Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

Abstract

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

1. Introduction

Vibration analysis of rotation rings in different structures, such as gears, rotors, bearings, etc., is of great interest for complex dynamic characteristics. This study was motivated by planetary gear systems, where planet gears often have a thin rim [1], especially in aerospace applications, where weight is an important consideration. Planet gears do not only rotate around their axis supported by bearings but revolve around the center of planetary gear systems meshed with central gears. Meanwhile, they are supported by both discrete stiffnesses of mesh pairs and partially distributed stiffnesses of support bearings. These planet gears may undergo significant elastic continuum vibration during operation, for their compliance, combined movement, and complex elastic supports.

Compliance of gears has received attention in some references. Talbert and Gockel [2] found elastic vibration in aerospace gears experimentally. Bettaieb et al. [3] investigated the flexible gear vibration, using FEM method. Elastic deformations are also found in planetary gears by experiments [4]. Vinayak and Singh [5] studied the vibration modes of compliant gears by multibody dynamic formulation. Kahraman et al. [6] investigated the effect of flexibility of gears on the dynamic behavior of a planetary gear set. Their study shows that dynamic gear tooth bending stress values were found to be significantly higher for all of the gears for a flexible internal gear as compared to the rigid internal gear.

Vibrations of stationary or rotating rings with different types of supports have been investigated by many researchers. Bert and Chen [7] studied the in-plane and out-plane vibration of rotating rings on an elastic foundation. Free vibrations of rings on rigid radial supports were investigated by Rao and Sundararajan [8], where the ring was separated into several parts with supports located at the ends of each segment. Detinko [9] studied the free vibration of a thick ring on multiple radial supports by the Galerkin method. Bickford and Reddy [10] investigated rotating rings, where discrete stiffnesses are attached to and rotate with the ring. Allaei et al. [11] investigated the vibration of non-axisymmetric rings with lumped masses and stiffnesses added. Loy and Lam [12] studied the cylindrical shell with ring support. Metrikine and Tochilin [13] studied the steady state response of an elastic ring subjected to a uniformly moving load by the method of images. The results show that, for small velocities of the load, the ring pattern is almost perfectly symmetric with respect to the loading point. The pattern becomes slightly asymmetric due to the viscosity of the springs if the load velocity is small. Kim and Chung [14] investigated the free non-linear vibration of a rotating thin ring with in-plane and out-plane motions. Nackenhorst and Brinkmeier [15] used two observer systems, that is, Lagrangian and Eulerian systems, to derive rotating ring models. Wu and Parker [16] studied the vibration of rings on a general elastic foundation. There are fewer studies on the vibration of rotating rings with discrete supports. Afterward, they [17] studied the modal properties of equally spaced planetary gears with elastic ring gears. The results show that all eigenfunctions fall into four mode types, rotational, translational, planet, and purely ring modes. Parker and Wu [18] mathematically proved modal properties of planetary gears having diametrically opposed planets and an elastic ring gear. Two types of modes were found, that is, rotational and translational modes. Canchi and Parker [19,20] studied the parametric instability of rotating rings connected to moving discrete springs with time-varying stiffness. They analytically studied the effects of different system parameters on the instability boundaries and well-defined properties for the occurrence or suppression of instabilities are obtained as simple relations in the system parameters. Cooley and Parker [21] mathematically proved the structured properties of the critical speeds and associated critical speed eigenvectors of high-speed planetary gears. The results show that planetary gears have the only planet, rotational, and translational mode critical speeds. Divergence instability is possible at speeds adjacent to critical speeds, and flutter instabilities occur at extremely high speeds. Then, they [22] studied the vibration of high-speed rotating rings coupled to space-fixed stiffnesses. Zhao et al. [23] examined the parametric vibration of an elastic dual-ring structure by an analytical model including stationary supports and rotating loads. The results imply that there exist four types of excitations and that vibration can occur in a coupled, uncoupled, principal, or combination manner. Lu et al. [24] investigated the in-plane free vibration of high-speed rotating thin ring with an elastic foundation. Cooley and Parker [25] investigated eigenvalue sensitivity to model parameters and veering in gyroscopic systems by a perturbation approach. The results are shown to be accurate over a wide range of rotation speeds. A veering parameter is defined to analyze veering in high-speed planetary gears. Liu et al. [26] investigated the vibration of rotating elastic rings that are excited by an arbitrary number of space-fixed discrete stiffnesses with periodically fluctuating stiffnesses. The results show that the bandwidths of the instability regions correlate with the fractional strain energy stored in the discrete stiffnesses. For rings with multiple discrete stiffnesses, the phase differences between them can eliminate large amplitude response under certain conditions. Lu et al. [27] developed a new high-order model for in-plane vibrations of rotating rings. The study shows that the new high-order model is superior to the existing ring models in predicting the dynamic behavior of either stationary or rotating rings, and the model is applicable to both plane strain and plane stress configurations. In this paper, the through-thickness variation of the radial stress and geometrical nonlinearity were considered. As mentioned above, there has been no study on the vibration of planet rings, which have both rotating and revolving movements, with discrete stiffnesses and partially distributed stiffnesses.

In this study, the vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were investigated. The governing equations of motion for the radial and tangential deformations of planet rings were derived based on a rotating-basis system. This model considered the support zone variation and revolving movement of planet rings. Matrix differential operators were used to express the governing equations. The vibration properties of planet rings with and without stiffnesses were investigated by using the Galerkin method. The influence of parameters on vibration characteristics was studied. The forced responses excited by rotating and revolving movement were also investigated.

2. Analytical Model

The schematic of a planetary gear system and the vibration model of a planet ring are shown in Figure 1a. The rotating speed and revolving speed of the planet ring are ω and Ω, respectively, and the ratio of Ω and ω is i₀ = Ω/ω. The stationary basis is {i, j, k}; the origin O₁ of {e₁, e₂, e₃} basis is set to the center of the ring, where the direction of e₁ is along the extension of O₀O₁. The basis {e₁′, e₂′, e₃′} is a local coordinate system with the origin O₂ on the middle surface, and e₁′ is along the radial direction of the ring. u(θθ, t) and v(θ, t) are radial and tangential deformations, respectively. M is a material point on the ring and ϕ is the relative angle of e₁′ to the basis e₁, where ϕ = ωt + θ, θ is the specified angular coordinate of the point M around the planet ϕring. R₀ is the revolving radii, and R₁ is the radii of the middle surface. The density of the ring is р, its elastic modulus is E, and its section area is A with width B and thickness h. Moreover, k_m1 and k_m2 are mesh stiffnesses between planet ring with sun gear and ring gear, respectively; α is the pressure angle of k_mi on the ring, and apparently the mesh stiffnesses will affect both the radial and tangential deformations. When a planetary gear system runs, the contact zone between planet ring and rolling bearing usually concentrates on a part of the whole circle contact area [28] because of mesh forces. Hence, the planet ring is supported in the radial direction by discrete partially distributed stiffnesses, as shown in Figure 1b, and each spring denotes the position of a roller in the bearing and its position to e₁ is θ_bi. The support stiffness of the roller is k_bi, and the values of all k_bi are assumed to be the same. The positions of k_m1 and k_m2 to e₁ are θ_mi (i = 1, 2). Because the planet ring rotates, θ_bi and θ_mi are time dependent, and θ_bi = θ_bi0 − ωt, θ_mi = θ_mi0-ωt, respectively, where θ_bi0 and θ_mi0 are the original angles to e₁. θ_mi0 is determined by the geometry of mesh pairs in the planetary gear system, and θ_bi0 is determined by the selected bearing.

The velocity vector of the point M on the ring is obtained by the following kinetic formula [29]:

$$\begin{array}{ll}{v}_M& =v_{O_1}+\Omega e_3\times r_M+v_{M,r}\\ & =R_0\Omega e_2+\Omega e_3\times \left(R_1\cos \varphi e_1+R_1\sin \varphi e_2+u{e}_1^{\prime }+v{e}_2^{\prime }\right)+v_{M,r}\end{array}$$

(1)

where ϕ = ωt + θ, r_M is the vector of M in the coordinate {e₁, e₂, e₃}, and v_M,r is the relative velocity of M in the coordinate {e₁, e₂, e₃}, which can be expressed as follows:

$$v_{M,r}=v_{O_2}+\omega e_3^{\prime }\times r_{M1}+v_r=R_1\omega e_2^{\prime }+\omega e_3^{\prime }\times \left(u{e}_1^{\prime }+v{e}_2^{\prime }\right)+\left(\dot{u}{e}_1^{\prime }+\dot{v}{e}_2^{\prime }\right)$$

(2)

Consequently, v_M becomes the following:

$$v_M=R_0\Omega e_2+\Omega e_3\times \left(R_1\cos \varphi e_1+R_1\sin \varphi e_2+u{e}_1^{\prime }+v{e}^{\prime }\right)+\left(\dot{u}-\omega v\right)e_1^{\prime }+\left(R_1\omega +\omega u+\dot{v}\right)e_2^{\prime }$$

(3)

Since the velocity, $v_M$, in Equation (3) is represented by unit vectors of two coordinates, it is better to be represented in one coordinate. Thus, according to the coordinate transformation, the velocity of M can be represented in the coordinate {e₁′, e₂′, e₃′} as follows:

$$\begin{array}{l}{v}_M=v_r{e}_1^{\prime }+v_t{e}_2^{\prime }\\ =\left[\dot{u}-v\left(\Omega +\omega \right)+R_0\Omega \sin \varphi \right]e_1^{\prime }+\left[\dot{v}+u\left(\Omega +\omega \right)+R_1\left(\Omega +\omega \right)+R_0\Omega \cos \varphi \right]e_2^{\prime }\end{array}$$

(4)

where v_r is the radial velocity and v_t is the tangential velocity. Then the kinetic energy of the ring is as follows:

$$K_E=\frac{1}{2}\rho A{R}_1{\displaystyle {\int }_0^{2\pi }\left(v_r^2+v_t^2\right)d\theta }$$

(5)

Because the strain energy of the ring is independent of the rotating and revolving movement, the deformation of point M can be considered in the coordinates {e₁′, e₂′, e₃′}. The original length of a small segment of the middle surface of the ring is given as dS = R₁dθ. After deformation, the vector O₁M, which is denoted by r and r = (R₁ + u)e₁′+ve₂′, will increase by dr. Then the length of the segment becomes the following:

$$\mathrm{d}{S}^{\prime }=\sqrt{\mathrm{d}\mathbf{r}\cdot \mathrm{d}\mathbf{r}}=\sqrt{{\left[\left(R_1+u+\partial v/\partial \theta \right)\mathrm{d}\theta \right]}^2+{\left[\left(\partial u/\partial \theta -v\right)\mathrm{d}\theta \right]}^2}$$

(6)

Therefore, the strain of the segment on the middle surface is as follows:

$${\epsilon }_0=\frac{\mathrm{d}{S}^{\prime }-\mathrm{d}S}{\mathrm{d}S}=\frac{\sqrt{{\left[\left(R_1+u+\partial v/\partial \theta \right)\mathrm{d}\theta \right]}^2+{\left[\left(\partial u/\partial \theta -v\right)\mathrm{d}\theta \right]}^2}-R_1\mathrm{d}\theta }{R_1\mathrm{d}\theta }$$

(7)

Here, Equation (7) is approximated to second-order, and it gives the following:

$${\epsilon }_0=\frac{1}{R_1}\left(u+\frac{\partial v}{\partial \theta }\right)+\frac{1}{2R_1^2}{\left(v-\frac{\partial u}{\partial \theta }\right)}^2$$

(8)

The strain of any material point on the cross-section of the ring can be obtained by linear assumption:

$$\epsilon ={\epsilon }_0+\frac{r^{\prime }}{R_1^2}\left(\frac{\partial v}{\partial \theta }-\frac{{\partial }^{2}u}{\partial {\theta }^2}\right)$$

(9)

where r′ is the radial distance of a point on the cross-section to the middle surface. Therefore, the total strain energy of the ring is as follows:

$$S_e=\frac{1}{2}{\displaystyle \underset{A\theta }{\iint }\sigma \epsilon dA{R}_1\mathrm{d}\theta }$$

(10)

where σ is the stress. The actions of partially distributed and discrete stiffnesses are considered as external forces to the ring; therefore, the work of the stiffnesses is as follows:

$$W_s={\displaystyle \underset{\theta }{\int }\left[\left(F_b+F_{mr}\right)u+F_{mt}v\right]}d\theta$$

(11)

where F_b is elastic forces of partially distributed stiffnesses, and F_mr and F_mt are radial and tangential components of discrete stiffnesses acting on the ring, respectively.

Accordingly, the governing equations can be derived by Hamilton’s principle:

$$\delta {\displaystyle {\int }_{t_0}^{t_1}\left(K_E-S_e+W_s\right)}\mathrm{d}t=0$$

(12)

By substituting Equations (5), (10), and (11) into Equation (12) and ignoring the nonlinear terms for the small vibrations, we get the governing equations of vibration of the planet ring:

$$\begin{array}{l}\rho A{R}_1\frac{{\partial }^{2}u}{\partial t^2}-2\rho A{R}_1\left(\Omega +\omega \right)\frac{\partial v}{\partial t}-\rho A{R}_1{\left(\Omega +\omega \right)}^{2}u+N-\frac{{\partial }^{2}M}{R_1\partial {\theta }^2}+F_b+F_{mr}\\ =\rho A{R}_1^2{\left(\Omega +\omega \right)}^2+\rho A{R}_1{R}_0{\Omega }^2\cos \varphi \end{array}$$

(13a)

$$\begin{array}{l}\rho A{R}_1\frac{{\partial }^{2}v}{\partial t^2}+2\rho A{R}_1\left(\Omega +\omega \right)\frac{\partial u}{\partial t}-\rho A{R}_1{\left(\Omega +\omega \right)}^{2}v-\frac{\partial N}{\partial \theta }-\frac{\partial M}{R_1\partial \theta }+F_{mt}\\ =-\rho A{R}_1{R}_0{\Omega }^2\sin \varphi \end{array}$$

(13b)

where N and M are the vibration-induced force and moment resultants, respectively:

$$N=EA\left(\frac{u}{R_1}+\frac{\partial v}{R_1\partial \theta }\right)$$

(14a)

$$M=\frac{EI}{R_1^2}\left(\frac{\partial v}{\partial \theta }-\frac{{\partial }^{2}u}{\partial {\theta }^2}\right)$$

(14b)

and

$$F_b={\displaystyle \sum _{i=1}^{N_b}\left[F_{bi}\Delta \left(\theta -{\theta }_{bi}\right)\right]}={\displaystyle \sum _{i=1}^{N_b}\left[k_{bi}u\left({\theta }_{bi},t\right)\right]}\Delta \left(\theta -{\theta }_{bi}\right)$$

(15a)

$$F_{mr}={\displaystyle \sum _{i=1}^{N_m}\left[F_{mri}\Delta \left(\theta -{\theta }_{mi}\right)\right]}={\displaystyle \sum _{i=1}^{N_m}\left[k_{mi}u\left({\theta }_{mi},t\right)\sin \alpha \right]}\Delta \left(\theta -{\theta }_{mi}\right)$$

(15b)

$$F_{mt}={\displaystyle \sum _{i=1}^{N_m}\left[F_{mti}\Delta \left(\theta -{\theta }_{mi}\right)\right]}={\displaystyle \sum _{i=1}^{N_m}\left[k_{mi}v\left({\theta }_{mi},t\right)\cos \alpha \right]}\Delta \left(\theta -{\theta }_{mi}\right)$$

(15c)

F_bi, F_mri, and F_mti are forces of the ith partially distributed stiffness, radial, and tangential components of discrete stiffness, respectively. N_b and N_m are the numbers of partially distributed stiffnesses and discrete stiffnesses, respectively. θ_bi, θ_mi are the corresponding time dependent positions of partially distributed stiffnesses and discrete stiffnesses, respectively. Δ (⋅) is the Dirac delta function. It can be seen from the right side of Equation (13) that the rotation and revolving movements will result in constant centrifugal force, $\rho A{R}_1^2{\left(\Omega +\omega \right)}^2$, and time-dependent excitation, $\rho A{R}_1{R}_0{\Omega }^2\cos \varphi$, $\rho A{R}_1{R}_0{\Omega }^2\sin \varphi$, to the ring.

For the steady configuration, the ring has no tangential deflection because there is no steady force in this direction, as shown in Equation (13b). Therefore, the radial steady equations of the ring can be obtained from Equation (13a) by the elimination of the time derivatives:

$$\begin{array}{l}\frac{EI}{R_1^3}\frac{{\partial }^4{u}_e}{\partial {\theta }^4}+\left(\frac{EA}{R_1}-\rho A{R}_1{i}_0^{\prime }{\omega }^2\right)u_e+{\displaystyle \sum _{i=1}^{N_b}\left[k_{bi}{u}_e\left({\theta }_{bi}\right)\right]}\Delta \left(\theta -{\theta }_{bi}\right)\\ +{\displaystyle \sum _{i=1}^{N_m}\left[k_{mi}{u}_e\left({\theta }_{mi}\right)\sin \alpha \right]}\Delta \left(\theta -{\theta }_{mi}\right)=\rho A{R}_1^2{i}_0^{\prime }{\omega }^2\end{array}$$

(16)

where u_e denotes the steady radial deformation and $i_0^{\prime }={\left(i_0+1\right)}^2$. Moreover, u_e in the nonlinear Equation (16) can be solved by numerical method.

To estimate the critical speed analytically, the high-order differentials and stiffnesses of Equation (16) are ignored, and the estimated steady radial deformation can be obtained from Equation (16) as follows:

$$u_e=\frac{\rho A{R}_1^2{i}_0^{\prime }{\omega }^2}{\frac{EA}{R_1}-\rho A{R}_1{i}_0^{\prime }{\omega }^2}$$

(17)

It can be found that the steady ring radial deformation in Equation (17) becomes extremely large when the speed, ω, is close to ω_c:

$${\omega }_c=\sqrt{EA/\rho A{R}_1^2{i}_0^{\prime }}$$

(18)

When the speed, ω, is larger than ω_c, the steady status in Equation (17) is unstable. By applying the parameters in

Table 1. Parameters of a planet ring with discrete and partially distributed stiffnesses.

Ring radius R0 (mm)	100
Ring radius R1 (mm)	80
Cross section area A (mm2)	250
Area moment of inertia I (mm4)	2083
Pressure Angle α (°)	20
Elastic modulus E (GPa)	206
Density р (kg/m3)	7800
Mean mesh stiffness kmi (N/m)	5 × 108
Support stiffness kbi (N/m)	1 × 108
The ratio i0 of Ω to ω	0.1825

in Section 3, the critical speed, ω_c, calculated by Equation (16) and Equation (18), is 5.4339 × 10⁴ and 5.4324 × 10⁴, respectively. Therefore, it is reasonable to roughly estimate the critical speed of Equation (16) by the simplified Equation (18). In this work, the speed is restricted lower than that of Equation (18).

Equation (13) can be rewritten to be the equations of small vibrations about the steady status by substitution of u → u_e + u. After that, Equation (13) is normalized by the following variables, to investigate the general vibration properties of planet rings:

$${\omega }_0=\frac{1}{R_1^2}\sqrt{\frac{EI}{\rho A}},\overline{u},\overline{v}=\frac{u,v}{R_1},\overline{t}={\omega }_{0}t,\overline{\omega },\overline{\Omega }=\frac{\omega ,\Omega }{{\omega }_0},{\overline{k}}_{bi},{\overline{k}}_{mi}=\frac{k_{bi},k_{mi}{R}_1^3}{EI},\gamma =\frac{EA{R}_1^2}{EI}$$

(19)

The dimensionless equations are obtained by substitution of Equation(19):

$$\begin{array}{l}\frac{{\partial }^{2}u}{\partial t^2}-2\left(\Omega +\omega \right)\frac{\partial v}{\partial t}-{\left(\Omega +\omega \right)}^{2}u+\gamma \left(u+\frac{\partial v}{\partial \theta }\right)+\left(\frac{{\partial }^{4}u}{\partial {\theta }^4}-\frac{{\partial }^{3}v}{\partial {\theta }^3}\right)\\ +{\displaystyle \sum _{i=1}^{N_b}\left[k_{bi}u\left({\theta }_{bi},t\right)\right]}\Delta \left(\theta -{\theta }_{bi}\right)+{\displaystyle \sum _{i=1}^{N_m}\left[k_{mi}u\left({\theta }_{mi},t\right)\sin \alpha \right]}\Delta \left(\theta -{\theta }_{mi}\right)=\frac{R_0}{R_1}{\Omega }^2\cos \varphi \end{array}$$

(20a)

$$\begin{array}{l}\frac{{\partial }^{2}v}{\partial t^2}+2\left(\Omega +\omega \right)\frac{\partial u}{\partial t}-{\left(\Omega +\omega \right)}^{2}v-\gamma \left(\frac{\partial u}{\partial \theta }+\frac{{\partial }^{2}v}{\partial {\theta }^2}\right)+\left(\frac{{\partial }^{3}u}{\partial {\theta }^3}-\frac{{\partial }^{2}v}{\partial {\theta }^2}\right)\\ +{\displaystyle \sum _{i=1}^{N_m}\left[k_{mi}v\left({\theta }_{mi},t\right)\cos \alpha \right]\Delta \left(\theta -{\theta }_{mi}\right)}=-\frac{R_0}{R_1}{\Omega }^2\sin \varphi \end{array}$$

(20b)

The right parts of Equation (20) are external excitations resulting from rotating and revolving movement. In Equation (20), the bar on variables is removed for convenience, and Equation (20) is rewritten in matrix form by partial differential operators, as follows:

$$\mathbf{M}\ddot{\mathbf{x}}+\left(\Omega +\omega \right)\mathbf{G}\dot{\mathbf{x}}+\mathbf{K}x=\mathbf{F}$$

(21)

where

$$x={\left\{\begin{array}{cc}u& v\end{array}\right\}}^{\mathrm{T}},M=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],G=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right],F={\left\{\frac{R_0}{R_1}{\Omega }^2\cos \varphi ,-\frac{R_0}{R_1}{\Omega }^2\sin \varphi \right\}}^{\mathrm{T}},$$

$$K=\left[\begin{array}{cc}\begin{array}{l}\gamma +\frac{{\partial }^4}{\partial {\theta }^4}-{\left(\Omega +\omega \right)}^2\\ +{\displaystyle \sum _{i=1}^{N_b}\left[k_{bi}\Delta \left(\theta -{\theta }_{bi}\right)\Delta \left(\theta -{\theta }_{bi}\right)\right]}\\ +{\displaystyle \sum _{i=1}^{N_m}\left[k_{mi}\sin \alpha \Delta \left(\theta -{\theta }_{mi}\right)\Delta \left(\theta -{\theta }_{mi}\right)\right]}\end{array}& \gamma \frac{\partial }{\partial \theta }-\frac{{\partial }^3}{\partial {\theta }^3}\\ -\gamma \frac{\partial }{\partial \theta }+\frac{{\partial }^3}{\partial {\theta }^3}& \begin{array}{l}-(\gamma +1)\frac{{\partial }^2}{\partial {\theta }^2}-{\left(\Omega +\omega \right)}^2\\ +{\displaystyle \sum _{i=1}^{N_m}\left[k_{mi}\cos \alpha \Delta \left(\theta -{\theta }_{mi}\right)\Delta \left(\theta -{\theta }_{mi}\right)\right]}\end{array}\end{array}\right]$$

In Equation (21), the components of F are functions of θ and t. Obviously, they are the excitation forces in the radial and tangential directions, which are caused by rotating and revolving movement. At the left side of Equation (21), the rotating and revolving velocities are all involved. Therefore, the revolving movement will affect the free vibration of the ring.

2.1. Natural Characteristics

To obtain natural frequencies and modes of the planet ring, the external excitation, F, is set to 0. By using a separable solution $x=\chi {\mathrm{e}}^{\lambda t}$, the corresponding eigenvalue problem becomes the following:

$${\lambda }^{2}M\chi +\lambda \left(\Omega +\omega \right)G\chi +K\chi =0$$

(22)

For the non-axisymmetric discrete and partially distributed stiffnesses on the planet ring, the eigenvectors are not single nodal diameter (ND) modes anymore. Therefore, Galerkin’s method is applied to solve the natural frequencies and their modes for this non-axisymmetric case. The eigenvectors are given as a series of linearly independent vectors:

$$\chi ={\left\{\begin{array}{cc}{\displaystyle \sum _{n=-N}^N{c}_n{\varphi }_n}& {\displaystyle \sum _{m=-M}^M{d}_m{\phi }_m}\end{array}\right\}}^{\mathrm{T}}$$

(23)

where 2N + 1 and 2M + 1 are the numbers of orders of radial basis function, ϕ_n, and tangential base function, φφ_m, in Equation (23), respectively. The weight function ${\chi }^{\prime }={\left[\begin{array}{cc}\begin{array}{cc}{\overline{\varphi }}_j& 0;\end{array}& \begin{array}{cc}0& {\overline{\phi }}_k;\end{array}\end{array}\right]}^{\mathrm{T}}$ is used to construct the Galerkin target function, and the equations are expressed in matrix form, with a size of 2 (N + M + 1) × 2(N + M + 1):

$${\lambda }^2\left[M\right]x+\lambda \left(\Omega +\omega \right)\left[G\right]x+\left[K\right]x=0$$

(24)

where $x={\left\{c_{-N},\dots ,c_N,d_{-M},\dots ,d_M\right\}}^{\mathrm{T}}$,

$$\left[\mathbf{M}\right]=\left[\begin{array}{cc}\left[{\mathbf{M}}_u\right]& 0\\ 0& \left[{\mathbf{M}}_v\right]\end{array}\right],\left[\mathbf{G}\right]=\left[\begin{array}{cc}0& -\left[{\mathbf{G}}_{uv}\right]\\ \left[{\mathbf{G}}_{vu}\right]& 0\end{array}\right],\left[K\right]=\left[\begin{array}{cc}\left[K_u+K_b+K_{mr}\right]& \left[K_{uv}\right]\\ \left[K_{vu}\right]& \left[K_v+K_{mt}\right]\end{array}\right],$$

$$\begin{gathered} {\mathbf{M}}_u(j,n)={\displaystyle {\int }_0^{2\pi }{\overline{\varphi }}_j{\varphi }_n\mathrm{d}\theta ,{\mathbf{M}}_v(k,m)=}{\displaystyle {\int }_0^{2\pi }{\overline{\phi }}_k{\phi }_m\mathrm{d}\theta }, \\ {G}_{uv}(j,m)={\displaystyle {\int }_0^{2\pi }2{\overline{\varphi }}_j{\phi }_m\mathrm{d}\theta },G_{vu}(k,n)={\displaystyle {\int }_0^{2\pi }2{\overline{\phi }}_k{\varphi }_n\mathrm{d}\theta }, \end{gathered}$$

$$\begin{gathered} K_u(j,n)={\displaystyle {\int }_0^{2\pi }\left(\left[\gamma -{\left(\Omega +\omega \right)}^2\right]{\overline{\varphi }}_j{\varphi }_n+{\overline{\varphi }}_j{\varphi }_n^{⁗}-N_c^{\prime }{\overline{\varphi }}_j{\varphi }_n^{″}\right)\mathrm{d}\theta }, \\ {K}_{uv}(j,m)={\displaystyle {\int }_0^{2\pi }\left(\gamma {\overline{\varphi }}_j{\phi }_m^{\prime }-{\overline{\varphi }}_j{\phi }_m^{‴}\right)\mathrm{d}\theta },K_{vu}(k,n)={\displaystyle {\int }_0^{2\pi }\left(-\gamma {\overline{\phi }}_k{\varphi }_n^{\prime }+{\overline{\phi }}_k{\varphi }_n^{‴}\right)\mathrm{d}\theta }, \\ {K}_v(k,m)={\displaystyle {\int }_0^{2\pi }\left[-\left(\gamma +1\right){\overline{\phi }}_k{\phi }_m^{″}-{\left(\Omega +\omega \right)}^2{\overline{\phi }}_k{\phi }_m\right]\mathrm{d}\theta }, \end{gathered}$$

$$\begin{gathered} K_b(j,n)={\displaystyle {\int }_0^{2\pi }{\displaystyle \sum _{i=1}^{N_b}{k}_{bi}{\overline{\varphi }}_j\left({\theta }_{bi},t\right){\varphi }_n\left({\theta }_{bi},t\right)\mathrm{d}\theta }}, \\ {K}_{mr}(j,n)={\displaystyle {\int }_0^{2\pi }{\displaystyle \sum _{i=1}^{N_b}{k}_{mi}\sin \alpha {\overline{\varphi }}_j\left({\theta }_{mi},t\right){\varphi }_n\left({\theta }_{mi},t\right)\mathrm{d}\theta }}, \end{gathered}$$

$$K_{mt}(k,m)={\displaystyle {\int }_0^{2\pi }{\displaystyle \sum _{i=1}^{N_m}{k}_{mi}\cos \alpha {\overline{\varphi }}_k\left({\theta }_{mi},t\right){\varphi }_m\left({\theta }_{mi},t\right)\mathrm{d}\theta }}$$

After substituting Equation (23) into Equation (22), the order numbers of radial and circumferential basis functions are 2N + 1 and 2M + 1, respectively. According to the Galerkin method, each formula is multiplied by the conjugate function of each corresponding basis function and then integrated. Thus, 2 (N + M + 1) × 2 (N + M + 1) formulas can be obtained, namely the size of matrices [M], [G], and [K] are 2 (N + M + 1) × 2 (N + M + 1). Each element of these matrices can be calculated according to the element formula under Equation (24). Choosing proper basis functions ϕ_n (θ) and φ_m (θ), the eigenvalue problem of Equation (24) can be solved following the state space eigenvalue problem, as described in Reference [30], and the numerical calculation can be made by standard methods.

2.2. Forced Response

Equation (21) reveals that the rotating and revolving movement will result in external excitation to the planet ring. Accordingly, the forced response of the planet ring to rotating and revolving movement were investigated. The forced response is therefore assumed to be of the following form:

$$u(\theta ,t)={\displaystyle \sum _{n=0}^{N_f-1}\left[p_n(t)\cos n\theta +q_n(t)\sin n\theta \right]}$$

(25a)

$$v(\theta ,t)={\displaystyle \sum _{n=0}^{N_f-1}\left[{\xi }_n(t)\cos n\theta +{\eta }_n(t)\sin n\theta \right]}$$

(25b)

where $p_n(t),q_n(t),{\xi }_n(t),{\eta }_n(t)$ are the generalized coordinates, N_f is the number of truncated modal order. Equation (25) is inserted into Equation (21), and the orthogonality of cos(nθ) and sin(nθ) in a period interval of 2π is then used to discretize the equations. The equations are expressed in matrix form, as follows:

$$\tilde{\mathbf{M}}\ddot{\mathbf{y}}+\tilde{\mathbf{G}}\dot{\mathbf{y}}+\tilde{\mathbf{K}}\mathbf{y}=\tilde{\mathbf{Q}}$$

(26)

where $y={\left\{\begin{array}{cccc}{p}_n(t)& q_n(t)& {\xi }_n(t)& {\eta }_n(t)\end{array}\right\}}^{\mathrm{T}}$; $\tilde{\mathbf{M}},\tilde{\mathbf{G}},\tilde{\mathbf{K}}$ are the matrices with order 4N_f × 4N_f; and the excitation vector, $\tilde{\mathbf{Q}}$, is written in Appendix A. Accordingly, the forced response of Equation (21) can be obtained by Equation (26) with the solution of Equation (25).

3. Vibration Analysis

To analyze the natural vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses, basis functions ϕ_n = e^jnθ, φ_m = e^jmθ were chosen. Since this work is motivated by planetary gear systems, where the planet gear is an elastic thin ring acted by discrete gear meshes and support bearing, the parameters of the planet ring in a planetary gear system are given in Table 1. Since there is no reference to study the vibration of planet rings, the model in Figure 1 is degenerated by ignoring the revolving movement to validate the equations and programming of our work with Reference [22]. By using the parameters in Reference [22], eigenvalue loci of a free and rotating elastic ring for varying rotation speeds are calculated by our degenerated model, and the result is shown in Figure 2. It can be found that the result is the same as the result of Figure 4 in Reference [22].

3.1. Free Planet Ring

In practical applications, such as planetary gear systems, planet rings behave like free rings in idle status. Figure 3a shows the natural frequencies of a free planet ring, with parameters in Table 1 calculated by the Galerkin method. The natural frequencies have degeneracy two, which is typical in axisymmetric spinning systems, like spinning disks [31] and spinning disk-spindle systems [32]. Two natural frequencies with the same nodal diameter separate at non-zero rotation speed and become different as shown in Figure 3a. One represents vibration waves propagating in the positive θ direction, and the other represents vibration waves propagating in the negative θ direction, which corresponds to backward waves and forward waves, respectively. Moreover, the corresponding vibration mode for all the frequencies is dominated by only one nodal diameter. The representative vibration mode and its components at ω = 1 are shown in Figure 3b. The eigenvalues for the system are purely imaginary over the range of analyzed speeds. It can be found that 1ND backward frequencies intersect with abscissa at point C in Figure 3a. This indicates that possible unstable phenomena exist and the value at point C is the critical speed.

Frequency veering occurs in multiple regions with the increase of rotation speed, such as the areas V1, V2, V3, and V4 in Figure 3a, and V2 for 1ND and 2ND at ω = 3.5, etc. In these regions, the lower ND frequencies become larger than the higher ND frequencies, that is, the modes exchange here. Figure 4 shows the modes exchange with the frequency veering at V2 around ω = 3.5. All the other veering regions have similar behavior. This means that, with the increase of rotation speed, the high-order frequencies may be dominated by lower ND components in the vibration mode, and, conversely, the low-order frequencies may be dominated by high ND components.

3.2. Planet Ring with Stiffnesses

Figure 5a shows the natural frequencies of a planet ring with discrete stiffnesses and partially distributed stiffnesses in the original region from −π to 0. There are also frequency veering regions as the free planet ring shows. Since the discrete and partially distributed stiffnesses disrupt the axisymmetry of the ring, the eigenvalues have distinct properties at ω = 0, compared to Figure 3a. For instance, in the regions where ω = 0, there are no the same natural frequencies of any two vibration modes, which is quite different from that of free planet rings. The number of frequency veering regions is less than free planet rings. In addition, the natural frequencies are much higher than free planet rings because the discrete and partially distributed stiffnesses enhance the stiffness of the whole ring. Figure 5b illustrates the typical vibration modes (2nd to 6th order modes) of planet rings with discrete and partially distributed stiffnesses. It is obvious that the partially distributed stiffnesses partially constrain the vibration of planet rings, as shown in Figure 5b. Meanwhile, the vibration modes are dominated by several components, as shown in Figure 5b, instead of a single component in free planet rings.

3.3. Influences of Parameters

The influences of several parameters, such as the number of partially distributed stiffnesses (NPDS), area of partially distributed stiffnesses, and discrete and partially distributed stiffnesses, on the vibration properties of planet rings are investigated in the following.

Different types of bearings with the same outer diameter may have different numbers of rollers or balls. Therefore, the influence of numbers of balls or rollers, that is, the number of partially distributed stiffnesses, is analyzed. Figure 6 shows the natural frequencies of a planet ring with discrete stiffnesses and changing the number of partially distributed stiffnesses. With the increase of the number of partially distributed stiffnesses, the natural frequencies increase, as it increases the stiffness of the whole ring partially. Moreover, from Figure 6, it can be observed that the lower frequencies are affected slightly, while the higher frequencies are increasing sharply with NPDS. Therefore, it could be a method for designers to avoid some natural frequencies by changing the number of partially distributed stiffnesses.

In applications of planetary gear systems, the loads on the gears may change at different working status. The load changing on the planet gear rings may change the contact area with the bearings, that is, the number of partially distributed stiffnesses. To analyze the influence, the number of rolling elements is set to 36, and the change range of the area is set as 90°~180°. Figure 7 shows the natural frequency variation versus partially distributed area at a rotating speed of ω = 1. Figure 7 reveals that the support area will influence the natural frequencies heavily. The natural frequencies are all increase with the increase of support area, and it especially has a great influence on the higher-order natural frequencies. This means, in heavy load conditions, the natural frequencies will get lower, and in light load conditions, the natural frequencies will get higher.

Figure 8a shows the influence of partially distributed stiffnesses to natural frequencies. It can be seen that only higher-order frequencies are affected largely by partially distributed stiffnesses. Moreover, in the analysis range of partially distributed stiffnesses, there is no obvious frequency veering phenomenon in Figure 8a. To analyze the mode veering properties, a veering parameter, ξ_mn, is chosen, as done in Reference [25], and can be calculated as follows. Equation (24) can be rewritten in the state space form, as follows:

$$\lambda Ax+Bx=0$$

(27)

where

$$\chi ={[\lambda x,x]}^{\mathrm{T}},A=\left[\begin{array}{cc}M& 0\\ 0& K\end{array}\right],B=\left[\begin{array}{cc}\left(\Omega +\omega \right)G& K\\ -K& 0\end{array}\right],$$

then

$${\xi }_{mn}=\frac{〈{\chi }_m,{\lambda }_m{A}_1{\chi }_n+B_1{\chi }_n〉〈{\chi }_n,{\lambda }_n{A}_1{\chi }_n+B_1{\chi }_m〉}{{\lambda }_m-{\lambda }_n}$$

(28)

where m, n denotes the order of frequencies; <•,•> denotes the inner production; A₁ and B₁ are the first derivative of A and B, with respect parameters such as stiffness here. For clarity, only 5–10 order veering parameters are calculated as shown in Figure 8c. It can be found that mode veering can be indicated by the veering parameter. Moreover, mode veering mainly appears in the range of smaller partial distributed stiffnesses. Figure 8b shows the influence of discrete stiffnesses on natural frequencies. As seen in Figure 8b a majority of frequencies are affected slightly by the changing of discrete stiffnesses and only a few frequencies are affected comparatively large. Besides, since the discrete stiffnesses are time-varying in gear systems, the influence of time-varying of discrete stiffnesses can also be analyzed by Figure 8b. Since the time-varying part of discrete stiffnesses is very small compared to their mean value, its influence is only a small segment of abscissa in Figure 8b. It also can be found that there is no mode veering in a large range of discrete stiffnesses, and mode veering appears only when discrete stiffness is very small.

3.4. Forced Response

In a planetary gear set, the rotating speed and revolving speed of a planet ring are proportional. In this work, the ratio is set to i₀ = 0.1825, according to a practical application. Therefore, the excitation of rotating and revolving movement to the planet ring can be expressed by rotating or revolving speed. Here, the rotating speed is chosen; thus, the amplitude of excitation is Ω²R₀/R, and the excitation angular frequency is ω. It should be noted that the excitation only acts on the first order component, as shown in $\tilde{Q}$ in Equation (26), after carrying out the orthogonal operation. In addition, Equation (26) represents a nonlinear dynamic system, since θ_bi and θ_mi are time dependent, as described in Section 2; then the stiffness matrix, K, will change with respect to time, t. Therefore, Equation (26) will be solved by the classical numerical method.

Figure 9 shows the forced responses of the planet ring under the excitation of rotating and revolving movement at the rotating speed ω = 1. Here the forced responses of radial direction at different positions are illustrated. It is obvious that, although the forced responses are nonlinear, as shown in Figure 9e, the forced responses at different positions exhibit some periodic behavior with a quasi-period of rotating speed, ω, as shown in Figure 9a–d. The regions with small amplitudes in Figure 9a–d mean that the partially distributed stiffnesses are passing through this position, and the regions with large amplitude mean that the partially distributed stiffnesses are out of the position. These phenomena imply that the forced responses of the planet ring will vary with a quasi-period of its rotating speed, and the partially distributed stiffnesses will suppress its vibrations.

Figure 10 shows the forced responses of the planet ring in different moments in a period, T, of rotating speed at ω = 1. It can be found that the vibration of the planet ring is small in partially distributed stiffnesses area, and the vibration in other areas is large and irregular, that is, nonlinear. The forced responses of the planet ring almost rotate with its rotating movement.

4. Conclusions

This study investigated the vibration of high-speed planet rings with discrete and partially distributed stiffnesses. The governing equations were derived by Hamilton’s principle based on stationary and co-rotating coordinates. The governing equation contains the revolving movement and discrete and partially distributed stiffnesses boundary conditions, where the differences with other references were given. The eigenvalue problem of the governing equations was solved by using Galerkin’s method and state space matrix.

The vibration properties of a free planet ring based on parameters of a planetary gear set were investigated. The results show that there are frequency degeneracies when the rotation speed is not zero, and frequency veering occurs with the increase of rotation speed. In these regions, frequencies veer and exchange vibration mode. Each vibration mode of the free planet ring is dominated by a single nodal diameter component. When the partially distributed stiffness and discrete stiffness are acted on the planet ring, the vibration properties vary heavily. The frequencies have a distinct difference compared to the free planet ring. The phenomenon of frequency veering occurs with the increase of rotation speed. Each vibration mode is comprised of several nodal diameter components instead of a single component in free states. In addition, partially distributed stiffnesses will change the vibration mode heavily.

The influences of parameters on vibration properties were investigated. The results show that the distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. Changing the number of partially distributed stiffnesses can change the natural frequencies of the planet ring. In heavy load conditions, the natural frequencies will get lower, and in light load conditions, the natural frequencies will get higher. The discrete stiffnesses affect the natural frequencies slightly, while the partially distributed stiffnesses affect the high order natural frequencies heavily.

Finally, the forced responses of the planet ring caused by rotating and revolving movement were investigated. The results show that the forced responses are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.