BULLETIN OF THE POLISH ACADEMY OF SCIENCES
TECHNICAL SCIENCES, Vol. 71(4), 2023, A icle numbe : e145567
DOI: 10.24425/bpas s.2023.145567
MECHANICAL AND AERONAUTICAL
ENGINEERING, THERMODYNAMICS
In luence o ma e ial dis ibu ion and damping
on he dynamic s abili y o Be noulli-Eule beams
Sebas ian GARUS1∗
∗
∗, Jus yna GARUS1, Wojciech SOCHACKI1, Ma cin NABIAŁEK2, Jana PETR ˚
U3,
Wojciech BOREK4, Michal ŠOFER5, and Paweł KWIATO ´
N1
1Facul y o Mechanical Enginee ing and Compu e Science, Czes ochowa Uni e si y o Technology, Poland
2Facul y o P oduc ion Enginee ing and Ma e ials Technology, Depa men o Physics, Czes ochowa Uni e si y o Technology,
A mii K ajowej 19, 42-201 Czes ochowa, Poland
3Depa men o Machining, Assembly and Enginee ing Me ology, Facul y o Mechanical Enginee ing, VSB-Technical Uni e si y o Os a a,
70833 Os a a, Czech Republic
4Depa men o Enginee ing Ma e ials and Bioma e ials, Silesian Uni e si y o Technology,
Kona skiego 18A, 44-100 Gliwice, Poland
5Depa men o Applied Mechanics, Facul y o Mechanical Enginee ing, VSB—Technical Uni e si y o Os a a, 17. lis opadu 2172/15,
70800 Os a a, Czech Republic
Abs ac . The s udy analyzed he in luence o ma e ials and di e en ypes o damping on he dynamic s abili y o he Be noulli-Eule beam.
Using he mode summa ion me hod and applying an o hogonal condi ion o eigen unc ions and desc ibing he analyzed sys em wi h he Ma hieu
equa ion, he p oblem o dynamic s abili y was sol ed. By examining he in luence o in e nal and ex e nal damping and damping in he beam
suppo s, hei in luence on he egions o s abili y and ins abili y o he solu ion o he Ma hieu equa ion was de e mined.
Key wo ds: mechanical ib a ions; damping; beam; dynamic s abili y; Ma hieu equa ion.
1. INTRODUCTION
A he beginning o he second hal o he 19 h cen u y, du ing
he cons uc ion o ailway b idges, i was al eady no iced ha
he damage o he s uc u e esul ed om he loss o s abili y o
hin pla es, shells, and slende ba s. Examples o such damage
can be seen in he wo ks [1,2], and one o he mos amous
was he Tacoma Na ows B idge (TNB) om 1940 due o he
ilm showing he oscilla ion and des uc ion o he b idge [3].
These ypes o e en s ini ia ed in ensi e esea ch in he ield o
dynamic s abili y.
Al eady in 1744, Leona d Eule desc ibed he basics o s uc-
u al s abili y in his wo k [4]. He sol ed he p oblem o axial
comp ession o an ideal ba wi h di e en suppo me hods.
In 1773, Lag ange wo ked on he op imiza ion o he shape
o a column loaded wi h an axial o ce [5,6]. The p oblem is
impo an because he sea ch o he mos op imal geome ies
o gi en applica ions is s ill ongoing [7–9]. The i s wo k on
a non-conse a i e loading o ce was Beck’s wo k om 1952
[10], in which he in luence o loading he acking o ce on he
end o he column on i s s abili y was analyzed.
The e a e a ious heo ies desc ibing he kinema ics o col-
umn de o ma ion, unde s ood as a slende and simple s uc u al
∗
∗
∗e-mail: [email p o ec ed]
© 2023 The Au ho (s). This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/)
Manusc ip submi ed 2023-02-08, e ised 2023-03-04, ini ially
accep ed o publica ion 2023-03-10, published in Augus 2023.
elemen subjec ed o comp ession. They a e desc ibed in mo e
de ail in [11], and he mos impo an o hem a e he heo ies
o Be noulli-Eule [12], Timoshenko [13], o Reddy-Bick o d
[14,15]. The las o hem was u he de eloped in [16]. The
subjec o buckling o columns and s uc u es was desc ibed in
he li e a u e, in e alia, in [17], and dynamic s abili y in [18].
The compa ison o selec ed heo ies was p esen ed in [19]. In
his wo k, he columns acco ding o he Be noulli-Eule heo y
will be analyzed.
Using he dynamic s abili y c i e ion and de e mining he
eigen alues de ining he co ela ions o he eigen equencies
o he es ed mechanical sys em o he load pa ame e , esul -
ing om he solu ion o he di e en ial equa ion o mo ion ak-
ing in o accoun he bounda y condi ions, i allows he s abili y
o a gi en sys em o be es ed and i s ype o be de e mined.
Leiphol z in [20] showed di e gen and lu e ypes o s abili y,
while Sunda a ajan in [21] also showed hyb id sys ems com-
bining ea u es o bo h ypes. The s abili y o he columns is
also in luenced by o he speci ic ac o s, o example, Ko das
and ˙
Zyczkowski in [22] analyzed he in luence o he acking
coe icien on he c i ical o ce o he can ile e column, while
he na u al equency and s abili y o he Beck column we e
in es iga ed by Sunda a ajan in [23] when he end o he col-
umn was esilien ly suppo ed. The passage h ough he s abil-
i y limi is desc ibed in [24]. Kounadis in es iga ed he in lu-
ence o he s i ness coe icien o he sp ings used o suppo
he ba on he loss o s abili y [25]. In he wo k [26], he in lu-
Bull. Pol. Acad. Sci. Tech. Sci., ol. 71, no. 4, p. e145567, 2023 1
S. Ga us, J. Ga us, W. Sochacki, M. Nabiałek, J. Pe ˚u, W. Bo ek, M. Šo e , and P. Kwia o´
n
ence o he ine ia o o a ion and he shea o ce in he Beck
commune wi h he lu e - ype o ce was also conside ed. The
ins abili y o he column was also analyzed in [27].
A special case o pa ame ic ib a ions, in which s uc u es
a e loaded wi h a pe iodic unc ion o ime, is called dynamic
s abili y [28]. Pa ame ic esonances in he dynamic s abili y
analysis play a undamen al ole. In he egion o uns able so-
lu ions, he ampli ude o esonan ib a ions inc eases unlimi -
edly. Mos o he wo k on dynamic s abili y ocuses on simple
beam sys ems wi h in e connec ed disc e e elemen s [29–33],
as many mechanical sys ems can be modeled wi h hem.
Du ing he ope a ion o a ious ypes o machines, mechan-
ical ib a ions e y o en occu as an undesi able elemen .
This phenomenon ad e sely a ec s he s eng h o he wo k-
ing elemen s and may cause hei as e wea , bu mo e impo -
an ly, hey cause heal h p oblems o people ope a ing hese
machines, such as diso de s in he ascula , ne ous, and os-
eoa icula sys ems, and may also cause p oblems wi h isual
acui y o p oblems wi h mo o coo dina ion. The in luence o
noise is also impo an . The abo e ha m ul phenomena caused
by ib a ions ha e been desc ibed in [34,35]. To minimize he
nega i e e ec s o ib a ions as much as possible, he damping
phenomenon is used, in which he dissipa ion o mechanical
ene gy occu s due o he occu ence o ic ional o ces du ing
ib a ions. The phenomenon o damping and i s in luence on
mechanical sys ems exposed o ib a ions has been desc ibed
in he wo ks o Osi´
nski [36] and Gie giel [37]. I he na u al
me hods o damping (in e nal ic ion in he ma e ial, en i on-
men al impac , ic ion in join s and suppo s) a e no su icien ,
addi ional special ib a ion dampe s a e used, such as dynamic
and ac i e ib a ion elimina o s [38]. Mo eo e , ib a ion iso-
la ion is also used, placed be ween he insula ed sys em and he
ib a ing sys em, made o addi ional s uc u al elemen s o hei
assemblies [39]. Damping is a phenomenon ha always occu s
du ing mechanical ib a ions [34,39,40].
E en small ib a ions in he machine can cause he phe-
nomenon o esonance [41] and lead o i s des uc ion. The use
o damping will educe he isk o his phenomenon. Damping,
depending on i s mechanism and sou ce, can be di ided in o in-
e nal damping in a iscoelas ic ma e ial [42,43], design damp-
ing in mo able connec ions [44,45], design damping in ixed
connec ions (so-called d y ic ion) [46,47], iscous esis ance
(e.g. he ex e nal in luence o he medium) [48,49].
This s udy in es iga ed how di e en ypes o damping a -
ec he dynamic s abili y o Be noulli-Eule beams. Using he
pe u ba ion me hod, he s abili y egions o he solu ion o he
equa ion o mo ion ans o med in o he o m o he Ma hieu
equa ion we e de e mined.
2. MATHEMATICAL MODEL
Figu e 1shows a Be noulli-Eule beam, a icula ed a i s ends,
loaded wi h an axial comp essi e o ce desc ibed by he ela-
ion P( ) = P0+Scosν , whe e P0is he cons an componen o
he longi udinal o ce, Sis he a iable componen o he longi-
udinal o ce,W(x, )is he ans e se displacemen o he beam
a loca ion xand ime ,νis he equency o he exci ing o ce,
Fig. 1. A simple Be noulli-Eule beam loaded wi h an axial
comp essi e o ce ha changes cyclically, aking in o accoun
di e en ypes o damping
and is ime. The ma e ial p ope ies o he beam a e Young’s
modulus E, he momen o ine ia Jo he c oss-sec ion, he
c oss-sec ional a ea A, and he ma e ial densi y ρ. The added
damping is due o he esis ance o mo emen in he suppo s
– hey a e modeled wi h CR o a y dampe s and he damping
is caused by he ex e nal iscous esis ance CE. The iscosi y
index o he ma e ial was deno ed by EC.
The p oblem o ans e se ib a ions o a s aigh Be noulli-
Eule beam was sol ed by o mula ing he bounda y p oblem
using Hamil on’s a ia ional p inciple
2
Z
1
(δT−δV)d +
2
Z
1
(−δWN)d =0.(1)
Equa ion (1) akes in o accoun he a ia ion o he i ual wo k
o non-conse a i e o ces as
δWN=ECJ∂3W(l, )
∂x2∂ δ∂W(l, )
∂x−ECJ∂3W(0, )
∂x2∂ δ∂W(0, )
∂x
−ECJ∂4W(l, )
∂x3∂ δW(l, )+ECJ∂4W(0, )
∂x3∂ δW(0, )
+
l
Z
0
ECJ∂5W(x, )
∂x4∂ δW(x, )dx+
l
Z
0
CE
∂W(x, )
∂ δW(x, )dx
+CR
∂2W(0, )
∂x∂ δ∂W(0, )
∂x−CR
∂2W(l, )
∂x∂ δ∂W(l, )
∂x.(2)
The a ia ion o he bending sp ing ene gy is
δV1=EJ ∂2W(l, )
∂x2δ∂W(l, )
∂x−EJ ∂2W(0, )
∂x2δ∂W(0, )
∂x
−EJ ∂3W(l, )
∂x3δW(l, )+EJ ∂3W(0, )
∂x3δW(0, )
+
l
Z
0
EJ ∂4W(x, )
∂x4δW(x, )dx.(3)
2Bull. Pol. Acad. Sci. Tech. Sci., ol. 71, no. 4, p. e145567, 2023
In luence o ma e ial dis ibu ion and damping on he dynamic s abili y o Be noulli-Eule beams
On he o he hand, he a ia ion in ene gy om he ex e nal
load is
δV2=
l
Z
0
P( )∂2W(x, )
∂x2δW(x, )dx−P( )∂W(l, )
∂xδW(l, )
+P( )∂W(0, )
∂xδW(0, ).(4)
The a ia ion in kine ic ene gy is de ined as
δT=−
l
Z
0
ρA∂2W(x, )
∂ 2δW(x, )dx.(5)
Subs i u ing equa ion (2)–(5) o equa ion (1), he di e en ial
equa ion o he mo ion o he beam ans e se ib a ions was
de e mined as
∂4W(x, )
∂x4+EC
E
∂5W(x, )
∂x4∂ +ρA
JE
∂2W(x, )
∂ 2
+P( )
JE
∂2W(x, )
∂x2+CE
JE
∂W(x, )
∂ =0.(6)
The solu ion o equa ion (6) is p edic ed as a se ies o eigen-
unc ions
W(x, ) =
∞
∑
n=1
Wn(x)Tn( ),(7)
whe e Wn(x)is he n h na u al modes o ib a ions and Tn( )is
an unknown ime unc ion.
The i s o m o ib a ion is o majo impo ance, so he
displacemen W(x, )is w i en in he o m o he p oduc o
unc ions wi h a iables sepa a ed by ime and he coo dina e x
W(x, ) = W(x)T( ) = W(x)eiω .(8)
A e sepa a ing he space and ime a iables and subs i u ing
d2=P/(JE +iωJEC),Ω2=ρAω2−iωCE/(JE +iωJEC),
he equa ion o mo ion akes he o m
d4W(x)
dx4+d2d2W(x)
dx2−Ω2W(x) = 0.(9)
Fo he sys em shown in Fig. 1, he bounda y condi ions a e
sepa a ing he a iables a e as ollows
W(0) = W(l) = 0,(10)
J(E+iωEC)d2W(0)
dx2=iωCR
dW(0)
dx ,(11)
J(E+iωEC)d2W(l)
dx2=−iωCR
dW(l)
dx .(12)
The gene al solu ion o he displacemen equa ion (9) is a unc-
ion
W(x) = C1cosh(αx)+C2sinh(αx)+C3cos(βx)
+C4sin(βx),(13)
whe e
α=s−1
2d2+ 1
4d4+Ω2,(14)
β=s1
2d2+ 1
4d4+Ω2.(15)
Subs i u ing solu ion equa ion (13) in o equa ions (10)–(12), a
homogeneous sys em o equa ions Awas ob ained wi h espec
o unknown cons an s Ci. This ma ix sys em is w i en as
[A](ω)C=0,(16)
whe e [A](ω)=[apq];[p,q]=(1−4), and C= [Ci]T;i=1−4.
In he case when he de e minan o he ma ix o coe icien s
is equal o ze o o he cons an s Ci, he sys em has a non- i ial
solu ion
de A(ω) = 0.(17)
Equa ion (17) allows o de e mine he dependence o he eigen-
equencies o he sys em ωion he load Pand o de e mine he
alue o he c i ical load Pk.
To expand he eigen unc ions in o a se ies equa ion (7), hei
o hogonali y was assumed. Equa ion (9), a e sepa a ing he
a iables o he n- h and m- h eigen unc ions and aking in o
accoun he bounda y condi ions, akes he o m
ρAωn2−iωnCE
J(E+iωnEC)−ρAωm2−iωmCE
J(E+iωmEC)
×
l
Z
0
Wn(x)Wm(x)dx=0.(18)
Since ωm6=ωn, when m6=n, hen he o hogonali y condi ion
sough is de e mined by he o mula
l
Z
0
Wn(x)Wm(x)dx=








0m6=n,
γ2
m=
l
Z
0
W2
m(x)dx m =n.(19)
Equa ion (6) was subs i u ed wi h equa ion (7) and ob ained
∞
∑
n=1JE d4Wn(x)
dx4Tn( ) +JEC
d4Wn(x)
dx4
dTn( )
d
+P0d2Wn(x)
dx2Tn( ) +Scos(ν )d2Wn(x)
dx2Tn( )
+Wn(x)ρAd2Tn( )
d 2+Wn(x)CE
dTn( )
d =0.(20)
Bull. Pol. Acad. Sci. Tech. Sci., ol. 71, no. 4, p. e145567, 2023 3
S. Ga us, J. Ga us, W. Sochacki, M. Nabiałek, J. Pe ˚u, W. Bo ek, M. Šo e , and P. Kwia o´
n
Equa ion (20) mul iplied by m- h eigen unc ionWm(x) akes he
o m
∞
∑
n=1JE d4Wn(x)
dx4Wm(x)+P0d2Wn(x)
dx2Wm(x)Tn( )
+JEC
d4Wn(x)
dx4Wm(x)dTn( )
d
+Scos(ν )d2Wn(x)
dx2Wm(x)Tn( )+Wn(x)Wm(x)ρAd2Tn( )
d 2
+Wn(x)Wm(x)CE
dTn( )
d =0.(21)
Mul iplied by he unc ions Wm(x)equa ion (9) a e sepa a ing
he a iables o he n- h eigen unc ion and mino ans o ma-
ions akes he o m
JE d4Wn(x)
dx4Wm(x)+P0d2Wn(x)
dx2Wm(x)
=ρAω2−iωCEWn(x)Wm(x)
−iωJEC
d4Wn(x)
dx4Wm(x).(22)
Subs i u ing equa ion (22) in o equa ion (21) gi es
∞
∑
n=1ρAω2−iωCEWn(x)Wm(x)
−iωJEC
d4Wn(x)
dx4Wm(x)+Scos(ν )d2Wn(x)
dx2Wm(x)Tn( )
+JEC
d4Wn(x)
dx4Wm(x)+Wn(x)Wm(x)CEdT n( )
d
+ρAWn(x)Wm(x)d2Tn( )
d 2=0.(23)
Only he i s e m o he sum in equa ion (7) is o signi ican
impo ance, as shown in [50]. The wo k analyzes he pa ame -
ic esonance o he i s ( undamen al) ib a ion equency o
he sys em n=1, which, aking in o accoun he o hogonali y
condi ion (18), allows o ans o m equa ion (23) in o he o m
T( )
ρAω2−iωCE
l
Z
0
W2(x)dx
−iωJEC
l
Z
0
d4W(x)
dx4W(x)dx+Scos(ν )
l
Z
0
d2W(x)
dx2W(x)dx

+d2T( )
d 2ρA
l
Z
0
W2(x)dx
+dT( )
d 
JEC
l
Z
0
d4W(x)
dx4W(x)dx
+CE
l
Z
0
W2(x)dx
=0.(24)
Di iding bo h sides o equa ion (24) by −ρAZl
0W2(x)dxand
subs i u ing τ=ν we ge
d2T(τ)
dτ2+








CE
ρAν2+
JEC
l
Z
0
d4W(x)
dx4W(x)dx
ρAν2
l
Z
0
W2(x)dx








dT(τ)
dτ
+








ω2
ν2−iωCE
ρAν2−
iωJEC
l
Z
0
d4W(x)
dx4W(x)dx
ρAν2
l
Z
0
W2(x)dx
+
l
Z
0
d2W(x)
dx2W(x)dx
ρAν2
l
Z
0
W2(x)dx
Scos(ντ)








T(τ) = 0.(25)
The equa ion o mo ion (25) has he o m o he damped Ma h-
ieu equa ion p esen ed in [51] as
d2T(τ)
dτ2+cdT(τ)
dτ+(δ+εcosτ)T(τ) = 0,(26)
whe e
c=CE+b
ρAν2,(27)
δ=ω2
ν2−iω(CE+b)
ρAν2,(28)
ε=
S
l
Z
0
d2W(x)
dx2W(x)dx
ρAν2
l
Z
0
W2(x)dx
(29)
and
b=
JEC
l
Z
0
d4W(x)
dx4W(x)dx
l
Z
0
W2(x)dx
.(30)
In o de o de e mine he in luence o damping on he an-
si ion cu es in he Ma hieu equa ion (26), he wo- a iable
expansion me hod was used [52–55]. To apply he pe u ba-
ion me hod, he damping coe icien cwas scaled o O(ε)by
c=εµ, which, assuming small alues o εand subs i u ing
4Bull. Pol. Acad. Sci. Tech. Sci., ol. 71, no. 4, p. e145567, 2023
In luence o ma e ial dis ibu ion and damping on he dynamic s abili y o Be noulli-Eule beams
ξ=τand η=ετ o equa ion (26) and pe o ming app op i-
a e ans o ma ions, leads o
∂2T(ξ,η)
∂ξ2+2ε∂2T(ξ,η)
∂ξ∂η +ε2∂2T(ξ,η)
∂η2
+εµ ∂T(ξ,η)
∂ξ +ε∂T(ξ,η)
∂η 
+(δ+εcosξ)T(ξ,η) = 0.(31)
By expanding he unc ion T(ξ,η)and δin o powe se ies and
omi ing he ac o O(ε2)and hen p io i izing by εsuccessi e
powe s, he ollowing sys em o equa ion was ob ained
∂2T0(ξ,η)
∂ξ2+δT0(ξ,η) = 0,(32)
∂2T1(ξ,η)
∂ξ2+δT1(ξ,η)
=−2∂2T0(ξ,η)
∂ξ∂η −µ∂T0(ξ,η)
∂ξ −T0(ξ,η)cosξ,(33)
2∂2T1(ξ,η)
∂ξ∂η +µ∂T1(ξ,η)
∂ξ +T1(ξ,η)cosξ
=−∂2T0(ξ,η)
∂η2−µ∂T0(ξ,η)
∂η ,(34)
∂2T1(ξ,η)
∂η2+µ∂T1(ξ,η)
∂η =0.(35)
Equa ion (32) is an equa ion o mo ion o a simple ha monic
oscilla o and i s gene al solu ion akes he o m
T0(ξ,η) = A(η)cos√δξ +B(η)sin√δξ.(36)
I is wo h no ing ha he ampli udes o he gene al solu ion de-
pend on η. Subs i u ing equa ion (36) in o equa ion (33) ans-
o ming and con e ing he p oduc s o igonome ic unc ions
in o sums, we ob ained
∂2T1(ξ,η)
∂ξ2+δT1(ξ,η)
=2dA(η)
dη√δsin√δξ −2dB(η)
dη√δcos√δξ
+µA(η)√δsin√δξ −µB(η)√δcos√δξ
−A(η)
2cos√δ+1ξ+cos√δ−1ξ
−B(η)
2sin√δ+1ξ−sin√δ−1ξ.(37)
The i s wo e ms on he igh side o he equa ion ep esen
esonance condi ions and may cause he solu ion o become un-
s able. I dA(η)/dη=0 and dB(η)/dη=0 he cos e m o
Ma hieu’s equa ion does no a ec he solu ion and he e is no
pa ame ic esonance phenomenon. In he case o subs i u ing
δ=1/4 and µ=0 (no damping) o equa ion (37) and a e
ans o ma ions, he ollowing is ob ained
∂2T1(ξ,η)
∂ξ2+1
4T1(ξ,η)
=dA(η)
dη+B(η)
2sin ξ
2−dB(η)
dη+A(η)
2cos ξ
2
−A(η)
2cos 3ξ
2−B(η)
2sin 3ξ
2,(38)
which allowed o ob ain addi ional esonance condi ions de-
ined as dA(η)/dη=−B(η)/2, dB(η)/dη=−A(η)/2 which
leads o d2A(η)/dη2=A(η)/4. The alue o he pa ame e
δ=1/4 causes ins abili y, and A(η)and B(η)inc ease expo-
nen ially. In he p esen ed example, he e is a subha monic es-
onance in which he exci a ion equency is wice he na u al
equency.
A e inse ing he expansion in o he powe se ies δwi h
espec o ε o equa ion (33) and o µ=0 (no damping), he
ollowing was ob ained
∂2T1(ξ,η)
∂ξ2+1
4T1(ξ,η) = −2∂2T0(ξ,η)
∂ξ∂η
−δ1T0(ξ,η)−T0(ξ,η)cosξ(39)
and he esonan condi ions ake he o m o dA(η)/dη=
(δ1−1/2)B(η)and dB(η)/dη=−(δ1+1/2)A(η)which
leads o d2A(η)/dη2+ (δ1+1/4)A(η) = 0. The condi ions
ul ill he sine and cosine unc ions o A(η)and B(η), espec-
i ely, wi h δ2
1−1/4>0, i.e. when δ1>1/2 o δ1<−1/2.
The cu es p esen ed ep esen he s abili y cu es in he space
δ–εas
δ=1
4±ε
2+O(ε2).(40)
Equa ions (36) and (40) co espond o he egion o ins abili y
wi h he ze o poin a δ=1/4.In he case o he damped sys em
µ6=0, equa ion (39) akes he o m
∂2T1(ξ,η)
∂ξ2+1
4T1(ξ,η) = −2∂2T0(ξ,η)
∂ξ∂η
−µ∂T0(ξ,η)
∂ξ −δ1T0(ξ,η)−T0(ξ,η)cosξ.(41)
Sui able de i a i es ake he o m o









dA(η)
dη=−µ
2A(η)+δ1−1
2B(η),
dB(η)
dη=−δ1+1
2A(η)−µ
2B(η).
(42)
The abo e sys em o equa ions can be sol ed assuming solu-
ions A(η) = A0eλη and B(η) = B0eλη . A non- i ial solu ion
is ob ained o

−µ
2−λ−1
2+δ1
−1
2+δ1−µ
2−λ
=0,(43)
Bull. Pol. Acad. Sci. Tech. Sci., ol. 71, no. 4, p. e145567, 2023 5

S. Ga us, J. Ga us, W. Sochacki, M. Nabiałek, J. Pe ˚u, W. Bo ek, M. Šo e , and P. Kwia o´
n
om which i ollows ha
λ=−µ
2± −δ2
1+1
4.(44)
To de e mine he ansi ion be ween he s able and uns able
s a e, λ=0 should be assumed and he alue o δ1should be
de e mined as δ1=±p1−µ2/2. Fo he i s egion o ins a-
bili y, a ela ionship was es ablished
δ=1
4±εp1−µ2
2+Oε2
=1
4±√ε2−c2
2+Oε2.(45)
Using he ela ion (45), he in luence o he damping con-
s an con he i s uns able egion o he Ma hieu equa ion is
shown in Fig. 2. Wi h he inc ease o he damping coe icien ,
he egion o uns able solu ions o he equa ion dec eased. The
minimum alue o he coe icien εinc eased i s alue wi h he
inc ease o he damping cons an cand a la ge and la ge e-
gion o possible s able solu ions a ose unde he ansi ion cu e
di iding he s a ic and uns able egions.
Fig. 2. In luence o iscous damping con he uns able (shaded) egion
o he solu ion o he Ma hieu equa ion
3. NUMERICAL RESULTS
The pape sol es he p oblem o dynamic s abili y o an un-
damped s aigh Be noulli-Eule beam, a icula ed a i s ends
and loaded wi h cyclically changing axial comp essi e o ce.
The Wol am Ma hema ica package was used, o which p op i-
e a y so wa e was p epa ed o pe o m calcula ions and make
d awings. The beam leng h was assumed o be l=3 m and a
squa e c oss-sec ion wi h a side h=0.3 m and a su ace a ea
A=0.09 m2. The alue o he c i ical axial comp essi e o ce
o s eel was assumed o be Pk=9.32676 ×107N, he con-
s an load componen was P0=5%Pk, simila ly, he a iable
load componen was S=5%Pk. The a ea o ine ia o he c oss-
sec ion was de e mined om he ela ionship J=h4/12 =
675 ×10−6m4. The ma e ial p ope ies o he beams o he
a ious ma e ials used in he simula ion a e summa ized in Ta-
ble 1. The i s h ee eigen equencies o he analyzed beams
we e de e mined and collec ed in Table 2.
Table 1
P ope ies o ma e ials used o he analysis o dynamic s abili y
o Be noulli-Eule beams [56]
Ma e ial E[GPa] ρ[kg/m3]
S eel 210 7860
Coppe 125 8900
Aluminum 70 2700
Ti anium 116 4500
Table 2
De e mined eigen equencies o beams
Ma e ial ω1[ ad/s] ω2[ ad/s] ω3[ ad/s]
S eel 483.473 1956.19 4410.66
Coppe 346.831 1414.67 3194.26
Aluminum 461.292 1912.38 4330.27
Ti anium 468.905 1915.59 4326.52
Based on he de e mined shapes o he displacemen unc-
ions o he gi en ma e ials, he pa ame e s δand εo he
Ma hieu equa ion om equa ions (28) and (29) we e de e -
mined and plo ed on he S u cha in Fig. 3. The equen-
cies o he exci ing o ce we e de e mined o a gi en i s na u-
al equency o a gi en ma e ial acco ding o he dependence
νn=2ω1/n, whe e n∈{1,2,3,4}.
Fig. 3. Rela ions be ween he coe icien s εand δ o he analyzed
ma e ials plo ed on he S u cha
As can be seen in Fig. 3, he beam made o aluminum was
cha ac e ized by he lowes dynamic s abili y wi h he same ge-
ome ic dimensions and load. Beams made o i anium and cop-
pe , despi e la ge di e ences in na u al equencies, we e cha -
ac e ized by simila dynamic s abili y in a o o he column
made o coppe . On he o he hand, he mos a o able esul s
we e ob ained o he column made o s eel.
In o de o analyze he in luence o a ious ypes o damping
on he dynamic s abili y o he analyzed beam, dimensionless
damping coe icien s we e in oduced o he in e nal damping
Ch, he ex e nal damping Cnand he s uc u al damping in he
6Bull. Pol. Acad. Sci. Tech. Sci., ol. 71, no. 4, p. e145567, 2023
In luence o ma e ial dis ibu ion and damping on he dynamic s abili y o Be noulli-Eule beams
beam suppo s Cmin he o m
Ch=EC
E l4ρA
EJ
,(46)
Cn=CEl2
√ρAEJ ,(47)
Cm=CR
l√ρAEJ .(48)
The pape analyzes he in luence o a ious ypes o damping
on he dynamic s abili y o a s eel beam wi h a geome y co e-
sponding o he case unde conside a ion wi hou damping. The
damping pa ame e s o he conside ed cases a e summa ized
in Table 3, he de e mined alues o he eigen equencies ωi,
he damping coe icien cin luencing he shape o he ansi ion
cu e be ween he s able and uns able egions o he solu ion o
he Ma hieu equa ion and he de e mined coe icien ε o he
i s uns able egion o he ε–δsys em a e also p esen ed he e.
The ela ions be ween he coe icien s o he Ma hieu equa-
ion plo ed on he S u cha in he ε–δsys em o he an-
alyzed cases a e p esen ed in Fig. 4–8. The sys em wi hou
damping is shown in Fig. 4. An a ea was obse ed o which he
Fig. 4. Rela ions be ween he coe icien s εand δ o he analyzed
ma e ials plo ed on he S u cha ; no damping (case 1 in Table 3)
Fig. 5. Rela ions be ween he coe icien s εand δ o he analyzed
ma e ials plo ed on he S u cha ; conside ed damping in he beam
suppo s o Cm=0.05 (case 2 in Table 3)
Fig. 6. Rela ions be ween he coe icien s εand δ o he analyzed
ma e ials plo ed on he S u cha ; medium damping o Cn=18.5
is aken in o accoun (case 3 in Table 3)
solu ion o he Ma hieu equa ion has uns able solu ions (shaded
a ea). The addi ion o damping in he beam suppo s inc eased
he s abili y o he sys em (Fig. 5) bu did no elimina e he
uns able a ea. P ope ly selec ed medium damping educed he
uns able a ea (Fig. 6) and made he sys em in he en i e an-
Table 3
De e mined eigen equencies o beams made o s uc u al s eel o selec ed damping cases
Case 1 2 3 4 5
Ch0 0 0 0.001 0
Cn0 0 18.5 0 4.26
Cm0 0.05 0 0 0.017
ω1483.473 483.26+49.454i150.569+459.429i483.467+2.42246i467.957+122.511i
ω21956.19 1973.03+198.266i1901.48+459.429i1955.81+38.7593i1951.55+171.862i
ω34410.66 4486.25+443.051i4386.67+459.429i4406.29+196.219i4413.77+253.874i
c0 0 0.00792+0.00582i5.181×10−6−5.193×10−8i0.00889−0.00011I
σ0.25 0.25 0.20151−0.14797i0.24999−0.00251i0.22224−0.10602i
ε0.00773 0.00121+0.01004i0.00623−0.00458i0.00773−0.00008i0.00762−0.00036i
Bull. Pol. Acad. Sci. Tech. Sci., ol. 71, no. 4, p. e145567, 2023 7
S. Ga us, J. Ga us, W. Sochacki, M. Nabiałek, J. Pe ˚u, W. Bo ek, M. Šo e , and P. Kwia o´
n
alyzed ange o δhad s able solu ions. The in e nal damping
did no cause any signi ican changes in he s abili y in ela ion
o he undamped sys em (Fig. 7). The use o a combina ion o
he damping o he cen e and suppo s (Fig. 8) signi ican ly in-
luenced he achie emen o dynamic s abili y in he analyzed
a ea o solu ions, as well as allowed o educe o he damping
coe icien o suppo s.
Fig. 7. Rela ions be ween he coe icien s εand δ o he analyzed
ma e ials plo ed on he S u cha ; in e nal damping aken in o ac-
coun o Ch=0.001 (case 4 in Table 3)
Fig. 8. Rela ions be ween he coe icien s εand δ o he analyzed ma-
e ials plo ed on he S u cha ; conside ed damping o he medium
o Cn=4.26 and damping in he beam suppo s o Cm=0.017
(case 5 in Table 3)
4. CONCULSIONS
The s udy in es iga ed he in luence o a ious ma e ials on he
dynamic s abili y o he Be noulli-Eule beam, which showed
ha he dynamic s abili y o such beams inc eases wi h he in-
c ease o hei Young’s modulus (solu ions o speci ic ma e ial
da a on longe sec ions lie in he s able a eas).
The in luence o di e en ypes o damping on he dynamic
s abili y o a s eel homogeneous Be noulli-Eule beam, a ic-
ula ed wi h o a ional dampe s a i s ends and loaded wi h an
axial o ce ( a ying in ime), was also in es iga ed. The con-
duc ed es s showed ha he in e nal damping does no signi -
ican ly change he dynamic s abili y o he analyzed sys em.
The use o damping in beam suppo s inc eases he s abili y o
he sys em. The damping o he medium su ounding he beam
has he mos signi ican in luence on he dynamic s abili y o
he sys em unde conside a ion. This ype o damping na ows
he a eas o uns able solu ions and causes he sys em o ha e
s able solu ions in he en i e analyzed ange.
The use o a iscous dampe sys em sui ably a ached a di -
e en poin s on he beam could simula e a change in medium
damping and elimina e uns able a eas.
ACKNOWLEDGEMENTS
This publica ion was inanced by he Minis y o Educa ion and
Science o Poland as he s a u o y inancial g an o he De-
pa men o Mechanics and Machine Design Fundamen als o
Czes ochowa Uni e si y o Technology.
The p ojec is co- inanced by he Go e nmen s o Czechia,
Hunga y, Poland and Slo akia h ough Viseg ad G an s om
In e na ional Viseg ad Fund. The mission o he und is o ad-
ance ideas o sus ainable egional coope a ion in Cen al Eu-
ope.
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