Flapwise bending vibration analysis of double tapered rotating Euler Bernoulli beam by using the differential transform method

Transcription

1 Meccanica 2006) 41: DOI /s z Flapwise bending vibration analysis of double tapered rotating Euler Bernoulli beam by using te differential transform metod Ozge Ozdemir Ozgumus Metin O. Kaya Received: 12 November 2005 / Accepted: 27 February 2006 / Publised online: 31 October 2006 Springer Science+Business Media B.V Abstract In tis study, te out-of-plane free vibration analysis of a double tapered Euler Bernoulli beam, mounted on te peripery of a rotating rigid ub is performed. An efficient and easy matematical tecnique called te Differential Transform Metod DTM) is used to solve te governing differential equation of motion. Parameters for te ub radius, rotational speed and taper ratios are incorporated into te equation of motion in order to investigate teir effects on te natural frequencies. Calculated results are tabulated in several tables and figures and are compared wit te results of te studies in open literature were a very good agreement is observed. Keywords Differential Transform Metod Tapered Euler beam Nonuniform Euler beam Mecanics of solids and structures 1 Introduction Te dynamic caracteristics, i.e., natural frequencies and related mode sapes, of rotating tapered beams are very important for te design and performance evaluation in several engineering applications including rotating macinery, elicopter O. Ozdemir Ozgumus M. O. Kaya B) Istanbul Tecnical University, Faculty of Aeronautics and Astronautics, 34469, Maslak, Istanbul, Turkey kayam@itu.edu.tr blades, windmills, robot manipulators and spinning space structures because tey are required to determine resonant responses and to perform forced vibration analysis. As a result, rotating tapered beams ave been te subject of interest for many investigators. Tis study is an etension of te autors previous work 1]. Parameters for te ub radius, rotational speed and taper ratios are incorporated into te equation of motion in order to investigate teir effects on te natural frequencies. After solving te problem by DTM, calculated results tat can be used as reference values for te future studies are tabulated in several tables and figures. Te superiority of DTM over oter metods is its simplicity and accuracy in calculating te natural frequencies and plotting te mode sapes and also, te variety of te problems to wic it may be applied. Zou 2] introduced te concept of tis metod by using it to solve bot linear and nonlinear initial value problems in electric circuit analysis. Since te metod can deal wit nonlinear problems, Ciou and Tzeng 3] applied te Taylor transform to solve nonlinear vibration problems. Additionally, te metod may be used to solve bot ordinary and partial differential equations so Jang et al. 4] applied te two-dimensional differential transform metod to te solution of partial differential equations. Abdel and Hassan 5] adopted te metod to solve some eigenvalue problems. Since previous studies ave sown tat DTM is an

2 662 Meccanica 2006) 41: R Ω y O z O b 0 d a) b) Fig. 1 a) Top view b) Side view of a rotating double tapered Euler Bernoulli beam efficient tool to solve nonlinear or parameter varying systems, recently it as gained muc attention by several researcers Cen and Ju 6], Arikoglu and Ozkol 7], and Bert and Zeng 8]. 2 Formulation Te governing partial differential equation of motion is derived for te out-of-plane free vibration of a rotating tapered cantilever Euler Bernoulli beam represented by Fig. 1. Here, te cantilever beam of lengt is fied at point O to a rigid ub wit radius R and it is rotating at a constant angular velocity,. Te beam tapers linearly from a eigt 0 at te root to at te free end in te z plane and from a breadt b 0 to b in te y plane. Te eigt taper ratio, c and te breadt taper ratio, c b, wose descriptions are going to be given in te following sections must be c < 1 and c b < 1 because oterwise te beam tapers to zero between its ends. In te rigt-anded Cartesian co-ordinate system, te -ais coincides wit te neutral ais of te beam in te undeflected position, te z-ais is parallel to te ais of rotation but not coincident) and te y-ais lies in te plane of rotation. Terefore, te principal aes of te beam cross-section are parallel to y and z directions, respectively. 0 b Te following assumptions are made in tis study, a. Te out-of-plane displacement of te beam is small. b. Te cross-sections tat are initially perpendicular to te neutral ais of te beam remain plane and are perpendicular to te neutral ais during bending. c. Te beam material is omogeneous and isotropic. d. Coriolis effects are not included. Moreover, te beam considered ere ave doubly symmetric cross-sections suc tat te sear center and te centroid of eac cross-section are coincident. Terefore, tere is no coupling between bending vibrations and torsional vibration. 2.1 Governing differential equations of motion According to te Euler Bernoulli beam teory, te governing differential equation of motion for te out-of-plane bending motion is as follows ρa 2 w t EI y 2 w 2 ) T w ) = p w 1) were w is te out-of-plane bending displacement, EI y is te bending rigidity, ρa is te mass per unit lengt, T is te centrifugal force, p w is te applied force per unit lengt in te flapwise direction, is te spanwise position and t is te time. Since free vibration is considered in tis study, p w is taken to be zero. Te centrifugal force, T, tat varies along te spanwise direction of te beam is given by T) = ρa 2 R + )d. 2) Te boundary conditions for a cantilever Euler Bernoulli beam can be epressed as follows w = w = 0 at = 0 3) 2 w 2 = 3 w = 0 at =. 4) 3 A sinusoidal variation of w, t) wit a circular natural frequency ω is assumed and te displacement function is approimated as w, t) = w ) e iωt 5)

3 Meccanica 2006) 41: Substituting Eq. 5) into Eq. 1), te equation of motion can be rewritten as follows ) ω 2 ρa w + d2 d 2 w d 2 EI y d 2 d T d w ) = 0. 6) d d 2.2 Tapered beam formulation and dimensionless parameters Te general equations for te breadt b ), te eigt ), te cross-sectional area, A ) and te second moment of area, I y ) of a beam tat tapers in two planes are given by ) m b ) = b 0 1 c b 7a) ) n ) = 0 1 c 7b) ) m ) n A ) = A 0 1 c b 1 c 7c) ) m I y ) = I y0 1 c b 1 c ) 3n. 7d) Here te breadt taper ratio, c b and te eigt taper ratio, c can be given by c b = 1 b b 0 c = a) 8b) Knowing tat te subscript ) o denotes te values at te root of te beam, te following formulas can be introduced. A 0 = b 0 0 9a) I y0 = b b) Values of n = 1 and m = 1 are used in tis study to model te beam tat tapers linearly in two planes. Young s modulus E and density of te material, ρ are assumed to be constant so tat te mass per unit lengt ρa and te bending rigidity EI y vary according to te Eqs. 7c) 7d). Te dimensionless parameters tat are used to make comparisons wit te studies in open literature can be given as follows 9] ξ =, δ = R, w ξ) = w, η2 = ρa EI y0, µ 2 = ρa 0 4 ω 2. 10) EI y0 Here δ is te ub radius parameter, η is te rotational speed parameter, µ is te frequency parameter, ξ is te dimensionless distance and w is te dimensionless flapwise deformation. Using te first two dimensionless parameters and Eq. 7c), te dimensionless epression for te centrifugal force can be given by Tξ) = ρa c b c 4 +δ 1 2 c bδ+c δ 1) 1 3 c b+c δ c b c δ) ξδ+ ξ 2 2 c bδ+c δ 1) + ξ 3 3 c b +c δ c b c δ) ξ 4 ] 4 c bc 11) Substituting tapered beam formulas, dimensionless parameters and Eq. 11) into Eq. 6), te following dimensionless equation of motion is obtained for te linear taper case m = 1, n = 1). ] d 2 1 c dξ 2 b ξ)1 c ξ) 3 d2 w dξ 2 µ 2 1 c b ξ)1 c ξ) w η { 2 d c b c dξ 4 1 ξ 4 ) + δ 1 ξ) 12) c bδ c δ) 1 ξ 2) c b c + c b c δ) ] } 1 ξ 3) d w = 0. dξ Additionally, te dimensionless boundary conditions can be epressed as follows w = d w = 0 at ξ = 0 13) dξ d 2 w dξ 2 = d3 w = 0 at ξ = 1. 14) dξ 3 3 Te Differential transform metod Te Differential Transform Metod is a transformation tecnique based on te Taylor series epansion and it is a useful tool to obtain analytical solutions of te differential equations. In tis metod, certain transformation rules are applied and te governing differential equations

4 664 Meccanica 2006) 41: Table 1 DTM teorems for te equations of motion Original function Transformed functions f ) = g ) ± ) F k ] = G k ] ± H k ] f ) = λg ) F k ] = λg k ] f ) = g ) ) F k ] = k G k l ] H l ] f ) = dn g) F k ] = k+n)! d n k! G k + n ] f ) = n F k ] { 0 if k = n = δ k n) = 1 if k = n l=0 Table 2 DTM teorems for boundary conditions = 0 = 1 Original B.C. Transformed B.C. Original B.C. Transformed B.C. /f 0) = 0 F 0] = 0 f 1) = 0 df 0) = 0 F 1 ] df = 0 1) = 0 d d d 2 f 0) = 0 F 2] = 0 d 2 f 1) = 0 d 2 d 2 d 3 f 0) = 0 F 3] = 0 d 3 f 1) = 0 d 3 d 3 F k ] = 0 kf k ] = 0 kk 1)F k ] = 0 kk 1)k 2)F k ] = 0 and te boundary conditions of te system are transformed into a set of algebraic equations in terms of te differential transforms of te original functions and te solution of tese algebraic equations gives te desired solution of te problem wit great accuracy. It is different from ig-order Taylor series metod because Taylor series metod requires symbolic computation of te necessary derivatives of te data functions and is epensive for large orders. Consider a function f ) wic is analytic in a domain D and let = 0 represent any point in D. Te function f ) is ten represented by a power series wose center is located at 0. Te differential transform of te function f ) is given by F k ] = 1 k! d k f ) d k ) = 0 15) were f ) is te original function and F k ] is te transformed function. Te inverse transformation is defined as f ) = 0 ) k F k ] 16) Combining Eqs. 15) and 16), we get ) 0 ) k d k f ) f ) = k! d k. 17) = 0 Considering Eq. 17), it is noticed tat te concept of differential transform is derived from Taylor series epansion. However, te metod does not evaluate te derivatives symbolically. In actual applications, te function f ) is epressed by a finite series and Eq. 17) can be rewritten as follows ) q 0 ) k d k f ) f ) = k! d k 18) = 0 wic means tat f ) = k=q+1 0 ) k k! ) d k f ) d k = 0 is negligibly small. Here, te value of q depends on te convergence rate of te natural frequencies. Teorems tat are frequently used in te transformation of te differential equations and te boundary conditions are introduced in Tables 1 and 2, respectively.

5 Meccanica 2006) 41: Table 3 Variation of te natural frequencies of a nonrotating Euler Bernoulli beam wit different combinations of breadt and eigt taper ratios δ = 0) c c b a) Fundamental natural frequency a a a a a a a a a a a a b) Second natural frequency a a a a a a a a a a a a c) Tird natural frequency a a a a a a a a a a a a d) Fourt natural frequency a a a a a a a a a a a a

6 666 Meccanica 2006) 41: Table 3 c Continued c b e) Fift natural frequency a a a a a a a a a a a a f) Sit natural frequency a a a a a a a a a a a a g) Sevent natural frequency a a a a a a a a a a a a ) Eigt natural frequency a a a a a a a a a a a a a Results of Downs 10]

7 Meccanica 2006) 41: a) 16 b) 36 First Natural Frequency c b = c = 0.8 c b = c = 0.4 c b = c = Second Natural Frequency c b = c = 0 c b = c = 0.4 c b = c = c) 80 d) 140 Tird Natural Frequency Fourt Natural Frequency 80 c b = c = 0 c c c 0.4 b = c = 0 b = = c b = c = 0.4 c b = c = 0.8 c c b = = Fig. 2 Variation of te a) first b) second c) tird d) fourt natural frequency wit respect to te rotational speed parameter, η and taper ratios, c b and c.δ = 0) 4 Formulation wit DTM In te solution step, te differential transform metod is applied to Eq. 12) by using te teorems introduced in Table 1 and te following analytical epression is obtained. cb c 4 k + 1)k 2) η2 c b c µ 2] W k 2 ] c b c + c b c δ) η 2 k 1)k + 1) + c b + c ) µ 2] W k 1 ] + c b c 3 k 1) k k + 1)k + 2) + 1 ] 2 c bδ c δ + 1) k k + 1) η 2 µ 2 W k ] + c 2 3c b + c )k 2) k 1) k k + 1) + k + 1) 2 δη 2] W k + 1 ] 19) { + 3c c b + c )k + 1) 2 k + 2) 2 1 +η 2 k + 1)k + 2) c bδ + c δ) + 1 )]} 1 3 c b+c c b c δ) 4 c bc +δ W k+2 ] ] k + 1)k + 2) 2 k + 3)c b + 3c ) W k + 3 ] + k + 1)k + 2)k + 3)k + 4) W k + 4 ] = 0. Additionally, te differential transform metod is applied to Eqs. 13) and 14) at 0 = 0byusing te teorems introduced in Table 2 and te following transformed boundary conditions are obtained. W 0] = W 1 ] = 0 at ξ = 0 20)

8 668 Meccanica 2006) 41: Table 4 Variation of te first four natural frequency parameters, µ wit respect to te rotational speed parameter, η,andte ub radius parameter, δ c b = 0, c = 0.5) Natural Frequency Parameters η δ = 0 δ = a a a a a a b b b a a a a a a b b b a a a a a a b b b a a a a a a b b b η δ = 0.5 δ = a Özdemir and Kaya 1] b Hodges and Rutkowski 11] kk 1)W k ] = kk 1)k 2) k=2 k=3 W k ] = 0 at ξ = 1 21) In Eqs. 19) 21), W k ] is te differential transform of w ξ). Using Eq. 19), W k ] values for k = 4, 5... can now be evaluated in terms of c b, c, µ, η, d 2 and d 3. Tese values, acieved by using te Matematica computer package for δ = 0, are as follows W 2] = d 2 W 3] = d 3 W 4] = 1 2 c b + 3c ) d 3 1 { c 2 6η2 + 4η 2 c +c b 4η 2 3c η 2 24)] } d 2 W 5 ] = 1 { 12 c c b + c ) d 2 108c 2 + 2c 3) η c b + 3c ) 72c 2 + 6η2 4c η 2 4c b η 2 72c c b + 3c c b η )d 2 2 ]} + 216c + 72c b ) d 3 Te coefficients are obtained to numerical accuracy and te constants d 2 and d 3 tat appear in W k ] s are given by d 2 = W 2] = 1 2! d 3 = W 3] = 1 3! d 2 w dξ 2 d 3 w dξ 3 5 Results and discussions ) ) =0 =0, 22) Te computer package Matematica is used to write a computer code for te epressions obtained using DTM. In order to validate te calculated results, comparisons wit te studies in open literature are made and related grapics are plotted. Te effects of te taper ratios, c b and c, te rotational speed parameter, η and te ub radius parameter, δ, are investigated and te calculated results

9 Meccanica 2006) 41: a) Natural Frequencies 4 t Mode rd Mode 40 2 nd Mode 1 st Mode c b Natural Frequencies rd Natural Frequency 2 nd Natural Frequency 1 st Natural Frequency b) Natural Frequencies t Mode 3 rd Mode 2 nd Mode 1 st Mode c Fig. 3 Effect of te a) breadt taper ratio, c b b) te eigt taper ratio, c, on te first four natural frequencies. δ = 0; η = 0) tat can be used as reference values for te future studies are tabulated in several tables and figures. In Table 3a, variation of te first eigt natural frequencies of a nonrotating Euler Bernoulli beam wit different combinations of breadt and eigt taper ratios is introduced and te results are compared wit te ones in te study of Downs 10] and as it is seen in tese tables, tere is a very good agreement between te results. In Fig. 2a d, variation of te first four natural frequencies wit respect to bot te taper ratios and te rotational speed parameter is introduced. Here bot of te taper ratios ave te same value. As it is observed in Figs. 2a d, te rotational speed parameter as an increasing effect on te natural frequencies at every taper ratio because te cen- Fig. 4 Variation of te first tree natural frequencies wit respect to te ub radius parameter, δ and te rotational speed parameter, η δ = 1, ----; δ = 0.5,.. ; δ = 0, ) trifugal force tat is proportional to te rotational speed as a stiffening effect tat increases te natural frequencies. Additionally, wen te Figs. 2a d are eamined, it is seen tat te taper ratios ave an increasing effect on te fundamental natural frequency wile tey ave an decreasing effect on te oter natural frequencies. Moreover, in order to observe te effects of te taper ratios seperately, Fig. 3a b can be considered. As it is seen in tese figures, te breadt taper ratio, c b, as very little, even no influence on te flapwise bending frequencies wile te eigt taper ratio, c, as a linear decreasing effect on te natural frequencies ecept te fundamental natural frequency. Te rotational speed parameter, η, and te ub radius parameter, δ ave significant effects on te values of te natural frequencies as it is introduced in Table 4. Tese results are compared wit te ones in Özdemir and Kaya 1] and Hodges and Rutkowski 11]. Te values of te natural frequency parameter, µ, increase as te rotational speed parameter, η, increases and te rate of increase becomes larger wit te increasing ub radius parameter,δ because te centrifugal force is directly proportional to bot of tese parameters as it can be seen in Eq. 2). For a better insigt and also in order to establis te trend, tese effects are sown in Fig. 4 were te first tree natural

10 670 Meccanica 2006) 41: frequencies are plotted for tree different values of te ub radius parameter and for several values of te rotational speed parameter. 6 Conclusion A new and semi-analytical tecnique called te differential transform metod is applied to te problem of a rotating double tapered Euler Bernoulli beam in a simple and accurate way and te natural frequencies are calculated and te related grapics are plotted. Te effects of te ub radius, taper ratios and rotational speed are investigated. Te numerical results indicate tat te fletural natural frequencies increase wit te rotational speed and ub radius wile tey decrease wit te eigt taper ratio. Te calculated results are compared wit te ones in literature and great agreement is considered. References 1. Özdemir Ö, Kaya MO 2006) Flapwise bending vibration analysis of a rotating tapered cantilevered Bernoulli Euler beam by differential transform metod. J Sound and Vib 289: Zou JK 1986) Differential transformation and its application for electrical circuits. Huazong University Press, PR, Cina 3. Ciou JS, Tzeng JR 1996) Application of te Taylor transform to nonlinear vibration problems. Trans Am Soc Mec Eng J Vib Acoust 118: Jang MJ, Cen C, iu YC 2001) Two-dimensional differential transform for partial differential equations. Appl Mate Comput 121: Abdel IH, Hassan H 2002) On solving some eigenvalue-problems by using a differential transformation. Appl Mat Comput 127: Cen CK, Ju SP 2004) Application of differential transformation to transient advective dispersive transport equation. Appl Mat Comput 155: Arikoglu A, Ozkol I 2005) Solution of boundary value problems for integro-differential equations by using differential transform metod. Appl Mat Comput 1682): Bert CW, Zeng H 2004) Analysis of aial vibration of compound bars by differential transformation metod. J Sound Vib 275: Banerjee JR 2001) Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timosenko beams. J Sound Vib 2471): Downs B 1977) Transverse vibrations of cantilever beam aving unequal breadt and dept tapers. J Appl Mec 44: Hodges DH, Rutkowski MJ 1981) Free vibration analysis of rotating beams by a variable order finite element metod. Am Instit Aeronaut Astronaut J 19: