Study on the nonlinear stiffness variation and its effect on the dynamics of the rigid rotor-bearing system

Abstract

In this paper, to obtain the nonlinear stiffness behavior of ball bearing and its effect on the dynamics of the rotor-bearing system, a comprehensive theoretical study on the typical rigid rotor system supported on ball bearings is presented, and the stiffness matrix of the rotor-bearing system is derived based on a novel mathematical model of the supported ball bearings with the contact angles as the new unknown parameters. Then the change rules of the nonlinear stiffness behaviors and dynamics of the rotor system versus the initial preload, rotating speed and external load are analyzed. The results demonstrate that the rigid rotor-bearing system presents the stiffness change trend of the first “soft-spring” and then “hard-spring” under large preload and low speed condition, while presents the obvious nonlinear “hard-spring” stiffness characteristic under small preload and high speed condition.

Keywords

Rotor-bearing system nonlinear stiffness characteristic rotor dynamics natural frequency

1. Introduction

Because of the advantages of the low cost and high efficiency, ball bearings are widely used in various mechanical systems. As the core functional component, the mechanical performance characterization of the ball bearing also plays a key role on the rotor dynamics. In the early studies, ball bearings are usually simplified to linear spring-damper elements for the rotor modeling and dynamic performance prediction. Simple linear assumptions not only cause large errors in model calculations, but also further lead to the omission of key feature information of the system. With the continuous improvement of theoretical study and product performance, a larger number of scholars are engaged in the research of the nonlinear characteristics of ball bearings, and the researches mainly focus the nonlinear time-varying stiffness characteristic of ball bearings that is generated by the periodic change of the number and position distribution of the balls in the loaded area, and the nonlinear VC (varying-compliance) vibration phenomena of the rotor-ball bearing system have been extensively studied (Fang et al., 2021a; Han and Chu, 2015; Harsha, 2004; Harsha et al., 2004; Jin et al., 2021; Kankar et al., 2012; Lu et al., 2020, 2022; Miao et al., 2021; Nataraj and Harsha, 2008; Pan et al., 2021; Upadhyay et al., 2009). However, for the axial-preloaded ball bearing, the periodical stiffness fluctuation behavior caused by the internal clearance of ball bearing is effectively suppressed (Fang et al., 2019).

In addition to the nonlinear time-varying stiffness characteristic, the ball bearing also exhibits significant nonlinear time-invariant stiffness characteristic. And such nonlinear stiffness characteristic mainly exists in the rotor-bearing systems with good preload condition (Cao and Altintas, 2004; Cao et al., 2018; Li et al., 2020; Xu et al., 2020). The nonlinear time-invariant stiffness characteristic of the axially-preloaded ball bearing is determined by:

(1) Contact nonlinearity: Based on the Hertzian point-contact law, the contact loads and the deformations of ball-raceway contacts satisfy the fractional-stiffness characteristic ( $Q = K δ^{1.5}$ ).

(2) Geometric nonlinearity: The stiffness of the rolling bearing can be regarded as the superposition effect of the contact stiffness between the ball-inner/outer rings, and the contact loads of ball-inner/outer rings are not collinear due to the inertia force of balls (i.e., the centrifugal load and gyroscopic moment), that is, the linear superposition of contact stiffness cannot be performed, resulting in the geometric nonlinear characteristic.

On the basis, the time-invariant stiffness characteristic of ball bearings is simultaneously affected by the external load and rotating speed. Li et al. (2018, 2019) conducted series studies on the nonlinear axial stiffness characteristics of the rotor-ball bearing system by taking in account of the thermal deformation and preload. However, the model ignores the inertial force of the ball, so it is not suitable for the study of the rotor nonlinear stiffness variation at high-speed range. Based on the Jones model and finite element method, Cao et al. (2011) discussed the change law of radial nonlinear stiffness of the rotor-ball bearing system under two different preload mechanisms. Besides, the influence of the rotating speed on rotor stiffness variation is discussed, the results show that the rotor stiffness presents a significant attenuation behavior as the speed rises, and the above-mentioned stiffness softening phenomenon can be restrained to a certain extent by using the fix-position preloading method. Recently, the experimental studies on the spindle dynamic performance (Matsubara et al., 2013, 2014, 2015) shown that the nonlinear time-invariant stiffness characteristic of the ball bearings can change the amplitude-frequency response characteristics of the rotor system near the resonance condition. At the same time, research on nonlinear vibration and control methods of rotor system supported by the magnetic bearings has been widely carried out in recent years (Kandil et al., 2020, 2021, 2024; Saeed et al., 2021, 2022, 2023a, 2023b). Compared with ball bearing -rotor systems, they exhibit more diverse nonlinear response behaviors.

In summary, for the rotor systems under good preloading conditions such as the aero-engine and high-precision spindle, it is necessary to systematically study the nonlinear time-invariant stiffness characteristic of the ball bearing (Fang et al., 2021b; Liu et al., 2019; Noel et al., 2013; Xia et al., 2014; Yang et al., 2016, 2018; Zhang et al., 2020), so as to obtain its core the variation rules and the main influencing factors.

In this paper, an accurate theoretical model for a typical rigid rotor system supported on the ball bearings is proposed, and the complete stiffness matrix of the rotor system is analytically derived based on a novel mathematical model of the supported ball bearing with the contact angles as the unknown parameters. And then the change rules of the rotor stiffness versus the loads and speeds are discussed, and the detailed explanation of the intrinsic mechanism of the rotor nonlinear stiffness variation is given, then the effect of the nonlinear stiffness variation on the rotor low-order resonance frequencies is analyzed.

2. Theoretical modeling

2.1. Analytical determination of the rotor nonlinear stiffness

As shown in Figure 1, the typical rigid rotor system supported on the ball bearings is given, and this structure can be regarded as a simplification of the fixed-position preloading spindle system with the DB configuration. Then three different coordinate systems are built to determine the rotor nonlinear stiffness variation characteristic: (i) the central coordinate system O^M-X^MY^MZ^M, it is used to describe the loading and deformation of the central rotor; (ii) the left-local coordinate system O^L-X^LY^LZ^L, it is used to describe the loading and deformation of the left-hand side ball bearing; (iii) the right-local coordinate system O^R-X^RY^RZ^R, it is used to describe the loading and deformation of the right-hand side ball bearing.

On the basis, one can assume that the rotor is subjected to a combined loads F^M = [ $F_{x}^{M}$ , $F_{y}^{M}$ , $F_{z}^{M}$ , $M_{y}^{M}$ , $M_{z}^{M}$ ] at the coordinate origin O^M (If the external load does not act on the origin O^M, one can equivalently translate it to the origin position by a simple vector translation). As shown in Figure 1, the following expression is given:

{\begin{cases} F_{x}^{L} - F_{x}^{R} = F_{x}^{M} \\ F_{y}^{L} + F_{y}^{R} = F_{y}^{M} \\ F_{z}^{L} - F_{z}^{R} = F_{z}^{M} \\ M_{y}^{L} + M_{y}^{R} + F_{z}^{L} L^{L} + F_{z}^{R} L^{R} = M_{y}^{M} \\ M_{z}^{L} - M_{z}^{R} - F_{y}^{L} L^{L} + F_{y}^{R} L^{R} = M_{z}^{M} \end{cases}

(1)

where the superscripts L and R represent the relevant variables of the left- and right-side bearings, respectively.

F_{x}

F_{y}

F_{z}

M_{y}

M_{z}

, respectively, denote the nonlinear restoring forces of supporting ball bearings, and the L^L and L^R is the center distances between the points O^M and O^L or O^R. One can find that the external loads act on the central rotor are balanced by the supporting forces of the ball bearings, so the above expression can be further rewritten as:

F^{M} = T_{1} (L^{L}) F^{L} + T_{2} (L^{R}) F^{R}

(2)

with

\begin{array}{l} T_{1} (L^{L}) = [\begin{array}{l} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & L^{L} & 1 & 0 \\ 0 & - L^{L} & 0 & 0 & 1 \end{array}] & and & T_{2} (L^{R}) = [\begin{array}{l} - 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 0 \\ 0 & 0 & L^{R} & 1 & 0 \\ 0 & L^{R} & 0 & 0 & - 1 \end{array}] \end{array}

(3)

\begin{array}{l} F^{L} = [\begin{array}{l} F_{x}^{L} \\ F_{y}^{L} \\ F_{z}^{L} \\ M_{y}^{L} \\ M_{z}^{L} \end{array}] & and & F^{R} = [\begin{array}{l} F_{x}^{R} \\ F_{y}^{R} \\ F_{z}^{R} \\ M_{y}^{R} \\ M_{z}^{R} \end{array}] \end{array}

(4)

For the rotor system under arbitrary operating conditions, its stiffness matrix can be calculated by iterative operation of the above nonlinear equations, the detailed derivation process of the rotor stiffness is given:

Κ_{r o t o r} = \frac{\partial F^{M}}{\partial d^{M}} = N_{1} (d^{L}) \frac{\partial F^{L}}{\partial d^{M}} + N_{2} (d^{R}) \frac{\partial F^{R}}{\partial d^{M}}

(5)

where the vector d ^M denotes the displacements of the central rotor at the coordinate origin O^M. On this basis, the vectors d ^L and d ^R denotes the relative displacements of the supporting ball bearings at the coordinate origins O^L and O^R under the fix outer ring assumption:

{\begin{cases} d^{L} = T_{1}^{T} (d^{L}) d^{M} + {[\begin{array}{l} δ_{P}^{L} & 0 & 0 & 0 & 0 \end{array}]}^{T} \\ d^{R} = T_{2}^{T} (d^{R}) d^{M} + {[\begin{array}{l} δ_{P}^{R} & 0 & 0 & 0 & 0 \end{array}]}^{T} \end{cases}

(6)

where the axial displacements

δ_{P}^{L}

and

δ_{P}^{R}

are added to guarantee that the supporting ball bearings are in a certain initial preload condition. On the basis, the equation (5) can be rewritten as:

\begin{array}{l} Κ_{r o t o r} = T_{1} (d^{L}) \frac{\partial F^{L}}{\partial d^{L}} \frac{\partial d^{L}}{d^{M}} + T_{2} (d^{R}) \frac{\partial F^{R}}{\partial d^{R}} \frac{\partial d^{R}}{d^{M}} \\ = T_{1} (d^{L}) \frac{\partial F^{L}}{\partial d^{L}} T_{1}^{T} (d^{L}) + T_{2} (d^{R}) \frac{\partial F^{R}}{\partial d^{R}} T_{2}^{T} (d^{R}) \\ = T_{1} (d^{L}) K^{L} T_{1}^{T} (d^{L}) + T_{2} (d^{R}) K^{R} T_{2}^{T} (d^{R}) \end{array}

(7)

where K^R and K^L are the supporting stiffness matrices of ball bearings on the right-hand side and left-hand side, respectively.

Therefore, the nonlinear stiffness variation of the rotor-bearing system is simultaneously determined by the supporting stiffness of ball bearings on both sides.

2.2. The improved mathematical model for the stiffness calculation of the single-side ball bearing

As presented in Figure 2, to get the stiffness matrix of the single-side ball bearing operating at the combined loads (F = [F_x, F_y, F_z, M_y, M_z ]), it is assumed that the rings relative displacement is d (d = [ $δ_{x}$ , $δ_{y}$ , $δ_{z}$ , $θ_{y}$ , $θ_{z}$ ]). Then one can obtain:

K = \frac{\partial F^{T}}{\partial d}

(8)

In addition, since the effect of the ball inertia forces need be considered to predict the nonlinear stiffness characteristic of ball bearings at different rotating speeds, the improved quasi-static method without the Raceway Control Theory is chosen in the subsequence analysis (Fang et al., 2021b).

For the ball at any the position angle $ψ_{k}$ ( $ψ_{k} = 2 π (k - 1) / Z + ψ_{0}$ , $ψ_{0}$ is the static position angle for the 1^st ball), due to the combined action of the external loads and ball inertia forces, the center positions of the ball and inner-raceway contact curvature, respectively, change from points $O_{b}$ and $O_{i}$ to $O_{b}^{'}$ and $O_{i}^{'}$ . Therefore, the plane displacements of the inner-ring are written as:

{\begin{cases} δ_{a k} = δ_{x} - \frac{d_{m}}{2} \cos ψ_{k} θ_{z} + \frac{d_{m}}{2} \sin ψ_{k} θ_{y} \\ δ_{r k} = \sin ψ_{k} δ_{z} + \cos ψ_{k} δ_{y} \end{cases}

(9)

where d_m is the bearing pitch diameter. Besides, the curvature center distances between the raceways after loading are given as:

{\begin{cases} D_{1 k} = (r_{i} + r_{o} - D) \sin α^{0} + δ_{a k} \\ D_{2 k} = (r_{i} + r_{o} - D) \cos α^{0} + δ_{r k} \end{cases}

(10)

where r_i and r_o denote the contact curvature radii for the inner- and outer-raceway, the contact angle

α^{0}

at static is given as:

α^{0} = \arccos [1 - \frac{P_{d}}{2 (r_{i} + r_{o} - D)}]

(11)

where the radial clearance P_d is given as:

P_{d} = d_{o} - d_{i} - D

(12)

In addition, based on the trigonometric relations of the ball-raceway contacts as shown in Figure 2(b), the elastic contact deformation $δ_{i k}$ and $δ_{o k}$ are written as:

[\begin{array}{l} δ_{i k} \\ δ_{o k} \end{array}] = {[\begin{array}{l} \sin α_{i k} & \sin α_{o k} \\ \cos α_{i k} & \cos α_{o k} \end{array}]}^{- 1} \times [\begin{array}{l} A_{i k} \\ A_{o k} \end{array}] - [\begin{array}{l} r_{i} \\ r_{o} \end{array}] + [\begin{array}{l} 0.5 D \\ 0.5 D \end{array}]

(13)

Besides, the nonlinear restoring forces (P_ik and P_ok) can be written based on the Hertzian point-contact theory:

{\begin{cases} P_{i k} = K_{i k} δ_{i k}^{1.5} & δ_{i k} \geq 0 \\ P_{o k} = K_{o k} δ_{o k}^{1.5} & δ_{o k} \geq 0 \end{cases}

(14)

{\begin{cases} P_{i k} = 0 & δ_{i k} < 0 \\ P_{o k} = tol & δ_{o k} < 0 \end{cases}

(15)

where K_ik and K_ok denote the deformation-load coefficients, and the state of the ball-inner raceway separation (i.e.,

P_{i k} = 0

and

δ_{i k} < 0

) is considered to ensure the effectiveness of the algorithm.

In particular, the variables of the contact angles $α_{i k}$ and $α_{o k}$ are chosen as the basic unknown variables which is different from the traditional studies on ball bearings. It can be seen from Figure 3(a), the ball is subjected to the combined loads of the gyroscopic moment (M_gk), the centrifugal force (F_ck), the nonlinear restoring loads (P_ik, P_ok) and the friction forces (f_ik, f_ok), so the ball equilibrium equation under the vector form is given:

\vec{F_{c k}} + \vec{P_{i k}} + \vec{f_{i k}} + \vec{P_{o k}} + \vec{f_{o k}} = 0

(16)

Besides, the ball-raceway tangential friction forces are calculated by the gyroscopic moment M_gk of ball (Fang et al., 2021b):

{\begin{cases} f_{i k} = P_{i k} μ = \frac{2 M_{g k} P_{i k}}{D (P_{i k} + P_{o k})} \\ f_{o k} = P_{o k} μ = \frac{2 M_{g k} P_{o k}}{D (P_{i k} + P_{o k})} \end{cases}

(17)

Then, equation (16) is further simplified by the vector operations:

\vec{F_{i k}} + \vec{F_{o k}} + \vec{F_{c k}} = 0

(18)

with

{\begin{cases} \vec{F_{i k}} = \vec{P_{i k}} + \vec{f_{i k}} \\ \vec{F_{o k}} = \vec{P_{o k}} + \vec{f_{o k}} \end{cases}

(19)

Based on the sine theorem (Zhang et al., 2020):

\frac{\sin θ_{o k}}{F_{i k}} = \frac{\sin θ_{i k}}{F_{o k}} = \frac{\sin (θ_{i k} - θ_{o k})}{F_{c k}}

(20)

with

{\begin{cases} α_{i k} = θ_{i k} + Δ α_{i k} \\ α_{o k} = θ_{o k} + Δ α_{o k} \end{cases}

(21)

Combining with equation (19), one can obtain:

Δ α = Δ α_{i k} = Δ α_{o k} = \arctan μ = \arctan [\frac{2 M_{g k}}{D (P_{i k} + P_{o k})}]

(22)

and

θ_{i k} - θ_{o k} = α_{i k} - α_{o k}

(23)

On the basis, the nonlinear restoring loads between the raceways and balls are given as:

[\begin{array}{l} P_{i k} \\ P_{o k} \end{array}] = [\begin{array}{l} F_{i k} \\ F_{o k} \end{array}] \cos Δ α = [\begin{array}{l} \frac{\sin (α_{o k} - Δ α)}{\sin (α_{i k} - α_{o k})} \\ \frac{\sin (α_{i k} - Δ α)}{\sin (α_{i k} - α_{o k})} \end{array}] F_{c k} \cos Δ α

(24)

According to the above analysis, the unknown variables $α_{i k}$ and $α_{o k}$ of the local ball can be obtained by the iterative operation of equation (18) for any given vector d.

On the basis, as shown in Figure 2(a), one can further give the following equilibrium equations for the inner ring:

[\begin{array}{l} F_{x} \\ F_{y} \\ F_{z} \\ M_{y} \\ M_{z} \end{array}] = \sum_{1}^{Z} G [δ_{i k}] {[\begin{array}{l} 1 & 0 \\ 0 & \cos ψ_{k} \\ 0 & \sin ψ_{k} \\ \frac{d_{m}}{2} \sin ψ_{k} & 0 \\ - \frac{d_{m}}{2} \cos ψ_{k} & 0 \end{array}] \times [\begin{array}{l} \sin θ_{i k} \\ \cos θ_{i k} \end{array}] \times F_{i k}}

(25)

with

G [δ_{i k}] = {\begin{cases} 1 & δ_{i k} \geq 0 \\ 0 & δ_{i k} < 0 \end{cases}

(26)

Based on the combined explicit and implicit differential method (Xia et al., 2014), the bearing stiffness matrix (equation (8)) need be further expanded by the intermediate variables $α_{i k}$ and $α_{o k}$ :

K = \sum_{1}^{Z} G [δ_{i k}] {\begin{cases} [\begin{array}{l} 1 & 0 \\ 0 & \cos ψ_{k} \\ 0 & \sin ψ_{k} \\ \frac{d_{m}}{2} \sin ψ_{k} & 0 \\ - \frac{d_{m}}{2} \cos ψ_{k} & 0 \end{array}] \times (\frac{\partial [\begin{array}{l} \sin θ_{i k} \\ \cos θ_{i k} \end{array}]}{\partial υ_{k}} \times F_{i k} + [\begin{array}{l} \sin θ_{i k} \\ \cos θ_{i k} \end{array}] \times \frac{\partial F_{i k}}{\partial υ_{k}}) . . . \\ . . . \times [\begin{array}{l} \frac{\partial υ_{k}^{T}}{\partial δ_{x}} & \frac{\partial υ_{k}^{T}}{\partial δ_{y}} & \frac{\partial υ_{k}^{T}}{\partial δ_{z}} & \frac{\partial υ_{k}^{T}}{\partial θ_{y}} & \frac{\partial υ_{k}^{T}}{\partial θ_{z}} \end{array}] \end{cases}}

(27)

where the intermediate variable set $υ_{k}$ = { $α_{i k}$ , $α_{o k}$ , $D_{1 k}$ , $D_{2 k}$ } is introduced to simplify the calculation process, and the partial differential items $\frac{\partial υ_{k}}{\partial d}$ can be further expanded as:

[\begin{array}{l} \frac{\partial υ_{k}^{T}}{\partial δ_{x}} & \frac{\partial υ_{k}^{T}}{\partial δ_{y}} & \frac{\partial υ_{k}^{T}}{\partial δ_{z}} & \frac{\partial υ_{k}^{T}}{\partial θ_{y}} & \frac{\partial υ_{k}^{T}}{\partial θ_{z}} \end{array}] = [\begin{array}{l} \frac{\partial α_{1 k}}{\partial D_{1 k}} & \frac{\partial α_{1 k}}{\partial D_{2 k}} \\ \frac{\partial α_{2 k}}{\partial D_{1 k}} & \frac{\partial α_{2 k}}{\partial D_{2 k}} \\ 1 & 0 \\ 0 & 1 \end{array}] \times [\begin{array}{l} \frac{\partial D_{1 k}}{\partial δ_{x}} & \frac{\partial D_{1 k}}{\partial δ_{y}} & \frac{\partial D_{1 k}}{\partial δ_{z}} & \frac{\partial D_{1 k}}{\partial θ_{y}} & \frac{\partial D_{1 k}}{\partial θ_{z}} \\ \frac{\partial D_{2 k}}{\partial δ_{x}} & \frac{\partial D_{2 k}}{\partial δ_{y}} & \frac{\partial D_{2 k}}{\partial δ_{z}} & \frac{\partial D_{2 k}}{\partial θ_{y}} & \frac{\partial D_{2 k}}{\partial θ_{z}} \end{array}]

(28)

where the explicit differential items $\frac{\partial D_{1 k}}{\partial d}$ and $\frac{\partial D_{2 k}}{\partial d}$ can be obtained by direct differentiation, while the implicit items $\frac{\partial α_{1 k}}{\partial D_{1 k}}$ , $\frac{\partial α_{1 k}}{\partial D_{2 k}}$ , $\frac{\partial α_{2 k}}{\partial D_{1 k}}$ , $\frac{\partial α_{2 k}}{\partial D_{2 k}}$ are calculate by the implicit differential operations:

\begin{array}{l} \frac{\partial α_{i k}}{\partial D_{1 k}} = - \frac{1}{J} \frac{\partial (E q_{1 k}, E q_{2 k})}{\partial (D_{1 k}, α_{o k})} & \frac{\partial α_{i k}}{\partial D_{2 k}} = - \frac{1}{J} \frac{\partial (E q_{1 k}, E q_{2 k})}{\partial (D_{2 k}, α_{o k})} \\ \frac{\partial α_{o k}}{\partial D_{1 k}} = - \frac{1}{J} \frac{\partial (E q_{1 k}, E q_{2 k})}{\partial (α_{i k}, D_{1 k})} & \frac{\partial α_{o k}}{\partial D_{2 k}} = - \frac{1}{J} \frac{\partial (E q_{1 k}, E q_{2 k})}{\partial (α_{o k}, D_{2 k})} \end{array}

(29)

where the determinant J is written as

J_{m} = [\begin{array}{l} \frac{\partial E q_{1 k}}{\partial ε_{k}} \\ \frac{\partial E q_{2 k}}{\partial ε_{k}} \end{array}] = [\begin{array}{l} \frac{\partial E q_{1 k}}{\partial α_{i k}} & \frac{\partial E q_{1 k}}{\partial α_{i k}} \\ \frac{\partial E q_{2 k}}{\partial α_{o k}} & \frac{\partial E q_{2 k}}{\partial α_{o k}} \end{array}]

(30)

and

J = \det J_{m}

(31)

where

E q_{1 k}

and

E q_{2 k}

are the ball nonlinear equilibrium equations (equation (24)).

Above all, the variables of $α_{i k}$ and $α_{o k}$ are chosen as the basic unknown parameters instead of the contact deformations $δ_{i k}$ and $δ_{o k}$ , it has two main advantages: (1) the contact angles have a clear physical meaning, and the initial values of $α_{i k}$ and $α_{o k}$ can be inferred through the initial contact angle $α^{0}$ of ball bearing, which can greatly improve the convergence of the iteration algorithm; (2) The use of contact angles as the iteration variables can effectively reduce the number of local variables in local balls of ball bearing (i.e., the number of unknown variables is reduced from 4*Z to 2*Z), thereby improving the overall computational efficiency of the iteration algorithm.

3. Calculation results and discussion

Based on the above theoretical model, the nonlinear stiffness variation rules of the typical rotor system supported on the ball bearings can be obtained, and the detailed parameters of the supporting ball bearings and central rotor are given in Table 1.

Table 1.

The structural parameters of the supporting ball bearings and central rotor.

Parameters	Bearing	Parameters	Rotor
Bearing’ contact curvature radius r_i/mm	5.33	Rotor’s mass m_r/kg	24.3
Bearing’ contact curvature radius r_o/mm	5.05	Rotor’s diametral inertia moment I_d/kg m²	0.048
Bearing’ contact raceway diameter d_i/mm	80.45	Rotor’s polar inertia moment I_p/kg m²	0.096
Bearing’ contact raceway diameter d_o/mm	99.56	The center spacing between O^L and O^M d^L/m	0.20
Ball’s number Z	25	The center spacing between O^R and O^M d^R/m	0.20
Ball’s diameter D/mm	9.525	Rotor’s length/m	0.7
Bearing’ pitch diameter d_m/mm	90

3.1. Nonlinear stiffness variation of the axially-loaded rotor-bearing system

The nonlinear stiffness variation curves versus axial load of the rotor operating at 5000 r/min with three different preload displacements (i.e., 5 μm, 10 μm and 11.5 μm) are presented in Figure 4. One can find that the initial preload plays a key role on the change rules of the rotor nonlinear stiffness. First, when the preload force is relatively small (i.e., 5 μm), the rotor nonlinear stiffness shows an obvious increase with the axial force (i.e., the stiffness growth rate within the given load range exceeds 50% and the rotor system exhibits the nonlinear hard-spring stiffness characteristic). Then, as the preload displacement increases from 5 μm to 10 μm, although the rotor stiffness still increases with the axial load, the increase rate has slowed down (i.e., the stiffness growth rate within the given load range has reduced to around 5%). Finally, when the preload displacement is further increased to 11.5 μm, the variation law of the rotor stiffness changes significantly, showing the change rule that the stiffness decreases slowly first and then increases rapidly versus the axial force.

In order to further explain the above change rules of the rotor nonlinear stiffness, it is necessary to return back to analyze stiffness results of the single-row ball bearing. Considering that the change rules of the rotor axial and radial stiffness are similar, so only the axial stiffness variation is chosen to be studied. One can see from Figure 5, both the axial stiffness and its variation rate with axial load and displacement for the single-side ball bearing at 5000 r/min are presented. It can be found that the axial stiffness keeps increasing with the axial load or axial displacement, but the growth rate shows an interesting change rule of increasing first and then decreasing (i.e., the axial load at the inflection point is approximately 125 N).

Then, back to the proposed rotor system, when the rotor system is acted by an axial force along the X^M positive semi-axis, one can assumed that the axial displacement of the ball bearing on left-hand side increases by $Δ δ_{x}$ , while the ball bearing on right-hand side decreases with the equal value. The axial stiffness variation rate of the rotor system is:

{\dot{K}}_{x}^{r o t o r} = {\dot{K}}_{x}^{L} + {\dot{K}}_{x}^{R}

(32)

where

{\dot{K}}_{x}^{L}

and

{\dot{K}}_{x}^{R}

, respectively, denote the axial stiffness variation rates of the ball bearings on left- or right-hand side.

First, as shown in Figure 6(a), for the initial preload displacement is 5 μm, no matter the axial load ( $Δ δ_{x}$ ) is large or small, the value of ${\dot{K}}_{x}^{L}$ is always greater than the value of $- {\dot{K}}_{x}^{R}$ , it means that the stiffness growth rate of the left-hand side ball bearing is always greater than the stiffness reduction rate of the right-hand side ball bearing. So the rotor nonlinear stiffness resents a constant increasing trend with the axial load (Figure 4(a)). Then, as the preload further increases to 11.5 μm, when the axial force is small, the value of ${\dot{K}}_{x}^{L}$ is less than the values of $- {\dot{K}}_{x}^{R}$ , the stiffness growth rate of the left-hand side ball bearing is less than the stiffness reduction rate of the right-hand side ball bearing, so the rotor axial stiffness shows an downward trend at small axial load range, when the axial external force is large, the value of ${\dot{K}}_{x}^{L}$ becomes larger than the value of $- {\dot{K}}_{x}^{R}$ , the rotor axial stiffness becomes an upward trend at larger axial load range. In summary, the segmental variation law of the rotor axial stiffness is presented ((Figure 4(c)).

Then, the nonlinear stiffness variation for the axially-loaded rotor operating at different speeds is given in Figure 7. One can find that the rotating speed plays a significant role on the nonlinear stiffness variation through the geometrical nonlinearity due to the ball centrifugal force. First, when the rotating speed is relatively low at 2000 r/min, the rotor stiffness first increases and then decreases with the axial load. Then, when the speed rises from 2000 r/min to 4000 r/min, the attenuation trend of the rotor stiffness at the small axial load range become gradual (i.e., the axial load range corresponding to stiffness attenuation has decreased from 600 N to 400 N). Finally, when the rotating speed increases to 6000 r/min, no attenuation trend of the rotor stiffness at the small load range is presented, and the rotor stiffness keeps increasing with the axial load (i.e., the stiffness growth rate within the given load range exceeds 20%).

Accordingly, the axial stiffness variation rates of the single-side ball bearing with axial load or axial displacement are given Figure 8. Similar to the result curves as presented in Figure 5(b-2), the growth rate of the rotor axial stiffness showed a change rule of increasing first and then decreasing, and the axial load or axial displacement corresponding to the maximum stiffness increasing rate gradually increases with the rotating speed, which can be used to explain the different change trends of the rotor nonlinear stiffness at different speed conditions.

On this basis, the dynamic parameters of rotor natural frequencies are further analyzed. One can see from Figure 9, both the variation curves of the rotor nonlinear stiffness and its equivalent natural frequencies with the axial load are presented (4000 r/min, 12 μm). One can find that the rotor stiffness in both the axial and radial directions first increases then decreases with the axial load, while the radial stiffness presents the reduction trend in a larger axial load range. Besides, one can see from Figure 9(b), the 1^st-order equivalent resonant frequency corresponding to the rotor axial stiffness shows a change rule similar to that of the axial stiffness, but the 2^nd-order equivalent resonant frequency corresponding to the rotor radial stiffness is quite different from the radial stiffness change curve, which shows an obvious downward trend with the axial load. The reason for the rapid decrease of the 2^nd-order equivalent natural frequency may be related to the off-diagonal stiffness elements (Fang et al., 2021b).

3.2. Nonlinear stiffness variation of the radially-loaded rotor-bearing system

It can be seen from Figure 10, the variation curves of the nonlinear stiffness of the rotor system with different initial preload displacements and the radial loads are given. It is interesting that the variation trend of the rotor stiffness versus radial load is very similar to results of the rotor stiffness versus the axial load. First, when the rotor system is under a small initial preload displacement of 5 μm, the rotor stiffness in both the axial and radial directions show an obvious increase with the radial force (i.e., the stiffness growth rate within the given load range exceeds 60% and the rotor system also exhibits nonlinear hard-spring stiffness characteristic). Then, as the preload displacement increases from 5 μm to 10 μm, the rotor stiffness shows a slowly increasing trend with the radial load (i.e., the stiffness growth rate within the given load range has reduced to 5%). Finally, when the preload displacement is further increased to 11.5 μm, the rotor stiffness first decreases and then increases with radial load (i.e., the stiffness fluctuation rate within the given load range is less than 3%).

Then, the influence of the rotating speed on the rotor stiffness variation with the radial load is given in Figure 11. The result curves are also similar to those of the rotor stiffness variation results versus the axial load. First, when the rotating speed is relatively low (2000 r/min), the rotor stiffness first increases and then decreases with radial load. Then, when the speed increases from 2000 r/min to 4000 r/min, the attenuation trend of the rotor stiffness at the small radial load range become gradual (i.e., the radial load range corresponding to stiffness attenuation has decreased from 1300 N to 800 N). Finally, when the speed increases to 6000 r/min, no attenuation trend of the rotor stiffness at the small load range is presented, and the stiffness of the rotor system keeps increasing with radial load (i.e., the stiffness growth rate within the given load range exceeds 15%).

One can see from Figure 12, the results curves of the rotor nonlinear stiffness and equivalent linear resonant frequencies under 4000 r/min and 10 μm with the radial load are presented. One can find that the axial and dual radial stiffness first increases then decreases with radial load, and the change rules of the rotor radial stiffness corresponding to its radial load is similar to that of the rotor axial stiffness. Besides, as shown in Figure 12(b), the rotor 1^st-order equivalent resonant frequency corresponding to its axial stiffness shows an obvious attenuation trend with the radial load, and the 2^nd- and 3^rd-order equivalent resonant frequencies, respectively, corresponding to the dual radial stiffness show a consistent variation law as the rotor radial stiffness.

3.3. Nonlinear stiffness variation of the combined-loaded rotor-bearing system

At last, the nonlinear stiffness variation of the rotor system under the combined axial and radial loads condition is given. Without loss of generality, one can assume that the radial load keeps constant, and then the change curves of the rotor stiffness with the axial load are given. One can see from Figure 13, the stiffness attenuation trends at the small axial load range gradually disappear as the radial load rises. Besides, under the action of the constant radial load, the change law of the rotor equivalent resonant frequencies is no longer consistent to its corresponding rotor stiffness. Specifically, the 1^st-order equivalent resonant frequency for the rotor-bearing system subjected to the radial load (500 N) shows the first increases then decreases trend with the axial load which is different from the change trend of the corresponding axial stiffness, and the 2^nd-order equivalent resonant frequency of the rotor-bearing system under different radial loads all shows a decrease trend with the axial load which is also different from the change rule of its corresponding radial stiffness.

4. Conclusions

In this paper, a theoretical model on the typical rigid rotor supported on ball bearings is built, the nonlinear stiffness variation and its effect on the rotor dynamics is studied, and the rotor stiffness is calculated based on a novel mathematical model of the supporting ball bearing by adopting the contact angles as the unknown parameters. Then the change rules of the rotor nonlinear stiffness and dynamics versus the preload, rotating speed and external loads are analyzed.

The rotor system exhibits rich nonlinear stiffness characteristics, both the preload and rotating speed play a significant role on the change rules of the rotor nonlinear stiffness. First, for the rotor system under relatively large preload and low speed, the rotor stiffness presents the change rule of the first “soft-spring” and then “hard-spring.” Second, for the rotor system with the small preload and high speed, the rotor stiffness presents the obvious nonlinear “hard-spring” stiffness characteristic. Finally, the rotor equivalent linear resonant frequencies cannot be analyzed simply by the variation law of its main diagonal stiffness elements.

This article not only analyzes and explains the different nonlinear stiffness variation behaviors for the typical rotor-ball bearing system, but also further discussed the effect of the rotor nonlinear stiffness characteristics on its dynamics, that is, the equivalent linear resonant frequencies.

Footnotes

Author contributions

Bin Fang: Writing—original draft, methodology, and validation; Wenchao Li: Writing—review & editing and validation; Jinhua Zhang: Writing—review & editing and supervision; Ke Yan: Conceptualization; Jun Hong: Supervision.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (No. 52205281), Two-Chain Fusion High-End Machine Tool Projects of Shaanxi Province (No. 2021LLRh-01-02), and China Postdoctoral Science Foundation (No. 2021M62551).

ORCID iD

Bin Fang

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