Abstract
The dualrotor system is the core component of advanced aeroengine. Establishing a reasonable, accurate, and efficient dynamics model is the key to studying the dynamics and vibration of the rotor system for aeroengine. This manuscript takes a representative aircraft engine dualrotor system as a prototype, considers the rigidflexible coupling characteristics of different stiffness elastic supports and rotor structures, and establishes an analytical dynamic model of the dualrotor system. Based on the established dynamic model, the natural characteristics of the dualrotor system are analyzed. The model was validated using two different research methods: the rigidflexible coupling multibody system dynamics simulation platform ADAMS, and finite element analysis. The dynamic model of the dualrotor system established in this paper can meet the requirements of hierarchical rigidflexible coupling of system and structure, overall mass distribution, and stiffness distribution. In particular, it can also effectively realize the simulation of multifacet and multiphase unbalanced vibration of the rotor system. The research methods of this paper can further enrich the basic theory of dynamics and vibration of the aeroengine rotor system.
Highlights
 The dualrotor system is the core component of advanced aeroengine
 Accurate dynamic model is the key to correctly predict rotor system vibration
 Elastic strain energy will be excessively introduced for parts with high stiffness under elastic assumption
 Rigidflexible coupling model can more accurately express the dynamic characteristics of aeroengine dualrotor system
1. Introduction
The dualrotor system of aeroengine has multiple elastic supports, complex and special structures, and many factors are affecting its dynamic characteristics. Exploring a simplified model of the dualrotor system that can reflect the structural characteristics and dynamic behavior of the aeroengine is the key to establishing an accurate and efficient dynamic model of rotor system for aeroengine. The model of the double rotors and multisupports system used in the current research mainly includes the following types: the dualrotor system with one disc for the highpressure rotor and one disc for the lowpressure rotor [14], the dualrotor system with two discs for the highpressure rotor and two discs for the lowpressure rotor [57], and the dualrotor system with multiplediscs [4, 810]. The rigidflexible coupling of rotor system structure is not considered in dynamic modeling. When analyzing parts with large stiffness, the elastic strain energy will be too much included in the analysis based on the elastic hypothesis. The calculation error of the dynamics analysis and vibration prediction for the dualrotor system is too large, so it is difficult to realize the accurate analysis and vibration prediction of the dynamic characteristics of an aeroengine rotor.
Aiming at the theoretical analysis requirements of dynamics and vibration characteristics of the dualrotor system for aeroengine, considering multiple supports and the coupling of internal and external rotor structures, the energy method is used to establish an analytical dynamic model. The model has four lumped mass discs (lowpressure compressor, lowpressure turbine, highpressure compressor, and highpressure turbine) and five supports with different stiffness characteristics, and the lowpressure turbine shaft is considered as a flexible shaft. Based on the analytical model, the inherent characteristics of the dualrotor system are analyzed and compared with the finite element results and multibody dynamics simulation results, which verifies the effectiveness and accuracy of the analytical model. The accurate modeling of the dualrotor system for aeroengine is realized, which lays a foundation for the subsequent vibration characteristics analysis of the dualrotor system.
2. Dynamic modeling of the dualrotor system
2.1. Mechanical model of the dualrotor system
This section takes a typical aviation engine dualrotor system as a prototype and establishes a simplified mechanical model of the dualrotor system based on the principle of dynamic similarity. Through the qualitative study of the simplified model, this paper explores the important parameters that affect the regularity of the motion behavior of the system. According to the structure and working characteristics of the dual rotor system, a simplified model that can reflect the main dynamic behavior of the dualrotor system for aeroengine is established, as shown in Fig. 1. The modeling process follows the dynamic similarity principle, and the obtained model has the same mass ratio as the prototype rotor system.
Fig. 1Simplified mechanical model of the dualrotor system
In the simplified model of the dualrotor system, the compressor and turbine disc are simplified to a disc with equivalent mass and moment of inertia. The actual multidisc structure is simplified into a few disc structures. The specific simplification method is as follows: the fourstage fan discs in the lowpressure rotor are simplified as disc D1 (LPC), the lowpressure turbine disc is equivalent to disc D2 (LPT), the ninestage compressor discs of the highpressure rotor is simplified as disc D3 (HPC), and the highpressure turbine disc is equivalent to disc 4 (HPT).
The supports of the dualrotor are simplified into five springs with different stiffness. The support stiffness ${K}_{b1}$, ${K}_{b3}$ and ${K}_{b5}$ are the combined equivalent stiffness of the bearing, squirrel cage and support at the front fulcrum of the fan rotor, the front fulcrum of the highpressure rotor and the rear fulcrum of the lowpressure turbine respectively. ${K}_{b2}$ is the combined stiffness of support 2 and support 3. The intermediate fulcrum adopts the same roller bearing as the prototype, and the stiffness is ${K}_{b4}$.
2.2. Rigidflexible coupling dynamic model of the dualrotor system
Taking the simplified model of the dualrotor system shown in Fig. 1 as the research object, the motion analytical equation of the dualrotor system is deduced with the help of the Lagrange method. The support is considered to be elastic and simulated with equivalent stiffness. The rotating speed of the rotating shaft is considered to be constant, and the disc, fan shaft, and highpressure shaft are considered to be rigid. In particular, the elastic characteristics of the slender shaft of the lowpressure turbine are considered.
2.2.1. Kinetic energy of the dualrotor system
The kinetic energy of the dualrotor system includes the translational kinetic energy and rotational kinetic energy of the four discs, which can be recorded as follows:
where, ${T}_{ti}$and ${T}_{ri}$ ($i=1~4$) are the translational kinetic energy and rotational kinetic energy of disc respectively.
Based on the method of establishing a coordinate system and determining the generalized displacement and velocity of the rotor system, the kinetic energy of the dualrotor system is obtained. The mass, polar moment of inertia, and diameter moment of inertia of the rotor system are represented by ${m}_{i}$, ${J}_{pi}$, and ${J}_{di}$ ($i=1:4$) respectively. The translational and rotational kinetic energy of turntables 14 are:
2.2.2. Potential energy of the dualrotor system
The potential energy of the dualrotor system is shown in Eq. (4), including the elastic potential energy of five fulcrums, the potential energy of the coupling and the elastic potential energy of the lowpressure turbine shaft:
It includes the elastic potential energy of five supports and the potential energy of coupling:
where, ${\mathbf{q}}_{{\mathbf{B}}_{\mathrm{i}}}$ ($i=1~4$) is the displacement vector of the rotor pivot ${B}_{i}$ ($i=1~4$) in the generalized coordinate system. ${\mathbf{K}}_{{B}_{i}}$ is the stiffness matrix of support ${B}_{i}$ ($i=1~4$)
The elastic potential energy of the intermediate bearing is as follows:
Among them, ${\mathbf{q}}_{{C}_{H}}$ and ${\mathbf{q}}_{{C}_{L}}$ are the generalized displacement vectors of the intermediate support points on the highpressure and lowpressure rotors, respectively. ${\mathbf{K}}_{{B}_{5}}$ is the stiffness matrix of the intermediate support.
The elastic potential energy at the coupling is as follows:
where, ${\mathbf{q}}_{{C}_{1}}$ and ${\mathbf{q}}_{{C}_{2}}$ are the generalized displacement vectors at coupling connection points ${C}_{1}$ and ${C}_{2}$, respectively. ${\mathbf{K}}_{C}$ is the equivalent stiffness of the coupling.
Potential energy of the elastic shaft of a lowpressure turbine:
Among them, ${F}_{y}$ and ${F}_{z}$ are the forces acting on the lowpressure turbine shaft in the $y$ and $z$ directions. ${M}_{y}$ and ${M}_{z}$ are the torque of the lowpressure turbine shaft in the $y$ and $z$ directions. ${\delta}_{ij}$ ($i=1~4$, $j=$$1~4$) is the flexibility coefficient.
2.2.3. Dynamic equation of the dualrotor system
The dynamic model of the dualrotor system is established through the Lagrange equation:
$+\left({z}_{2}+{a}_{34}{\theta}_{y4}{z}_{4}\right){k}_{b4z}{a}_{41}+\left({k}_{b2z}{a}_{2}{k}_{b5z}{a}_{5}\right){z}_{2}{k}_{b4\theta y}{\theta}_{y4}$
$+\frac{\left\{\begin{array}{c}12{E}_{c}{c}_{2}{I}_{yc}\left({\theta}_{y1}{c}_{1}+{\theta}_{y2}{c}_{2}{z}_{1}+{z}_{2}\right)+6{E}_{c}{I}_{c}\left({\theta}_{y1}{c}_{1}+{\theta}_{y1}{c}_{2}+2{\theta}_{y2}{c}_{2}{z}_{1}+{z}_{2}\right){L}_{c}\\ +2{E}_{c}{I}_{c}\left({\theta}_{y1}+2{\theta}_{y2}\right){{L}_{c}}^{2}\end{array}\right\}}{{{L}_{c}}^{3}}$
$+\frac{3I{E}_{s}{L}_{l}{z}_{2}(2{a}_{2}{L}_{l})}{{{a}_{2}}^{2}{\left({a}_{2}{L}_{l}\right)}^{2}}\frac{3I{E}_{s}{L}_{l}{\theta}_{y2}}{{a}_{2}\left({a}_{2}{L}_{l}\right)}=0,$
$+{a}_{41}\left({y}_{2}+{a}_{34}{\theta}_{z4}+{y}_{4}\right){k}_{b4y}+\left({a}_{2}{k}_{b2y}+{a}_{5}{k}_{b5y}\right){y}_{2}{k}_{b4\theta z}{\theta}_{z4}$
$+\frac{\left\{\begin{array}{c}12{E}_{c}{I}_{c}{c}_{2}\left({c}_{1}{\theta}_{z1}+{c}_{2}{\theta}_{z2}+{y}_{1}{y}_{2}\right)+\\ 2{E}_{c}{I}_{c}\left[\left({\theta}_{z1}+2{\theta}_{z2}\right){{L}_{c}}^{2}+\left(3{c}_{1}{\theta}_{z1}+3{c}_{2}{\theta}_{z1}+6{c}_{2}{\theta}_{z2}+3{y}_{1}3{y}_{2}\right){L}_{c}\right]\end{array}\right\}}{{{L}_{c}}^{3}}$
$+\frac{3I{E}_{s}{L}_{l}{y}_{2}}{2{{a}_{2}}^{2}{\left({a}_{2}{L}_{l}\right)}^{2}}\left(2{a}_{2}{L}_{l}\right)+\frac{3I{E}_{s}{L}_{l}{y}_{2}\left(2{a}_{2}{L}_{l}\right)}{{{a}_{2}}^{2}{\left({a}_{2}{L}_{l}\right)}^{2}}\frac{3I{E}_{s}{L}_{l}{\theta}_{z2}}{{a}_{2}\left({a}_{2}{L}_{l}\right)}=0,$
where, ${a}_{i}$ ($i=1~5$) and ${c}_{i}$ ($i=1\uff0c2$) represents the distance between two points in Fig. 1, in units of m. ${a}_{1}={B}_{1}{D}_{1}$, ${a}_{2}={B}_{2}{D}_{2}$, ${a}_{3}={B}_{3}{D}_{3}$, ${a}_{4}={B}_{4}{D}_{4}$, ${a}_{5}={B}_{5}{D}_{2}$, ${c}_{1}={C}_{1}{D}_{1}$, ${c}_{2}={B}_{5}{C}_{2}$. Additionally, ${a}_{41}$ is the distance between two points ${B}_{4}$and${D}_{2}$. ${a}_{h}$ is the distance between two points ${B}_{4}$ and ${D}_{4}$, expressed as ${a}_{h}=\left({a}_{4}{a}_{34}\right)$. ${L}_{c}$ is the length of the coupling, expressed as ${L}_{c}={C}_{1}{C}_{2}$. ${l}_{l}$ is the total length of the lowpressure turbine shaft, expressed as ${l}_{l}={B}_{2}{B}_{5}$. ${E}_{s}$and ${E}_{c}$ is the elastic modulus, in units of kg/(m·s^{2}). ${I}_{s}$ and ${I}_{c}$ is the moment of inertia, in units of kg·m·s^{2}.
3. The inherent characteristics of the dualrotor system
Mastering the natural frequency and vibration mode is the basis of studying the dynamic response of the dualrotor system. The inherent characteristics of the dualrotor system are obtained based on the above dynamic Eq. (9).
The first three natural frequencies and vibration modes of the simplified model are shown in Fig. 2. Campbell diagram is shown in Fig. 3, and the first two critical speeds are shown in Table 1.
Fig. 2Analytical results for the first three orders natural frequencies and vibration modes of the dualrotor system
a) The first order (58.8 Hz)
b) The second order (75.4 Hz)
c) The third order (127 Hz)
Table 1The first two critical speeds of the dualrotor system
Order  The first order  The second order 
Value of the critical speeds  3528 r/min  4524 r/min 
It can be seen from Fig. 2 that the first vibration mode is represented by the bending of the lowpressure turbine rotor and the translation of the highpressure rotor. The second vibration mode is characterized by the lowpressure turbine rotor bending and the highpressure rotor pitching. The thirdorder modes are lowpressure fan rotor pitch, lowpressure turbine rotor bending, and highpressure rotor pitch.
Fig. 3Campbell diagram for the dualrotor system
4. Model validation
4.1. Model verification of the dualrotor system based on multibody dynamics
In this section, the analytical dynamic model of the dualrotor system obtained in Section 2 is verified by the multibody dynamics simulation analysis method. The rigidflexible coupling simulation model of the dualrotor system is carried out based on the software ADAMS, and the natural characteristics are analyzed. The results obtained by simulation are compared with the analytical results.
Based on the theory of multibody dynamics, ADAMS can combine the mechanical boundary, load boundary, and geometric conditions of complex rotor systems for aeroengine in a concise simulation analysis model. ADAMS can realize the rigidflexible coupling modeling of the aeroengine rotor system and can simulate the system motion and dynamic characteristics under actual working conditions. The rigidflexible coupling simulation model of the aeroengine dualrotor system is shown in Fig. 4.
Fig. 4Rigidflexible coupling dynamic model of the dualrotor system based on ADAMS
Where the lowpressure turbine shaft is set as a flexible body, and other components adopt the rigid assumption. The supports are defined by stiffness and damping, simulated by the "bushing" element. The coupling is simplified to a short shaft structure. The rotation of the dualrotor system is realized by adding a drive to the center of the rotating shaft of the lowpressure rotor and the highpressure rotor respectively.
The inherent characteristics of the dualrotor system are calculated by the simulation method. The first three natural frequencies and vibration modes of the dualrotor system are shown in Fig. 5.
Fig. 5The first three orders natural frequencies and vibration modes of the dualrotor system based on ADAMS
a) The first order (56.4 Hz)
b) The second order (79 Hz)
c) The third order (117.1 Hz)
The first vibration mode is represented by the bending of the lowpressure turbine rotor and the translation of the highpressure rotor; The vibration mode is characterized by the lowpressure turbine rotor bending and the highpressure rotor pitching. The thirdorder modes are lowpressure fan rotor pitch, lowpressure turbine rotor bending, and highpressure rotor pitch. The first three modes are consistent with the results obtained in the previous Section 3 based on the analytical dynamic model.
4.2. Model verification of the dualrotor system based on finite element method
In this section, the analytical dynamic model of the dualrotor system obtained in section 2 is verified by the finite element method. The finite element model of the dualrotor system is carried out, and the natural characteristics are analyzed. The results obtained by simulation are compared with the analytical results. In addition, the results of the dualrotor system with four discs and five supports are compared with the prototype model.
Fig. 6Finite element model of the dualrotor system with four discs and five supports
4.2.1. Finite element model of the dualrotor system with four discs and five supports
During the establishment of the finite element model of the dualrotor system, the following simplifications are made: (1) The shaft section is considered to be elastic. The mass and moment of the inertia of the shaft section are placed on the nodes at both ends. (2) The supports are considered as spring elements without cross stiffness and damping. (3) The fan disc, lowpressure turbine disc, highpressure compressor disc, and highpressure turbine disc are simplified as hollow discs, which are regarded as rigid bodies with rotation effects. It is concentrated at the center of the mass of each disc in the form of concentrated mass and moment of inertia.
The finite element model of the dualrotor system is established, as shown in Fig. 6. The lowpressure rotor of the dualrotor system with four discs and five supports is discretized with 30 nodes and 29 shaft segments. The highpressure rotor system is discretized with 12 nodes and 11 shaft segments.
The first three natural frequencies and vibration modes of the lowpressure rotor system are calculated as shown in Fig. 7. The first vibration mode is the bending of the lowpressure turbine rotor. The second vibration mode is the bending of the fan rotor and the lowpressure turbine rotor. The third vibration mode is the bending of the lowpressure turbine section.
Fig. 7The first three orders natural frequencies and modes of the lowpressure rotor obtained by finite element method
a) The first order (76.9 Hz)
b) The second order (122.6 Hz)
c) The third order (150.6 Hz)
Fig. 8The first three orders natural frequencies and modes of the highpressure rotor obtained by finite element method
a) The first order (60.4 Hz)
b) The second order (96.4 Hz)
c) The third order (388.2 Hz)
The first three natural frequencies and vibration modes of the highpressure rotor are calculated as shown in Fig. 8. The first vibration mode is translational of the highpressure rotor, the second vibration mode is the pitching of the highpressure rotor, and the third vibration mode is the bending of the highpressure rotor.
The first three natural frequencies and vibration modes of the dualrotor system with four discs and five supports are obtained, as shown in Fig. 9. The first vibration mode is the bending of the lowpressure turbine rotor and the translational of the highpressure rotor. The second vibration mode is the bending of the lowpressure turbine rotor and the pitching of the highpressure rotor. The third vibration mode is the bending of the fan rotor and the lowpressure turbine rotor, the pitching of the highpressure rotor.
Fig. 9The first three orders natural frequencies and modes of the dualrotor system obtained by finite element method
a) The first order (55.9 Hz)
b) The second order (75.5 Hz)
c) The third order (122.0 Hz)
4.2.2. Finite element model of the dualrotor system for the prototype
The finite element model of the dualrotor system of the prototype is established, as shown in Fig. 10. The lowpressure rotor system of the prototype is discretized with 54 nodes and 53 shaft segments. The highpressure rotor is discretized with 47 nodes and 46 shaft segments.
Fig. 10Finite element model for the dualrotor system of the prototype
Fig. 11The first three orders natural frequencies and modes for the lowpressure rotor of the prototype
a) The first order (89.6 Hz)
b) The second order (122.3 Hz)
c) The third order (156.8 Hz)
The first three natural frequencies and vibration modes of the lowpressure rotor system of the prototype are obtained as shown in Fig. 11. The first vibration mode of the lowpressure rotor of the prototype is the bending of the lowpressure turbine rotor. The second vibration mode is the pitching of the fan rotor. The third vibration mode is the pitching of the fan rotor and the bending of the lowpressure turbine rotor.
The first three natural frequencies and vibration modes of the highpressure rotor of the prototype are shown in Fig. 12. The first vibration mode is translational of the highpressure rotor, the second vibration mode is the pitching of the highpressure rotor, and the third vibration mode is the bending of the highpressure rotor.
Fig. 12The first three orders natural frequencies and modes for the highpressure rotor of the prototype
a) The first order (59.8 Hz)
b) The second order (106.4 Hz)
c) The third order (587.7 Hz)
The first three natural frequencies and vibration modes of the dualrotor system for the prototype are obtained as shown in Fig. 13. The first vibration mode is the bending of the lowpressure turbine rotor and the translational of the highpressure rotor. The second vibration mode is the bending of the lowpressure turbine rotor and the pitching of the highpressure rotor; The third vibration mode is the pitching of the fan rotor, bending of the lowpressure turbine rotor, and translation of the highpressure rotor.
4.2.3. Comparative analysis of results
The first three natural frequencies and vibration modes obtained from the two finite element models of the dualrotor system are compared, and the following conclusions are obtained: 1) Among the first three vibration modes of the lowpressure rotor system, the lowpressure turbine rotor are the same, and the fan rotor shows bending and pitching in the simplified and prototype model respectively. The relative errors of the first three natural frequencies are 14 %, 0.2 %, and 3.9 % respectively. 2) The first three modes of the highpressure rotor system are consistent, and the relative errors of the first three natural frequencies are 1 %, 9 %, and 33 % respectively. 3) Among the first three vibration modes of the dualrotor system, the vibration modes of the lowpressure turbine rotor and the highpressure rotor are the same. The fan rotor shows the bending vibration mode in the simplified model and the pitching vibration mode in the prototype model. The relative errors of the first three natural frequencies are 0.3 %, 5.2 %, and 8.6 % respectively.
The first three vibration modes of the dualrotor system obtained in Section 2 based on the analytical dynamic model consist with the results of the prototype model. It further shows that it is reasonable to consider the fan rotor as rigid and the lowpressure turbine shaft as flexible in the rigidflexible dynamic model of the dualrotor system.
Fig. 13The first three orders natural frequencies and modes for the dualrotor system of the prototype
a) The first order (55.7 Hz)
b) The second order (79.7 Hz)
c) The third order (133.5 Hz)
5. Conclusions
In response to the urgent demand for a rotor system model that can reasonably and accurately express the dynamics and vibration characteristics of the rotor system in the study of vibration in aircraft engine rotor systems, this article takes a representative aeroengine dualrotor system as the research object, established a simplified mechanical model which can reflect the structural characteristics and motion behavior of the prototype model is introduced. Considering the rigidflexible coupling characteristics of an elastic support with different stiffness and rotor structure, and analytical dynamic model of the dualrotor system is established. The model contains the kinetic energy of four equivalent discs and the elastic potential energy of five supports and a lowpressure turbine shaft, with 16 degrees of freedom.
The dynamic model of the rigidflexible coupled dualrotor system is verified and confirmed by the rigidflexible coupling multibody dynamics simulation method and finite element method. The feasibility of using the analytical dynamic model of the dualrotor system to study the vibration is clarified, which lays a model foundation for the subsequent research on the vibration of the dualrotor system. It shows that it is reasonable to consider the fan rotor as rigid and the lowpressure turbine shaft as flexible in the rigidflexible dynamic model of the dualrotor system.
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About this article
Study on Dynamic Optimization Method of Large Squeeze Film Damper Driven by Data (12072069).
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Pingping Ma is responsible for providing main ideas and writing articles. Yuehui Dong provides information collection and data analysis. Muge Liu is responsible for revising the format and grammar of the paper. Qingkai Han is responsible for reviewing the content and results.
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