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1 Copyright 2009 by ASME

Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers andInformation in Engineering ConferenceIDETC/CIE 2009August 30 - September 2, 2009, San Diego, California, USA

DETC2009/VIB- 87843FREE VIBRATION ANALYSIS OF ROTATING TAPERED BRESSE-RAYLEIGH BEAMS USING

THE DIFFERENTIAL TRANSFORMATION METHOD

Dominic R. Jackson

Dynamic and Aeroelasticity Group,School of Mechanical, Aerospace and Civil Engineering,

The University of Manchester,Manchester M60 1QD, UK

S. Olutunde Oyadiji

Dynamic and Aeroelasticity Group,School of Mechanical, Aerospace and Civil Engineering,

The University of Manchester,Manchester M60 1QD, UK

Author for correspondence, Phone: (0044) 161-2754348, Email: [email protected].

ABSTRACT

The free vibration characteristics of a rotating taperedRayleigh beam is analysed in this study. First, the strain-displacement relationship for the rotating beam is formulatedand used to derive the kinetic and strain energies in explicitanalytical form. Second, Hamiltons variational principle isused to derive the governing differential equation of motionand the associated boundary conditions. Third, the Differential

Transformation Method (DTM) is applied to reduce thegoverning differential equations of motion and the boundaryconditions to a set of algebraic equations from which thefrequency equation is derived. Next, a numerical algorithmimplemented in the software package Mathematica is used tocompute the natural frequencies of vibration for a few pairedcombinations of clamped, pinned and free end conditions ofthe beam. Also, the variation of the natural frequencies ofvibration with respect to variations in the rotational speed, hubradius, taper ratio and the slenderness ratio is studied. Theresults obtained from the Bresse-Rayleigh theory arecompared with results obtained from the Bernoulli-Euler and

Timoshenko theories to demonstrate the accuracy andrelevance of their application.

Nomenclature

EIxx bending rigidityF0 outboard forceL total length of the beam

T centrifugal forceCx taper ratio in the xoy planeCz taper ratio in the zoy planedy length of a beam elementm mass per unit length

ri non-dimensional hub radius parameters non-dimensional rotational speed parameterw transverse displacement of the beamUK kinetic energyUS strain energyy distance of a differential element measured from the root

of the beam rotational speed in rad/sec variational operator

non-dimensional natural frequencyrx rotary inertia parameterIxx second moment of area about the x axis

1 INTRODUCTION

Many structural elements of rotating machinery likehelicopters, wind turbines, aircraft propellers, satellitemanipulators and turbo-machinery are modelled as rotatingflexible beams. The dynamic behaviour of these structures isquite different from their non-rotating behaviour due theinduced centrifugally stiffening effect as they rotate. Also, thecoefficients of the differential equations governing thedynamics of rotating beams are variable whilst for non-

rotating beams, they are constants. Due to the variablecoefficients, an exact solution of this type of problem is notalways available. This challenging problem has always beenof interest to researchers and a variety of exact analyticalsolutions, approximate and Finite Element Methods (FEM)have been used to study the modal characteristics of rotatingbeams. Most of these studies are either based on the Bernoulli-Euler or the Timoshenko beam theories.

The Bernoulli-Euler theory is the most elementary of the beamtheories used in vibration analysis of beams. It is known that

Proceedings of the ASME 2009 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference

IDETC/CIE 2009August 30 - September 2, 2009, San Diego, Cali fornia, USA

DETC2009-87843


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the theory is accurate enough to analyse beams with verysmall cross-sectional dimensions to bending wavelength ratios(slender beams or beams with high aspect ratios). The theoryignores the effects of rotary inertia and shear deformation.However, for thick beams where the ratio of the cross-sectional dimensions to the bending wavelength is significant,the Bernoulli-Euler theory is known to be inadequate

especially at high rotational speeds or when higher modes ofvibration are considered.

This error is partially corrected by the Bresse-Rayleigh theorywhich basically adds the rotary inertia effect to the Bernoulli-Euler theory. The kinetic energy of the beam in this case isassumed to be the sum of the kinetic energy due to thetransverse displacement and also the rotary kinetic energy dueto the angular displacement or the sectional rotation of adifferential element of the beam. The addition of the rotaryinertia to the Bernoulli-Euler theory according to manypublications was proposed by William Strut, Lord Rayleigh[1] in 1877 and has since acquired the name, the Rayleighbeam theory. Contrary to the general view expressed in theliterature that Rayleigh introduced the concept of the rotaryinertia in the equations of flexural vibrations, it has beenestablished that Bresse [2] made this correction earlier in1866. The authors of this paper would henceforth, refer to theRayleigh theory as the Bresse-Rayleigh theory. The theory isbasically equivalent to the Bernoulli-Euler theory plus therotary inertia effect. This acknowledgment of Bresse as beingthe first to propose rotary inertia correction has been madepreviously, for example, in [3].

Most often, the rotary inertia or the shear deformation effectsare addressed as secondary effects or special cases whenapplying the Timoshenko theory [4]. The literature available

shows that this effect is rarely treated as a special effect on itsown or as a separate theory. A literature survey on the subjectof rotating beam vibration reveals that exact analyticalsolutions for the rotating Bernoulli-Euler and Timoshenkobeams were obtained by Banerjee et al. [4,5] using theDynamic Stiffness Method (DSM). Kaya [6], zdemir andKaya [7], Ozgumus and Kaya [8,9] and Mei [10] obtainedanalytical solutions for rotating beams using the Differential

Transformation Method (DTM). Naguleswaran [11] and Du[12] applied a Power Series Method (PSM) to obtain exactfrequencies and mode shapes of the rotating beam. There is avast literature covering the subject using the Finite ElementMethod (FEM) and other approximate solutions. Amongstthem are the works of Khulief [13], using the FEM, and

Hunter [14], using an Integrating Matrix Method (IMM). Aconcise literature review of the rotating beam vibrationproblem is given by Bazoune [15].

In this study, the rotating beam problem is formulated byincluding the rotary kinetic energy in the strain-displacementrelationship and energy equations. References [16], [17] areamong the few available in the literature on the analysis ofvibration of rotating beams considering only the rotary inertiaeffect. There are of course a few applications of the rotaryinertia effect in the literature but for non-rotating and spinningbeams [18]. The authors of this article wish to contribute tothe existing literature of rotating beams by including only the

rotary inertia effect in addition to the Bernoulli-Euler model asa unique theory known as the Bresse-Rayleigh theory. It isalso assumed that, the rotary inertia does not induce any sheardeformations during the flexural oscillations of the rotatingbeam. The rotary kinetic energy is therefore integrated into thekinetic energy formulation but the strain energy remains thesame as that of the Bernoulli-Euler case. In addition to that,

the DTM, which has not been applied specifically to thisproblem according to the literature, is used to compute thenatural frequencies of the rotating Bresse-Rayleigh (B-R)beam with a linear span-wise taper variation. The followingsection gives a summary of the mathematical formulationsused in deriving the kinematic parameters, energyformulations and the differential equation of motion associatedwith the rotating tapered B-R beam.

2 MATHEMATICAL FORMULATIONS

Figure 1. Rotating tapered beam with a constant width in the xoyplane and a linear taper variation in the zoy plane.

Figure 1 represents a rotating tapered beam with a linear span-wise taper variation in the zoy plane. The Y axis (global) isassumed to coincide with the local y axis of the beam which is

also coincidental with the elastic axis of the beam. The beamrotates uniformly in the XOY plane about the Z axis (global)which is offset rH units from the root and performs harmonicoscillations in the ZOY plane only. The oscillations in theXOY planes can easily be evaluated if the rotational speed andthe oscillations in the ZOY plane are known [11]. I t isassumed that, the cross section profile of the beam issymmetric. This implies that the locus of the centroids andshear centres coincide along the span of the beam and.therefore, there are no bending and torsion coupling effects.Also, it is assumed that the material properties of the beam areisotropic and homogeneous. Finally, the magnitudes of thetransverse and angular displacements are assumed to be smallto allow the application of the small deformation theory. The

differential equation of motion is derived as follows.

Geometry

The definitions of the cross sectional area A(y) and secondmoment of area Ixx(y) variation of the rotating tapered Bresse-Rayleigh beam are given below as:

)1)(0()1)(0()( CzzCxxyA = (1)

{ }12

)1)(0()1)(0()(

3 CzzCxxyI xx

= (2)


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where 0 Cx


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the governing differential equation of motion can be expressedgenerally in a compact symbolic and non-dimensional formas:

0Wc )n(4

1n

4

0m

mnm =

= =

(17)

where the coefficients cm n given in the Appendix B arefunctions of the geometric, kinematic and circular frequencyparameters of the beam. In the above equation, n representsthe order of the derivatives of W and m represents the powersof the variable .This equation can also be expanded andexpressed generally as:

(

) (

) ( )

( ) )18(0Wccccc

WcccccWcc

cccWccc

ccWccccc

444

343

2424140

434

333

2323130

424

323

2222120

414

313

212

11104

043

032

020100

=++++

++++++

++++++

+++++++

4 APPLICATION OF THE DIFFERENTIALTRANSFORMATION METHOD

The Differential Transformation Method (DTM) is a semi-analytic technique derived from the Taylor series expansion.

The DTM can be used to solve both ordinary and partialdifferential equations. It provides an iterative procedure toobtain the Taylor series expansion for solving differentialequations in the form of polynomials without the computationof the actual derivatives. Convergence of the DTM to theexact analytical solution of the governing differential equationof the rotating tapered Bresse-Rayleigh beam is obtainedthrough very simple computational routines which in thisanalysis are implemented in Mathematica.

The concept of the DTM was first introduced by Zhou [19] in1986 for studies on electrical circuits and it has since beenapplied to vibration problems. Malik et al. [20, 21] and Zengand Bert [22] are among the early researchers to apply the

DTM to beam and plate vibration problems. The applicationof the DTM involves the transformation of each term in thegoverning differential equation of motion and the associatedboundary conditions to a set of algebraic equations. Theinverse transform of the solution of the algebraic expressionsis then the required solution. Some basic definitions of theDTM are given below:

Let w(y) be analytic in a given domain R* and y =y0 be asubset of this domain then, the Taylor series expansion of w(y)about the point y =y0 by definition is given by

0yyk

k

0k

0

yd

)y(wd!k

)yy()y(w

=

=

= (18)

If we define the differential transformation W[k] of w(y) as

0yyk

k

dy

)y(wd!k

1]k[W

=

= (19)

and the inverse transformation w(y) of W[k] as

]k[W)yy()y(w0k

k0

=

= (20)

then, a careful observation of equation (18) reveals that acombination of equations (19) and (20) is in fact, the Taylorseries expansion of the function w(y) about the point y =y0. Inthe DTM technique, the inverse function w(y) can beexpressed as the sum of two series expansions;

)21(dy

)y(wd

!k

)yy(

dy

)y(wd

!k

)yy()y(w

0yyk

k

1nk

k0

0yyk

kn

0k

k0

=

+===

+

=

In practical computations, the DTM solution is expressed as afinite series. It is therefore assumed that the second term ofequation (21) is negligibly small and can be dropped. The firstterm of equation (21) therefore converges to the exactanalytical solution with an acceptable degree of accuracy andthe value of n determines the convergence of the solution.Applying the definitions in equations (18) to (21) togetherwith the basic operations of the DTM in Appendix A, the

governing differential equation of motion of the rotatingBresse-Rayleigh beam can be transformed to the following:

=

=

=

==

==

===

++++

++++

+++++++

++++++

+++++++

+++

k

0r22

k

0r21

20

k

0r14

k

0r13

k

0r12

k

0r1110

k

0r04

k

0r03

k

0r02

k

0r0100

]r2k[W)r2k)(r1k](2r[c

]r2k[W)r2k)(r1k](1r[c

]2k[W)2k)(1k(c]r1k[W)r1k](4r[c

]r1k[W)r1k](3r[c]r1k[W)r1k](2r[c

]r1k[W)r1k](1r[c]1k[W)1k(c]rk[W]4r[c

]rk[W]3r[c]rk[W]2r[c]rk[W]1r[c]k[Wc

............]3[)3)(2)(1](1[

]3[)3)(2)(1(

]2[)2)(1](4[

]2[)2)(1](3[.

031

30

024

023

contrkWrkrkrkrc

kWkkkc

rkWrkrkrc

rkWrkrkrc

k

r

k

r

k

r

=

=

=

+++++

+++++

++++

++++


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=

=

=

+++++

+++++

+++++

k

r

k

r

k

r

rkWrkrkrkrc

rkWrkrkrkrc

rkWrkrkrkrc

0

34

033

032

]3[)3)(2)(1](4[

]3[)3)(2)(1](3[

]3[)3)(2)(1](2[.

)21(....0.]r4k[W)r4k)(r3k)(r2k)(r1k](4r[c

]r4k[W)r4k)(r3k)(r2k)(r1k](3r[c



]4k[W)4k)(3k)(2k)(1k(c

k

0r44

k

0r43

k

0r42

k

0r41

40

=++++++

++++++

++++++

++++++

++++++

=

=

=

=

Solving equation (21) with respect to W[k+4] and applying the

definitions in Appendix A, the eigenvalue equations fromwhich the frequency equations for any paired combination ofthe clamped (C), pinned (P) and free (F) boundary conditionscan be derived. For example the frequency equation for therotating tapered cantilever or Clamped-Free (C-F) Bresse-Rayleigh beam is given as:

0)3(W

)2(W

aa

aa

2221

1211 =

(22)

where the elements aij are given by the following expressions:

,.....2 002624400000 )1620000(9720000 )810000(900112422

2 ++= ++ a

,.....6006123600000

)1620000(16200000

)810000(90012

24222++= ++ a

,.....04374000000

)1620000(2430000

)810000(45021

24222+= ++ a (23)

,.....608745000000

)1620000(3240000

)810000(30022

24222++= ++ a

Analogically, the frequency equation for the rotating tapered

Pinned-Free (P-F) Bresse-Rayleigh beam is given as:

0)3(W

)1(W

bb

bb

2221

1211 =

(24)

where the elements bij are given by the following:

,.....16200000

)150(006123600000

))810000(150(900

15011

24222++=

++b

,.....6616200000

)810000(006123600000

)1620000(290012

24242++=

++b

,.....16200000

)150(006123600000

))810000(150(900

15021

4242++=

+b (25)

,.....616200000

)810000(006123600000

)1620000(90022

22242 += ++b

The natural frequencies of vibration i of the rotating taperedBresse-Rayleigh beam for the given boundary conditions arecomputed by equating the determinant of the matrices inequations (22) and (24) to zero and evaluating the roots of theresulting polynomial equations. In this study, a numericalalgorithm implemented in the software package Mathematicais used to compute the natural frequencies or eigenvalues ofequations (22) and (24).

5 RESULTS AND DISCUSSIONS

In this section, the results obtained from the rotating taperedBresse-Rayleigh (B-R) beam are presented in tabular andgraphical form for selected geometric and kinematicparameters. Since there are no results available for the rotatingtapered B-R beam in the present literature, results obtainedusing the Bernoulli-Euler (B-E) and Timoshenko theoriesfrom selected publications as well as B-E and Timoshenkofrequencies computed using the Differential TransformationMethod (DTM) are included in the tables for comparison.

In Table 1, the natural frequencies for a non-rotating Bresse-Rayleigh (B-R) cantilever beam are given with respect to thetaper variation Cz for a fixed slenderness ratio rx = 1/30.

Results obtained using the Bernoulli-Euler theory based on theDTM are also provided by the authors for comparison inaddition to results obtained from references [5] and [18] whichare also based on the Bernoulli-Euler theory. The relative error between the Bresse-Rayleigh (B-R) and the Bernoulli-Euler(B-E) results are also given in the last column.

Table 1. Fundamental natural frequencies of vibration i for non-rotating tapered cantilever Bresse-Rayleigh (B-R) beams with respectto the taper ratio Cz with rx =1/30, ri =0 and Cx =0.

B-R (DTM) B-E (DTM) Ref [5] Ref[18]Cz

s =0 s =0 s =0

0.0 3.517694 3.516015 - 3.516 0.000480.1 3.559286 3.558702 3.55870 3.559 0.00016

0.2 3.607873 3.608275 3.60827 3.608 0.00011

0.3 3.665473 3.666749 3.66675 3.667 0.00035

0.4 3.735036 3.737076 3.73708 3.737 0.00055

0.5 3.821091 3.823785 3.82379 3.824 0.00070

0.6 3.931038 3.934277 3.93428 3.934 0.00082

0.7 4.078085 4.081712 4.08171 4.082 0.00089

0.8 4.293157 4.292495 4.29249 4.292 0.00015

0.9 4.692242 4.632978 4.63073 4.631 0.01279


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Table 2 Fundamental natural frequencies of vibration i of non-rotating uniform cantilever Bresse-Rayleigh beams as a function ofthe slenderness ratio rx with ri =0.

Ref[13] Ref [14] B-E (DTM) rx=1/1000 rx =1/30

3.51531 3.51602 3.51602 3.51602 3.51769

22.0337 22.03449 22.03449 22.03435 21.87322

61.6970 61.69721 61.69721 61.69580 60.18079120.9050 120.90192 120.90190 120.89594 114.75634

199.8750 199.85953 199.85953 199.84239 183.15483

298.6090 298.55553 298.55553 298.51613 262.51759

417.1370 416.99079 416.99079 416.91238 350.18300

555.5100 555.16525 555.16525 555.02428 443.86650

713.8030 713.07892 712.84384 541.73299

892.1290 890.73180 890.36191 642.37872

1090.6400 1088.12400 1087.56820 744.76942

1309.5500 1304.72590 1304.45120 848.16524

Table 3. Natural frequencies of vibration i of rotating taperedcantilever Bresse-Rayleigh beams as a function of the rotationalspeed parameter s in comparison with the Bernoulli-Euler andTimoshenko theories. ri =0, Cx =0, Cz =0.5 and rx =0.01, 0.02and 0.1. i denotes percentage deviation.

rx =0.01 rx =0.02 rx =0.1s rx =

0.01 1 2 3

BE[23] 3.8238 3.8238 3.8238

B-R 3.8235 0.01 3.8228 0.03 3.7998 0.631Tim[23] 3.8312 0.20 3.8176 0.17 3.4462 10.96

BE[23] 18.3170 18.3170 18.3170

B-R 18.3089 0.04 18.2838 0.18 17.5283 4.502Tim[23] 18.2620 0.30 17.9140 2.25 12.0370 52.17

BE[23] 47.2650 47.2650 47.2650

B-R 47.2016 0.13 47.0133 0.54 41.9616 12.643Tim[23] 46.7270 1.15 44.8210 5.45 25.0210 88.90

BE[23] 90.4505 90.4505 90.4505

0

4B-R 90.2020 0.28 89.4682 1.11 72.5711 24.64

BE[23] 6.7434 6.7434 6.7434

B-R 6.74270 0.01 6.74062 0.04 6.6745 1.031Tim[23] 6.75350 0.15 6.72720 0.24 6.1753 9.20

BE[23] 21.9050 21.9050 21.9050

B-R 21.8950 0.05 21.8640 0.19 20.9311 4.652Tim[23] 21.8650 0.18 21.5130 1.82 16.3600 33.89

BE[23] 50.9340 50.9340 50.9340

B-R 50.8653 0.14 50.6615 0.54 45.1815 12.733Tim[23] 50.4180 1.02 48.5540 4.90 30.9530 64.55

BE[23] 94.2064 94.2064 94.2064

5

4B-R 93.9474 0.28 93.1828 1.10 75.5500 24.68

Table 2 represents the first ten natural frequencies for a non-rotating uniform (B-R) beam for two relatively extremeslenderness ratios rx = 1/1000 and rx = 1/30. Frequenciesbased on the Bernoulli-Euler (B-E) theory using the DTM arealso included for comparison in column 3. Results taken fromreferences [13] and [14] which are based on the (B-E) theoryand on the Finite Element Method (FEM) and Integrating

Matrix Method (IMM), respectively, are also given in columns1 and 2. As expected, when rx, the (B-R) converges to the(B-E) model. This explains why the (B-R) frequencies for rx =1/1000 are closer to the (B-E) frequencies than the non-slendercase rx =1/30. In both cases, the Bresse-Rayleigh frequenciesare lower than the results taken from references [13] and [14]except the first mode for the non-slender case (rx = 1/30)

which is slightly higher than the results from reference [14]with a relative deviation of 0.00167.

In Table 3, the first four natural frequencies of vibration of arotating tapered B-R beam with respect to rotational speedparameter s at given slenderness ratios rx =0.01, 0.02 and 0.1are given in comparison with Timoshenko and B-E takenfrom reference [23]. The DTM is used by the authors toestimate the fourth mode of the B-E theory in order tocomplete the table. The percentage deviation i between theB-E and the B-R and Timoshenko theories are also included inthe table. The relative deviation increases as the mode numberand also the slenderness ratio increases.

In Tables 4 the frequency variation of the rotating taperedBresse-Rayleigh beam are given for three rotational speeds 0,5 and 10. Results from reference [5] based on the Bernoulli-Euler theory for s =10 are used for comparison. The resultsshow that the Bresse-Rayleigh frequencies are more accuratethan the Bernoulli-Euler frequencies.

Table 4 Natural frequencies of vibration i of rotating taperedcantilever Bresse-Rayleigh beams as a function of the rotationalspeed parameter s with ri =0, Cx =0, Cz =0.5 and rx =1/30.

Rotating Tapered Bresse-RayleighFrequencies (DTM)

Ref [5]B-E

s =0 s =5 s =10 s =10

3.821091 6.735684 11.485601 11.5015

18.224596 21.791126 30.023222 30.1827

46.575780 50.187698 59.673719 60.5639

87.797402 91.441341 101.542288 104.612

140.819265 144.446277 154.786506 162.677

204.224610 207.802697 218.165535 _

276.507481 280.013857 290.266925 _

356.167260 359.597223 369.657483

Figure 2 represents the natural frequencies of vibration of arotating tapered Bresse-Rayleigh beam with respect to thevariation of the hub radius parameter ri. The frequenciesincrease with the increase in ri as expected because the

centrifugal force distribution is a function of the hub radius.Thus, the stiffening effect of the rotating beam increases as thehub radius increases. The frequency therefore increases as aresult of the enhanced rigidity of the beam.

In Figure 3, the percentage deviation of the natural frequenciesof the rotating Bresse-Rayleigh and Bernoulli-Euler beams isshown. It is evident from the figure that, as the slendernessratio increases, the Rayleigh frequencies converge to theBernoulli-Euler frequencies. As mentioned before, the Bresse-Rayleigh theory is more accurate in predicting the naturalfrequencies of rotating beams than the Bernoulli-Euler beam.


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The figure shows that the deviation in the frequenciesincreases as the mode number increases.

Figure 4 shows the trend in the variation of the first six naturalfrequencies of vibration of the rotating beam based on theBresse-Rayleigh and Bernoulli-Euler theories. The graphshows that, the frequencies diverge further from each other as

the mode number increases. The natural frequencies alsoincrease when the rotational speed increase. This is obviouslydue to the increase in the centrifugal force which tends tostretch the beam and as a result enhances its bending rigidity

0

25

50

75

100

125

150

175

200

225

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ri

1

2

3

4

5

6

Figure 2. Variation of the natural frequencies of vibration of arotating tapered Bresse-Rayleigh beam as a function of the hub radius(ri) with Cx = 0, Cz =0.5, ri =0 and s =0

0

4

8

12

16

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

rx

1

2

3

4

6

5

Figure 3. Variation of the slenderness ratio (rx) with respect to thepercentage deviation () of the first six natural frequencies betweenthe Rayleigh and Bernoulli Euler tapered beams with Cx = Cz =0.5, ri =0 and s =0.

0

50

100

150

200

250

300

350

0 3 6 9 12 15s

i

6

1

4

5

3

2

----- Bernoulli - Euler thoery

___ Rayleigh theory

Figure 4 Natural frequencies of vibration I of rotating uniformRayleigh beams as a function of rotational speed parameter s withri =0 and rx =1/30

Although this paper has focused on out-of-plane vibrations ofthe rotating beam, the in-plane vibrations can be evaluated ifthe rotational speed and the out-of-plane vibrations are knownusing the following expression [11],

222

sopip = (26)where ip and op are the in-plane and the out-of-plane non-dimensional frequencies of corresponding modes of vibration,respectively, ands is the non-dimensional rotational speed.

6 CONCLUDING REMARKS

The Differential Transformation Method (DTM) is used tosolve the free vibration problem of a rotating tapered beambased on the Bresse-Rayleigh beam theory. The theory is validfor beams with different taper configurations and also for thespecial case of a uniform beam with a wide range of

applications in rotating machinery design.

The derivations of the governing equation of motion and allthe algebraic manipulations are achieved by simple symboliccodes written in Mathematica. A numeric algorithm is alsoimplemented in Mathematic to compute the naturalfrequencies of vibration with very accurate results incomparison with the selected references in the literatureavailable. It is also demonstrated in the numerical routines thatthe DTM technique is quite simple and converges quickly tothe exact solution with very minimal computational effort andresources.


8/9


Furthermore, the Bresse-Rayleigh theory is proved to givemore accurate results as compared to results obtained usingthe Bernoulli-Euler theory without requiring any significantextra effort in the solution procedure. The study thereforedemonstrates the reliability and convenience of the applicationof the Bresse-Rayleigh theory. This investigation is alsointended to form the foundation for the application of the B-R

theory to other rotating beam problems.

7 ACKNOWLEDGEMENTS

The award of a scholarship award to D. R. Jackson by theSchool of Mechanical, Aerospace and Civil Engineering isgratefully acknowledged.

8 REFERENCES

[1] Strutt, J. W., 1877, Theory of Sound. London, MacmillanPublications Co., Inc.

[2] Bresse, J.A.C., 1866, Cours de Mecanique Applique, Part1, Gauther-Villars, Second Edition.

[3] Oyadiji, S.O. and Tomlinson, G.R., 1985, VibrationTransmissibility Characteristics of ReinforcedViscoelastic Pipes Employing Complex Moduli MasterCurves, Journal of Sound and Vibration, 102, 347-367.

[4] Banerjee, J .R., 2001, Dynamic stiffness formulation andfree vibration analysis of centrifugally stiffened

Timoshenko beams, Journal of Sound and Vibration,Vol. 247 (1), pp 97-115.

[5] Banerjee, J .R., Su, H., Jackson, D.R., 2006, Freevibration of rotating tapered beams using the dynamicstiffness method, J ournal of Sound and Vibration, Vol.284, pp. 455-466.

[6] Kaya, M.O., 2006, Free vibration analysis of rotatingTimoshenko beams by differential transform method,Aircraft Engineering & Aerospace Technology, 78(3),194203.

[7] zdemir, . and Kaya, M.O., 2006, Flapwise bendingvibration analysis of a rotating tapered cantileverBernoulli-Euler beam by differential transform method,

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AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics, and Materials Conference, Honolulu, Hawaii.

9 APPENDICES

A Basic operations using the DTM

Functional Form

fHxL = wHxLfHxL = wHxL + gHxL

fHxL = wHxL*gHxL

fHxL = dmwHxLdym

fHxL =xm

Transformed Function

F@kD = W@kDF@kD = W@kD + G@kD

F@kD = r=0

k

W@rD G@k - rD

F@kD = Hk + mL!k !

W@k +mD

F@kD = d@k -mD where d is theKronecker delta

Boundary Conditions

x = 0

wH0L =0 W@0D = 0

dwH0Ldy

=0 W@1D = 0

d2wH0Ldy2

= 0 W@2D = 0

d3wH0Ldy3

= 0 W@3D = 0

x = 1

wH1L =0 k=0

W@kD= 0

dwH1Ldy

= 0 k=0

k W @kD = 0

d2wH0Ldy2

= 0 k=0

k Hk -1L W@kD = 0

d3wH0Ldy3

=0 k=0

k Hk -1LHk -2L W@kD = 0

B Coefficients of the differential equation of motion

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