Influence of material distribution and damping on the dynamic stability of Bernoulli-Euler beams

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. 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