[Mechanisms and Machine Science] Vibration Engineering and Technology of Machinery Volume 23 || Large Amplitude Free Vibration Analysis of Axially Functionally Graded Tapered Rotating

Large Amplitude Free Vibration Analysisof Axially Functionally Graded TaperedRotating Beam by Energy Method

Saurabh Kumar and Anirban Mitra

Abstract Geometrically nonlinear free vibration behaviour of axially functionallygraded non-uniform rotating beams are investigated following variational form ofenergy method. Nonlinear strain displacement relations are employed to account forgeometric nonlinearity present in the system. The static solution of the displacementfield of the beam under centrifugal loading is obtained first and then the dynamicproblem is formulated as an Eigenvalue problem based on the known static solu-tion. The displacement fields are approximated by linear combination of orthogo-nally admissible functions and undetermined parameters. The method is validatedsuccessfully and results are presented in non-dimensional plane.

Keywords Geometric nonlinearity � Centrifugal stiffening � Axially functionallygraded material (AFGM) � Loaded natural frequencies

1 Introduction

Functionally graded materials (FGMs) are the new generation of engineeringmaterials with various advantages over traditional homogeneous and contemporarylaminated composite materials due to the continuous transition of properties.Gradual variation of properties in FGMs may occur through depth-wise or length-wise directions. These advanced class of materials offer unique properties such asthermal resistance, high toughness, and low density and are currently used in

S. KumarDepartment of Mechanical Engineering, National Institute of Technology, Rourkela 769008,Odisha, Indiae-mail: [email protected]

A. Mitra (&)Department of Mechanical Engineering, Jadavpur University, Jadavpur, Kolkata 700032,West Bengal, Indiae-mail: [email protected]

© Springer International Publishing Switzerland 2015J.K. Sinha (ed.), Vibration Engineering and Technology of Machinery,Mechanisms and Machine Science 23, DOI 10.1007/978-3-319-09918-7_42

473

various engineering structures such as, turbine blades, helicopter rotor blades, shippropellers etc. The simplest representation of such a rotating elastic structure is aslender clamped-free beam under rotation with constant angular velocity. Hence,dynamic analysis of rotating functionally graded beams is an area of considerableresearch interest.

As already mentioned, the gradient variation in FG beams may be oriented alongthe transverse and/or axial direction. Majority of the work done in the corre-sponding field is concentrated on rotating FG beams with depth-wise propertyvariation. Banerjee and Jackson [1] developed a dynamic stiffness method andapplied the Wittrick-Williams algorithm to investigate the free vibration charac-teristics of rotating tapered Rayleigh beam. Piovan and coworkers [2, 3] studied thedynamic behaviour of such FG beams using finite element formulation and non-linear normal modes. Li et al. [4] investigated the free vibration characteristics ofFG beams with dynamic stiffening effect using rigid–flexible coupled dynamicstheory. Few literatures are also available for FG beams with axial gradation ofproperties. Shahba and coworkers [5, 6] analyzed the free vibration behaviour ofcentrifugally stiffened AFG beams by FEM approach and introduced new elementsin terms of basic displacement functions (BDFs). Rajasekaran [7, 8] studied the freevibration of AFG Euler-Bernoulli and Timoshenko tapered beams with differentboundary conditions using differential transformation and quadrature methods.

A review of the recent literature reveals that number of works on AFG rotatingtapered beams are considerably fewer than those on depth-wise graded beams.Hence, the objective of the present work is to investigate the free vibrationbehaviour of rotating non-uniform AFG beam with clamped-free boundarycondition.

2 Mathematical Formulation

An AFG non-uniform beam of length L with constant width (b) but varyingthickness t(x) is shown in Fig. 1. Three different taper profiles, namely, linear,parabolic and exponential, are considered for the present analysis and thicknessvariations along the longitudinal x-axis is given by the following expressions—Linear taper: t(ξ) = t0(1 − αξ); Parabolic taper: t(ξ) = t0(1 − αξk); Exponential taper:t(ξ) = t0 exp(−αξk). Here t0 is the thickness at the root of the beam, α and k aregeometric parameters. Though the width is kept constant for present analysis themethodology is flexible enough to account for varying width as well. The model ismade realistic by providing an offset between the axis of rotation and initiation ofthe beam at the fixed end. In Fig. 1 R is the offset distance of the root of the beamfrom the axis of rotation and Ω is the constant angular velocity of rotation. Materialproperties, namely, elastic modulus (E(x)) and density (ρ(x)), are also considered tobe varying along the longitudinal x-axis as shown in Fig. 1.

The thickness of the beam being very small compared to its length, the effect ofshear deformation and rotary inertia has been neglected. It is to be noted that the

474 S. Kumar and A. Mitra

effect of non-uniformity at the root of the beam is neglected. The computations arecarried out in normalized axial coordinate ξ which is given by ξ = (x − R)/L.

The large amplitude free vibration problem of axially functionally graded (AFG)tapered rotating beams is mathematically formulated following variational princi-ple. The analysis is performed in two separate but interrelated steps. First the staticanalysis under centrifugal loading is carried out to obtain the deformed configu-ration. In the second phase, the dynamic problem, formulated as a standardEigenvalue problem on the basis of known static displacement fields, is solved.Thus the calculated natural frequencies refer to the statically loaded configuration ofthe rotating beam arising from centrifugal forces.

2.1 Static Analysis

The mathematical formulation of the static analysis is based on the principle ofminimum total potential energy, which is mathematically stated as,

dðpÞ ¼ 0; ð1Þ

where, π (= U + V) is the total potential energy of the system. U is total strainenergy stored in the system which consists of strain energies due to bending andstretching. Nonlinear strain-displacement relations are considered in order toincorporate stretching effect due to geometric nonlinearity in the system. Theexpression for strain energy is given by, U ¼ 1

2

Rvol EðxÞ e2xdv, where, ex is the axial

strain of a fibre at a distance of z from the mid-plane and it is made up of both

bending and stretching of mid-plane, as shown. ex ¼ �z d2wdx2 þ du

dx þ 12

dwdx

� �2.

Fig. 1 Axially functionally graded tapered rotating beam

Large Amplitude Free Vibration Analysis of Axially Functionally… 475

Substituting the expression for ex, total strain energy in the normalized computa-tional domain is obtained as,

U ¼ð1�2L3Þ Z1

0

EðnÞ IðnÞ d2w

dn2

� �2

d n

þ ð1=2LÞEðnÞ AðnÞZ1

0

dudn

� �2

þ 1�4L2

� � dwdn

� �4

þ 1=Lð Þ dudn

� �dwdn

� �2" #

d n

ð2Þ

V is the potential of the external centrifugal force, calculated corresponding toconstant angular speed of Ω and expressed as,

V ¼ bX2Z1

0

qðnÞ tðnÞuðnÞ ndn ð3Þ

Here, u and w denote the displacements along x- and z- directions, respectively.It is to be noted that u is stretching displacement whereas w is transverse deflection.These displacement fields, w and u, are expressed by linear combinations ofunknown coefficients (ci) and orthogonal admissible functions as follows,

w nð Þ ¼Xnwi¼1

ci/i nð Þ and u nð Þ ¼Xnwþnu

i¼nwþ1

ciwi nð Þ: ð4Þ

The start functions of these orthogonal sets of functions are selected by satisfyingthe flexural and membrane boundary conditions of the beam. Gram-Schmidtorthogonalization procedure is used to form appropriate sets of higher ordercoordinate functions. Substitution of the expressions of energy functionals andapproximate displacement fields gives the governing set of equations as,

K½ � cf g ¼ ff g ð5Þ

where, [K] is the stiffness matrix, {f} is the force vector and {c} is a vector ofunknown coefficients. The set of governing equation of the static problem isnonlinear in nature and solved through an iterative direct substitution method,employing an appropriate relaxation technique.

2.2 Dynamic Analysis

The governing set of equations for the dynamic analysis is derived followingHamilton’s principle, which is mathematically expressed as,


dZs2s1

T � Uð Þ ds0@

1A ¼ 0 ð6Þ

Here, T denotes the total kinetic energy of the system given by the expression,

T ¼ 12

Z1

0

dwdn

� �2

þ dudn

� �2" #

tðnÞqðnÞAðnÞ dn ð7Þ

The unknown dynamic displacement fields are assumed to be separable in timeand space and the spatial part of the fields are approximated by finite linear com-bination of admissible orthogonal functions identical to those utilized for staticanalysis,

w n; sð Þ ¼Xnwi¼1

di/i nð Þejxs and u n; sð Þ ¼Xnwþnu

i¼nwþ1

diwiejxs nð Þ ð8Þ

where, ω is the natural frequency of the system and j ¼ ffiffiffiffiffiffiffi�1p

. di represents a newset of unknown coefficients to be evaluated. nw and nu are the number of functionsfor w and u respectively. Substituting Eqs. (2) and (7) along with the dynamicdisplacement fields in Eq. (6) gives the governing set of equations for the beam inthe following form,

�x2 M½ � df g þ K½ � df g ¼ 0; ð9Þ

where, [M] is mass matrix. In the above expression, [K] is the stiffness matrix of thesystem at the deflected configuration. This implies that the unknown parametersobtained from the converged static solution are used to compute the starting valuesfor solving the dynamic problem. Equation (9) is a standard eigenvalue problemwhich is solved numerically through a Matlab program. The square roots of thecomputed eigenvalues give the free vibration frequencies of the beam at thedeflected configuration.

3 Results and Discussions

The present analysis is carried out for axially functionally graded non-uniformclamped-free beams under the action of centrifugal loading, while three differenttaper profiles (linear, parabolic and exponential) as well as three different materialmodels have been considered. The elastic modulus and density are regarded asvarying following various continuous functions as shown, Material 1: E(ξ) = E0,ρ(ξ) = ρ0; Material 2: E(ξ) = E0(1 + ξ), ρ(ξ) = ρ0(1 + ξ + ξ2) and Material 3:


E(ξ) = E0eξ, ρ(ξ) = ρ0e

ξ, where E0 is the modulus of elasticity and ρ0 is the massdensity at the root of the cantilever beam i.e. at x = 0 (Fig. 1). It should be noted thatmaterial 1 refers to a homogeneous material.

For clamped-free beam, start function for the definition of transverse deflectionis chosen as ϕ1 = ξ2 (ξ2 − 4ξ + 6). The number of functions for each of thedisplacements is taken as 8. The tolerance value of the error limit for the numerical

Table 1 Non-dimensional natural frequency of a linearly tapered cantilever homogeneous beamunder different rotation speeds (α = 0.5, R/L = 0)

Non-dimensional load, λ Work done by Non-dimensional frequency, μ

Mode 1 Mode 2 Mode 3

0 Present 3.8248 18.3217 47.2660

Ref. [7] 3.8238 18.3173 47.2648

1 Present 3.9873 18.4781 47.4183

Ref. [7] 3.9866 18.4740 47.4173

2 Present 4.4369 18.9399 47.8721

Ref. [7] 4.4368 18.9366 47.8716

3 Present 5.0921 19.6859 48.6186

Ref. [7] 5.0927 19.6839 48.6190

4 Present 5.8774 20.6855 49.6442

Ref. [7] 5.8788 20.6852 49.6456

5 Present 6.7412 21.9038 50.9310

Ref. [7] 6.7434 21.9053 50.9338

6 Present 7.6523 23.3057 52.4590

Ref. [7] 7.6551 23.3093 52.4633

7 Present 8.5920 24.8590 54.2065

Ref. [7] 8.5956 24.8647 54.2124

8 Present 9.5498 26.5358 56.1519

Ref. [7] 9.5540 26.5437 56.1595

9 Present 10.5191 28.3127 58.2740

Ref. [7] 10.5239 28.3227 58.2833

10 Present 11.4961 30.1706 60.5529

Ref. [7] 11.5015 30.1827 60.5639

11 Present 12.4784 32.0943 62.9702

Ref. [7] 12.4845 32.1085 62.9829

12 Present 13.4645 34.0714 65.5093

Ref. [7] 13.4711 34.0877 65.5237


iteration scheme used to solve the static problem is taken as 0.50 % and therelaxation parameter is 0.50. Following geometrical dimensions and materialproperties are used for validation as well generation of results: L = 1.0 m,b = 0.02 m, t0 = 0.01 m, E0 = 210 GPa, ρ0 = 7850 kg/m3.

The present methodology and solution technique are validated by comparisonwith published results of Rajasekaran [7] for a non-uniform homogeneous rotatingbeam. The results are generated for a beam with uniform width and linearly varyingthickness with α = 0.50 and comparison of the non-dimensional natural frequencyparameters (l ¼ xi L2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq0 A0=E0I0

p, i = 1, 2, 3…) corresponding to non-dimen-

sional load (k ¼ XL2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq0 A0=E0I0

p) is furnished in Table 1. The results were found

to be in excellent agreement.The variation of non-dimensional vibration frequencies (μ) with non-dimen-

sional rotational speed (λ) for the first four modes are shown in Figs. 2 and 3 fordifferent combinations of taper profile and material models. The figures are gen-erated by keeping the R/L ratio and value of k constant at 0 and 0.5, respectively,while four different values of the taper parameter (α = 0.0, 0.2, 0.4 and 0.6) areconsidered in order to show its effect on the dynamic behaviour of tapered rotatingbeams.

The figures indicate that the natural frequencies increase with increase in rota-tional speed (hardening type nonlinearity) which is obviously due to the centrifugalstiffening effect. It is also noted from these set of figures that there is almost noeffect of α on the first natural frequencies for all the cases, but the natural fre-quencies for other higher modes tend to decrease with increasing taper parameter.Moreover, the extent of decrease in the natural frequencies are greater for highermodes.

The effect of R/L ratio on the variation of non-dimensional out-of-plane vibrationfrequencies with non-dimensional speed of rotation for the first four modes isshown in Fig. 4a, b. The taper parameter is taken as 0.5 and kept constant while R/Lratio is varied from 0.0 to 0.3. The results show that for non-rotating beam the R/Lratios do not affect the natural frequencies but for rotating beam the natural fre-quencies increase with the increase in R/L ratios and the difference is larger for highrotating speeds.

To visualize the effect of centrifugal stiffening on amplitude of vibration, modeshape plots for first four modes are presented in Fig. 5. The figure is generated forMaterial-2 with linear taper (α = 0.4) and R/L = 0 for both λ = 0 and λ = 12. Thedifference in mode shape for non-rotating (λ = 0) and rotating (λ = 12) beam isclearly evident from the figure and it underlines the effect of centrifugal stiffeningon the vibration amplitude.


Fig. 2 Effect of taper parameters on dynamic behaviour of parabolic tapered AFG beam: a E(ξ) = E0, ρ(ξ) = ρ0. b E(ξ) = E0(1 + ξ), ρ(ξ) = ρ0(1 + ξ + ξ2). c E(ξ) = E0e

ξ, ρ(ξ) = ρ0eξ


Fig. 3 Effect of taper parameters on dynamic behaviour of exponentially tapered AFG beam: a E(ξ) = E0, ρ(ξ) = ρ0. b E(ξ) = E0(1 + ξ), ρ(ξ) = ρ0(1 + ξ + ξ2). c E(ξ) = E0e

ξ, ρ(ξ) = ρ0eξ


4 Conclusion

The present work deals with the large displacement free vibration of axiallyfunctionally graded non-uniform rotating beams by variational approach. Threedifferent material property variations and three different types of taper profiles areconsidered for the analysis. Results for varying taper parameters and different R/Lratios are provided in a non-dimensional load-frequency plane and the effect oftaper parameter and offset distance is discussed. Mode shape plots are also shown toelucidate the effect of centrifugal stiffening on amplitude of vibration. The results

Fig. 4 Effect of R/L ratios on dynamic behaviour of linearly tapered AFG beam (E(ξ) = E0(1 + ξ),ρ(ξ) = ρ0(1 + ξ + ξ2))

Fig. 5 First four modeshapes for rotating linearlytapered AFG beam (E(ξ) = E0(1 + ξ),ρ(ξ) = ρ0(1 + ξ + ξ2))


are validated with the benchmark results provided by other researchers. The presentmethod can be extended to various other material property variations and taperpatterns of rotating beam.

References

1. Banerjee JR, Jackson DR (2013) Free vibration of a rotating tapered Rayleigh beam: a dynamicstiffness method of solution. Comput Struct 124(2013):11–20

2. Piovan MT, Sampaio R (2009) A study on the dynamics of rotating beams with functionallygraded properties. J Sound Vib 327(2009):134–143

3. Machado SP, Piovan MT (2013) Nonlinear dynamics of rotating box FGM beams usingnonlinear normal modes. Thin-Walled Struct 62(2013):158–168

4. Li L, Zhang DG, Zhu WD (2014) Free vibration analysis of a rotating hub–functionally gradedmaterial beam system with the dynamic stiffening effect. J Sound Vib 333(5):1526–1541

5. Shahba A, Attarnejad R, Zarrinzadeh H (2013) Free vibration analysis of centrifugally stiffenedtapered functionally graded beams. Mech Adv Mater Struct 20(2013):331–338

6. Zarrinzadeh H, Attarnejad R, Shahba A (2012) Free vibration of rotating axially functionallygraded tapered beams. J Aerosp Eng 226(4):363–379

7. Rajasekaran S (2013) Differential transformation and differential quadrature methods forcentrifugally stiffened axially functionally graded tapered beams. Int J Mech Sci 74(2013):15–31

8. Rajasekaran S (2013) Free vibration of centrifugally stiffened axially functionally gradedtapered Timoshenko beams using differential transformation and quadrature methods. ApplMath Model 37(2013):4440–4463


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[Mechanisms and Machine Science] Vibration Engineering and Technology of Machinery Volume 23 || Large Amplitude Free Vibration Analysis of Axially Functionally Graded Tapered Rotating