1 Introduction

Rotating structures are key components in many dynamic engineering systems, e.g., propellers, compressor blades, and turbine blades. With the involvement of additional centrifugal force and Coriolis effects, the rotating beams differ significantly from the non-rotating beams in fundamental theory and application conditions. For these reasons, it is of paramount importance to properly and accurately evaluate the modal characteristics of these rotating structures for engineering design purposes.

Earlier related works all focused on the centrifugal forces caused by the rotating velocity ^[1-4]. With the increase in the rotating velocity of modern high speed machines, the effects of the Coriolis forces become more and more important. In fact, the Coriolis forces provide gyroscopic couplings in different directions. Surace et al. ^[5] studied the bending-bending-torsion vibration of rotating blades with the Euler beam model. Liu et al. ^[6] discussed the axial-torsional coupled vibration of a pre-twisted beam. Ghafarian and Ariaei ^[7] investigated a rotating tapered Timoshenko beam with the differential transform method, and focused on the bending vibrations. With the Galerkin method, Huo and Wang ^[8] obtained the natural frequencies of a rotating double-tapered Timoshenko beam. Huang and Zhu ^[9] studied the nonlinear dynamics of a rotating Euler-Bernoulli beam.

The governing equations of the rotating slender bodies involve the position-dependent parameters due to the non-uniform distribution of the centrifugal forces along the axial direction. Hence, the exact solutions are not available. Approximate solutions and numerical results are other options to study the dynamics of such rotating materials. The Galerkin method ^[10-11] is one of the most used methods for the partial differential equations of rotating bodies. Banerjee et al. ^[12], Banerjee and Kennedy ^[13], and Banerjee ^[14-15] verified that the dynamic stiffness method was an efficient method for the study of rotating beams. Chung and Yoo ^[16] and Hashemi and Richard ^[17] used the finite element method to study rotating beams. Du et al. ^[18] proposed the power series method for the study of rotating materials. Compared with other methods, the power series expansion method is more suitable to solve the differential equations with variable coefficients.

In this study, we first construct a full mathematical model of the rotating tapered Timoshenko beam with preset and pre-twist angles. Then, with the power series method, we investigate the natural frequencies and mode shapes, and focus on the centrifugal and gyroscopic effects.

2 Mathematical modeling

In this study, a tapered Timoshenko beam with the preset angle α and the pre-twist angle β(x) fixed on a rotating hub with the radius r_h is investigated (see Fig. 1). To well describe the beam deformation, several coordinate frames are introduced. The non-inertial frame OXYZ is rotating with respect to the central line of the hub OZ by the velocity Ω. OX denotes the neutral line of the beam. The other rotating frame oxyz originates on the fixed end of the beam. There exists an angle α between OXYZ and oxyz, which exhibits the preset angle of the beam. For an arbitrary cross-section in the beam, a local rotating frame o₀x₀y₀z₀ is used to show the normal directions of the rectangular section. The pre-twisted angle between oxyz and o₀x₀y₀z₀ is assumed to vary linearly with the beam length, i.e.,

(1)

It is assumed that both the width and the height of the beam are tapering linearly, i.e.,

(2)

where b₀ and h₀ are the width and height at the beam root, respectively, and c_b and c_h are the tapering ratios.

The degrees of freedom of the system comprise 6 elements, i.e., the axial displacement u in the x-direction, two orthogonal lateral deflections of the beam centroid v and w which are defined on the y₀-and z₀-directions, respectively, the twist angle θ, and the rotational angles ψ and φ due to bending and transverse shearing along the y₀-and z₀-directions, respectively.

First, we use a sequence of Euler angles to measure the three deformed angles of an arbitrary cross-section on the rotating frame o₀x₀y₀z₀. The cross section rotations, which are described by sequence of the Euler angles as θ(x, t), φ (x, t), and ψ (x, t), are presented in Fig. 2. With the undeformed plane, the cross section first rotates about the x₀-axis by θ (twist), then by φ about the updated z₀-axis (with flapwise bending), and finally by ψ about the updated y₀-axis (with edgewise bending). Hence, the normal line to the cross section is mapped to the plane perpendicular to the deformed cross section on the local orthogonal curvilinear coordinate system x₃y₃z₃. The unit vectors on different coordinates in Fig. 2 are related by

(3)

It should be noted that the angle η is a sum of two components, i.e., the pre-twisted angle β (x) and the twist Euler angle θ.

Therefore, the transformation matrix Q can be obtained as follows:

(4)

which gives the relation between the oxyz-frame of the undeformed cross-section and the o₀x₃y₃z₃ -frame of the deformed cross-section, i.e.,

(5)

The relative angular velocity of the x₁y₁z₁-frame with respect to x₀y₀z₀ is , and the relative angular velocities and are x₂y₂z₂ to x₁y₁z₁ and x₃y₃z₃ to x₂y₂z₂, respectively. With Eq. (3) and by some manipulations, we obtain the angular velocity of the local frame x₃y₃z₃ to the inertial frame as follows:

(6)

Then, the linear displacements of an arbitrary point in the beam are studied. The position of an arbitrary point is denoted by a vector R, which is the sum of R₀ on the rotating xyz-frame and R₁ on the local x₃y₃z₃-frame, i.e.,

(7)

where u, v, and w denote the axial, flapwise, and edgewise deformations of the central line, respectively.

The velocity of this point can be obtained by the first derivative of Eq. (7) with respect to time, i.e.,

(8)

where ω=(ω₁, ω₂, ω₃)^T, and

(9)

Since the deformed angles θ(x, t), φ (x, t), and ψ (x, t) are assumed to be small, in the subsequent manipulations, the following linear approximations are used:

(10)

Then, the overall kinetic energy is

(11)

where the density of the beam material is ρ.

Moreover, the potential energy of the Timoshenko beam is

(12)

It should be noted that the cross-sectional area and the inertia moments vary with the beam length due to the tapering and pre-twist effects. The cross-sectional area and the inertia moments normal to the rectangular cross section are

(13)

where A₀, I₁₀, and I₂₀ are the values measured on the root of the beam.

Since the vibrations are measured on the oxyz-frame, the local inertia moments should be transferred from o₀x₃y₃z₃ to oxyz, i.e.,

(14)

The following geometric equations are assumed:

(15)

where ε, κ₁, κ₂, γ₁, and γ₂ represent the membrane strain, the flapwise bending curvature, the edgewise bending curvature, the flapwise shear strain, and the edgewise shear strain, respectively.

Substituting Eq. (15) into Eq. (12) and substituting the resulted potential energy and kinetic energy (11) into the Hamilton principle

(16)

yield the following six nonlinear partial differential equations governing the three central line displacements in the axial, flapwise, and edgewise directions and the three cross-sectional angles of torsion, flapwise bending, and edgewise bending:

(17)

(18)

(19)

(20)

(21)

(22)

3 Static deformations and linearized governing equations

The constant terms appear in Eqs. (17) and (20) governing the axial displacement and twist may result in static deformation in the final solutions. The distributed static axial deformation u_s(x) and distributed static twist angle θ_s(x) are introduced in the solutions, i.e.,

(23)

where u_d(x) and θ_d(x) denote the dynamic parts of the solutions, which account for the vibrations of the corresponding directions.

The static deformations can be obtained from Eqs. (17) and (20) by neglecting the time-dependent terms. Substituting Eq. (23) back into Eqs. (17) -(22) and inserting the dimensionless variables and parameters

(24)

yield the following equations governing the vibrations of six degrees of freedom:

(25)

(26)

(27)

(28)

(29)

(30)

where the over bars are omitted without causing any ambiguity. When k≪1, the axial static deformation related tension N is

(31)

Because the static deformations are known, some nonlinear terms in Eqs. (17) -(22) present as the linear terms in Eqs. (25) -(30), which makes the linear methods available. In Eqs. (25) -(30), the terms proportional to the rotating speed k are gyroscopic terms, and those proportional to the square of the rotating speed are centrifugal terms. From Eq. (25), we can conclude that if the preset angle α equals 0, the axial motion u is gyroscopically coupled with the flapwise bending motion v. If the preset angle α equals 90°, the axial motion u is gyroscopically coupled with the edgewise bending motion w. From Eq. (28), we can conclude that if the sum of the preset and pre-twist angles equals 0°, the torsional motion is gyroscopically coupled with the edgewise bending motion. If the sum of the preset and pre-twist angles equals 90°, the torsional motion is gyroscopically coupled with the flapwise bending motion. It should be noted that, without the introduction of the pre-twist angle β, two separated sets (u, v, φ) and (θ, w, ψ) are gyroscopically coupled, respectively, for α =0. For a 90° preset angle, (u, w, ψ) and (θ, v, φ) are gyroscopically coupled, respectively. When the pre-twist angle β is introduced, the six degrees will be coupled altogether.

4 Natural frequencies and mode shapes by power series method

In this section, the power series method will be used to study the governing equations with six degrees of freedom. Since the governing equations are coupled by the gyroscopic terms, the complex modes should be considered. It is assumed that the solutions are

(32)

where K is the dimensionless natural frequency, and

(33)

Substituting Eq. (32) into Eqs. (25) -(30) and separating the real and imaginary parts lead to the following two equations in vector form:

(34)

(35)

where

(36)

Approximate power series are used in the mathematical manipulations for the case of c₀ =c_h =c_b. It can be found that

(37)

Hence, it is concluded that the transverse deflections v and w have a π/2 phase difference with respect to the axial deflection due to gyroscopic coupling. Similarly, the rotating deformations φ and ψ also have a π/2 phase difference with respect to the twist θ.

Either Eq. (34) or Eq. (35) can be used to yield all the solutions. Based on Eq. (34), the matrices in the power series form are

(38)

where P₀, P₁, P₂, Q₀, Q₁, Q₂, Q₃, S₀, S₁, S₂, S₃, and S₄ are constant real-value matrices. The solution to Eq. (34) is

(39)

where C_1n, C_2n, C_3n, C_4n, C_5n, and C_6n are coefficients to be determined. In the above equation, C₀ and C₁ are independent matrices, and C_n(n≥ 2) can be derived by a sequence of recurrences as follows:

(40)

where the matrices with negative subscripts are assumed to be zero.

Since only C₀ and C₁ are independent matrices, C_n (n≥ 2) can be rewritten as follows:

(41)

Moreover, considering the boundary conditions of the fixed end

(42)

in Eq. (41), we have C₀=0.

Substituting Eq. (41) and C₀=0 into Eq. (39) yields

(43)

where

(44)

In the above equation,

(45)

Now, the free end boundary conditions

(46)

require

(47)

Based on the zero determinant of the coefficient matrix of Eq. (47) and the solutions of C₁, we can determine the natural frequencies and mode functions.

4.1 Natural frequencies

The first two orders of flapwise bending frequencies K₁^v and K₂^v, the first two orders of edgewise bending frequencies K₁^w and K₂^w, the first torsional frequency K₁^θ, and the first axial frequency K₁^u of the rotating tapped Timoshenko beam with the preset and pre-twist angles are listed in Tables 1 and 2 for different parameters.

Set

Then, we can obtain the natural frequencies with varying rotating velocity k for different preset angles (see Fig. 3). The twist frequency decreases slightly with the increase in the rotating velocity. The frequencies of other directions increase due to the centrifugal effect.

The contribution of the tapering ratio to the natural frequencies is drastic. It can be found from Table 1 that the tapering increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequencies.

The effect of the pre-twist angle appears complicated although it is weak (see Table 2). From the governing equations (25) -(30), we can see that the pre-twist angle changes the cross-sectional inertia and leads to gyroscopic couplings among the angles, even the preset angle is not considered.

4.2 Mode shapes

The mode shapes of every direction for different natural frequencies are plotted in Fig. 4, where the parameters are

Because of the involvement of the preset angle and the pre-twist angle, all the degrees of vibrations are gyroscopically coupled. However, on a particular natural frequency, the vibration in one direction dominates, which is also the reason why the corresponding direction term is used to denote such modes. For example, for the first flapwise mode, the flapwise motion is the maximum motion among the others.

5 Conclusions

In this study, the mathematical modeling of a rotating tapered Timoshenko beam with the preset and pre-twist angles is constructed. The partial differential equations governing the six degrees, i.e., three displacements in the axial, flapwise, and edgewise directions and three cross-sectional angles of torsion, flapwise bending, and edgewise bending, are obtained by the Euler angle manipulations. The power series method is then used to investigate the natural frequencies and the corresponding complex mode functions. Some interesting conclusions are made based on the numerical results.

(ⅰ) Due to the rotating velocity, most of the natural frequencies are increased by the centrifugal stiffening. However, the twist frequency is slightly decreased.

(ⅱ) The tapering increases the first transverse, torsional, and axial frequencies, while decreases the second transverse frequency.

(ⅲ) Because of the pre-twist, all the directions are gyroscopically coupled. The phase differences among the six degrees are located.