Modeling and Dynamical Behavior of Rotating Composite Shafts with SMA Wires

Abstract

A dynamical model is developed for the rotating composite shaft with shape-memory alloy (SMA) wires embedded in. The rotating shaft is represented as a thin-walled composite of circular cross-section with SMA wires embedded parallel to shaft’s longitudinal axis. A thermomechanical constitutive equation of SMA proposed by Brinson is employed and the recovery stress of the constrained SMA wires is derived. The equations of motion are derived based on the variational-asymptotical method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin method. The model incorporates the transverse shear, rotary inertia, and anisotropy of composite material. Numerical results of natural frequencies and critical speeds are obtained. It is shown that the natural frequencies of the nonrotating shaft and the critical rotating speed increase as SMA wire fraction and initial strain increase and the increase in natural frequencies becomes more significant as SMA wire fraction increases. The initial strain of SMA wires appears to have marginal effect on dynamical behaviors of the shaft. The actuation performance of SMA wires is found to be closely related to the ply-angle.

1. Introduction

Composite materials have found the increased applications for replacement of the conventional metallic materials in the rotating flexible shaft employed for drive shafts of helicopters, steam, and gas turbines. This is likely attributed to high stiffness and strength/weight ratios of composite shaft compared with its metallic counterparts. The development trend in design of light-weight composite shafts is towards higher operating speeds, which gives rise to the problems of high vibration amplitude and stability. Seeking the solution of these problems has caused great research effort [1] in the dynamic of composite rotor.

A review on the literature in this area has shown that composite shafts have high whirling resistance capability and are less susceptible to dynamic instability associated with metallic shafts [2]. Several attempts to develop mathematical models of spinning composite shafts are reported in the literature. These models include the shaft models based on shell theories [3], or beam theories combined with the strain—displacement relations of the shell theories [4], or a thin-walled beam theory [5]. Song et al. [5] developed the composite thin-walled shaft model based on a thin-walled beam theory of Rehfield [6]. This model was used to investigate the natural frequencies and stability of the system subject to the variation of the axial edge load and the lamination angle of the composite layer. However, since Rehfield’s formulation is known to be nonasymptotically correct [7], there is no guarantee for consistent accuracy on the results.

SMA composites are a new class of materials that have the ability to change both their stiffness and their elastic properties [8]. SMA composites consist of SMA actuators embedded into a matrix material or in a fiber-reinforced composite. This stiffness modification occurs as a result of thermally induced martensite phase transformation of SMA actuators embedded in composite structures. Several studies have been done on [9–15] combining the advantages of both the composite material and the SMA to build smart composite shafts. Baz and Chen [9] have proposed a smart shaft, which can actively stiffen in response to increased rotational speed or increased amplitudes of vibration. Their results showed activating the Nitinol wires results in reducing the amplitude of its vibration amplitudes by about 50%. Gupta [10] investigated the combined effect of embedding the SMA wires in a rotor shaft and change of support stiffness using the SMA on rotor critical speeds. Sawhney and Jain [11] carried out fabrication and experimental investigations on the fiber-reinforced composite shaft embedded with the SMA wires. Baz and Chen [12] studied the static and thermal characteristics of SMA-reinforced composite drive shafts. They found that the Nitinol wires can play a role in enhancing the torsional stiffness. Tylikowski [13] studied the dynamic stability of globally activated simply supported hybrid shells consisting of symmetrical balanced angle-ply laminated classical plies and symmetrically laminated active plies with axially oriented SMA fibers. Their results indicate that the activation significantly increases the critical value of angular velocity. Tylikowski and Hetnarski [14] analyzed the stability criterion of the shaft equilibrium. The shaft is treated as a thin symmetrically laminated shell containing both the conventional fibers and the activated SMA fibers. The results indicated that the SMA activation significantly increases the stability domains of the shaft. Gupta et al. [15] designed an experimental setup to embed prestrained SMA wires in the fiber-reinforced composite shaft. Experimental results showed a noticeable increase in the natural frequency of the composite shaft due to activation of SMA wires.

In our dynamics study, the rotating shaft is represented as a thin-walled composite of circular cross-section embedded with SMA wires. It is the purpose of the present work to study the effect of increase in stiffness and tension in wires due to phase recovery stresses when wires are activated on the rotor dynamic characteristics such as natural frequencies and the critical speeds. The modeling of the rotating shaft is based on the VAM by Berdichevsky et al. [7]. The thermomechanical constitutive equation of SMA proposed by Brinson [16] is employed to establish the constitutive equation of the SMA reinforced composite shaft. To determine the rotating shaft’s dynamical characteristics, the Galerkin approach is carried out here to approximate the motion equations by a system of ordinary differential equations. Based on these approximate equations the dynamical characteristics of the rotating shaft systems are then calculated. Finally, the influence of SMA activation on dynamical behaviors of the rotating composite shaft has been examined.

2. Theoretical Formulations

2.1. Coordinate Systems and Basic Assumptions

The composite shaft of length 𝐿, thickness ℎ, and the radius of curvature 𝑟 rotating with constant rate Ω. As seen in Figure 1, (𝑥,𝑦,𝑧) is the rotating coordinate system, (𝑋,𝑌,𝑍) is the inertial coordinate system, and (𝑥,𝑠,𝜉) is the local coordinate system. The coordinate systems (𝑥,𝑦,𝑧) and (𝑋,𝑌,𝑍) have the common origin 𝑜 located in the geometric center. It is assumed that at 𝑡=0, the axes of the two systems coincide.

(a) Global view of the shaft

(b) Cross-section of the shaft

Figure 1 

Schematic drawing of the composite thin-walled beam of a circular cross-section.

In our study, the following assumptions are made. (1) The composite shaft is characterized as a slender thin-walled beam [7]; the geometric dimensions are such that 𝑑≪𝐿, ℎ≪𝑑, ℎ≪𝑟; (2) transverse shear effects are considered, where 𝐿, ℎ, 𝑟, and 𝑑 denote the length, the thickness, the radius of curvature, and the maximum cross-sectional dimension of the cylinder, respectively; (3) the SMA wires are embedded at an interlayer of the shaft and arranged parallel to its longitudinal axis.

2.2. Equations of Motion

Berdichevsky et al. [7] have shown that VAM is an asymptotically correct theory which can be used effectively for the analysis of tubular composite thin-walled. However, VAM does not account for the effects of transverse shear because of which it may generate inaccurate predictions of the rotating composite shaft. In the present work, VAM has been refined to include effects of transverse shear.

The displacement field incorporating shear deformation in the local frame denoted (𝑥,𝑠,𝜉) is assumed in the form 𝑢₁𝑥,𝑦,𝑧,𝑡=𝑈₁(𝑥,𝑡)−𝑦(𝑠)𝜃_𝑦()(𝑠)𝜃𝑥,𝑡−𝑧_𝑧()(),𝑢𝑥,𝑡+𝑔𝑠,𝑥,𝑡₂𝑥,𝑦,𝑧,𝑡=𝑈₂()(),𝑢𝑥,𝑡−𝑧𝜙𝑥,𝑡₃𝑥,𝑦,𝑧,𝑡=𝑈₃(𝑥,𝑡)+𝑦𝜙(𝑥,𝑡), (1) where 𝑈₁(𝑥,𝑡), 𝑈₂(𝑥,𝑡), 𝑈₃(𝑥,𝑡) denote the rigid-body translations along the 𝑥-, 𝑦-, and 𝑧-axes, while 𝜙(𝑥,𝑡), 𝜃_𝑦(𝑥,𝑡), 𝜃_𝑧(𝑥,𝑡) denote the twist about the 𝑥-axis and rotations about the 𝑦- and 𝑧-axes, respectively.

The warping functions 𝑔(𝑠,𝑥,𝑡) can be assumed as follows: 𝑔(𝑠,𝑥,𝑡)=𝐺(𝑠)𝜙^′(𝑥,𝑡)+𝑔₁(𝑠)𝑈^′₁(𝑥,𝑡)+𝑔₂(𝑠)𝜃^′_𝑦(𝑥,𝑡)+𝑔₃(𝑠)𝜃^′_𝑧(𝑥,𝑡). (2)

In the above equation, the functions 𝑔₁(𝑠), 𝑔₂(𝑠), 𝑔₃(𝑠), 𝐺(𝑠) are associated with physical behavior for the axial strain, the bending curvatures, and the torsion twist rate, respectively. The primes in (2) denote differentiation with respect to 𝑥.

In (1) and (2), the expressions of 𝜃_𝑦(𝑥,𝑡), 𝜃_𝑧(𝑥,𝑡) are in the following form: 𝜃_𝑦(𝑥,𝑡)=𝑈^′₂(𝑥,𝑡)−2𝛾_𝑧𝑥𝜃_𝑧(𝑥,𝑡)=𝑈^′₃(𝑥,𝑡)−2𝛾_𝑦𝑥. (3)

Based on the displacement representations (1), (2), and (3), and using the linear strain-displacement relations [7], while referring to [17], the strain of the composite shaft can be expressed as 𝛾_𝑥𝑥=𝑈^′₁(𝑥,𝑡)−𝑦𝜃^′_𝑦(𝑥,𝑡)−𝑧𝜃^′_𝑧(𝑥,𝑡),2𝛾_𝑥𝑠=𝑑𝑔𝑑𝑠+𝑟_𝑛𝜙^′+𝑈^′₂(𝑥,𝑡)−𝜃_𝑦(𝑥,𝑡)𝑑𝑦+𝑈𝑑𝑠_′3(𝑥,𝑡)−𝜃_𝑧(𝑥,𝑡)𝑑𝑧,𝑑𝑠2𝛾_𝑥𝜉=𝑈^′₂(𝑥,𝑡)−𝜃_𝑦(𝑥,𝑡)𝑑𝑧−𝑈𝑑𝑠_′3(𝑥,𝑡)−𝜃_𝑧(𝑥,𝑡)𝑑𝑦.𝑑𝑠 (4)

The position vector of an arbitrary point on the cross-section of the deformed shaft is 𝐫=(𝑦+𝑢₂)𝐢+(𝑧+𝑢₃)𝐣+(𝑥+𝑢₁)𝐤. Here, 𝐢, 𝐣, 𝐤 are unit vectors of the coordinate systems (𝑥,𝑦,𝑧). By taking the time derivatives of unit vectors and adopting the assumption of constant rate 𝑊, the velocity vector of an arbitrary point is 𝐕=𝐫̇=[𝑢̇₂−Ω(𝑧+𝑢₃)]𝐢+[𝑢̇₃+Ω(𝑦+𝑢₂)]𝐣+𝑢̇₁𝐤.

The equations of motion for the rotating composite shaft are derived based on Hamilton’s principle, which can be expressed as [18] ^𝑡1_𝑡0(𝛿𝑈−𝛿𝑇−𝛿𝑊)𝑑𝑡=0, (5) where 1𝑈=2_𝜏𝜎_𝑖𝑗𝜀_𝑖𝑗1𝑑𝑠𝑑𝜉𝑑𝑥,𝐾=2_𝜏𝜌(𝐫̇⋅𝐫̇)𝑑𝑠𝑑𝜉𝑑𝑥 (6) are the strain energy and the kinetic energy of the composite shaft, respectively.

Here 𝑡₀ and 𝑡₁ are two arbitrary instants of time, 𝜎_𝑖𝑗 and 𝜀_𝑖𝑗 are the stress components and the strain components, respectively, 𝑊 is the virtual work of the external forces, 𝜌 is the mass density, and 𝛿 is the variation operator.

In order to display the elastic coupling between vertical bending and horizontal bending, a special ply-angle distribution referred to as circumferentially uniform stiffness-CUS configuration [19], achieved by skewing angle plies with respect to the longitudinal beam axis meeting the condition 𝜃(𝑦)=𝜃(−𝑦), 𝜃(𝑧)=𝜃(−𝑧), is considered.

By employing Hamilton’s principle and taking into account the results with a longitudinal compressive force in [20], which is generated by SMA wires activation, the equations of motion involving CUS configuration in terms of displacements can be written as −𝑘₃₅𝜃_𝑧^′′−𝑘₅₅𝑈₂^′′−𝜃^′_𝑦+𝑁^SMA−𝑁^Δ𝑇𝑈₂^′′+𝑏₁𝑈̈₂−2Ω𝑈̇₃−Ω²𝑈₂=0,−𝑘₄₆𝜃_𝑦^′′−𝑘₆₆𝑈₃^′′−𝜃^′_𝑧+𝑁^SMA−𝑁^Δ𝑇𝑈₃^′′+𝑏₁𝑈̈₃+2Ω𝑈̇₂−Ω²𝑈₃=0,−𝑘₄₄𝜃_𝑦^′′−𝑘₄₆𝑈₃^′′−𝑘₅₅𝑈^′₂−𝜃_𝑦−𝑏₄𝜃̈_𝑦=0,−𝑘₃₃𝜃_𝑧^′′−𝑘₃₅𝑈₂^′′−𝑘₆₆𝑈^′₃−𝜃_𝑧−𝑏₅𝜃̈_𝑧=0, (7)

in which 𝑘₃₃=_Γ𝐵𝐴−²𝐶𝑧²∮𝑑𝑠+^Γ(𝐵/𝐶)𝑧𝑑𝑠²∮_Γ(),𝑘1/𝐶𝑑𝑠₄₄=_Γ𝐵𝐴−²𝐶𝑦²∮𝑑𝑠+^Γ(𝐵/𝐶)𝑦𝑑𝑠²∮_Γ(),𝑘1/𝐶𝑑𝑠₃₅1=−2_Γ𝐵𝑧𝑑𝑦𝑘𝑑𝑠𝑑𝑠,₄₆1=−2_Γ𝐵𝑦𝑑𝑧𝑘𝑑𝑠𝑑𝑠,₅₅=_Γ12𝐶𝑑𝑦𝑑𝑠²+𝐷𝑑𝑧𝑑𝑠²𝑘𝑑𝑠,₆₆=_Γ12𝐶𝑑𝑧𝑑𝑠²+𝐷𝑑𝑦𝑑𝑠²𝑑𝑠. (8)(9) Herein, 𝐴=𝐴₁₁−𝐴₁₂²𝐴₂₂,𝐵=2𝐴₁₆−𝐴₁₂𝐴₂₆𝐴₂₂,𝐶=4𝐴₆₆−𝐴₂₆²𝐴₂₂,𝐷=𝐴₄₄−𝐴₄₅²𝐴₅₅,𝐴_𝑖𝑗=^𝑁_𝑘=1𝑄^(𝑘)_𝑖𝑗𝑧_𝑘−𝑧_𝑘−1,𝑖,𝑗=1,2,6;𝑖,𝑗=4,5, (10) where 𝐴_𝑖𝑗 denotes the local stretching stiffness. 𝑄^(𝑘)_𝑖𝑗 denotes the element of the transformed stiffness matrix of the 𝑘th layer.

In the case of CUS configuration, it can be seen that (7) reveals different elastic couplings which consist of the coupling between flapwise bending and chordwise bending, the coupling between flapwise transverse shear and chordwise bending, and the coupling between chordwise transverse shear and flapwise bending.

And 𝐐=𝐓𝐐(𝐓)^𝑇, (11) in which 𝑛𝐓=²𝑚²𝑚−2𝑚𝑛₂𝑛²2𝑚𝑛𝑚𝑛−𝑚𝑛𝑛²−𝑚²𝑄,𝐐=¹¹𝑄₁₂0𝑄₂₁𝑄₂₂000𝑄₆₆ (12)(13) and 𝐐 is the stiffness matrix, 𝑛=cos𝜃, 𝑚=sin𝜃, and 𝜃 is the fiber ply-angle.

And 𝑄₁₁=𝐸₁1−𝑣₁₂𝑣₂₁,𝑄₁₂=𝑄₂₁=𝑣₁₂𝐸₂1−𝑣₁₂𝑣₂₁,𝑄₂₂=𝐸₂1−𝑣₁₂𝑣₂₁,𝑄₆₆=𝐺₁₂. (14)

In addition, the mass terms 𝑏₄, 𝑏₅, and 𝑏₆ in (7) are expressed as 𝑏₄=_𝐴𝜌𝑦²𝑑𝐴,𝑏₅=_𝐴𝜌𝑧²𝑑𝐴,𝑏₆=_𝐴𝜌𝑦𝑧𝑑𝐴. (15)

The modulus and Poisson’s ratio of the shaft can be determined by considering the mixture rule of composition 𝐸₁=𝐸_1𝑚1−𝑉_𝑠+𝐸_𝑠𝑉_𝑠,𝐸₂=𝐸_2𝑚𝐸_𝑠𝐸_2𝑚𝑉_𝑠+𝐸_𝑠1−𝑉_𝑠,𝐺₁₂=𝐺_12𝑚𝐺_𝑠𝐺_12𝑚𝑉_𝑠+𝐺_𝑠1−𝑉_𝑠,𝜈₁₂=𝜈_12𝑚1−𝑉_𝑠+𝜈_𝑠𝑉_𝑠,𝑣₁₂𝐸₁=𝑣₂₁𝐸₂, (16) where the subscript 𝑚 and 𝑠 denote the composite matrix and SMA wire, respectively, and 𝑉 is volume fraction.

Herein, 𝑉_𝑠=𝐴_sma𝐴_Total, (17) where 𝐴_sma, 𝐴_Total express the total cross-sectional areas of the embedded SMA wires and the composite shaft, respectively, and 𝐴_sma=𝜋𝑑²4𝑛, (18) where 𝑛 and 𝑑 represent the number and diameter of the SMA wires, respectively.

The terms 𝑁^SMA and 𝑁^Δ𝑇 appearing in (7) denote the axial tension due to thermal expansion and SMA wires activation, respectively.

The axial tensions 𝑁^SMA and 𝑁^Δ𝑇 are given by 𝑁^Δ𝑇=2𝜋𝑟^𝑛_𝑘=1𝑄_11𝑘𝛼_𝑥𝑘+𝑄_12𝑘𝛼_𝑦𝑘+𝑄_16𝑘𝛼_𝑥𝑦𝑘⋅𝑧_𝑘−𝑧_𝑘−1Δ𝑇, (19) where 𝛼_𝑥𝑘, 𝛼_𝑦𝑘, and 𝛼_𝑥𝑦𝑘 denote the reduced thermal expansion coefficients in the 𝑘th layer of composite medium, and 𝑁^SMA=𝜋𝑑²4𝑛𝜎_𝑟=𝐴_sma𝜎_𝑟, (20) where 𝜎_𝑟 represents the recovery stress which can be determined analytically.

By eliminating from (7) the quantities 𝑘₅₅(𝑈₂^′′−𝜃^′_𝑦) and 𝑘₆₆(𝑈₃^′′−𝜃^′_𝑧), using the relations 𝜃_𝑦(𝑥,𝑡)=𝑈^′₂(𝑥,𝑡) and 𝜃_𝑧(𝑥,𝑡)=𝑈^′₃(𝑥,𝑡), the equations of motion for unshearable shaft can be obtained. The results are not presented in this paper for the sake of simplicity.

3. The Recovery Stress of the Constrained SMA Wires

Based on the one-dimensional model of SMA proposed by Brinson [16] and assuming all SMA wires are fully constrained, the expressions for the recovery stress of SMA wires during heating and cooling can be written as follows, respectively:

(i) during heating,𝜎_𝑟=Θ𝑇−𝑇₀+𝜎₀,0≤𝑇≤𝐴^𝜎_𝑠𝐸(𝜉)𝜀₀−𝐸𝜉₀𝜀₀+Ω(𝜉)𝜉_𝑠−Ω𝜉₀𝜉_𝑠0+Θ𝑇−𝐴^𝜎_𝑠+𝜎₁,𝐴^𝜎_𝑠≤𝑇≤𝐴^𝜎_𝑓Θ𝑇−𝐴^𝜎_𝑓+𝜎₂,𝑇≥𝐴^𝜎_𝑓, (21a)

(ii) during cooling, 𝜎_𝑟=Θ𝑇−𝐴^𝜎_𝑓+𝜎₂,𝑀^𝜎_𝑠≤𝑇≤𝐴^𝜎_𝑓𝐸(𝜉)𝜀₀−𝐸𝜉₀𝜀₀+Ω(𝜉)𝜉_𝑠+Θ𝑇−𝑀^𝜎_𝑠+𝜎₃,𝑀^𝜎_𝑓≤𝑇≤𝑀^𝜎_𝑠, (21b)where 𝜉 denotes the martensite fraction, 𝐸(𝜉) denotes the elastic modulus of SMA, Θ denotes the thermal elastic modulus, 𝑇 denotes temperature, 𝑇₀ is the reference temperature, and Ω(𝜉) denotes the phase transformation coefficient. The subscript 0 denotes initial state. 𝐴^𝜎_𝑠 and 𝐴^𝜎_𝑓 denote the start and finish temperatures of Austenite in stress, and 𝑀^𝜎_𝑠 and 𝑀^𝜎_𝑓 denote the start and finish temperatures of martensite in stress. 𝜉_𝑠 is the martensite fraction induced by stress. The first expression in (21a) is used for SMA in the initial martensite state, while the third expression among them is used for SMA in 100% austenite state, and the second one among them is used for SMA in the phase transformation state from martensite to Austenite.

The transformation processes from martensite to austenite (𝑀→𝐴) and from austenite to martensite are given as follows (𝐴→𝑀), respectively: 𝜉_𝑀→𝐴=𝜉_𝑠02𝜋cos𝐴_𝑓−𝐴_𝑠𝑇−𝐴_𝑠−𝜎^𝑟𝐶_𝐴,𝜉+1_𝐴→𝑀=1−𝜉_𝑠02𝜋×cos𝜎^cr_𝑠−𝜎^cr_𝑓𝜎^𝑟−𝜎^cr_𝑓−𝐶_𝑀𝑇−𝑀_𝑠+1+𝜉_𝑠02, (22) where 𝑀_𝑠 and 𝑀_𝑓 denote the start and finish temperature points during the phase transformation from austenite to martensite and 𝐴_𝑠 and 𝐴_𝑓 denote the start and finish temperature points during the phase transformation from martensite to austenite, respectively. 𝐶_𝐴 and 𝐶_𝑀 are material constant of SMA wires which determine the influence of the stress on the transformation temperature. 𝜎^cr_𝑠 and 𝜎^cr_𝑓 denote the start and finish stress points of phase transformation, respectively, and the subscript “s0” denotes martensite fraction induced by stress corresponding to initial state.

In (21a) and (21b), 𝜎₁ and 𝜎₂ denote the start and finish stress for the transformation (𝑀→𝐴); 𝜎₃ denotes the start stress for the transformation (𝐴→𝑀).

The phase transformation coefficient Ω(𝜉) and the elastic modulus 𝐸_𝑠(𝜉) can be expressed, respectively, as 𝐸(𝜉)=𝐸_𝐴+𝜉𝐸_𝑀−𝐸_𝐴,Ω(𝜉)=−𝜀_𝐿𝐸(𝜉), (23) where 𝐸_𝑀 and 𝐸_𝐴 denote Young’s modulus in the martensite and austenite phase, respectively, and 𝜀_𝐿 is recovery strain limit.

4. Solution Method of Motion Equations

In order to find the approximate solution of the rotating composite shaft, the quantities 𝑈₂(𝑥,𝑡), 𝑈₃(𝑥,𝑡), 𝜃_𝑦(𝑥,𝑡), and 𝜃_𝑧(𝑥,𝑡) are assumed in the form 𝑈₂(𝑥,𝑡)=^𝑁_𝑗=1𝑈_2𝑗(𝑡)𝛼_𝑗(𝑥),𝑈₃(𝑥,𝑡)=^𝑁_𝑗=1𝑈_3𝑗(𝑡)𝛼_𝑗(𝑥),𝜃_𝑦(𝑥,𝑡)=^𝑁_𝑗=1Θ_𝑦𝑗(𝑡)𝜓_𝑗(𝑥),𝜃_𝑧(𝑥,𝑡)=^𝑁_𝑗=1Θ_𝑧𝑗(𝑡)𝜓_𝑗(𝑥), (24) where 𝛼_𝑗=sin𝑗𝜋𝐿𝜓𝑥,_𝑗=cos𝑗𝜋𝑥𝐿. (25)

Substituting (24) into the governing (7) and applying Galerkin procedure, the following governing equations in matrix form can be found: [𝑀]𝑋̈+[𝐶]𝑋̇+𝐾_Ω+𝐾₁{𝑋}={0}, (26) where𝑏𝐌=¹𝐸_𝑖𝑗0000𝑏₁𝐸_𝑖𝑗0000−𝑏₄𝐹_𝑖𝑗0000−𝑏₅𝐹_𝑖𝑗,𝐂=0−2𝑏₁Ω𝐸_𝑖𝑗002𝑏₁Ω𝐸_𝑖𝑗00000000000,𝐊_Ω=−𝑏₁Ω²𝐸_𝑖𝑗0000−𝑏₁Ω²𝐸_𝑖𝑗,,𝐊0000000000𝑖,𝑗=1,2,…,𝑁₁=−𝑘₅₅+𝑁^SMA−𝑁^Δ𝑇𝐼_𝑖𝑗0𝑘₅₅𝐽_𝑖𝑗−𝑘₃₅𝐾_𝑖𝑗0−𝑘₆₆+𝑁^SMA−𝑁^Δ𝑇𝐼_𝑖𝑗−𝑘₄₆𝐾_𝑖𝑗𝑘₆₆𝐽_𝑖𝑗−𝑘₅₅𝑂_𝑖𝑗−𝑘₄₆𝑃_𝑖𝑗−𝑘₄₄𝑄_𝑖𝑗+𝑘₅₅𝐹_𝑖𝑗0−𝑘₃₅𝑃_𝑖𝑗−𝑘₆₆𝑂_𝑖𝑗0−𝑘₃₃𝑄_𝑖𝑗+𝑘₆₆𝐹_𝑖𝑗,𝑖,𝑗=1,2,…,𝑁, (27)𝐸_𝑖𝑗=^𝐿₀𝛼_𝑖𝛼_𝑗𝑑𝑥,𝐹_𝑖𝑗=^𝐿₀𝜓_𝑖𝜓_𝑗𝐼𝑑𝑥,_𝑖𝑗=^𝐿₀𝛼_𝑖𝛼_𝑗^′′,𝐽_𝑖𝑗=^𝐿₀𝛼_𝑖𝜓^′_𝑗𝐾𝑑𝑥,_𝑖𝑗=^𝐿₀𝛼_𝑖𝜓_𝑗^′′𝑑𝑥,𝑂_𝑖𝑗=^𝐿₀𝜓_𝑖𝛼^′_𝑗𝑃𝑑𝑥,_𝑖𝑗=^𝐿₀𝜓_𝑖𝛼_𝑗^′′𝑑𝑥,𝑄_𝑖𝑗=^𝐿₀𝜓_𝑖𝜓_𝑗^′′.𝑑𝑥,𝑖,𝑗=1,2,…,𝑁 (28)

Herein, the generalized displacements can be defined as 𝐗^𝑇=𝑈_2𝑗(𝑡),𝑈_3𝑗(𝑡),Θ_𝑦𝑗(𝑡),Θ_𝑧𝑗(𝑡)_1×4𝑁𝑗=1,2…,𝑁. (29)

From the matrix Equation (26), the eigenvalue problem of the rotating composite shaft can be expressed as 𝐀𝐘=𝜆𝐘, (30) where 𝐘=𝐗𝐗̇^𝐓,𝐴=𝟎𝐈−𝐌⁻¹𝐊_Ω+𝐊₁−𝐌⁻¹𝐂. (31)

Lower frequencies are primarily concerned in order to investigate the dynamic behavior of rotating composite shaft system. If there is a value of 𝑁 for which any increase in 𝑁 will not significantly change the eigenvalues associated with these lower frequencies, then the eigenvalues are considered to be converged at this value of 𝑁.

In (26) the stiffness matrix is made up of three components: the elastic stiffness of the shaft with stiffness coefficients 𝑘_𝑖𝑗 (see 𝐊₁ in (27)), the geometric stiffness that accounts for the axial tension due to thermal expansion and SMA wires activation, and the stiffness 𝐊_Ω due to the shaft rotation. It should be remarked that the stiffness matrix of the shaft is reduced by the rotational stiffness matrix 𝐊_Ω. However, on the other hand, if the total axial tension of the shaft, which is the sum of 𝑁^SMA and 𝑁^Δ𝑇, is high enough such that (𝑁^SMA−𝑁^Δ𝑇) is positive then the stiffness of the shaft can been enhanced. Thus, it can be seen that the SMA wires activation plays an important role in controlling stiffness of the shaft and maintaining its stability at higher rotating speed.

5. Numerical Results and Discussion

The numerical calculations are performed by considering the shaft made of graphite-epoxy whose elastic characteristics are listed in Table 1. The shaft has geometrical characteristics as 𝑟=0.127 m, 𝐿=2.023 m, ℎ=0.381 mm. The SMA wires used in the numerical simulations are listed in Table 2.

Table 1 

Mechanical properties of composite material [5].

Table 2 

Material properties of SMA wires [21].

The numerical results will be given in terms of the normalized natural frequencies and rotating rate that are defined by 𝜔^∗=𝜔/𝜔₀, Ω^∗=Ω/𝜔₀, where the normalizing factor 𝜔₀=138.85 rad/s is the fundamental frequency of the nonrotating shaft with 𝜃=0^∘ and corresponds to the case of the absence of the SMA activation (𝑁^SMA=0) and of thermal effect (𝑁^Δ𝑇=0).

The natural frequencies of a cantilever composite shaft obtained for without shear deformation using the present model together with those obtained in [22] are shown in Table 3 for different rotating speeds. A perfect agreement of numerical results with those in [22] can be seen.

Table 3 

Comparison of the natural frequencies of a cantilever composite shaft without shear deformation.

In order to examine the influence of the number of mode shape functions used in the solution of the equation on the accuracy of the results, the numerical results of natural frequency are shown in Table 4 for an increasing number of mode shape functions. From Table 4, it can be seen that to obtain an accurate result of the first three natural frequencies, no more than six mode shape functions are required. This indicates clearly that the convergence of the present model is quite good.

Table 4 

Effect of model number 𝑁 on natural frequencies (Ω^∗=0 and 𝜃=30^∘).

Based on mode convergence examination, it is found that 𝑁=6 gives suitably converged eigenvalues. So for all results given in this paper, 𝑁=6 unless otherwise noted.

Figure 2 shows the variations of the first three natural frequencies versus rotating speed for without SMA activation and with SMA activation (𝜃=30°, 𝑇=50°C, 𝑉_𝑠=0.24, 𝜀₀=0.067). From this figure it clearly appears that when Ω^∗=0, a single zero-speed mode natural frequency is obtained. The reason is that for the circular cross-sectional shaft, the frequencies in transversal and lateral bending of each mode coincide. When Ω^∗≠0, the natural frequency “splits” into two distinct branches of bending vibration due to the gyroscopic effect, namely, the upper and lower frequency branches. The branch corresponding to the higher frequencies is associated with the up-whirling frequency (UWF) motion. Similarly, the branch corresponding to the lower frequencies is associated with the low-whirling frequency (LWF) motion. The minimum rotating rate at which the LWF becomes zero is referred to as the critical rotating speed that corresponds to the dynamical instability of the rotating shaft. In addition, this figure clearly indicates that SMA actuation can play a significant role in increasing the frequencies of the nonrotating shaft and postponing the occurrence of the whirling instability.

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Figure 2 

The first three natural frequencies versus rotating speeds of the shaft without SMA and with SMA (𝜃=30∘; 𝑇=50°C; 𝑉𝑠=0.24; 𝜀0=0.067).

Figure 3 shows the variations of the first three natural frequencies versus rotating speed for different fractions of SMA wires at a temperature above 𝐴_𝑓 (𝜃=30^∘; 𝑇=50°C; 𝜀₀=0.067). It is obviously seen that the natural frequencies of the nonrotating shaft and the critical rotating speed increase with increasing SMA wire fraction, because increasing of the fraction (or number) of the SMA wires results in generating a higher axial tension 𝑁^SMA inside the rotating shaft. It thus is apparent that the increase of the SMA wires which induces the increase of bending stiffness brings the increase of overall natural frequencies of the shaft.

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Figure 3 

The first three natural frequencies versus rotating speed for different SMA wire fractions (𝜃=30∘; 𝑇=50°C; 𝜀0=0.067).

Figure 4 shows that the variation of the first three natural frequencies as function of the temperature for different SMA wire fractions (Ω^∗=5 rad/s; 𝜃=30^∘; 𝜀₀=0.067). Figure 4 is plotted for both UWF and LWF.

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Figure 4 

The first three natural frequencies versus temperature for different SMA wire fractions (Ω∗=5 rad/s; 𝜃=30∘; 𝜀0=0.067).

The temperature is increased from 0°C to 100°C and then decreased from 100°C to 0°C. The phase transformation from the martensite to the austenite (𝑀→𝐴) is induced in the SMA wires during heating, and the phase transformation from the austenite to the martensite (𝐴→𝑀) is induced during cooling. From Figure 4, the following phenomena can be observed: (1) the curves of natural frequencies and temperature are typical hysteresis loops in a thermal cycle, and the development of the natural frequencies can be divided into two stage, that is, the increase stage and the decline stage; (2) the natural frequencies increase with the increase in temperature during the phase transformation from the martensite to the austenite but decline with the decrease in temperature during the phase transformation from the austenite to the martensite; (3) the increase of the SMA wire fraction is accompanied by the shift of these hysteresis loops towards higher natural frequencies.

Figure 5 shows the variation of the first three natural frequencies with the ply-angle for different SMA wire fractions (Ω^∗=5 rad/s; 𝑇=50°C; 𝜀₀=0.067). The results show that the effect of SMA wire fraction on the natural frequencies is substantial when the ply-angle is located near 90°. In addition, from Figure 5 it can be found that the trend of variation of natural frequencies with ply-angle is the same as that of the shaft without SMA actuation. Figure 5 also shows that as the SMA wire fraction increases, the natural frequencies increase.

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Figure 5 

The first three natural frequencies versus ply-angle for different SMA wire fractions (Ω∗=5 rad/s; 𝑇=50°C; 𝜀0=0.067).

Figure 6 shows the variations of the first three natural frequencies versus rotating speed for the different initial strains of SMA wire (𝜃=30^∘, 𝑇=50°C, 𝑉_𝑠=0.36). It can be seen that the natural frequencies of the rotating shaft increase with increasing of the initial strain of SMA wire.

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Figure 6 

The first three natural frequencies versus rotating speed for the different initial strains of SMA wire (𝜃=30∘; 𝑇=50°C; 𝑉𝑠=0.36).

Figure 7 shows the variation of the first three natural frequencies and temperature for the different initial strains of SMA wire (𝜃=30^∘;  𝑇=50°C; 𝑉_𝑠=0.36). The results show that there is an obvious change in the shapes of natural frequencies and temperature curves as the initial strain of SMA wire is increased.

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Figure 7 

The first three natural frequencies versus temperature for the different initial strains of SMA wire (𝜃=30∘; 𝑇=50°C; 𝑉𝑠=0.36).

Figure 8 shows the variation of the first three natural frequencies with the ply-angle for the different initial strains of SMA wire (Ω^∗=5 rad/s; 𝑇=50°C; 𝑉_𝑠=0.36). It can be noted that the effect of the initial strain of SMA wire on the natural frequencies is similar to that previously presented for the SMA wire fraction. However, the SMA wire fraction has a clear influence on the modes, while the initial strain of SMA wire appears to have a marginal effect on the modes.

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Figure 8 

The first three natural frequencies versus ply-angle for the different initial strains of SMA wire (𝜃=30∘; 𝑇=50°C; 𝑉𝑠=0.36).

Figure 9 shows the variation of the first three natural frequencies with the ply-angle for different rotating speeds (𝑉_𝑠=0.24; 𝜃=30^∘; 𝑇=50°C; 𝜀₀=0.067). The results show that the increase of rotating speed is accompanied by both an upward shift of the UWF branches and by a downward shift of the LWF branches, which are similar to that in connection with the results of Figures 2, 3, and 6.

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Figure 9 

The first three natural frequencies versus ply-angle for different rotating speeds (𝑉𝑠=0.24; 𝜃=30∘; 𝑇=50°C; 𝜀0=0.067).

Figure 10 shows the variation of the first three natural frequencies versus ply-angle during heating and cooling at 50°C (𝑉_𝑠=0.12; Ω^∗=5 rad/s; 𝜀₀=0.067). For the same temperature, there are two natural frequencies which correspond to the heating and cooling processes, respectively (see Figure 4). Moreover, when rotating speed Ω^∗≠0 two different frequencies are produced. Thus, we can obtain totally four different curves at temperature 𝑇=50°C.

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Figure 10 

The first three natural frequencies versus ply angle during heating and cooling processes at 50°C (𝑉𝑠=0.12; Ω∗=5 rad/s; 𝜀0=0.067).

The effect of the fraction and the initial strain of SMA wires on the critical rotating speed during heating at 50°C is presented in Tables 5 and 6, respectively.

Table 5 

The critical rotating speed corresponding to the different SMA wire fractions (𝑇=50°C, 𝜀₀=0.067).

Table 6 

The critical rotating speed corresponding to the different initial strains of SMA wire (𝑇=50°C, 𝑉_𝑠=0.36).

The significant effects of the SMA wire fraction on the critical rotating speed can be observed in Table 5. However, as it appears from Table 6, the initial strain of SMA wires does not influence the critical rotating speed significantly.

6. Conclusion

A model was presented for the study of the vibration and stability of rotating thin-walled composite shaft embedded with SMA wires. The presented model was used to predict the natural frequencies and dynamical stability. Emphasis is placed on the study of the effects of SMA activation on the above dynamical behavior. From the present analysis and the numerical results, the following main conclusions can be obtained.(1)The developed model provides means of predicting the natural frequency and critical rotating speed of rotating composite thin-walled shaft with SMA wires actuation.(2)SMA wires activation can significantly postpone the occurrence of the whirling instability and increase the critical rotating speed through SMA phase transformation.(3)The SMA wire fraction and ply-angle affect the actuation performance of SMA significantly. There is an obvious increase in the natural frequencies and the critical rotating speed as the SMA wire fraction is increased.(4)The initial strain of SMA wire appears to have marginal effect on the critical rotating speed of the rotating shaft during heating.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is funded by the National Natural Science Foundation of China (Grant no. 11272190) and Shandong Provincial Natural Science Foundation of China (Grant no. ZR2011EEM031).