Free Vibration Analysis of a Rotating Composite Shaft Using the 𝑝-Version of the Finite Element Method

Abstract

This paper is concerned with the dynamic behavior of the rotating composite shaft on rigid bearings. A 𝑝-version, hierarchical finite element is employed to define the model. A theoretical study allows the establishment of the kinetic energy and the strain energy of the shaft, necessary to the result of the equations of motion. In this model the transverse shear deformation, rotary inertia and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. A hierarchical beam finite element with six degrees of freedom per node is developed and used to find the natural frequencies of a rotating composite shaft. A program is elaborate for the calculation of the eigenfrequencies and critical speeds of a rotating composite shaft. To verify the present model, the critical speeds of composite shaft systems are compared with those available in the literature. The efficiency and accuracy of the methods employed are discussed.

1. Introduction

The application of composite shafts has come a long way from early low-speed automotive driveshafts to helicopter tail rotors operating above the second critical speed. With operation at supercritical speeds, a substantial amount of payoffs and net system weight reductions are possible. At the same time, the rotordynamic aspects assume more importance, and detailed analysis is required. There are some technological problems associated with implementation, such as joints with bearings, affixing of lumped masses, couplings, provision of external damping, and so forth. The solutions proposed are just adequate, but require substantial refinements, which might explain some of the differing experiences of various authors.

Zinberg and Symonds [1] described a boron/epoxy composite tail rotor driveshaft for a helicopter. The critical speeds were determined using equivalent modulus beam theory (EMBT), assuming the shaft to be a thin-walled circular tube simply supported at the ends. Shear deformation was not taken into account. The shaft critical speed was determined by extrapolation of the unbalance response curve which was obtained in the subcritical region.

Dos Reis et al. [2] published analytical investigations on thin-walled layered composite cylindrical tubes. In part III of the series of publications, the beam element was extended to formulate the problem of a rotor supported on general eight coefficient bearings. Results were obtained for shaft configuration of Zinberg and Symmonds. The authors have shown that bending-stretching coupling and shear-normal coupling effects change with stacking sequence, and alter the frequency values. Gupta and Singh [3] studied the effect of shear-normal coupling on rotor natural frequencies and modal damping. Kim and Bert [4] have formulated the problem of determination of critical speeds of a composite shaft including the effects of bending-twisting coupling. The shaft was modelled as a Bresse-Timoshenko beam. The shaft gyroscopics have also been included. The results compare well with Zinberg's rotor [1]. In another study, Bert and Kim [5] have analysed the dynamic instability of a composite drive shaft subjected to fluctuating torque and/or rotational speed by using various thin shell theories. The rotational effects include centrifugal and Coriolis forces. Dynamic instability regions for a long span simply supported shaft are presented.

M.-Y. Chang et al. [6] published the vibration behaviours of the rotating composite shafts. In the model, the transverse shear deformation, rotary inertia, and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. The model based on a first-order shear deformable beam theory (continuum-based Timoshenko beam theory).

C.-Y. Chang et al. [7] published the vibration analysis of rotating composite shafts containing randomly oriented reinforcements. The Mori-Tanaka mean-field theory is adopted here to account for the interaction at the finite concentrations of reinforcements in the composite material.

Additional recent work on composite shafts dealing with both the theoretical and experimental aspects was reported by Singh [8], Gupta and Singh [3], and Singh and Gupta [9, 10]. Rotordynamic formulation based on equivalent modulus beam theory was developed for a composite rotor with a number of lumped masses and supported on general eight coefficient bearings. A layerwise beam theory was derived by Gupta and Singh [3] from an available shell theory, with a layerwise displacement field, and was then extended to solve a general composite rotordynamic problem. The conventional rotor dynamic parameters as well as critical speeds, natural frequencies, damping factors, unbalance response, and threshold of stability were analysed in detail and results from the formulations based on the two theories, namely, the equivalent modulus beam theory (EMBT) and layerwise beam theory (LBT) were compared [9, 10]. The experimental rotordynamic studies carried by Singh and Gupta [11, 12] were conducted on two filament wound carbon/epoxy shafts with constant winding angles (±45^∘ and ±60^∘). Progressive balancing had to be carried out to enable the shaft to traverse through the first critical speed. Inspite of the very different shaft configurations used, the authors' have shown that bending-stretching coupling and shear-normal coupling effects change with stacking sequence, and alter the frequency values.

Some practical aspects such as effect of shaft disc angular misalignment, interaction between shaft bow, which is common in composite shafts and rotor unbalance, and an unsuccessful operation of a composite rotor with an external damper were discussed and reported by Singh and Gupta [11]. The Bode and cascade plots were generated and orbital analysis at various operating speeds was performed. The experimental critical speeds showed good correlation with the theoretical prediction.

This paper deals with the 𝑝-version, hierarchical finite element method applied to free vibration analysis of rotating composite shafts. The hierarchical concept for finite element shape functions has been investigated during the past 25 years. Babuska et al. [13] established a theoretical basis for 𝑝-elements, where the mesh keeps unchanged and the polynomial degree of the shape functions is increased; however, in the standard ℎ-version of the finite element method the mesh is refined to achieve convergence and the polynomial degree of the shape functions remains unchanged. Since then, standard forms of the hierarchical shape functions have been represented in the literature elsewhere; see for instance [14, 15].

Meirovitch and Baruh [16] and Zhu [17] have shown that the hierarchical finite element method yields a better accuracy than the ℎ-version for eigenvalues problems. The hierarchical shape functions used by Bardell [18] are based on integrated Legendre orthogonal polynomials; the symbolic computing is used to calculate the mass and stiffness matrices of beams and plates. Côté and Charron [19] give the selection of 𝑝-version shape functions for plate vibration analysis.

In the presented composite shaft model, the Timoshenko theory will be adopted. It is the purpose of the present work to study dynamic characteristics such as natural frequencies, whirling frequencies, and the critical speeds of the rotating composite shaft. In the model, the transverse shear deformation, rotary inertia, and gyroscopic effects, as well as the coupling effect due to the lamination of composite layers have been incorporated. To determine the rotating shaft system's responses, the hierarchical finite element method with trigonometric shape functions [20, 21] is used here to approximate the governing equations by a system of ordinary differential equations.

2. Equations of Motion

2.1. Kinetic and Strain Energy Expressions

The shaft is modelled as a Timoshenko beam, that is, first-order shear deformation theory with rotary inertia and gyroscopic effect is used. The shaft rotates at constant speed about its longitudinal axis. Due to the presence of fibers oriented than axially or circumferentially, coupling is made between bending and twisting. The shaft has a uniform, circular cross-section.

The following displacement field of a rotating shaft is assumed by choosing the coordinate axis 𝑥 to coincide with the shaft axis: 𝑈(𝑥,𝑦,𝑧,𝑡)=𝑈₀(𝑥,𝑡)+𝑧𝛽_𝑥(𝑥,𝑡)−𝑦𝛽_𝑦(𝑥,𝑡),𝑉(𝑥,𝑦,𝑧,𝑡)=𝑉₀(𝑥,𝑡)−𝑧𝜙(𝑥,𝑡),𝑊(𝑥,𝑦,𝑧,𝑡)=𝑊₀(𝑥,𝑡)+𝑦𝜙(𝑥,𝑡), (1) where 𝑈, 𝑉, and 𝑊 are the flexural displacements of any point on the cross-section of the shaft in the 𝑥, 𝑦, and 𝑧 directions (see Figure 1), the variables 𝑈₀, 𝑉₀, and 𝑊₀ are the flexural displacements of the shaft's axis, while 𝛽_𝑥 and 𝛽_𝑦 are the rotation angles of the cross-section, about the 𝑦 and 𝑧 axis, respectively. The 𝜙 is the angular displacement of the cross-section due to the torsion deformation of the shaft.

Figure 1 

The cylindrical coordinate system.

The strain components in the cylindrical coordinate system (as shown in Figures 1-2) can be written in terms of the displacement variables defined earlier as 𝜀_𝑥𝑥=𝜕𝑈₀𝜕𝑥+𝑟sin𝜃𝜕𝛽_𝑥𝜕𝑥−𝑟cos𝜃𝜕𝛽_𝑦,𝜀𝜕𝑥_𝑟𝑟=𝜀_𝜃𝜃=𝜀_𝑟𝜃𝜀=0,_𝑥𝜃=𝜀_𝜃𝑥=12(𝛽_𝑦sin𝜃+𝛽_𝑥cos𝜃−sin𝜃𝜕𝑉₀𝜕𝑥+cos𝜃𝜕𝑊₀𝜕𝑥+𝑟𝜕𝜙𝜀𝜕𝑥),_𝑥𝑟=𝜀_𝑟𝑥=12(𝛽_𝑥sin𝜃−𝛽_𝑦cos𝜃−sin𝜃𝜕𝑊₀𝜕𝑥+cos𝜃𝜕𝑉₀𝜕𝑥). (2)

Figure 2 

𝑘-layers of composite shaft.

Let us consider a composite shaft consists of 𝑘 layered (Figure 2) of fiber inclusion reinforced laminate. The constitutive relations for each layer are described by {𝜎_𝑥𝑥𝜎_𝜃𝜃𝜎_𝑟𝑟𝜏_𝑟𝜃𝜏_𝑥𝑟𝜏_𝑥𝜃𝐶}=^′11𝐶^′₁₂𝐶^′₁₃00𝐶^′₁₆𝐶^′₁₂𝐶^′₂₂𝐶^′₂₃00𝐶^′₂₆𝐶^′₁₃𝐶^′₂₃𝐶^′₃₃00𝐶^′₃₆000𝐶^′₄₄𝐶^′₄₅0000𝐶^′₄₅𝐶^′₅₅0𝐶^′₁₆𝐶^′₂₆𝐶^′₃₆00𝐶^′₆₆{𝜀_𝑥𝑥𝜀_𝜃𝜃𝜀_𝑟𝑟𝛾_𝑟𝜃𝛾_𝑥𝑟𝛾_𝑥𝜃}, (3) where 𝐶^′_𝑖𝑗 are the effective elastic constants, they are related to lamination angle 𝜂 (as shown in Figures 3-4) and the elastic constants of principal axes.

Figure 3 

A typical composite lamina and its principal axes.

Figure 4 

The definitions of the principal coordinate axes on an arbitrary layer of the composite.

The stress-strain relations of the 𝑛th layer expressed in the cylindrical coordinate system (Figure 5) can be expressed as 𝜎_𝑥𝑥=𝐶^′_11𝑛𝜀_𝑥𝑥+𝑘_𝑠𝐶^′_16𝑛𝛾_𝑥𝜃,𝜏_𝑥𝜃=𝜏_𝜃𝑥=𝑘_𝑠𝐶^′_16𝑛𝜀_𝑥𝑥+𝑘_𝑠𝐶^′_66𝑛𝛾_𝑥𝜃,𝜏_𝑥𝑟=𝜏_𝑟𝑥=𝑘_𝑠𝐶^′_55𝑛𝛾_𝑥𝑟, (4) where 𝑘_𝑠 is the transverse shear correction factor.

(a)

(b)

Figure 5 

The stress components: (a) in the coordinate axes (𝑥,𝑦,𝑧), (b) in the coordinate axes (𝑥,𝑟,𝜃).

The formula of the strain energy is 𝐸_𝑑=12_𝑉(𝜎_𝑥𝑥𝜀_𝑥𝑥+2𝜏_𝑥𝑟𝜀_𝑥𝑟+2𝜏_𝑥𝜃𝜀_𝑥𝜃)𝑑𝑉. (5)

The various components of strain energy come from the shaft: 𝐸_𝑑=12𝐴₁₁^𝐿₀(𝜕𝑈₀)𝜕𝑥²+1𝑑𝑥2𝐵₁₁[^𝐿₀(𝜕𝛽_𝑥)𝜕𝑥²𝑑𝑥+^𝐿₀(𝜕𝛽_𝑦)𝜕𝑥²+1𝑑𝑥]2𝑘_𝑠𝐵₆₆^𝐿₀(𝜕𝜙)𝜕𝑥²1𝑑𝑥+2𝑘_𝑠𝐴₁₆×[2^𝐿₀𝜕𝜙𝜕𝑥𝜕𝑈₀𝜕𝑥𝑑𝑥+^𝐿₀𝛽_𝑦𝜕𝛽_𝑥𝜕𝑥𝑑𝑥−^𝐿₀𝛽_𝑥𝜕𝛽_𝑦−𝜕𝑥𝑑𝑥_𝐿0𝜕𝑉₀𝜕𝑥𝜕𝛽_𝑥𝜕𝑥𝑑𝑥−^𝐿₀𝜕𝑊₀𝜕𝑥𝜕𝛽_𝑦+1𝜕𝑥𝑑𝑥]2𝑘_𝑠(𝐴₅₅+𝐴₆₆)×[^𝐿₀(𝜕𝑉₀)𝜕𝑥²𝑑𝑥+^𝐿₀(𝜕𝑊₀)𝜕𝑥²𝑑𝑥+^𝐿₀𝛽²_𝑥+𝑑𝑥_𝐿0𝛽²_𝑦𝑑𝑥+2^𝐿₀𝛽_𝑥𝜕𝑊₀𝜕𝑥𝑑𝑥−2^𝐿₀𝛽_𝑦𝜕𝑉₀𝜕𝑥𝑑𝑥], (6) where 𝐴_𝑖𝑗 and 𝐵_𝑖𝑗 are given in the appendix.

The kinetic energy of the rotating composite shaft, including the effects of translatory and rotary inertia, can be written as 𝐸_𝑐=12^𝐿₀[𝐼_𝑚(𝑈̇²₀+𝑉̇²₀+𝑊̇²₀)+𝐼_𝑑(𝛽̇²_𝑥+𝛽̇²_𝑦)−2Ω𝐼_𝑝𝛽_𝑥𝛽̇_𝑦+2Ω𝐼_𝑝𝜙̇+𝐼_𝑝𝜙̇²+Ω²𝐼_𝑝+Ω²𝐼_𝑑(𝛽²_𝑥+𝛽²_𝑦)]𝑑𝑥, (7) where Ω is the rotating speed of the shaft which is assumed constant, 𝐿 is the length of the shaft, the 2Ω𝐼_𝑝𝛽_𝑥𝛽̇_𝑦 term accounts for the gyroscopic effect, and 𝐼_𝑑(𝛽̇²_𝑥+𝛽̇²_𝑦) represent the rotary inertia effect. The mass moments of inertia 𝐼_𝑚, the diametrical mass moments of inertia 𝐼_𝑑, and polar mass moment of inertia 𝐼_𝑝 of rotating shaft per unit length are defined in the appendix. As the Ω²𝐼_𝑑(𝛽²_𝑥+𝛽²_𝑦) term is far smaller than Ω²𝐼_𝑝, it will neglected in further analysis.

2.2. Hierarchical Beam Element Formulation

The spinning flexible beam is descretised into one hierarchical finite element is shown in Figure 6. The element's nodal d.o.f. at each node are 𝑈₀, 𝑉₀, 𝑊₀, 𝛽_𝑥, 𝛽_𝑦, and 𝜙. The local and nondimensional coordinates are related by 𝜉=𝑥/𝐿,(0≤𝜉≤1). (8) The vector displacement formed by the variables 𝑈₀, 𝑉₀, 𝑊₀, 𝛽_𝑥, 𝛽_𝑦, and 𝜙 can be written as 𝑈₀=^𝑝𝑈_𝑚=1𝑥_𝑚(𝑡)⋅𝑓_𝑚𝑉(𝜉),₀=^𝑝𝑉_𝑚=1𝑦_𝑚(𝑡)⋅𝑓_𝑚𝑊(𝜉),₀=^𝑝𝑊_𝑚=1𝑧_𝑚(𝑡)⋅𝑓_𝑚𝛽(𝜉),_𝑥=^𝑝𝛽𝑥_𝑚=1𝛽_𝑥𝑚(𝑡)⋅𝑓_𝑚𝛽(𝜉),_𝑦=^𝑝𝛽𝑦_𝑚=1𝛽_𝑦𝑚(𝑡)⋅𝑓_𝑚(𝜉),𝜙=^𝑝𝜙_𝑚=1𝜙_𝑚(𝑡)⋅𝑓_𝑚(𝜉). (9) And it can be expressed as {𝑈₀𝑉₀𝑊₀𝛽_𝑥𝛽_𝑦𝜙}=[𝑁]{𝑞}, (10) where [𝑁] is the matrix of the shape functions, given by [𝑁]=[𝑁_𝑈]000000[𝑁_𝑉]000000[𝑁_𝑊]000000[𝑁_𝛽𝑥]000000[𝑁_𝛽𝑦]000000[𝑁_𝜙], (11)[𝑁_{𝑈,𝑉,𝑉,𝛽𝑥,𝛽𝑦,𝜙}]=[𝑓₁𝑓₂⋯𝑓_{𝑝𝑈,𝑝𝑉,𝑝𝑊,𝑝𝛽𝑥,𝑝𝛽𝑦,𝑝𝜙}], (12) where 𝑝_𝑈, 𝑝_𝑉, 𝑝_𝑊, 𝑝_𝛽𝑥, 𝑝_𝛽𝑦, and 𝑝_𝜙 are the numbers of hierarchical terms of displacements (are the numbers of shape functions of displacements).

Figure 6 

Beam element with two nodes.

In this work, 𝑝_𝑈=𝑝_𝑉=𝑝_𝑊=𝑝_𝛽𝑥=𝑝_𝛽𝑦=𝑝_𝜙=𝑝.

The vector of generalised coordinates given by {𝑞}={𝑞_𝑈,𝑞_𝑉,𝑞_𝑊,𝑞_𝛽𝑥,𝑞_𝛽𝑦,𝑞_𝜙}^𝑇, (13) where {𝑞_𝑈}={𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑝𝑈}^𝑇exp(𝑗𝜔𝑡),{𝑞_𝑉}={𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑝𝑉}^𝑇exp(𝑗𝜔𝑡),{𝑞_𝑊}={𝑧₁,𝑧₂,𝑧₃,…,𝑦_𝑝𝑊}^𝑇exp(𝑗𝜔𝑡),{𝑞_𝛽𝑥}={𝛽_𝑥1,𝛽_𝑥2,𝛽_𝑥3,…,𝛽_{𝑥𝑃𝛽𝑥}}^𝑇exp(𝑗𝜔𝑡),{𝑞_𝛽𝑦}={𝛽_𝑦1,𝛽_𝑦2,𝛽_𝑦3,…,𝛽_{𝑦𝑃𝛽𝑦}}^𝑇exp(𝑗𝜔𝑡),{𝑞_𝜙}={𝜙₁,𝜙₂,𝜙₃,…,𝜙_𝑝𝜙}^𝑇exp(𝑗𝜔𝑡). (14)

The group of the shape functions used in this study is given by 𝑓₁𝑓=1−𝜉,₂𝑓=𝜉,_𝑟+2=sin(𝛿_𝑟𝛿𝜉),_𝑟=𝑟𝜋;𝑟=1,2,3,…. (15)

The functions (𝑓₁, 𝑓₂) are those of the finite element method necessary to describe the nodal displacements of the element, whereas the trigonometric functions 𝑓_𝑟+2 contribute only to the internal field of displacement and do not affect nodal displacements. The most attractive particularity of the trigonometric functions is that they offer great numerical stability. The shaft is modeled by only one element called hierarchical finite element.

By applying the Euler-Lagrange equations, the equations of motion of free vibration of spinning flexible shaft can be obtained: [𝑀]{𝑞}̈+[𝐺]{𝑞}̇+[𝐾]{𝑞}={0}. (16)

The system obtained is linear of which equations are coupled once by the gyroscopic effect, represented by the matrix [𝐺].

[𝑀] and [𝐾] are the conventional hierarchical finite element mass and stiffness matrix, [𝐺] is the gyroscopic matrix (the different matrices of the system of equation are given in the appendix).

3. Results

A program based on the formulation proposed was developed for the resolution of (16).

3.1. Convergence

First, the material properties for boron-epoxy are listed in Table 1, and the geometric parameters are 𝐿=2.47m, 𝐷=12.69cm, 𝑒=1.321mm, 10 layers of equal thickness (90^∘, 45^∘, −45^∘, 0^∘₆, 90^∘). The shear correction factor 𝑘_𝑠=0.503 and the rotating speed Ω=0.

Table 1 

Properties of composite materials [22].

The results of the five bending modes for various boundary conditions of the composite shaft as a function of the number of hierarchical terms are shown in Figures 7 and 8. Figures clearly show that rapid convergence from above to the exact values occur as the number of hierarchical terms is increased.

Figure 7 

Convergence of the frequency ω for the 5 bending modes of the simply-supported (S-S) shaft as a function of the number of hierarchical terms 𝑝.

Figure 8 

Convergence of the frequency ω for the 5 bending modes of the clamped-clamped (C-C) shaft as a function of the number of hierarchical terms 𝑝.

3.2. Numerical Examples and Discussions

In all the following examples, 𝑝=10.

In the first example, the properties of the boron-epoxy composite shaft are given by Table 1. The results obtained using the present model are shown in Table 2 together with those of referenced papers. As can be seen from the table, our results are close to those predicted by other beam theories. Since in the studied example the wall of the shaft is relatively thin, models based on shell theories [4] are expected to yield more accurate results. In the present example, the critical speed measured from the experiment however is still underestimated by using Sander shell theory, while overestimated by Donnell shallow shell theory. When the material of the shaft is changed to the graphite-epoxy given in Table 1 with other conditions left unchanged, the critical speed obtained from the present model is shown in Table 3. In this case, the result from the present model is compatible to that of the Bresse-Timoshenko beam theory of M.-Y. Chang et al. [6].

Table 2 

The critical speed of the boron-epoxy composite shaft.

Table 3 

The critical speed of the graphite-epoxy composite shaft.

Next, comparisons are made with those of [6, 22] for different length to mean diameter ratios 𝐿/𝐷. The shafts being analysed are made of the graphite-epoxy material given in Table 1 and all have the same lamination [90^∘/45^∘/45^∘/0^∘₆/90^∘]. The mean diameter and the wall thickness of the shaft remain the same as the previous examples. The shear correction factor being used is again 0.503. The results are listed in Table 4. Further results being compared are for generalized orthotropic composite tube of different lamination angles 𝜂. The results are shown in Table 5.

Table 4 

The critical speed (rpm) of the graphite-epoxy composite shaft for various lengths to mean diameter ratio.

Table 5 

The critical speed (rpm) of the graphite-epoxy composite shaft for various lamination angles.

In our work, the shaft is modelled by only one element with two nodes, but in the model of [6] the shaft is modelled by 20 finite elements of equal length (ℎ-version). The rapid convergence while taking only one element and a reduced number of shape functions shows the advantage of the method used.

From the thin-walled shaft systems studied above, the present shaft model yields result in all cases close to those of the model of Bert and Kim [22] based on the Bresse-Timoshenko theory. One should stress here that the present model is not only applicable to the thin-walled composite shafts as studied above, but also to the thick-walled shafts as well as to the solid ones.

In the following example, the frequencies responses of a graphite-epoxy composite shaft system are analysed. The material properties are those listed in Table 1. The lamination scheme of the shaft remains the same as previous examples, while its geometric properties, the Campbell diagram containing the frequencies of the first five pairs of bending whirling modes of the above composite system is shown in Figure 9. The intersection point of the line (Ω = ω) with the whirling frequency curves indicate the speed at which the shaft will vibrate violently (i.e., the critical speed). The first 10 eigenvalues correspond to 5 forward (F) and 5 backward (B) whirling bending modes of the shaft.

Figure 9 

The Campbell diagram of a composite shaft (F) forward modes, (B) backward modes).

The gyroscopic effect causes a coupling of orthogonal displacements to the axis of rotation, and by consequence separates the frequencies in two branches: backward precession mode and forward precession mode. In all cases, the backward modes increase with increasing rotating speed however the forward modes decrease.

Figure 10 shows the variation of the bending fundamental frequency ω as a function of the rotating speed Ω (Campbell diagram) for different boundary conditions and different rotating speeds.

Figure 10 

The first backward (1B) and forward (1F) bending mode of a composite shaft for different boundary conditions and different rotating speeds (S: simply supported; C: clamped; F: free).

The gyroscopic effect inherent to rotating structures induces a precession motion. The backward precession modes (1B) increase with increasing the rotating speed, however the forward precession (1F) modes decrease.

4. Conclusion

This paper has presented the free vibration analysis of spinning composite shaft using the 𝑝-version, hierarchical finite element method with trigonometric shape functions. Results obtained using the method has been evaluated against those available in the literature and the agreement has been found to be good. The main conclusions have emerged from this work, these are itemised below.(1) Monotonic and uniform convergence is found to occur as the number of hierarchical functions is increased.(2)The dynamic characteristics of rotating composite shaft are influenced significantly by varying the fiber orientation, the rotating speed, and boundary conditions.(3)The gyroscopic effect causes a coupling of orthogonal displacements to the axis of rotation, and by consequence separates the frequencies in two branches: backward and forward precession modes. In all cases, the backward modes increase with increasing rotating speed however the forward modes decrease.

Appendix

The terms 𝐴_𝑖𝑗, 𝐵_𝑖𝑗 of (6) and 𝐼_𝑚, 𝐼_𝑑, 𝐼_𝑝 of (7) are given as follows: 𝐴₁₁=𝜋^𝑘_𝑛=1𝐶^′_11𝑛(𝑅²_𝑛−𝑅²_𝑛−1𝐴),₅₅=𝜋2^𝑘_𝑛=1𝐶^′_55𝑛(𝑅²_𝑛−𝑅²_𝑛−1𝐴),₆₆=𝜋2^𝑘_𝑛=1𝐶^′_66𝑛(𝑅²_𝑛−𝑅²_𝑛−1𝐴),₁₆=2𝜋3^𝑘_𝑛=1𝐶^′_16𝑛(𝑅³_𝑛−𝑅³_𝑛−1𝐵),₁₁=𝜋4^𝑘_𝑛=1𝐶^′_11𝑛(𝑅⁴_𝑛−𝑅⁴_𝑛−1𝐵),₆₆=𝜋2^𝑘_𝑛=1𝐶^′_66𝑛(𝑅⁴_𝑛−𝑅⁴_𝑛−1𝐼),_𝑚=𝜋^𝑘_𝑛=1𝜌_𝑛(𝑅²_𝑛−𝑅²_𝑛−1𝐼),_𝑑=𝜋4^𝑘_𝑛=1𝜌_𝑛(𝑅⁴_𝑛−𝑅⁴_𝑛−1𝐼),_𝑝=𝜋2^𝑘_𝑛=1𝜌_𝑛(𝑅⁴_𝑛−𝑅⁴_𝑛−1), (A.1) where 𝑘 is the number of the layer, 𝑅_𝑛−1 is the 𝑛th layer inner radius of the composite shaft, and 𝑅_𝑛 is the 𝑛th layer outer of the composite shaft. 𝐿 is the length of the composite shaft and 𝜌_𝑛 is the density of the 𝑛th layer of the composite shaft.

Whereas the various matrices of (16) are expressed as follows: [𝑀]=[𝑀_𝑈]000000[𝑀_𝑉]000000[𝑀_𝑊]000000[𝑀_𝛽𝑥]000000[𝑀_𝛽𝑦]000000[𝑀_𝜙],[𝐾]=[𝐾_𝑈]0000[𝐾₁]0[𝐾_𝑉]0[𝐾₂][𝐾₃]000[𝐾_𝑊][𝐾₄][𝐾₅]00[𝐾₂]^𝑇[𝐾₄]^𝑇[𝐾_𝛽𝑥][𝐾₆]00[𝐾₃]^𝑇[𝐾₅]^𝑇[𝐾₆]^𝑇[𝐾_𝛽𝑦]0[𝐾₁]^𝑇0000[𝐾_𝜙],[𝐺]=0000000000000000000000[𝐺₁]0000−[𝐺₁]^𝑇,00000000[𝑀_𝑈]=𝐼_𝑚𝐿¹₀[𝑁_𝑈]^𝑇[𝑁_𝑈]𝑑𝜉,[𝑀_𝑉]=𝐼_𝑚𝐿¹₀[𝑁_𝑉]^𝑇[𝑁_𝑉]𝑑𝜉,[𝑀_𝑊]=𝐼_𝑚𝐿¹₀[𝑁_𝑊]^𝑇[𝑁_𝑊]𝑑𝜉,[𝑀_𝛽𝑥]=𝐼_𝑑𝐿¹₀[𝑁_𝛽𝑥]^𝑇[𝑁_𝛽𝑥]𝑑𝜉,[𝑀_𝛽𝑦]=𝐼_𝑑𝐿¹₀[𝑁_𝛽𝑦]^𝑇[𝑁_𝛽𝑦]𝑑𝜉,[𝑀_𝜙]=𝐼_𝑝𝐿¹₀[𝑁_𝜙]^𝑇[𝑁_𝜙]𝑑𝜉,[𝐾_𝑈1]=𝐿𝐴₁₁¹₀[𝑁^′_𝑈]^𝑇[𝑁^′_𝑈]𝑑𝜉,[𝐾_𝑉1]=𝐿𝑘_𝑠(𝐴₅₅+𝐴₆₆)¹₀[𝑁^′_𝑉]^𝑇[𝑁^′_𝑉]𝑑𝜉,[𝐾_𝑊1]=𝐿𝑘_𝑠(𝐴₅₅+𝐴₆₆)¹₀[𝑁^′_𝑊]^𝑇[𝑁^′_𝑊]𝑑𝜉,[𝐾₁1]=𝐿𝑘_𝑠𝐴₁₆¹₀[𝑁^′_𝜙]^𝑇[𝑁^′_𝑈]𝑑𝜉,[𝐾₂1]=−𝑘2𝐿^𝑠𝐴₁₆¹₀[𝑁^′_𝑉]^𝑇[𝑁^′_𝛽𝑥]𝑑𝜉,[𝐾₃]=−𝑘_𝑠(𝐴₅₅+𝐴₆₆)¹₀[𝑁_𝛽𝑦]^𝑇[𝑁^′_𝑉]𝑑𝜉,[𝐾₄]=𝑘_𝑠(𝐴₅₅+𝐴₆₆)¹₀[𝑁_𝛽𝑥]^𝑇[𝑁^′_𝑊]𝑑𝜉,[𝐾₅1]=−𝑘2𝐿^𝑠𝐴₁₆¹₀[𝑁^′_𝑊]^𝑇[𝑁^′_𝛽𝑦]𝑑𝜉,[𝐾₆1]=[2𝑘_𝑠𝐴₁₆¹₀[𝑁_𝛽𝑦]^𝑇[𝑁^′_𝛽𝑥]𝑑𝜉]1−[2𝑘_𝑠𝐴₁₆¹₀[𝑁_𝛽𝑥]^𝑇[𝑁^′_𝛽𝑦]𝑑𝜉],[𝐾_𝛽𝑥1]=[𝐿𝐵₁₁¹₀[𝑁^′_𝛽𝑥]^𝑇[𝑁^′_𝛽𝑥]𝑑𝜉]+[𝐿𝑘_𝑠(𝐴₅₅+𝐴₆₆)¹₀[𝑁_𝛽𝑥]^𝑇[𝑁_𝛽𝑥]𝑑𝜉],[𝐾_𝛽𝑦1]=[𝐿𝐵₁₁¹₀[𝑁^′_𝛽𝑦]^𝑇[𝑁^′_𝛽𝑦]𝑑𝜉]+[𝐿𝑘_𝑠(𝐴₅₅+𝐴₆₆)¹₀[𝑁_𝛽𝑦]^𝑇[𝑁_𝛽𝑦]𝑑𝜉],[𝐾_𝜙1]=𝐿𝐵₆₆¹₀[𝑁^′_𝜙]^𝑇[𝑁^′_𝜙]𝑑𝜉,[𝐺₁]=Ω𝐼_𝑝𝐿¹₀[𝑁_𝛽𝑥]^𝑇[𝑁_𝛽𝑦]𝑑𝜉, (A.2)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25) where [𝑁^′_𝑖]=𝜕[𝑁_𝑖]/𝜕𝜉, with (𝑖=𝑈,𝑉,𝑊,𝛽_𝑥,𝛽_𝑦,𝜙).

The terms of the matrices are a function of the integrals, 𝐽^𝛼𝛽_𝑚𝑛=∫¹₀𝑓^𝛼_𝑚(𝜉)𝑓^𝛽_𝑛(𝜉)𝑑𝜉, (𝑚,𝑛) indicate the number of the shape functions used, where (𝛼,𝛽) is the order of derivation.

Nomenclature

𝑈(𝑥,𝑦,𝑧):	Displacement in 𝑥 direction
𝑉(𝑥,𝑦,𝑧):	Displacement in 𝑦 direction
𝑊(𝑥,𝑦,𝑧):	Displacement in 𝑧 direction
𝛽𝑥:	Rotation angles of the cross-section on 𝑦 axis
𝛽𝑦:	Rotation angles of the cross-section on 𝑧 axis
𝜙:	Angular displacement of the cross-section due to the torsion deformation of the shaft
(𝑥,𝑦,𝑧):	Cartesian coordinates
(→𝑖,→𝑗,→𝑘):	Axes of the Cartesian coordinates
(𝑥,𝑟,𝜃):	Cylindrical coordinates
(→𝑖,→𝑒𝑟,→𝑒𝜃):	Axes of the cylindrical coordinates
(1, 2, 3):	Principal axes of a layer of laminate
𝐶′𝑖𝑗:	Elastic constants
𝐸:	Young modulus
𝐺:	Shear modulus
𝑣:	Poisson coefficient
𝜌:	Masse density
𝑘𝑠:	Shear correction factor
𝐿:	Length of the shaft
𝐷:	Mean radius of the shaft
𝑒:	Wall thickness of the shaft
𝑘:	Number of the layer of the composite shaft
𝑅𝑛−1:	The 𝑛th layer inner radius of the composite shaft
𝑅𝑛:	The 𝑛th layer outer radius of the composite shaft
𝑅0:	Inner radius of the composite shaft
𝑅𝑘:	Outer radius of the composite shaft
𝜂:	Lamination angle
𝜃:	Circumferential coordinate
𝜉:	Local and nondimensional coordinates
𝜀𝑖𝑗:	Strain tensor
𝜎𝑖𝑗:	Stress tensor
𝜔:	Frequency, eigenvalue
Ω:	Rotating speed
𝑓(𝜉):	Shape functions
[𝑁]:	Matrix of the shape functions
𝑝:	Number of the shape functions or number of hierarchical terms
𝑡:	Time
𝐸𝑐:	Kinetic energy
𝐸𝑑:	Strain energy
{𝑞𝑖}:	Generalised coordinates, with (𝑖=𝑈,𝑉,𝑊,𝛽𝑥,𝛽𝑦,𝜙)
[𝑀]:	Masse matrix
[𝐾]:	Stiffness matrix
[𝐺]:	Gyroscopic matrix.