Issue 
Mechanics & Industry
Volume 25, 2024



Article Number  29  
Number of page(s)  15  
DOI  https://doi.org/10.1051/meca/2024026  
Published online  25 October 2024 
Original Article
Vibrations of a dualrotor system in aeroengine induced by the support misalignment
^{1}
School of Machinery and Automation, Weifang University, 261000 Weifang, China
^{2}
School of Mechanical Engineering, Liaoning University of Technology, 121000 Jinzhou, China
^{3}
Northeast Agricultural University, 150006 Heilongjiang, China
^{4}
China Agricultural University, 100000 Beijing, China
^{5}
Dongfeng Equipment Manufacturing Co., Ltd., 442022 Shiyan, China
^{*} email: xiaoma9203@163.com
Received:
5
March
2024
Accepted:
13
September
2024
In this paper, an analytical model of a rotorbearingrotor coupling system is established for aeroengine from the Lagrange equation, based on the dynamics theory of the dualrotor system and combined with its stiffness, damping, and misalignment characteristics. In the model, the misalignment fault simulation method is introduced. The coupling effect between the lowpressure rotor and highpressure rotor and the influence of misalignment of the support on lowpressure coupling are considered. The vibration characteristics of the dualrotor system with the misalignment fault of the lowpressure turbine rear support are studied. The timedomain responses, the frequency spectra, and the shaftcenter trajectory of the dualrotor system with different misalignments are obtained. The influence of unbalance on the vibration characteristics of the lowpressure rotor and the highpressure rotor with misalignment fault is analyzed. Finally, test verification is carried out by the designed dualrotor test rig. The experimental results are consistent with the analytical results, confirming the established model's feasibility and simulation method in this paper. The results may provide a theoretical basis for research on the misalignment fault in aeroengine dualrotor systems.
Key words: Dualrotor system / misalignment faults / vibration characteristics / theoretical analysis / experiment
© P. Ma et al., Published by EDP Sciences 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The vibration fault forms of aeroengine rotor systems are various, such as unbalance, rubimpact, and misalignment [1]. The misalignment fault is one of the most common faults of the aeroengine rotor system. In actual production, the support misalignment fault of the rotor system is caused by the manufacturing error, installation error thermal expansion of the bearing base, etc. Misalignment fault will lead to excessive stress of the rotor shaft and instability of system motion. It can also cause other faults in the rotor system. The research on the influence of misalignment faults on dualrotor system vibration response characteristics occupies an important role in the diagnosis and identification of aeroengine misalignment faults.
In recent years, research on the dynamic characteristics of the rotor system with misalignment has been conducted by relevant scholars. Wang and Jiang [2], Qingkai et al. [3], Hongxian and Xuejun [4], Zhansheng et al. [5] and Pachpor et al. [6] summarized the misalignment of the rotor system. The rotor misalignment fault is mainly divided into coupling misalignment and support misalignment. As early as 1976, Gibbon [7] deduced and obtained the calculation formula of the force and torque generated at the connection point between the coupling and the rotating shaft. Sekhar and Prabhu [8] derived the calculation formula of angular misalignment force based on the Gibbons formula. Lee and Lee [9] derived the formula of excitation force decomposition when the coupling on the rotor system is misalignment, and the vibration response of the rotor system was analyzed. AlHussain and Redmond [10] derived the kinematics differential equation of the rotor system with parallel misalignment of rigid coupling using the basic kinematic principle, based on the Lagrange equation. Guo and Xingyan [11] studied the vibration characteristics of rotor systems under sleeve coupling misalignment faults. Some researchers [12] studied the misalignment excitation when the bearing of the rotor is misalignment, and further analyzed the influence of the misalignment of the bearing on the dynamics of the rotor system. Guoquan et al. [13] took the double rotor system of an aeroengine as the research object and considered the misalignment of outer rotor bearing, established the dynamic model of the dualrotor system by using the finite element method, and studied the vibration response characteristics of the rotor under different rotating speeds and misalignment through numerical simulation. Xiao [14] deduces the dynamic models of misalignment of supporting bearings and intermediate bearings in the dualrotor system using the finite element method. Meipeng et al. [15] equated the eccentricity of the highpressure rotor support with the additional misalignment moment at the support of the highpressure rotor and deduced the equation of the dualrotor system considering the eccentricity of the highpressure rotor support by using the Lagrange equation method.
Research on rotor misalignment mainly focuses on coupling misalignment in the rotor system [5,16–19]. The team [20,21] established a simplified model of a dual rotor system with four supports and four disks. The proper orthogonal decomposition method is used to reduce the dimension of the dualrotor system for the first time. With the help of the finite element method, the nonlinear response caused by the misalignment of the coupling between the rotor and the motor was studied and verified by the test method. Research on the misalignment of the support is aimed at the misalignment of bearing, but misalignment of bearing is only a part of the misalignment of the support, and lack of research on a misalignment of rolling bearing. The research on a misalignment of support has theoretical and practical engineering needs.
Although many scholars have done a lot of research on the vibration of rotor systems with misalignment faults, in the current research, the modeling of coupling misalignment and bearing misalignment is relatively mature, and the impact on rotor system vibration is mainly concentrated on a single rotor. For the dualrotor system, the vibration coupling problem of the dualrotor system considering the support misalignment fault and the influence of the fault on the coupling is seldom involved. In this paper, a typical aeroengine model tester is taken as the research object, considering the load torque force of the lowpressure caused by the misalignment of the rear support of the lowpressure turbine, the excitation mechanism of support misalignment is analyzed. The motion differential equation of the dualrotor system with misalignment faults is established, and the effects of misalignment, unbalanced mass, and load torque on the vibration response of the dualrotor are studied. Finally, test verification is carried out by the designed dualrotor test rig.
2 The dynamic model for a dualrotor system
With the increase of the thrustweight ratio of the aeroengine, the structure of the rotor system changes greatly, and the high speed of the aeroengine makes the vibration problem of the aeroengine more prominent. The coupling characteristics of aeroengine rotor dynamics and the transfer law of rotor misalignment vibration between the highpressure rotor and the lowpressure rotor are still unclear and need to be studied urgently. In this paper, an advanced aeroengine rotor in reference [22] was simplified as a rigid rotor with elastic supports, as shown in Figure 1. The inner rotor is also called a lowpressure rotor, and the outer rotor is also called the highpressure rotor. The lowpressure rotor consists of one fan (referred to as LPC) and one turbine (referred to as LPT), supported by bearings 1, 2, 4, and 5. The highpressure rotor is composed of one compressor disc (referred to as HPC) and a highpressure turbine (referred to as HPT), which is supported by bearings 3 and 4.
The parameters of the dualrotor system are shown in Tables 1–4, among which the parameters of four equivalent discs are shown in Table 1, the equivalent parameters of the rotating shaft are shown in Table 2, and the distance parameters are shown in Table 3. According to the principle of dynamic similarity, the first two natural frequencies and vibration modes of the simplified dualrotor system are kept consistent with the prototype by adjusting the support stiffness. Thus achieving the goal of dynamic similarity between the simplified model and the prototype's dual rotor system [22]. The designed stiffness parameters are shown in Table 4.
Fig. 1 Structural sketch of a dualrotor system. 
Disc parameters of the dynamic model.
Shaft parameters of the dualrotor system.
Distance parameters of the dualrotor system.
Equivalent stiffness of five supports.
3 Motion equations of the dualrotor system
The motion equations of the dualrotor system are as follows.
$$\frac{d}{dt}\frac{\partial T}{\partial {{\displaystyle \stackrel{q}{}}}_{j}}\frac{\partial T}{\partial {q}_{j}}+\frac{\partial U}{\partial {q}_{j}}={Q}_{j}(t),\text{}j=1,2,3$$(1)
where the kinetic energy of the system can be recorded as follows.
$$T={T}_{t1}+{T}_{r1}+{T}_{t2}+{T}_{r2}+{T}_{t3}+{T}_{r3}+{T}_{t4}+{T}_{r4}.$$(2)
T_{ti} and T_{ri} are the translational kinetic energy and rotational kinetic energy of disc respectively.
The total potential energy of the system can be recorded as follows.
$$U={U}_{1}+{U}_{2}+{U}_{3}+{U}_{4}+{U}_{c}+{U}_{5}.$$(3)
It includes the elastic potential energy of five supports and the potential energy of coupling.
$${U}_{i}=\frac{1}{2}{q}_{{B}_{i}}^{T}K{q}_{{B}_{}}=\frac{1}{2}{\left\{{B}_{i}{q}_{i}\right\}}^{T}K\left\{{B}_{i}{q}_{i}\right\},i=14.$$(4)
The elastic potential energy of the intermediate bearing is as follows.
$${U}_{5}=\frac{1}{2}{\left({q}_{{C}_{H}}{q}_{{C}_{L}}\right)}^{T}{K}_{C}\left({q}_{{C}_{H}}{q}_{{C}_{L}}\right).$$(5)
The elastic potential energy at the coupling is as follows.
$${U}_{C}=\frac{1}{2}{\left({q}_{{C}_{2}}{q}_{{C}_{1}}\right)}^{T}{K}_{C}\left({q}_{{C}_{2}}{q}_{{C}_{1}}\right)=\frac{1}{2}{\left\{{B}_{5}{q}_{2}{B}_{4}{q}_{1}\right\}}^{T}{K}_{C}\left\{{B}_{5}{q}_{2}{B}_{4}{q}_{1}\right\}.$$(6)
The analytic model of the dualrotor system is established by using the Lagrange equation as shown in euation (7). Please refer to reference [22] for the specific modeling process.
$${m}_{1}{{\displaystyle \dot{y}}}_{1}\left({\text{k}}_{b1y}{\theta}_{z1}{a}_{1}{k}_{b1y}{y}_{1}\right)\text{+}\frac{12{E}_{C}{I}_{yc}}{{L}_{c}{3}^{}}\left(\left(\frac{\theta z1}{2}+\frac{\theta z2}{2}\right){L}_{c}+{\theta}_{z1}{c}_{1}+{\theta}_{z2}{c}_{2}+{y}_{1}{y}_{2}\right)=0$$(7.a)
$${m}_{1}{{\displaystyle \dot{z}}}_{1}+{k}_{b1z}{\theta}_{y1}{a}_{1}+{k}_{b1z}{z}_{1}+\frac{12{E}_{C}{I}_{yc}}{{L}_{c}{3}^{}}\left(\left(\frac{{\theta}_{y1}}{2}+\frac{{\theta}_{y2}}{2}\right){L}_{c}+{\theta}_{y1}{c}_{1}+{\theta}_{y2}{c}_{2}{z}_{1}+{z}_{2}\right)=0$$(7.b)
$$\begin{array}{l}{J}_{d1}{{\displaystyle \dot{\theta}}}_{y1}+{J}_{p1}{\text{\Omega}}_{1}{{\displaystyle \dot{\theta}}}_{z1}+\frac{4{E}_{C}{I}_{yc}\left({\theta}_{y1}+{\theta}_{y2}/2\right)}{{L}_{c}}+\frac{12{E}_{C}{I}_{yc}{c}_{1}\left({\theta}_{y1}{c}_{1}+{\theta}_{y2}{c}_{2}{z}_{1}+{z}_{2}\right)}{{L}_{c}{3}^{}}\\ \text{+}\frac{12{E}_{C}\left(\left({\theta}_{y1}+{\theta}_{y2}/2\right){c}_{1}+1/2{\theta}_{y2}{c}_{2}{z}_{1}/2+{z}_{2}/2\right){I}_{yc}{L}_{c}}{{L}_{c}{3}^{}}+{k}_{b1z}{\theta}_{y1}{a}_{1}{2}^{}+{k}_{b1z}{z}_{1}{a}_{1}=0\end{array}$$(7.c)
$$\begin{array}{l}{J}_{d1}{{\displaystyle \dot{\theta}}}_{z1}{J}_{p1}{\text{\Omega}}_{1}{{\displaystyle \dot{\theta}}}_{y1}+\frac{4\left({\theta}_{z1}+{\theta}_{z2}/2\right){E}_{C}{I}_{yc}}{{L}_{c}}\text{+}\frac{12{E}_{C}{I}_{yc}{c}_{1}\left({\theta}_{z1}{c}_{1}+{\theta}_{z2}{c}_{2}+{y}_{1}{y}_{2}\right)}{{L}_{c}{3}^{}}\\ +\frac{12{E}_{C}{I}_{yc}{L}_{c}\left(\left({\theta}_{z1}+{\theta}_{z2}/2\right){c}_{1}+1/2{\theta}_{z2}{c}_{2}+{y}_{1}/2{y}_{2}/2\right)}{{L}_{c}{3}^{}}+{k}_{b1y}{\theta}_{z1}{a}_{1}{2}^{}{k}_{b1y}{y}_{1}{a}_{1}=0\end{array}$$(7.d)
$$\begin{array}{l}{m}_{2}{{\displaystyle \dot{y}}}_{2}+\left({k}_{b2y}{a}_{2}{k}_{b4y}{a}_{41}+{k}_{b5y}{a}_{5}\right){\theta}_{z2}+\left({y}_{2}+\left({a}_{4}+{a}_{34}\right){\theta}_{z4}{y}_{4}\right){k}_{b4y}2\\ \frac{6{E}_{C}{I}_{yc}\left({\theta}_{z1}+{\theta}_{z2}\right){L}_{c}+12{E}_{C}{I}_{yc}\left({\theta}_{z1}{c}_{1}+{\theta}_{z2}{c}_{2}+{y}_{1}{y}_{2}\right)}{{L}_{c}{3}^{}}+\left({k}_{b5y}+{k}_{b2y}\right)y=0\end{array}$$(7.e)
$$\begin{array}{l}{m}_{2}{{\displaystyle \dot{z}}}_{2}+\left({k}_{b2z}{a}_{2}+{k}_{b4z}{a}_{41}{k}_{b5z}{a}_{5}\right){\theta}_{y2}+\left({z}_{2}+\left({a}_{4}{a}_{34}\right){\theta}_{y4}{z}_{4}\right){k}_{b4z}\\ +\frac{6{E}_{C}{I}_{yc}\left({\theta}_{y1}+{\theta}_{y2}\right){L}_{c}+12{E}_{C}{I}_{yc}\left({\theta}_{y1}{c}_{1}+{\theta}_{y2}{c}_{2}{z}_{1}+{z}_{2}\right)}{{L}_{c}{3}^{}}+\left({k}_{b5z}+{k}_{b2z}\right){z}_{2}=0\end{array}$$(7.f)
$$\begin{array}{l}{J}_{d2}{{\displaystyle \dot{\theta}}}_{y2}+{J}_{p2}{\text{\Omega}}_{1}{{\displaystyle \dot{\theta}}}_{z2}+\left({k}_{b2z}{a}_{2}{2}^{}+{k}_{b4z}{a}_{41}{2}^{}+{k}_{b5z}{a}_{5}{2}^{}+{k}_{b4\theta y}\right){\theta}_{y2}\\ +\left({z}_{2}+\left({a}_{4}{a}_{34}\right){\theta}_{y4}z4\right){k}_{b4z}{a}_{41}\\ +\left({k}_{b2z}{a}_{2}{k}_{b5z}{a}_{5}\right){z}_{2}{k}_{b4\theta y}{\theta}_{y4}\text{+}\frac{+6{E}_{C}{I}_{yc}\left({\theta}_{y1}{c}_{1}+{\theta}_{y1}{c}_{2}+2{\theta}_{y2}{c}_{2}{z}_{1}+{z}_{2}\right){L}_{c}}{{L}_{c}{3}^{}}\\ +\frac{2{E}_{C}{I}_{yc}\left({\theta}_{y1}+2{\theta}_{y2}\right){L}_{c}{2}^{}+12{E}_{C}{c}_{2}{I}_{yc}+{\theta}_{y1}{c}_{1}+{\theta}_{y2}{c}_{2}{z}_{1}+{z}_{2})}{{L}_{c}{3}^{}}=0\end{array}$$(7.g)
$$\begin{array}{l}{J}_{d2}{{\displaystyle \dot{\theta}}}_{z2}{J}_{p2}{\text{\Omega}}_{1}{{\displaystyle \dot{\theta}}}_{y2}+\left({a}_{2}{2}^{}{k}_{b2y}+{a}_{5}{2}^{}{k}_{b5y}+{a}_{41}{2}^{}{k}_{b4y}+{k}_{b4\theta z}\right){\theta}_{z2}\\ +{a}_{41}\left({y}_{2}+\left({a}_{4}{a}_{34}\right){\theta}_{z4}+{y}_{4}\right){k}_{b4y}+\left({a}_{2}{k}_{b2y}+{a}_{5}{k}_{b5y}\right){y}_{2}{k}_{b4\theta z}{\theta}_{z4}\\ +\frac{2{E}_{c}{I}_{yc}\left(\left({\theta}_{z1}+2{\theta}_{z2}\right){L}_{c}{2}^{}+\left(3{c}_{1}{\theta}_{z1}+3{c}_{2}{\theta}_{z1}+6{c}_{2}{\theta}_{z2}+3{y}_{1}3{y}_{2}\right){L}_{c}\right)}{{L}_{c}{3}^{}}\\ \text{+}\frac{12{E}_{c}{I}_{yc}{c}_{2}\left({c}_{1}{\theta}_{z1}+{c}_{2}{\theta}_{z2}+{y}_{1}{y}_{2}\right)}{{L}_{c}{3}^{}}=0\end{array}$$(7.h)
$${m}_{3}{{\displaystyle \dot{y}}}_{3}{a}_{3}{\theta}_{z3}{k}_{b3y}+{y}_{3}{k}_{b3y}+\frac{6{E}_{g}{I}_{y}\left(\left({\theta}_{z3}+{\theta}_{z4}\right)L+2{y}_{3}2{y}_{4}\right)}{{L}^{3}}=0$$(7.i)
$${m}_{3}{{\displaystyle \dot{z}}}_{3}+{a}_{3}{\theta}_{y3}{k}_{b3z}+{z}_{3}{k}_{b3z}\frac{6{E}_{g}{I}_{y}\left(\left({\theta}_{y3}+{\theta}_{y4}\right)L2{z}_{3}+2{z}_{4}\right)}{{L}^{3}}=0$$(7.j)
$${J}_{d3}{{\displaystyle \dot{\theta}}}_{y3}+{J}_{p3}{\text{\Omega}}_{3}{{\displaystyle \dot{\theta}}}_{z3}+{a}_{3}{2}^{}{k}_{b3z}{\theta}_{y3}+{a}_{3}{k}_{b3z}{z}_{3}+\frac{4{E}_{g}{I}_{y}\left(\left({\theta}_{y3}+1/2{\theta}_{y4}\right)L3/2{z}_{3}+3/2{z}_{4}\right)}{{L}^{2}}=0$$(7.k)
$${J}_{d3}{{\displaystyle \dot{\theta}}}_{z3}{J}_{p3}{\text{\Omega}}_{3}{\displaystyle \dot{\theta}}y\mathrm{\_3}+{a}_{3}{2}^{}{k}_{b3y}{\theta}_{z3}{a}_{3}{k}_{b3y}{y}_{3}+\frac{4{E}_{g}{I}_{y}\left(\left({\theta}_{z3}+1/2{\theta}_{z4}\right)L+3/2{y}_{3}3/2{y}_{4}\right)}{{L}^{2}}=0$$(7.l)
$${m}_{4}{{\displaystyle \dot{y}}}_{4}+\left({y}_{4}+\left({a}_{4}{a}_{34}\right){\theta}_{z4}+{\theta}_{z2}{a}_{41}{y}_{2}\right){k}_{b4y}\frac{6{E}_{g}{I}_{y}\left(\left({\theta}_{z3}+{\theta}_{z4}\right)L+2{y}_{3}2{y}_{4}\right)}{{L}^{3}}=0$$(7.m)
$${m}_{4}{{\displaystyle \ddot{z}}}_{4}\left(\left({a}_{4}{a}_{34}\right){\theta}_{y4}+{\theta}_{y2}{a}_{41}+{z}_{2}{z}_{4}\right){k}_{b4z}+\frac{6{E}_{g}{I}_{y}\left(\left({\theta}_{y3}+{\theta}_{y4}\right)L2{z}_{3}+2{z}_{4}\right)}{{L}^{3}}=0$$(7.n)
$$\begin{array}{l}{J}_{d4}{{\displaystyle \dot{\theta}}}_{y4}+{J}_{p4}{\text{\Omega}}_{3}{{\displaystyle \dot{\theta}}}_{z4}+\left({a}_{4}{a}_{34}\right)\left(\left({a}_{4}{a}_{34}\right){\theta}_{y4}+{\theta}_{y2}{a}_{41}+{z}_{2}{z}_{4}\right){k}_{b4z}\\ {k}_{b4\theta y}{\theta}_{y2}+{k}_{b4\theta y}{\theta}_{y4}+\frac{2{E}_{g}{I}_{y}\left(\left({\theta}_{y3}+2{\theta}_{y4}\right)L3{z}_{3}+3{z}_{4}\right)}{{L}^{2}}=0\end{array}$$(7.o)
$$\begin{array}{l}{J}_{d4}{{\displaystyle \dot{\theta}}}_{z4}{J}_{p4}{\text{\Omega}}_{3}{{\displaystyle \dot{\theta}}}_{y4}+\left({a}_{4}{a}_{34}\right)\left({y}_{4}+\left({a}_{4}{a}_{34}\right){\theta}_{z4}+{\theta}_{z2}{a}_{41}{y}_{2}\right){k}_{b4y}\\ {k}_{b4\theta z}{\theta}_{z2}+{k}_{b4\theta z}{\theta}_{z4}+\frac{2{E}_{g}{I}_{y}\left(\left({\theta}_{z3}+2{\theta}_{z4}\right)L+3{y}_{3}3{y}_{4}\right)}{{L}^{2}}=0\end{array}.$$(7.p)
4 The excitation mechanism of the support misalignment
The characteristics of the slender shaft of the lowpressure rotor make the assembly more difficult, which increases the possibility of support misalignment. The misalignment of the rear support of the lowpressure turbine rotor in the dualrotor system will not only change the potential energy of the rotor system but also affect the coupling that connects the fan rotor and the lowpressure turbine rotor. The structure diagram of the lowpressure turbine rear support misalignment is shown in Figure 2.
When the support has a misalignment of Δy in the ydirection and a misalignment of Δz in the zdirection, the support has a deformation in the equilibrium state, the expression of the elastic potential energy of the support becomes:
$${U}_{6}=\frac{1}{2}({q}_{{B}_{6}}^{T}{\text{\Delta}}^{T}){K}_{{B}_{6}}({q}_{{B}_{6}}\text{\Delta})$$(8)
where Δ = (0 Δy Δz 0 0) ^{T} . K_{B6} is the stiffness matrix of misalignment support. K_{B6} is the generalized displacement of the rear support of the lowpressure turbine disc.
When the lowpressure rotor is a misalignment, the sleeve coupling on the lowpressure rotor is bent, and there is a misalignment angle α between the fan rotor and the lowpressure turbine rotor. The rotational angular velocities of the fan rotor and the lowpressure turbine rotor are Ω_{1} and Ω_{2} respectively. Fan rotor drives the lowpressure turbine rotor through the coupling when the angular displacement of the fan rotor is ϕ_{1}, the angle of rotation of the coupling gear sleeve is ϕ_{2}, and the relationship between ϕ_{1} and ϕ_{2} is as follows.
$$\text{tg}{\varphi}_{1}=\text{tg}{\varphi}_{2}\mathrm{cos}\alpha .$$(9)
Considering the small misalignment angle α, i.e tan α ≅ α.
Derivatives on both sides of the above equation (8).
$$\frac{{\text{\Omega}}_{2}}{{\text{\Omega}}_{1}}=\frac{C}{1+D\mathrm{cos}2{\varphi}_{1}}$$(10)
where $C=\frac{4\mathrm{cos}\alpha}{3+\mathrm{cos}2\alpha},\text{}D=\frac{\mathrm{cos}2\alpha 1}{3+\mathrm{cos}2\alpha}$. The calculation method refers to Wang Meiling [23]. The angular acceleration of the lowpressure turbine rotor can be obtained by derivation of this formula:
$${{\displaystyle \dot{\text{\Omega}}}}_{2}=\frac{2CD{{\text{\Omega}}_{1}}^{2}\mathrm{sin}2{\varphi}_{1}}{{(1+\text{Dcos}2{\varphi}_{1})}^{2}}.$$(11)
The torque transmitted from the fan rotor shaft to the lowpressure turbine rotor shaft through the coupling is mainly used for two parts, one part is the accelerated motion of the lowpressure turbine rotor, and the other part is used to balance the resistance suffered by the lowpressure turbine rotor, namely:
$${T}_{x}={T}_{a}+{T}_{f}$$(12)
where ${T}_{a}=I{{\displaystyle \dot{\text{\Omega}}}}_{2}\text{,}{T}_{f}=c{\text{\Omega}}_{2}$. I is the moment of inertia of the lowpressure turbine rotor. c is the damping of the rotor, and the torque of the fan rotor is in the horizontal direction. When torque passes through the coupling, it divides into two parts:
$${T}_{x}=T\mathrm{cos}\alpha ,\text{}{T}_{s}=T\mathrm{sin}\alpha .$$(13)
Combining equation (11) and equation (12), it can be concluded that:
$$T\mathrm{cos}\alpha =I{{\displaystyle \dot{\text{\Omega}}}}_{2}+c{\text{\Omega}}_{2}$$(14)
where T_{s} is in the axial direction of the lowpressure turbine rotor. Further, decomposes into y and z directions.
$${T}_{y}=T\mathrm{sin}\alpha \mathrm{sin}\beta ,\text{}{T}_{z}=T\mathrm{sin}\alpha \mathrm{cos}\beta $$(15)
where β is the intersection angle between the misalignment vector and the zaxis. The torque forces of the lowpressure turbine rotor in the y and z directions at the coupling can be obtained.
Fig. 2 Schematic diagram of rear support misalignment of the lowpressure turbine. 
5 Misalignment excitation force
In the case of misalignment, the speed of the lowpressure turbine rotor has changed, and its excitation force also changes accordingly. The expression of the misalignment excitation force as follows
$${Q}_{2u}=\left\{\begin{array}{c}\hfill \alpha {m}_{2}{e}_{2}{\left[\frac{C{\text{\Omega}}_{1}}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}\right]}^{2}\mathrm{cos}(\frac{C{\text{\Omega}}_{1}t}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}+{\phi}_{20})\hfill \\ \hfill {m}_{2}{e}_{2}{\left[\frac{C{\text{\Omega}}_{1}}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}\right]}^{2}\mathrm{cos}(\frac{C{\text{\Omega}}_{1}t}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}+{\phi}_{20})\hfill \\ \hfill {m}_{2}{e}_{2}{\left[\frac{C{\text{\Omega}}_{1}}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}\right]}^{2}\mathrm{sin}(\frac{C{\text{\Omega}}_{1}t}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}+{\phi}_{20})\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right\}.$$(16)
From equation (15), it can be concluded that the excitation frequency of the lowpressure turbine disc is $\frac{C}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}$ times of rotating frequency of the fan rotor. The amplitude of excitation force is directly proportional to the sum of unbalance m_{2} e_{2} of the lowpressure turbine disc are ${\left[\frac{C{\text{\Omega}}_{1}}{1+D\mathrm{cos}(2{\text{\Omega}}_{1}t)}\right]}^{2}$.
6 The vibration of the dualrotor excited by both unbalanced and support misalignment
6.1 Misalignment in Zdirection
The misalignment of the rear support of the lowpressure turbine included the ydirection misalignment, zdirection misalignment, and comprehensive misalignment in the two directions. Four measuring points are selected on the lowpressure and the highpressure rotor respectively, including 1# support (No. 1), the centroid of LPC (No. 2), the centroid of LPT (No. 3), 6# support (No. 4), 4# support (No. 5), the centroid of HPC (No. 6), the centroid of HPT (No. 7) and the intermediate bearing (No. 8). The rotational speed of the lowpressure rotor and the highpressure rotor are 5000 r/min (N1) and 6000r/min (N2) respectively, and the initial unbalance of the fan disc is 3.0 × 10^{−3} kg m.
The situation that misalignment of 0.5 mm exists in the zdirection at 6# supporting was analyzed, and the timedomain responses, the frequency spectra, and the shaftcenter trajectories of each measuring point for the dualrotor system are obtained, as shown in Figure 3.
It can be seen from the frequency spectra that the vibration in the ydirection at 1#, 2#, 3#, 4#, 7#, and 8# measuring points has two obvious frequencies, one is the rotational frequency of the lowpressure rotor (N1), the other one is 2 times of rotational frequency of the lowpressure rotor (2N1). The orbit of the axis at the center is not a regular circle, but an approximate ‘concave’ shape, which indicates that the vibration response in the ydirection and the zdirection is coupled. The measuring points 5# and 6# on the highpressure rotor are less affected by the misalignment, and the amplitude of the 2 times rotational frequency of the lowpressure rotor (2N1) in the spectrum diagram is very small, which can be ignored, and the orbit of the axis center is a regular circle.
Fig. 3 Vibration response of measuring points 18 with zdirection misalignment. 
Fig. 3 (Continued). 
6.2 Ydirection and Zdirection misalignment
The situation that the misalignment of 0.5mm exists in the zdirection and ydirection at 6# supporting was analyzed, and the timedomain responses, the frequency spectra, and the shaftcenter trajectories of each measuring point for the dualrotor system are obtained, as shown in Figure 4.
According to the vibration response of 8 measuring points, the following conclusions can be drawn:
It can be seen from the spectrum diagram that the misalignment of the rear support of the lowpressure turbine will cause the 2 times rotational frequency of the lowpressure rotor.
It can be seen from the axle center trajectory in the figure that ‘concave’ axle center trajectory appears at measuring point 1 and point 2; The axial trajectories of measuring point 3, point 4, point 7 and point 8 are an external ‘8’ shape. And axial trajectories of measuring points 5 and 6 are close to circular. This is because measuring points 3, 4, 7 and 8 are close to the rear support of the misalignment lowpressure turbine while measuring points 5 and 6 are coupled with the lowpressure rotor to generate frequency 2N1 and are far away from the rear support of the misalignment lowpressure turbine, so they are least affected by misalignment.
It can be seen from the frequency spectra that the magnitude of frequency N1 of 5# and 6# is larger, which indicates that the two measuring points are greatly affected by unbalanced excitation.
Fig. 4 Vibration response of measuring points 18 with zdirection and ydirection misalignment. 
Fig. 4 (Continued). 
6.3 Influence of misalignment on the vibration response of the dualrotor system
In this section, the variation law of the dualrotor system vibration with misalignment of the support is studied and the fitting curves of vibration amplitude of each measuring point of the dualrotor system vary with misalignment are obtained, as showed in Figure 5 and Table 5. Among them, Figure 5a is a threedimensional spectrum diagram of the dualrotor system vibration with different misalignment.
Figure 5b is a fitting curve of the vibration amplitude of frequency 2N1 varying with the size of misalignment. The fitting curve equation of the vibration amplitude of frequency 2N1 changing with misalignment of the dualrotor system is shown in Table 5.
It can be seen from Figure 5a that with the increase of the misalignment of the support, the amplitude of frequency N1 has no obvious change, while the amplitude of frequency 2N1 increases. From Figure 5b, it can be seen that the vibration amplitude of frequency 2N1 is approximately quadratic with the misalignment. And the fitting curve equation and fitting error are obtained by fitting the quadratic curve, as showed in Table 5. Through analysis, it is found that the fitting error is very small. It can be considered that the amplitude of the rotor vibration frequency 2N1 caused by the misalignment of the rear support of the lowpressure turbine has a quadratic relationship with the misalignment.
Fig. 5 Influence of misalignment on vibration response of the dualrotor system. 
Fitting curve equation of vibration amplitude with misalignment.
6.4 Influence of rotor load torque on the vibration response of the dualrotor system
The influence of load torque on the response of the dualrotor support misalignment is studied in this section. The threedimensional spectrum of vibration amplitude for every test point varying with load torque are obtained, as showed in Figure 6a. The relationship between the vibration response amplitude of the dualrotor system and the load torque of the rotor is analyzed, and the fitting curve is obtained, as showed in Figure 6b. And the fitting equation and error are shown in Table 6.
As can be seen from Figure 6a, the amplitude of vibration frequency N1 of the rotor system does not change with the increase of rotor load torque. And the amplitude of the vibration frequency 2N1 is approximately linearly related to the load torque as showed in Figure 6b. The linear fitting results between the amplitude of frequency 2N1 and the load torque are shown in Table 6. It can be seen that the error of the fitting equation is within 0.1%. It can be considered that the amplitude of the rotor vibration frequency 2N1 caused by the misalignment of the rear support of the lowpressure turbine is linearly related to the load torque.
Fig. 6 Influence of rotor load torque on vibration response of the dualrotor system. 
Fitting curve equation of vibration amplitude with load torque.
6.5 Influence of unbalance on vibration response of the dualrotor system
In this section, the influence of unbalanced mass on the vibration response of the dualrotor system with support misalignment faults is studied. The threedimensional spectrum of the dualrotor vibration response varying with the unbalance of the fan disc is plotted, as showed in Figure 7a. The fitting curve of the amplitude of frequency N1 with unbalance in the vibration response of the dualrotor system is obtained, as showed in Figure 7b. And the fitting curve equation and fitting error between the amplitude of the vibration frequency N1 and the magnitude of the unbalance are obtained by fitting the quadratic curve, as showed in Table 7.
As can be seen from Figure 7a, the amplitude of the vibration response frequency N1 of the dualrotor system gradually increases with the increase of the unbalanced mass, the amplitude of 2N1 does not change. Furthermore, the fitting curve of the vibration amplitude of the frequency N1 with the unbalanced mass is given in Figure 7b, it is found that the amplitude of the frequency N1 has a linear relationship with the magnitude of the unbalanced mass. From the fitting results in Table 7, it can be seen that the error of the fitting function gradually decreases with the increase of unbalanced mass. It can be considered that the amplitude of the frequency N1 is linearly related to the unbalanced mass.
Fig. 7 Influence of unbalance on vibration response of the dualrotor system. 
Fitting curve equation of vibration amplitude with unbalance.
7 Experimentation research
By arranging four measuring points on the dualrotor system as shown in Figure 8, the vibration characteristics of the dualrotor are studied. The 6# support of the test rig has the function of adjusting the misalignment of the support, as shown in Figure 9a. The misalignment adjusting device mainly consists of 6# support, outer adjusting ring and inner adjusting ring of support misalignment. The misalignment of the 6# support can be adjusted by adjusting the angles of the two adjusting rings, and the adjustment range is 0–3 mm. Figure 9b shows the adjustment principle of the misalignment adjustment device. There is eccentricity H between the inner adjustment ring and the outer adjustment ring. When the inner adjustment ring is rotated, the center O_{2} of the inner ring rotates through the angle θ along the center O1 of the outer adjustment ring, so that the inner adjustment ring is nonconcentric concerning the outer adjustment ring, so that the rotor support is out of alignment. An annular groove is arranged on the outer ring of the adjusting device to control and adjust the eccentric angle. The different mean values (horizontal direction and vertical direction) of the rotor support are as follows.
$${\delta}_{z}=H\mathrm{sin}(\theta ),\text{}{\delta}_{y}=H\left[1\mathrm{sin}(\theta )\right].$$
Vibration response of the dualrotor system with 6# support misalignment was tested on the test rig. Rotational speed of the inner and outer rotors is 1000 r/min and 0 r/min respectively. The timedomain responses, the frequency spectra and the shaftcenter trajectories at the rear support of the lowpressure turbine are obtained employing eddy current sensors with the 2mm misalignment, as showed in Figures 10a and 10b. And the analytical results are also given, as showing in Figure 10c. The analytical results also conform to the practical vibration measurement due to support misalignment, which confirms the feasibility of the established model and an analytical method.
As can be seen from Figure 10a, miscellaneous edges appear at the peaks and valleys of the timedomain displacement curve, which are caused by the influence of highfrequency components in the signal. The main components of the spectrum are frequencies N1 and 2N1, the peak value of component frequency is relatively low. And N1 is the rotating frequency of the lowpressure rotor, 2N1 is mainly caused by the misalignment of the support. The displacement time domain curve and spectrum after filtering out highfrequency components are shown in Figure 10b, which is regular waveforms. The measured axis trajectory is relatively messy due to the influence of highfrequency signals caused by gear meshing, bearing defects and other factors. And the axis trajectory after filtering out highfrequency components presents an obvious “waterdrop” shape, which is consistent with the simulation results of analytical calculation.
The vibration time domain and frequency spectrum of different measuring points with the same misalignment is obtained are shown in Figure 11a. By adjusting the angle θ, the displacement vibration time domain and frequency domain of the lowpressure turbine shaft under four kinds of misalignment are obtained as shown in Figure 11b. The vibration time domain and frequency domain of the lowpressure turbine shaft under four different unbalances are obtained as shown in Figure 11c by changing the unbalance of the fan disc. The magnitude of the four imbalances from case 1–4 are 20 g, 26 g, 42 g, and 63 g respectively.
As can be seen from Figure 11a, the magnitude order of the four vibration measuring points is 3 < 1 < 2 < 4. It shows that the vibration decreases on the lowpressure rotor transmission path: LPTLPCFan shaft. The vibration of the lowpressure rotor causes the vibration of N1 as the main frequency at the measuring point 3 of the highpressure rotor, which indicates that there is coupling between vibration of the highpressure rotor and the lowpressure rotor. It can be seen from Figure 11b that during the change of θ from 90° to 0°, the amplitude of frequency N1 basically has no change, and the amplitude of frequency 2N1 gradually decreases. It can be seen from Figure 11c that the amplitude of frequency N1 is linear with the magnitude of unbalance.
Fig. 8 Measuring points on the dualrotor system testrig. 
Fig. 9 6# fulcrum misalignment adjusting device. 
Fig. 10 The timedomain responses, the frequency spectra and the shaftcenter trajectories of dualrotor. 
Fig. 11 The displacement vibration of the lowpressure turbine shaft under four kinds of misalignment. 
8 Conclusions
In this paper, the vibration characteristics of the dualrotor system with support misalignment faults are studied by analytical method. Based on the selfdesigned testrig of the dualrotor system, the vibration characteristics of the dualrotor system under the misalignment fault of support are studied. After calculation and analysis, the following conclusions are obtained:
The 2 times of rotational frequency of the lowpressure rotor (2N1) will occur in the vibration of the dualrotor when the support is misalignment. And coupling will occur in the misalignment of supports in horizontal and vertical directions.
The amplitude of the frequency 2N1 in the vibration response of the dualrotor system is quadratic with the misalignment.
The amplitude of the frequency 2N1 in the vibration response of the dualrotor system is linearly related to the load torque.
The amplitude of the rotor frequency in the vibration response of the dualrotor system has a linear relationship with the unbalanced mass, while the frequency 2N1 does not correlate with the unbalanced mass.
The simplified model of the dualrotor system eliminates the interference of many factors on the vibration characteristics of the dualrotor system, and clarifies the feasibility of using the analytical dynamic model of the dualrotor system to study the vibration problem under the rotor misalignment fault. The establishment of the dual rotor system testrig with a misalignment adjustment device can simulate the misalignment of support. The theoretical results are verified by experimental tests. The conclusion of this paper has important value for vibration prediction, evaluation and control of aeroengine internal and external dualrotor systems unbalance and misaligned support faults.
Funding
Key Laboratory of Vibration and Control of AeroPropulsion System Ministry of Education, Northeastern University(VCAME201801). Study on Dynamic Optimization Method of Large Squeeze Film Damper Driven by Data (12072069).
Conflicts of interest
The authors declare no conflict of interest in preparing this article.
Data availability statement
All data, models, and code generated or used during the study appear in the submitted article.
Author contribution statement
Pingping Ma's contribution to the work can be summarized in the following roles: Writing − Review and Editing, WritingOriginal Draft Preparation Software, Conceptualization, and Methodology. Muge Liu's contribution to the work can be summarized in the following roles: Visualization and formal Analysis. Junkai Guo's contribution to the work can be summarized in the following roles: Data Curation and investigation. Haoyu Wang's contribution to the work can be summarized in the following roles: Data Curation, Formal Analysis, and Resources. Yuehui Dong's contribution to the work can be summarized in the following roles: Visualization, Data Curation, and Investigation.
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All Tables
All Figures
Fig. 1 Structural sketch of a dualrotor system. 

In the text 
Fig. 2 Schematic diagram of rear support misalignment of the lowpressure turbine. 

In the text 
Fig. 3 Vibration response of measuring points 18 with zdirection misalignment. 

In the text 
Fig. 3 (Continued). 

In the text 
Fig. 4 Vibration response of measuring points 18 with zdirection and ydirection misalignment. 

In the text 
Fig. 4 (Continued). 

In the text 
Fig. 5 Influence of misalignment on vibration response of the dualrotor system. 

In the text 
Fig. 6 Influence of rotor load torque on vibration response of the dualrotor system. 

In the text 
Fig. 7 Influence of unbalance on vibration response of the dualrotor system. 

In the text 
Fig. 8 Measuring points on the dualrotor system testrig. 

In the text 
Fig. 9 6# fulcrum misalignment adjusting device. 

In the text 
Fig. 10 The timedomain responses, the frequency spectra and the shaftcenter trajectories of dualrotor. 

In the text 
Fig. 11 The displacement vibration of the lowpressure turbine shaft under four kinds of misalignment. 

In the text 
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