# Free Vibration Analysis of Rotating Pretwisted Functionally Graded Sandwich Blades

Authors: L. T. Liu; Y. X. Hao; W. Zhang; J. Chen
Journal: International Journal of Aerospace Engineering
 (2018)

---

## Abstract

A new structural dynamic model for the free vibration characteristic analysis of rotating pretwisted functionally graded (FG) sandwich blades is developed. The sandwich blade is made up of two functionally graded skins and a homogeneous material core. The thick shell theory is applied to derive the basic equations of motion of the rotating FG sandwich blade by considering the effects of centrifugal and Coriolis forces. The mode shapes are expanded in terms of two-dimensional algebraic polynomials in the Rayleigh–Ritz method, and the static and dynamic natural frequencies of the blade are obtained. The convergence analysis is studied, and the accuracy of the proposed model is verified by comparing with the literature results and ANSYS data. The effects of frequency parameters such as the twist angle, the thickness ratio, the aspect ratio, the layer thickness ratio, the scalar parameter of volume fraction, the stagger angle, and the rotation velocity on the vibration characteristics for pretwist FG sandwich blade are investigated in detail. In addition, the phenomena of frequency locus veering and mode shape exchanging occur in the static and dynamic states. Frequency locus veering is essentially caused by the coupling between different modes.


---

## Body

## 1. Introduction

The aero gas turbine engine is the key technology of the aircraft; blade is one of the most important parts of the system. Vibration, especially resonance of the blade, will produce larger stress, which will lead to fatigue failure. Thus, the dynamic behavior of rotating blades has been studied by numerous researchers.Some simplified blade models such as beams, plates, and shells are used to investigate the vibration characteristics of the blades in the published literatures. Leissa and Jacob [1] investigated the free vibration of pretwisted, cantilevered beams and plates by using the Ritz method. The work was the first three-dimensional study of the problem. Yoo et al. [2] derived the equations of motion with a concentrated mass by using a modeling method with hybrid deformation variables and investigated the effects of the nondimensional parameters on the vibration characteristics of the pretwisted rotating blade through numerical analysis. Chandiramani et al. [3] simplified the pretwisted rotating blade as a laminated composite, hollow uniform box-beam model, which considers the centrifugal and Coriolis effects, transverse shear flexibility, and restrained warping, and studied the free and forced vibration by using HSDT and Galerkin method. Carrera et al. [4] used the Carrera unified formulation and FEM to study the free vibration analysis of rotating blades; the Coriolis and centrifugal force fields were included in their work.The beam models are quite suitable for blades with large aspect ratio and low width thickness ratio. However, flexible rotating structures such as blades with low aspect ratios are also widely used in actual engineering applications. Therefore, more and more literatures were found to study the vibration of blade with plate and shell models.Qatu and Leissa [5] were the first to study the effect of plate parameters such as twist angle, stacking sequence, and lamination angle on the natural frequency and mode shapes of laminated composite pretwisted cantilever plates by using the shallow shell theory and Ritz method. Nabi and Ganesan [6] analyzed the vibration characteristics of metal matrix composite pretwisted blades by using beam and plate theories, respectively, and summed up the quantitative comparison of natural frequency. Yoo and Chung [7] developed a linear dynamic modeling method for plates by using two in-plane stretch variables and analyzed and studied the transient characteristics of rotating plates. Hu et al. [8] proposed a numerical procedure for the free vibration analysis of pretwisted thin plates based on the thin shell theory and studied the vibration characteristics considering different twist rates and aspect ratios. Hashemi et al. [9] used the Mindlin plate theory and Kane dynamic method to develop a finite element formulation for vibration analysis of rotating thick plates. The coupling between in-plane and out-of-plane deformations and Coriolis effect was considered in the study. Sinha and Turner [10] derived the governing partial differential equation of motion for the rotating pretwisted plate by using the thin shell theory and studied the free vibration of a turbomachinery cantilevered airfoil blade with the Rayleigh–Ritz technique by considering the centrifugal force filed as a quasi-static load. Sun et al. [11] presented a dynamic model based on CLPT, investigated the vibration behavior of a rotating blade by using Hamilton’s principle, and studied the point and distribution forced response using a proportional damping model.Rao and Gupta [12] studied the vibration of natural frequencies with various parameters of a rotating pretwisted blade with a small aspect ratio by using the classical bending theory of thin shells. Sun et al. [13] developed a novel dynamic model for pretwisted rotating compressor blades by using the general shell theory and studied the eigenfrequencies and damping properties of the pretwisted rotating blade. Sinha and Zylka [14] derived the governing partial differential equation of motion based on the thin shell theory and formulated the free vibration of a turbomachinery cantilevered airfoil by considering it as an anisotropic shell in a centrifugal force field. Kielb et al. [15] gave the complete theoretical and experimental results of a joint research study on the vibration characteristics of pretwisted cantilever plates.Claassen and Thorne [16] were the first to find out and discuss the curve veering in the plate vibration based on CLPT. Afolabi and Mehmed [17] studied the curve veering and flutter of rotating blades based on the theory of algebraic curves and catastrophe theory. They found that the frequency loci of undamped rotating blades do not cross, but must either repel each other or attract each other. Yoo and Pierre [18, 19] studied the vibration characteristics of rotating cantilever rectangular plates and composite plates, respectively, and found out and discussed the natural frequency locus veering, crossing, and associated mode shape variations in detail.Most of the early studies were focused on homogeneous materials, and in recent years, some researchers have extended the range to composite materials because of their excellent properties. The structure was developed from single layer to multilayer and composite layer, and more and more literature has been published.Oh et al. [20] studied the vibration of turbomachinery rotating blades made up of functionally graded materials and investigated the influence of parameters on the frequency of rotating blades. A refined dynamic theory involving the coupling between flapping, lagging, and transverse shear was used in their work. Ramesh and Rao [21] studied the natural frequencies of a pretwisted rotating FG cantilever beam by using Lagrange’s equation and the Rayleigh–Ritz method. The influences of different parameters and coupling between chordwise and flapwise bending modes on the natural frequency were investigated. Oh and Yoo [22] presented a new structural dynamic model to study the vibration characteristics of rotating blades. The blade was modeled as a functionally graded material pretwisted beam by using the Rayleigh–Ritz and Kane method. Mantari and Granados [23] studied the free vibration of FG plates by using a new novel FSDT with 4 unknowns. Li and Zhang [24] developed a dynamic model for FG plates undergoing large overall motions, studied the free vibration of rotating cantilever FGM rectangular plates, and found frequency locus veering and associated mode shift phenomena. Sun et al. [25] developed a novel dynamic model for multilayer blades by using the quadratic layerwise theory and studied the structural dynamics, particularly the damping properties of the rotating blade. Frequency locus veering with rotation speed was also found. Cao et al. [26] developed a rotating cantilever pretwisted sandwich plate model with a prestagger angle and investigated the vibrational behavior of a turbine blade with thermal barrier coating layers by using FSDT, von Karman plate theory, and Chebyshev–Ritz method.In the present paper, a new structural dynamic model is developed to study the free vibration characteristics of pretwisted rotating FG sandwich blades. The thick shell theory is applied to derive the basic equations of motion by considering the effects of centrifugal and Coriolis forces. The accuracy of the proposed model is verified by comparing with the literature and ANSYS. The effects of frequency parameters such as the twist angle, the thickness ratio, the aspect ratio, the layer thickness ratio, the scalar parameter of volume fraction, the stagger angle, and the rotation velocity on the vibration characteristics are investigated in detail. Furthermore, frequency locus veering and mode shape exchanging phenomena are found both in both static and dynamic states.


## 2. Theoretical Analysis

### 2.1. Basic Equations

As shown in Figure1, the functionally graded sandwich blade is modeled as a cantilever pretwisted thick plate with the twist angle θ at the free end, which is clamped to the rigid disk with a radius R, mounted with a stagger angle φ. The geometric parameters of the blade are the span (length) dimension L, the chord (width) dimension b, the total thickness h, and the angular rotating velocity about the rigid disk axis Ω.Figure 1
A typical pretwisted rotating blade model.In this paper, three coordinate systems are established for dynamic modeling. One is theXYZ-coordinate system (with unit vectors (iX,iY,iZ)), where the x-axis is along the spanwise direction of the blade, Y is the rotation axis, and the Z-axis is perpendicular to the XY-plane following the right-hand rule. The other is the xyz-coordinate system (with unit vectors (ix,iy,iz)), where the origin lies in the root of the mid-surface of the blade and the x-axis is parallel to the X-axis; the y-axis can be obtained by rotating around the x-axis with the stagger angle φ starting from the Y-axis direction. Figure 2 shows the last xη coordinate, a dynamic surface coordinate where the η-axis is perpendicular to the x-axis and equal to the y-axis at x=0, rotating at a uniform twist rate k=θ/L. Thus, the rotating angle at x can be expressed as
(1)θ′x=kx,x∈0,L.Figure 2
The dynamic coordinate system of the pretwisted rotating blade.The position vectorr00x,η of a typical point on the mid-surface of the pretwisted blade in the xη-coordinate can be written as
(2)r00x,η=xηcosθ′ηsinθ′Tixiyiz.Based on the Frenet–Serret formula [28], the corresponding unit base vectors can be expressed as
(3)e0xe0ηe0ζ=1A1−kηsinθ′kηcosθ′0Acosθ′Asinθ′−kη−sinθ′cosθ′ixiyiz,where A is the Lame parameter of the middle surface in the x-direction, given by
(4)A=1+k2η2.Then, an arbitrary point of the blade before deformation can be expressed by a position vectorr0 as
(5)r0=r00+ζe0ζ=x−kηζAηcosθ′−ζsinθ′Aηsinθ′+ζcosθ′ATixiyiz.As illustrated in Figure3, the core of the blade (h1≤ζ≤h2) is fully metal (isotropic) and two skins are composed of functionally graded material (FGM) in the thickness direction. The top skin varies from ceramic-rich surface (ζ=h3) to metal-rich surface (ζ=h2), while the bottom skin varies from ceramic-rich surface (ζ=h0) to metal-rich surface (ζ=h1). There are no interfaces between the core and skins of the functionally graded sandwich blade, and two skins are symmetrical about the middle surface in this paper. The volume fraction of the metal phase is obtained as [27]
(6)Vmζ=h+2ζh−hmnζ∈h0,h1,1ζ∈h1,h2,h−2ζh−hmnζ∈h2,h3.Figure 3
The section of the FG sandwich blade.As shown in Figure4, n is the scalar parameter that allows users to define the gradation of material properties in the thickness direction. The n=0 case corresponds to a fully metal blade. The volume fraction for the ceramic phase is given as
(7)Vcζ=1−Vmζ.Figure 4
Effect of the scalar parameter of volume fraction in the FG sandwich blade.
(a)
hm:h=0.5
(b)
hm=0In the present work, the homogenization procedure and the law of mixtures are used. The variation of the parameters in the thickness direction is given by(8)Pζ=PmVmζ+PcVcζ,where Pζ is the physical parameter, which refers to density ρ, elastic modulus E, and Poisson’s ratio μ. In this paper, Pm and Pc are the physical parameters of metal and ceramic phase.


### 2.2. The Strain Energy

According to the first-order shear deformation theory, the displacements can be written as(9)ux,η,ζ=u0x,η+ζϕxx,η,vx,η,ζ=v0x,η+ζϕηx,η,wx,η,ζ=w0x,η,where ϕx and ϕη are mid-surface rotations and u0, v0, and w0 are mid-surface displacements of the blade along the x, η, and ζ directions.According to the thick shell theory [29] and the improved version of the Novozhilov nonlinear shell theory [30], the strains at any point in the blade can be written as
(10a)εxεηγxη=εx0εη0γxη0+2Rxη00w0+ζκx1κη1κxη1,(10b)γxζγηζ=γxζ0γηζ0−1Rxηv0u0,where
(11a)εx0εη0γxη0=1A∂u0∂x+v0A∂A∂η+121A∂w0∂x2¯∂v0∂η+12∂w0∂η2¯∂u0∂η+1A∂v0∂x−u0A∂A∂η+1A∂w0∂x∂w0∂η¯,(11b)κx1κη1κxη1=1A∂ϕx∂x+ϕηA∂A∂η∂ϕη∂η∂ϕx∂η+1A∂ϕη∂x−ϕxA∂A∂η,(11c)γxζ0γηζ0=1A∂w0∂x+ϕx∂w0∂η+ϕη,where Rxη is the radius of the twist, which can be derived as [31]
(12)1Rxη=−kA2.The nonlinear parts, which are underlined in (11), will be ignored in the free vibration analysis of the blade.The stress–strain equations can be written as(13)σxσησxζσηζσxη=Q11ζQ12ζ000Q21ζQ22ζ00000Q44ζ00000Q55ζ00000Q66ζεxεηγxζγηζγxη,where the elastic constants Qijζ are functions of thickness ζ, which is defined as
(14a)Q11ζ=Q22ζ=Eζ1−μ2ζ,(14b)Q12ζ=Q21ζ=μζEζ1−μ2ζ,(14c)Q44ζ=Q55ζ=KsEζ21+μζ,(14d)Q66ζ=Eζ21+μζ,where Ks is the shear correction coefficient. Since the shear correction coefficient depends on the blade material properties and boundary conditions, it is difficult to obtain the exact value. In this paper, the shear correction coefficient Ks will be selected uniformly as 5/6 [32].The strain energy of the blade is defined as(15)US=12∫0L∫−b/2b/2∫−h/2h/2σxεx+σηεη+σxζγxζ+σηζγηζ+σxηγxηAdζdηdx=∫0L∫−b/2b/212D11κx12+D22κη12+D66κxη12+2D12κx1κη1+B11εx0κx1+B22εη0κη1+B66γxη0κxη1+B12εx0κη1+B21εη0κx1+12A11εx02+A22εη02+A66γxη02+2A12εx0εη0+A44γxζ02+A55γηζ02+B66′w0γxη0+B44′v0γxζ0+B55′u0γηζ0+A66′w02+A44′v02+A55′u02Adηdx,where Aij, Bij, and Dij are the extensional, bending-extensional coupling, and bending stiffness [32], Aij′ and Bij′ are the stiffness due to the initial twist of the blade. They are defined as
(16a)AijBijDij=∫−h/2h/2Qijζ1ζζ2dζ=∑m=13∫hm−1hmQijζ1ζζ2dζi,j=1,2,6,(16b)Aij=∫−h/2h/2Qijζdζ=∑m=13∫hm−1hmQijζdζi,j=4,5,(16c)Bij′=−1RxηBiji=j=4,5,2RxηBiji=j=6,(16d)Aij′=12Rxη2Aiji=j=4,5,2Rxη2Aiji=j=6.It should be noted that the stiffnessBij and Bij′ will be zero while the structure is symmetrical about the middle surface in the thickness direction. Figure 5 shows the variations of the nondimensional stiffness coefficients (A44′/A44 and A66′/A66 at η=0) with the initial twist angle. It can be found that both the nondimensional stiffness coefficients increase with increase in the initial twist angle.Figure 5
Variations of nondimensional stiffness coefficients with the initial twist angle (η=0).


### 2.3. The Potential Energy of Centrifugal Force

The coordinate system transformation relationship between thexyz-coordinate and XYZ-coordinate is given by
(17)ixiyiz=1000cosφsinφ0−sinφcosφiXiYiZ.Then, the angular velocityω in the xyz-coordinate of the pretwisted plate is expressed as
(18)ω=0Ωcosφ−ΩsinφTixiyiz.The centrifugal forceFC per unit volume in the blade is given by
(19)FC=ρω×r0+Rix×ω=ρΩ2R+x−kηζAηsinΦsinφ+ζcosΦsinφAηsinΦcosφ+ζcosΦcosφATixiyiz,where Φ is the sum of the stagger angle and the twist angle of the point, given by
(20)Φ=φ+θ′.The rotation of the blade about the turbine axial direction produces a significant amount of membrane stresses, which are essentially acting in the radial direction (the spanwise and chordwise directions of the blade). The component of centrifugal force along the spanwise directionσcs and the chordwise direction σcc can be written as [10, 14]
(21a)σcs=∫XLFC⋅e0xAdx=ρΩ2R−12AkηζL−x+12L2−x2+14η2cos2Φ−cos2Θ+14Aηζsin2Φ−sin2Θ,(21b)σcc=∫b/20FC⋅e0ηdη=ρΩ212kζsin2Φsinh−112kb−18b2sin2Φ,where Θ is the sum of the stagger angle and the total twist angle, given by
(22)Θ=φ+θ.The centrifugal potential energy can be derived as(23)UCF=12∫0L∫−b/2b/2∫−h/2h/2σcsεcs+σccεccAdζdηdx.According to [26], εcs and εcc are given by
(24)εcs=∂v0∂x2+∂w0∂x2,εcc=∂u0∂η2+∂w0∂η2.Substituting (21) and (24) into (23), the potential energy can be written as
(25)UCF=∫0L∫−b/2b/2I18bsin2Φ−I016b2sin22Φ∂u0∂η2+∂w0∂η2+I1η8Asin2Φ−sin2Θ−kηL−x4A∂v0∂x2+∂w0∂x2+I012RL−x+14L2−x2+η28cos2Φ−cos2Θ∂v0∂x2+∂w0∂x2Adηdx.The total potential energy can be written as(26)U=US+UCF.


### 2.4. The Kinetic Energy

The displacement vectoru can be written as
(27)u=uvwe0xe0ηe0ζ=1Au−kηwAvcosθ′−kηu+wsinθ′Avsinθ′+kηu+wcosθ′Tixiyiz.An arbitrary point of the blade after deformation can be expressed by the position vectorr as
(28)r=r0+u=1Au+Ax−kηw+ζAv+ηcosθ′−kηu+w+ζsinθ′Av+ηsinθ′+kηu+w+ζcosθ′Tixiyiz.The velocityVr of the point due to the rotation can be derived as
(29)Vr=ω×r+Rix=ΩAAv+ηsinΦ+kηu+w+ζcosΦkηw+ζ−AR+x−usinφkηw+ζ−AR+x−ucosφTixiyiz.The kinetic energy reads(30a)T=12∫0L∫−b/2b/2∫−h/2h/2ρu̇+Vr⋅e0x2+v̇+Vr⋅e0η2+ẇ+Vr⋅e0ζ2Adζdηdx=Tin+Tgy+Tro,where
(30b)Tin=∫0L∫−b/2b/212I2ϕ̇x2+ϕ̇η2+I1u̇0ϕ̇x+v̇0ϕ̇η+12I0u̇02+v̇02+ẇ02Adηdx,(30c)Tgy=Ω∫0L∫−b/2b/2ϕηsinΦ+AcosΦI2ϕ̇x+I1u̇0+kη−ϕxI2ϕ̇η+I1v̇0sinΦ+η+v0sinΦ+Aw0−kηR+xcosΦI1ϕ̇x+I0u̇0+kηw0−AR−Ax−u0I1ϕ̇η+I0v̇0sinΦ−I0kη2+kηv0sinΦ+Au0+R+xcosΦẇ0−I1AφxcosΦ+kηφηsinΦẇ0dηdx,(30d)Tro=Ω2∫0L∫−b/2b/212I2+I0ϕηsinΦ+1A1+kηϕxcosΦ2+12A2I2ϕx−kη2+12I0R+x+1Au0−1Akηw02+I1η+v0sinΦ+1Aw0+kηu0cosΦϕηsinΦ+1A1+kηϕxcosΦ+1A2I1AR+x+u0−kηw0ϕx−kηAdηdx,where “·” is the first-order derivation of time; I0, I1, and I2 in (25) and (30) are defined in terms of the density ρζ as
(31)I0I1I2=∫−h/2h/2ρζ1ζζ2dζ=∑m=13∫hm−1hmρζ1ζζ2dζ.


## 2.1. Basic Equations

As shown in Figure1, the functionally graded sandwich blade is modeled as a cantilever pretwisted thick plate with the twist angle θ at the free end, which is clamped to the rigid disk with a radius R, mounted with a stagger angle φ. The geometric parameters of the blade are the span (length) dimension L, the chord (width) dimension b, the total thickness h, and the angular rotating velocity about the rigid disk axis Ω.Figure 1
A typical pretwisted rotating blade model.In this paper, three coordinate systems are established for dynamic modeling. One is theXYZ-coordinate system (with unit vectors (iX,iY,iZ)), where the x-axis is along the spanwise direction of the blade, Y is the rotation axis, and the Z-axis is perpendicular to the XY-plane following the right-hand rule. The other is the xyz-coordinate system (with unit vectors (ix,iy,iz)), where the origin lies in the root of the mid-surface of the blade and the x-axis is parallel to the X-axis; the y-axis can be obtained by rotating around the x-axis with the stagger angle φ starting from the Y-axis direction. Figure 2 shows the last xη coordinate, a dynamic surface coordinate where the η-axis is perpendicular to the x-axis and equal to the y-axis at x=0, rotating at a uniform twist rate k=θ/L. Thus, the rotating angle at x can be expressed as
(1)θ′x=kx,x∈0,L.Figure 2
The dynamic coordinate system of the pretwisted rotating blade.The position vectorr00x,η of a typical point on the mid-surface of the pretwisted blade in the xη-coordinate can be written as
(2)r00x,η=xηcosθ′ηsinθ′Tixiyiz.Based on the Frenet–Serret formula [28], the corresponding unit base vectors can be expressed as
(3)e0xe0ηe0ζ=1A1−kηsinθ′kηcosθ′0Acosθ′Asinθ′−kη−sinθ′cosθ′ixiyiz,where A is the Lame parameter of the middle surface in the x-direction, given by
(4)A=1+k2η2.Then, an arbitrary point of the blade before deformation can be expressed by a position vectorr0 as
(5)r0=r00+ζe0ζ=x−kηζAηcosθ′−ζsinθ′Aηsinθ′+ζcosθ′ATixiyiz.As illustrated in Figure3, the core of the blade (h1≤ζ≤h2) is fully metal (isotropic) and two skins are composed of functionally graded material (FGM) in the thickness direction. The top skin varies from ceramic-rich surface (ζ=h3) to metal-rich surface (ζ=h2), while the bottom skin varies from ceramic-rich surface (ζ=h0) to metal-rich surface (ζ=h1). There are no interfaces between the core and skins of the functionally graded sandwich blade, and two skins are symmetrical about the middle surface in this paper. The volume fraction of the metal phase is obtained as [27]
(6)Vmζ=h+2ζh−hmnζ∈h0,h1,1ζ∈h1,h2,h−2ζh−hmnζ∈h2,h3.Figure 3
The section of the FG sandwich blade.As shown in Figure4, n is the scalar parameter that allows users to define the gradation of material properties in the thickness direction. The n=0 case corresponds to a fully metal blade. The volume fraction for the ceramic phase is given as
(7)Vcζ=1−Vmζ.Figure 4
Effect of the scalar parameter of volume fraction in the FG sandwich blade.
(a)
hm:h=0.5
(b)
hm=0In the present work, the homogenization procedure and the law of mixtures are used. The variation of the parameters in the thickness direction is given by(8)Pζ=PmVmζ+PcVcζ,where Pζ is the physical parameter, which refers to density ρ, elastic modulus E, and Poisson’s ratio μ. In this paper, Pm and Pc are the physical parameters of metal and ceramic phase.


## 2.2. The Strain Energy

According to the first-order shear deformation theory, the displacements can be written as(9)ux,η,ζ=u0x,η+ζϕxx,η,vx,η,ζ=v0x,η+ζϕηx,η,wx,η,ζ=w0x,η,where ϕx and ϕη are mid-surface rotations and u0, v0, and w0 are mid-surface displacements of the blade along the x, η, and ζ directions.According to the thick shell theory [29] and the improved version of the Novozhilov nonlinear shell theory [30], the strains at any point in the blade can be written as
(10a)εxεηγxη=εx0εη0γxη0+2Rxη00w0+ζκx1κη1κxη1,(10b)γxζγηζ=γxζ0γηζ0−1Rxηv0u0,where
(11a)εx0εη0γxη0=1A∂u0∂x+v0A∂A∂η+121A∂w0∂x2¯∂v0∂η+12∂w0∂η2¯∂u0∂η+1A∂v0∂x−u0A∂A∂η+1A∂w0∂x∂w0∂η¯,(11b)κx1κη1κxη1=1A∂ϕx∂x+ϕηA∂A∂η∂ϕη∂η∂ϕx∂η+1A∂ϕη∂x−ϕxA∂A∂η,(11c)γxζ0γηζ0=1A∂w0∂x+ϕx∂w0∂η+ϕη,where Rxη is the radius of the twist, which can be derived as [31]
(12)1Rxη=−kA2.The nonlinear parts, which are underlined in (11), will be ignored in the free vibration analysis of the blade.The stress–strain equations can be written as(13)σxσησxζσηζσxη=Q11ζQ12ζ000Q21ζQ22ζ00000Q44ζ00000Q55ζ00000Q66ζεxεηγxζγηζγxη,where the elastic constants Qijζ are functions of thickness ζ, which is defined as
(14a)Q11ζ=Q22ζ=Eζ1−μ2ζ,(14b)Q12ζ=Q21ζ=μζEζ1−μ2ζ,(14c)Q44ζ=Q55ζ=KsEζ21+μζ,(14d)Q66ζ=Eζ21+μζ,where Ks is the shear correction coefficient. Since the shear correction coefficient depends on the blade material properties and boundary conditions, it is difficult to obtain the exact value. In this paper, the shear correction coefficient Ks will be selected uniformly as 5/6 [32].The strain energy of the blade is defined as(15)US=12∫0L∫−b/2b/2∫−h/2h/2σxεx+σηεη+σxζγxζ+σηζγηζ+σxηγxηAdζdηdx=∫0L∫−b/2b/212D11κx12+D22κη12+D66κxη12+2D12κx1κη1+B11εx0κx1+B22εη0κη1+B66γxη0κxη1+B12εx0κη1+B21εη0κx1+12A11εx02+A22εη02+A66γxη02+2A12εx0εη0+A44γxζ02+A55γηζ02+B66′w0γxη0+B44′v0γxζ0+B55′u0γηζ0+A66′w02+A44′v02+A55′u02Adηdx,where Aij, Bij, and Dij are the extensional, bending-extensional coupling, and bending stiffness [32], Aij′ and Bij′ are the stiffness due to the initial twist of the blade. They are defined as
(16a)AijBijDij=∫−h/2h/2Qijζ1ζζ2dζ=∑m=13∫hm−1hmQijζ1ζζ2dζi,j=1,2,6,(16b)Aij=∫−h/2h/2Qijζdζ=∑m=13∫hm−1hmQijζdζi,j=4,5,(16c)Bij′=−1RxηBiji=j=4,5,2RxηBiji=j=6,(16d)Aij′=12Rxη2Aiji=j=4,5,2Rxη2Aiji=j=6.It should be noted that the stiffnessBij and Bij′ will be zero while the structure is symmetrical about the middle surface in the thickness direction. Figure 5 shows the variations of the nondimensional stiffness coefficients (A44′/A44 and A66′/A66 at η=0) with the initial twist angle. It can be found that both the nondimensional stiffness coefficients increase with increase in the initial twist angle.Figure 5
Variations of nondimensional stiffness coefficients with the initial twist angle (η=0).


## 2.3. The Potential Energy of Centrifugal Force

The coordinate system transformation relationship between thexyz-coordinate and XYZ-coordinate is given by
(17)ixiyiz=1000cosφsinφ0−sinφcosφiXiYiZ.Then, the angular velocityω in the xyz-coordinate of the pretwisted plate is expressed as
(18)ω=0Ωcosφ−ΩsinφTixiyiz.The centrifugal forceFC per unit volume in the blade is given by
(19)FC=ρω×r0+Rix×ω=ρΩ2R+x−kηζAηsinΦsinφ+ζcosΦsinφAηsinΦcosφ+ζcosΦcosφATixiyiz,where Φ is the sum of the stagger angle and the twist angle of the point, given by
(20)Φ=φ+θ′.The rotation of the blade about the turbine axial direction produces a significant amount of membrane stresses, which are essentially acting in the radial direction (the spanwise and chordwise directions of the blade). The component of centrifugal force along the spanwise directionσcs and the chordwise direction σcc can be written as [10, 14]
(21a)σcs=∫XLFC⋅e0xAdx=ρΩ2R−12AkηζL−x+12L2−x2+14η2cos2Φ−cos2Θ+14Aηζsin2Φ−sin2Θ,(21b)σcc=∫b/20FC⋅e0ηdη=ρΩ212kζsin2Φsinh−112kb−18b2sin2Φ,where Θ is the sum of the stagger angle and the total twist angle, given by
(22)Θ=φ+θ.The centrifugal potential energy can be derived as(23)UCF=12∫0L∫−b/2b/2∫−h/2h/2σcsεcs+σccεccAdζdηdx.According to [26], εcs and εcc are given by
(24)εcs=∂v0∂x2+∂w0∂x2,εcc=∂u0∂η2+∂w0∂η2.Substituting (21) and (24) into (23), the potential energy can be written as
(25)UCF=∫0L∫−b/2b/2I18bsin2Φ−I016b2sin22Φ∂u0∂η2+∂w0∂η2+I1η8Asin2Φ−sin2Θ−kηL−x4A∂v0∂x2+∂w0∂x2+I012RL−x+14L2−x2+η28cos2Φ−cos2Θ∂v0∂x2+∂w0∂x2Adηdx.The total potential energy can be written as(26)U=US+UCF.


## 2.4. The Kinetic Energy

The displacement vectoru can be written as
(27)u=uvwe0xe0ηe0ζ=1Au−kηwAvcosθ′−kηu+wsinθ′Avsinθ′+kηu+wcosθ′Tixiyiz.An arbitrary point of the blade after deformation can be expressed by the position vectorr as
(28)r=r0+u=1Au+Ax−kηw+ζAv+ηcosθ′−kηu+w+ζsinθ′Av+ηsinθ′+kηu+w+ζcosθ′Tixiyiz.The velocityVr of the point due to the rotation can be derived as
(29)Vr=ω×r+Rix=ΩAAv+ηsinΦ+kηu+w+ζcosΦkηw+ζ−AR+x−usinφkηw+ζ−AR+x−ucosφTixiyiz.The kinetic energy reads(30a)T=12∫0L∫−b/2b/2∫−h/2h/2ρu̇+Vr⋅e0x2+v̇+Vr⋅e0η2+ẇ+Vr⋅e0ζ2Adζdηdx=Tin+Tgy+Tro,where
(30b)Tin=∫0L∫−b/2b/212I2ϕ̇x2+ϕ̇η2+I1u̇0ϕ̇x+v̇0ϕ̇η+12I0u̇02+v̇02+ẇ02Adηdx,(30c)Tgy=Ω∫0L∫−b/2b/2ϕηsinΦ+AcosΦI2ϕ̇x+I1u̇0+kη−ϕxI2ϕ̇η+I1v̇0sinΦ+η+v0sinΦ+Aw0−kηR+xcosΦI1ϕ̇x+I0u̇0+kηw0−AR−Ax−u0I1ϕ̇η+I0v̇0sinΦ−I0kη2+kηv0sinΦ+Au0+R+xcosΦẇ0−I1AφxcosΦ+kηφηsinΦẇ0dηdx,(30d)Tro=Ω2∫0L∫−b/2b/212I2+I0ϕηsinΦ+1A1+kηϕxcosΦ2+12A2I2ϕx−kη2+12I0R+x+1Au0−1Akηw02+I1η+v0sinΦ+1Aw0+kηu0cosΦϕηsinΦ+1A1+kηϕxcosΦ+1A2I1AR+x+u0−kηw0ϕx−kηAdηdx,where “·” is the first-order derivation of time; I0, I1, and I2 in (25) and (30) are defined in terms of the density ρζ as
(31)I0I1I2=∫−h/2h/2ρζ1ζζ2dζ=∑m=13∫hm−1hmρζ1ζζ2dζ.


## 3. Frequency Solving

The Rayleigh–Ritz method is used to solve the natural frequency of the blade; the expressions of the displacements and rotations are assumed(32a)u0x,η,t=∑N=1∞UNx,ηeiωNt,(32b)v0x,η,t=∑N=1∞VNx,ηeiωNt,(32c)w0x,η,t=∑N=1∞WNx,ηeiωNt,(32d)ϕxx,η,t=∑N=1∞ΦxNx,ηeiωNt,(32e)ϕηx,η,t=∑N=1∞ΦηNx,ηeiωNt.The functionsUNx,η, VNx,η, WNx,η, ΦxNx,η, and ΦηNx,η represent the N order mode shape, and ωN is the corresponding circular frequency. The algebraic polynomials can form a mathematically complete set of functions, which guarantees convergence to the exact solution as the number of terms taken increases [5]. Thus, in this paper, the mode shapes are expanded in terms of two-dimensional algebraic polynomials in terms of the nondimensional coordinates χ and ξ as
(33a)UNχ,ξ=∑i=1I∑j=0JUijχiξj,(33b)VNχ,ξ=∑k=1K∑l=0LVklχkξl,(33c)WNχ,ξ=∑m=2M∑n=0NWmnχmξn,(33d)ΦxNχ,ξ=∑i=1I∑j=0JΦxijχiξj,(33e)ΦηNχ,ξ=∑k=1K∑l=0LΦηklχkξl,where χ=x/L and ξ=2η/b. Uij, Vkl, Wmn, Φxij, and Φηkl are undetermined coefficients. The Rayleigh–Ritz method imposes a necessary condition that trial functions must satisfy the geometric boundary conditions arbitrarily. The cantilever blade is clamped at χ=0 (u=v=w=dw/dx=0) along the spanwise direction and is completely free at χ=1, ξ=−1, and ξ=1. The indices i, k, and m in (33), (46), and (47) begin with i=1, k=1, and m=2; it can be ensured that all terms of polynomials satisfied the geometric boundary conditions.To solve the free frequency of the blade, (32) and (33) are substituted into (26) and (30) in order to obtain the maximum strain energy (Umax) and kinetic energy (Tmax) by setting t=0. The Rayleigh–Ritz method requires the minimization of the function ∏=Tmax−Umax, which can be achieved by taking the derivatives
(34a)∂∏Uij=0,∂∏Φxij=0i=1,2,…,I;j=0,1,…,J,(34b)∂∏Vkl=0,(34c)∂∏Φηkl=0k=1,2,…,K,l=0,1,…,L,∂∏Wmn=0m=2,3,…,M,n=0,1,…,N.There are a total of2×I×J+1+2×K×L+1+M−1×N+1 equations, which can be described by
(35)K−ωN2MΔ=0,where K and M are the stiffness and mass matrices, respectively; Δ is the vector of undetermined coefficients, given by
(36)Δ=UijVklWmnΦxijΦηklT.The generalized eigenvalues (natural frequency) can be obtained by setting the coefficient matrix of (35) equal to zero, and the corresponding mode shape can be determined by substituting the eigenvector Δ back into (33).


## 4. Numerical Results and Analyses

### 4.1. Convergence Analysis

A typical example of pretwisted blade with chord lengthb=0.254m, aspect ratio l¯=L/b from 1 to 5, thickness ratio h¯=b/h from 10 to 100, twist angle θ from 0° to 60°, layer thickness ratio h¯l=h−hm/2hm from 0.25 to 4, scalar parameter of volume fraction of FGM n from 0.1 to 10, stagger angle φ from 0° to 90°, and disk radius ratio r¯=R/b is used.All the numerical results are presented in the form of the nondimensional frequency parameterω¯=ωNL2ρh/D=ωNL2/h12ρ1−μ2/E for the homogeneous material blade, where D=Eh3/121−μ2 is the blade flexural rigidity per unit length and ω¯=ωNL2/h12ρm1−μm2/Em for the functionally graded sandwich blade in the following section, and the nondimensional rotation velocity is Ω¯=Ω/ω1, where ω1 is the first static circular frequency (Ω=0) of the functionally graded sandwich blade.According to [5], the number of terms chosen for each component of the displacements is the same. The results of the homogeneous material blade and functionally graded sandwich blade are given in Tables 1 and 2. The maximum difference between the 5×9×9=405 term and the 5×8×8=320 term solutions for the first six modes of the homogeneous material blade is no more than 0.28%, and 0.21% for the FG sandwich blade. But the time spent in computing has multiplied. Moreover, the eigenvalues are more likely to be ill-conditioned with the increase in the size of the matrix in numerical calculation. Therefore, it has been decided to use 5×8×8=320 terms in the subsequent analysis.Table 1
Convergence of nondimensional frequency parameters for the first six modes of the homogeneous material blade (l¯=3, h¯=20, and θ=15°).
Mode
ANSYS
5 × 5 × 5
5 × 6 × 6
5 × 7 × 7
5 × 8 × 8
5 × 9 × 9
Value
Error
Value
Error
Value
Error
Value
Error
Value
Error
1st
3.4225
3.4217
−0.02%
3.4102
−0.36%
3.4077
−0.43%
3.4194
−0.09%
3.4124
−0.30%
2nd
20.7177
20.7606
0.21%
20.7156
−0.01%
20.7222
0.02%
20.7540
0.18%
20.7270
0.04%
3rd
21.9608
22.2650
1.39%
21.9718
0.05%
22.1278
0.76%
22.0861
0.57%
22.0340
0.33%
4th
58.8524
59.6322
1.33%
54.3477
−7.65%
58.1334
−1.22%
58.1917
−1.12%
58.0295
−1.40%
5th
64.5829
66.5017
2.97%
64.0481
−0.83%
65.6567
1.66%
65.4872
1.40%
65.6406
1.64%
6th
68.0114
68.9640
1.40%
68.1982
0.27%
68.6168
0.89%
68.4144
0.59%
68.3678
0.52%Table 2
Convergence of nondimensional frequency parameters for the first six modes of the FG sandwich blade (l¯=3, h¯=20, n=2, h¯l=1/2, and θ=15°).
Mode
ANSYS
5 × 5 × 5
5 × 6 × 6
5 × 7 × 7
5 × 8 × 8
5 × 9 × 9
Value
Error
Value
Error
Value
Error
Value
Error
Value
Error
1st
4.5272
4.5320
0.11%
4.5209
−0.14%
4.5228
−0.10%
4.5211
−0.13%
4.5215
−0.13%
2nd
27.3966
27.5019
0.38%
27.4741
0.28%
27.4639
0.25%
27.4684
0.26%
27.4665
0.26%
3rd
29.0406
29.5393
1.72%
29.3030
0.90%
29.3432
1.04%
29.2158
0.60%
29.1542
0.39%
4th
76.2376
75.8763
−0.47%
70.8471
−7.07%
74.8356
−1.84%
74.8922
−1.76%
75.0109
−1.61%
5th
80.3331
84.3843
5.04%
78.8139
−1.89%
82.1073
2.21%
82.2715
2.41%
82.2392
2.37%
6th
89.9847
91.5581
1.75%
90.9359
1.06%
91.0424
1.18%
90.7128
0.81%
90.5917
0.67%The data used for comparison below “ANSYS” in all tables are computed by the finite element model in ANSYS Workbench (version 17.2). The middle surface model of the blade is established, and the finite element analysis is performed by using the SHELL181 linear quadrilateral element type. The layered section object in ANSYS mechanical is used to define the layered section of composite modeling. In order to get accurate results as much as possible, 20 layers in each skin (total of 41 layers in the thickness direction) are defined to simulate the cross section of the functionally graded sandwich blade.


### 4.2. Comparative Studies

The homogeneous material blade is used first to demonstrate the numerical accuracy of the analytical model in this paper, and the first six nondimensional frequency parameters are compared with ANSYS and reference data [15] in Table 3. The results are in good agreement when the twist angle θ=0° to 45° in the case of the aspect ratio l¯=3. The results match well also when the twist angle θ=0° to 30° in the case of the aspect ratio l¯=1, but the maximum error (occurring at the 2nd order) is about 8.86% when θ increases to 45°. The reason is that the Lame parameter A is approximated to 1 in order to improve the calculation speed in the process of numerical calculation. The error is smaller in the case of a larger aspect ratio and smaller twist angle, but the error becomes larger as the aspect ratio decreases and the twist angle increases. Thus, the aspect ratio l¯=3 is used to obtain more accurate results.Table 3
Comparison of nondimensional frequency parameters for the first six modes of homogeneous material blade (h¯=20, μ=0.3).
θ
Mode
l¯=3
l¯=1
1st
2nd
3rd
4th
5th
6th
1st
2nd
3rd
4th
5th
6th
0°
Present
3.4205
20.8483
21.3316
59.6783
62.2910
65.4353
3.4757
8.3462
20.9625
26.6650
30.1067
43.5409
ANSYS
3.4223
20.8230
21.3590
59.8793
62.2508
65.3690
3.4814
8.3793
21.0993
26.7267
30.2352
43.5819
Ref [15]
3.4140
21.2200
21.3600
60.1200
61.8200
66.6800
3.4620
8.4870
21.2900
27.4700
30.9800
43.5200
15°
Present
3.4194
20.7540
22.0861
58.1917
65.4872
68.4144
3.4479
10.4233
20.4526
26.6975
31.5336
44.6423
ANSYS
3.4225
20.7177
21.9608
58.8524
64.5829
68.0114
3.4540
10.3394
20.3668
26.6264
31.3218
44.4679
Ref [15]
3.4150
20.6700
22.2800
58.1500
65.2900
69.3200
3.4450
10.4000
20.5700
27.3500
32.1100
44.5800
30°
Present
3.4035
19.2320
25.4184
54.4781
70.6496
76.3746
3.4030
14.8939
18.9948
27.1437
35.2177
46.9877
ANSYS
3.4153
18.9950
25.0621
55.7643
71.6994
75.9291
3.3794
14.3509
18.4374
26.7434
34.0572
45.9808
Ref [15]
3.4170
19.0000
25.1500
55.9500
71.4000
76.5700
3.3970
14.3300
18.6900
27.3900
34.9000
46.5200
45°
Present
3.4163
17.4172
29.7580
54.0239
80.8963
87.6963
3.3387
20.0628
17.1876
28.2967
40.1453
48.6332
ANSYS
3.4176
16.9599
29.4234
53.3063
79.0586
87.0233
3.2668
18.4291
16.0421
27.1718
37.3254
46.1459
Ref [15]
3.4200
16.9900
29.2000
53.5800
78.8200
87.0500
3.3230
18.3200
16.3900
27.7100
38.1700
47.3000The functionally graded sandwich blade is composed of metal and ceramic materials; their properties are given in Table4. In Table 5, the first six nondimensional frequency parameters ω¯ of the FG sandwich blade are compared with results in ANSYS Workbench, both showing good agreements, respectively.Table 4
The materials and properties of the FG sandwich blade.
Materials
Material properties
Metal (alloy C-276)
Em=2.08×1011Pa,μm=0.32,ρm=8900kg/m3
Ceramic (Al2O3)
Ec=3.50×1011Pa,μc=0.26,ρc=3970kg/m3Table 5
Comparisons of nondimensional frequency parameters for the first six modes of the FG sandwich blade (l¯=3, h¯=20, n=2, and h¯l=1/2).
Mode
0°
15°
30°
Present
ANSYS
Error
Mode shape
Present
ANSYS
Error
Mode shape
Present
ANSYS
Error
Mode shape
1st
4.5233
4.5270
−0.08%
1B
4.5206
4.5272
−0.15%
1B
4.4729
4.5290
−1.24%
1B
2nd
27.9633
27.7729
0.69%
1 T
27.4621
27.3966
0.24%
2B
25.4167
25.0992
1.26%
2B
3rd
28.3106
28.2605
0.18%
2B
29.1565
29.0406
0.40%
1T
33.0633
32.5549
1.56%
1T
4th
75.9386
75.8281
0.15%
1 EB
74.8841
76.2376
−1.78%
3B
73.0937
72.9145
0.25%
3B
5th
79.3035
79.1454
0.20%
3B
82.0284
80.3331
2.11%
1EB
89.2792
88.7755
0.57%
1EB
6th
87.5852
87.0515
0.61%
2 T
90.5827
89.9847
0.66%
2T
99.5237
98.9166
0.61%
2TThe mode shapes corresponding to the first six orders of frequencies are also listed in Table5. It should be noted that spanwise bending modes are divided into flapwise bending modes (whose displacements are predominantly in the direction normal to the middle surface) and edgewise bending modes (whose displacements are predominantly in the direction tangent to the surface) [15] in this paper. The modes 1B and 2B represent the first and second flapwise bending modes, respectively, and 1EB represents the first edgewise bending mode. The modes 1T and 2T represent the first and second torsional modes, respectively. The chordwise bending modes (CB, having two or more nodal lines approximately parallel to the pretwisted blade axis) are also found in the successive research. The curves such as 1B, 1T, and 2B and so on in Figures 6–8 are all obtained by using ANSYS Workbench. The curves such as 1st, 2nd, and 3rd are all obtained through numerical calculation.Figure 6
Variations of nondimensional frequency parameters with the twist angle (l¯=3, h¯=20, n=2, and h¯l=1/2).
(a)
1st to 3rd modes
(b)
4th to 6th modesFigure 7
Variations of nondimensional frequency parameters with the thickness ratio (l¯=3, n=2, h¯l=1/2, and θ=30°).Figure 8
Variations of nondimensional frequency parameters with the aspect ratio (h¯=20, n=2, h¯l=1/2, θ=30°).
(a)
1st to 3rd modes
(b)
4th to 6th modesThe first six mode shapes of the FG sandwich blade withl¯=3, h¯=20, n=2, h¯l=1/2, and θ=30∘ are given in Figure 9. It can be seen that the results obtained by the two methods are in good agreement. The results in Table 5 show that that the 1st modes are both of 1B mode shape when θ=0°, 15°, and 30°, but the 2nd mode is of 1T mode shape when θ=0° and 2B mode shape when θ=15° and 30°. The same happens with the 1EB mode shape. The reason for this phenomenon is that the stiffness of the FG sandwich blade is changed due to the initial twist in (15).Figure 9
Mode shapes of the FG sandwich blade. (A1) 1st mode shape by Matlab; (B1) 2nd mode shape by Matlab; (C1) 3rd mode shape by Matlab; (D1) 4th mode shape by Matlab; (E1) 5th mode shape by Matlab; (F1) 6th mode shape by Matlab; (A2) 1st mode shape by ANSYS; (B2) 2nd mode shape by ANSYS; (C2) 3rd mode shape by ANSYS; (D2) 4th mode shape by ANSYS; (E2) 5th mode shape by ANSYS; (F2) 6th mode shape by ANSYS.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.3. Free Vibrations of the FG Sandwich Blade

The effect of the twist angle on the vibration characteristics for the pretwist FG sandwich blade with geometric parameters asl¯=3, h¯=20, n=2, and h¯l=1/2 is studied. Table 6 gives the first six nondimensional frequency parameters ω¯ of the FG sandwich blade with varying twist angles (θ from 0° to 60°), and the relationship between the nondimensional frequency parameters and twist angles for the FG sandwich blade is shown in Figure 6. It can be seen that the resulting curves are very close to the curves obtained by ANSYS in Figure 6(a). The first mode corresponds to the 1B vibration mode for all twist angles. The 2nd and 3rd nondimensional frequency parameter curves represent the 1T and 2B mode shapes when θ=0°, and they come close to each other and separate gradually with the twist angle increasing. The two curves are up to the nearest when the twist angle θ is about 5°; the two modes exchange with each other, and the variation tendencies of nondimensional frequency parameters are also interchanged. The variations for 4th to 6th can be found in Figure 6(b).Table 6
Nondimensional frequency parameters for the first six modes of FG sandwich blade with varying twist angle (l¯=3, h¯=20, n=2, and h¯l=1/2).
Mode
0°
5°
10°
15°
20°
25°
30°
45°
50°
60°
1st
4.5233
4.5342
4.5342
4.5206
4.5041
4.4935
4.4743
4.4020
4.3605
4.3245
2nd
27.9633
28.0799
27.9178
27.4621
26.8716
26.1715
25.4167
22.9643
22.1860
20.7649
3rd
28.3106
28.2050
28.5154
29.1565
29.9955
31.3439
33.0633
38.0040
39.9970
44.6692
4th
75.9386
75.7447
75.3807
74.8841
74.5120
73.8517
73.0937
70.8379
70.1630
68.9817
5th
79.3035
79.7009
80.7576
82.0284
84.0329
86.7774
89.2792
98.6081
101.9785
107.9219
6th
87.5852
87.9114
88.9680
90.5827
92.8839
95.9725
99.5237
112.9271
118.0741
130.0957It should be noted that the FG sandwich blade designed by the geometric parameters corresponding to the frequency turning points will be prone to internal resonance.The effect of the thickness ratio on the vibration characteristics for the pretwist FG sandwich blade is studied also. The first six nondimensional frequency parametersω¯ of the FG sandwich blade with varying thickness ratios (h¯ from 10 to 100) is given in Table 7. Figure 7 shows the relationship between the nondimensional frequency parameters and the thickness ratio. It can be found that the nondimensional frequency parameters of 1B, 2B, 3B, and 4B modes remain essentially unchanged when the thickness ratio varies from 10 to 100. The nondimensional frequency parameters of 1T, 2T, and 1EB modes tend to increase with increasing thickness ratio. On the other side, the 3rd and 4th frequency parameter loci have an obvious veering at h¯=85, and the mode shapes exchanged between 1T and 3B. The same situation can be seen on the 5th and 6th modes, and higher-order frequency parameter locus veering happens more frequently.Table 7
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying thickness ratio (l¯=3, n=2, h¯l=1/2, and θ=30°).
Mode
10
20
30
40
50
60
70
80
90
100
1st
4.4578
4.4743
4.4745
4.4759
4.4903
4.5064
4.5022
4.5115
4.5146
4.5213
2nd
24.2636
25.4167
25.5518
25.6696
25.7090
25.7337
25.7686
25.8024
25.8433
25.8342
3rd
28.2468
33.0633
38.1941
44.8357
51.5288
59.2127
67.3645
74.2099
77.6186
77.2360
4th
43.9320
73.0937
75.9033
76.1792
76.4980
76.7083
76.7618
76.3814
82.7484
91.1187
5th
76.6391
89.2792
113.8445
130.9914
147.7835
151.9612
156.271
159.8808
159.7313
159.6019
6th
87.7677
99.5237
126.202
147.8068
155.1631
169.1361
191.4968
207.7979
230.7914
250.1723The nondimensional frequency parametersω¯ for the first six modes of the FG sandwich blade with varying aspect ratios (l¯ from 1 to 5) are given in Table 8 to study the effect of the aspect ratio on the vibration characteristics for the pretwist FG sandwich blade. Figure 8 shows the relationships between the first six frequency parameters, the corresponding mode shapes, and the aspect ratio. It is obvious that the 1st mode corresponds to the 1B vibration mode for all aspect ratios, and the value of the frequency parameter varies little. The 2nd and 3rd modes exchange with each other, and the vibration tendencies of their frequency loci also exchange near l¯=2. The change of the 4th to 6th nondimensional frequency parameters is relatively complicated. The mode shapes include 3B, 2T, 1EB, and 1CB. For example, the 4th to 6th mode shapes are 1CB, 2T, and 1EB when l¯=1; 2T, 1EB, and 1CB when l¯=1.5; and 1EB, 2T, and 3B when l¯=1.75. All in all, the mode shape corresponding to the same order frequency is very sensitive to the variation of small aspect ratios, especially when l¯<2. The tendencies of their frequency parameters become relatively stable until l¯>2.5.Table 8
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying aspect ratio (h¯=20, n=2, h¯l=1/2, and θ=30°).
Mode
1.00
1.25
1.50
1.75
2.00
2.25
2.50
3.00
4.00
5.00
1st
4.4970
4.5187
4.4946
4.4879
4.4857
4.4847
4.4762
4.4743
4.4643
4.4475
2nd
18.7284
20.8402
22.6114
24.2429
25.4037
25.4146
25.4190
25.4167
25.4069
25.3682
3rd
25.0639
25.2854
25.3453
25.3845
25.9349
27.5549
29.2911
33.0633
40.4134
48.5665
4th
35.6083
49.9937
61.8109
68.2541
70.4597
71.6661
72.3847
73.0937
73.8139
73.9446
5th
45.7901
53.9883
63.0533
69.0347
75.8721
82.0019
87.1364
89.2792
91.5483
93.0731
6th
58.3492
65.8222
76.2076
82.7443
84.4782
85.9505
88.1750
99.5237
123.1095
147.6281Table9 and Figure 10 show the variations of the first six nondimensional frequency parameters of the FG sandwich blade with the layer thickness ratio. Table 10 and Figure 11 show the relationships between the first six nondimensional frequency parameters and the scalar parameter of the volume fraction. It can be found that the content of ceramic increases with increasing layer thickness ratio and scalar parameter of the volume fraction, and the frequency parameters of all modes increase, respectively.Table 9
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying layer thickness ratios (l¯=3, h¯=20, n=2, and θ=30°).
Mode
1/4
1/3
1/2
1/1
2/1
3/1
4/1
1st
4.2021
4.3068
4.4743
4.7574
4.9764
5.0854
5.1378
2nd
23.6280
24.3536
25.4167
27.0889
28.3771
28.9429
29.2352
3rd
30.4576
31.3180
33.0633
35.0677
36.7356
37.6719
38.4396
4th
68.0698
70.1619
73.0937
78.1350
81.7252
83.5669
84.7660
5th
83.1451
85.6584
89.2792
95.5154
100.6206
103.1477
104.3400
6th
92.4498
95.1984
99.5237
106.5238
111.4889
114.2924
116.4530Figure 10
Variations of nondimensional frequency parameters with the layer thickness ratio (l¯=3, h¯=20, n=2, and θ=30°).Table 10
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying scalar parameters of volume fraction (l¯=3, h¯=20, h¯l=1/2, and θ=30°).
Mode
0.0
0.1
0.2
0.3
0.4
0.5
1.0
2.0
3.0
4.0
5.0
1st
3.3606
3.5795
3.6493
3.8619
3.8665
3.9753
4.2248
4.4743
4.5659
4.6931
4.7617
2nd
19.1226
20.0849
20.7619
21.4358
21.9111
22.3903
23.8825
25.4167
26.1362
26.5972
26.8850
3rd
24.9882
26.0961
26.9426
27.6920
28.2425
28.7613
30.7357
33.0633
33.7313
34.2049
34.8816
4th
55.8597
58.2781
60.3344
62.0908
63.5612
64.7114
68.9569
73.0937
75.1927
76.5550
77.6652
5th
71.5372
73.7012
75.7393
77.3010
78.8343
80.0439
84.5755
89.2792
91.5699
93.1665
93.7638
6th
75.5421
79.2376
81.6705
84.0076
85.8903
86.6503
93.4294
99.5237
102.7955
104.0268
106.0757Figure 11
Variations of nondimensional frequency parameters with the scalar parameter of volume fraction (l¯=3, h¯=20, h¯l=1/2, and θ=30°).Table11 shows the first six nondimensional frequency parameters of the FG sandwich blade for the nondimensional rotation velocity with the range of Ω¯ 0 to 11 (about 15,000 rpm). A typical Campbell diagram is plotted to study the variation trend of the nondimensional frequency parameters for varying rotation velocities in Figure 12. Frequency locus veering is observed as expected: the 1st and 2nd frequency modes exchange between 1B and 1T around the rotation velocity Ω¯=11; the 2nd and 3rd frequency modes exchange between 2B and 1T around Ω¯=2; the 5th and 6th frequency modes exchange between 1EB and 2T around Ω¯=3; and the 7th and 8th frequency modes exchange between 4B and 3T around Ω¯=1 and exchange between 3T and 1CB again around the rotation velocity Ω¯=9. In addition, the 8th frequency locus can be seen veering at Ω¯=4 and the mode shape is changed from 4B to 1EB. It can be seen clearly that higher-order frequencies tend to have multiple frequency veering with the mode shapes exchanging and coupling.Table 11
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying nondimensional rotation velocities (l¯=3, h¯=20, n=2, h¯l=1/2, θ=30°, r¯=1, and φ=45°).
Mode
0
1
2
3
4
5
6
7
8
9
10
11
1st
4.4743
6.9030
11.2340
15.8419
20.5025
25.1652
29.8098
34.4146
38.9622
43.4424
47.8358
52.1334
2nd
25.4167
28.5235
34.0383
35.1206
36.4308
38.1374
40.2129
42.4223
44.6896
47.3581
50.0889
52.8240
3rd
33.0633
33.0979
35.8589
45.2978
55.2322
64.5633
72.3156
78.0880
82.3864
85.9246
89.1536
92.3135
4th
73.0937
76.2806
80.9215
84.9748
88.5518
92.9797
99.3391
108.0164
118.4814
130.0075
142.1141
154.5548
5th
89.2792
92.7844
96.5525
106.6897
114.5673
121.7465
129.8789
138.6539
147.8984
157.7082
167.8623
178.1525
6th
99.5237
100.9778
103.9269
108.5284
120.9391
137.5384
155.5090
174.2894
193.5633
213.0626
232.5021
250.6452Figure 12
A Campbell diagram of the FG sandwich blade (l¯=3, h¯=20, n=2, h¯l=1/2, θ=30°, r¯=1, and φ=45°).Figure13 shows the effects of the stagger angle for the first six nondimensional frequency parameters with varying rotation velocities. It can be found that the stagger angle has a predominant influence in the 1st and 2nd modes, and the effect is very marginal in higher modes. The effect of the stagger angle has little influence on all frequencies when the rotation velocity is low. The reason is that the centrifugal force has little effect on the stiffening and softening when the rotating speed is low, and the influence becomes more and more obvious with the rotating speed increasing.Figure 13
Variations of the first six frequency parameters with the rotation velocity and stagger angles (l¯=3, h¯=20, n=2, h¯l=1/2, θ=30°, and r¯=1). (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.1. Convergence Analysis

A typical example of pretwisted blade with chord lengthb=0.254m, aspect ratio l¯=L/b from 1 to 5, thickness ratio h¯=b/h from 10 to 100, twist angle θ from 0° to 60°, layer thickness ratio h¯l=h−hm/2hm from 0.25 to 4, scalar parameter of volume fraction of FGM n from 0.1 to 10, stagger angle φ from 0° to 90°, and disk radius ratio r¯=R/b is used.All the numerical results are presented in the form of the nondimensional frequency parameterω¯=ωNL2ρh/D=ωNL2/h12ρ1−μ2/E for the homogeneous material blade, where D=Eh3/121−μ2 is the blade flexural rigidity per unit length and ω¯=ωNL2/h12ρm1−μm2/Em for the functionally graded sandwich blade in the following section, and the nondimensional rotation velocity is Ω¯=Ω/ω1, where ω1 is the first static circular frequency (Ω=0) of the functionally graded sandwich blade.According to [5], the number of terms chosen for each component of the displacements is the same. The results of the homogeneous material blade and functionally graded sandwich blade are given in Tables 1 and 2. The maximum difference between the 5×9×9=405 term and the 5×8×8=320 term solutions for the first six modes of the homogeneous material blade is no more than 0.28%, and 0.21% for the FG sandwich blade. But the time spent in computing has multiplied. Moreover, the eigenvalues are more likely to be ill-conditioned with the increase in the size of the matrix in numerical calculation. Therefore, it has been decided to use 5×8×8=320 terms in the subsequent analysis.Table 1
Convergence of nondimensional frequency parameters for the first six modes of the homogeneous material blade (l¯=3, h¯=20, and θ=15°).
Mode
ANSYS
5 × 5 × 5
5 × 6 × 6
5 × 7 × 7
5 × 8 × 8
5 × 9 × 9
Value
Error
Value
Error
Value
Error
Value
Error
Value
Error
1st
3.4225
3.4217
−0.02%
3.4102
−0.36%
3.4077
−0.43%
3.4194
−0.09%
3.4124
−0.30%
2nd
20.7177
20.7606
0.21%
20.7156
−0.01%
20.7222
0.02%
20.7540
0.18%
20.7270
0.04%
3rd
21.9608
22.2650
1.39%
21.9718
0.05%
22.1278
0.76%
22.0861
0.57%
22.0340
0.33%
4th
58.8524
59.6322
1.33%
54.3477
−7.65%
58.1334
−1.22%
58.1917
−1.12%
58.0295
−1.40%
5th
64.5829
66.5017
2.97%
64.0481
−0.83%
65.6567
1.66%
65.4872
1.40%
65.6406
1.64%
6th
68.0114
68.9640
1.40%
68.1982
0.27%
68.6168
0.89%
68.4144
0.59%
68.3678
0.52%Table 2
Convergence of nondimensional frequency parameters for the first six modes of the FG sandwich blade (l¯=3, h¯=20, n=2, h¯l=1/2, and θ=15°).
Mode
ANSYS
5 × 5 × 5
5 × 6 × 6
5 × 7 × 7
5 × 8 × 8
5 × 9 × 9
Value
Error
Value
Error
Value
Error
Value
Error
Value
Error
1st
4.5272
4.5320
0.11%
4.5209
−0.14%
4.5228
−0.10%
4.5211
−0.13%
4.5215
−0.13%
2nd
27.3966
27.5019
0.38%
27.4741
0.28%
27.4639
0.25%
27.4684
0.26%
27.4665
0.26%
3rd
29.0406
29.5393
1.72%
29.3030
0.90%
29.3432
1.04%
29.2158
0.60%
29.1542
0.39%
4th
76.2376
75.8763
−0.47%
70.8471
−7.07%
74.8356
−1.84%
74.8922
−1.76%
75.0109
−1.61%
5th
80.3331
84.3843
5.04%
78.8139
−1.89%
82.1073
2.21%
82.2715
2.41%
82.2392
2.37%
6th
89.9847
91.5581
1.75%
90.9359
1.06%
91.0424
1.18%
90.7128
0.81%
90.5917
0.67%The data used for comparison below “ANSYS” in all tables are computed by the finite element model in ANSYS Workbench (version 17.2). The middle surface model of the blade is established, and the finite element analysis is performed by using the SHELL181 linear quadrilateral element type. The layered section object in ANSYS mechanical is used to define the layered section of composite modeling. In order to get accurate results as much as possible, 20 layers in each skin (total of 41 layers in the thickness direction) are defined to simulate the cross section of the functionally graded sandwich blade.


## 4.2. Comparative Studies

The homogeneous material blade is used first to demonstrate the numerical accuracy of the analytical model in this paper, and the first six nondimensional frequency parameters are compared with ANSYS and reference data [15] in Table 3. The results are in good agreement when the twist angle θ=0° to 45° in the case of the aspect ratio l¯=3. The results match well also when the twist angle θ=0° to 30° in the case of the aspect ratio l¯=1, but the maximum error (occurring at the 2nd order) is about 8.86% when θ increases to 45°. The reason is that the Lame parameter A is approximated to 1 in order to improve the calculation speed in the process of numerical calculation. The error is smaller in the case of a larger aspect ratio and smaller twist angle, but the error becomes larger as the aspect ratio decreases and the twist angle increases. Thus, the aspect ratio l¯=3 is used to obtain more accurate results.Table 3
Comparison of nondimensional frequency parameters for the first six modes of homogeneous material blade (h¯=20, μ=0.3).
θ
Mode
l¯=3
l¯=1
1st
2nd
3rd
4th
5th
6th
1st
2nd
3rd
4th
5th
6th
0°
Present
3.4205
20.8483
21.3316
59.6783
62.2910
65.4353
3.4757
8.3462
20.9625
26.6650
30.1067
43.5409
ANSYS
3.4223
20.8230
21.3590
59.8793
62.2508
65.3690
3.4814
8.3793
21.0993
26.7267
30.2352
43.5819
Ref [15]
3.4140
21.2200
21.3600
60.1200
61.8200
66.6800
3.4620
8.4870
21.2900
27.4700
30.9800
43.5200
15°
Present
3.4194
20.7540
22.0861
58.1917
65.4872
68.4144
3.4479
10.4233
20.4526
26.6975
31.5336
44.6423
ANSYS
3.4225
20.7177
21.9608
58.8524
64.5829
68.0114
3.4540
10.3394
20.3668
26.6264
31.3218
44.4679
Ref [15]
3.4150
20.6700
22.2800
58.1500
65.2900
69.3200
3.4450
10.4000
20.5700
27.3500
32.1100
44.5800
30°
Present
3.4035
19.2320
25.4184
54.4781
70.6496
76.3746
3.4030
14.8939
18.9948
27.1437
35.2177
46.9877
ANSYS
3.4153
18.9950
25.0621
55.7643
71.6994
75.9291
3.3794
14.3509
18.4374
26.7434
34.0572
45.9808
Ref [15]
3.4170
19.0000
25.1500
55.9500
71.4000
76.5700
3.3970
14.3300
18.6900
27.3900
34.9000
46.5200
45°
Present
3.4163
17.4172
29.7580
54.0239
80.8963
87.6963
3.3387
20.0628
17.1876
28.2967
40.1453
48.6332
ANSYS
3.4176
16.9599
29.4234
53.3063
79.0586
87.0233
3.2668
18.4291
16.0421
27.1718
37.3254
46.1459
Ref [15]
3.4200
16.9900
29.2000
53.5800
78.8200
87.0500
3.3230
18.3200
16.3900
27.7100
38.1700
47.3000The functionally graded sandwich blade is composed of metal and ceramic materials; their properties are given in Table4. In Table 5, the first six nondimensional frequency parameters ω¯ of the FG sandwich blade are compared with results in ANSYS Workbench, both showing good agreements, respectively.Table 4
The materials and properties of the FG sandwich blade.
Materials
Material properties
Metal (alloy C-276)
Em=2.08×1011Pa,μm=0.32,ρm=8900kg/m3
Ceramic (Al2O3)
Ec=3.50×1011Pa,μc=0.26,ρc=3970kg/m3Table 5
Comparisons of nondimensional frequency parameters for the first six modes of the FG sandwich blade (l¯=3, h¯=20, n=2, and h¯l=1/2).
Mode
0°
15°
30°
Present
ANSYS
Error
Mode shape
Present
ANSYS
Error
Mode shape
Present
ANSYS
Error
Mode shape
1st
4.5233
4.5270
−0.08%
1B
4.5206
4.5272
−0.15%
1B
4.4729
4.5290
−1.24%
1B
2nd
27.9633
27.7729
0.69%
1 T
27.4621
27.3966
0.24%
2B
25.4167
25.0992
1.26%
2B
3rd
28.3106
28.2605
0.18%
2B
29.1565
29.0406
0.40%
1T
33.0633
32.5549
1.56%
1T
4th
75.9386
75.8281
0.15%
1 EB
74.8841
76.2376
−1.78%
3B
73.0937
72.9145
0.25%
3B
5th
79.3035
79.1454
0.20%
3B
82.0284
80.3331
2.11%
1EB
89.2792
88.7755
0.57%
1EB
6th
87.5852
87.0515
0.61%
2 T
90.5827
89.9847
0.66%
2T
99.5237
98.9166
0.61%
2TThe mode shapes corresponding to the first six orders of frequencies are also listed in Table5. It should be noted that spanwise bending modes are divided into flapwise bending modes (whose displacements are predominantly in the direction normal to the middle surface) and edgewise bending modes (whose displacements are predominantly in the direction tangent to the surface) [15] in this paper. The modes 1B and 2B represent the first and second flapwise bending modes, respectively, and 1EB represents the first edgewise bending mode. The modes 1T and 2T represent the first and second torsional modes, respectively. The chordwise bending modes (CB, having two or more nodal lines approximately parallel to the pretwisted blade axis) are also found in the successive research. The curves such as 1B, 1T, and 2B and so on in Figures 6–8 are all obtained by using ANSYS Workbench. The curves such as 1st, 2nd, and 3rd are all obtained through numerical calculation.Figure 6
Variations of nondimensional frequency parameters with the twist angle (l¯=3, h¯=20, n=2, and h¯l=1/2).
(a)
1st to 3rd modes
(b)
4th to 6th modesFigure 7
Variations of nondimensional frequency parameters with the thickness ratio (l¯=3, n=2, h¯l=1/2, and θ=30°).Figure 8
Variations of nondimensional frequency parameters with the aspect ratio (h¯=20, n=2, h¯l=1/2, θ=30°).
(a)
1st to 3rd modes
(b)
4th to 6th modesThe first six mode shapes of the FG sandwich blade withl¯=3, h¯=20, n=2, h¯l=1/2, and θ=30∘ are given in Figure 9. It can be seen that the results obtained by the two methods are in good agreement. The results in Table 5 show that that the 1st modes are both of 1B mode shape when θ=0°, 15°, and 30°, but the 2nd mode is of 1T mode shape when θ=0° and 2B mode shape when θ=15° and 30°. The same happens with the 1EB mode shape. The reason for this phenomenon is that the stiffness of the FG sandwich blade is changed due to the initial twist in (15).Figure 9
Mode shapes of the FG sandwich blade. (A1) 1st mode shape by Matlab; (B1) 2nd mode shape by Matlab; (C1) 3rd mode shape by Matlab; (D1) 4th mode shape by Matlab; (E1) 5th mode shape by Matlab; (F1) 6th mode shape by Matlab; (A2) 1st mode shape by ANSYS; (B2) 2nd mode shape by ANSYS; (C2) 3rd mode shape by ANSYS; (D2) 4th mode shape by ANSYS; (E2) 5th mode shape by ANSYS; (F2) 6th mode shape by ANSYS.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.3. Free Vibrations of the FG Sandwich Blade

The effect of the twist angle on the vibration characteristics for the pretwist FG sandwich blade with geometric parameters asl¯=3, h¯=20, n=2, and h¯l=1/2 is studied. Table 6 gives the first six nondimensional frequency parameters ω¯ of the FG sandwich blade with varying twist angles (θ from 0° to 60°), and the relationship between the nondimensional frequency parameters and twist angles for the FG sandwich blade is shown in Figure 6. It can be seen that the resulting curves are very close to the curves obtained by ANSYS in Figure 6(a). The first mode corresponds to the 1B vibration mode for all twist angles. The 2nd and 3rd nondimensional frequency parameter curves represent the 1T and 2B mode shapes when θ=0°, and they come close to each other and separate gradually with the twist angle increasing. The two curves are up to the nearest when the twist angle θ is about 5°; the two modes exchange with each other, and the variation tendencies of nondimensional frequency parameters are also interchanged. The variations for 4th to 6th can be found in Figure 6(b).Table 6
Nondimensional frequency parameters for the first six modes of FG sandwich blade with varying twist angle (l¯=3, h¯=20, n=2, and h¯l=1/2).
Mode
0°
5°
10°
15°
20°
25°
30°
45°
50°
60°
1st
4.5233
4.5342
4.5342
4.5206
4.5041
4.4935
4.4743
4.4020
4.3605
4.3245
2nd
27.9633
28.0799
27.9178
27.4621
26.8716
26.1715
25.4167
22.9643
22.1860
20.7649
3rd
28.3106
28.2050
28.5154
29.1565
29.9955
31.3439
33.0633
38.0040
39.9970
44.6692
4th
75.9386
75.7447
75.3807
74.8841
74.5120
73.8517
73.0937
70.8379
70.1630
68.9817
5th
79.3035
79.7009
80.7576
82.0284
84.0329
86.7774
89.2792
98.6081
101.9785
107.9219
6th
87.5852
87.9114
88.9680
90.5827
92.8839
95.9725
99.5237
112.9271
118.0741
130.0957It should be noted that the FG sandwich blade designed by the geometric parameters corresponding to the frequency turning points will be prone to internal resonance.The effect of the thickness ratio on the vibration characteristics for the pretwist FG sandwich blade is studied also. The first six nondimensional frequency parametersω¯ of the FG sandwich blade with varying thickness ratios (h¯ from 10 to 100) is given in Table 7. Figure 7 shows the relationship between the nondimensional frequency parameters and the thickness ratio. It can be found that the nondimensional frequency parameters of 1B, 2B, 3B, and 4B modes remain essentially unchanged when the thickness ratio varies from 10 to 100. The nondimensional frequency parameters of 1T, 2T, and 1EB modes tend to increase with increasing thickness ratio. On the other side, the 3rd and 4th frequency parameter loci have an obvious veering at h¯=85, and the mode shapes exchanged between 1T and 3B. The same situation can be seen on the 5th and 6th modes, and higher-order frequency parameter locus veering happens more frequently.Table 7
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying thickness ratio (l¯=3, n=2, h¯l=1/2, and θ=30°).
Mode
10
20
30
40
50
60
70
80
90
100
1st
4.4578
4.4743
4.4745
4.4759
4.4903
4.5064
4.5022
4.5115
4.5146
4.5213
2nd
24.2636
25.4167
25.5518
25.6696
25.7090
25.7337
25.7686
25.8024
25.8433
25.8342
3rd
28.2468
33.0633
38.1941
44.8357
51.5288
59.2127
67.3645
74.2099
77.6186
77.2360
4th
43.9320
73.0937
75.9033
76.1792
76.4980
76.7083
76.7618
76.3814
82.7484
91.1187
5th
76.6391
89.2792
113.8445
130.9914
147.7835
151.9612
156.271
159.8808
159.7313
159.6019
6th
87.7677
99.5237
126.202
147.8068
155.1631
169.1361
191.4968
207.7979
230.7914
250.1723The nondimensional frequency parametersω¯ for the first six modes of the FG sandwich blade with varying aspect ratios (l¯ from 1 to 5) are given in Table 8 to study the effect of the aspect ratio on the vibration characteristics for the pretwist FG sandwich blade. Figure 8 shows the relationships between the first six frequency parameters, the corresponding mode shapes, and the aspect ratio. It is obvious that the 1st mode corresponds to the 1B vibration mode for all aspect ratios, and the value of the frequency parameter varies little. The 2nd and 3rd modes exchange with each other, and the vibration tendencies of their frequency loci also exchange near l¯=2. The change of the 4th to 6th nondimensional frequency parameters is relatively complicated. The mode shapes include 3B, 2T, 1EB, and 1CB. For example, the 4th to 6th mode shapes are 1CB, 2T, and 1EB when l¯=1; 2T, 1EB, and 1CB when l¯=1.5; and 1EB, 2T, and 3B when l¯=1.75. All in all, the mode shape corresponding to the same order frequency is very sensitive to the variation of small aspect ratios, especially when l¯<2. The tendencies of their frequency parameters become relatively stable until l¯>2.5.Table 8
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying aspect ratio (h¯=20, n=2, h¯l=1/2, and θ=30°).
Mode
1.00
1.25
1.50
1.75
2.00
2.25
2.50
3.00
4.00
5.00
1st
4.4970
4.5187
4.4946
4.4879
4.4857
4.4847
4.4762
4.4743
4.4643
4.4475
2nd
18.7284
20.8402
22.6114
24.2429
25.4037
25.4146
25.4190
25.4167
25.4069
25.3682
3rd
25.0639
25.2854
25.3453
25.3845
25.9349
27.5549
29.2911
33.0633
40.4134
48.5665
4th
35.6083
49.9937
61.8109
68.2541
70.4597
71.6661
72.3847
73.0937
73.8139
73.9446
5th
45.7901
53.9883
63.0533
69.0347
75.8721
82.0019
87.1364
89.2792
91.5483
93.0731
6th
58.3492
65.8222
76.2076
82.7443
84.4782
85.9505
88.1750
99.5237
123.1095
147.6281Table9 and Figure 10 show the variations of the first six nondimensional frequency parameters of the FG sandwich blade with the layer thickness ratio. Table 10 and Figure 11 show the relationships between the first six nondimensional frequency parameters and the scalar parameter of the volume fraction. It can be found that the content of ceramic increases with increasing layer thickness ratio and scalar parameter of the volume fraction, and the frequency parameters of all modes increase, respectively.Table 9
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying layer thickness ratios (l¯=3, h¯=20, n=2, and θ=30°).
Mode
1/4
1/3
1/2
1/1
2/1
3/1
4/1
1st
4.2021
4.3068
4.4743
4.7574
4.9764
5.0854
5.1378
2nd
23.6280
24.3536
25.4167
27.0889
28.3771
28.9429
29.2352
3rd
30.4576
31.3180
33.0633
35.0677
36.7356
37.6719
38.4396
4th
68.0698
70.1619
73.0937
78.1350
81.7252
83.5669
84.7660
5th
83.1451
85.6584
89.2792
95.5154
100.6206
103.1477
104.3400
6th
92.4498
95.1984
99.5237
106.5238
111.4889
114.2924
116.4530Figure 10
Variations of nondimensional frequency parameters with the layer thickness ratio (l¯=3, h¯=20, n=2, and θ=30°).Table 10
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying scalar parameters of volume fraction (l¯=3, h¯=20, h¯l=1/2, and θ=30°).
Mode
0.0
0.1
0.2
0.3
0.4
0.5
1.0
2.0
3.0
4.0
5.0
1st
3.3606
3.5795
3.6493
3.8619
3.8665
3.9753
4.2248
4.4743
4.5659
4.6931
4.7617
2nd
19.1226
20.0849
20.7619
21.4358
21.9111
22.3903
23.8825
25.4167
26.1362
26.5972
26.8850
3rd
24.9882
26.0961
26.9426
27.6920
28.2425
28.7613
30.7357
33.0633
33.7313
34.2049
34.8816
4th
55.8597
58.2781
60.3344
62.0908
63.5612
64.7114
68.9569
73.0937
75.1927
76.5550
77.6652
5th
71.5372
73.7012
75.7393
77.3010
78.8343
80.0439
84.5755
89.2792
91.5699
93.1665
93.7638
6th
75.5421
79.2376
81.6705
84.0076
85.8903
86.6503
93.4294
99.5237
102.7955
104.0268
106.0757Figure 11
Variations of nondimensional frequency parameters with the scalar parameter of volume fraction (l¯=3, h¯=20, h¯l=1/2, and θ=30°).Table11 shows the first six nondimensional frequency parameters of the FG sandwich blade for the nondimensional rotation velocity with the range of Ω¯ 0 to 11 (about 15,000 rpm). A typical Campbell diagram is plotted to study the variation trend of the nondimensional frequency parameters for varying rotation velocities in Figure 12. Frequency locus veering is observed as expected: the 1st and 2nd frequency modes exchange between 1B and 1T around the rotation velocity Ω¯=11; the 2nd and 3rd frequency modes exchange between 2B and 1T around Ω¯=2; the 5th and 6th frequency modes exchange between 1EB and 2T around Ω¯=3; and the 7th and 8th frequency modes exchange between 4B and 3T around Ω¯=1 and exchange between 3T and 1CB again around the rotation velocity Ω¯=9. In addition, the 8th frequency locus can be seen veering at Ω¯=4 and the mode shape is changed from 4B to 1EB. It can be seen clearly that higher-order frequencies tend to have multiple frequency veering with the mode shapes exchanging and coupling.Table 11
Nondimensional frequency parameters for the first six modes of the FG sandwich blade with varying nondimensional rotation velocities (l¯=3, h¯=20, n=2, h¯l=1/2, θ=30°, r¯=1, and φ=45°).
Mode
0
1
2
3
4
5
6
7
8
9
10
11
1st
4.4743
6.9030
11.2340
15.8419
20.5025
25.1652
29.8098
34.4146
38.9622
43.4424
47.8358
52.1334
2nd
25.4167
28.5235
34.0383
35.1206
36.4308
38.1374
40.2129
42.4223
44.6896
47.3581
50.0889
52.8240
3rd
33.0633
33.0979
35.8589
45.2978
55.2322
64.5633
72.3156
78.0880
82.3864
85.9246
89.1536
92.3135
4th
73.0937
76.2806
80.9215
84.9748
88.5518
92.9797
99.3391
108.0164
118.4814
130.0075
142.1141
154.5548
5th
89.2792
92.7844
96.5525
106.6897
114.5673
121.7465
129.8789
138.6539
147.8984
157.7082
167.8623
178.1525
6th
99.5237
100.9778
103.9269
108.5284
120.9391
137.5384
155.5090
174.2894
193.5633
213.0626
232.5021
250.6452Figure 12
A Campbell diagram of the FG sandwich blade (l¯=3, h¯=20, n=2, h¯l=1/2, θ=30°, r¯=1, and φ=45°).Figure13 shows the effects of the stagger angle for the first six nondimensional frequency parameters with varying rotation velocities. It can be found that the stagger angle has a predominant influence in the 1st and 2nd modes, and the effect is very marginal in higher modes. The effect of the stagger angle has little influence on all frequencies when the rotation velocity is low. The reason is that the centrifugal force has little effect on the stiffening and softening when the rotating speed is low, and the influence becomes more and more obvious with the rotating speed increasing.Figure 13
Variations of the first six frequency parameters with the rotation velocity and stagger angles (l¯=3, h¯=20, n=2, h¯l=1/2, θ=30°, and r¯=1). (a) 1st mode, (b) 2nd mode, (c) 3rd mode, (d) 4th mode, (e) 5th mode, and (f) 6th mode.
(a)
(b)
(c)
(d)
(e)
(f)


## 5. Conclusion

In this paper, a dynamic model based on the thick shell theory is developed to investigate the free vibration behavior of functionally graded sandwich pretwisted blades. The validation of the homogeneous material and FG sandwich blade is performed by comparison to the literature and ANSYS results, both showing good agreement.Interesting frequency veering and the associated mode shift phenomena are found and discussed for different twist angles, thickness ratios, and aspect ratios in static natural frequency analysis. The results show that the fundamental mode is flapwise bending in every case. The mode shapes of the 2nd to 6th orders are changing when different geometric parameters are used; especially, the thickness ratio is 15–50 and the aspect ratio is 1.0–2.5. Higher-order frequency loci (≥4) may veer more than once between the bending and torsional vibrations of the FG sandwich blade.The stagger angle is considered to study the dynamic frequency with varying rotation velocities. The results show that the frequency modes have more complicated shapes and stronger coupling in the rotating FG sandwich blade system. The Campbell diagram reveals that the rotating velocity, especially high rotating velocity, has a significant impact on the variations of frequency loci and mode shapes.



---

*Source: 2727452-2018-09-16.xml*