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Journal of Sound and Vibration

Journal of Sound and Vibration 405 (2017) 68–85

http://d0022-46

n CorrE-m

journal homepage: www.elsevier.com/locate/jsvi

On dynamic analysis of variable thickness disks and complexrotors subjected to thermal and mechanical prestresses

A. Entezari, M. Filippi n, E. CarreraDepartment of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

a r t i c l e i n f o

Article history:Received 16 January 2017Received in revised form11 May 2017Accepted 19 May 2017Handling Editor: W. Lacarbonara

Keywords:RotordynamicsPrestressFinite element methodCarrera unified formulation

x.doi.org/10.1016/j.jsv.2017.05.0390X/& 2017 Published by Elsevier Ltd.

esponding author.ail addresses: ayoob.entezari@polito.it (A. En

a b s t r a c t

This paper is focused on the analysis of complex rotating structures subjected to plausibleoperating conditions. Effects of the temperature gradient and the centrifugal stiffeningcontribution have been evaluated. In particular, the analyses have been performed con-sidering disks with constant and variable thicknesses, which have been assumed eitherclamped at the bore or supported by a deformable shaft. The prestress contributions havebeen obtained through the integration of the three-dimensional stress state, which isgenerated by centrifugal and thermal loads, multiplied by the non-linear terms of thestrain field. The weak form of the governing equations has been solved using the FiniteElement method. High-fidelity one-dimensional elements have been developed accordingto the Carrera Unified Formulation. The comparisons between the current results andthose obtained from solid finite element solutions have demonstrated that the proposedmethodology can represent a valuable alternative to the three-dimensional modellingtechnique by ensuring a comparable accuracy with a lower computational effort.

& 2017 Published by Elsevier Ltd.

1. Introduction

As one of the advanced engineering systems, turbomachines are widely used in aerospace and power plant industries. Inthese machines, vibration phenomena are the primary cause of failure of rotating parts such as blades, disks and shafts. Thusthe dynamic characteristics must be carefully taken into account in structural analysis and design of such structures.

On the other hand, in real applications such as aero-engines and industrial turbine engines, a rotor assembly is geo-metrically complex and is usually formed as a combination of individual parts. For instance, several disks with variablethickness, spacers, and air seals with a shaft may be axially tied together by bolts or a special welding method can be used toattach adjacent disks and spacers to each other. In other cases, as well as, a forged integrated drum with rings to carry theblades may be employed as a rotor assembly. However, due to the dynamic interaction effects between the constituentcomponents, dynamic characteristics of each single component may change significantly in the rotor assembly.

Furthermore, in operation, the rotors may be concurrently subjected to external mechanical and thermal loads, in ad-dition to the centrifugal force. Prestresses induced by these loads can considerably affect the dynamic behavior of the rotor.Thermal loads, as well as prestress, can lead to variation of the material properties, and consequently, the dynamic char-acteristics of the rotor.

The dynamic behavior of rotating structures has been investigated by many researchers since the second half of

tezari), matteo.filippi@polito.it (M. Filippi), erasmo.carrera@polito.it (E. Carrera).

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 69

nineteenth century so that a comprehensive history of rotor dynamic has been presented by Rao [1], as well. Moreover,theoretical basis and different methods relating to the rotor dynamic analysis can be easily found in the textbooks [2–5].

In the traditional rotordynamic analysis, simplified mathematical models are used on the basis of one- and two-dimen-sional assumptions. According to one-dimensional (1D) modeling, the rotor shafts are modeled as beams, while other com-ponents such as disks are considered as a series of concentrated masses with specified moments of inertia. Bauer [6], Chen andLiao [7] and Banerjee [8] employed the classical beam theory to study vibration of rotors. It is well known that the classicaltheory cannot correctly model short and thick beams. Thus, refined theories of beam have been extensively used by re-searchers to take into account shear deformations for the beams with a low slenderness. For instance, vibrations of a rotatingTimoshenko beamwas investigated by Zu and Han [9] and Choi et al. [10]. Tai and Chan [11] applied Legendre polynomials toimprove Timoshenko beam element. They validated this model in static, dynamic and rotordynamic analyses.

It is recalled that in the 1D rotordynamic, a rotor is reduced to deformable beams with rigid mass points, so that flex-ibility of disks and other components of the rotor which extend along the radius is not concerned. Disregarding the flex-ibility may lead to unrealistic solutions, especially when deformation of a disk is considerable. Chivens and Nelson [12]studied analytically the effect of disk flexibility on the dynamic characteristics in a rotating shaft-disk problem. Genta et al.[13,14] developed a finite element (FE) formulation to study the vibration of rotors containing flexible thin disks and bladeddisks. They modeled the shaft as a beamwhile disks and blades were assumed to be annular elements whose displacementswere approximated by Fourier series in the circumferential direction. The gyroscopic effects as well as influence of thecentrifugal and the thermal prestresses in disks under the plane stress conditions were taken into account in this studies.This approach can only be employed for axisymmetric thin disks. Furthermore, Liu et al. [15] investigated free vibration ofrotating annular plates by using a modified axisymmetric FE method. All the components of prestress induced by thecentrifugal force were computed and taken into account in their analysis. In addition, the vibration of a mechanical clutchsystem with rotating disks was studied by Awrejcewicz et al. [16–18]. These authors investigated effects of the wear pro-cesses, the flexibility of the disks and the heat transfer between the disks and environment on the dynamic behavior of thesystem.

Owing to the geometrical intricacies, complicated loads, non-classical boundary conditions, and intrinsic limitations of1D models, three-dimensional (3D) models have been used in practical applications. Nowadays, the FE Method is the mostcommon numerical technique to study the dynamic behavior of rotating structures. Rao and Sreenivas [19] investigated theinteraction of flexible supports and casing on the dynamics behavior of a twin rotor system. They used solid elements tomodel these structures in ANSYS software. The Stress stiffening and spin softening were taken into account in this study.Hong et al. [20] developed MSC/NASTRAN to study dynamic behavior of rotor assembly and casing in an aero-engine model.They considered effects of prestress due to centrifugal force and gyroscopic moment using 3D solid elements in theiranalysis. Moore et al. [21] employed several available 3D FE codes such as ANSYS to study the effect of foundation and casingdynamics on dynamic response of the rotor in a typical air compressor. Taplak and Parlak [22] used Dynrot program toevaluate the dynamic characteristics of a gas turbine rotor. Zhuo et al. [23] studied dynamic behavior of a rotor-bearingsystem subjected prestress effects using 3D FE ANSYS. The prestress distributions caused by centrifugal and steady-statethermal loads were considered in their study.

It is obvious that the 3D FE modeling leads to more accurate results than the traditional models. However, dramaticincrease of degrees of freedom (DOF) and, consequently, computational efforts is a notable drawback of the 3D models.Indeed, a 3D rotordynamic model with huge DOF reduces the computational competence especially in an iterative designprocess of a complex rotor assembly. To lessen the computational costs without loss of accuracy, reduced FE techniques maybe used.

A class of refined 1D FE methods with 3D capabilities has been presented by Carrera et al. [24]. In fact, they developed aunified formulation for the FE analysis, the so-called CUF (Carrera Unified Formulations), to overcome the limitations ofbeam theories. According to CUF, the displacement field over the beam cross-section can be approximated by arbitraryexpansions such as polynomials, harmonics, and exponentials. Carrera et al. successfully employed the unified formulationto study the dynamic behavior of a rotor made of isotropic [25] and composite materials [26] including the shaft and thethin constant thickness disk. In these papers, they used Taylor-like polynomial approximations to describe the cross-sectiondeformations. Applying Lagrange-like polynomial expansions, the CUF was then presented for the vibration analysis of thinconstant thickness rings and disks; and bladed flexible shafts by Carrera and Filippi in [27]. In this study, effects of thegeometrical stiffening due to centrifugal plane stresses on the dynamic behavior were considered. Moreover, the CUF ap-proach has been recently adopted by the authors, Entezari et al. [28,29], to perform thermoelastic analysis of rotors androtating disks. Therefore, the refined FE method in the context of the CUF may be its ability to handle more complexrotordynamic problems subjected to the combined mechanical and thermal loads.

This paper is aimed, following previous works [28,25–27,29], at extending the CUF approach for dynamic analysis ofvariable thickness disks and complex rotors considering thermal and mechanical prestress effects.

2. Variational formulation of rotating structures

To obtain the equations of motion of rotating structures in the variational form, Hamilton's principle can be used. Thisprinciple, in the absence of non-conservative forces, is written as

Fig. 1. Sketch of the rotating reference frame.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8570

( )∫δ − ( + ) =( )

K U U td 01t

t

g0

1

where δ denotes the first variation of a function in terms of its variables, K is the total kinetic energy of the system, and U isthe total potential energy that includes the strain energy and potential of any conservative external forces. The term Ug isdue to the pre-stress σ0 (or pre-strain ϵ0) field, which is assumed to be generated by centrifugal or thermal effects.

The total kinetic energy of the rotor can be written as

∫ ρ= ( ) ( )K Vv v12

d 2V

T

where ρ is the material density, v is absolute velocity vector of a material point P, as shown in Fig. 1, V is the volume of thebody, and the superscript (T) denotes the transpose of the vector. The absolute velocity v of the point P can be expressed as

Ω= ̇ + ( + ) ( )v u r u 3

where = { }u u uu x y zT is the displacement vector while the superscript dot (.) indicates the time differentiation. Moreover,

in Fig. 1, = { }x zr 0P PT is the position vector of the material point P with respect to rotational axis and Ω stands for the

angular velocity of the rotor in the xz-plane so that the matrix Ω is given as

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Ω

ΩΩ =

− ( )

0 00 0 0

0 0 4

Substituting Eq. (3) into Eqs. (2), (5) is obtained

()

∫ ρ

Ω

Ω Ω Ω

Ω Ω Ω Ω

= ̇ ̇ + ̇ +

+ ̇ + + ( )

K

V

u u u u u u

u r u r r r

12

2

2 2 d 5

V

T T T T T

T T T T T

The total potential energy containing the elastic energy and potential of conservative external forces can be written as

∫ ∫ ∫ ∑ϵ σ= − − −( )

U V V Su f u f u f12

d d d6V V S k

eT T B T S T Ck

here the surface area of the body is denoted by S. Furthermore, ϵe and σ stand for the elastic strain and stress vectors,respectively. = { }f f ff x y z

B B B B T stands for the vector of all body forces except the inertial forces per unit volume. fS is the

vector of surface forces per unit surface area, and { }= f f ff x y zC C C C T

k k k k is the vector of concentrated loads where k denotes

the load application point. It is noted that the components of fB and fS may be a function of the coordinates x, y and z,however the specific x, y and z coordinates of S are considered for f Ck [30].

It should be noted that the centrifugal forces caused by the rotation of a disk can be generally considered as a type ofbody forces. However, since they are taken into account for a rotor in the kinematic energy relation, i.e. in the fifth term of

Eq. (5) as ρΩ ΩrT , the rotation-induced body forces is not included in the vector fB. Therefore, fB contains other types of bodyforces such as gravitational-weight forces and magnetic forces.

The Hooke law for a linear thermoelastic material is

σ ϵ= ( )C 7e

where C is the forth order tensor of elastic moduli. In the linear thermoelasticity, the elastic strain vector ϵe is equal to

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 71

ϵ ϵ ϵ= − ( )8e t

where ϵ denotes the total strain vector and ϵt is the strain vector caused by the temperature change Δ = −T T T0, that is

ϵ α= (Δ ) ( )T 9t

where T0 is the reference temperature. The steady-state temperature distribution Tmay be, in general, a function of all threespace coordinates. α stands for the vector of linear thermal expansion coefficients. Moreover, the linear strain-displacementrelations can be written as

ϵ = ( )Du 10

where D is the matrix of linear differential operators.Thus, substituting Eqs. (7) and (8) into Eq. (6), the total potential energy becomes

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥∫ ∫

∫ ∫∫ ∑

α

α α

= ( ) ( ) − ( ) Δ

+ (Δ Δ ) −

− −( )

U V T V

T T V V

S

u D C Du u D C

C u f

u f u f

12

d d

12

d d

d11

VV

V V

S k

T T T T

T T B

T S T Ck

The geometric strain energy, Ug, determines significant variations in the dynamic characteristics of a structure. In the lin-earized theory, this contribution derives from the linearization of the nonlinear geometric relations

∫ σ ϵ= ( )U Vd 12gV

0T nl

where σ0 is the initial stress vector, and ϵnl is the non-linear part of strains

⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦

ϵ = ( ) + ( ) + ( )

ϵ = ( ) + ( ) + ( )

ϵ = ( ) + ( ) + ( )

ϵ = [ + + ]

ϵ = [ + + ]

ϵ = [ + + ] ( )

u u u

u u u

u u u

u u u

u u u

u u u

121212121212 13

xx x x y x z x

yy x y y y z y

zz x z y z z z

xy x xy y xy z xy

xz x xz y xz z xz

yz x yz y yz z yz

nl,

2,

2,

2

nl,

2,

2,

2

nl,

2,

2,

2

nl, , ,

nl, , ,

nl, , ,

Depending on the s0 distribution, the geometric strain energy can either supply or subtract stiffness at the structure. Forexample, the centrifugal forces typically generate tensile stresses, which tend to rigidify the structure with a consequentincreasing of natural frequencies. On the contrary, the contribution produced by thermal stresses can soften the structurereducing, in such a way, the natural frequencies.

Substituting Eqs. (5), (11) and (12) into Hamilton's principle (1) gives

⎡⎣⎢

⎤⎦⎥

∫ ∫ ∫

∫ ∫

∫ ∫

∫ ∫

∫ ∫∫

∫

∑

α

α α

σ ϵ

δ ρ ρ

ρ ρ

ρ Ω ρ

Ω

Ω Ω Ω

Ω Ω Ω

( ̇ ̇ ) + ( ̇ )

+ ( ) + ( ̇ )

+ ( ) + ( )

− [( ) ( )] + [( ) Δ )]

− (Δ Δ ) +

+ +

− =( )

V V

V V

V V

V T V

T T V V

S

V t

u u u u

u u u r

u r r r

u D C Du u D C

C u f

u f u f

12

d d

12

d d

d12

d

12

d d

12

d d

d

d d 014

t

t

V V

V V

V V

V V

V V

S k

V

T T T

T T T

T T T T

T T T T

T T B

T S T C

0T nl

k

0

1

Fig. 2. A sample 1D FE model of a disk with arbitrary profile in the CUF framework.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8572

3. One-dimensional FEM through the unified formulation

Eq. (14) is being written within the framework of the FE method and according to the Carrera Unified Formulation. Thestructure of Fig. 2 is assumed as a beam along its axis (y) so that each cross section of the beam ( )A is defined in anyorthogonal −x z plane. The structure is discretized into a finite number of 1D beam elements in the y-direction, while theLagrange-type expansions are employed to assume the model kinematics.

The displacement vector is assumed as

( ) = ( ) ( ) ( )ττx y z N y F x zu q, , , 15i

i

where { }=τ τ τ τq q qqixi

yi

zi T

is the nodal displacement vector, ( )N yi are the shape functions, and τF are Lagrange polynomial

expansions. In Eq. (15), τ τ( = … )1, 2, , M and ( = … )i i 1, 2, , Nnodes indicate summation, according to the generalized Ein-stein's notation. Here, M and Nnodes are the number of terms of the expansion and the number of element nodes, respectively.

Three types of beam elements, two nodes (B2), three nodes (B3) and four nodes (B4), may be adopted to provide a linear,a quadratic and a cubic interpolation of the displacement variable along the longitudinal direction, respectively. Moreover,cross sections can be discretized by using different types of Lagrange elements (LEs) such as linear three-point (denoted asL3), bilinear four-point (L4), quadratic nine-point (L9) and cubic sixteen-point (L16) elements.

The Lagrange polynomial expansions for L4 and L9 elements in the natural coordinate system ξ η( ), are reported in [31].Fig. 3 shows the distribution of Lagrange points in the actual and natural coordinate systems. In general, the coordinate

transformation from an arbitrary cross section referred to (x,z) coordinates to the natural square ξ η( ), is trivial and moredetails can be found in Ref. [30].

The nodal displacement vector q contains all the nodal degrees of freedom (DOF) of the structural model, which can becalculated as

( )∑= ×( )=

DOF 3 N16i

i

1

N

LN

BN

Fig. 3. Sample scheme of L9 element in the actual (a) and the natural (b) coordinate systems.

Fig. 4. Mesh for the constant thickness disk, (a) 1B3 along the axis; (b) 2�16 L16.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 73

where NiLN is the total number of Lagrange nodal points on each cross section and NBN stands for the total number of beam

nodes along the longitudinal axis Fig. 4.According to 1D-CUF theory, the components of strain, ϵ, stress, s, and thermal expansion coefficient, α, vectors may be

grouped as it follows

ϵ ϵ= {ϵ ϵ ϵ } = {ϵ ϵ ϵ } ( )2 2 2 17p zz xx xz n yy yz xyT T

σ σσ σ σ σ σ σ= { } = { } ( )18p zz xx xz n yy yz xyT T

α αα α α α α α= { } = { } ( )19p zz xx xz n yy yz xyT T

where the subscript p denotes the in-plane components over a cross-section, while n indicates the normal components tothe cross-section. Therefore, strain-displacement relations (10) can be decomposed into the two following expressions

ϵ

ϵ

=

= ( + ) ( )

D u

D D u 20

p p

n ny np

where the matrices Dp, Dnp and Dny are the linear differential operators

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=∂ ∂

∂ ∂∂ ∂ ∂ ∂

= ∂ ∂∂ ∂

=∂ ∂

∂ ∂∂ ∂ ( )

z

x

z x

z

x

y

y

y

D D

D

0 0 /

/ 0 0

/ 0 /

,0 0 00 / 0

0 / 0

0 / 0

0 0 /

/ 0 0 21

p np

ny

In a similar manner, the grouped stresses and stress-temperature moduli are obtained as

σ ϵ ϵ= + ( )C C 22p pp p pn n

σ ϵ ϵ

β α α

β α α

= +

= +

= + ( )

C C

C C

C C 23

n np p nn n

p pp p pn n

n np p nn n

in which β α= C stands for the vector of stress-temperature moduli. The expanded expressions for components of the matrixC for anisotropic materials can be found in Ref. [26]. For isotropic materials with the thermal expansion coefficient equal toα, the vectors αp and αn are α α= { }1 1 0p

T, and α α= { }1 0 0nT, respectively. The matrices Cpp, Cpn, and Cnn are

Fig. 5. Frequencies (Hz) vs. rotational speed (rpm) for the rotating clamped thin disk with prestress effects and different diametral nodal lines, m. ‘Solidlines’: prestress computed with Eq. (37); ‘Dashed lines’: analytical prestress distributions Eqs. (40).

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8574

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

λ μ λλ λ μ

λλ

μ

λ μ

μ

=+

+ =

˜ =+

( )

C C

C

2 02 0

0 0 0

,0 00 0

0 0

2 0 00 0 00 0 24

pp pn

nn

in which λ and μ are Lame constants

λ νν ν

μν

=( + )( − )

=( + )

E E1 1 2

,2 1

Fig. 6. First six mode shapes of the thin disk with different diametral nodal lines, m, and n¼0. (a): m¼0; (b): m¼1; (c): m¼2; (d): m¼3; (e): m¼4;(f): m¼5.

Fig. 7. Frequencies (Hz) vs. rotational speed (rpm) for the rotating thin disk and different diametral nodal lines, m, with 3D pre-stress effects due tocentrifugal forces. ‘Solid lines’: fixed boundary condition; ‘Dashed lines’: simply-supported boundary condition [27]; ‘Symbols’: experimental data [27].

Table 1Effect of thermal load on natural frequencies of the stationary disk computed with the CUF model 2�16 L16.

m Δ =−T 0o i Δ =−T 5o i Δ = °−T 10 Co i

0 81.30 ( )89.94 10.68 ( )97.79 20.35

1 82.97 ( )88.78 7.05 ( )94.20 13.58

2 91.73 (− )89.28 2.66 (− )86.68 5.49

3 115.5 (− )102.4 11.31 (− )87.30 24.42

4 159.0 (− )138.0 13.17 (− )113.1 28.88

5 222.0 (− )197.2 11.34 (− )167.9 24.51

(): % change w.r.t. the disk without thermal load.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 75

4. Governing equations in CUF form

Using Eqs. (15), (14) becomes

( )∫ δ ¨ + ̇ + − =( )

τ τ τ τ τ tq M q G q K q F d 025t

ti ij s js ij s js ij s js iT

0

1

where

= + + +

= + − [ ] ( )

τ τ τ τ τ

τ τσ

τ τΩ

F F F F F

K K K K 26

iTi i i i

ij s ij s ij s ij s

B S C

st 0

The superscripts s and j are similar to τ and i, respectively, and they indicate summation based on Einstein's notation. τFTi is

the thermal load vector, which represents artificial forces for modeling thermal expansion, τFiB is the vector of all types of

body forces, τFiS is the force vector due to traction applied on the surface area, and τFi

C is the force vector due to concentratedexternal loads.

The superscripts τ, s, i and j are exploited to assemble the global matrices and load vectors. In fact, the CUF presents acondensed notation that leads to the so-called fundamental nucleus (FN) of all mathematical operators involved. The ex-pressions of fundamental nuclei of Eq. (25) are

Fig. 8. Frequencies (Hz) vs. rotational speed (rpm) for the rotating thin disk, and different diametral nodal lines, m, with 3D thermal pre-stress effects. Solidcurves: Δ = °−T 5 Co i ; Dashed curves Δ = °−T 10 Co i .

Fig. 9. 1D-CUF models of variable thickness disk. (a) discretization along the y-axis into 8 2-node beam elements; (b) ( ) ×1/2/3/4 32 L4; (c): ( ) ×2/4/6/8 32 L4;(d): ( ) ×5/7/9/14 32 L4; (e) ( ) ×1/3/4/6 24 L9; (f) ( ) ×1/2/3/4 8 L16; (g) ( ) ×1/2/3/4 12 L16; (h): ( ) ×2/4/6/8 8 L16.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8576

∫ ∫ ρ= ( ) ( ) ( )τ

τN N y F F AM Id d 27ij s

li j

As

Ω= ( )Ωτ τG M2 28ij s ij s

Fig. 10. First eight mode shapes (n¼0) of the variable thickness disk obtained with a 1D-CUF model. (a): m¼0; (b): m¼1; (c): m¼2; (d): Torsion mode;(e): m¼3; (f): m¼4; (g): m¼5; (h): Shear mode.

Table 2Natural frequencies (Hz) of the stationary variable thickness disk computed with the 1D-CUF FE models.

Model 1D FE CUF 3D FE (ANSYS)

Mesh (1) (2) (3) (4) (5) (6) (7) (I) (II)DOF 3168 5472 8928 11088 5832 8748 11016 11088 166320

=m 1 1806.88 1713.04 1696.63 ( )1659.07 1.47 1655.45 1655.92 ( )1648.44 0.81 [ ]1635.15.10 1555.7

=m 0 1918.47 1826.97 1811.52 ( )1753.14 2.41 1749.77 1750.20 ( )1742.62 1.79 [ ]1711.9 4.82 1633.1

=m 2 1977.97 1889.68 1874.50 ( )1826.56 0.79 1823.67 1823.42 ( )1817.29 1.29 [ ]1841.17.25 1716.6

Torsion 2171.46 2148.44 2143.12 ( )2121.45 0.45 2119.95 2120.20 ( )2118.53 0.58 [ ]2131.10.81 2113.8

=m 3 3074.81 2986.35 2968.32 ( )2771.40 2.45 2778.93 2767.16 ( )2772.83 2.40 [ ]2841.18.77 2611.9

=m 4 4840.04 4727.56 4697.11 ( )4226.07 3.53 4212.01 4217.48 ( )4205.28 4.00 [ ]4380.5 9.14 4013.3

Shear 4360.53 4318.00 4307.90 ( )4273.95 0.38 4271.33 4271.71 ( )4269.04 0.49 [ ]4290.10.52 4267.9

=m 5 7045.51 6900.09 6851.59 ( )5970.414.46 6180.24 5956.85 ( )6171.86 1.23 [ ]6249.2 9.61 5700.8

(1): ( ) ×1/2/3/4 32 L4, (2): ( ) ×2/4/6/8 32 L4, (3): ( ) ×5/7/9/14 32 L4, (4): ( ) ×1/3/4/6 24 L9,(5): ( ) ×1/2/3/4 8 L16, (6): ( ) ×1/2/3/4 12 L16, (7): ( ) ×2/4/6/8 8 L16.(I): Step-wise model with 2688 SOLID185, (II): Hyperbolic-profile model with 6240 SOLID185 ( × ×32 80 20).( ): % difference with respect to the model (I).[ ]: % difference with respect to the model (II).

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 77

⎡⎣ ⎤⎦⎡⎣ ⎤⎦

⎡⎣ ⎤⎦( ))

(∫ ∫

∫ ∫

= ( ) ( ) ( ) + ( ) +

+ ( ) ( ) + ( ) +

+ ( ) ( ) + ( ) + ( )

ττ

τ

τ τ

N N y F F F

F F F A

N N y F F F A

K D I C D I C D I

D I C D I C D I

D I D I C

d

d

d d29

ij s

li j

Anp np p s nn np s

p pp p s pn np s

li j

Anp p pn s

stT

T

,T T

y

(

( )( )

∫ ∫∫ ∫

∫ σ σ σ

+ ( ) [ ( ) + ( )] +

+ ( )

= + + ( )

τ

τ

στ

τ τ τ

N N y F F F A

N N y F F A

F N F N F N F N F N F N

I C D I C D I

I C I

K I I I

d d

d d

30

li j

AAy np p s nn np s

li j

AAy nn Ay s

ij s

Vi s j i s j i s j

,T

, ,T

0 0 0

y

y y

x xx x y xy x z xz x0 , , , , , ,

)σ σ σ

σ σ σ

Ω Ω

+ + +

+ + +

= ( )

τ τ τ

τ τ τ

Ωτ τ

F N F N F N F N F N F N

F N F N F N F N F N F N V

I I I

I I I

K M

d

31

i s j i s j i s j

i s j i s j i s j

ij s ij s

0 0 0

0 0 0

T

x xy y y yy y z yz y

x xz z y yz z z zz z

, , , , , ,

, , , , , ,

Fig. 11. 3D ANSYS models of the variable thickness disk (a) Step-wise model with 2688 SOLID185 elements (DOF¼11088), (b) Hyperbolic-profile modelwith 6240 SOLID185 elements (mesh × ×32 80 20, DOF¼166320).

Table 3Effect of the thermal load on natural frequencies of the stationary variable thickness disk.

Mode 1D FE CUF Model

Model (3) Model (4)

=m 1 ( )1703.35 0.40 ( )1665.76 0.40

=m 0 ( )1825.33 0.76 ( )1767.16 0.80

=m 2 (− )1858.24 0.87 (− )1810.41 0.88

Torsion ( )2150.29 0.33 ( )2127.55 0.29

=m 3 (− )2936.30 1.08 (− )2738.69 1.18

=m 4 (− )4658.95 0.81 (− )4186.27 0.94

Shear ( )4313.34 0.13 ( )4278.68 0.11

=m 5 (− )6809.54 0.61 (− )5926.11 0.74

(): % change w.r.t. the disk without thermal load.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8578

⎜ ⎟⎛⎝

⎡⎣ ⎤⎦⎞⎠( ) ( )∫ ∫ β β= ( ) Δ +

( )τ

τ τN y T F F AF D Dd d32T

i

li

Ap p np nT T

⎜ ⎟

⎡⎣ ⎤⎦⎛⎝

⎞⎠

∫ ∫

∫ ∫ ∫( )

β

ρΩ Ω

+ ( ) (Δ )

= ( ) +( )

τ

ττ τ

N y TF A

N y F A NF V

I

F r f

d d

d d d33

li

AAy n

i

li

A ViB

T B

y,

( )∫= ( )τ

τNF SF f d 34i

SiS

S

( )∑=( )

ττNFF f

35i

kiC

Ck

Fig. 12. Frequencies (Hz) vs. rotational speed (rpm) in inertia coordinate system for the variable thickness disk. Solid curves: L4 model; Dashed curves: L9model.

Fig. 13. 3D model of a complex rotor.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 79

where

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣⎢⎢

⎤

⎦⎥⎥= =

( )I I

0 0 11 0 00 1 0

1 0 00 1 00 0 1 36

A y

The fundamental nucleus related to the Coriolis term is skew-symmetric, while those related to the spin softening andcentrifugal stiffening are diagonal matrices. More details about assembly technique of these matrices and vectors can befound in Ref. [32].

Neglecting the external force vectors f Ck, fS and fB, static analyses must be carried out in order to determine the stressesinduced by the centrifugal and steady-state thermal forces. As far as rotation-induced stresses σ( )Ω0 are concerned, Eq. (37) issolved for Ω = 1 rad s�1 and =f 0B as

( )− = ( )τ τ

Ω

τ

ΩΩ

= = =K K q F

37ij s ij s js i

fst

1B

1, 0B

while, for the prestress due to thermal loads σ T0 , the three-dimensional stress state is computed with Eq. (38)

= ( )τ τK q F 38ij s jsTi

st

The natural frequencies and the corresponding modal shapes of the rotor are obtained from the system of Eq. (39)

⎡⎣ ⎤⎦ω ω ΩΩ Ω Ω¯ − + + + + − = ( )ω

σ σΩeq M i M K K K M2 0 39

tT

i 2st

2 T0 0

where ω is the eigenvalue, q̄ is the corresponding eigenvector, σ ΩK

0and σK

T0are the matrices due to centrifugal and thermal

effects, respectively. The quadratic eigenvalue problem of Eq. (39) is solved using the state-space transformation technique(see for example [26]).

Fig. 14. 1D-CUF FE models of the complex rotor: (a) 32 B2, (b) 40 B2 and (c) 50 B2 elements in the axial direction.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8580

5. Numerical applications

5.1. Thin disk with constant thickness

A disk with constant thickness made of steel, with the Young's modulus E¼210 GPa, Poisson's ratio ν = 0.3 and density

ρ = 7800 ( )−kgm 3 has been considered. The inner and outer radii, and the thickness are assumed as 0.1016, 0.2032 and1.016�10�3 m, respectively. The inner boundary of the disk (hub) is assumed to be fully fixed (mounted on a rigid shaft),while the outside boundary is traction free. This structure was previously analysed using the 1D-CUF theory in [27], wherethe capabilities of Lagrange-type elements has been evaluated. The mathematical model used for the simulations is shownin Fig. 11, and it consisted of a single 3-node beam element along the y-axis, and 2�16 cross-sectional mesh of L16 ele-ments. As pointed out in [27], this model provides accurate results for the considered structure.

Fig. 5 shows the Campbell diagram related to the first 6 mode shapes (see Fig. 6) of the disk characterized bym diametraland n circumferential nodal lines. The centrifugal stiffening have been obtained using Eq. (37), and the following analyticalexpressions under the plane stress assumptions

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

( )

( )

( ) ( ) ( )

( ) ( ) ( )

σν

ν ν ρΩν

ν

σν

ν ν ρΩν

ν

( ) =−

−+ −

+ + +−

( ) =−

−+ −

+ + +−

θ

rE r

Ec

c

r

rE r

Ec

c

r

1

3 1

81

1

1

1 3 1

81

1

r 2

2 2 2

12

2

2

2 2 2

12

2

The above equations for the considered disk and fixed boundary conditions σ( ( ) = ( ) = )u r r0 0i r o reduce to

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

σ Ω

σ Ω

( ) = − + +

( ) = − − +( )θ

r rr

r rr

3217.5 0.58671

118.6416

1852.5 0.58671

118.641640

r2 2

2

2 22

It is noteworthy that the two approaches have provided similar results. The slight differences for =m 0 and 1 are due tothe accuracy of the adopted model in the stress computation. Likely, the number of Lagrange elements along the radialdirection should be increased to perfectly reproduce the analytical distributions [29]. Moreover, Fig. 7 compares the current

Fig. 15. Cross-sectional meshes: (a) Uniform L4; (b) Refined L4; (c) ( × )7 10 L9; (d) ( × )7 5 L16.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 81

solutions with the results presented in [27], where the stiffening contribution was included in advance, by assuming that noforces were exchanged between the disk and the support at the inner radius. For the umbrella and the first bending modeshapes (Fig. 6-a and -b), prestress distributions related to the fixed boundary condition have determined a more significantincreasing of the corresponding frequencies with the rotational speed with respect to the simply-supported boundarycondition. On the other hand, for the mode shapes characterised by 3, 4 and five nodal diameters, the critical speedsoccurred at lower values for the clamped disk (about at 3000 rpm). Furthermore, it should be noted that the solution relatedto the simply-supported condition is in good agreement with the reported experimental data (symbols).

The effects of thermal loads on the dynamic behaviour of the disk have been evaluated considering a parabolic radialsteady-state temperature distribution (Eq. (41))

( ) = + (Δ )( − ) ( − ) ( )−T r T T r r r r/ 41o iin in2

out in2

where Δ = −−T T To i out in. The temperature at the inner radius is assumed to be equal to the reference temperature = °T C20in .Table 1 shows the natural frequencies for two different values of Δ −To i.

Unlike the centrifugal effect, only the frequencies related to the mode shapes with =m 0 and 1 increased with theconsidered thermal load. The reductions of the other frequencies can be ascribed to the negative values of the circumfer-ential stresses along the radius. Furthermore, it is noteworthy that the order of appearance of the modal deformations isstrongly affected by the value of Δ −To i. Fig. 8 shows the Campbell diagrams related to the two loading conditions. Besides the

“modal swapping” phenomenon, it should be noted that the critical speed values are lower for the case with Δ = °−T 10 Co i .

Table 4Natural frequencies (Hz) corresponding to the first 15 mode shapes of the complex rotor.

Model 1D FE CUF 3D FE (ANSYS)

Mesh (1) (2) (3) (4) (5) (I)DOF 27936 27072 30528 16500 17820 37293

1 619.62 638.57 636.90 635.61 ( )634.03 0.95 640.12

2 598.26 663.94 652.04 661.91 ( )659.82 3.34 682.60

3 756.03 771.60 770.10 768.57 ( )766.74 1.75 753.57

4 973.99 1023.59 1005.08 1016.34 ( )1011.60 2.19 1034.20

5 1409.99 1421.09 1410.36 1384.21 ( )1369.42 1.03 1355.40

6 1580.10 1650.87 1618.98 1617.92 ( )1606.68 2.91 1561.20

7 1629.55 1666.13 1647.42 1630.49 ( )1624.49 3.74 1565.90

8 1732.16 1776.91 1737.66 1696.70 ( )1678.59 3.43 1623.00

9 1774.38 1797.42 1703.46 1762.24 ( )1742.012.21 1704.30

10 1880.98 1891.29 1855.72 1785.82 ( )1765.57 2.05 1730.10

11 1851.36 1848.64 1828.22 1799.13 ( )1786.70 2.24 1747.50

12 1897.56 1898.53 1885.35 1812.63 ( )1791.65 2.43 1749.20

13 1897.63 1915.96 1823.47 1868.06 ( )1845.57 2.72 1796.70

14 1968.38 1963.17 1935.57 1879.19 ( )1857.211.10 1837.00

15 2336.70 2356.83 2229.60 2334.89 ( )2316.60 0.59 2303.10

(1): 32B2/Uniform L4, (2): 32B2/Refined L4, (3): 50B2/Refined L4,(4): 32B2/L9, (5): 32B2/L16, (I): 10368 Solid185(): % difference with respect to the 3D ANSYS model.

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8582

5.2. Disk with variable thickness

An annular disk with variable thickness subjected to centrifugal and thermal loads has been analyzed. The inner and

outer radii of the disk, whose thickness varies according to the hyperbolic function ( ) = −h r r0.013 0.5, have been assumed tobe =r 0.05 min and =r 0.2 mout . The material properties of the disk have been assumed to be =E 174 GPa, ν = 0.3,

ρ = −8200 kg m 3 and α = × °−16.36 10 1/ C6 . The inner radius has been considered fully fixed, while other boundaries are freeof surface traction and concentrated loads. The hyperbolic disk profile has been modelled using 8 cross sections withdifferent thicknesses and diameters. A single 2-node beam element has been used for each cross-section along the long-itudinal direction (see Fig. 9-a). Regarding the cross-sectional discretization, various distributions of Lagrange-type elementshave been considered and graphically shown in Fig. 9. Values in brackets (e.g. (1/2/3/4) of Fig. 9-b) refer to the number ofLagrange elements along the radial direction to model each cross-section. The second value (e.g. 32) indicates the number ofsubdivisions along the peripheral direction.

The natural frequencies, which are related to the mode shapes shown in Fig. 10, have been computed with the different1D-CUF models and listed in Table 2. Contrary to the previous case, the mode shapes involve out-of-plane and in-planedeformations.

Two different 3D ANSYS models, which are shown in Fig. 11, have been provided to validate the current results. Model(I) has been built using the step-wise approach adopted for creating the CUF models. For Model (II), the disk profilesmoothly varies along the radius according to the prescribed hyperbolic law. The variations between the frequenciescomputed with the ANSYS models (last two columns of Table 2) are ascribed to the different descriptions of the diskgeometry. It is expected that the frequency discrepancies can be lessened by increasing the number of cross-sections inModel (I). It is clear that the longitudinal refinement would lead to an increasing of the number of degrees of freedom, aspointed out in [29]. It is observed that the CUF models detected all mode shapes with acceptable accuracies, especially withrespect to the first ANSYS model. In fact, the maximum discrepancies between the L9 and L16 models (columns (4) and (7))compared to the ANSYS Model (I) are below the 5%.

Similarly to the previous example, the disk has been subjected to a parabolic temperature distribution such that =T 537in

and = °T 614 Cout , with the reference temperature, = °T 20 C0 . Frequencies related to the first 8 mode shapes and computedwith the third and fourth CUF models are shown in Table 3. Unlike the thin constant disk, the thermal prestress does notsignificantly affect the frequency values. In fact, the maximum variations with respect to the unloaded case are below 2% forboth models. However, it is possible to observe that, also in this case, the thermal pre-stress has led to increments andreductions of the frequencies. Fig. 12 shows the Campbell diagrams obtained with the L4 and L9 CUF models. Although thestarting points of L9 curves are at lower frequencies, the trends of branches predicted by both models are in strongagreement for all mode shapes. In particular, the curves related to the in-plane deformations are essentially overlapped.

Fig. 16. First fifteen natural frequencies (in Hz) and the corresponding exaggerated mode shapes of the complex rotor for the 1D-CUF model 32B2/L16. (a):ω = 634.01 ; (b): ω = 659.82 ; (c): ω = 766.73 ; (d): ω = 1011.64 ; (e): ω = 1369.45 ; (f): ω = 1606.76 ; (g): ω = 1624.57 ; (h): ω = 1678.68 ; (i): ω = 1742.09 ;(j): ω = 1765.610 ; (k): ω = 1786.711 ; (l): ω = 1791.612 ; (m): ω = 1845.613 ; (n): ω = 1857.214 ; (o): ω = 2316.615 .

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 83

5.3. Complex rotor

The dynamics of the complex rotor considered in [29] has been here evaluated. The structure that is shown in Fig. 13consisted of one turbine disk and two compressor disks, which are mounted on a flexible hollow shaft fixed at its ends. Theprofile of the turbine disk has been assumed to be hyperbolic, while those of the compressor disks were of web-type with

Fig. 17. Campbell diagram of the complex rotor, (Ω Ω* = /60).

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–8584

smaller radii compared to the turbine disk. The material of the rotor is assumed to be steel with =E 207 GPa, ν = 0.28,ρ = −7860 kg m 3 and α = × °−13 10 1/ C6 .

Various combinations of longitudinal and cross-sectional discretizations have been considered to generate the mathe-matical models for the dynamic analyses. Fig. 14 shows the three meshes adopted along the y-axis, in which each disk hasbeen modelled using eight 2-node beam elements. The coarsest (Fig. 14-a) and the finest mesh (Fig. 14-c) consisted of 32 and50 finite elements for modelling the structure, while the other discretization that is shown in Fig. 14-b represents anintermediate solution with 40 beam elements. Regarding the cross-sectional discretizations, Fig. 15 shows the adoptedLagrange models. The first two models have been created with bilinear elements, whose distributions have been assumeduniform (Fig. 15-a) and refined at the disk boundaries (Fig. 15-b). The remaining meshes have been built using Lagrangeelements with 9 (Fig. 15-c) and 16 (Fig. 15-d) nodes. Table 4 lists the natural frequencies (atΩ¼0 rad s�1) corresponding tothe 15 exaggerated mode shapes shown in Fig. 16. These mode shapes involve bending, torsional, shell-type and coupleddeformations. Comparisons with a converged 3D ANSYS solution have revealed that the CUF models provided accurateestimations of the considered frequencies. Moreover, it is noteworthy that the use of higher-order LE elements requiredlower numbers of DOF than the L4 models for obtaining comparable results.

The L9 model has been adopted to draw the Campbell diagram shown in Fig. 17. As expected, the gyroscopic effect issignificant for the branches that correspond to bending mode shapes of the shaft and the disks. The centrifugal stiffeningdetermines a slight increasing of the frequencies corresponding to the umbrella mode shapes (for example, ω6 and ω8 ofFig. 16) while, for the torsional deformations (ω1,ω3 andω7 of Fig. 16), the frequencies are almost constant within the speedrange.

6. Conclusion

This paper aimed at analysing the dynamic behaviour of complex-shaped rotating structures subjected to centrifugal andthermal loads. To this end, high-fidelity beam formulations have been used to ensure a satisfactory accuracy with a lowercomputational effort with respect to the full 3D solutions. The Finite Element method has been used to solve the governingequations by including all contributions due to the rotation, namely the spin-softening and the Coriolis terms. The three-dimensional stress state generated by thermal and centrifugal loads has been computed through static analyses. The effectsof these prestress distributions on the natural frequencies have been evaluated by including the corresponding geometricalstiffening matrices into the dynamic system. Firstly, the analyses have been performed on a thin, constant thickness disk anda disk with a hyperbolic profile. These structures have been considered clamped at the bore and subjected to parabolictemperature distributions. The results revealed that the thermal effects are significant for the thin disk while, conversely,they can be considered negligible for the thicker configuration. However, the “mode swapping” phenomenon has beenobserved for both cases. Secondly, the dynamics of a rotor, which is constituted of a flexible shaft and three disks withdifferent profiles, has been studied. The frequencies related to bending, torsional, shell-type and coupled mode shapes havebeen computed as functions of the rotational speed. The effects of gyroscopic and stiffening terms on the different mode

A. Entezari et al. / Journal of Sound and Vibration 405 (2017) 68–85 85

shapes have been discussed. The comparisons demonstrated that, also for complex-shaped rotors, the proposed formulationcan reproduce 3D results with lower/comparable computational efforts.

Acknowledgement

"E. Carrera has been partially supported by the Russian Science Foundation (Grant No. 15-19-30002).

References

[1] J. Rao, History of Rotating Machinery Dynamics, 20, Springer Science & Business Media, 2011.[2] A.D. Dimarogonas, S.A. Paipetis, T.G. Chondros, Analytical Methods in Rotor Dynamics, Springer, 2013.[3] G. Genta, Dynamics of Rotating Systems, Springer, New York, 2007 (URL: ).[4] C.-W. Lee, Vibration analysis of rotors, 21, Springer, 1993.[5] J. Vance, F. Zeidan, B. Murphy, Machinery Vibration and Rotordynamics, Wiley Online Library, 2010.[6] H.F. Bauer, Vibration of a rotating uniform beam, part i: orientation in the axis of rotation, J. Sound Vib. 72 (2) (1980) 177–189.[7] M. Chen, Y. Liao, Vibrations of pretwisted spinning beams under axial compressive loads with elastic constraints, J. Sound Vib. 147 (3) (1991) 497–513.[8] J. Banerjee, H. Su, Development of a dynamic stiffness matrix for free vibration analysis of spinning beams, Comput. Struct. 82 (23) (2004) 2189–2197.[9] J.W.-Z. Zu, R.P. Han, Natural frequencies and normal modes of a spinning timoshenko beam with general boundary conditions, J. Appl. Mech. 59 (2S)

(1992) S197–S204.[10] S.-T. Choi, J.-D. Wu, Y.-T. Chou, Dynamic analysis of a spinning timoshenko beam by the differential quadrature method, AIAA J. 38 (5) (2000) 851–856.[11] C.Y. Tai, Y.J. Chan, A hierarchic high-order timoshenko beam finite element, Comput. Struct. 165 (2016)

48–58, http://dx.doi.org/10.1016/j.compstruc.2015.12.002.[12] D.R. Chivens, H. Nelson, The natural frequencies and critical speeds of a rotating, flexible shaft-disk system, J. Eng. Ind. 97 (3) (1975) 881–886.[13] G. Genta, C. Feng, A. Tonoli, Dynamics behavior of rotating bladed discs: a finite element formulation for the study of second and higher order

harmonics, J. Sound Vib. 329 (25) (2010) 5289–5306, http://dx.doi.org/10.1016/j.jsv.2010.07.015.[14] G. Genta, A. Tonoli, A harmonic finite element for the analysis of flexural, torsional and axial rotordynamic behaviour of discs, J. Sound Vib. 196 (1)

(1996) 19–43.[15] C.-F. Liu, J.-F. Lee, Y.-T. Lee, Axisymmetric vibration analysis of rotating annular plates by a 3d finite element, Int. J. solids Struct. 37 (41) (2000)

5813–5827.[16] J. Awrejcewicz, D. Grzelczyk, Modeling and analytical/numerical analysis of wear processes in a mechanical friction clutch, Int. J. Bifurc. Chaos 21 (10)

(2011) 2861–2869.[17] J. Awrejcewicz, D. Grzelczyk, Modeling and analysis of thermal processes in mechanical friction clutchnumerical and experimental investigations, Int.

J. Struct. Stab. Dyn. 13 (07) (2013) 1340004.[18] D. Grzelczyk, J. Awrejcewicz, Wear processes in a mechanical friction clutch: theoretical, numerical, and experimental studies, Mathematical Problems

in Engineering 2015.[19] J. Rao, R. Sreenivas, Dynamics of a three level rotor system using solid elements, in: ASME Turbo Expo 2003, collocated with the 2003 International

Joint Power Generation Conference, American Society of Mechanical Engineers, pp. 601–606.[20] J. Hong, M. Chen, S. Liu, Application of whole engine finite element models in aero-engine rotordynamic simulation analysis, ASME.[21] J.J. Moore, G. Vannini, M. Camatti, P. Bianchi, Rotordynamic analysis of a large industrial turbocompressor including finite element substructure

modeling, J. Eng. Gas. Turbines Power 132 (8) (2010) 082401.[22] H. Taplak, M. Parlak, Evaluation of gas turbine rotor dynamic analysis using the finite element method, Measurement 45 (5) (2012) 1089–1097.[23] M. Zhuo, L. Yang, L. Yu, The steady-state thermal effect on rotordynamics of a rod-fastened rotor-bearing system, in: ASME 2014 International

Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, pp. V04BT04A014–V04BT04A014.[24] E. Carrera, G. Giunta, M. Petrolo, Beam Structures: classical and Advanced Theories, Wiley, 2011 (URL: ).[25] E. Carrera, E.M. Filippi, E. Zappino, Analysis of rotor dynamic by one-dimensional variable kinematic theories, J. Eng. Gas. Turbines Power 135 (9)

(2013) 092501, http://dx.doi.org/10.1115/1.4024381.[26] E. Carrera, M. Filippi, Variable kinematic one-dimensional finite elements for the analysis of rotors made of composite materials, J. Eng. Gas. Turbines

Power 136 (9) (2014) 092501, http://dx.doi.org/10.1115/1.4027192.[27] E. Carrera, M. Filippi, A refined one-dimensional rotordynamics model with three-dimensional capabilities, J. Sound Vib. 366 (2016)

343–356, http://dx.doi.org/10.1016/j.jsv.2015.12.036.[28] E. Carrera, A. Entezari, M. Filippi, M.A. Kouchakzadeh, 3d thermoelastic analysis of rotating disks having arbitrary profile based on a variable kinematic

1d finite element method, J. Therm. Stress. (2016) 1–16, http://dx.doi.org/10.1080/01495739.2016.1207119.[29] A. Entezari, M.A. Kouchakzadeh, E. Carrera, M. Filippi, A refined finite element method for stress analys is of rotors and rotating disks with variable

thickness, Acta Mech.(2016)1–20, http://dx.doi.org/10.1007/s00707-016-1727-x. (URL: http://dx.doi.org/10.1007/s00707-016-1727-x).[30] K. Bathe, Finite element procedure, Prentice hall.[31] E. Carrera, M. Petrolo, Refined one-dimensional formulations for laminated structure analysis, AIAA J. 50 (1) (2012) 176–189.[32] E. Carrera, M. Cinefra, M. Petrolo, E. Zappino, Finite Element Analysis of Structures through Unified Formulation, Wiley, 2014 (URL: ).