Maganti and Nalluri International Journal of Mechanicaland Materials Engineering (2015) 10:21 DOI 10.1186/s40712-015-0040-0

ORIGINAL ARTICLE Open Access

Flapwise bending vibration analysis offunctionally graded rotating double-tapered beamsN. V. Ramesh Maganti1* and Mohan Rao Nalluri2

Abstract

Background: During the past decade, many researchers have focused on the vibration characteristics of stationaryand rotating functionally graded beams employing various methods of analysis. Hitherto, the studies have beenmainly confined to uniform beams. Hence, the present study is aimed at studying the free vibration characteristicsof functionally graded double-tapered rotating cantilever beam with symmetric cross section.

Material and Method: This paper presents a method which considers the deformation variables to determineflapwise bending vibrations of rotating functionally graded doubletapered beam attached to a rigid hub. Thetapered beam is characterized by continuously variable Young’s modulus along the thickness direction according toa power law. The equations of motions are derived using hybrid deformation variables employing Lagrange’sapproach. Rayleigh-Ritz method is used to evaluate the frequencies of the beam.

Results: The effect of several parameters such as material composition gradient, taper ratios, hub radius ratio androtational speed on flapwise bending natural frequencies of tapered beams are investigated.

Conclusions: Variation in taper ratios affects the frequency parameters of the beam. The frequency parameters areobserved to increase with an increase in hub radius. The power law index has negligible influence on the trend offlapwise bending natural frequencies with an increase in angular speed for different taper ratios.

Keywords: Functionally graded beam; Rotating beam; Rayleigh-Ritz method; Flapwise vibration; Double-tapered beam

BackgroundGenerally, to increase the strength as well as stability ofthe structure and to reduce the weight, tapered beamshave practical importance at the design stage of structures,such as the helicopter blades, rotating machinery, windturbine blades, etc. The design as well as analysis and esti-mation of dynamic characteristics of these structures isvery difficult. Centrifugal inertia force due to rotation isone of the critical factors in evaluating the dynamic char-acteristics of the rotating structures. Many researchershave investigated the dynamic characteristics of the homo-geneous rotating beams in the past. Tsai et al. (2011) in-vestigated the free vibrations for inclined Euler beamrotating at constant angular speed using the co-rotationalfinite element method combined with floating framemethod. In this work, the equations of motion were de-rived in terms of an inertial global coordinate system using

* Correspondence: ramesh.mnv@gmail.com1Department of Mechanical Engineering, Nalla Malla Reddy EngineeringCollege, Hyderabad, Telangana 500088, IndiaFull list of author information is available at the end of the article

© 2015 Maganti and Nalluri. Open Access ThAttribution 4.0 International License (http://cdistribution, and reproduction in any mediusource, provide a link to the Creative Comm

d’Alembert principle and principle of virtual work. Amethod based on the power series solution was proposedby Huang et al. (2010) for a slender beam rotating at highangular velocity, in which the beam was divided into equalsegments for estimating flapwise vibration and coupledlagwise and axial vibration. New shape functions thatinclude the rotating speed, element position, and centrifu-gal stiffening effect were considered to derive new finiteelement to evaluate free vibrations of beams rotating athigh speeds by Gunda et al. (2009). Using p-version finiteelement method, the vibration characteristics of a rotatingbeam were obtained by Cheng et al. (2011). Fracture me-chanics and Lagrange equation were used to obtain crackelement stiffness matrix and p-version finite elementmodel respectively in the analysis. The vibrations and reli-ability of rotating beams were studied considering therandom properties under random excitations by Hosseiniand Khadem (2007). Finite element method and spacestate analysis were applied to obtain the equations ofmotion. Free vibrations of tapered rotating beam in elastic

is article is distributed under the terms of the Creative Commonsreativecommons.org/licenses/by/4.0/), which permits unrestricted use,m, provided you give appropriate credit to the original author(s) and theons license, and indicate if changes were made.

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 2 of 10

and post elastic areas were evaluated using variational princi-ples to derive the equations of motions in the works of Daset al. (2009). Assuming the centrifugal force as constant, thedifferential equation of motion of rotating beam was reducedas stiff string and natural frequencies of uniform and taperedbeams were evaluated by Ganesh and Ganguli (2013) usingRayleigh-Ritz method. A new finite element was developedby Babu Gunda and Ganguli (2008) satisfying the governingequation of Euler-Bernoulli rotating beams to study the staticand dynamic behavior of rotating beam. Lin et al. (2006)evaluated bending-bending vibrations of rotating dampedbeam with pre-twist and an elastically restrained root andtheir effects on the complex frequencies of rotating beamswere observed. Banerjee and Jackson (2013) studied onfree vibration characteristics of a rotating tapered Rayleighbeam considering centrifugal stiffening, outboard force,arbitrary hub radius, and rotary inertia.Smooth and continuous variation in material properties

in one or more directions in case of structures like beamsplates and shells, which are made from functionallygraded materials (FGMs), exhibits several advantages overhomogeneous and laminated composite materials. Duringthe past one decade, many researchers have turned theirinterest to study the vibration and dynamic characteristicsof FGM structures. Sankar (2001) developed a solution fora functionally graded beam subjected to transverse loads,whose Young’s modulus was assumed to vary in thicknessdirection, and found that the stress concentration dependson the distribution of constituent material of the beam.Based on two-dimensional theory of elasticity, Ying et al.(2008) proposed an exact solution for free vibrations offunctionally graded beams resting on elastic foundationwherein material properties were assumed to vary inthickness direction as per exponential law. Higher ordershear deformation theory was implemented by Kadoliet al. (2008) to study the behavior of the FG beam withvariation of material composition based on power lawunder uniformly distributed load with different end condi-tions. Differential equation and its solution for deflectionof FGM beams resting on elastic foundation usingsecond-order theory were obtained by Murin et al. (2013)and used in free vibration analysis of the beams. Consider-ing the fourth-order differential equation of the second-order beam theory, modal analysis was carried out forbeams by Aminbaghai et al. (2012). Using the first-ordershear deformation theory, dynamic stiffness matrix wasderived by Ziane et al. (2013) to evaluate the free vibra-tions of functionally graded box beam.The above studies chiefly focused on the evaluation of dy-

namic characteristics of homogeneous stationary and rotat-ing uniforms and FGM beams that are stationary. For thedesign and analysis of tapered rotating structures like windturbine blades, helicopter blades, rotating machinery, etc., vi-bration characteristics are vital parameters. The knowledge

on these characteristics play a very important role in de-sign stage for studying the dynamic behavior and toavoid structural resonance. The flapwise and chordwisedynamic behavior of the beam depends on the nature ofthe transverse loads, geometry of the component, andthe boundary conditions. Hitherto, the studies have beenmainly confined to uniform beams. Hence, the presentstudy is aimed at studying the free vibration characteris-tics of double-tapered rotating cantilever beam withsymmetric cross section. The beam material consideredis FGM and the variables considered are taper ratio,composition variation of material as per power lawindex, hub radius ratio, and rotational speed. The resultsso obtained are compared with those reported in theearlier works based on different approaches.

Material and MethodFunctionally graded beamConsider a doubly symmetric cross-section functionallygraded beam with length, L, width, w0, and height, h0, atthe root of the beam and width, w, and height, h, at the tipof the beam as shown in Fig. 1. The graded material prop-erties vary symmetrically along thickness direction fromthe core towards the surface according to a power law:

P zð Þ ¼ P mð Þ þ P cð Þ−P mð Þ� � 2� z

h

��������n

ð1Þwhere P(z) represents an effective material property (i.e.,density, P(z), or Young’s modulus, E(z)) and P(m) and P(c)are metallic and ceramic properties, respectively.The volume fraction exponent or power law index, n, is a

variable whose value is greater than or equal to zero andthe variation in properties of the beam depends on its mag-nitude. Structure is constructed with functionally gradedmaterial with ceramic rich at top and bottom surfaces (at z= h/2 and −h/2) with protecting metallic core (at z = 0).

Equations of motionFor the beam considered in this study, the equations ofmotion are obtained with the following assumptions:

1. There is a smooth and continuous variation inproperties along thickness direction.

2. The neutral and centroidal axes in the cross sectionof the rotating beam coincide so that effects due toeccentricity and torsion can be negligible.

3. Cross section of the beam varies uniformly along itslength with the uniform taper.

4. Shear and rotary inertia effects of the beam areneglected due to slender shape of the beam.

Figure 1 shows the deformation of the neutral axis of abeam fixed to a rigid hub rotating about the axis z. The ta-pered beam is attached to a rigid hub which rotates with

Fig. 1 Configuration of the functionally graded double-tapered rotating beam

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 3 of 10

constant angular speed and no external force acts on thebeam. The beam rotates at an angular speed of Ω (t) aboutthe z-axis. In the present work, a hybrid set of Cartesianvariable w and a non-Cartesian variable s is approximatedby spatial functions and corresponding coordinates areemployed to derive the equations of motion. The positionof a generic point on the neutral axis of the double-tapered beam before deformation located at P0 changes toP after deformation and its elastic deformation is denoted

as d that has two components in two-dimensional space.

Approximation of deformation variablesUsing the Rayleigh-Ritz assumed mode method, the de-formation variables are approximated as

s x; tð Þ ¼Xμ1j¼1

ϕ1j xð Þq1j tð Þ ð2Þ

and

w x; tð Þ ¼Xμ3j¼1

ϕ3j xð Þq3j tð Þ ð3Þ

In the above equations, ϕ1j and ϕ3j are the assumedmodal functions (test functions) for s and w, respectively.Any compact set of functions which satisfies the essentialboundary conditions of the cantilever beam can be used asthe test functions. The qijs are the generalized coordinatesand μ1 and μ3 are the number of assumed modes used fors and w, respectively. The total number of modes, μ, isequal to the sum of individual modes, i.e., μ = μ1 + μ3.

The geometric relation between the arc length stretchs and Cartesian variables u and w presented by Yoo andShin (1998) as

s ¼ uþ 12

Zx0

w0ð Þ2� �dσ ð4Þ

where a symbol with a prime (′) represents the partialderivative of the symbol with respect to the integral do-main variable.

Strain energy of the systemThe total elastic strain energy of the beam consideringthe assumptions given in “Functionally graded beam”section can be written as

Es ¼ 12E zð ÞA xð Þ

ZL

dsdx

� �2

dxþ 12E zð ÞIy xð Þ

ZL

d2wdx2

� �2

dx i ¼ 1; 2; μð Þ

ð5Þwhere Iy(x) is the second moment of area of the taperedbeam.For the beam tapers in two planes, the general parame-

ters of the beam are given by

h xð Þ ¼ ho 1−τhxL

ð6Þ

w xð Þ ¼ wo 1−τwxL

ð7Þ

Table 1 Comparison of the non-dimensional lowest threeflapwise bending natural frequencies of a homogenousstationary beam with different taper ratios (δ = 0)

τh τb 0.2 0.4 0.6 0.8

0.0

(a) Fundamental frequency

0.0 3.15601 3.76285 4.09697 4.58531 5.39758

3.51602a 3.76286a 4.09698a 4.58531a 5.39759a

0.2 3.60827 3.85511 4.18932 4.67815 5.49265

3.60828a 3.85512a 4.18932a 4.67816a 5.49266a

0.4 3.73707 3.98419 4.31877 4.80842 5.62551

3.73708a 3.98419a 4.31878a 4.80842a 5.62552a

0.6 3.93427 4.18234 4.51799 5.00903 5.82882

3.93428a 4.18235a 4.51799a 5.00903a 5.82882a

0.8 4.29250 4.54369 4.88244 5.37615 6.19641

4.29249a 4.54368a 4.88244a 5.37614a 6.19639a

(b) Second natural frequency

0.0 22.0344 22.5018 23.1186 24.0210 25.6558

22.0345a 22.5018a 23.1186a 24.0211a 25.6558a

0.2 20.6210 21.0567 21.6327 22.4774 24.0153

20.6210a 21.0567a 21.6327a 22.4774a 24.0153a

0.4 19.1138 19.5166 20.0500 20.8343 22.2710

19.1138a 19.5166a 38.4920a,b 20.8343a 22.2710a

0.6 17.4879 17.8557 18.3441 19.0649 20.3953

17.4879a 17.8557a 18.3441a 19.0649a 20.3952a

0.8 15.7473 16.0731 16.5132 17.1668 18.3871

15.7427a 16.0725a 16.5123a 17.1657a 18.3855a

(c) Third natural frequency

0.0 61.6972 62.1525 62.7763 63.7515 65.7470

61.6972a 62.1525a 62.7763a 63.7515a 65.7470a

0.2 56.1923 56.6304 57.2258 58.1467 60.0096

56.1923a 56.6303a 57.2257a 58.1466a 60.0094a

0.4 50.3537 50.7715 51.3348 52.1965 53.9177

50.3537a 50.7714a 51.3346a 52.1963a 53.9173a

0.6 44.0252 44.4180 44.9437 45.7391 47.3066

44.0248a 44.4175a 44.9432a 45.7384a 47.3051a

0.8 36.8955 37.2558 37.7358 38.4555 39.8568

36.8846a 37.2439a 37.7223a 38.4392a 39.8336a

aResults of Ozgumus and Kaya (2006)bThe trends in the second frequency reported by Ozgumus and Kaya (2006) shouldexhibit decreasing trend with an increase in taper ratio. However, at τb= τh= 0.4,a higher value is reported. It is opened that it is a typographical error

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 4 of 10

A xð Þ ¼ A0 1−τwxL

1−τh

xL

ð8Þ

Iy xð Þ ¼ I0y 1−τwxL

1−τh

xL

3ð9Þ

where τw and τh are the width taper ratio and heighttaper ratio respectively and are defined as

τw ¼ 1−ww0

� �ð10Þ

τh ¼ 1−hh0

� �ð11Þ

where A0 and I0y are area of cross section and area mo-ment of inertia of the beam at the root of the taperedbeam respectively and are defined as

A0 ¼ w0h0 ð12Þ

I0y ¼ b0w30

12ð13Þ

Kinetic energy of the systemThe velocity of a generic point P can be obtained as

v→P ¼ v

→O þAd→pdt

þ ω→A� p

→ ð14Þ

where v→O is the velocity of point O that is a reference

point identifying a point fixed in the rigid frame A; ω→ o

vector P→

in the reference frame A and the terms P→, v→o,

and ω→A can be expressed as follows

p→¼ xþ uð Þi þ vj ð15Þ

v→O ¼ rΩj; ð16Þ

ω→A ¼ Ωk ð17Þ

v→p ¼ _u−Ωvð Þi þ _v þΩ r þ xþ uð Þ½ �j ð18Þ

where ι , ĵ, and k are orthogonal unit vectors fixed in A, r isthe distance from the axis of rotation to point O (i.e., radiusof the rigid frame), and Ω is the angular speed of the rigidframe. Using Eq. 14, the kinetic energy of the rotating beamis derived as

Ek ¼ 12

ZL0

ρ zð ÞA xð Þv→ p:v→ pdx ð19Þ

In which, A(x) is the area of cross section of the ta-pered beam, P(z) is the mass density, and L is the lengthof the beam.

Equation of motionSubstituting Eqs. 2 and 3 into Eqs. 5 and 19 and takingpartial derivatives of Ek and Es with respect to qij and qij,neglecting the higher order non-linear terms, the equationof motion for the tapered beam is formulated using the

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 5 of 10

Lagrange’s method. The Lagrange’s equation of motion forfree vibration of distributed parameter system can be writ-ten as

ddt

∂Ek

∂ _qi

� �−∂Ek

∂qiþ ∂Es

∂qi¼ 0 i ¼ 1; 2; 3 μ ð20Þ

The linearized equations of motion can be obtained asfollows

Xμ1j¼1

"ρ zð Þ

ZL0

A0 1−τwxL

1−τh

xL

ϕ1iϕ1jdx

0@

1A

€q1j−2Ω ρ zð Þ

ZL0

A0 1−τwxL

1−τh

xL

ϕ1iϕ2jdx

0@

1A _qij

−Ω2 ρ zð Þ

ZL0

A0 1−τwxL

1−τh

xL

ϕ1iϕ1jdx

0@

1Aq1j

þ E zð Þ

ZL0

I0z 1−τbxL

31−τh

xL

ϕ

01iϕ

01jdx

0@

1Aq1j

¼ Ω2ρ zð Þ

ZL0

A0 1−τwxL

1−τh

xL

xϕ1idx

þΩ2ρ zð ÞZL0

A0 1−τwxL

1−τh

xL

ϕ1idx

ð21Þ

Table 2 Variation of the first three non-dimensional natural frequenhub radius parameter, δ (τb = 0, τh = 0.5)

δ γ First frequency Second frequency

Presentapproach

Ref. (Ozgumus and Kaya2006)

Presentapproach

Ref200

0.0 1 4.017 3.986 18.501 18.

2 4.547 4.436 19.045 18.

3 5.308 5.092 19.917 19.

4 6.211 5.878 21.077 20.

2.0 1 4.482 4.386 18.974 18.

2 6.033 5.742 20.819 20.

3 7.955 7.452 23.562 22.

4 10.027 9.310 26.918 25.

0.5 1 4.138 4.090 18.621 18.

2 4.961 4.797 19.504 19.

3 6.082 5.777 20.892 20.

4 7.359 6.905 22.689 22.

1.0 1 4.256 4.191 18.739 18.

2 5.343 5.132 19.953 19.

3 6.766 6.386 21.821 21.

4 8.346 7.793 24.187 23.

Xμ3j¼1

"ρ zð Þ

ZL0

A0 1−τwxL

1−τh

xL

ϕ3iϕ3jdx

0@

1A€q3j

þ E zð ÞI0zZL0

1−τbxL

1−τh

xL

3ϕ

003iϕ

003jdx

0@

1Aq3j

þΩ2

(r ρ zð Þ

ZL0

A0 1−τwxL

1−τh

xL

L−xð Þϕ 0

3iϕ03jdx

0@

1Aq3j

þ ρ zð Þ

ZL0

A0 1−τwxL

1−τh

xL

12

L2−x2� �

ϕ03iϕ

03jdx

0@

1Aq3j

)#¼ 0

ð22Þ

where a symbol with double prime (″) represents thesecond derivative of the symbol with respect to the inte-gral domain variable.

Dimensionless transformationFor the analysis, Eqs. 21 and 22 may be obtained in di-mensionless form by introducing following dimension-less variables in to the equations:

τ ¼ tT

ð23Þ

ξ ¼ xL

ð24Þ

θj ¼qjL

ð25Þ

cies, with respect to the rotational speed parameter, γ, and the

Third frequency

. (Ozgumus and Kaya6)

Presentapproach

Ref. (Ozgumus and Kaya2006)

474 47.448 47.417

936 47.995 47.871

683 48.891 48.619

685 50.117 49.645

879 47.939 47.830

473 49.902 49.486

879 52.995 52.120

866 57.013 55.581

576 47.571 47.521

332 48.480 48.281

531 49.953 49.520

099 51.940 51.201

677 47.694 47.624

720 48.959 48.686

343 50.990 50.403

425 53.692 52.706

Fig. 2 Effect of taper ratio and power law index on flapwisebending natural frequencies

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 6 of 10

δ ¼ rL

ð26Þ

γ ¼ TΩ ð27Þwhere τ, δ, and γ refer to dimensionless time, hub radiusratio, and dimensionless angular speed, respectively.

Analysis of flapwise bending natural frequenciesEquation 22 governs the flapwise bending vibration ofthe double-tapered rotating beam which is not coupledwith Eq. 21. With the assumption that the first stretch-ing natural frequency of an Euler beam far separatedfrom the first natural frequency, Eq. 21 is ignored inpresent analysis. Equation 22 involves the parameters L,Ω, x, E, and P , which are the properties of the beam.After introducing the dimensionless variable from

Eqs. 23, 24, 25, 26, and 27 in Eq. 22, the equation results

Xμ3j¼i

" Z10

1−τwξð Þ 1−τhξð Þψ3iψ3jdζ

0@

1A€θ2j

þZ10

1−τbξð Þ 1−τhξð Þ3ψ 003iψ

003jdζ

0@

1Aθ3j

þγ2

( Z10

δ 1−τwξð Þ 1−τhξð Þ 1−ξð Þψ 03iψ

03jdζ

!θ3j

þ 12

Z10

1−τwξð Þ 1−τhξð Þ 1−ξ2� �

ψ03iψ

03jdζÞ

!θ3j

)#¼ 0

ð28Þwhere

T ¼ ρA0L4

E zð ÞI0y

� �12

ð29Þ

Equation 28 can be written as

Xμ3j¼1

M33ij€θ3j þ KB3

ij θ3j þ γ2 KG3ij −M33

ij

θ3j

h i¼ 0 ð30Þ

where

Mabij ¼

Z10

1−τwξð Þ 1−τhξð Þψaiψbjdξ ; ð31Þ

Table 3 Properties of metallic (steel) and ceramic (alumina)materials

Properties of materials Steel Alumina (Al2O3)

Young’s modulus E (GPa) 214.00 390.00

Material density ρ (kg/m3) 7800.00 3200.00

KBaij ¼

Z10

1−τbξð Þ3 1−τhξð Þψ00aiψ

00bjdξ ð32Þ

and

KGaij ¼

Z10

δ 1−τwξð Þ 1−τhξð Þ 1−ζð Þψ 0aiψ

0bjdζ

þ 12

Z10

1−τwξð Þ 1−τhξð Þ 1−ζ2� �

ψ0aiψ

0bjdζ

ð33Þ

where ψai is a function of ξ that has the same functionalvalue of x.From Eq. 30, an eigenvalue problem can be derived by

assuming that θs are harmonic functions of τ expressed as

Fig. 3 Effect power law index on lowest three flapwise bendingnatural frequencies versus width taper ratio

Fig. 5 Effect of taper ratio and power law index on flapwise bending natu

Fig. 4 Effect of power law index on lowest three flapwise bendingnatural frequencies versus height taper ratio

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 7 of 10

θ ¼ ejωτΘ ð34Þ

where j is the imaginary number, ω is the ratio of theflapwise bending natural frequency to the reference fre-quency, and Θ is a constant column matrix characteriz-ing the deflection shape for synchronous motion andthis yields

ω2MΘ ¼ KCΘ ð35Þ

where M is mass matrix and KC is stiffness matrix,whose elements are defined as

Mij ¼ M33ij ; ð36Þ

KCij ¼ KB3

ij þ γ2 KG3ij −M33

ij

ð37Þ

Results and discussionThe procedure presented above for free vibration ana-lysis of a double-tapered beam is demonstrated throughnumerical examples.

ral frequencies versus angular speed (τw > τh)

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 8 of 10

The results from the present modeling method havebeen obtained considering ten assumed modes com-pared with those reported by Ozgumus and Kaya (2006)using the differential transform method. The influenceof taper ratios on the first three fundamental frequenciesis presented in Tables 1 and 2 and the results arecompared.From the data, it has been observed that there is

nearly one-to-one correspondence between the resultsobtained from the current approach and those reportedbased on the differential transform method.In view of this observation, it is opined that the current

approach is in order and has been adopted for further ana-lysis in this study.Further analysis has been carried out on double-tapered

functionally graded beam made of steel as metallic con-stituent and alumina as ceramic constituent whose mech-anical properties are given in Table 3. The dimensionalparameters length = 1000 mm, breadth = 20 mm, andheight = 10 mm are considered for the analysis.The frequencies are evaluated with ten assumed modes

to obtain the converged bending natural frequencies. As

Fig. 6 Effect of taper ratio and power law index on flapwise bending natu

the frequencies obtained with number of modes greaterthan ten did not show any significant improvement inthe results, the number of assumed modes is restrictedto ten only.The effect of power law index and taper ratios on flap-

wise bending natural frequencies of a stationary func-tionally graded double-tapered beam is presented inFig. 2. It is observed that the three frequencies decreasewith an increase in power law index, n, and when thebeam composition approached a more metallic state, therate of decrease in these frequencies is predominant upto a certain n value, thereafter the frequencies are rela-tively un-effected by increase in n value at all taper ratios.The variation in flapwise bending natural frequencies withthe increase in taper ratios is decreasing at all power lawindices in all modes of vibration, as the change in taper ra-tio alters the stiffness of the beam.The effect of taper and power law index on frequencies

of a single-tapered beam is studied. The variation of flap-wise bending natural frequencies of the functionallygraded rotating stationary beam with respect to width andheight taper ratios has been presented in Figs. 3 and 4.

ral frequencies versus angular speed (τw < τh)

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 9 of 10

For all the values of power law index, all frequenciesare observed to increase with the increase in heighttaper ratio and the first flapwise frequency is observedto increase while a decreasing trend has been observedin case of higher frequencies with an increase in widthtaper ratio. It is also observed that, at higher values ofpower law index, the variation of flapwise bending nat-ural frequencies with a variation in width and heighttaper ratios tends to reduce.The influence of angular speed on flapwise bending

natural frequency for different taper ratios and powerlaw indices is presented in Figs. 5 and 6. It has been ob-served that the first frequency has a lower magnitude foruniform beam compared to that for tapered beam and areverse trend has been observed in respect of higherorder frequencies for all the taper ratios of the beam. Itis further observed that the variation in frequencies ispredominant when τw < τh as compared to the case ofτw > τh though the area of beam cross section is thesame in both cases.

Fig. 7 Effect of hub radius ratio and taper ratio on flapwise bending natura

It has been observed that for different values of taperratios and power law indices, the fundamental frequen-cies are diverging with the increase in angular speed andall higher order frequencies are observed to be conver-ging. However, at a given speed, the frequencies decreasewith an increase in power law index.It is also observed that the influence of taper ratio on

fundamental flapwise bending natural frequency is in-creasing with the increase in angular speed and the in-fluence is less on higher modes of vibration. The powerlaw index has negligible influence on the trend of flap-wise bending natural frequencies with the increase in an-gular speed for different taper ratios which is evidentfrom similarity of variation pattern for solid (n = 0) andthat for dashed (n = 2.0) and dotted (n = 4.0) lines for allmodes of vibration. These trends should be expectedthat the tapered beams under consideration would be-have differently, as the stiffness at the root and loadvariation along the length of the beam depend on thetaper ratio.

l frequency versus angular speed

Maganti and Nalluri International Journal of Mechanical and Materials Engineering (2015) 10:21 Page 10 of 10

The effect of hub radius ratio and beam taper ratio onflapwise bending natural frequencies of the rotating ta-pered beam is presented in Fig. 7. In general, the flap-wise bending natural frequencies increase with theincrease in angular speed. From the figure, it is observedthat the difference in frequencies for the two geometriesof the beam is more at lower speeds and is decreasing asthe speed increases. It may be also noted that the rate ofincrease in frequencies increase with an increase in hubradius ratio and the increase in ratio of τb/τh. The hubradius ratio could not influence the rate of increase offlapwise bending natural frequency with the increase ofangular speed. This is evident from uniformity in thegap between the solid line and dashed lines. Thesetrends should be expected from the increase in angularspeed, and hub radius leads to stiffening effect of rota-tion and increase in taper ratio results to softening effectresulting from the decrease of the cross-sectional area.

ConclusionsThe free flapwise bending vibrations of a rotating ta-pered beam are investigated in this work. The equationsof motions are derived using hybrid deformation variablesemploying Lagrange’s approach. Rayleigh-Ritz method isused to evaluate the frequencies of the beam. Linear taper-ing in breadth and depth directions is considered for theanalysis. Variation in taper ratios affects the frequencyparameters of the beam. The frequency parameters areobserved to increase with the increase in hub radiusparameter. The power law index has negligible influenceon the trend of flapwise bending natural frequencies withthe increase in angular speed for different taper ratios.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsNVRM carried out the analytical solution and MRN conceived the study andparticipated in concept checking and MS preparation. Both the authors readand approved the final MS.

Author details1Department of Mechanical Engineering, Nalla Malla Reddy EngineeringCollege, Hyderabad, Telangana 500088, India. 2Department of MechanicalEngineering, JNTUK College of Engineering Vizianagaram, Vizianagaram,Andhra Pradesh 535003, India.

Received: 24 March 2015 Accepted: 29 June 2015

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