NONLINEAR NATURAL FREQUENCIES OF A TAPERED CANTILEVER BEAM

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1 Advanced Steel Constructon Vol. 5, No., pp (9) 59 NONLINEAR NATURAL FREQUENCIES OF A TAPERED CANTILEVER BEAM M. Abdel-Jaber, A.A. Al-Qasa,* and M.S. Abdel-Jaber Department of Cvl Engneerng, Faculty of Engneerng, Appled Scence Unversty, Amman, Jordan Department of Mechancal Engneerng, Faculty of Engneerng and Technology, Unversty of Jordan, Amman, Jordan Department of Cvl Engneerng, Faculty of Engneerng and Technology, Unversty of Jordan, Amman, Jordan *(Correspondng author: E-mal: alqasa@ju.edu.jo) Receved: 5 November 7; Revsed: 7 January 8; Accepted: January 8 ABSTRACT: The non-lnear natural frequences of the frst three modes of a clamped tapered beam are nvestgated. The mathematcal model s derved usng the Euler-Lagrange method and the contnuous system s dscretzed usng the assumed mode method. The resulted un-modal nonlnear equaton of moton was solved usng the harmonc balance (HB) to obtan approxmate analytcal expressons for the nonlnear natural frequences. Results were obtaned for two types of taper; double taper,.e. the beam wdth and thckness are vared lnearly along the beam axs and sngle taper wedge shaped beams,.e. the varaton s n thckness only. The effects of vbraton ampltude and taper rato on the nonlnear natural frequences for the frst three modes are obtaned and presented n non-dmensonal form. Keywords: Nonlnear, free vbraton, harmonc balance, tapered beam, cantlever beam. INTRODUCTION It s known that a lot of engneerng structures can be modeled as beam. Some can be modeled as tapered beams, such as ples, fxed-type platforms, tower structures, hgh buldngs and robot arms. In general, due to varous exctaton loads wnd and waves, hgh aspect rato and flexblty such structures mght have large deformatons and deflectons. The predcton of the dynamc behavor s extremely mportant durng the desgn process. The lnear vbraton theory predcts the natural frequences to be ndependent of the ampltude. But n many cases, the deflecton n structures may reach large values and consequently, usng the lnear vbraton assumpton s not vald. In order to take nto consderaton the nonlneartes arsed due to large deformatons, the nonlnear vbraton theory must be used to predct wth hgh accuracy the dynamc behavor lke; natural frequences and dynamc responses. In ths paper, the non-lnear planar large ampltude free vbraton of a tapered cantlever beam s studed for two cases; double tapered beam and sngle tapered wedge shaped beam. Most of the pertnent lterature s drected towards the calculaton of lnear natural frequences and mode shapes [-6] wth dfferent end condtons and wth attached nerta elements at the free end of the beam. In [7], a smple formulaton for the large ampltude free vbratons of tapered beams was presented. The method s based on an teratve numercal scheme to obtan results for tapered beams wth rectangular and crcular cross sectons. The objectve of the present work s to extend the results and analyss obtaned n [8] to study the non-lnear planar large ampltude free vbratons of a cantlever tapered beam for the cases of a double taper beam and a sngle taper wedge shaped beam. The mathematcal model s derved The Hong Kong Insttute of Steel Constructon

2 6 Nonlnear Natural Frequences of a Tapered Cantlever Beam usng the Lagrange method and the resultng contnuous equaton s dscretzed usng the assumed mode method [9, ]. The nextensblty condton [] s used to relate the axal shortenng due to transverse deflecton n the formulaton of the knetc energy of the beam and the nonlnear curvature s used n the potental energy expresson.. MATHEMATICAL MODEL. System Descrpton and Assumptons A schematc drawng of the beam under study s shown n Fgure. The physcal propertes, modulus of elastcty E and densty, of the beam are constants. Whle the beam thckness and wdth are vared lnearly along the beam axs. The beam s clamped at one end and free at the other, the cross sectonal area and moment of nerta at the large (Clamped end) are A b and I b, respectvely. The thckness of the beam s assumed to be small compared to the length of the beam, so that the effects of rotary nerta and shear deformaton can be gnored. The beam transverse vbraton can be consdered to be purely planar and the ampltude of vbraton may reach large values. I A b b I A b b b b Fgure. A Schematc Drawng of the Tapered Cantlever Beam

3 M. Abdel-Jaber, A.A. Al-Qasa and M.S. Abdel-Jaber 6. Dervaton of the Equaton of Moton Usng the deformed beam, see Fgure, the potental energy of the beam can be wrtten as V El I( ) R d ) () where s / l I s the varable second moment of area and R s the curvature of the beam neutral axs and can expressed as [8, ] R, () where /l, the prme s the dervatve wth respect to the dmensonless length,, and s the change of the slope along the beam (see Fgure ). In order to express the exact curvature n terms of the transverse deflecton, v, t s noted that cos sn. Ths mples that sn d v/ ds v (as n Fgure ). Dfferentatng sn v wth respect to, usng the above trgonometrc denttes, expandng the resulted term n a power seres and retanng the terms up to the fourth order, the nonlnear curvature R can be wrtten as R 4 v" v" v' () The knetc energy T of the beam can be wrtten as ( ) d (4) T l A u v where u s the axal shortenng due to bendng deformaton as can be seen n Fgure. The nextensblty condton dctates that a total axal shortenng u s gven by [] u cosd v d (5) v Expandng the radcal term n a power seres, assumng that be represented as, the axal shortenng can 4 u v v d 4 (6) Dfferentatng Eq. 6 wth respect to tme yelds d u v d (7) dt

4 6 Nonlnear Natural Frequences of a Tapered Cantlever Beam y ds dv s v x u s Fgure. The Deformed Inextensble Beam The Lagrangan of the beam under consderaton can be expressed as L T V (8) It s clear that the contnuous system n Eq. 8 does not admt a closed form soluton. The nterest here s n the case where the beam moton s governed by sngle actve mode. The Lagrangan of the system L can be dscretzed by usng the assumed mode method and substtutng v t q t (9), s the normalzed, self-smlar (.e. ndependent of the moton ampltude) assumed where mode shape of the beam and for a double tapered beam s (see reference [6]):. In the present work q t s an unknown tme modulaton of the assumed deflecton mode ) C J ( Z) C Y ( Z) C I ( Z) C K ( Z) () ( 4 where A( ) and A b I( 4 ) I b, and for wedge-type beams (sngle taper) ( ) CJ( Z) CY ( Z) CI( Z) C4K( Z) () And A( ) and A b I( ) I b For both cases Z, A L 4 4 b l, l EIb s the lnear frequency of vbraton, J and Y

5 M. Abdel-Jaber, A.A. Al-Qasa and M.S. Abdel-Jaber 6 are Bessel functons of the frst and second knd, respectvely, and I and K are modfed Bessel functons of the frst and second knd, respectvely. C, C, C and C 4, are arbtrary constants to be determned by mposng the followng boundary condtons to both ends of the beam; zero bendng moment and zero shear force at the free end and zero deflecton and zero slope at the clamped end. EI ( ) d d ( ) EI ( ) = ( ) () Usng Eqs. 7, 9 and or the Lagrangan expresson of the tapered beam under consderaton can be expressed as, for the -th mode of vbraton L l ( q q q q q ) () 4 4 where * A d (4) * A d d (5) I * d (6) * 4 I d (7) For the double tapered beam; wedge beam * A and A b I * A Ab and * Ib. I * 4 Ib and for sngle tapered beam, the Applyng the Euler-Lagrangan equaton to the system Lagrangan d L L dtq q (8) the followng non-lnear, non-dmensonal un-modal equaton of moton s obtaned:

6 64 Nonlnear Natural Frequences of a Tapered Cantlever Beam q q q qq q q (9) 4 Due to the fact that, some of the coeffcents, defned by Eqs. 4-7, may have large values, Eq. (9) for convenence s scaled to the form; q q q q qq q () * A dot s used to denote a dervatve wth respect to the non-dmensonal tme. t t and 4 are dmensonless coeffcents., Eq. descrbes the non-lnear non-dmensonal planar flexural free vbraton of the nextensble tapered beam. In ths equaton, the terms q q and qq are nerta non-lneartes arsed from usng the nextensblty condton n the knetc energy and they are of softenng type (.e., they lead to a decrease n the natural frequency when the vbraton ampltude ncreases). The non-lnear term q s due to the potental energy stored n bendng and arses as a result of usng non-lnear curvature and t s of hardenng statc type (.e., t leads to an ncrease n the natural frequency when the vbraton ampltude ncreases). The nonlnear natural frequences of the beam are domnated by the two competng non-lneartes mentoned above, and the behavour of the tapered beam consdered n ths work s ether hardenng or softenng dependng on the rato [].. METHOD OF SOLUTION The calculatons of the coeffcents n Eqs. 4-7, and ndcate that the non-lnear oscllator descrbed n Eq. s strongly nonlnear, and the nonlnear natural frequences are calculated usng the Harmonc Balance method (HB). The ntal condtons are taken to be q( ) A and q ( ) where A s the ampltude of the moton. Accordng to the HB method, an approxmate sngle term soluton (SHB) [8, 9] takes the form * * qt ( ) Acos( t ) () where s the non dmensonal nonlnear natural frequency,.e. the rato of the nonlnear frequency to the lnear one. Substtutng Eq. and ts dervatves nto Eq. and equatng coeffcents, one obtans ( 4) A ( ) A () To mprove the accuracy of the assumed soluton, more terms can be added and a two term soluton s sought (THB), such that qt ( ) Acos( t ) Acos( t ) () * * * As one can see, the added term s of order three and ths s due to the fact that the nonlnear terms

7 M. Abdel-Jaber, A.A. Al-Qasa and M.S. Abdel-Jaber 65 n Eq. q, q q and q q are odd and of order three. Usng the above mentoned ntal condtons, yelds A A A (4) Substtutng Eq. () and ther dervatves nto Eq. () and equatng the coeffcent of each of the assumed harmoncs, one obtans A ( 4)( A A ) ( /)( A 9 A ) ( ( A / ) (9 5 A ) (5) ( /4)( A AA A ) ( /)( ) A AA A (6) Eqs. 5 and 6 are solved numercally for a gven ampltude A, usng an teratve technque wth 6 an accuracy of. 4. RESULTS AND DISCUSSION The derved non-lnear non-dmensonal un-modal equaton of moton gven n () s vald for vbratons wth large ampltudes and small rotatons,.e. v, whch s the case n structures wth hgh slender rato. The coeffcents of the terms gven n Eq. 9 are calculated by ntegratng numercally the coeffcents gven n Eqs Also, t s worth mentonng that the range of moton ampltudes to be consdered n the present work, (.e., the values of vbraton ampltude A ), s assumed to be up to. for the frst mode,. 4 for the second mode and. for the thrd mode, to be consstent wth the assumpton of large ampltude vbraton. For example, a vbraton ampltude of corresponds to a rato of tp dsplacement/length of the beam. The accuracy of the calculated nonlnear natural frequences was frst examned by comparng the results obtaned usng: the Harmonc Balance method usng sngle (SHB) and two terms (THB) gven n Eqs. and 6, for the double tapered beam and b b., as shown n Fgure. Results were obtaned and presented n Fgures -5, for the frst three modes. As one, can see the SHB fals to predct the correct nonlnear natural frequency, specally for the second and thrd mode, and the THB method s more accurate. Consequently, all the remanng results were obtaned usng the method of Harmonc Balance method wth two terms (THB). In Fgures 6-8, results were obtaned for the double tapered beam and for dfferent values of the taper rato b b. Results have shown that the behavor of the frst and second modes s changed from hardenng to softenng when the taper rato s ncreased, whle the thrd mode s of a softenng type regardless the value of the taper rato. Ths s due to the fact that when the taper rato ncreases the mode shape s modfed accordngly, whch n turn affects the values of the calculated coeffcents gven n Eq. 9 and the values of and.

8 66 Nonlnear Natural Frequences of a Tapered Cantlever Beam st Mode,. Nonlnear Natural Frequency, Fgure. Nonlnear Natural Frequency Versus Ampltude of the Frst Mode, Double Tapered Beam and for. SHB, THB. nd Mode, Nonlnear Natural Frequency, Fgure 4. Same as n Fgure, but for the Second Mode

9 M. Abdel-Jaber, A.A. Al-Qasa and M.S. Abdel-Jaber 67 rd Mode,. Nonlnear Natural Frequency, Fgure 5. Same as n Fgure, but for the Thrd Mode. st Mode, Nonlnear Natural Frequency, Fgure 6. Nonlnear natural frequency versus Ampltude of the frst mode, double tapered..,.,.,.4,. 5

10 68 Nonlnear Natural Frequences of a Tapered Cantlever Beam nd Mode,. Nonlnear Natural Frequency, Fgure 7. Same as n Fgure 6, but for the second mode rd Mode,. Nonlnear Natural Frequency, Fgure 8. Same as n Fgure 6, but for the thrd mode

11 M. Abdel-Jaber, A.A. Al-Qasa and M.S. Abdel-Jaber 69 The nonlnear equaton of moton gven n Eq., as mentoned before, s domnated by the non lneartes; ( q q and qq ) and q, nerta and statc nonlneartes, respectvely and the behavor s of hardenng type when the rato ( ). 6 and of softenng type when ( ).6 []. In Fgures 9- a comparson between the double tapered and wedge type beams for dfferent values of taper rato b b s presented also. Results have shown that for a gven value of taper rato, the nonlnear natural frequency of a double tapered beam s hgher than that of a wedge type sngle taper beam.. st Mode, Nonlnear Natural Frequency, Fgure 9. A comparson between the double tapered beam (D) and wedge sngle tapered (S) of the Nonlnear natural frequency versus Ampltude of the frst mode.. (D),. (S). (D),. (S).5 (D),. 5 (S)

12 7 Nonlnear Natural Frequences of a Tapered Cantlever Beam nd Mode,. Nonlnear Natural Frequency, Fgure. Same as n Fgure 9, but for the second mode. rd Mode,. Nonlnear Natural Frequency, Fgure. Same as n Fgure 9, but for the thrd mode.

13 M. Abdel-Jaber, A.A. Al-Qasa and M.S. Abdel-Jaber 7 5. CONCLUSIONS A mathematcal model for calculatng the nonlnear natural frequences of a tapered cantlever beam s derved. The axal shortenng due to transverse deflecton and the nonlnear curvature are used n the formulaton of the knetc and potental energy, respectvely. The assumed mode method s used to dscretze the contnuous Lagrangan of the system and the resulted un-modal nonlnear dfferental equaton of moton s solved usng the Harmonc Balance method (HB) to calculate the nonlnear natural frequences for the frst three modes of vbratons and for dfferent values of the taper rato b b for two types of beams; double taper and sngle taper wedge shaped. Results have shown that for the frst and second modes the behavor s changed from hardenng to softenng type when the taper rato s ncreased, whle the thrd mode s of a softenng type regardless the value of the taper rato. Also, for a gven value of a taper rato, the nonlnear natural frequency of a double tapered beam s hgher than that of a sngle tapered beam. From the results presented for the effect of taper rato on the nonlnear natural frequences, a qualtatve change was notced,.e. when the nonlnear natural frequency changes from hardenng type to a softenng type. Ths would requre a more detaled analyss to study the dynamc response of the beam under a gven exctaton load whch s currently under consderaton and beyond the scope of the present work. ACKNOWLEDGMENT Dr. Ahmad AL-Qasa acknowledges the support of the Deanshp of Academc research at the Unversty of Jordan. REFERENCES [] Aucello, N.M. and Nole, G., Vbratons of a Cantlever Tapered Beam wth Varyng Secton Propertes and Carryng a Mass at the Free End, Journal of Sound and Vbraton, 998, Vol. 4, pp [] Nagaya, K. and Ha, Y., Sesmc Response of Underwater Members of Varable Cross Secton, Journal of Sound and Vbraton, 985, Vol. 9, pp [] Laura, P.A. and Guterrez, R.H., Vbratons of an Elastcally Restraned Cantlever Beam of Varyng Cross Sectons wth Tp Mass of Fnte Length, Journal of Sound and Vbraton, 986, Vol. 8, pp. -. [4] Shong, J.W. and Chen, C.T., An Exact Soluton for the Natural Frequency and Modes Shapes of An Immersed Elastcally Wedge Beam Carryng an Eccentrc Tp Mass wth Mass Moment of Inerta, Journal of Sound and Vbraton, 5, Vol. 86, pp [5] Chen, D.W. and Wu, J.S., The Exact Solutons for the Natural Frequency and Modes Shapes of Non-Unform Beams wth Multple Sprng-Mass Systems, Journal of Sound and Vbraton,, Vol. 55, pp [6] Goorman, D.J., Free Vbratons of Beams and Shafts, John-Wley & Sons, 975, pp. 65. [7] Rao, B.N. and Rao, G.V., Large Ampltude Vbratons of a Tapered Cantlever Beam, Journal of Sound and Vbraton, 988, Vol. 7, pp

14 7 Nonlnear Natural Frequences of a Tapered Cantlever Beam [8] Al-Qasa, A.A., Shatnaw, A., Abdel-Jaber, M.S., Abdel-Jaber M. and Sadder, S., Non-Lnear Natural Frequences of a Tapered Cantlever Beam,. Proceedngs of the Sxth Internatonal Conference on Steel and Alumnum Structures (ICSAS'7), Oxford, UK, July 4-7, 7, pp [9] Al-Qasa, A.A., Effect of Flud Mass on Non-Lnear Natural Frequences of a Rotatng Beam, Proceedngs of ASME Pressure Vessels and Ppng Conference PVP, Cleveland, OHIO, USA, July -4, PVP--78, pp [] Al-Qasa, A. A. and Hamdan, M. N., On the Steady State Response of Oscllators wth Statc and Inerta Non-Lneartes, Journal of Sound and Vbraton, 999, Vol., pp [] Al-Qasa, A.A. and Hamdan, M.N., Bfurcaton and Chaos of an Immersed Cantlever Beam n a Flud and Carryng an Intermedate Mass, Journal of Sound and Vbraton,, Vol. 5, pp