1 Introduction. Minho Lee 1 Jihoon Lee 1 Gunhee Jang 1

Transcription

1 DOI /s TECHNICAL PAPER Stability analysis of a whirling rigid rotor supported by stationary grooved FDBs considering the five degrees of freedom of a general rotor bearing system Minho Lee 1 Jihoon Lee 1 Gunhee Jang 1 Received: 30 September 01 / Accepted: 7 March 015 Springer-Verlag Berlin Heidelberg 015 Abstract This paper proposes a method to determine the stability of a whirling rotor supported by stationary grooved fluid dynamic bearings (FDBs), considering the five degrees of freedom of a general rotor-bearing system. Dynamic coefficients are calculated by using the finite element method and the perturbation method, and they are represented as periodic harmonic functions. Because of the periodic time-varying dynamic coefficients, the equations of motion of the rotor supported by FDBs can be represented as a parametrically excited system. The solution of the equations of motion can be assumed to be a Fourier series, allowing the equations of motion to be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Hill s infinite determinant is calculated by using these algebraic equations to determine the stability. Increasing rotational speed increases K xx and K θx θ x, which decreases the stability of the stationary grooved FDBs; increasing whirl radius increases the stability of the FDBs because the resulting increases in the averages and variations of C xx and C θx θ x increase the stability faster than the corresponding increases of K xx and K θx θ x decrease the stability. The proposed method was verified by investigating the convergence and divergence of the whirl radius after the equations of motion were solved using the fourth-order Runge Kutta method. * Gunhee Jang ghjang@hanyang.ac.kr 1 PREM, Department of Mechanical Convergence Engineering, Hanyang University, 17 Haengdang dong, Seongdong gu, Seoul , Republic of Korea 1 Introduction Fluid dynamic bearings (FDBs) generally have less vibration and noise than ball bearings because they not only prevent direct contact between the rotating and stationary parts through an oil film but also provide a damping effect. A rotor supported by FDBs generally has a whirling motion caused by centrifugal force due to mass unbalance, or caused by the flexibility of shaft. This whirling motion generates a periodic time-varying oil-film reaction and dynamic coefficients even in the case of stationary grooved FDBs such as FDBs with grooved sleeves. Many research works on FDBs have been conducted without including time-varying dynamic coefficients. However, the periodic time-varying dynamic coefficients affect the dynamic performance and stability of FDBs. Several researchers have studied the stability of a rotor supported by FDBs and have proposed several methods to investigate this stability. Kakoty and Majumdar (000) studied the stability of journal bearings considering the effect of fluid inertia with increasing eccentricity ratio by introducing a mass parameter. However, their study was limited only to the stability analysis of a journal bearing. Yoon and Jang (003) applied Hill s infinite determinant to investigate the stability of a rotor supported by rotating grooved journal bearings with the assumption that the rotor rotates concentrically, but did not study the effects of whirling motion and stationary grooves upon stability, and their study was limited to translational two degrees of freedom. Kim et al. (010) evaluated the stability of FDBs by using a critical mass with five degrees of freedom; however, their study was restricted to a special case of rotating groove that generates constant dynamic coefficients with respect to a rotating coordinate, and they did not include whirling motion. Lee et al. (011)

2 Fig. 1 Mechanical structure of the disk spindle system supported by FDBs investigated the stability of rotating grooved FDBs by using a critical mass in five degrees of freedom, including utilization of the fact that the dynamic coefficients are constant in the rotating coordinate system. However, they did not analyze the stability of stationary grooved FDBs. Chen et al. (01) performed static, dynamic and stability analyses of a journal bearing with a herringbone grooved sleeve. However, their research was limited to stability analysis of the journal bearing, and they did not include whirling motion. This paper proposes a method to determine the stability of a whirling rotor supported by stationary grooved FDBs, such as coupled journal and thrust bearings with a grooved sleeve, considering the five degrees of freedom of a general rotor-bearing system. Dynamic coefficients are calculated by using the finite element method and the perturbation method, and they are represented as periodic harmonic functions by considering whirling motion. Because of the periodic time-varying dynamic coefficients, the equations of motion of the rotor supported by FDBs can be represented as a parametrically excited system. The solution of the equations of motion can be assumed to be a Fourier series, allowing the equations of motion to be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients; then, to determine the stability, Hill s infinite determinant is calculated by using these algebraic equations. The proposed method is validated by determining the convergence of the locus of the whirl radius after solving the equations of motion numerically. Method of analysis.1 Determinant of dynamic coefficients Figure 1 shows the mechanical structure of a disk spindle system supported by FDBs. The FDBs are composed of lower and upper grooved journal bearings and lower and upper grooved thrust bearings, in which the grooves are inscribed in a stationary sleeve. The governing equations of the stationary grooved journal bearing and the thrust bearing can be written in a coordinate system fixed to the sleeve as shown in Eqs. (1) and (), respectively. ( h 3 R θ 1µ 1 r r (r h3 1µ p R θ ) + ( h 3 z 1µ ) p + ( h 3 r r θ 1µ ) p = R θ z ) p r θ = r θ h R θ + h t h r θ + h t where R, θ, h, p and µ are the radius, rotating speed, film thickness, pressure and coefficient of viscosity, respectively. And r, θ and z are the radial, circumferential and axial coordinates, respectively. Figure shows the coordinate system of the journal bearing with stationary herringbone grooves. Perturbation equations are obtained by substituting a first-order Taylor series of pressure, film thickness and rate change of film thickness with respect to small displacement and velocities into the Reynolds equation, and by applying the separation of variables with respect to each perturbed displacement or velocity. The perturbation equations are solved by using the finite element method and the dynamic coefficients can be obtained by integrating the perturbed pressure in bearing regions (Jang and Kim 1999). Equation (3) shows the relationship between reaction force and the dynamic coefficients. F X x ẋ F Y y ẏ F Z = {F 0 } + [K] z + [C] ż (3) F θx θ x θ x F θy θ y θ y (1) ()

3 G = θ z I z 000 θ z I z 0 x y x = z θ x θ y (8) (9) where I x, I y and I z are the rotor s mass moments of inertia (010). If we assume that a rotor moves along a circular trajectory, the dynamic coefficients can be defined as time-varying components as shown in Eqs. (10) (15): C ij = C ij + C ij cos(χt + φ ij ) i, j = x, y, θ x, θ y (10) Fig. Coordinate system of the journal bearing with stationary herringbone grooves K ij = K ij + K ij cos(χt + ϕ ij ) i, j = x, y, θ x, θ y (11) where {F 0 }, [K] and [C] are the load capacity, stiffness coefficients and damping coefficients, respectively.. Stability analysis The equations of motion of a rigid rotor with mass m a supported by FDBs can be written as follows. Mẍ + (C + G)ẋ + Kx = 0 where each coefficient matrix can be represented in the following forms: m a m a M = 0 0 m a 0 0 (5) I x I y C xx C xy C xz C xθx C xθy C yx C yy C yz C yθx C yθy C = C zx C zy C zz C zθx C zθy C θx x C θx y C θx z C θx θ x C θx θ y C θy x C θy y C θy z C θy θ x C θy θ y K xx K xy K xz K xθx K xθy K yx K yy K yz K yθx K yθy K = K zx K zy K zz K zθx K zθy K θx x K θx y K θx z K θx θ x K θx θ y K θy x K θy y K θy z K θy θ x K θy θ y () (6) (7) C ij = C ij + C ij cos ( χ t + φ ij or i = x, y, θ x, θ y, j = z K ij = K ij + K ij cos ( χ t + ϕ ij or C zz = C zz K zz = K zz i = x, y, θ x, θ y, j = z (1) where C ij, K ij, C ij, K ij, φ ij and ϕ ij are the averages, variations and phase angles of damping and stiffness coefficients, respectively. Because reaction force in the journal bearing is always toward the center of the journal, C ij and K ij of the journal bearing (i, j = x, y, θ x, θ y ) change twice per revolution. On the other hand, C ij and K ij coupled between the journal and thrust bearings (i = z, j = x, y, θ x, θ y or i = x, y, θ x, θ y, j = z) change once per revolution. When parametrically excited frequencies of the dynamic coefficients in the journal bearing are assumed to be χ, those corresponding to dynamic coefficients coupled between journal and thrust bearings can be expressed as χ. C ij and K ij of the thrust bearing (i, j = z) are assumed to be constant because the flying height is assumed to be constant. Also, Eq. () can be transformed into the eigenvalue problem by using the following matrix. ) ) i = z, j = x, y, θ x, θ y i = z, j = x, y, θ x, θ y (13) (1) (15)

4 β = 1 K xx K xx K xy K xz K xθx K xθy K yx K yy K yz K yθx K yθy K zx K zy K zz K zθx K zθy K θx x K θx y K θx z K θx θ x K θx θ y K θy x K θy y K θy z K θy θ x K θy θ y (16) After substituting Eqs. (10) (15) into Eq. () and multiplying by the inverse matrix of Eq. (16), Eq. () can be rewritten in the following form: where, each coefficient matrix can be represented in the following forms: mẍ + pẋ + qẋ cos χt Qẋ sin χt + uẋ cos χ t Uẋ sin χ t + K ax + rx cos χt Rx sin χt + vx cos χ t Ux sin χ t = 0 m = β 1 M K a = β 1 K p = β 1 ( C + G) (17) (18) (19) (0) U = β 1 C z sinφ (8) where K and C are the matrices composed of the average value of the dynamic coefficients, and Kcosϕ, Ksinϕ, Ccosφ, Csinφ, K z cosϕ, K z sinϕ, C z cosφ and C z sinφ are the matrices calculated by the multiplication of sine and cosine values of phase angles and variations of dynamic coefficients. In Eq. (17), the average stiffness matrix contains only the diagonal terms. Because of the parametric excitation frequency in Eq. (17), the solution of Eq. (17) has the period of T = π/χ with the following form: x(t + T) = σ x(t) (9) where the constant σ determines the boundary between the stable and unstable regions. If σ = 1, the solution exists on the boundary between the stable and unstable regions. When σ =+1, the period of the solution is T. When σ = 1, the period of the solution is T. Because the characteristic equation in the σ = 1 case contains all the eigenvalues of the σ =+1 case, the characteristic equation is derived when σ = 1 (003). If the period of the solution is T, Eq. (9) can be written in the following form. r = β 1 Kcosϕ R = β 1 Ksinϕ q = β 1 Ccosφ Q = β 1 Csinφ v = β 1 K z cosϕ V = β 1 K z sinϕ u = β 1 C z cosφ (1) () (3) () (5) (6) (7) Table 1 Design parameters of the FDBs Design variable Grooved journal bearing Bearing width (mm) Upper: 1.85 Lower: 1.5 Grooved thrust bearing Upper: 0.5 Lower: 0.50 Radial clearance (μm) 1.85 Axial total clearance (μm) 30 Groove pattern Herringbone Spiral Number of grooves 8 Upper: 10 Lower: 0 Groove depth (μm) 5 Upper: 8 Lower: 15 Fig. 3 a Finite element model and b pressure distribution of FDBs with stationary herringbone grooves

5 Fig. Stiffness coefficients and corresponding frequency spectra for the stationary grooved FDB with the whirl radius of µm and rotational speed of 700 rpm

6 x(t + T) = x(t) (30) The solutions of Eq. (17) can be assumed to have the following forms by using the Fourier series with the period T: ( x(t) = A 0 + A n cos nχt (31) + B n sin nχt ) y(t) = C 0 + n=1 n=1 ( C n cos nχt + D n sin nχt ) (3) z(t) = E 0 + θ x (t) = G 0 + θ y (t) = I 0 + n=1 n=1 n=1 ( E n cos nχt + F n sin nχt ) ( G n cos nχt + H n sin nχt ) ( I n cos nχt + J n sin nχt ) (33) (3) (35) where A n, B n, C n, D n, E n, F n, G n,h n, I n and J n are the Fourier coefficients. Fig. 5 Dynamic coefficients of stationary grooved FDB versus whirl radius for the rotational speed of 700 rpm; average values of the dynamic coefficients are connected by the trend lines, and vertical bars show the maximum and minimum values

7 Fig. 6 Dynamic coefficients of stationary grooved FDB versus rotational speed for the whirl radius of µm Substituting Eqs. (31) (35) into the equations of motion in Eq. (17) and equating the coefficients of the constant, the sine and cosine terms become zero, linear algebraic equations can be obtained in terms of Fourier coefficients. The linear algebraic equations obtained from the first equation in Eq. (17) are given in the Appendix. In the same way, similar linear algebraic equations can be derived from the other equations in Eq. (17). The equations of motion can be written in matrix form as follows: [P]{U} =0 (36) where the matrix [P] includes the coefficients of five linear algebraic equations, and the matrix {U} is composed of the Fourier coefficients. {U} ={A 0,..., A n, B 1,..., B n, C 0,..., C n, D 1,..., D n, E 0,..., E n, F 1,..., F n, G 0,..., G n, H 1,..., H n, I 0,..., I n, J 1,..., J n } T (37) The matrix [P] is transformed to dimensionless components as follows: λ = K xx /m 11 χ M ij = m ij /m 11 η ij = p ij /m 11 χ ζ ij = r ij /m 11 χ (38) (39) (0) (1)

8 ξ ij = V ij /m 11 χ (6) ρ ij = u ij /m 11 χ (7) τ ij = U ij /m 11 χ For Eq. (36) to have a nontrivial solution, the determinant of the matrix [P] must be zero, which results in the characteristic equation. Because only the diagonal elements of matrix [P] have a dimensionless variable λ(= K xx /m 11 χ ), λ is the eigenvalue of the matrix [P] with the following form: [P λ=0 ]+λ[i] = 0 where [P λ=0 ] is the matrix [P] at λ = 0 and [I] is the identity matrix. The solution of Eq. (9) corresponds to the boundary between the stable and unstable regions. The unstable region is determined by the following condition, based on Hill s infinite determinant (Newland 1989): det[p] < 0. 3 Results and discussion 3.1 Determination of dynamic coefficients (8) (9) (50) Fig. 7 Stability chart: Hill s infinite determinant versus a rotational speed and b whirl radius ν ij = R ij /m 11 χ µ ij = q ij /m 11 χ κ ij = Q ij /m 11 χ ε ij = v ij /m 11 χ () (3) () (5) A computer program developed by Jang and Lee (006) was used to calculate the dynamic coefficients of the FDBs. Table 1 shows the design parameters of the FDBs in the spindle motor of the.5 hard disk drive design depicted in Fig. 1. The mass of the rotor, including a.5 disk, is g. FDBs consist of coupled journal and thrust bearings. They only allow an axial rotational motion by constraining remaining five degrees of freedom, so the dynamic characteristics of coupled journal and thrust bearings have to be analyzed in five degrees of freedom. In this chapter, the dynamic characteristics and the stability are investigated with the consideration of five degrees of freedom. Figure 3 shows a finite element model of the stationary grooved FDBs, and the pressure distribution of the Fig. 8 Whirl orbit in a stable and b unstable conditions

9 stationary grooved FDBs with the whirl radius of µm at the rotational speed of 700 rpm. This finite element model consists of two herringbone-grooved journal bearings, four plain journal bearings, two spiral-grooved thrust bearings and one plain thrust bearing. The fluid film was discretized into 610 isoparametric bilinear elements with four nodes. Figure shows K xx, K xz and K zz and their frequency spectra for the stationary grooved FDBs with the whirl radius of µm at the rotational speed of 700 rpm. The dynamic coefficients are sinusoidal periodic functions, and the periodic function of the dynamic coefficients was derived by using the Fourier transformation. Because reaction force in the journal bearing is always toward the center of the journal, it varied once every 180 ; thus, the dynamic coefficients of the journal bearing varied twice per revolution (Fig. a). On the other hand, the dynamic coefficients coupled between journal and thrust bearings changed once per revolution (Fig. b), because the trajectory of whirling motion repeats every 360. The dynamic coefficients of pure axial direction were constant (Fig. c) because the flying height was assumed to be constant in steady states. Figure 5 shows the averages and variations of K xx, K zz, K θx θ x and C xx, C zz, C θx θ x of the stationary grooved FDBs versus whirl radius. With increasing whirl radius, the average values of K xx, K θx θ x, C xx and C θx θ x increased, and their variations rapidly increased because increasing whirl radius corresponded to increasing eccentricity of the FDBs. K zz and C zz were almost constant because the flying height of the FDBs was assumed to be constant for constant rotational speeds. Figure 6 shows the averages and variations of K xx, K zz, K θx θ x and C xx, C zz, C θx θ x of the stationary grooved FDBs versus rotating speed. With increasing rotational speed, the average values of K xx and K θx θ x increased, whereas the average values of K zz, C xx, C zz and C θx θ x were almost constant regardless of rotational speed because their perturbation equations are not functions of rotational speed. The variations of the dynamic coefficients were very small because the whirl radius was very small (0.185 µm). 3. Stability analysis Stability analysis of the rotating and stationary grooved FDBs was performed by using the dynamic coefficients calculated as shown in the previous section. Figure 7 shows the stability of the FDBs calculated by using the proposed Hill s infinite determinant including the five degrees of freedom of a general rotor-bearing system. The stability of the FDBs decreased with increasing rotational speed, because this increased the averages of the stiffness coefficients, which played a major role in increasing the unstable region of the grooved FDBs. And the increased average values of the damping coefficients played a major role in decreasing the unstable region of the grooved FDBs (003). The stability of the FDBs increased with increasing whirl radius because the resulting increases in the averages and variations of C xx and C θx θ x increased the stability faster than the corresponding increases of K xx and K θx θ x decreased the stability. To validate the proposed stability analysis, the equations of motion in Eq. () were solved numerically by using the fourth-order Runge Kutta method with a time step of 10 s to determine the locus of the whirl radius. The initial position of the rotor was assumed to be at 0.0 µm in x, y and z coordinates and at 0.0 in θ x and θ y coordinates. When the FDBs supported the rotor (m = kg, I x = I y = kg m ) at 700 rpm, Hill s infinite determinant was positive and the whirl radius converged (Fig. 8a). When the mass of the rotor was 5. kg and I x and I y were kg m, Hill s infinite determinant was and the whirl radius diverged (Fig. 8b). These simulated loci of the whirl radius validate the proposed stability analysis. Conclusions This paper proposed a method to determine the stability of a whirling rigid rotor supported by stationary grooved FDBs by using Hill s infinite determinant and including the five degrees of freedom of a general rotor-bearing system. As calculated by this method, increasing rotational speed increases K xx and K θx θ x, which decreases the stability of the stationary grooved FDBs; increasing whirl radius increases the stability of the FDBs because the resulting increases in the averages and variations of C xx and C θx θ x increase the stability faster than the corresponding increases of K xx and K θx θ x decrease the stability. The proposed method was verified by investigating the convergence and divergence of the whirl radius after the equations of motion were solved using the fourth-order Runge Kutta method. Acknowledgments This research was supported by the Basic Science Research Program of the National Research Foundation of Korea, funded by the Ministry of Education, Science and Technology ( ).

10 Appendix Constant: K xx A 0 + χ (q 11B + q 1 D + q 13 F + q 1 H + q 15 J + Q 11 A + Q 1 C + Q 13 E + Q 1 G + Q 15 I ) + 1 (r 11A + r 1 C + r 13 E + r 1 G + r 15 I R 11 B R 1 D R 13 F R 1 H R 15 J ) + χ (u 11B 1 + u 1 D 1 + u 13 F 1 + u 1 H 1 + u 15 J 1 + U 11 A 1 + U 1 C 1 + U 13 E 1 + U 1 G 1 + U 15 I 1 ) + 1 (v 11A 1 + v 1 C 1 + v 13 E 1 + v 1 G 1 + v 15 I 1 V 11 B 1 V 1 D 1 V 13 F 1 V 1 H 1 V 15 J 1 ) = 0 cos χt : K xx A 1 χ (m 11A 1 + m 1 C 1 + m 13 E 1 + m 1 G 1 + m 15 I 1 ) + χ (p 11B 1 + p 1 D 1 + p 13 F 1 + p 1 H 1 + p 15 J 1 ) + χ (v 11A 0 + v 1 C 0 + v 13 E 0 + v 1 G 0 + v 15 I 0 ) + χ (q 11B 1 + q 1 D 1 + q 13 F 1 + q 1 H 1 + q 15 J 1 + Q 11 A 1 + Q 1 C 1 + Q 13 E 1 + Q 1 G 1 + Q 15 I 1 ) + 1 (r 11A 1 + r 1 C 1 + r 13 E 1 + r 1 G 1 + r 15 I 1 R 11 B 1 R 1 D 1 R 13 F 1 R 1 H 1 R 15 J 1 ) + 3χ (q 11B 3 + q 1 D 3 + q 13 F 3 + q 1 H 3 + q 15 J 3 + Q 11 A 3 + Q 1 C 3 + Q 13 E 3 + Q 1 G 3 + Q 15 I 3 ) + 1 (r 11A 3 + r 1 C 3 + r 13 E 3 + r 1 G 3 + r 15 I 3 R 11 B 3 R 1 D 3 R 13 F 3 R 1 H 3 R 15 J 3 ) + χ (u 11B + u 1 D + u 13 F + u 1 H + u 15 J + U 11 A + U 1 C + U 13 E + U 1 G + U 15 I ) + 1 (v 11A + v 1 C + v 13 E + v 1 G + v 15 I V 11 B V 1 D V 13 F V 1 H V 15 J ) = 0 cos χt : K xx A + (r 11 A 0 + r 1 C 0 + r 13 E 0 + r 1 G 0 + r 15 I 0 ) χ (m 11 A + m 1 C + m 13 E + m 1 G + m 15 I ) + χ(p 11 B + p 1 D + p 13 F + p 1 H + p 15 J ) + χ (u 11B 1 + u 1 D 1 + u 13 F 1 + u 1 H 1 + u 15 J 1 U 11 A 1 U 1 C 1 U 13 E 1 U 1 G 1 U 15 I 1 ) + 1 (v 11A 1 + v 1 C 1 + v 13 E 1 + v 1 G 1 + v 15 I 1 + V 11 B 1 + V 1 D 1 + V 13 F 1 + V 1 H 1 + V 15 J 1 ) + 3χ (u 11B 3 + u 1 D 3 + u 13 F 3 + u 1 H 3 + u 15 J 3 + U 11 A 3 + U 1 C 3 + U 13 E 3 + U 1 G 3 + U 15 I 3 ) + 1 (v 11A 3 + v 1 C 3 + v 13 E 3 + v 1 G 3 + v 15 I 3 V 11 B 3 V 1 D 3 V 13 F 3 V 1 H 3 V 15 J 3 ) + χ(q 11 B + q 1 D + q 13 F + q 1 H + q 15 J + Q 11 A + Q 1 C + Q 13 E + Q 1 G + Q 15 I ) + 1 (r 11A + r 1 C + r 13 E + r 1 G + r 15 I R 11 B R 1 D R 13 F R 1 H R 15 J ) = 0

11 cos Nχt (N 3): K xx A N N χ (m 11 A N + m 1 C N + m 13 E N + m 1 G N + m 15 I N ) + Nχ (p 11B N + p 1 D N + p 13 F N + p 1 H N + p 15 J N ) (N + )χ + (q 11 B N+ + q 1 D N+ + q 13 F N+ + q 1 H N+ + q 15 J N+ + Q 11 A N+ + Q 1 C N+ + Q 13 E N+ + Q 1 G N+ + Q 15 I N+ ) + 1 (r 11A N+ + r 1 C N+ + r 13 E N+ + r 1 G N+ + r 15 I N+ R 11 B N+ R 1 D N+ R 13 F N+ R 1 H N+ R 15 J N+ ) + (N )χ (q 11 B N + q 1 D N + q 13 F N + q 1 H N + q 15 J N Q 11 A N Q 1 C N Q 13 E N Q 1 G N Q 15 I N ) + 1 (r 11A N + r 1 C N + r 13 E N + r 1 G N + r 15 I N + R 11 B N + R 1 D N + R 13 F N + R 1 H N + R 15 J N ) + (N 1)χ (u 11 B N 1 + u 1 D N 1 + u 13 F N 1 + u 1 H N 1 + u 15 J N 1 U 11 A N 1 U 1 C N 1 U 13 E N 1 U 1 G N 1 U 15 I N 1 ) + 1 (v 11A N 1 + v 1 C N 1 + v 13 E N 1 + v 1 G N 1 + v 15 I N 1 + V 11 B N 1 + V 1 D N 1 + V 13 F N 1 + V 1 H N 1 + V 15 J N 1 ) + (N + 1)χ (u 11 B N+1 + u 1 D N+1 + u 13 F N+1 + u 1 H N+1 + u 15 J N+1 + U 11 A N+1 + U 1 C N+1 + U 13 E N+1 + U 1 G N+1 + U 15 I N+1 ) + 1 (v 11A N+1 + v 1 C N+1 + v 13 E N+1 + v 1 G N+1 + v 15 I N+1 V 11 B N+1 V 1 D N+1 V 13 F N+1 V 1 H N+1 V 15 J N+1 ) = 0 sin χt : K xx B 1 (V 11 A 0 + V 1 C 0 + V 13 E 0 + V 1 G 0 + V 15 I 0 ) χ (m 11B 1 + m 1 D 1 + m 13 F 1 + m 1 H 1 + m 15 J 1 ) χ (p 11A 1 + p 1 C 1 + p 13 E 1 + p 1 G 1 + p 15 I 1 ) + χ (q 11A 1 + q 1 C 1 + q 13 E 1 + q 1 G 1 + q 15 I 1 Q 11 B 1 Q 1 D 1 Q 13 F 1 Q 1 H 1 Q 15 J 1 ) 1 (r 11B 1 + r 1 D 1 + r 13 F 1 + r 1 H 1 + r 15 J 1 + R 11 A 1 + R 1 C 1 + R 13 E 1 + R 1 G 1 + R 15 I 1 ) 3χ (q 11A 3 + q 1 C 3 + q 13 E 3 + q 1 G 3 + q 15 I 3 Q 11 B 3 Q 1 D 3 Q 13 F 3 Q 1 H 3 Q 15 J 3 ) + 1 (r 11B 3 + r 1 D 3 + r 13 F 3 + r 1 H 3 + r 15 J 3 + R 11 A 3 + R 1 C 3 + R 13 E 3 + R 1 G 3 + R 15 I 3 ) + χ ( u 11A u 1 C u 13 E u 1 G u 15 I + U 11 B + U 1 D + U 13 F + U 1 H + U 15 J ) + 1 (v 11B + v 1 D + v 13 F + v 1 H + v 15 J + V 11 A + V 1 C + V 13 E + V 1 G + V 15 I ) = 0 sin χt : K xx B (R 11 A 0 + R 1 C 0 + R 13 E 0 + R 1 G 0 + R 15 I 0 ) χ (m 11 B + m 1 D + m 13 F + m 1 H ++m 15 J ) χ(p 11 A + p 1 C + p 13 E + p 1 G + p 15 I ) χ(q 11 A + q 1 C + q 13 E + q 1 G + q 1 I Q 11 B Q 1 D Q 13 F Q 1 H Q 1 J ) + 1 (r 11B + r 1 D + r 13 F + r 1 H ++r 15 J R 11 A + R 1 C + R 13 E + R 1 G + R 15 I ) χ (u 11A 1 + u 1 C 1 + u 13 E 1 + u 1 G 1 + u 15 I 1 + U 11 B 1 + U 1 D 1 + U 13 F 1 + U 1 H 1 + U 15 J 1 ) + 1 (v 11B 1 + v 1 D 1 + v 13 F 1 + v 1 H 1 + v 15 J 1 V 11 A 1 V 1 C 1 V 13 E 1 V 1 G 1 V 15 I 1 ) + 3χ ( u 11A 3 u 1 C 3 u 13 E 3 u 1 G 3 u 15 I 3 + U 11 B 3 + U 1 D 3 + U 13 F 3 + U 1 H 3 + U 15 J 3 ) + 1 (v 11B 3 + v 1 D 3 + v 13 F 3 + v 1 H 3 + v 15 J 3 + V 11 A 3 + V 1 C 3 + V 13 E 3 + V 1 G 3 + V 15 I 3 ) = 0

12 sin Nχt (N 3): K xx B N N χ (m 11 B N + m 1 D N + m 13 F N + m 1 H N ) Nχ (p 11A N + p 1 C N + p 13 E N + p 1 G N ) (N + )χ (q 11 A N+ + q 1 C N+ + q 13 E N+ + q 1 G N+ + q 15 I N+ Q 11 B N+ Q 1 D N+ Q 13 F N+ Q 1 H N+ Q 15 J N+ ) + 1 (r 11B N+ + r 1 D N+ + r 13 F N+ + r 1 H N+ + r 15 J N+ + R 11 A N+ + R 1 C N+ + R 13 E N+ + R 1 G N+ + R 15 I N+ ) (N )χ (q 11 A N + q 1 C N + q 13 E N + q 1 G N + q 15 I N + Q 11 B N + Q 1 D N + Q 13 F N + Q 1 H N + Q 15 J N ) + 1 (r 11B N + r 1 D N + r 13 F N + r 1 H N + r 15 J N R 11 A N R 1 C N R 13 E N R 1 G N R 15 I N ) (N 1)χ (u 11 A N 1 + u 1 C N 1 + u 13 E N 1 + u 1 G N 1 + u 15 I N 1 + U 11 B N 1 + U 1 D N 1 + U 13 F N 1 + U 1 H N 1 + U 15 J N 1 ) + 1 (v 11B N 1 + v 1 D N 1 + v 13 F N 1 + v 1 H N 1 + v 15 J N 1 V 11 A N 1 V 1 C N 1 V 13 E N 1 V 1 G N 1 V 15 I N 1 ) + (N + 1)χ ( u 11 A N+1 u 1 C N+1 u 13 E N+1 u 1 G N+1 u 15 I N+1 + U 11 B N+1 + U 1 D N+1 + U 13 F N+1 + U 1 H N+1 + U 15 J N+1 ) + 1 (v 11B N+1 + v 1 D N+1 + v 13 F N+1 + v 1 H N+1 + v 15 J N+1 + V 11 A N+1 + V 1 C N+1 + V 13 E N+1 + V 1 G N+1 + V 15 I N+1 ) = 0 References Chen SK, Chou HC, Kang Y (01) Stability of hydrodynamic bearing with herringbone grooved sleeve. Tribol Int 55:15 8 Jang GH, Kim YJ (1999) Calculation of dynamic coefficients in a hydrodynamic bearing considering five degrees of freedom for a general rotor-bearing system. J Tribol 11(3): Jang GH, Lee SH (006) Determination of the dynamic coefficients of the coupled journal and thrust bearings by the perturbation method. Tribol Lett (3):39 6 Kakoty SK, Majumdar BC (000) Effect of fluid inertia on stability of oil journal bearings. J Tribol 1:71 75 Kim MG, Jang GH, Kim HW (010) Stability analysis of a disk-spindle system supported by coupled journal and thrust bearings considering five degrees of freedom. Tribol Int 3: Lee JH, Jang GH, Jung KM, Ha HJ (011) Stability analysis of a whirling disk-spindle system supported by FDBs with rotating grooves. Microsyst Technol 17: Newland DE (1989) Mechanical vibration analysis and computation. Longman Scientific and Technical, Harlow Yoon JW, Jang GH (003) Stability analysis of a hydrodynamic journal bearing with rotating herringbone grooves. J Tribol 15:91 300