A refined one-dimensional rotordynamics model with three-dimensional capabilities

Journal of Sound and Vibration 366 (2016) 343–356

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A refined one-dimensional rotordynamics model with three-dimensional capabilities E. Carrera, M. Filippi n Department 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

abstract

Article history: Received 3 June 2015 Received in revised form 15 December 2015 Accepted 16 December 2015 Handling Editor: H. Ouyang Available online 2 January 2016

This paper evaluates the vibration characteristics of various rotating structures. The present methodology exploits the one-dimensional Carrera Unified Formulation (1D CUF), which enables one to go beyond the kinematic assumptions of classical beam theories. According to the component-wise (CW) approach, Lagrange-like polynomial expansions (LE) are here adopted to develop the refined displacement theories. The LE elements make it possible to model each structural component of the rotor with an arbitrary degree of accuracy using either different displacement theories or localized mesh refinements. Hamilton's Principle is used to derive the governing equations, which are solved by the Finite Element Method. The CUF one-dimensional theory includes all the effects due to rotation, namely the Coriolis term, spin softening and geometrical stiffening. The numerical simulations have been performed considering a thin ring, discs and bladeddeformable shafts. The effects of the number and the position of the blades on the dynamic stability of the rotor have been evaluated. The results have been compared, when possible, with the 2D and 3D solutions that are available in the literature. CUF models appear very practical to investigate the dynamics of complex rotating structures since they provide 2D and quasi-3D results, while preserving the computational effectiveness of one-dimensional solutions. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Rotordynamics FEM Component-wise approach Higher-order beam theories Carrera Unified Formulation

1. Introduction The study of the dynamics of rotors is of primary importance for the design of several structural components such as compressors, turbines, and propellers. These structures are generally analyzed through simplified mathematical models that have been established on the basis of one- and two-dimensional assumptions. One-dimensional models yield accurate results for rotors with compact cross-sections. For instance, using Euler–Bernoulli and Timoshenko's beam theories, Bauer [1] and Curti et al. [2,3] analytically evaluated the critical speeds and instabilities of metallic shafts. Bauchau [4] and Chen et al. [5] considered the same problems for thin-walled shafts made of composite materials. Bert and Kim [6] compared the critical speeds of laminated cylinders with shell and experimental results taking into account the effect of bending–twisting coupling. In order to overcome the limitations of classical beam theories, Song et al. [7,8] proposed refined formulations that included non-classical effects such as primary and secondary warping effects.

n

Corresponding author. E-mail addresses: [email protected] (E. Carrera), matteo.fi[email protected] (M. Filippi).

http://dx.doi.org/10.1016/j.jsv.2015.12.036 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

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When the cross-section deformability is no longer negligible, the effect of centrifugal stiffening becomes predominant. Therefore, 2D formulations are typically adopted to model thin-walled structures, such as discs and cylinders. Saito and Endo [9], for instance, used Flügge-type basic equations to evaluate the effects of different boundary conditions on the dynamics of finite-length rotating shells. Later, Chen et al. [10] adopted the Novoshilov theory to develop shell finite elements for vibratory problems of high speed rotating cylinders. Similarly, Duo et al. proposed a nonlinear two-dimensional finite element to analyze thin [11] and thick [12] shells, in which the radial stress contribution was overlooked. Furthermore, Sun et al. [13] performed interesting parametric studies using an analytical solution for any boundary condition, based on the Fourier series expansion method and the Sander shell theory. Lam and Loy [14] evaluated the accuracy of Donnell, Flügge, Love and Sanders shell theories for simply supported laminated cylinders through a unified formulation. In [15], the authors considered different boundary conditions using the most efficient theory for thin cylinders, namely the Love theory. The same displacement formulation was used in [16], where Lee and Kim studied the dynamic behavior of orthotropic cylinders with axial and circumferential stiffeners. Considering the Reissner–Mindlin kinematical hypothesis, Chatelet et al. [18] proposed reduction techniques for dynamic analyses of cyclically symmetric multilayered structures such as cylinders and disc-shaft assemblies. On the other hand, several rotor configurations have been modeled through combinations of one-, two- and three-dimensional approaches. For instance, Jang and Lee [19] combined the FEM with substructure synthesis to describe the dynamic behavior of a spinning disc-spindle system. The disc was modeled using 2D elements based on the Kirchhoff theory, while the spindle and stationary were discretized using Rayleigh and Euler beam elements, respectively. Later, Jang and Han [20] extended this approach to a disc-spindle system supported by flexible structures, such as a stator core, housing and base plate. The supports were modeled with three-dimensional elements to ensure the fulfilment of geometric compatibility with the beam elements. Genta et al. [21] developed an annular element to describe the dynamics of rotating bladed discs on flexible shafts. The structural components (shaft, disc and blades) were linked using “transition” elements. The displacement field for the disc was obtained using a Fourier series and polynomial shape functions along the circumferential and radial direction, respectively. Combescure and Lazarus [22] combined 2D Fourier elements with 3D FE to perform analyses of large rotating machines. The main limitation of the Fourier approach is that the rotors must be axisymmetric. Therefore, in order to study asymmetric structures and ensure an accurate description of their kinematics, the 3D FE modeling techniques are often used [23]. Despite the significant advances in computing power, large and sophisticated 3D models still represent complex computational problems. These problems have driven the derivation of reducing techniques, which are commonly found in the control, optimization, and structural mechanics fields, especially over the last few years. A reduction model, in a nutshell, is a mathematical tool that is devoted to limiting the number of computations, starting from information obtained from computationally expensive 3D models [24]. As far as computing efficiency is concerned, the benefits of these approaches are significant but their use generally requires great capabilities in the interpretation of the physical system in order to preserve the characteristics of the real structure. An interesting review on reduction models can be found in [25]. The present paper presents a class of refined one-dimensional beam models obtained through the Carrera Unified Formulation [26]. CUF enables one, at least theoretically, to derive an infinite number of sophisticated displacement models. As far as the rotor dynamics problem is concerned, the 1D-CUF approach has led to encouraging results for shafts made of isotropic [27] and orthotropic materials [28] and for thin/thick rotating shells [29]. In these works, Taylor-like expansions of n-th order (TEn) were adopted to obtain the displacement theories. The results have demonstrated that TE models are efficient for prismatic structures, but they can show limitations when the components of the rotor present different deformability. In order to overcome this shortcoming, the CW method has been extended, for the first time, within the CUF framework, to the rotordynamics problem. The CW approach exploits the inherent capability of Lagrange-type expansions (LE) to assemble CUF beam models at the cross-section level. The CW methodology allows only beam elements to be used to model each component of the rotor (shaft, disc, blade, etc.) with arbitrary kinematic assumptions, which depend on the characteristics of the components (deformability, mechanical properties, etc.). Moreover, only physical surfaces are employed to build the mathematical model. The enhanced capabilities of the CWCUF approach have already been observed in static and dynamic analyses of aeronautical, civil and composite structures. The proposed formulation includes all the effects due to the rotation, namely the Coriolis, spin softening and stress stiffening contribution. Numerical analyses have been carried out on both axisymmetric and asymmetric rotating structures, and the related results have been compared with solutions available in the literature and with those obtained by means of an FE commercial software. The article is organized as follows. Section 2 presents the basic notions of the LE formulation; Section 3 contains the rotordynamics equations in CUF form; Section 4 is devoted to the numerical results, and the concluding remarks are given in Section 5.

2. One-dimensional theories through the unified formulation: the Lagrange element Within the CUF framework, the displacement field is an expansion of generic cross-sectional functions, F τ : uðx; y; zÞ ¼ F τ ðx; zÞuτ ðyÞ

τ ¼ 1; 2; …; M

(1)

where uτ is the vector of the generalized displacements, M is the number of terms of the expansion and, in accordance with the generalized Einstein's notation, τ indicates summation. The functions, F τ ðx; zÞ, are Lagrange-like polynomials and the isoparametric formulation is used to deal with arbitrary shape geometries. For the nine-point element (denoted as L9)

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shown in Fig. 1, the interpolation functions are

 τ ¼ 1; 3; 5; 7 F τ ¼ 14 r 2 þ r r τ s2 þ s sτ



 2 1 2 2 1 2 2 F τ ¼ 2 sτ s  s sτ 1  r þ 2 r τ r  r r τ 1  s2 τ ¼ 2; 4; 6; 8 F τ ¼ ð1  r 2 Þð1  s2 Þ

τ¼9

(2)

where r and s vary from  1 to þ 1, whereas rτ and sτ are the coordinates of the nine points whose locations in the natural coordinate frame are shown in Fig. 1. The displacement field of a L9 element is therefore ux ¼ F 1 ux1 þ F 2 ux2 þ F 3 ux3 þ ⋯ þ F 9 ux9 uy ¼ F 1 uy1 þ F 2 uy2 þ F 3 uy3 þ ⋯ þ F 9 uy9 uz ¼ F 1 uz1 þF 2 uz2 þ F 3 uz3 þ⋯ þ F 9 uz9

(3)

in which ux1 ; …; uz9 are the displacement variables of the problem and they represent the translational displacement components of each of the nine points of the L9 element. The beam cross-section can be discretized by using several Lagrange-type elements for further refinements as shown in Fig. 2. For more details on Lagrange 1D models, the authors suggest to refer to [30,31], where Lagrange elements with 3, 4, 6 and 16 nodes are described. The stresses and the strains are grouped as follows:

ϵp ¼ fϵzz ϵxx ϵxz gT σ p ¼ fσ zz σ xx σ xz gT ϵn ¼ fϵzy ϵxy ϵyy gT σ n ¼ fσ zy σ xy σ yy gT

(4)

The subscript p refers to the terms lying on the cross-section, while n refers to those lying on the orthogonal planes to the cross-section. The linear strain–displacement relations and Hooke's law are, respectively,

ϵp ¼ Dp u ϵn ¼ ðDny þDnp Þu σ p ¼ C~ pp ϵp þ C~ pn ϵn

Fig. 1. L9 element in the natural coordinate system.

Fig. 2. Two assembled L9 elements in actual geometry.

(5)

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σ n ¼ C~ np ϵp þ C~ nn ϵn in which Dp, Dny and Dnp are the material k are 2 k C~ 6 11 k 6 k C~ pp ¼ 6 C~ 12 4 k C~ 14

(6)

linear differential operators reported in [26]. The coefficient matrices of the generic k C~ 12 k C~ 22 k C~ 24

k C~ 14

3

2

k C~ 6 15 k 6 k C~ pn ¼ 6 C~ 25 4 k C~ 45

7 k 7 C~ 24 7; 5 k C~ 44

k C~ 16

k C~ 13

3

2

k C~ 6 55 k 6 k C~ nn ¼ 6 C~ 56 4 k C~ 35

7 k 7 C~ 23 7; 5 k C~ 43

k C~ 26 k C~ 46

k C~ 56 k C~ 66 k C~ 36

k C~ 35

3

7 k 7 C~ 36 7 5 k C~ 33

(7)

Explicit forms of the coefficients of the C~ matrices are reported in [28]. Since a classical Finite Element technique is adopted, the generalized displacement vector is uτ ðy; tÞ ¼ N i ðyÞqτi ðtÞ

(8)

in which Ni(y) are the shape functions and qτi ðtÞ is the nodal displacement vector n oT quy quz qτi ðtÞ ¼ qux τi

τi

τi

(9)

3. Rotordynamics equations in CUF form In order to obtain the equations of motion of a structure that is rotating about its longitudinal axis with a constant speed

Ω, Hamilton's Principle is used:

δ

Z

t1




t0

 T  ðU þ U σ 0 Þ dt ¼ 0

(10)

where T and U are, respectively, the kinetic and the potential energies in the rotating reference frame (see [27]). The contribution U σ 0 is due to the pre-stress σ 0 (or pre-strain ϵ0 ) field, which may be generated by centrifugal or thermal effects. The expression of U σ is Z U σ 0 ¼ σ T0 ϵnl dV (11) V

where ϵnl is the nonlinear part of strains. The pre-stress field for axial-symmetric structures is typically provided in terms of circumferential ðσ δδ Þ, radial (σrr) and axial (σyy) contributions that are in a cartesian coordinate system

σ 0xx ¼ σ δδ sin ðδÞ2 þ σ rr cos ðδÞ2 σ 0yy ¼ σ yy σ 0xz

σ 0zz ¼ σ δδ cos ðδÞ2 þ σ rr sin ðδÞ2 ð ¼ σ 0zx Þ ¼ σ rr sin ðδÞ cos ðδÞ  σ δδ sin ðδÞ cos ðδÞ

The expressions of the initial stress are [23]

 Disc (no forces are exchanged in radial direction with the support):   r2 r2 ð3 2νÞ 2 r o þ r 2i  i 2 o  r 2 8ð1  νÞ r   r 2i r 2o ð1 þ 2 νÞ 2 2 ð3  2νÞ 2 2 r o þr i þ 2  r σ δδ ¼ ρΩ 8ð1  νÞ ð3 2νÞ r

σ rr ¼ ρΩ2

 Blade (cantilevered beam):



σ rr ¼ 12 ρΩ2 ðro þ lb Þ2  r2



where ri and ro are the inner and outer radii, respectively, lb is the length of the blade, r is the generic radius, and material density. Therefore, the geometrical potential energy is Z nl nl nl U σ 0 ¼ σ 0xx ϵnl xx þ σ 0zz ϵzz þ σ 0xz ϵxz þ σ 0zx ϵzx dV V

(12)

(13)

ρ is the (14)

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Using Eqs. (1) and (8), Eq. (10) becomes Z t1     δqTτi Mijτs q€ sj þ δqTτi GijΩτs q_ sj þ δqTτi Kijτs þKijστs  Kijτs qsj þ δqTτi ½FiΩτ r dt ¼ 0

(15)

Ω

0

t0

The matrices written in terms of fundamental nuclei are Mijτs ¼ I ijl ◁ðF τ ρk IF s Þ▷

GijΩτs ¼ I ijl ◁ðF τ ρk IF s Þ▷2Ω h i h k i h k i k k k Kijτs ¼ I ijl ◁DTnp ðF τ IÞ C~ np Dp ðF s IÞ þ C~ nn Dnp ðF s IÞ þDTp ðF τ IÞ C~ pp Dp ðF s IÞ þ C~ pn Dnp ðF s IÞ ▷ þ I ij;y ◁ DTnp ðF τ IÞ þDTp ðF τ IÞC~ pn F s ▷ l h k i k k ITAy ◁F τ C~ np Dp ðF s IÞ þ C~ nn Dnp ðF s IÞ ▷ þI i;yj;y ITAy IAy ◁F τ C~ nn F s ▷ þ I i;yj l l Kijστ0s ¼ I ijl ◁ðF τ;x σ 0xx IF s;x Þ þ ðF τ;z σ 0zz IF s;z Þ þ ðF τ;x σ 0xz IF s;z Þ þ ðF τ;z σ 0zx IF s;x Þ▷ KijΩτs ¼ I ijl ◁ðF τ ρk IF s Þ▷Ω Ω T

iτ FΩ ¼ I il ◁F τ ρr▷Ω Ω T

where

2

0 0 Ω¼6 4 Ω

0 0 0

Ω

3

2

0 6 IAy ¼ 4 1 0

7 05 0

0 0 1

1

3

7 05 0

(16) 2

1 6 I¼40 0

0 1 0

0

3

7 05 1

(17)

Z ◁…▷¼

A

… dA

(18)

  Z   ij; i; j i; j; N i ; N i Nj ; Ni N j;y ; N i;y Nj ; N i;y N j;y dy I il ; I ijl ; I l y ; I l y ; I l y y ¼

(19)

l

and r ¼ fxP ; 0; zP gT is the distance of a generic point P belonging to the cross-section from the neutral axis. The nine components of the fundamental nucleus of the matrix Kijτs are written in an explicit form in [28]. The homogeneous equations are solved assuming a periodic solution q ¼ qeiωt in order to obtain natural frequencies and normal modes of the rotor: 
 qeiωt K þKσ 0  KΩ þ ðGΩ Þiω  ðMÞω2 ¼ 0 (20) The quadratic eigenvalue problem of Eq. (20) is solved as previously done in [27,28].

4. Numerical results This section has the aim of showing how various rotors can be modeled using only the 1D-CUF approach. In order to demonstrate the enhanced features of LE models, highly deformable structures have been considered, in which the effects due to rotation can only be predicted properly using 2D and 3D approaches. Table 1 Frequencies (Hz) of the ring shown in Fig. 3. Mode

7,8a

9,10b

11,12a

13,14a

15,16b

17,18a

19,20b

DOF

40L9 1B3 2B3 3B3 4B3 5B3

51.46 51.46 51.46 51.46 51.46

94.92 92.52 92.11 91.99 91.94

145.9 145.9 145.9 145.9 145.9

280.8 280.8 280.8 280.7 280.7

300.7 295.5 294.6 294.4 294.3

456.1 456.1 456.0 456.0 456.0

605.9 598.0 596.7 596.3 596.3

2160 3600 5040 6480 7920

20L16 1B3 2B3 3B3 4B3 5B3 HEXA8c HEXA20c

50.79 50.78 50.78 50.78 50.78 50.64 50.80

94.13 91.67 91.29 91.18 91.14 94.43 91.06

143.7 143.7 143.7 143.7 143.7 143.8 143.0

275.9 275.9 275.9 275.9 275.9 275.8 273.7

298.7 293.3 292.5 292.3 292.2 298.3 291.3

447.2 447.2 447.2 447.2 447.2 446.5 441.5

602.2 593.9 592.8 592.4 592.3 599.6 589.3

2160 3600 5040 6480 7920 2160 7200

a b c

In-plane mode. Out-of-plane mode. 80  2  2 brick finite elements.

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4.1. Rotating ring A thin ring, which is free to rotate about its center, has been considered here. The mean radius is 0.5 m, and the rectangular cross-section has a width and height equal to 0.05 and 0.02 m, respectively. The structure is made of aluminum, whose mechanical properties are Young's Modulus, E ¼73.1 GPa; density, ρ ¼2770 kg m  3 and Poisson's ratio, ν ¼0.33. With the purpose of evaluating the accuracy of the proposed approach, the frequencies obtained from various LE models are compared in Table 1 with two converged NASTRAN finite element solutions. Fig. 3 schematically shows the 1D discretizations. The 3-node beam elements (denoted as B3) have been used along the longitudinal direction, while Lagrange elements with either 9 or 16 nodes (denoted as L16) have been adopted to model the cross-section. The degrees of freedom (DOF) required by the models, which are reported in the last column of Table 1, are equal to: DOF ¼ N LE  N B  3

(21)

where NLE and NB are the total numbers of nodes on the cross-section and along the longitudinal axis, respectively. The comparisons have revealed that the 1D-CUF results are in good agreement with the reference solutions. For example, for the

Fig. 3. Mesh for the thin ring.

Fig. 4. Frequencies (Hz) vs. spin speed (rad s  1) for a rotating ring; ‘—’: 20L16 and 1B3, ‘- -’: theoretical solution [23], and ‘’: solid FE model [23].

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Fig. 5. Mesh for the thin disc. (a) 5  16 L9, (b) 4  30 L9, and (c) 2  16 L16.

Table 2 Frequencies (Hz) computed with the LE models shown in Fig. 5 of the non-rotating disc. m DOF

5  16 (a) 3168

4  30 (b) 4860

2  16 (c) 3024

NASTRAN 6000

0

81:69ð1:68Þ

82:37ð2:53Þ

81:30ð1:19Þ

80.34

1

83:56ð2:01Þ

84:07ð2:64Þ

82:97ð1:29Þ

81.91

2

92:76ð2:71Þ

92:89ð2:86Þ

91:73ð1:57Þ

90.31

3

118:4ð4:31Þ

117:1ð3:17Þ

115:5ð1:76Þ

113.5

4

169:2ð8:46Þ

162:3ð4:04Þ

159:0ð1:92Þ

156.0

5

249:9ð14:9Þ

229:6ð5:61Þ

222:0ð2:12Þ

217.4

m: number of nodal diameters. (): percentage error with respect to the shell solution.

Fig. 6. Frequencies (Hz) vs. spin speed (rev/min) for a rotating disc; ‘—’: LE9, ‘- - -’ numerical solution [34], and ‘◯’, ‘▵’, ‘□’: experimental data [34].

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Fig. 7. Mesh for the thin disc on the flexible shaft.

Fig. 8. Frequencies vs. spin speed for a rotating disc on a flexible shaft; ‘—’: LE model, ‘- - -’: TE7 model, ‘’: solid FE model [22].

20 L16 model with a single B3, the maximum error with respect to the finest FE reference model is about 3.3 percent. The Campbell diagram obtained with the coarsest L16 model is shown in Fig. 4. The related backward and forward frequencies have been compared with theoretical and FE results proposed in [23]. It has been observed that the agreement between the 1D-CUF and the closed-form solutions is significant for all the considered modal shapes. On the other hand, the 3D FE solution (40 20-node brick elements corresponding to 1440 DOF) overestimated the eighth in-plane frequency, thus demonstrating that a finer mesh would be required for this kind of structure. 4.2. Rotating disc Since the LE beam elements enable non-classical boundary conditions to be imposed a hollow disc, fixed at its bore, has been studied. The outer and inner radii, and the uniform thickness are 203.2, 101.6 and 1.016 mm, respectively. The disc is made of steel, whose material properties are E¼210 GPa, ρ ¼7800 kg m  3, and ν ¼ 0.3. A single 3-node beam element (1B3) has been used along the longitudinal direction, while three different meshes have been adopted to model the cross-section (see Fig. 5). The frequencies of the non-rotating disc are compared in Table 2 with those obtained from a NASTRAN solution. This model has been built using 20 and 60 QUAD4 elements along the radial and peripheral directions, respectively. It can be observed that the discrepancy between the LE results and the shell solution increases when the number of nodal diameters (m) increases. As for the L9 elements, the maximum error related to the 5  16 mesh is about 14 percent. As expected, a significant improvement has been achieved by

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Table 3 Frequencies, f (rad s  1), at different speed values, Ω (rad s  1), related to the mode shapes shown in Fig. 8. Ω

f1

f2

f3

f4

0 500 698 1000 1500 2000

637.9 187.5 0.0 296.2 786.7 1278.8

637.9 978.5 1111.8 1338.1 1760.3 2192.3

1001.7 1081.8 1185.4 1386.0 1780.5 2216.0

1001.7 1295.0 1473.1 1761.2 2254.4 2754.9

Fig. 9. First mode shapes of four blades on the flexible shaft in the non-rotating state.

enriching the discretization along the circumferential direction. In fact, the maximum relative error decreases to below 6 percent with the 4  30 L9 mesh (see Fig. 5b). Since the mesh refinement technique determines an increase in the computational cost, a higher-order of the Lagrange expansions has been evaluated with the purpose of reducing the number of DOFs and preserving accuracy. It is worth noting that the use of cubic expansions within the element is more effective than further mesh refinements; the maximum error is in fact about 2 percent with only 32 L16 elements. Fig. 6 shows Campbell's diagram obtained with the L16 model for the mode shapes characterized by different numbers of nodal diameters. The results are reported in the inertial reference frame in order to compare them with the experimental and numerical data presented in [34], in which Wilson and Kirkhope proposed a finite annular element based on Mindlin's plate theory for the study of axisymmetric structures. The 1D-CUF results are close to the reference solutions, despite the significant effect of rotation on the disc frequencies.

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Fig. 10. Frequencies (Hz) as functions of the spin speed (Hz) of a flexible shaft with blades.

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Table 4 Critical speeds (Hz) of various structural configurations for blades on a flexible shaft. 1 blade

2 perp. blades

2 paral. blades

3 blades

4 blades

378

378

364

364

357

Fig. 11. Damping (Hz) as a function of the spin speed (Hz) of a flexible shaft with blades.

4.3. Rotating disc on a flexible shaft The rotor is constituted by a thin disc, with the radius and the thickness equal to rd ¼0.15, hd ¼0.005 m, respectively, fixed at one-third of a shaft with length, Ls ¼0.4 m, and radius, rs ¼0.01 m. The rotor has been considered simply supported at each extremity and made of steel (E¼207 GPa, ρ ¼7860 kg m  3, and ν ¼0.28). The used mathematical model consisted of 12 L3 for the shaft cross-section, 24 L9 for the disc and, 7 4-node beam elements along the longitudinal axis (see Fig. 7). The Campbell diagram, related to the first two mode shapes, is shown in Fig. 8. The first mode shape is dominated by the bending of the shaft, whereas the second one involves a significant deformation of the disc. The analyses have been performed, within the CUF context, using both Taylor- and Lagrange-type expansions. The related results are compared with a 3D finite element solution that was proposed in [22]. Although the TE7 approximation requires a higher number of degrees of freedom (2376 DOF) than the LE model (2010 DOF), discrepancies are evident in the computation of the frequency variations with respect to the 3D solution. The slow convergence of the TE approach is because both the compact shaft and the deformable disc are modeled with the same approximation order (7th order), despite a significant difference in their deformability. This limitation can be overcome by adopting different displacement theories for the various rotor components, thus ensuring displacement compatibility through Lagrange multipliers. On the other hand, the enhanced capabilities of the CW approach lead to results very close to the solid solution. To facilitate future comparisons, Table 3 lists forward and backward frequencies at different speeds computed with the LE model. 4.4. Blades on a flexible shaft The rotor is constituted by the same shaft as that of the previous case, and by a variable number of long blades. The rotor, which is made of steel, is clamped and simply supported at its ends. The blades are fixed at 0.14 m from the clamped extremity, and their length, width, and thickness are 0.14, 0.02, and 0.001 m, respectively. The finite element model consists of 6 L9 for each blade, 20 L3 for the shaft and 7 4-node beam elements along the neutral axis. Fig. 9 shows four mode shapes of the structure in the non-rotating state, where it is possible to observe that the blades behave similar to cantilevered beams with bending and torsional deformations. In order to evaluate the effects of blade deformability on the dynamics of the structure, Fig. 10 shows the Campbell diagrams of the shaft with 1, 2, 3 and 4 blades. Owing to the number and the position of the blades, the moments of inertia, with respect to the two cross-sectional axes, are only equal for the configurations in Fig. 10b and d. The critical speed is unique for these cases, while, for the other structures, it is possible to observe an interval of critical values. Table 4 lists the first critical speeds of the considered

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Fig. 12. Graphical representations of auto-MACX values.

structures. Moreover, it should be observed that the curves related to the bending frequencies of the blades veer when they approach the global bending branches (Fig. 9c). In the case of multiple blades, only one curve interacts with the global bending mode. Furthermore, dynamic instability, which is generated by the interaction between the backward bending curve and the blade frequencies, occurs after the range of critical speeds. The instability regions are shown in Fig. 11, where the real part of the complex eigenvalues is plotted as functions of the spin speed. The graph points out that the instability region increases when the number of blades increases. Moreover, regarding the structures with two blades, the perpendicular configuration exhibits unstable behavior within a speed range that is almost twice that of the interval of the structure with parallel blades. In order to evaluate the dependency of the mode shapes on the rotational speed, a generalization of the Modal Assurance Criterion (MAC) has been used. This criterion, called MACX, was proposed by Vacher et al. [33] to measure the degree of linearity (consistency) of complex vectors. The expression of the MACX is:
 MACX ψ Ω1 ; ψ Ω2 ¼

ðjψ Ω1 ψ Ω2 j þ jψ TΩ1 ψ Ω2 jÞ2 ðψ Ω1 ψ Ω1 þ jψ TΩ1 ψ Ω1 jÞðψ Ω2 ψ Ω2 þjψ TΩ2 ψ Ω2 jÞ 

(22)

where ψ Ω1 and ψ Ω2 are the eigenvectors computed at different rotational speeds (Ω1 and Ω2). Besides the Hermitian inner product ðψ Ω1 ψ Ω2 Þ, the criterion requires the computation of the scalar product ðψ TΩ1 ψ Ω2 Þ between the two vectors. First,

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Fig. 13. Graphical representations of MACX values at different rotational speeds.

the auto-MACX ðΩ1¼ Ω2Þ values shown in Fig. 12 were computed at different speeds for the one-bladed (a, b) and fourbladed (c, d) configurations. It has been observed that, for Ω ¼ 0, the MACX criterion only provides high correlations for mode pairs on the diagonal whereas, when the structure is rotating, non-negligible correlations are also predicted for offdiagonal pairs. Examples of correlated modal shapes are shown in Fig. 12b–d. Second, the correlation values at different speeds are shown in Fig. 13. As expected, the MACX matrices show sparse correlations due to crossing and veering phenomena. Regarding Fig. 13b and d, it is worth noting that the fourth (at Ω ¼0 rad/s) and the tenth (at Ω ¼100 rad/s) eigenvectors have little correspondence with the others, thus demonstrating that the rotation has an important effect on the deformation modes.

5. Conclusion In this paper, Carrera Unified Formulation has been used to study the dynamics of various kinds of spinning structures. The equations of motion have been derived from Hamilton's Principle and solved with the Finite Element Method. Effects of

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the initial stress have been included in the present one-dimensional formulation. Analyses have been performed on thin discs, rings and bladed-shafts made of metallic material. The main results show the following:

 the proposed beam elements enable the natural frequencies of deformable structures to be predicted with a high-level of accuracy;

 unlike classical beam theories, the 1D-CUF models enable the initial stress and the related geometrical stiffness matrix to be included;

 unlike the TE approach, the LE elements enable different structural components to be modeled with arbitrary orders of accuracy;

 the LE elements enable non-conventional boundary conditions (for example the disc clamped at the bore) to be modeled;  the LE approach enables the stability of non-axisymmetric configurations to be evaluated. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [33] [34]

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