+2𝛾𝑙2𝑅)𝐼𝑛
′′(𝛾𝑙𝑟)−2𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+𝐵
~𝑠2∫𝑅
0(𝛽22+𝑅2𝛽3+𝛽2𝛽3)𝑑𝑟+
+𝐶
~𝑠2∫𝑅
0
×𝑅𝐼𝑛(𝛾𝑡𝑟)−𝐼𝑛
′′(𝛾𝑡𝑟)𝑑𝑟 +𝐴
~𝑠𝐶
~𝑠∫𝑅
0
×𝐼𝑛(𝛾𝑡𝑟)+𝑘𝛾𝑙2𝛾𝑡2
𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′′(𝛾𝑙𝑟)+𝛾𝑡2𝑘3
𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)𝐼𝑛
′′(𝛾𝑡𝑟)+2𝛽4𝛽5+ (11)
+𝛽4(𝛾𝑙2𝑅𝐼𝑛
′′(𝛾𝑙𝑟)−𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟))+𝛽5𝑘𝛾𝑡2
𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)−𝑅𝐼𝑛
′′(𝛾𝑡𝑟)𝑑𝑟+
+𝐴
~𝑠𝐵
~𝑠∫𝑅
0
−𝑘2𝑅𝛽3𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+𝐵
~𝑠𝐶
~𝑠∫𝑅
0
×𝐼𝑛
′′(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝛽2
𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝑅𝛽3
𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑅𝛽2𝛽4𝑑𝑟}𝑠𝑖𝑛2𝜔𝑡
𝛽1(𝑟)=−𝑛2
𝑟𝐼𝑛(𝛾𝑙𝑟)+𝛾𝑙𝐼𝑛
′(𝛾𝑙𝑟);𝛽2(𝑟)=𝑛𝛾𝑡
𝑟𝐼𝑛
′(𝛾𝑡𝑟)−𝑛
𝑟2𝐼𝑛(𝛾𝑡𝑟);
𝛽3(𝑟)=−𝛾𝑡𝐼𝑛
′(𝛾𝑡𝑟)+𝑛
𝑟𝐼𝑛(𝛾𝑡𝑟);𝛽4(𝑟)=−𝑘
𝜇𝑡(𝑛2
𝑟𝐼𝑛(𝛾𝑡𝑟)−𝛾𝑡𝐼𝑛
′(𝛾𝑡𝑟));
𝛽5(𝑟)=−𝑛2
𝑟𝐼𝑛(𝛾𝑙𝑟)+𝛾𝑙𝐼𝑛
′(𝛾𝑙𝑟);
𝑉𝑘𝑠 =𝜋𝑙𝜔2𝜌𝑠
8{𝐴
~𝑠
2∫𝑅
0[𝑘2𝐼𝑛
2(𝛾𝑙𝑟)+𝑛2
𝑟2𝐼𝑛
2(𝛾𝑙𝑟)+𝛾𝑙2𝐼𝑛
′2(𝛾𝑙𝑟)]𝑑𝑟+
+𝐵
~𝑠2∫𝑅
0[𝛾𝑡2
𝑛2𝐼𝑛
′2(𝛾𝑙𝑟)+𝑛2
𝑟2𝐼𝑛
2(𝛾𝑡𝑟)]𝑑𝑟+𝐶
~𝑠2∫𝑅
0
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+𝑘2𝛾𝑡2
𝜇𝑡2𝐼𝑛
′2(𝛾𝑡𝑟)𝑑𝑟+𝐴
~𝑠𝐶
~𝑠∫𝑅
0
−2𝑘𝛾𝑙𝛾𝑡
𝜇𝑡𝐼𝑛
′(𝛾𝑙𝑟)𝐼𝑛
′(𝛾𝑡𝑟)𝑑𝑟+𝐴
~𝑠𝐵
~𝑠∫𝑅
0
+2𝑛𝛾𝑙
𝑟𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑙𝑟)𝑑𝑟+𝐵
~𝑠𝐶
~𝑠∫𝑅
0
−2𝑛𝑘𝛾𝑡
𝑟𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑡𝑟)𝑑𝑟}𝑠𝑖𝑛2𝜔𝑡
𝐴𝑞=−𝜋𝑙𝑅
4
−𝐴
~𝑠𝐶
~𝑠2𝑘𝛾𝑙𝛾𝑡
𝜇𝑡∫𝑅
0𝐼𝑛
′(𝛾𝑙𝑟)𝐼𝑛
′(𝛾𝑡𝑟)𝑑𝑟+ 𝐴
~𝑠𝐵
~𝑠∙2𝛾𝑙𝑛∫𝑅
0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑙𝑟)
𝑟𝑑𝑟−
−𝐵
~𝑠𝐶
~𝑠2𝑛𝑘𝛾𝑡
𝜇𝑡∫𝑅
0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑡𝑟)
𝑟𝑑𝑟(𝑘𝜗−𝜔2𝑚𝑑)𝑠𝑖𝑛2𝜔𝑡
2
0sin
2
n e n
ss
p s n
I R I R
p l C k B n
A A I R
R
%%
%
By subs i u ing all exp essions om o mulas (11) in o (2) o he 𝐽 and aking 𝑡′=0,𝑡′′ =
𝜋
𝜔, and applying he Hamil on–Os og adsky a ia ional p inciple (1), he ollowing exp ession is
ob ained:
𝑊={𝜋𝑙𝜔2𝜌𝑠
8{𝐴
~𝑠
2∫𝑅
0[𝑘2𝐼𝑛
2(𝛾𝑙𝑟)+𝑛2
𝑟2𝐼𝑛
2(𝛾𝑙𝑟)+𝛾𝑙2𝐼𝑛
′2(𝛾𝑙𝑟)]𝑑𝑟+
+𝐵
~𝑠2∫𝑅
0[𝛾𝑡2
𝑛2𝐼𝑛
′2(𝛾𝑙𝑟)+𝑛2
𝑟2𝐼𝑛
2(𝛾𝑡𝑟)]𝑑𝑟+𝐶
~𝑠2∫𝑅
0
+𝑘2𝛾𝑡2
𝜇𝑡2𝐼𝑛
′2(𝛾𝑡𝑟)𝑑𝑟+𝐴
~𝑠𝐶
~𝑠∫𝑅
0
−2𝑘𝛾𝑙𝛾𝑡
𝜇𝑡𝐼𝑛
′(𝛾𝑙𝑟)𝐼𝑛
′(𝛾𝑡𝑟)𝑑𝑟+𝐴
~𝑠𝐵
~𝑠∫𝑅
0
+2𝑛𝛾𝑙
𝑟𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑙𝑟)𝑑𝑟+𝐵
~𝑠𝐶
~𝑠∫𝑅
0
−2𝑛𝑘𝛾𝑡
𝑟𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑡𝑟)𝑑𝑟−𝜋𝑙(𝜆𝑠+2𝜇𝑠)
16 {𝐴
~𝑠
2∫𝑅
0
−2𝑘2𝛾𝑙2𝐼𝑛(𝛾𝑙𝑟)𝐼𝑛
′′(𝛾𝑙𝑟)+𝛽1(𝑅2+2𝛾𝑙2𝑅)𝐼𝑛
′′(𝛾𝑙𝑟)−2𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+
+𝐵
~𝑠2∫𝑅
0(𝛽22+𝑅2𝛽3+𝛽2𝛽3)𝑑𝑟+𝐶
~𝑠2∫𝑅
0
−𝑘2𝛾𝑡4
𝜇𝑡2𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′′(𝛾𝑡𝑟)+𝛽4𝑘𝛾𝑡2
𝜇𝑡×𝑅𝐼𝑛(𝛾𝑡𝑟)−𝐼𝑛
′′(𝛾𝑡𝑟)𝑑𝑟+
+𝐴
~𝑠𝐶
~𝑠∫𝑅
0
+𝑘𝛾𝑙2𝛾𝑡2
𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′′(𝛾𝑙𝑟)+𝛾𝑡2𝑘3
𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)𝐼𝑛
′′(𝛾𝑡𝑟)+2𝛽4𝛽5+ (12)
+𝛽4(𝛾𝑙2𝑅𝐼𝑛
′′(𝛾𝑙𝑟)−𝑘2𝑅𝐼𝑛(𝛾𝑙𝑟))+𝛽5𝑘𝛾𝑡2
𝜇𝑡𝐼𝑛(𝛾𝑙𝑟)−𝑅𝐼𝑛
′′(𝛾𝑡𝑟)𝑑𝑟+
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+𝐴
~𝑠𝐵
~𝑠∫𝑅
0
−𝑘2𝑅𝛽3𝐼𝑛(𝛾𝑙𝑟)𝑑𝑟+𝐵
~𝑠𝐶
~𝑠∫𝑅
0
×𝐼𝑛
′′(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝛽2
𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑘𝛾𝑡2𝑅𝛽3
𝜇𝑡𝐼𝑛(𝛾𝑡𝑟)+𝑅𝛽2𝛽4𝑑𝑟+
+𝜋𝑙𝑅
4
−𝐴
~𝑠𝐶
~𝑠2𝑘𝛾𝑙𝛾𝑡
𝜇𝑡∫𝑅
0𝐼𝑛
′(𝛾𝑙𝑟)𝐼𝑛
′(𝛾𝑡𝑟)𝑑𝑟+ 𝐴
~𝑠𝐵
~𝑠∙2𝛾𝑙𝑛∫𝑅
0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑙𝑟)
𝑟𝑑𝑟−
−𝐵
~𝑠𝐶
~𝑠2𝑛𝑘𝛾𝑡
𝜇𝑡∫𝑅
0𝐼𝑛(𝛾𝑡𝑟)𝐼𝑛
′(𝛾𝑡𝑟)
𝑟𝑑𝑟(𝑘𝜗−𝜔2𝑚𝑑)+
0
2
n e n
ss
s n
I R I R
p l C k B n
A I R
R
%%
%
∙𝜋
2𝜔
F om exp essions (12) o 𝑊, i ollows ha he unc ional ep esen s a quad a ic o m wi h
espec o he cons an s 𝐴
~𝑠,𝐵
~𝑠,𝐶
~𝑠. By a ying he unc ional wi h espec o hese unknowns
𝐴
~𝑠,𝐵
~𝑠,𝐶
~𝑠, a sys em o non-homogeneous algeb aic equa ions is ob ained:
123
1) ;2) ;3) .
s s s
W W W
A B C
%%
%
(13)
In expanded o m, he sys em o equa ions is as ollows:
{𝜑11𝐴𝑠+𝜑12𝐵𝑠+𝜑13𝐶𝑠=−𝑙𝑝0
2𝜑21𝐴𝑠+𝜑22𝐵𝑠+𝜑23𝐶𝑠
=−𝑙𝑝0
2(14)𝜑31𝐴𝑠+𝜑32𝐵𝑠+𝜑33𝐶𝑠=−𝑙𝑝0
2𝑛𝐼𝑛(𝛾𝑡𝑅)
𝑅
whe e he coe icien s 𝜑𝑖𝑗(𝑖,𝑗=1,2,3) a e ob ained om he a ia ional unc ional 𝑊
a e a ia ion as he coe icien s o he cons an s 𝐴
~𝑠,𝐵
~𝑠,𝐶
~𝑠. Using C ame ’s ule, hese cons an s
a e de e mined as ollows:
𝐴
~𝑠=∆1
∆,𝐵
~𝑠=∆2
∆,𝐶
~𝑠=∆3
∆ (15)
He e, ∆ is he main de e minan o he sys em, and ∆𝑖(𝑖=1,2,3) a e he auxilia y
de e minan s o he sys em (14). By subs i u ing he alues o 𝐴
~𝑠,𝐵
~𝑠,𝐶
~𝑠 om (15) in o exp ession
(10) o 𝑠𝑟, he inal o mula o he cushion displacemen is ob ained:
cos sin sin
2
n e n
ss
s n
I R I R
C k B n n
s A I R kx
R
%%
%
The esonance equencies o he cushion a e de e mined om he condi ion ∆=0. The
oo s o he equa ion ∆=0 a e ound nume ically. Fo he calcula ions, he ollowing pa ame e s
cha ac e izing he ma e ial o he semi-cylind ical cushion a e adop ed:
𝑘𝜗=50𝑀𝑃𝑎
𝑚,𝑚𝑑=1000𝑀𝑃𝑎
𝑚,
The dependence o he na u al equencies o he semi-cylind ical cushion on i s leng h is
shown in Fig. 2. The no mal displacemen s o su ace poin s o he cushion as unc ions o he
coo dina es o di e en alues o he ounda ion’s speci ic weigh a e shown in Fig. 3. In all
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INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 11 NOVEMBER 2025
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82
g aphs: cu e 1 co esponds o 𝑚𝑑=1000𝑀𝑃𝑎
𝑚, cu e 2 o 𝑚𝑑=1500𝑀𝑃𝑎
𝑚, and cu e 3 o 𝑚𝑑=
2000𝑀𝑃𝑎
𝑚.
Fig. 2. Dependence o he ib a ion equency o he semi-cylind ical cushion on i s leng h
Fig. 3. Dependence o he de lec ion o he semi-cylind ical cushion on he coo dina e x
Model Valida ion
To e i y he accu acy and applicabili y o he de eloped analy ical model, a h ee-
dimensional ini e elemen (FEM) model was c ea ed in ABAQUS. The FEM model ep oduced
he exac geome y o he semi-cylind ical cushion, including all bounda y condi ions and ma e ial
pa ame e s used in he analy ical o mula ion.
The cushion was disc e ized using 8-node linea b ick elemen s (C3D8), which a e sui able
o hick elas ic solids. Mesh con e gence analysis was pe o med, and he inal mesh size ensu ed
nume ical s abili y and accu acy. Na u al equencies we e compu ed using he Lanczos
eigen alue ex ac ion me hod. A compa ison be ween he analy ical esul s and FEM simula ions
showed e y close ag eemen . Fo he i s h ee ib a ion modes, he di e ence did no exceed
2.8%, con i ming he co ec ness o he assumed displacemen unc ions and he de i ed
a ia ional equa ions. This alida ion demons a es ha he p oposed analy ical model is eliable
and can be con iden ly used o pa ame e s udies, op imiza ion, and design applica ions in
ib a ion isola ion sys ems.
Conclusions
1. Inc easing he leng h o he semi-cylind ical cushion leads o a dec ease in i s na u al
ib a ion equencies.
2. An inc ease in he speci ic weigh o he suppo esul s in highe na u al ib a ion
equencies.
3. An inc ease in he speci ic weigh o he ounda ion causes g ea e de lec ion o he
cushion.
4. The ine ial p ope ies o he semi-cylind ical cushion lead o a educ ion in i s na u al
equencies.
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