Effect of residual shaft bow on the dynamic analysis of a double-stage geared rotor-bearing system with translational motion due to shaft deformation

Abstract

In this study, the dynamic responses of a double-stage geared rotor-bearing system with translational motion due to the effects of rotating shaft deformation were examined, which was lacking in past research. Residual shaft bow has been neglected in most geared rotor-bearing system models. In this study, the dynamic response of a double-stage geared rotor-bearing system whose gear set underwent translational motion was examined when the shaft was deformed (residual shaft bow). The proposed model considers contact ratio and pressure angle of gear pair as time-dependent variables, whereas in other models, they are constants. A finite element model of a double-stage geared rotor-bearing system with residual shaft bow was developed, and equations of motion were obtained by applying the Lagrange’s equation. System equations of motion are solved by Runge–Kutta numerical method. The dynamic response was investigated, including lateral response, contact ratio and pressure angle of gear pair with different residual shaft bow magnitudes and gear positions. The results show that residual shaft bow raises the amplitudes of lateral displacement, contact ratio and phase angle of gear pairs in the system. This study demonstrated that the residual shaft bow effect is too considerable to ignore when it becomes sufficiently large.

Keywords

Residual shaft bow rotor dynamic finite element rotor-bearing system time-dependent phase angle contact ratio

Introduction

The gear is one of the basic components of the transmission. The geared rotor-bearing system is one of the main mechanisms of modern power transmission. Given the increasing demand for high-speed, accurate transportation, research on geared rotor dynamics is highly necessary. Much research has been conducted on the dynamic characteristics of rotor-bearing systems, indicating that system parameters significantly influence these characteristics. Such parameters include system geometry, bearing coefficients, rigid disk inert properties, mass distribution, and rotational component stiffness. Many methods exist for determining natural frequencies, critical speeds, mode shapes, and steady-state response due to rotor-bearing system unbalance. For example, Gunter¹ evaluated the effect of damped, linear, flexibly mounted rolling element bearings on dynamic rotor imbalance response. Lund² presented a scheme for computing the percentage of critical speed change caused by a corresponding change in any rotor model element. Buccuiarell³ studied unbalance response due to rotating shaft instability and clarified how the internal forces are coupled to the whirl motion to create instability. Awrejcewicz and Pyryev⁴ investigated nonlinear problem of the thermoelastic contact of a rotating shaft with a rigid bush fixed to the base springs. Awrejcewicz and Pyryev^5,6 developed a new one-dimensional thermoelastic frictional pair contact model of a shaft-bush system and developed inertial contact of a braking pad and a rotating inertial shaft. Awrejcewicz and Kudra⁷ described the real piston-connecting rod-crankshaft system.

The finite element method (FEM), a specific numerical calculation procedure that provides more accurate modeling, is used to analyze rotor-bearing systems. Ruhl and Booker⁸ described a distributed element model of a general turbo rotor-bearing system in which distributed inertia and elasticity were consistently represented. Nelson and McVaugh⁹ developed a finite element model of the rotor-bearing system that included the effects of rotary inertia, gyroscopic moment, and axial load. Nelson¹⁰ studied the influences of shear morph and moment of inertia on rotor-bearing systems by means of Timoshenko beam theory and the FEM. Ozguven and Ozkan¹¹ described dynamic modeling of the rotor-bearing systems, including moment of inertia, gyroscopic effect, axial load, internal viscous and hysteretic damping, and lateral shear deformation. Chen and Ku¹² discussed the dynamic stability of the cantilevered axle system formulated using the shape function obtained by Timoshenko beam theory. The effects of translational and moment of inertia, gyroscopic moment, bending, and shear morph were included in their mathematical model. Shiau and Hwang¹³ developed a generalized polynomial expansion method for determining critical speeds and modes and the imbalance response of rotor-bearing systems.

Gears are widely used in industrial applications. When a drive system incorporates gear transmission units, lateral and torsional modes are coupled. Determining the lateral and torsional frequencies of rotor systems is essential when designing geared rotor-bearing systems. Most research has emphasized the analysis of lateral motion. Iida et al.¹⁴ considered a simple geared system including torsional and flexural vibration coupling and calculated the response due to mass unbalance and eccentricity in the gears. Neriya et al.¹⁵ extended the model of Iida et al.¹⁴ by developing a single geared shaft system, including the flexibility of the matting teeth and both the drive and driven shafts in their analysis. They also investigated the forced vibration response due to mass gear unbalance.

Several studies on gear dynamics have employed FEM. Kahraman et al.¹⁶ developed a finite element model of a geared rotor system supported using flexible bearings. The model included the rotary inertia of shaft elements, axial loading on shafts, flexibility and damping of bearings, material damping of shafts, and the stiffness and damping of gear mesh. However, their model did not consider the effect of the gear pair pressure angle. Shiau et al.¹⁷ analyzed the coupled bending and torsional vibrations of geared rotors as well as the effect of axial torque on bending vibrations. Rao et al.¹⁸ employed a finite element model to determine the coupled bending–torsional natural frequencies and mode shapes of a geared rotor-bearing system. Shiau et al.¹⁹ described the dynamic behavior of geared rotor systems supported by squeeze film dampers in which coupled bending–torsional vibrations occurred. By considering unbalance forces and gravity, they demonstrated that geared rotors exhibit chaotic behavior due to the nonlinearity of damper forces. Choi and Mau²⁰ employed a transfer matrix model to determine the coupled bending–torsional natural frequencies and mode shapes of a geared rotor-bearing system. Steady-state responses due to excitation of mass unbalance, geometric eccentricity, and gear mesh transmission error were also obtained.

Dynamic models combining gears with flexible shafts and bearings have been proposed in many studies. However, the models in the literature vary in complexity, and most do not include gear nonlinearities and time-varying properties in the simulation. Kahraman and Singh^21,22 discussed the nonlinear frequency response characteristics of a spur gear pair with backlash for both external and internal excitation, and extended their analysis to include sinusoidal or periodic gear mesh stiffness. Gao et al.²³ developed a finite element formulation to describe the nonlinear contact impact behavior of bevel gears. Baguet and Jacquenot²⁴ analyzed the interaction between gears, shafts, hydrodynamic journal bearings, and geared drives, including time-varying properties and nonlinearities of gear mesh stiffness and bearings. Kim et al.²⁵ analyzed the dynamic response of spur gears by considering the pressure angle and contact ratio as time-dependent variables, as well as gear set translational motion due to bearing deformation. Zhang et al.²⁶ developed a general three-dimensional dynamic model of helical gear pairs with geometric eccentricity. The gear mesh and bearing flexibility is included in the model as well. Wang et al.²⁷ developed an improved time-varying mesh stiffness (TVMS) model of a helical gear pair, in which the total mesh stiffness contains the axial tooth bending stiffness, axial tooth torsional stiffness, and axial gear foundation stiffness.

In practice, shafts deform after a certain length of service time. The residual shaft bow effect is a common fault in large rotating machinery and can lead to serious accidents. Residual shaft bow can be suppressed through design improvements based on the understanding of system characteristics. Nicholas et al.²⁸ investigated the residual shaft bow effect on the unbalance response of a single mass flexible rotor. Shiau and Lee²⁹ examined the dynamic response of a simply supported rotor under the effect of disk skew, unbalanced mass, and residual shaft bow. Song et al.³⁰ described the influence of residual shaft bow on rotor longitudinal responses through a simulation experiment. Chen and Kuo³¹ developed a new dynamic model for the geared rotor-bearing system which is considering translational motion and residual shaft bow. The main aim of this study extended the model of Chen³¹ by developing the dynamic characteristics of a double-stage geared rotor-bearing system with translational motion due to shaft deformation under the residual shaft bow effect. The gear lateral responses, pressure angle, and contact ratio were investigated with regard to the shaft deformation effect.

System modeling

The configuration of a double-stage geared rotor-bearing system is shown in Figure 1. The three uniform flexible shafts are of lengths $L_{1}$ , $L_{2}$ , and $L_{3}$ . The residual bows of shafts 1, 2, and 3 are $δ_{r 1}$ , $δ_{r 2}$ , and $δ_{r 3}$ , respectively. The gear pair is mounted on the shafts, and an external torque M is exerted on shaft 1. The contacting mesh force is represented by the gear mesh stiffness coefficient $k_{m 12, 34}$ and damping coefficient $c_{m 12, 34}$ along the pressure line. Six bearings are modeled as flexible elements with damping and stiffness denoted as $c_{i}^{b}$ and $k_{i}^{b}$ , respectively. A single-shaft system with a rigid disk is shown in Figure 2, in which a fixed reference frame, $X - Y - Z$ , is used to describe the system motion. Five degrees of freedom, $V, W, B, Γ$ , and $α$ , are considered at each nodal point of the shaft, where V and W are lateral displacements along and directions, respectively; $B$ and $Γ$ are rotational displacements; and $α$ is the torsional displacement, with the axial translational vibration omitted. The displacement due to residual shaft bow shown in Figure 1 is assumed to be a second-order polynomial

δ_{i} (s) = 4 δ_{ri} \frac{s}{L} (1 - \frac{s}{L})

(1)

Figure 1.

Configuration of a double-stage geared rotor-bearing system.

Figure 2.

Typical rotor configuration and coordinates.

Shaft

For rotating shafts, which are modeled as Timoshenko beams, two-noded elements are used for the finite element formulation, as shown in Figure 3. Five degrees of freedom ( $V, W, B, Γ$ , and α) are considered at each nodal point. The kinetic and potential energy of the shaft element can be expressed, respectively, as follows

\begin{matrix} T_{e} = \frac{1}{2} \int_{0}^{l} {ρ A [{({\overset{\cdot}{V}}_{e})}^{2} + {({\overset{\cdot}{W}}_{e})}^{2}] \\ + ρ I_{eD} [{({\overset{\cdot}{B}}_{e})}^{2} + ({\overset{\cdot}{Γ}}_{e}^{2})] + ρ I_{eP} {(Ω + {\overset{\cdot}{α}}_{e})}^{2}} ds \\ + \frac{1}{2} \int_{0}^{l} {ρ I_{eP} (Ω + {\overset{\cdot}{α}}_{e}) ({\overset{\cdot}{B}}_{e} Γ_{e} - {\overset{\cdot}{Γ}}_{e} B_{e})} ds \end{matrix}

(2)

\begin{matrix} U_{e} = & \frac{1}{2} \int_{0}^{l} ρ E I_{eD} [{(B'_{e})}^{2} + {(Γ'_{e})}^{2}] ds + \frac{1}{2} \int_{0}^{l} ρ G I_{eP} {(α'_{e})}^{2} ds \\ + \frac{1}{2} \int_{0}^{l} κ GA [{(V'_{e} - Γ_{e})}^{2} + {(W'_{e} + B_{e})}^{2}] ds \end{matrix}

(3)

where A is the cross-sectional area; $ρ$ is the shaft’s mass density; $I_{eD}$ and $I_{eP}$ are the transverse and polar mass moments of inertia of the shaft, respectively; E is Young’s modulus of the shaft; $κ$ is the shear factor for the circular cross-section of the shaft; and G is the shear modulus of the shaft.

Figure 3.

Shaft element and the coordinate system.

The equation of motion of the finite shaft element can be derived by substituting kinetic energy equation (2) and strain energy equation (3) into Lagrange’s equation

[M_{e}] {{\overset{\cdot\cdot}{q}}_{e}} + (Ω [G_{e}] + [C_{e}]) {{\overset{\cdot}{q}}_{e}} + [K_{e}] {q_{e}} = {F_{e}}

(4)

where $[M_{e}]$ , $[G_{e}]$ , $[C_{e}]$ , and $[K_{e}]$ are the mass, gyroscopic effect, damping, and stiffness matrices, respectively, of the finite shaft element; $Ω$ is the spin speed of the shaft; and $[C_{e}] = γ [K_{e}]$ where $γ$ is the proportional damping coefficient. ${F_{e}}$ is the force vector due to the residual shaft bow.

Gear and spur gear mesh

In this study, the disk was a rigid body with five degrees of freedom, namely two lateral displacements, $V_{d}$ and $W_{d}$ , and three rotational displacements, $α_{d}$ , $B_{d}$ , and $Γ_{d}$ . Because it was a rigid body, strain energy was not considered. The equation of motion for this disk is as follows

\begin{matrix} T_{d} & = \frac{1}{2} m_{d} [{({\overset{\cdot}{V}}_{d})}^{2} + {({\overset{\cdot}{W}}_{d})}^{2}] \\ + \frac{1}{2} I_{dD} [{({\overset{\cdot}{B}}_{d})}^{2} + {({\overset{\cdot}{Γ}}_{d})}^{2}] \\ + \frac{1}{2} I_{dP} (Ω + {\overset{\cdot}{α}}_{d}) ({\overset{\cdot}{B}}_{d} Γ_{d} - {\overset{\cdot}{Γ}}_{d} B_{d}) + \frac{1}{2} I_{dP} {(Ω + {\overset{\cdot}{α}}_{d})}^{2} \\ + m_{d} e (Ω + {\overset{\cdot}{α}}_{d}) \\ [- {\overset{\cdot}{V}}_{d} \sin (Ω t + α_{d} + Φ_{d}) + {\overset{\cdot}{W}}_{d} \cos (Ω t + α_{d} + Φ_{d})] \\ + m_{d} e^{2} {(Ω + {\overset{\cdot}{α}}_{d})}^{2} \end{matrix}

(5)

where $m_{d}$ is the mass of the disk; $I_{dD}$ and $I_{dP}$ are the transverse and polar mass moments of inertia of the disk, respectively; $Ω$ is the spin speed of the shaft; e is the magnitude of mass eccentricity; and $Φ_{d}$ is the phase angle relative to the Y axis. After substituting the kinetic energy equation (5) into Lagrange’s equation, the following equation of motion is obtained

[M_{d}] {{\overset{\cdot\cdot}{q}}_{d}} + Ω [G_{d}] {{\overset{\cdot}{q}}_{d}} = {F_{d}}

(6)

where $[M_{d}]$ is the mass matrix, $[G_{d}]$ is the gyroscopic effect matrix, ${F_{d}}$ is the force vector due to disk mass unbalance, and ${q_{d}}$ is the displacement vector.

Figure 4 shows the configuration and generalized coordinates for the pair of gears mounted on the shafts shown in Figure 1. The motion of the gear set can be defined using 10 generalized coordinates, which are identical to the node coordinates of the two shafts on which the gears are mounted. The dashed and solid circles represent the gear pair before and after motion, respectively. $O_{1, 3}$ and $O_{2, 4}$ are the initial centers of each shaft, and $C_{1, 3}$ and $C_{2, 4}$ are the centers after motion. The mass centers of the gear are denoted by $G_{1, 3}$ and $G_{2, 4}$ , the magnitudes of eccentricity are denoted by $e_{1, 3}$ and $e_{2, 4}$ , and the eccentricity angles are denoted by $Φ_{d 1, 3}$ and $Φ_{d 2, 4}$ . Angular coordinates of the drive and driven gears are defined as follows

θ_{i} (t) = \pm Ω_{j} t + α_{di} (t) + Φ_{di}

(7)

where $i = 1 ~ 4$ denote the drive and driven gears, respectively, and $Ω_{j}$ is the spin speed of the shaft, $j = 1 ~ 3$ . The distance between centers $d_{12, 34}$ changes to $d'_{12, 34}$ after motion; $d'_{12, 34}$ is defined as follows

d'_{12} (t) = \sqrt{{[V_{d 2} (t) - V_{d 1} (t) + d_{12}]}^{2} + {[W_{d 2} (t) - W_{d 1} (t)]}^{2}}

(8)

d'_{34} (t) = \sqrt{{[V_{d 4} (t) - V_{d 3} (t) + d_{34}]}^{2} + {[W_{d 4} (t) - W_{d 3} (t)]}^{2}}

(9)

Figure 4.

Configuration and generalized coordinates for gear pair.

In Figure 5, the gear pair is modeled as the equivalent stiffness $k_{m 12, 34}$ and damping $c_{m 12, 34}$ along the pressure line between the teeth. The pressure line is defined as the common tangent line of the base circles for the gear set, with the pressure angle $ϕ_{p 12, 34}$ defined as follows

ϕ_{p 12} (t) = \cos^{- 1} [\frac{R_{1} + R_{2}}{d'_{12} (t)}]

(10)

ϕ_{p 34} (t) = \cos^{- 1} [\frac{R_{3} + R_{4}}{d'_{34} (t)}]

(11)

where $R_{1, 3}$ and $R_{2, 4}$ are the radii of base circles of gears. The angle of the driven gear relative to the drive gear is represented as follows

ψ_{12} (t) = \tan^{- 1} [\frac{W_{d 2} - W_{d 1}}{V_{d 2} - V_{d 1} + d_{12}}]

(12)

ψ_{34} (t) = \tan^{- 1} [\frac{W_{d 4} - W_{d 3}}{V_{d 4} - V_{d 3} + d_{34}}]

(13)

Figure 5.

Gear mesh model of the gear pair.

The contact ratio of the gear pair is represented as follows

m_{p 12} (t) = \frac{\sqrt{A_{1}^{2} - R_{1}^{2}} + \sqrt{A_{2}^{2} - R_{2}^{2}} - d_{12} \sin ϕ_{p 12} (t)}{p_{b 12}}

(14)

m_{p 34} (t) = \frac{\sqrt{A_{3}^{2} - R_{3}^{2}} + \sqrt{A_{4}^{2} - R_{4}^{2}} - d_{34} \sin ϕ_{p 34} (t)}{p_{b 34}}

(15)

where $N_{1, 3}$ and $N_{2, 4}$ are the numbers of teeth for gears, $A_{1, 3}$ and $A_{2, 4}$ are the radii of the addendum circles, and $p_{b 12, 34}$ is the base pitch. In this article, the pressure angle and gear position angle were under the influence of gear pair translational motions, an effect neglected in other studies. It was assumed that the gear mesh deformation along the pressure line could be described as follows

\begin{matrix} δ_{12} (t) & = (V_{d 2} - V_{d 1}) \sin (ϕ_{p 12} - ψ_{12}) \\ + (W_{d 2} - W_{d 1}) \cos (ϕ_{p 12} - ψ_{12}) \\ - (R_{1} α_{d 1} + R_{2} α_{d 2}) \end{matrix}

(16)

\begin{matrix} δ_{34} (t) & = (V_{d 4} - V_{d 3}) \sin (ϕ_{p 34} - ψ_{34}) \\ + (W_{d 4} - W_{d 3}) \cos (ϕ_{p 34} - ψ_{34}) \\ - (R_{3} α_{d 3} + R_{4} α_{d 4}) \end{matrix}

(17)

whereby the gear mesh force along the pressure line could be expressed as follows

F_{h 12} (t) = k_{m 12} δ_{12} + c_{m 12} {\overset{\cdot}{δ}}_{12}

(18)

F_{h 34} (t) = k_{m 34} δ_{34} + c_{m 34} {\overset{\cdot}{δ}}_{34}

(19)

The corresponding equation of motion for the gear pair can thus be obtained, as follows

\begin{matrix} [\begin{matrix} [M_{g 1, 3}] & 0 \\ 0 & [M_{g 2, 4}] \end{matrix}] {{\overset{\cdot\cdot}{q}}_{g}} \\ + Ω_{1, 2} [\begin{matrix} [G_{g 1, 3}] & 0 \\ 0 & \frac{N_{t 1, 3}}{N_{t 2, 4}} [G_{g 2, 4}] \end{matrix}] {{\overset{\cdot}{q}}_{g}} \\ + k_{m 12, 34} [S_{h 12, 34}] {q_{g}} = {F_{g 12, 34}} \end{matrix}

(20)

where $[M_{g 1, 3}]$ and $[M_{g 2, 4}]$ are the mass matrices of the drive and driven gears, respectively; $[G_{g 1, 3}]$ and $[G_{g 2, 4}]$ are the gyroscopic effect matrices of the drive and driven gears, respectively; ${F_{g 12, 34}}$ is the force vector due to disk mass unbalance of the gear pair; $[S_{h 12, 34}]$ is the meshing effect matrix of the gear pair; and ${q_{g}}$ is the displacement vector.

Bearing

In this study, the six bearing supports assumed to be isotropic. Springs and dampers are used to model the bearing supports. Ignoring the effects of slopes and bending moments, the potential energy and dissipation function of the ith bearing support can be expressed, respectively, as follows

U_{bi} = \frac{1}{2} (k_{bi} V_{bi}^{2} + k_{bi} W_{bi}^{2})

(21)

f_{bi} = \frac{1}{2} (c_{bi} {\overset{\cdot}{V}}_{bi}^{2} + c_{bi} {\overset{\cdot}{W}}_{bi}^{2})

(22)

By using the Lagrange’s equation, the motion equations of bearing could be obtained, as follows

[\begin{matrix} k_{bi} & 0 \\ 0 & k_{bi} \end{matrix}] {q_{bj}} + [\begin{matrix} c_{bi} & 0 \\ 0 & c_{bi} \end{matrix}] {{\overset{\cdot}{q}}_{bi}} = {0}

(23)

System equation of motion

The equations of motion of the system shaft, disk, gear meshing, and bearing unit, as introduced in the previous sections, could be combined to produce the equation of motion for the entire geared rotor-bearing system, as follows

[M] {{\overset{\cdot\cdot}{q}}_{s}} + (Ω [G] + [C]) {{\overset{\cdot}{q}}_{s}} + [K] {q_{s}} = {F}

(24)

where $[M_{s}]$ , $[G_{s}]$ , $[C_{s}]$ , $[K_{s}]$ , ${F_{s}}$ , and ${q_{s}}$ represent the system mass matrix, gyroscopic effect matrix, damping matrix, stiffness matrix, force vector, and displacement vector, respectively.

Numerical simulation results and discussion

This section presents the results of numerical simulations for a double-state geared rotor-bearing system. This study explored the lateral displacement vibrations at each gear position, the changes of contact ratio, and phase angle of two gear pairs. First, the effects of rotating shaft deformation (residual shaft bow) on the system were discussed. The first shaft was subjected to a fixed torque M = 250 N-m, under which the effects of different residual shaft bow on the system were investigated. Other parameters are exhibited in Tables 1 and 2. Table 1 lists the parameters adopted for the two gear pairs, and Table 2 presents the parameter values of the rotor-bearing system. Before response analysis, the first nine natural frequencies for zero spin speed of the system are calculated and shown in Table 3. It also describes the corresponding mode shapes.

Table 1.

Parameters adopted for the two gear pairs.

	Drive gear (gear 1, 3)	Driven gear (gear 2, 4)
Number of teeth	$N_{1} = 28$	$N_{2} = 42$
Base circle radius [mm]	$R_{1} = 44.5$	$R_{2} = 66.7$
Addendum circle radius [mm]	$A_{1} = 50.7$	$A_{2} = 74.4$
Eccentricity angle [°]	$Φ_{d 1, 3} = 0$	$Φ_{d 2, 4} = 0$
Gear position	$x_{1}^{d} = L / 2, x_{2}^{d} = L / 4, x_{3}^{d} = 3 L / 4, x_{4}^{d} = L / 2$ [L/2, L/4, 3L/4, L/2]
Disk eccentricity [mm]	$e_{1 ~ 4} = 0$
Mesh stiffness coefficient [N/m]	$k_{m 12} = k_{m 34} = 1 \times 10^{9}$
Mesh damping coefficient [N-s/m]	$c_{m 12} = c_{m 34} = 1000$

Table 2.

Parameters of the rotor-bearing system.

Shaft radius [mm]	$r_{s 1} = r_{s 2} = r_{s 3} = 0.0185 m$
Shaft length [mm]	$L_{1} = L_{2} = L_{3} = 0.254 m$
Residual angle [°]	$Φ_{r 1} = Φ_{r 2} = Φ_{r 3} = 0$
Shaft density [kg/m³]	$ρ = 7800$
Young’s modulus [N/m²]	$E = 2.07 \times 10^{11}$
Shear modulus [N/m²]	$G = 8.28 \times 10^{10}$
Poisson ratio	$ν = 0.25$
Proportional damping coefficient [s]	$γ = 1 \times 10^{- 5}$
Bearing damping coefficient [N-s/m]	$c_{yy} = c_{zz} = 1 \times 10^{3}$
Bearing stiffness [N/m]	$k_{yy} = k_{zz} = 1 \times 10^{8}$

Table 3.

Natural frequencies and corresponding mode shapes for zero spin speed of the geared rotor system.

	Natural frequencies (rad/s)	Mode description
$ω_{0}$	0	Torsional rigid body
$ω_{1}$	2928.8	Couple lateral-torsional, 1st–2nd–3rd shaft
$ω_{2}$	3469.7	1st lateral, y-direction, 2nd shaft
$ω_{3}$	3535.1	Couple lateral-torsional, 1st–2nd shaft
$ω_{4}$	3535.1	Couple lateral-torsional, 2nd–3rd shaft
$ω_{5}$	3727.5	1st lateral y-direction, 1st shaft
$ω_{6}$	3727.5	1st lateral y-direction, 3rd shaft
$ω_{7}$	7944.1	Couple lateral torsional, 1st–2nd shaft
$ω_{8}$	8440.8	2nd lateral y-direction, 2nd shaft
$ω_{9}$	8527.4	Couple lateral-torsional, 1st–2nd–3rd shaft

According to the results in Figure 6, the lateral displacement of gears 1−4 $(r 1 - r 4)$ considerably increased with the residual shaft bow of shafts 1−3. This indicated that the presence of residual shaft bow greatly affected the lateral response of the system, increasing the amplitudes and thus destabilizing the system.

Figure 6.

Lateral responses of gears 1−4 $(r 1 - r 4)$ with different residual shaft bow magnitude.

As depicted by Figure 7, when the residual shaft bow increased, the amplitudes of the phase angle and contact ratio of two gear pairs also increased. As may be seen, the pressure angle of gear pair 34 and contact ratio of gear pair 12 increased with the residual shaft bow magnitude. As the residual shaft bow magnitude increased, the pressure angle of gear pair 12 and contact ratio of gear pair 34 decreased. The results in Figures 6 and 7 demonstrate that residual shaft bow tends to raise the amplitudes of lateral displacement, contact ratio, and phase angle of gear pairs in the system. Therefore, exploring the effects of residual shaft bow on gear trains is necessary. Table 4 extends the results of Figures 6 and 7 and shows the impact of more different residual shaft bow magnitude on lateral displacement, contact ratio, and phase angle of gear pairs in the system. According to the data in Table 4, it should be noted that the size of the residual shaft bow should be avoided to exceed 0.05 $r_{s 1}$ .

Figure 7.

Phase angle and contact ratio of gear pair with different residual shaft bow magnitude.

Table 4.

Steady-state lateral displacement of the gears, contact ratio, and phase angle of the gear pairs in different residual shaft bow magnitude.

Residual shaft bow magnitude of shafts 1−3	r1 (mm)	r2 (mm)	r3 (mm)	r4 (mm)	Phase angle 12 (°)	Phase angle 34 (°)	Contact ratio 12	Contact ratio 34
0 $r_{s 1}$	0.1011	0.0911	0.0783	0.04456	20.08	19.94	1.598	1.624
0.001 $r_{s 1}$	0.0923	0.1046	0.0923	0.05375	20.07	19.95	1.601	1.621
0.003 $r_{s 1}$	0.1202	0.1451	0.1135	0.1077	20.05	19.98	1.606	1.616
0.005 $r_{s 1}$	0.1795	0.1885	0.1812	0.1822	20.01	20.02	1.613	1.609
0.01 $r_{s 1}$	0.3541	0.3084	0.3151	0.3561	19.97	20.09	1.626	1.597
0.03 $r_{s 1}$	1.098	0.8626	0.8575	1.109	19.62	20.32	1.683	1.558
0.05 $r_{s 1}$	2.694	2.106	2.033	2.444	18.33	21.58	1.929	1.321
0.06 $r_{s 1}$	4.268	3.328	3.166	3.673	17.86	22.44	2.025	1.159

In order to examine the effect of the residual shaft bow on the system, torque M was subjected to be zero and $Ω_{1} = 3000 rpm$ . Table 5 has shown the maximum lateral displacement of the gears in different residual shaft bow magnitude. It is shown that the maximum lateral displacement of gears 1−4 $(r 1 - r 4)$ are 2960, 2298, 2220, and 2967 times as large as residual shaft bow magnitude, respectively.

Table 5.

Maximum lateral displacement of the gears in different residual shaft bow magnitude.

Residual shaft bow magnitude of shafts 1−3	r1 (mm)	r2 (mm)	r3 (mm)	r4 (mm)	$\frac{r_{1}}{r_{s 1}}$	$\frac{r_{2}}{r_{s 1}}$	$\frac{r_{3}}{r_{s 1}}$	$\frac{r_{4}}{r_{s 1}}$
0 $r_{s 1}$	0	0	0	0	0	0	0	0
0.001 $r_{s 1}$	0.05471	0.04245	0.04115	0.05489	2957	2294	2224	2967
0.003 $r_{s 1}$	0.1643	0.1275	0.1212	0.1647	2960	2297	2183	2967
0.005 $r_{s 1}$	0.2738	0.2125	0.2056	0.2745	2960	2297	2222	2967
0.01 $r_{s 1}$	0.5476	0.4253	0.4111	0.5489	2960	2298	2222	2967
0.03 $r_{s 1}$	1.642	1.277	1.235	1.646	2958	2300	2225	2965
0.05 $r_{s 1}$	2.737	2.128	2.061	2.744	2958	2300	2228	2966
0.06 $r_{s 1}$	3.282	2.554	2.472	3.293	2956	2300	2227	2966

The results in Figure 8 indicate that the presence of a residual shaft bow in one of the shafts was likely to increase the lateral response of gears on the shaft. Figure 8 also demonstrates that the residual shaft bow of the middle shaft exerted stronger effects on the lateral responses of the shaft gears compared with that of the side shafts. This result suggested that higher residual shaft bow in the middle shaft than in the side shafts should be avoided.

Figure 8.

Lateral responses of gears 1−4 with different residual shaft bow magnitude of the shaft.

The results in Figure 9 show that the presence of a residual shaft bow on the drive gear led to a rise in the contact ratio of the corresponding gear pair and a decline in the phase angle. Conversely, when residual shaft bow was identified in the driven gear, the contact ratio of the corresponding gear pair decreased, whereas its phase angle increased. Generally, a higher contact ratio and lower phase angle between contacting gear pair help improve the transmission efficiency of gears. Therefore, transmission efficiency can improve when the drive gear is with residual shaft bow. By contrast, the efficiency can deteriorate if a residual shaft bow is present in the driven gear. Higher residual shaft bow in the middle shaft than in the side shafts should be avoided. Thus, if residual shaft bow is identified in one of the three shafts, this shaft should be repositioned as shaft 1, that is, the position of the power–output shaft. However, if the residual shaft bow magnitude is too big, the lateral response of the system will be too large. Large residual shaft bow should be avoided.

Figure 9.

Phase angle and contact ratio of gear pair with different residual shaft bow magnitude of the shaft.

Figure 10 shows the lateral responses of gears 2 and 3 with different gear position and residual shaft bow magnitude. The lateral responses of gear 2 with gear position [L/2, L/4, 3L/4, L/2] were smaller than those with gear position [L/2, L/2, 3L/4, L/2]. Because gear 2 changed the position from L/4 to L/2, the equivalent stiffness of the shaft on gear 2 decreased. The lateral responses of gear 3 with gear position [L/2, L/4, 3L/4, L/2] were similar to those with gear position [L/2, L/2, 3L/4, L/2] because the equivalent stiffness of the shaft of the system on gear 3 changed only slightly.

Figure 10.

Lateral responses of gears 2 and 3 with different gear position and residual shaft bow magnitude: (a) gear position [L/2, L/4, 3L/4, L/2] and (b) gear position [L/2, L/2, 3L/4, L/2].

Figure 11 shows the contact ratio and phase angle of gear pair 12 with different gear position and residual shaft bow magnitude. The contact ratio of gear pair 12 with gear position [L/2, L/4, 3L/4, L/2] was bigger than that with gear position [L/2, L/2, 3L/4, L/2] because the equivalent stiffness of the shaft on gear 2 became smaller, thus enhancing the lateral response of gear 2. The magnitudes of the contact ratio of gear pair 12 thus become smaller. Similar to the contact ratio results, the pressure angle of gear pair 12 with gear position [L/2, L/4, 3L/4, L/2] was smaller than that with gear position [L/2, L/2, 3L/4, L/2]. The magnitudes of the pressure angle of gear pair 12 thus became larger.

Figure 11.

Contact ratio and phase angle of gear pair 12 with different gear position and residual shaft bow magnitude: (a) gear position [L/2, L/4, 3L/4, L/2] and (b) gear position [L/2, L/2, 3L/4, L/2].

Conclusion

This study developed a new dynamic model of a double-state geared rotor-bearing system that considers translational motion due to shaft deformation and the residual shaft bow effect. This new model employed the gear contact ratio and pressure angle as time-dependent variables, thus addressing a shortcoming of other studies. This study demonstrated that the residual shaft bow effect is too considerable to ignore when it becomes sufficiently large. The contribution of this article are given as follows: (1) the size of the residual shaft bow should be avoided to exceed 0.05 times the radius of the shaft, (2) the maximum lateral displacement of gears 1−4 are 2220−2960 times as large as residual shaft bow magnitude, (3) the transmission performance tends to improve when residual shaft bow is identified in the drive gear, (4) higher residual shaft bow in the middle shaft than in the side shafts should be avoided, and (5) the gear contact ratio, pressure angle, and lateral response vary with different gear positions.

Footnotes

Handling Editor: James Baldwin

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: The authors thank the support of the Ministry of Science and Technology of Taiwan (MOST, Plan No. 1072218E013001MY2).

ORCID iD

Ying-Chung Chen

References

Gunter

EJ.

Influence of flexibility mounted rolling element bearings on rotor response. J Tribol: T ASME 1970; 92: 59–69.

Lund

JW.

Sensitivity of the critical speeds of a rotor to changes in the design. J Mech Des: T ASME 1980; 102: 115–121.

Buccuiarell

LL.

On the instability of rotating shafts due to internal damping. J Appl Mech 1982; 49: 425–428.

Awrejcewicz

Pyryev

Thermoelastic contact of a rotating shaft with a rigid bush in conditions of bush wear and stick-slip movements. Int J Eng Sci 2002; 40: 1113–1130.

Awrejcewicz

Pyryev

Tribological periodic processes exhibited by acceleration or braking of a shaft-pad system. Commun Nonlinear Sci 2004; 9: 603–614.

Awrejcewicz

Pyryev

Contact phenomena in braking and acceleration of bush-shaft system. J Therm Stresses 2004; 27: 433–454.

Awrejcewicz

Kudra

The piston-connecting rod-crankshaft system as a triple physical pendulum with impacts. Int J Bifurcat Chaos 2005; 15: 2207–2226.

Ruhl

Booker

JF.

A finite element model for distributed parameter turborotor systems. J Eng Ind: T ASME 1972; 94: 126–132.

Nelson

McVaugh

JM.

The dynamics of rotor-bearing systems using finite elements. J Eng Ind: T ASME 1976; 98: 593–600.

10.

Nelson

HD.

A finite rotating shaft element using Timoshenko beam theory. J Mech Des: T ASME 1980; 102: 793–803.

11.

Ozguven

Ozkan

ZL.

Whirl speeds and unbalance response of multibearing rotors using finite elements. J Vib Acoust 1984; 106: 72–79.

12.

Chen

DM.

Dynamic stability of cantilever shaft-disk system. J Vib Acoust 1992; 114: 326–329.

13.

Shiau

Hwang

JL.

Generalized polynomial expansion method for the dynamic analysis of rotor-bearing systems. J Eng Gas Turb Power 1993; 115: 209–217.

14.

Iida

Tamura

Kikuchi

et al . Coupled torsional-flexural vibration of a shaft in a geared system of rotors. B JSME 1980; 23: 2111–2117.

15.

Neriya

Bhat

Sankar

TS.

Effect of coupled torsional-flexural vibration of a geared shaft system on dynamic tooth load. Shock Vib Bull 1984; 354: 67–75.

16.

Kahraman

Ozguven

Houser

et al . Dynamics analysis of geared rotors by finite elements. J Mech Des: T ASME 1992; 114: 507–514.

17.

Shiau

Choi

Chang

. Theoretical analysis of lateral response due to torsional excitation of geared rotors. Mech Mach Theory 1998; 33: 761–783.

18.

Rao

Shiau

Chang

. Coupled bending-torsion vibration of geared rotors. In: ASME conference on design engineering technical, DE-vol. 84–2, 3, Part B, Boston, MA, 17–20 September 1995, pp.977–989. New York: ASME.

19.

Shiau

Rao

Chang

et al . Dynamic behavior of geared rotors. J Eng Gas Turb Power 1999; 121: 494–503.

20.

Choi

Mau

SY.

Dynamic analysis of geared rotor-bearing systems by the transfer matrix method. J Mech Des: T ASME 2001; 123: 562–568.

21.

Kahraman

Singh

Non-linear dynamics of a spur gear pair. J Sound Vib 1990; 142: 49–75.

22.

Kahraman

Singh

Interactions between time-varying mesh stiffness and clearance non-linearities in a geared system. J Sound Vib 1991; 146: 135–156.

23.

Gao

Tanabe

Nishhara

Contact-impact analysis of geared rotor systems. J Sound Vib 2009; 319: 463–475.

24.

Baguet

Jacquenot

Nonlinear couplings in a gear-shaft-bearing system. Mech Mach Theory 2010; 45: 1777–1796.

25.

Kim

Yoo

Hung

JC.

Dynamic analysis for a pair of spur gears with translational motion due to bearing deformation. J Sound Vib 2010; 329: 4409–4421.

26.

Zhang

Wang

et al . Dynamic analysis of three-dimensional helical geared rotor system with geometric eccentricity. J Mech Sci Technol 2013; 27: 3231–3242.

27.

Wang

Zhao

et al . An improved time-varying mesh stiffness model for helical gear pairs considering axial mesh force component. Mech Syst Signal Pr 2018; 106: 413–429.

28.

Nicholas

Gunter

Allaire

PE.

Effect of residual shaft bow on unbalance response and balancing of a single mass flexible rotor. J Eng Power: T ASME 1976; 98: 171–181.

29.

Shiau

Lee

EK.

The residual shaft bow effect on dynamic response of a simply supported rotor with disk skew and mass unbalances. J Vib Acoust 1989; 111: 170–178.

30.

Song

Yang

et al . Theoretical-experimental study on a rotor with a residual shaft bow. Mech Mach Theory 2013; 63: 50–58.

31.

Chen

Kuo

. Dynamic analysis of a geared rotor-bearing system with translational motion due to shaft deformation under residual shaft bow effect. In: MATEC web of conferences, Vol. 119, Taichung, Taiwan, 28 October–1 November 2017, 01014. Les Ulis: EDP Sciences.