0 Introduction

According to the difference of contact area，the rubbing can be divided into the following four categories：Fixed point rubbing，partial rubbing，full annual rubbing and partial rubbing mixed with fixed point rubbing［1］.Researches on rubbing rotor system have been studied by many scholars［2－7］.Based on a dynamic model of rotorstator rubbing proposed by Muszynska［1］，Yan et al.［8］developed a simplified model of rotor－stator rub－impact for a kind of dual－rotor engine.The vibration characteristics for some types of rubbing for the dual－shaft engine are analyzed，the theoretical results are consistent with data measured on a dual－shaft aircraft engine.Yang et al.［9－10］established a kinetic model of aero engine considering imbalances and fixed point rubbing，furthermore，a new model of rubbing force with coating painted on the discs and casing is established based on the Lankarani－Nikravesh model.Xu et al.［11］simplified the stator in a dual－rotor system into a flexible sheet from a new perspective，taking into account the effect of the elastic deformation of the sheet and the amount of intrusion caused by the contact.Collision force and Friction were calculated by utilizing Hertz contact theory and Coulomb model.The transient dynamic response of the whole aero－engine under the rubbing fault is analyzed.Zhou et al.［12］established the coupling dynamic model of the dual rotor－ball bearing－stator，considering the rolling bearing gap and the nonlinear Hertz contact force.The vibration responses of the system were obtained from numerical integral method，and the phenomenon of beat vibration of the dual rotor system is analyzed.Finally，the influence of different speed ratios and rubbing stiffness on the dynamic characteristics of the system is discussed.

Most of the scholars mentioned earlier use numerical simulation to analyze the dual－rotor system rubbing response［13－15］，and some scholars simulate the rubbing response by commercial software.Han et al.［16］established the local rubbing model of the dual－rotor system using MSC.ADAMS software based on the rigid－flexible multi－body model，and analyzed the nonlinear dynamic characteristics of the system.Aiming at the shortcomings of the traditional model of the impact force model，Wang et al.［17］established the finite element（FE）model of the rotor shaft and obtained the first six－order natural frequencies of the rotor system by ANSYS.By the methods of spectrum and cepstrum analysis，the rubbing characteristics of the casing vibration acceleration time series data are analyzed.The results show that the casing vibration acceleration has obvious impact characteristics.

综合上述各种因素，向威达认为，未来一个较长的时间内A股还会处于存量资金博弈和反复震荡的格局。建议投资者还是从稳基建、稳增长和科技创新自主可控两条线挖掘个股，特别是中央经济工作会议明确提到的5G、人工智能和物联网概念。

From the literatures listed above，conclusions can be made that most researches concentrate on the dynamic characteristics of a dual－rotor system without casing.Nevertheless，the presence of the casing will have a great impact on the vibration response of the system.In terms of structural integrity，the dual－rotor－casing coupling system is closer to the aero－engine.In addition，the vibration responses （displacement/acceleration）of the stator such as casing have been paid more attention because of the convenience of signal collection.In this paper，the casing is modeled using Timoshenko beam element and rubbing－induced vibration responses in dual－rotorcasing system are analyzed.

1 Dynamic Model of Dual－Rotor－Casing System

1.1 FE model of dual－rotor－casing system

In this paper，the rubbing in dual－rotor－casing system is analyzed.Schematic of the model is shown in Fig.1.The low－pressure（LP）rotor and high－pressure（HP）rotor subsystems include one rigid compressor disc and one rigid turbine disc.The bearing is simulated by the linear spring and damping elements.LP and HP rotors are connected by inter－shaft bearing，the coupling of two－rotor subsystems could be considered.By FE discretization，the rotors are divided into 19elements and 21nodes while the casing is discretized into 15elements.The inner rotor，outer rotor and the casing are modeled by Timoshenko beam element.In order to consider the torsional vibration，each node of LP/HP rotors has five degrees of freedom （DOFs）including two lateral DOFs（ux，uy）and three rotational DOFs（rot x，rot y，rotz），while each node of the casing has two lateral DOFs （ux，uy）and two rotational DOFs（rot x，rot y）.All the disks are simulated by lumped mass elements.

Fig.1 Schematic of dual－rotor－casing system

In Fig.1，kby1，kby4and cby1，cby4are the supporting stiffness and damping of the Y direction between the LP rotor and the casing，respectively.kby2and cby2are the supporting stiffness and damping of the Ydirection between the HP rotor and the casing.kby3and cby3are the supporting stiffness and damping of the Ydirection of interbearing.kby5，kby6and cby5，cby6are the supporting stiffness and damping of the Ydirection between the casing and the base，respectively.

1.2 Rubbing force model

（2）The amplitude of torsional vibration under rubbing conditions is larger than that of lateral vibration.

本研究使用的问卷分为两部分：第一，教师基本资料。包括性别、年龄、学历、教龄、职称。第二，乡村教师教育信念问卷。包含5个维度：教育目的、学生、课程、教学、教师角色五个方面的信念，共计27题。问卷中的27题均采用李克特五点计分方式。问卷回收后，采用 SPSS17.0统计软件对调查数据进行统计分析。

where xmand ymare the components of the eccentriccity e in the xand y directions.

A FE model of the dual－rotor system with casing is established.Based on shaft center orbits，time－domain waveform，and frequency spectrum，the local rubbing induced nonlinear characteristics of dual－rotor－casing bearing system were analyzed.Some conclusions are summarized as follows：

In this section，the effects of rotational speed and speed ratio on rubbing－induced vibration responses of the dual－rotor－casing system are discussed in detail.Time－domain waveform，spectrum cascades，and frequency spectrum are used to analyze the vibration responses.It is worth noting that only local rubbing between disk 1and casing（see Fig.1）is concerned in this paper.

如图3所示，下部装置本体内侧圆弧与动力钻具紧密贴合，且均匀分布有增摩带和钻井液通道，本体上侧与角差下刻度盘为一体，两侧各有一个固定圆环用于连接链条；上刻度盘与装置本体配合后，再通过限位螺栓固连。下部固定装置结构同上部。

The rubbing force can be written as

Fig.2 Schematic diagram of rubbing between disk－casing

where FNis the normal rubbing force，FTthe tangential rubbing force，krthe contact stiffness，andμthe friction coefficient.

1.3 Model validation

The natural frequencies obtained from ANSYS and MATLAB are listed in Table 1.Comparing the first six natural frequencies，the errors are less than 3%.It demonstrates the validity of the model developed by FE method.Research on the dynamic characteristics in both co－rotation and counter－rotation was conducted.As shown in Fig.3，Campbell diagrams of these two cases were obtained by calculation of natural frequencies.The forward whirl frequency increases with spin speed while the backward whirl frequencies decreases.One common phenomenon that can be found in two Campbell diagrams is that the torsional frequencies do not change with the increase of spin speed.

The minimum gap c between the disk and casing，and the eccentricity e can be written as

1.要正确把握教学起点和课堂容量。教学起点应是绝大部分学生能够达到的，不能盲目拔高；例如大纲在词汇要求上分为四会、三会和二会，在语法要求上分为理解、掌握和运用，这些都是教师把握教学起点的依据。课堂容量也应适度，容量太大或太小都会影响课堂教学效果。许多心理学家和语言学家提出的“Comprehensible input(i+1)”，都肯定了输入语言信息要合理性、针对性。调查表明，34%的学生认为，初中与高中最大区别是课堂上，高中要求记大量笔记，教学内容多；50%的学生认为，高中英语课难度大，听课吃力。可见教学起点过高和课堂容量过多是造成学生不适应的主要因素。

Table 1 Natural frequencies comparison of dual rotor（Ω＝0rev/min）

Order Natural frequency/Hz ANSYS MATLAB Error/%1 24.273 24.277 0.016 2 35.102 35.002 －0.285 3 35.102 35.002 －0.285 4 74.697 74.714 0.023 5 94.265 94.122 －0.152 6 94.265 94.122 －0.152

Fig.3 Campbell diagram of dual－rotor system

2 Rubbing－Induced Vibration Responses in Dual－Rotor－Casing System

Penetration depthδbetween the disk and casing is obtained from

2.1 Effects of rotational speed

Rotational speed has a significant influence on the vibration responses of the dual－rotor－casing system.The spectrum cascades under the case of different rotational speeds（800—5 800rev/min）are shown in the rubbing process （see Fig.4）.From Fig.4，the phenomenon can be distinguished by the separation of the rotational speed of LP rotor.Firstly，when the range of rotational speed is low （800—2 000rev/min），the rubbing is minor and occurs infrequently.It is easy to found the peak values of f1and f2are prominent.Secondly，whenΩ＝1 700rev/min，rubbing appears frequently and the amplitudes of combined frequency components become obvious.In addition，the spectral line of HP rotor generates resonance（point A）firstly and amplitude amplifies because of the excitation of 1st order natural frequency.After the first critical speed，spectrum cascades are filled with abundant frequency components such as f2－f1，1/2f2，2f1－f2，2f1，f1＋f2，2f2.The response values of combined frequency components become larger with the increasing rotational speed（2 000—4 000rev/min）.When rotational speed is 2 300rev/min，the spectral line of LP rotor generates resonance （point B）and amplitude amplification occurs due to the excitation of 1st order natural frequency.At the interval of［4 000，5 800］rev/min，rubbing becomes very strong，more combined frequencies occur and the 2nd order natural frequency is excited（points Cand D）.

Fig.4 Spectrum cascades of inner rotor rubbing under different rotational speeds

Vibration responses of LP rotor and casing at Ω＝2 300rev/min are shown in Fig.5，the red broken lines in time domain waveform correspond to the time when the rubbing begins.The orbits of the inner rotor and casing are asymmetric and both compressed seriously.In the spectrum，the response frequencies contain more components and amplitude of combined frequencies are dominant.Meanwhile，time domain waveform of torsional displacement at the compressor disc of LP rotor and HP rotor are shown in Fig.6，respectively.The amplitude of torsional displacement is larger than lateral displacement under the same condition.In order to compare the effect of the casing on the rubbing induced vibration responses，the vibration responses which the casing is simulated as lumped mass are obtained under the same case（see Fig.7）.

那一次是我第一次去外婆出生的地方，也是唯一一次。从此，没有人再在外婆的面前提“山里老太太”，这是个家里永久的禁忌词汇。不提，也许可以假装遗忘；提起，就是一次残忍的撕裂。

Fig.5 Rubbing－induced responses atΩ＝2 300rev/min

Fig.6 Torsional displacement atΩ＝2 300rev/min

Fig.7 Rubbing－induced responses of mass point casing atΩ＝2 300rev/min

In Fig.7，the elastic supports between the dual－rotor and the casing are ignored.It can be seen that the vibration amplitude in Fig.5is larger than that in Fig.7，which indicates that the coupling between rotor and casing has significant effects on the dynamics of the system and should be considered.

2.2 Effects of speed ratio

The dual－rotor system has two operation modes including co－rotation and counter－rotation.Fig.8indicates the LP rotor motion trajectory with rubbing under different speed ratios.From the Fig.8，it can be seen that the motion trajectory under counter－rotation shows petal－like and the number of petals implies the period of motion.When the speed ratio is positive，the motion trajectories are multiple circles.It′s worth noting that the response amplitudes under counter－rotation are less than that under co－rotation because the gyro－moment is offset partially. What′s more，when 10ηis odd number the motion trajectories have more cycles than that when 10ηis even.

Fig.9shows the rubbing response underη＝－1.7.From the time－domain waveform，it can be found that the change of waveform is not obvious both in LP rotor and casing.In the figure of rotor orbit，petals are crossed.Two rotationalfrequencies of dual rotor are prominent and there are also many combined frequency components.Corresponding simulation parameters are listed in Table 2.

研究采用GF-1 WFV遥感影像，基于分区决策树面向对象分类方法，采用多尺度分割方法对预处理后的遥感影像进行分割处理，在得到一个个分割对象的前提下，对这些对象进行特征提取，然后利用决策树算法通过选择的训练样本建立决策树，根据建立的决策树，对图像中已经提取的对象的特征属性进行分类，最后生成分类结果图，得到黑龙江省旱地、水田分布图。

Fig.8 Motion trajectories of disk 1under different speed ratios

Fig.9 Rubbing－induced responses withη ＝－1.7

Table 2 Simulation parameters

Invariant parameter Value Rotational speedΩ/（rev·min－1）Rubbing stiffness/（N·m－1）Friction coefficient Initial minimum gap/m Unbalance/（kg·m）Eccentricity/m Speed ratio 3 000 kr＝8×106 μ＝0.3 c＝1×10－4 fu1＝fu2＝fu3＝fu4＝156×10－6 e＝8×10－5 η＝－1.7

3 Conclusions

In actual working conditions，Od′and Oc′，the center of rotor and casing，will deviate from the original position in plane because of the installation error and rubbing.And the radial relative displace mentξbetween rotor and casing meets

（1）Besides two unbalanced excitation frequencies and their multiple frequency components，different combined frequency components appear in the spectrum.

In order to constrain the rigid casing，the linear spring and damping elements are connected to the casing，the local－contact elastic deformation caused by contact force is ignored.Fig.2illustrates the principle of rubbing between disk and casing.When the system remains stationary，Od and Ocis the geometric center of LP rotor and casing，respectively.In addition，the casing center Occoincides with the coordinate system origin O.Rdand Rcdenote the radius of rubbing disk and the casing，respectively.e is eccentricity.kcx，kcyand ccx，ccyare the supporting stiffness and damping of the x and y directions between the casing and the base，respectively.

在美娟和梁诚的大床上，安安躺在梁诚怀里心满意足地看着一张刚刚出炉的遗嘱——是安安的父亲，银行家凌宇生今天刚刚修改的。

（3）The speed ratio has a great impact on the periodicity of the dual－rotor system.The rubbing induced response amplitudes under counter－rotation are less than that under co－rotation with the same parameters.

Acknowledgements

This work was supported by the National Natural Science Foundation of China（No.11772089），the Fundamental Research Funds for the Central Universities （Nos.N160312001and N160313004），and the Research Project of State Key Laboratory of Mechanical System and Vibration（No.MSV201707）.

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