Dynamic response analysis for the aero-engine dual-rotor-bearing system with flexible coupling misalignment faults

2.1. Dual-rotor system model

A finite element model of the aero-engine dual-rotor system is presented in Fig. 1, which is composed of a high-pressure rotor (rotor 1, node 1 to node 4) with a higher rotation speed ${\mathrm{\Omega }}_1$ and a low-pressure rotor (rotor 2, node 5 to node 11) with a lower rotation speed ${\mathrm{\Omega }}_2$. Rotor 2 is composed of low-pressure compressor rotor (node 5 to node 7) and low-pressure turbine rotor (node 8 to node 11), with are coupled by a flexible coupling. There are a total of 5 supports in the model, which are respectively located at node 5, node 7, node 8, node 1, node 11, having the linear stiffness. In addition, the inter-shaft bearing is located at node 4 (node 9), having the nonlinear stiffness denoted by the restoring forces. The disks 1-4, located respectively at node 2, node 3, node 6 and node 10, represent the concentrations of high-pressure compressor disks, high-pressure turbine disks, low-pressure compressor disks and low-pressure turbine disks.

Fig. 1Finite element model of a dual-rotor system coupling by the cylindrical roller inter-shaft bearing

2.2. Model of flexible coupling misalignment faults

Misalignment in gear coupling includes parallel, angular and the combined conditions as shown in Fig. 2(b)-(d)) [19]. The coupling consists of two half parts, each with the driving shaft and driven shaft connection. The two half-couplings rotate around their own center lines and engage with the shell separately. The shell is forced to revolve around both center lines of half-couplings during a whirl process and its center may move along a planar circle.

According to the study of Han [8], central axis of the shell goes along a cylindrical surface as the coupling runs with parallel misalignment. When angular misalignment appears in the coupling, central axis of the shell spins and swings continuously between two half-couplings and its motion trace is a double-cone surface. If the coupling works with both angular and parallel misalignment, the central-axis trace of the shell is a hemi double-cone surface which is between cylindrical and double-cone surfaces. Different motion styles of the shell in various misalignment conditions are shown in Fig. 2(f)-(h).

Fig. 2Misalignment conditions and the relevant central-axis traces of the shell

a) Alingment

b) Parallel misalignment

c) Angular misalignment

d) Combined misalignment

e) Straight line

f) Cylindrical surface

g) Double-cone surface

h) Hemi double-cone surface

For parallel misalignment fault, the inertial centrifugal force acting on right and left side of the rotor system can be expressed as:

where $F_{cxl}$ and $F_{cyl}$ are the inertial centrifugal force acting on left side of the rotor system in two directions, $F_{cxr}$ and $F_{cyr}$ are the inertial centrifugal force acting on right side of the rotor system in two directions, $M$ is the mass of the shell, $\mathrm{\Delta }y$ is the amount of misalignment for parallel misalignment, $\omega$ is the rotating speed of rotor, $\varphi$ is the initial angle.

For angular misalignment fault, the inertial centrifugal force acting on left side ($F_{cxl}$, $F_{cyl}$) of the rotor system can be expressed as:

And the inertial centrifugal force acting on right side ($F_{cxr}$, $F_{cyr}$) of the rotor system can be expressed as:

where $M$ is the mass of the shell, $\mathrm{\Delta }\alpha$ is the amount of misalignment for angular misalignment, $\mathrm{\Delta }L$ is the assemble distance between two half-couplings, $\omega$ is the rotating speed of rotor, $\varphi$ is the initial angle.

For combined misalignment fault, the inertial centrifugal force acting on the rotor system can be obtained by the accumulation of the Eq. (1) and (2).

2.3. The inter-shaft bearing model and the supports model

The inter-shaft bearing is a kind of bearing which located between the high-pressure rotor and the low-pressure rotor. Most of the inter-shaft bearings are the cylindrical roller bearings for the aero-engine dual-rotor system. Its main function is to transmit the force of low-pressure rotor to high-pressure rotor. Generally, the inner race of the inter-shaft bearing is built on the shaft of the low-pressure rotor, and the outer race is built on the shaft of the high-pressure rotor. In this paper, we consider that the two rotors rotate in the same direction. Fig. 3 shows the real object and the schematic diagram of the cylindrical roller inter-shaft bearing, in which, the considered bearing has equispaced elements rolling on the surface of the inner and outer races.

Fig. 3The schematic diagram of the inter-shaft bearing

For the inter-shaft bearing, the total restoring force is the sum of restoring force from each of rolling elements and its components in the $x$ and $y$ directions are [24]:

where $C_b$ is the Hertz contact stiffness related to the shape and the material of contact objects, $H\left(•\right)$ is a Heaviside function $H\left({\delta }_i\right)=\left\{\begin{array}{c}1,{\delta }_i\ge 0\\ 0,{\delta }_i<0\end{array}\right.$, $N_b$ is the number of the rolling elements. ${\delta }_i$ is the normal contact deformation between the $i$th element and the races, is given by ${\delta }_i=x\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_i+y\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_i-{\gamma }_0\text{.}$${\theta }_i$the location of the $i$th element, is assumed as ${\theta }_i={\omega }_{cage}\times t+2\pi (i-1)/N_b$, $i=\mathrm{1,2},\cdots N_b$, and ${\omega }_{cage}$ is the angular velocity of the cage, can be expressed as ${\omega }_{cage}=\left({\omega }_i\times r_i+{\omega }_o\times r_o\right)/\left(r_i+r_o\right)$, where ${\omega }_i$ and $r_i$ are, respectively, the rotating angular velocity and the radius of the inner race, ${\omega }_o$ and $r_o$ represent, respectively, the rotating angular velocity and the radius of the outer race.

For the supports 1-4 and support 6, the following relationship can be obtained through the assumption of the linear restoring force:

where $F_x$ and $F_y$ are the elastic forces, $x$ and $y$ are the elastic deformations, $k_{xx}$ and $k_{yy}$ are the supporting stiffness in $x$ direction and $y$ direction.

2.4. Motion equations of the dual-rotor coupling system with misalignment faults

Based on the finite element method of rotor dynamic [25, 26], the rotor system model can be divided into rigid disks, elastic shaft segments and linear bearing supports as shown in Fig. 1. For the $i$th elastic shaft element, the nodal displacements in an inertial coordinate system are expressed as:

where $x_i$, $y_i$ represent node $i$’s displacements and $x_{i+1}$, $y_{i+1}$ represent node $\left(i+1\right)$’s displacements. And we have:

The equations for the rigid disks are:

where:

The equations for the elastic shaft element are:

In which, ${\mathbf{M}}_R^{es}$, ${\mathbf{M}}_R^{es}$ and ${\mathbf{K}}_T^{es}$ are given as follows:

The equations for the linear bearings are:

where ${\mathbf{C}}^b=\left[\begin{array}{llll}{c}_{xx}& c_{xy};& c_{yx}& c_{yy}\end{array}\right]$, ${\mathbf{K}}^b=\left[\begin{array}{llll}{k}_{xx}& k_{xy};& k_{yx}& k_{yy}\end{array}\right]$.

According to Eqs. (6) to (9), and with the consideration of the factor of the inter-shaft bearing, the motion equations of the assembled dual-rotor system coupling by the cylindrical roller inter-shaft bearing can be denoted as:

where ${\mathbf{M}}_1$, ${\mathbf{M}}_2$ are the assembled mass matrixes, ${\mathbf{C}}_1$, ${\mathbf{C}}_2$ are the damping matrixes (the proportional damping method is used in this paper, so we have ${\mathbf{C}}_1={\alpha }_1{\mathbf{M}}_1+{\beta }_1{\mathbf{K}}_1\text{,}$${\mathbf{C}}_2={\alpha }_2{\mathbf{M}}_2+{\beta }_2{\mathbf{K}}_2$). ${\mathbf{K}}_1$, ${\mathbf{K}}_2$ are the stiffness matrixes. ${\mathbf{F}}_{u1}$, ${\mathbf{F}}_{u2}$ are the vectors of the unbalance excitation of the rotor 1 and rotor 2, ${\mathbf{F}}_{g1}$, ${\mathbf{F}}_{g2}$ are the gravities vectors. ${\mathbf{F}}_{c1}$, ${\mathbf{F}}_{c2}$ are the vectors of the inertial centrifugal excitation of the rotor 1 and rotor 2, can be expressed as:

And ${\mathbf{F}}_{b1}$, ${\mathbf{F}}_{b2}$ are the restoring forces vectors caused by the inter-shaft bearing, given as:

Then from Eq. (10), the motion equations of the dual-rotor system coupling by the cylindrical roller inter-shaft bearing can be expressed as:

Letting:

then Eq. (13) can be transformed into:

And the unbalance excitation vectors ${\mathbf{F}}_u$ can be expressed as:

In which, ${\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2$ are the rotation speeds of rotor 1 and rotor 2. ${\mathbf{F}}_1\mathrm{c}\mathrm{o}\mathrm{s}\left({\mathrm{\Omega }}_{1}t\right)$, ${\mathbf{F}}_2\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathrm{\Omega }}_{1}t\right)$are the unbalance excitation vectors of rotor 1, where disk 1 has an unbalance mass with an eccentricity $e_1$. ${\mathbf{F}}_3\mathrm{c}\mathrm{o}\mathrm{s}\left({\mathrm{\Omega }}_{2}t\right)$, ${\mathbf{F}}_4\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathrm{\Omega }}_{2}t\right)$ are the unbalance excitation vectors of rotor 2, where disk 3 and disk 4 have an unbalance mass with a eccentricity $e_3$ and $e_4$ respectively. By letting $\beta ={\mathrm{\Omega }}_2/{\mathrm{\Omega }}_1$, represents the speed ratio, and substituting Eq. (15) into Eq. (14), we obtain the motion equations of the dual-rotor system coupling by the cylindrical roller inter-shaft bearing:

In which, $\mathbf{M}$, $\mathbf{C}$ and $\mathbf{K}$ are 22×22 matrixes, $\mathbf{q}\left(t\right)$, ${\mathbf{F}}_b$, ${\mathbf{F}}_c$, ${\mathbf{F}}_1$, ${\mathbf{F}}_2$, ${\mathbf{F}}_3$, ${\mathbf{F}}_4$ and ${\mathbf{F}}_g$ are 22×1 vectors.

3.1. Effect of amount of the parallel misalignment ($\mathbf{\Delta }\mathit{y}$) on unbalanced vibration signal

In order to give a deep and detailed insight into the properties the unbalanced vibration, the Fast Fourier Transform (FFT) is used to analyze the nonlinear response signal.

Vibration amplitudes power spectrum versus different amount of the parallel misalignment ($\mathrm{\Delta }y$) ($\mathrm{\Delta }\alpha =$ 0: $\mathrm{\Delta }y=$ 0, 0.1 mm, 0.6 mm, 1.1 mm, 1.6 mm) at rotation speed of ${\mathrm{\Omega }}_1=$ 600 rad/s, ${\mathrm{\Omega }}_2=$ 480 rad/s are analyzed detailedly for the nonlinear responses of the dual-rotor-bearing system with parallel misalignment faults, as shown from Fig. 6 to Fig. 8. In Fig. 6, and Fig. 8, the super-harmonic frequency components and the combined frequency components can be seen except the basic frequency components of ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$, and with the increase of $\mathrm{\Delta }y$ (from 0 mm to 1.6 mm), the frequency components 2${\mathrm{\Omega }}_2$ appear, besides, the amplitudes of the responses at the frequency components 2(${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$), 2${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$, 3${\mathrm{\Omega }}_1$-2${\mathrm{\Omega }}_2$, 2${\mathrm{\Omega }}_2$, 2${\mathrm{\Omega }}_2+{\mathrm{\Omega }}_1$ get larger for node 2 in Fig. 6(b) and node 10 in Fig. 10(b). The frequency components are only ${\mathrm{\Omega }}_2$ in at node 6 in Fig. 7 for $\mathrm{\Delta }\alpha =$ 0, $\mathrm{\Delta }y=$ 0, but with the increase of $\mathrm{\Delta }y$ (from 0.1 mm to 1.6 mm), the frequency components 2${\mathrm{\Omega }}_2$ appears and the amplitudes of the responses get larger. It is also indicated the vibration characteristics of misalignment faults in rotor 2 can be transmitted to the rotor 1, which has already been shown in [23], but the increase of the amplitudes at the combined frequency components related to $\mathrm{\Delta }y$ is a new phenomenon found in this study.

Fig. 5The whirling orbits and the amplitude power spectrum for Ω1= 600 rad/s, Ω2= 480 rad/s: a)-c) the whirling orbits, d)-f) the amplitude power spectrum

a) Node 2

b) Node 6

c) Node 10

d) Node 2

e) Node 6

f) Node 10

Fig. 6The amplitude power spectrum for different amount of the parallel misalignment (Δy) of x2

a) Global view

b) Enlarged view

In addition, the whirling orbits for low-pressure compressor rotor (node 6) are inner-looped (two loops) at rotation speed of ${\mathrm{\Omega }}_1=$ 600 rad/s, ${\mathrm{\Omega }}_1=$ 480 rad/s in Fig. 9, and this phenomenon is confirmed in Ref. [16] (Fig. 11), Ref. [19] (Fig. 11) and Ref. [28] (Fig. 3). And accompanied by an increase in $\mathrm{\Delta }y$ from 0.6 mm to 1.6 mm, the whirling orbits increased significantly, it means the misalignment signal is increased.

Fig. 7The amplitude power spectrum for different amount of the parallel misalignment (Δy) of x6

Fig. 8The amplitude power spectrum for different amount of the parallel misalignment (Δy) of x10

a) Global view

b) Enlarged view

Fig. 9The whirling orbits for Δα= 0, Ω1= 600 rad/s, Ω2= 480 rad/s: a) Δy= 0.6 mm, b) Δy= 1.1 mm, c) Δy= 1.6 mm

a)

b)

c)

3.2. Effect of amount of the angular misalignment ($\mathbf{\Delta }\mathit{\alpha }$) on unbalanced vibration signal

Vibration amplitudes power spectrum versus different amount of the angular misalignment ($\mathrm{\Delta }\alpha$) ($\mathrm{\Delta }y=$ 0: $\mathrm{\Delta }\alpha =$ 0, 0.1, 0.2, 0.4, 0.6) at rotation speed of ${\mathrm{\Omega }}_1=$ 600 rad/s, ${\mathrm{\Omega }}_2=$ 480 rad/s are analyzed for the nonlinear responses of the dual-rotor-bearing system with angular misalignment faults, as shown from Fig.10 to Fig. 12. In Fig. 10, and Fig. 12, the super-harmonic frequency components and the combined frequency components can be seen except the basic frequency components of ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$, and with the increase of $\mathrm{\Delta }\alpha$ (from 0 mm to 0.6 mm), the frequency components 2${\mathrm{\Omega }}_2$ appear, besides, the amplitudes of the responses at the frequency components ${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$, 2${\mathrm{\Omega }}_1$-${\mathrm{\Omega }}_2$, 2${\mathrm{\Omega }}_2$, 2${\mathrm{\Omega }}_2+{\mathrm{\Omega }}_1$ get larger for node 2 in Fig. 10(b) and node 10 in Fig. 12(b). The frequency components are only ${\mathrm{\Omega }}_2$ in at node 6 in Fig. 11 for $\mathrm{\Delta }y=$ 0, $\mathrm{\Delta }\alpha =$ 0, but with the increase of $\mathrm{\Delta }\alpha$ (from 0 mm to 0.6 mm), the frequency components 2${\mathrm{\Omega }}_2$ appears and the amplitudes of the responses get larger.

Fig. 10The amplitude power spectrum for different amount of the angular misalignment (Δα) of x2

a) Global view

b) Enlarged view

Fig. 11The amplitude power spectrum for different amount of the angular misalignment (Δα) of x6

Fig. 12The amplitude power spectrum for different amount of the angular misalignment (Δα) of x10

a) Global view

b) Enlarged view

Fig. 13The whirling orbits for Δy= 0, Ω1= 600 rad/s, Ω2= 480 rad/s: a) Δα= 0.1, node 2, b) Δα= 0.1, node 6, c) Δα= 0.1 node 10, d) Δα= 0.2, node 2, e) Δα= 0.2, node 6, f) Δα= 0.2 node 10, g) Δα= 0.4, node 2, h) Δα= 0.4, node 6, i) Δα= 0.4 node 10, j) Δα= 0.6, node 2, k) Δα= 0.6, node 6, l) Δα= 0.6 node 10

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

Besides, the whirling orbits of node 2, node 6 and node 10 are given in Fig. 13 at rotation speed of ${\mathrm{\Omega }}_1=$ 600 rad/s, ${\mathrm{\Omega }}_2=$ 480 rad/s ($\mathrm{\Delta }y=$ 0: $\mathrm{\Delta }\alpha =$ 0.1, 0.2, 0.4, 0.6). Accompanied by an increase in $\mathrm{\Delta }\alpha$, the whirling orbits (inner-looped) increased significantly for low-pressure compressor rotor (node 6), and the same phenomenon can be seen in Ref. [16], Ref. [19] and Ref. [28]. Besides, in this study, we find with an increase in $\mathrm{\Delta }\alpha$, the whirling orbits for node 2 and node 10 are all become more complex.

From Fig. 5 to Fig. 13, we can conclude that, the multiplier excitation (2$\omega t$) are induced by the misaligned faults, and which will lead the complex nonlinear phenomenon for the dual-rotor system coupled by inter-shaft bearing. The nonlinear phenomenon caused by inter-shaft bearing can be seen more clearly at node 10 (high-pressure turbine rotor in rotor 1) which has a near distance with the inter-shaft bearing. And the complex nonlinear phenomenon caused by misaligned faults can be seen more clearly at node 6(the low-pressure compressor rotor system in rotor 2) which has a near distance with the flexible coupling misalignment faults between node 7 and node 8. And with the increase of the amount of parallel misalignment (angular misalignment) $\mathrm{\Delta }y$ ($\mathrm{\Delta }\alpha$), not only the super-harmonic frequency component 2${\mathrm{\Omega }}_2$ representing the misalignment signal increases for both rotor 2 and rotor 1, but also the responses at the combined frequency components $\text{2}{\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2$, $\text{2}{\mathrm{\Omega }}_2+{\mathrm{\Omega }}_1$ etc. also get larger, which indicates that the vibration characteristics of misalignment faults in rotor 2 can be transmitted to the rotor 1.