Dynamic Response to Harmonic Transverse Excitation of Cantilever Euler-Bernoulli Beam Carrying a Point Mass

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1 Ante Skoblar Teachng Assstant Unversty of Rjeka aculty of Engneerng Croata Roberto Žgulć ull Professor Unversty of Rjeka aculty of Engneerng Croata Sanjn Braut ull Professor Unversty of Rjeka aculty of Engneerng Croata Sebastjan Blaževć Teachng Assstant Unversty of Rjeka aculty of Engneerng Croata Dynamc Response to Harmonc Transverse Exctaton of Cantlever Euler-Bernoull Beam Carryng a Pont ass In ths paper, the calculaton of the dynamc response to harmonc transverse exctaton of cantlever Euler-Bernoull beam carryng a pont mass wth the mode superposton method s presented. Ths method uses mode shapes and modal coordnate functons whch are calculated by separaton of varables and aplace transformatons. The accuracy of defned expressons s confrmed on examples wth and wthout Raylegh dampng. All calculatons are n good agreement wth the results of commercal software based on fnte element methods. Keywords: dynamc response, Euler-Bernoull beam, pont mass, mode superposton method, Raylegh dampng. 1. INTRODUCTION Dynamc analyss of a structure as contnuum (where nerta, stffness and dampng are evenly dstrbuted throughout the doman has lmted use n practce because of ts mathematcal complexty. However, the analyss of contnuum systems, such as e.g. beam, gves us useful nformaton of complete dynamc behavour of more complex structures, so papers wth dfferent beam boundary condtons as [1], [] and [], beam exctatons (n pont [4], [5], base exctaton [6], etc. or beam materals (composte [7], [8], etc. are publshed. Transverse vbratons of unform slender beam can be calculated usng the Euler-Bernoull theory. Ths paper descrbes specfc combnaton of boundary condtons (cantlever beam wth pont mass and exctaton (harmonc exctaton n one pont whch can be found n practce as n [9] and [1] but ths calculaton procedure can be used for any boundary condtons. In ths paper, natural frequences and mode shapes are calculated usng the method of separaton of varables and aplace transformatons of homogeneous part of dfferental equaton of moton of cantlever beam carryng a pont mass wthout dampng. Then dynamc response of undamped and damped beam, where dampng s defned wth Raylegh theory, s defned by the mode superposton method whch uses mode shapes and modal coordnate functons. Calculaton of modal coordnates functons s based on the orthogonalty characterstc of mode shapes. Analytcal results, calculated by atlab1a on Wndows 8, are compared to numercal results, calculated by software SC.Nastran for Wndows [11] where beam fnte element, based on Tmoshenko beam theory, s used for creaton of fnte element model.. DYNAIC RESPONSE TO HARONIC TRANSVERSE EXCITATION Beam wth the constant rectangular cross-secton wth pont mass on poston x and under the acton of transversal harmonc force on poston x s analysed. gure 1. Sketch of cantlever Euler-Bernoull beam carryng a pont mass wth harmonc transverse exctaton n one pont A dfferental equaton of forced vbratons of beam (see [1] and [1] s defned by usng the Hamltons varatonal prncple: 4 5 cd ρ 4 4 w w w w EI EI c A x ω x t t t w δ x x = Ωt x x t ( sn ( δ ( Receved: September 16, Accepted: November 16 Correspondence to: D. Sc. Ante Skoblar aculty of Engneerng, Unversty of Rjeka Vukovarska 58, 51 Rjeka, Croata E-mal: askoblar@rteh.hr do:1.597/fmet1767s aculty of echancal Engneerng, Belgrade. All rghts reserved E Transactons (17 45, (1 where E s Youngs modulus, I second moment of area, w transversal dsplacement, x beam longtudnal coordnate, c d stffness proportonal and c mass proportonal Raylegh dampng coeffcent, ρ beam materal densty, A cross-secton area, value of pont mass, x coordnate of pont mass, exctaton force ampltude and Ω angular frequency of harmonc exctaton. Parts of (1 are flexural stffness, nner dampng, external dampng [1] and nerta of beam, nerta of pont mass and external force, respectvely. It s known that undamped lnear dynamc systems have natural mode shapes, and that n every mode shape

2 dfferent parts of the system are vbratng n phase, passng through equlbrum n the same moment. In the damped systems, however, ths characterstc generally does not apply [14]. Raylegh [1] has shown that there s a group of damped systems whch have classcal mode shapes. In ths paper a Raylegh defnton of dampng s used and uncoupled functons of modal coordnates are defned as a basc part of mode superposton method..1 Natural frequences and mode shapes The objectve of ths chapter s to defne natural frequences and mode shapes from homogeneous part of dfferental equaton of cantlever beam carryng a pont mass wthout dampng []: 4 w w w EI ρ A δ ( x x 4 =. ( x t t Boundary condtons for cantlever beam are: ( ( ( ( w, t = w, t = w, t = w, t = ( where s dervatve wth respect to x. Condton w(,t= s vald f the free beam end s not loaded wth external force or pont mass. Dfferental equaton ( s solved by the method of separaton of varables by whch dsplacement s defned wth the product of two separated functons of poston and tme: (, ψ ( sn( ω w x t = x t, (4 where ψ(x s mode shape and sn(ωt represents harmonc vbratons wth angular natural frequency ω. After ncluson of (4 nto ( and dvson of the whole expresson wth sn(ωt an expresson s obtaned: ( ( ε ( x EIψ ρ A δ x x ψω =, (5 where functon ε(x s so called weghtng functon. Boundary condtons has to be satsfed after applcaton of method of separaton of varables whch s possble f: ( ( ( ( ψ = ψ = ψ = ψ =. (6 Equatons (5 and (6 defne egenvalue problem whch s solved by aplace transformatons wth respect to x coordnate so equaton (5 now has the form: ( A ( EI ψ ρ ψ ω. (7 ω ( ψδ ( x x = aplace transformaton for functon ψ (x s: 4 ( ψ ( = ( ψ ( ψ ( x s x s ( s ( ( s ψ ψ ψ (8 where boundary condtons ψ ( and ψ ( are equal to zero (see (6, whle aplace transformaton for equaton part ψδ ( x x s: sx ( ψ ( δ ( ψ ( x x x e x =. (9 After ncluson of (8 and (9 nto (7 we obtan expresson: s 1 ( ψ ( x = ψ ( ψ ( s k s k, (1 ω ψ sx ( x e EI 4 4 s k where wave number k s calculated from expresson: ω ρ A k = EI Inverse aplace follows: ( ψ = ψ cosh.5 ( kx cos( kx k. (11 ( kx sn ( kx ω ψ ( x snh ψ ( k snh U ( x x k ( k ( x x sn ( k ( x x and wth the rest of the boundary condtons ψ (=ψ (= functons ψ ( and ψ ( are calculated and mode shape functon s defned as: ( x EI ω ψ ψ ( x = ( A kx kx 4EIk B kx kx u x x ( sn ( snh ( ( cos( cosh ( ( ( kx kx snh ( kx kx ( sn (1, (1 where values of A and B are calculated from expressons [9]: cos A = sn B = ( k kx cosh ( k kx sn ( k snh ( k sn ( k kx snh ( k kx cos( k cosh ( k cos( k cosh ( k sn ( k snh ( k sn ( k snh ( k cos( k cosh ( k ( kx snh ( kx cos( k kx snh ( k cosh ( k kx sn ( k sn ( k kx cosh ( k snh ( k kx cos( k cos( k cosh ( k 1, (14. (15 By nsertng condton x=x n (1, the expresson for calculaton of natural frequences ω s obtaned: ( sn snh ( ( ( ( ( ( ω A kx kx. (16 B cos kx cosh kx 4EIk = 68 VO. 45, No, 17 E Transactons

3 It s observed that equaton (16 has the form f(k= after ncluson of (11 so ts zero ponts (see fgure defne wave numbers k. Then natural frequences from (11 and mode shapes from (1 can be obtaned. gure. Zero ponts of equaton (16 The accuracy of calculated mode shapes are confrmed wth (5.. unctons of modal coordnates The equaton (1 s solved by mode superposton method where the soluton s assumed n form:, n n n= 1 ( ( ψ ( w x t = q t x (17 where q n (t are the functons of modal coordnates and ψ n (x are the mode shapes (1. After ncluson of (17 nto (1 and defnton of stffness proportonal (c d =βω and mass proportonal Raylegh dampng (c=αε(x [1], equaton (1 has a form ɺ ( ɺ ( x qɺɺ sn ( t ( x x EIψ q EI βψ q αε x ψ q ε ψ = Ω δ. (18 Now, an orthogonalty characterstc of normalzed mode shapes s used to get uncoupled functons of modal coordnates. Normalzed mass of mode shape s defned wth expresson: ( ψ ( = ε x x dx (19 and mass normalzed mode shape can be determned from expresson: ψ ψ =. ( ass normalzed mode shape functons satsfy expressons: and ( ε ( x ψ ( x d x = 1 (1 ε ( x ψ ( x ψ j ( x d x = ( whch makes them orthogonal wth respect to the weghtng functon ε (x. After ncluson of expresson (5 nto (18, multplyng expresson (18 wth mode shape functon ψ j and dvdng t wth the product of square roots of normalzed masses ( j an dfferental equaton s obtaned: ( x jq ( x jqɺ αε ( x ψ ψ jqɺ ε ( x ψ ψ jqɺɺ ψ j ( x = sn ( Ωt δ ( x x ε ω ψ ψ βε ω ψ ψ. ( After ntegraton of eq. ( over the beam doman for all mode shapes the followng expresson s obtaned: ( jd ɺ ( j ω q ε x ψ ψ x βω q ε x ψ ψ dx ( d ɺɺ ( αqɺ ε x ψ ψ x q ε x ψ ψ dx j j ( x ψ j = sn ( Ωt δ x x (, (4 where all combnatons of ndexes and j of normalzed mode shapes are under ntegral. Integrals n expresson (4 wth dfferent ndexes and j are equal to zero whle ntegrals wth equal ndexes =j are equal to one. As a result, a number of uncoupled expressons for functons of modal coordnates are gven: ( ( ɺ ( ω q t α βω q t ( x ψ qɺɺ ( t = sn ( Ωt. (5 urthermore, modal dampng rato ζ of -th mode shape based on Raylegh theory [1] s defned wth: so expresson (5 has a new form: wth a soluton: α βω ζ = (6 ω ( ζ ωɺ ( ω q t q t ( x ψ qɺɺ ( t = sn ( Ωt ζ ωt qɺ ζ ωq ( sn ( ω q t = e t ω ψ ( x q ( ωt ω cos t ζ ωτ sn ( τ e sn ( ω ( t τ dτ Ω (7 (8 E Transactons VO. 45, No, 17 69

4 where ω = ω 1 ζ s natural damped frequency of -th mode shape.. EXAPE 1 In ths example, natural frequences and mode shapes for a beam whch can be modelled by Euler-Bernoull theory are verfed by Nastran. Also, numercal overflow whch appears at hgher natural frequences s analysed. Beam dmensons are.46m*.m*.7m and beam materal s galvanzed steel wth Youngs modulus E=.1*1 11 Pa, the densty ρ=778 kg/m and the Possons coeffcent ν=.. The accuracy of the analytcal results wll be confrmed usng the software Nastran [11] by comparson of natural frequences and mode shapes. Beam fnte elements wth two nodes and unconnected fnte element degrees of freedom T y (dsplacement parallel to x axs and R z (rotaton parallel to z axs, see fgure 1 and an opton of coupled mass are used. After confrmaton of convergence of the results, 5 fnte elements of the same sze s used for modellng of the beam whch gves, approxmately, 1 fnte elements along one mode shape peak of the hghest mode shape. Pont mass wth.1 kg s postoned at 6. node (on coordnate x=. m and natural frequences are calculated. In table 1, a good agreement between analytcal results calculated from zero ponts of equaton (16 and expresson (11 (see fgure and Nastran results can be seen. Table 1. Natural frequences Natural frequency No. Analytc expressons, Hz Nastran, Hz In fgures and 4 a good agreement between mode shapes calculated wth (1 and (9 and wth Nastran can be seen. The descrbed approach gves exact natural frequences and mode shapes for 1-st to 5-th mode shape, but the error s observed for 6-th and hgher mode shapes. The values of equaton (16 whose zero ponts gve natural frequences start to oscllate (see fgure 5 whch grows for hgher natural frequences. Ths error s caused by a numercal overflow [9] whch appears n expressons (14 and (15 because hyperbolc functons wth hgh values are n ratos. gure 4. ode shapes calculated wth expressons (1 and (9 ( and wth Nastran ( *. gure 5. Value of functon (16 around zero pont of 5-th natural frequency As a result a group of ncorrect natural frequences can be calculated n the vcnty of the exact natural frequency (see fgure 6. gure. ode shapes 1.-. calculated wth expressons (1 and (9 ( and wth Nastran ( *. gure 6. Value of functon (16 around zero pont for 6-th natural frequency 7 VO. 45, No, 17 E Transactons

5 urthermore, t was shown that wth reducton of the pont mass the natural frequency of the beam wth pont mass converge toward natural frequency of beam wthout mass (see table. nfluence on response because ts frequency (44.65 Hz s close to exctaton frequency (45 Hz. Table. Values of natural frequences wth reducton of pont mass Natural frequency Natural frequences of beam wth pont mass, Hz No..1 kg.1kg.1kg Beam natural freq., Hz EXAPE The objectve of example s the comparson of numercal and analytcal results of equaton (7.e. the functon of modal coordnates and comparson of dynamc response wthout dampng calculated by mode superposton method and by drect transent method wth Nastran [11]. The harmonc transversal force have the ampltude 1 N and frequency of 45 Hz whch s more than four tmes lower than the hghest calculated 6-th natural frequency (17 Hz n order to get more accurate results [9]. orce s actng at. node (on coordnate x=. m. Good agreement of results for functon of modal coordnate can be seen on fgure 7 whch confrms the accuracy of analytc soluton. An overflow also occurs n the functon of modal coordnates but t can be avoded by the procedure explaned n [9]. gure 8. Dynamc response of undamped beam at node 46. calculated wth expresson (17 ( and wth software Nastran (* gure 9. Dynamc response of damped beam calculated wth mode superposton method ( and wth Nastran( * 5. EXAPE gure 7. Comparson of 1.(,.(o and.( * functon of modal coordnates calculated by numercal soluton of dfferental equaton (7 and by expresson (8 ( In ths example beam dynamc response n node 46. (at coordnate x=.41 m s also calculated usng mode superposton method and by drect transent method n Nastran [11] and calculated results are n good agreement whch can be seen on fgure 8. uncton of modal coordnate for. mode shape have largest The objectve of example s the comparson of results of dynamc response for damped beam by usng mode superposton method wth results of software Nastran. Overall structural dampng coeffcent G s used n Nastran for defnton of dampng n drect transent method. Structural dampng s connected wth dsplacement and t has constant value e.g. t does not depend up on frequency. In presented theory of mode superposton method we use Raylegh dampng whch depends up on frequency. Wth the am of comparng analytc results wth Nastran, we wll use the equvalent vscous dampng to defne G. Equvalent vscous dampng grows lnearly wth the frequency and t can be equal wth Raylegh dampng f Raylegh mass proportonal coeffcent s equal to zero. In that case overall structural coeffcent G can be calculated from the value of Raylegh stffness proportonal coeffcent E Transactons VO. 45, No, 17 71

6 (e.g. β=.1 n ths example and frst natural angular frequency ω 1 [11] from expresson: G = ξ = βω. (9 The setup parameters n drect transent analyss n Nastran are f 1 =W=.549 Hz and G=.16. At fgure 9 an transversal dynamc response at node 46. can be seen. 6. CONCUSION In ths paper an analytc expresson for dynamc response to transverse harmonc force of Euler-Bernoull cantlever beam wth the pont mass s defned. A procedure for defnng natural frequences, mode shapes and functons of modal coordnates s descrbed and results are used n the modal superposton method. Expressons for natural frequences are tested by reducng the pont mass, whereby natural frequences converged toward natural frequences of cantlever beam wthout pont mass, whch s a confrmaton of expressons accuracy. Durng the analyss of natural frequences t s observed that error occurs when hgher natural frequences are calculated. Errors are caused by numercal overflow whch appears because hyperbolc functons wth hgh values are n ratos. In ths paper dampng s defned wth Rayleghs theory whereby t s possble to calculate uncoupled functons of modal coor dnates, whch s descrbed n detal. or confrmaton of the results a software based on fnte element method was used (Nastran. Structural dampng for the drect transent method n Nastran was defned from equvalent vscous dampng whch correspond to Raylegh stffness proportonal dampng whle mass proportonal dampng was set to zero. All calculated results are n good agreement. ACKNOWEDGENT AdraHub s a collaboratve project funded by the European Unon (EU nsde the Adratc IPA CBC Programme, an Instrument for the Pre-Accesson Assstance (IPA of neghbourng countres of the Western Balkans, thanks to the nvestments n Cross- Border Cooperaton (CBC amng at jont economc and socal development. Detals n Savoa et al. [15]. Ths study s also supported by Unversty of Rjeka Grants (No and No ovng oads, Structural Engneerng and echancs, Vol. 44, No. 5, pp , 1. [5] Hamdan,.N. et al.: orce and orced Vbratons of a Restraned Cantlever Beam Carryng a Concentrated ass, Eng. Sc., Vol., pp. 71-8, [6] Erturk, A., Inman, D.J.: On echancal modelng of Cantlevered Pezoelectrc Vbraton Energy Harvesters, Journal of Intellgent ateral Systems and Structures, Vol. 19, p.p , 8. [7] Žvkovć, I., Pavlovć, A., ragassa, C.: Improvements n Wood Thermoplastc atrx Composte aterals Propertes by Physcal and Chemcal Treatments, Int J Qual Res, Vol. 1, No., pp. 5-18, 16. [8] Saghaf, H., Brugo, T.., Zucchell, A., ragassa, C., nak, G.: Comparson of the Effect of Preload and Curvature of Composte amnate under Impact oadng, E Transactons, Vol. 44, No. 4, p.p. 5-57, 16. [9] Skoblar, A., Žgulć, R., Braut, S.: Numercal Illcondtonng n evaluaton of the dynamc response of structures wth mode superposton method, Proc IechE Part C: J echancal Engneerng Scence, n press, DOI: / , 16. [1] Pavlovć, A., ragassa, C.: Analyss of lexble Barrers Used as Safety Protecton n Wood workng, Int J Qual Res, Vol. 1, No. 1, pp , 16. [11] SC. Nastran Verson 68. Basc dynamc analyss user s gude. SC. Software Corporaton, 4. [1] Banks, H.T., Inman, D.J.: On dampng mechansms n beams, Journal of Appled echancs, Vol. 58, pp , [1] Goel, R.P.: Vbratons of a Beam Carryng a Concentrated ass, Journal of Appled echancs, pp. 81-8, September 197. [14] Caughey, T.K., OKelly,.E.J.: Classcal Normal odes n Damped near Dynamc Systems, J. Appled echancs, p.p , September [15] Savoa,., Stefanovc,. and ragassa, C.: ergng techncal competences and human resources wth the am at contrbutng to transform the Adratc area n a stable hub for a sustanable technologcal development, Int J Qual Res, 1, pp. 1 1, 16. REERENCES [1] Han S, Benaroya H and We T. Dynamcs of transversely vbratng beams usng four engneerng theores. J Sound Vbrat, 5, pp , [] Chen, Y.: On the vbraton of beams or rods carryng a concentrated mass. J Appl ech,, pp. 1 11, 196. [] Gest, B.: The Effect of Structural Dampng on Nodes for the Euler-Bernoull Beam: A Specfc Case Study, Appl. ath. ett., Vol. 7, No., pp , [4] Pccardo, G., Tubno,.: Dynamc Response of Euler-Bernoull beams to Resonant Harmonc ДИНАМИЧКИ ОДГОВОР НА ХАРМОНИЧНУ ПОПРЕЧНУ ПОБУДУ КОНЗОЛЕ EUER- BERNOUI СА КОНЦЕНТРИЧНИМ МАСАМА А. Скоблар, Р. Жигулић, С. Браут, С. Блажевић У овом раду приказан је прорачун динамичког одговора на хармоничну попречну побуду конзоле Euler-Bernoull са концентричним масама, методом режима суперпозиције. Метод суперпозиције користи режиме облика и координата функције које су изра 7 VO. 45, No, 17 E Transactons

7 чунате одвајањем променљивих и Лаплас трансфор мацијама. Тачност дефинисаних израза су потврђени на примерима са и без Ralegh пригушења. Сви резу лтати су у сагласности са резултатима комерцијал них софтвера на основу метода коначних елемената. E Transactons VO. 45, No, 17 7