effect of the aero-engine mounting stiffness on the whole

M. J. QuCollege of Civil Aviation,

Nanjing University of Aeronautics

and Astronautics,

Nanjing 211106, China

e-mail: [email protected]

G. ChenCollege of Civil Aviation,

Nanjing University of Aeronautics

and Astronautics,

Nanjing 211106, China

e-mail: [email protected]

Effect of the Aero-EngineMounting Stiffness on the WholeEngine Coupling VibrationA finite element (FE) model of the rotor tester of an aero-engine, having a thin-walledcasing structure, mounted with the way of an actual engine, is developed to simulate theintrinsic vibration characteristics under actual engine-mounting condition. First, a modalexperiment of the rotor tester for the whole aero-engine is conducted, and the FE modelis modified and validated based on the modal experimental results. Second, the first threeorders of natural frequencies and the modal shapes are evaluated using the modified FEmodel under three different types of mounting stiffness, namely, a fixed mounting bound-ary, a free mounting boundary, and a flexible mounting boundary. Subsequently, theinfluences of the mounting stiffness on the coupling vibration of the rotor and stator arestudied via a new rotor–stator coupling factor, which is proposed in this study. Theresults show that the higher the rotor–stator coupling degree of the modal shape, thegreater the influence of the mounting condition on the modal shape. Moreover, the influ-ence of the mounting stiffness on the rotor–stator coupling degree is nonlinear. The cou-pling phenomena of the rotor and stator exist in many modal shapes of actual largeturbofan engines, and the effect of mounting stiffness on the rotor–stator coupling cannotbe ignored. Hence, the mounting stiffness needs to be considered carefully while model-ing the whole aero-engine and simulating the dynamic characteristics of the whole aero-engine. [DOI: 10.1115/1.4038542]

Keywords: aero-engine, the whole aero-engine vibration, stator-rotor coupling,mounting stiffness, finite element modeling

1 Introduction

The modeling and simulation analysis of the vibration of anentire aero-engine is one of the important approaches that helpevaluate its vibration characteristics. In recent years, thin casingsand elastic supports have been widely used, and the studies on thecoupling vibration of the rotor-bearing-casing-mounting systemhave been receiving considerable attention [1,2]. The vibrationcharacteristics of the entire aero-engine were calculated using the“impulsive receptance method” and the “receptance harmonic bal-ance method” [3–5]. A dynamic model of a general, complexrotor-bearing-stator coupling was developed, wherein the rotorsand casings were modeled using the finite element (FE) method,and the supports were modeled using the lumped parametermethod [6,7]. A finite element model of the compressor of a tur-bine with solid elements was developed and its dynamic charac-teristics were evaluated [8]. Moreover, a finite element model ofan aero-engine with solid elements was established [9]. A finiteelement model was constructed for a dual-rotor structure based ona certain aero to study the dynamic characteristics of the system[10]. A high-speed rotor FE model with certain geometrical andmechanical properties is carried out, and the Campbell diagram,critical speeds, operational deflection shapes, and unbalanceresponse of the rotor are computed [11]. The dynamic characteris-tics of BR710 were evaluated and analyzed using finite elementmethod [12].

Since many years, the studies on the coupling vibration ofentire aero-engines were conducted by only considering the cou-pling dynamic characteristics of the rotor-bearing-stator system.In the vibration models of the entire aircraft engine, the boundaryconditions of the mountings were mostly either simply free orfixed. Simple mounting boundary conditions are lack of theory

supports. The models with simple mounting boundary conditionscannot reflect the actual engineering situation, and the vibrationanalysis of the whole engine was never performed effectively forengineering applications.

A whole aircraft engine model is built, which includes mount-ing, stator, bearing, and rotor to study rotor-stator contact phe-nomena [13]. A representative model of aero-engine test rig isbuilt, comprising the following basic components: rotor, shaft,casing, flexible support structure, and bearing housings [14].These studies considered the influence of the mountings while cal-culating the dynamic characteristics; however, they did not ana-lyze the influence of the installation conditions on the dynamiccharacteristics of the entire engine and the coupling degree of therotor and stator systems. In addition, the current studies do notemploy effective evaluation methods to calculate the couplingdegree between the rotor and stator systems. Hence, it was notadvantageous to study the coupling dynamic characteristics of theentire system under such circumstances.

Hence, in this study, the rotor tester of an aero-engine having athin-walled casing is chosen as the research object. Moreover, thecoupling effect of the mountings, stator, supports, and rotor sys-tem is considered in the analysis, and a finite element model ofthe vibration of the entire aero-engine is developed. The model isthen modified and verified based on the results of the modal testof the entire engine. To calculate the coupling degree of the rotorand stator systems quantitatively, an indicator for evaluating therotor–stator coupling degree, namely, the rotor–stator couplingfactor, is proposed. The dynamic characteristics of the mounting-stator-bearing-rotor system are studied, and the influences of themounting stiffness on the dynamic characteristics of the engineand the stator–rotor coupling degree are analyzed.

2 Rotor-Bearing-Stator Tester

The rotor-bearing-stator tester of the aero-engine was theresearch object, which was designed and manufactured by

Contributed by the Structures and Dynamics Committee of ASME for publicationin the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript receivedSeptember 7, 2016; final manuscript received October 3, 2017; published onlineApril 10, 2018. Assoc. Editor: Alexandrina Untaroiu.

Journal of Engineering for Gas Turbines and Power JULY 2018, Vol. 140 / 072501-1Copyright VC 2018 by ASME

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Shenyang aero-engine institute. The stator-shape of the tester issimilar to the shape of a real, core aero-engine casing. The size isreduced to 1/3 of the original based on the geometric similarityprinciple. The size design mainly considers the installation of thetester in lab. Meanwhile, the manufacturing cost difficulty and thedismounting difficulty are taken into account. The internal struc-tures are simplified, which include the combustion chamber that isremoved; the disk structures of the multistage compressor are sim-plified to a single-stage disk structure, wherein the blades are sim-plified to have inclined planes; the disk structures of themultistage turbo are simplified using the same method as that ofthe multistage compressor. The shaft is a solid, rigid shaft. Thedisk and shaft of the turbine are connected on a conical matingsurface and a 180 deg double bond. There are two bearingsbetween compressor disk and turbine disk, one is a rolling bearingand it is near the compressor disk, the other one is a ball bearingand it is near the turbine disk. Moreover, there are two squirrelcage elastic supports together with the bearings. The rotor tester isdriven by an electric motor. The motor rated power is 37 kW, withthe maximum speed of 2980 rpm. The motor and the tester areconnected by a gear accelerator. A nylon rope coupling is usedbetween the gearbox and the tester, which has some tolerance formisalign of rotors. The maximum of the rotor tester is 7000 rpm.The tester is a typical single-rotor engine system model. Figure1(a) shows the actual image of the tester and Fig. 1(b) shows itsprofile.

The rotor tester is fixed via rigid supports and installed on a testplatform. The front mounting sections are located on both sides ofthe compressor casing. The bolt structures are used to fix the testeron the bracket structure, which is shown in Fig. 2(a). The rear-mounting section is located at the left side of the turbo casing(with respect to the axis), on a hanging hinge that lifts the tail ofthe tester, which is shown in Fig. 2(b).

3 Finite Element Modeling and Experimental

Verification of the Rotor Tester

3.1 Modal Test of the Whole Tester

3.1.1 Test Scheme. The single-point excitation and multipointmeasurement methods are adopted in the experiment. A total of13 test points are selected on the tester in the horizontal direction(Fig. 3 shows the positions of the measuring points), six of the 13points are located on the surface of the rotor and the remainingseven points are located on the surface of the casing. In the test,the accelerations of the test points are measured usingB&K4508ICP acceleration sensors. The sensitivity of the sensoris 9.782 mV/g, the measuring range is 0–714 g, and the frequency

response range is 0.1 Hz–8 kHz. The sine excitation method isadopted in the experiment. The exciting point is near to the point1. The HEV-500 high-energy electric vibration exciter is installed.The maximum excitation force of the exciter is 500 N, the fre-quency range of the exciter is 0–2 kHz, and the maximum ampli-tude of the exciter is 610 mm. In addition, the sinusoidalexcitation force is measured using an impedance head, which ismounted between the rod of the exciter and the tester.

3.1.2 Test Results and Analysis. A column in the matrix ofthe frequency response function is obtained using the single-pointexcitation and multipoint measurement methods. The first threemodal parameters of the tester (listed in Table 1) can be obtainedusing the vibration modal identification software. The first-ordernatural frequency is 38.2 Hz, and the modal shape of the rotor andstator coupling exhibits a rigid body vibration, wherein both therotor and the stator exhibit pitching behavior. The second-ordernatural frequency is 46.6 Hz, and the modal shape of the rotorexhibits a rigid body movement. The third-order natural frequencyis 113.3 Hz, and the modal shape of the rotor exhibits bending.Figure 4 shows the first-, second-, and third-order modal shapes.The modal shape of the stator is presented on the upper half ofFig. 4, and the lower half of the figure shows the modal shape ofthe rotor, position of the measurement points are shown in figures.Clearly, in the first three orders of tester modal shapes of the tes-ter, only the modal shape of the first-order shows the couplingvibration of the stator and rotor. In the second and third modalshapes, the vibration in the stator casing is considerably lowerthan the vibration in the rotor; hence, the coupling degree of therotor and stator is low.

The stator-casing thickness of the rotor tester is 4 mm, whichimplies that the stiffness of the stator is high. The phenomenon ofcoupling vibration of the stator and rotor is not caused by theirstructures alone. The vibration in the stator in the horizontal direc-tion is due to the mounting method. The rear mounting of the tes-ter is a hanging hinge that lifts the tail of the tester, implying thatthe rear of the tester is not constrained in the horizontal direction.Hence, the rigid modal shape of the first-order stator and rotorcoupling is due to the mounting method of the tester. The first-order critical speed is an important dynamic characteristic amongthe vibration characteristics of the engine. Therefore, it is veryimportant to set up reasonable mounting section stiffness in wholeaero-engine vibration modeling and analysis.

3.2 Finite Element Modeling of the Tester

3.2.1 Three-Dimensional Modeling Based on the Software ofUnigraphics. The rotor tester is a complex structure comprisingobservation holes, incentive holes, bolts, keyways, and other

Fig. 1 Aero-engine rotor tester: (a) the actual aero-engine rotor tester and (b) the profile of the rotor tester

072501-2 / Vol. 140, JULY 2018 Transactions of the ASME


small-size structures. These structures have little effect on thedynamic characteristics of the whole tester. However, these struc-tures will increase the scale of the finite element model whenmeshing the geometrical model, which increases the computationsteps. Hence, the structure is simplified for the modeling process,and some small-size structures are neglected. Figure 5 shows thehalf-profile of three-dimensional (3D) model of the rotor testerbased on the Unigraphics.

3.2.2 Finite Element Modeling Based on HYPERMESH. TheUnigraphics model was imported from the HYPERMESH in the initialgraphics exchange specification format to develop a finite elementmodel. HYPERMESH software provides many meshing methods,which can control the scale of the finite element models and thequality of the elements precisely, thereby achieving efficient andhigh quality finite element modeling. The rotor material is30CrMnSiA, and the casing material is 1Cr18Ni9Ti. Table 2 liststhe material parameters.

The 3D eight-node solid element (Solid185) is selected to meshthe rotor and stator systems of the rotor tester. The element hasplasticity, hyperelasticity, stress stiffening, creep, large deflection,and large strain capabilities. Conbin14 is selected to simulate thebearings of the rotor tester. Conbin14 has longitudinal or torsionalcapability in one-dimensional, two-dimensional, or 3D applica-tions. The longitudinal spring-damper option is a uniaxial tension-compression element with up to three degrees-of-freedom at eachnode: translations in the nodal x, y, and z directions. No bendingor torsion is considered. The torsional spring-damper option is apurely rotational element with three degrees-of-freedom at eachnode: rotations about the nodal x, y, and z axes. No bending oraxial loads are considered. Combin14 can be used to simulatebearing stiffness exactly and easily. There are 57,379 nodes and181,340 elements in the FE model of the tester.

The stiffness values of the supports of the front and rear areconsistent with the series stiffness values of the bearings andsquirrel cage elastic supports. Because the stiffness of elastic sup-ports is weak, the supports stiffness values are small. The finiteelement model is updated based on the experimental results. Table3 lists the stiffness values of the supports of the model, where k1

is the front support-stiffness (the series stiffness values of the roll-ing bearing and squirrel cage elastic support) value, and k2 is therear-support stiffness (the series stiffness values of the ball bear-ing and squirrel cage elastic support) value. k3x and k3y are thestiffness values of the front mounting in the horizontal and verti-cal directions, respectively. k4x and k4y are the stiffness values ofthe rear mounting in the horizontal and vertical directions, respec-tively. Figure 3(a) shows the location of k1, k2, k3x, k3y, k4x, k4y.Figure 6 shows the finite element model of the whole tester.

3.3 Verification of the Finite Element Model. The finiteelement model is exported in the coded database format, and thenimported into the ANSYS software for analysis. Figure 7 showsthe first three order modes. The first-order natural frequency is38.3 Hz, and the modal shape of the rotor and stator couplingexhibits a rigid-body vibration, wherein both the rotor and the sta-tor exhibit pitching behavior. The second-order natural frequencyis 45.6 Hz, and the modal shape of the rotor exhibits a rigid bodymovement. The third-order natural frequency is 113.0 Hz, and themodal shape of the rotor exhibits bending.

The simulation results are compared with the experimental firstthree order natural frequencies. Table 4 lists the results. In the table,the relative error in the calculation is based on the test results.

The harmonic response analysis of the simulation model is con-ducted. The points corresponding to the test points are selected inthe finite element model. The acceleration–frequency responsefunctions of each selected point are calculated, and the frequencyresponse functions are compared with the test results. Figure 8shows the frequency response curves of the test and the simulationfor points 1, 3, 4, 5, 7, and 13.

The first three order modal shapes obtained from the simulationand the experiment are similar. The first-order modal shape ofboth the rotor and stator coupling exhibits rigid-body vibration,wherein both the rotor and the stator exhibit pitching behavior.The second modal shape of the rotor exhibits rigid-body vibration.The third-order modal shape of the rotor exhibits bending.

The relative error in the first-order natural frequencies obtainedfrom the simulation calculation and the experimental analysis isonly �0.26%. The relative error in the second-order natural fre-quencies obtained from the simulation calculation and the experi-mental analysis is only �2.14%. The relative error of the third-order natural frequencies obtained from the simulation calculationand the experimental analysis is only �0.35%. The simulationand experimental results are in good agreement with the first threeorder natural frequencies.

The acceleration–frequency response functions obtained fromthe simulation and experimental analysis for points 1, 3, 4, and 5look similar. The simulation and test curves of points 7 and 13have a little differences and it is mainly because the points arelocated on casing. The real supports are nonlinear structures,which include bearing, bearing housing, and elastic support struc-ture. However, we simulated the supports as linear springs. Thereare some errors between test and simulation. Generally, it is diffi-cult to simulate the points on casing exactly the same.

In summary, the finite element model developed in this study canreflect the dynamic characteristics of actual testers. The modelcan be used to calculate and predict the characteristics of a realtester.

Fig. 2 The mounting structures of the rotor tester mounted in the lab: (a) the front mountingand (b) the rear mounting

Journal of Engineering for Gas Turbines and Power JULY 2018, Vol. 140 / 072501-3


4 The Influence of the Mounting Stiffness on the

Dynamic Characteristics of the Whole Tester

This study analyzes the rotor-tester model, which is a dynamicfinite element model of the rotor-bearing-stator-mounting coupling.The influence of the stiffness of the mounting sections on the entirecoupling vibration of the tester is studied. The following three cases

of the mounting sections are considered—free mounting boundary,fixed mounting boundary, and elastic mounting boundary—implyingthat the mountings have stiffness. Free boundary means the stator isfree, there are no any constraint at the mounting sections. Fixedboundary means there are fixed constraints at the mounting sections.Elastic boundary considers the mounting stiffness, there are somesprings to simulate the mounting stiffness.

Fig. 3 Schematic of the modal experiment of the tester: (a) the schematic of excitation and measure points position, (b) mea-surement points 1–2, (c) measurement points 3–5, (d) measurement point 6, and (e) measurement points 7–13



4.1 The Dynamic Characteristics of the Whole TesterUnder Free Mounting Boundary. Without considering the stiff-ness values of the mountings, the mountings are unconstrained.The dynamic characteristics of the whole tester are computed

Table 1 Experimental modal results

Order First Second Third

Natural frequency (Hz) 38.2 46.6 113.3Damping ratio 0.0113 0.0167 0.0134

Fig. 4 The first three order modal shapes of the rotor tester: (a) the first-order, (b) the second-order, and (c) the third-order

Fig. 5 Half-profile of geometric model of the rotor tester

Table 3 Supports parameters of the rotor tester

Location k1 k2 k3x k3y k4x k4y

Stiffness/106 (N/m) 1.0 1.0 50 50 5.0 5.0

Fig. 6 Finite element model of the rotor tester

Table 2 Material parameters of the tester model

Elastic modulus (Pa) Density (kg/m3) Poisson’s ratio (l)

2.11� 1011 7800 0.3



under the free mounting boundary. Figure 9 shows the first threeorder modal frequencies and modal shapes.

The first-order natural frequency is 55.06 Hz. The modal shapeof the rotor exhibits pitching behavior. The stator also pitches andinverses with the rotor. The vibration amplitudes of the stator androtor have similar sizes.

The second-order natural frequency is 56.02 Hz. The modalshape of the rotor exhibits translational rigid vibration. The ampli-tude of the front end of the rotor is slightly larger than that of therear. The compressor stator pitches with the rotor. The vibrationamplitudes of the stator are high compared to that of the rotor.

The third-order natural frequency is 112.97 Hz. The modalshape of the rotor exhibits bending. The structure of the stator hasalmost no vibration compared to that of the rotor.

4.2 The Dynamic Characteristics of the Whole TesterUnder Fixed Mounting Boundary. The mounting sections areconstrained under the fixed mounting boundary. The dynamiccharacteristics of the whole tester are computed. Figure 10 showsthe first three order natural frequencies and modal shapes. Thismethod is the most commonly used research method in the three-dimensional modeling and analysis of aircraft engines.

The first-order natural frequency is 42.59 Hz. The modal shapeof the rotor exhibits pitching behavior. The structure of the statorhas almost no vibration compared to that of the rotor.

The second-order natural frequency is 47.09 Hz. The modalshape of the rotor exhibits translational rigid vibration. The struc-ture of the stator has almost no vibration compared to that of therotor.

The third-order natural frequency is 112.97 Hz. The modalshape of the rotor exhibits bending. The structure of the stator hasalmost no vibration compared with rotor vibration.

In the first three order vibration modal shapes, the vibrationamplitudes of the structure of the stator were small compared tothat of the rotor. This was mainly because the installations wereall fixed constraints, which limited the vibration of the statorstructure. Compared to the first-order modal shapes obtained fromthe simulation and test, the stator-structure vibration was notexhibited, and the first-order natural frequencies of the simulation

Fig. 7 The first three order modal shapes in the mounting condition in test room: (a) the first-order, (b) the second-order, and(c) the third-order

Table 4 The comparisons between the first three orders natu-ral frequencies of simulation and that of experiment

First-order Second-order Third-order

Experimental results (Hz) 38.3 46.6 113.3Simulation results (Hz) 38.2 45.6 112.9Relative error (%) �0.26 �2.14 �0.35



increased. This was mainly because the stator structure was con-strained excessively, and the vibrating mass of the first-order modereduced, thereby increasing the first-order natural frequency.

To measure the coupling degree of the rotor and stator struc-tures, a new evaluation index for the rotor–stator coupling degreeis defined, which is named the rotor–stator coupling factor. Therotor–stator coupling factor of the ith order modal shape Ci isdefined as follows:

Ci ¼jSijmax

jRijmax

(1)

where jRijmax is the maximum absolute value of modal displace-ment of all rotor nodes in this modal shape and jSijmax is the maxi-mum absolute value of modal displacement of all stator nodes inthis modal shape.

4.3 Comparison of the Characteristics of the WholeMachine Coupling Vibration Under Different Conditions. Themass-matrix normalization method is adopted because of theANSYS modal calculation; hence, the displacement of the modalshapes can be compared with each other for the same structure.The data of the first three order modal shapes under the free, fixed,

Fig. 8 Comparisons between the frequency response functions of the experiment and the simulation: (a) test point 1 (rotor),(b) test point 3 (rotor), (c) test point 4 (rotor), (d) test point 5 (rotor), (e) test point 7 (casing), and (f) test point 13 (casing)



and elastic boundaries are exported. The first-, second-, and third-order modal shapes under different boundary conditions are com-pared. Figure 11 shows the first three order modes. In Fig. 11,“free boundary rotor” means the rotor modal shape under mount-ing free boundary condition, “free boundary casing” means thecasing modal shape under mounting free boundary condition, sodo others. There are three boundary conditions. Every boundarycondition has rotor and casing modal shape, respectively.

The rotor–stator coupling factors of the first three modal shapesunder the free installation, fixed, and elastic boundaries are calcu-lated. Table 5 lists the results.

The first modal shape of the rotor tester strongly influences themounting boundary. The modal shapes and the rotor–stator cou-pling factors under different mounting boundaries change consid-erably. Under the free boundary, the modal shape of the rotorexhibits pitching behavior; the stator also pitches and inverseswith the rotor. The rotor–stator coupling factor of the first modalshape under the free boundary is 0.496, and the rotor–stator cou-pling degree is high. Under the fixed boundary, the first modalshape of the rotor exhibits pitching behavior; the structure of thestator has almost no vibration compared to that of the rotor.The rotor–stator coupling factor of the first modal shape under thefixed boundary is 0.012, and the rotor–stator coupling degree islow. Under the flexible boundary, the first modal shape of therotor and stator coupling exhibits rigid-body vibration, wherein

both the rotor and the stator show pitching behaviors. Therotor–stator coupling factor of the first modal shape under the flex-ible boundary is 0.211, and the rotor–stator coupling degree ishigh. The modal shape is similar to the modal shape of the first-order experiment.

The second modal shape of the rotor tester has a lesser influ-ence on the mounting boundary. The modal shapes under differentmounting boundaries are similar. The modal shape of the rotorexhibits translational rigid vibration. Moreover, the rotor–statorcoupling factors under both the flexible and fixed boundaries areless than 0.1. Under the free boundary, the modal shape of boththe rotor and the stator exhibits translational rigid vibration. Therotor–stator coupling factor of the modal shape under the freeboundary is 0.384, and the rotor–stator coupling degree is high.Under the fixed boundary, the modal shape of the rotor exhibitstranslational rigid vibration. The stator structure has almost novibration compared to that of the rotor. The rotor–stator couplingfactor of the modal shape under the fixed boundary is 0.015, andthe rotor–stator coupling degree is low. Under the flexible bound-ary, the modal shape of the rotor exhibits translational rigid vibra-tion, and the stator structure has almost no vibration compared tothat of the rotor. The rotor–stator coupling factor of the modalshape under the fixed boundary is 0.078, and the rotor–stator cou-pling degree is low. The modal shape is similar to modal shape ofthe second-order experiment.

Fig. 9 The first three order modal shapes under the free mounting boundary: (a) the first-order, (b) the second-order, and (c)the third-order



The third modal shape is largely unaffected by the mountingsituation. Moreover, the rotor–stator coupling factors under thefree, fixed, and flexible boundaries are less than 0.003, and therotor–stator coupling degree is low. The vibration displacementcurves of the stator and rotor are the same under different mount-ing conditions. The vibration transferred to the stator structurefrom the rotor structure is very limited.

Under the fixed boundary, the first three order modal shapes ofthe vibration displacement of the stator are very small, showingalmost no vibration with respect to the centerline of the statorstructure. The rotor–stator coupling-degree factors were less than0.02. The experimental modal observation shows the vibrationdisplacement of the stator in the first-order natural frequency.Compared with the rotor, the vibration displacement is not negli-gible. Under the fixed boundary, the vibration in the casing struc-ture is limited. In the study of the coupling vibration in therotor–stator of the whole engine, the fixed boundary model is notsuitable for the simulation of the coupled vibration, particularlyfor the evaluation of the vibration characteristics of the casing.

Hence, the first-order modal shape of the rotor–stator couplingdegree is high. The installation conditions have significant effecton the first-order modal shape. The second-order modal shape ofthe rotor–stator coupling degree is low, and the influence of theinstallation condition is insignificant. The third-order modal shapeof the coupling degree is very low, almost unaffected by the

installation conditions. The installation condition has significantinfluence on the modal shapes for the high rotor–stator couplingdegree. Reasonable installation conditions should be set in thestudy of specific modal shapes.

4.4 Analysis of the Influence of the Mounting Stiffness onthe Dynamic Characteristics of the Whole Engine. Under theelastic installation boundary, the stiffness values of the rear andfront mountings stiffness are set to 5� 106 N/m and 5� 107 N/m,respectively. Another mounting stiffness is changed. Figure 12shows the first three order natural frequencies of the tester.

The first-order natural frequency of the tester increases with theincrease in the stiffness of the front mounting. When the stiffnessin the front mounting increases to 2� 106 N/m, the first-order nat-ural frequency remains stable at approximately 38 Hz. With theincrease in the stiffness of the rear mounting, the first-order natu-ral frequency of the tester increases. When the stiffness of the rearmounting increases to 1� 107 N/m, the first-order natural fre-quency remains stable at approximately 38 Hz.

With the increase in the stiffness of the front mounting, thesecond-order natural frequency changes slightly. The stiffness ofthe front mounting is in between 1� 104 N/m and 1� 106 N/m,and the second-order natural frequency remains stable at approxi-mately 39 Hz. The stiffness of the front mounting is in between

Fig. 10 The first three order modal shapes under the fixed mounting boundary: (a) the first-order, (b) the second-order, and(c) the third-order



1� 106 N/m and 1� 107 N/m, and the second-order natural fre-quency increases with the increase in the front-mounting stiffness.The second-order natural frequency remains stable at approxi-mately 46 Hz when the front-mounting stiffness is greater than1� 107 N/m. The second natural frequency remains stable atapproximately 46 Hz, changing only slightly with the increase inthe rear-mounting stiffness.

The third-order natural frequency changes slightly with theincrease in the stiffness of both the front and rear mountings.

The rotor–stator coupling degree of the first-order modal shapeis high, and the first-order natural frequency is sensitive to themounting stiffness. The rotor-stator coupling degrees of thesecond- and third-order modal shapes are low. The second- andthird-order natural frequencies are affected slightly by the mount-ing stiffness.

The rotor–stator coupling degree of the first modal shape ishigh, and the mounting stiffness significantly affects the first

natural frequency. The rear and front mounting stiffness valuesare set to 5� 106 N/m and 5� 107 N/m, respectively, anothermounting stiffness is changed to calculate. Under different condi-tions, the first natural frequency is calculated, and the displace-ment data are obtained. The effects of the front and rear mountingstiffness on the rotor–stator coupling degrees of the first-ordermodal shapes are analyzed. The rotor and the stator both exhibitpitching vibration. The modal shape of the rotor can be indicatedby the displacement of test points 1 and 6, and the modal shape ofthe stator can be indicated by the displacement of test point 7 (thefirst test point on the casing) and test point 13 (the last test pointon the casing). The curves of the dimensionless vibration dis-placement of the selected test points with respect to the mountingsstiffness are obtained; Fig. 13 shows the results. To make theresult more clear and the trend more evident, the absolute dimen-sionless vibration displacements of the test points are considered;Fig. 14 shows the results.

The modal displacements of test points 1 and 6 are stable whenthe stiffness of the front-mounting section is less than 1� 105 N/m.The rotor pitch is reversed when the front-mounting stiffness is inbetween 3� 106 N/m and 4� 106 N/m. The modal displacementsof the test points 1 and 6 no longer change, and the modal shape ofthe rotor no longer changes when the front-mounting stiffness ismore than 3� 107 N/m.

The modal displacements of test points 7 and 13 are stablewhen the front mounting stiffness is less than 1� 105 N/m. The

Fig. 11 Simulation of the dimensionless mode displacements under different mounting boundaries: (a) the first-order, (b) thesecond-order, and (c) the third-order

Table 5 Coupling factor of the first three order modal shapesunder different mounting boundaries

Coupling degree Free boundary Fixed boundary Elastic boundary

C1 0.496 0.012 0.211C2 0.384 0.015 0.078C3 0.015 0.003 0.021



casing pitch is reversed when the front-mounting stiffness is inbetween 3� 106 N/m and 4� 106 N/m. The modal displacementsof test points 7 and 13 no longer change. The modal shape of therotor no longer changes when the front-mounting stiffness isgreater than 3� 107 N/m.

The modal displacements of test points 1 and 6 are stable whenthe rear-mounting stiffness is less than 1� 105 N/m. The rotorpitch is reversed when the rear-mounting stiffness is in between1� 107 N/m and 2� 107 N/m. The modal shape of the rotor nolonger changes when the rear mounting stiffness is more than1� 108 N/m.

The trends in the modal displacements of test points 7 and 13are the same with the increase in the rear-mounting stiffness. Thedisplacements are stable at first and then reduce; they finally stabi-lize at 0, implying that the casing is no longer vibrating. When therear-mounting stiffness is less than 1� 105 N/m, the modal dis-placements of test points 7 and 13 become stable. The casing pitchis reversed when the rear-mounting stiffness is in between 1� 107

N/m and 2� 107 N/m. The modal shape of the casing nolonger changes when the rear-mounting stiffness is more than1� 108 N/m.

The rotor–stator coupling factors of the first-order modal shapeare calculated under different mounting stiffness values. FromFig. 14,

C1¼ j Test point 7 (casing) modal displacement j / j Test point 1(rotor) modal displacement j (k3< 2� 106 N/m)

C1¼ j Test point 13 (casing) modal displacement j / j Testpoint 6(rotor) modal displacement j (k3 � 2� 106 N/m)

C1¼ j Test point 13 (casing) modal displacement j / j Test point6(rotor) modal displacement j (k4 < 3� 107 N/m)

C1¼ j Test point 7 (casing) modal displacement j / j Test point 1(rotor) modal displacement j (k4 � 3� 107 N/m)

The values of C1 are calculated for different front and rearmounting stiffness values. Figure 15 shows the curve C1 with thefront and rear mounting stiffness values.

With the increase in the front-mounting stiffness, C1 decreasesand the rotor–stator coupling degree decreases continuously.When the front-mounting stiffness is less than 1� 105 N/m, C1

changes slightly. The front-mounting stiffness slightly affects therotor–stator coupling degree. When the front-mounting stiffness isin between 1� 105 N/m and 2� 106 N/m, C1 decreases rapidly,and the rotor–stator coupling degree clearly increases with theincrease in the front-mounting stiffness. C1 remains stable, andthe rotor–stator coupling degree is no longer affected by the front-mounting stiffness when the front mounting-stiffness is more than2� 106 N/m.

With the increase in the rear-mounting stiffness, C1 decreasescontinuously until reaching 0. When the rear-mounting stiffness isless than 1� 105 N/m, C1 changes slightly. The rear-mountingstiffness slightly affects the rotor–stator coupling degree. Whenthe rear-mounting stiffness is in between 1� 105 N/m and1� 108 N/m, C1 decreases rapidly, and the rotor–stator coupling

Fig. 12 The relationship between the first three order natural frequencies and stiffness valuesof the mountings: (a) the front mounting stiffness changes and (b) the rear mounting stiffnesschanges

Fig. 13 The relationships between the first-order mode displacement of test points 1, 6, 7, and 13 and the stiffness values ofthe mountings: (a) the front mounting stiffness changes and (b) the rear mounting stiffness changes



degree clearly decreases with the increase in the rear-mountingstiffness. C1 remains stable, and the rotor–stator coupling degreeis no longer affected by the rear-mounting stiffness when the rear-mounting stiffness is more than 1� 108 N/m.

When the mounting-stiffness values are either considerablyhigh or considerably small, the modal shapes of the vibration ofthe whole engine and rotor–stator coupling degrees are affectedslightly by the mounting stiffness. The mounting-stiffness valuessignificantly affect the rotor–stator coupling degrees when themounting stiffness is in a specific range. For the rotor tester, themounting stiffness significantly influences the first-order modalshape and the rotor–stator coupling degree, when the front andrear mounting stiffness are in the ranges 1� 105–2� 106 N/m and1� 105–1� 108 N/m, respectively.

5 Conclusions

In this study, a finite element model of the rotor tester of anentire aero-engine under different mounting conditions was devel-oped. The model was modified and validated using the results ofthe modal test. A dimensionless evaluation factor—the rotor–stator coupling degree—was proposed to evaluate the rotor–statorcoupling behavior quantitatively. The effect of the mounting

stiffness on the first three order natural frequencies was analyzed.Moreover, the effect of the front and rear mounting stiffness val-ues on the first-order modal shape and the rotor–stator couplingdegree of this modal shape was simulated. The proposedrotor–stator coupling factor helps in evaluating the couplingdegree between the rotor and the stator quantitatively; moreover,the coupling degree of different modal shapes can be compared.The conclusions are as follows:

The simulation of the first three natural frequencies and modalshapes of the finite element model developed in this study and theexperimental results are similar. In addition, the calculated fre-quency response functions at the test points of the rotor and casingare in good agreement with the experimental results. The finiteelement model of the whole tester can reflect the dynamic charac-teristics of the tester, providing a basis for further calculation andcomparison.

Under the free, fixed, and elastic boundaries, the first-ordermodal shapes have significant differences, the second-order modalshapes are the same in general, and the third-order modal shapesare almost identical. The higher the rotor–stator coupling degreeof the modal shape, the greater the influence of the mounting con-ditions on the modal shape. The results show that the stiffness ofthe mountings cannot be ignored when the finite element calcula-tion of the entire machine is conducted. Employing the fixed

Fig. 14 The relationship between the first mode absolute displacements of test points 1, 6, 7, and 13 and the stiffness valuesof the mountings: (a) the front mounting stiffness changes and (b) the rear mounting stiffness changes

Fig. 15 The relationship between the first modal coupling factors and mounts stiffness values: (a) the front mounting stiff-ness changes and (b) the rear mounting stiffness changes



constraints on the installation of the boundary setting will result ina considerable error in the calculation results.

The influence of the mounting stiffness on the rotor–stator cou-pling degree is nonlinear. The influences of the modal shape of thevibration and the rotor–stator coupling degree of the whole engineare negligible when the installation section stiffness is either consid-erably large or considerably small. For the first-order modal shapeof the rotor tester, the influence of the mounting stiffness on therotor–stator coupling degree is significant when the front and rearmounting stiffness values are in the ranges 1� 105–2� 106 N/mand 1� 105–1� 108 N/m, respectively. Outside this range, the influ-ence becomes less evident. During the design, tweaking the stiffnessof the mountings can adjust the coupling degree of the rotor and sta-tor, but only when the mounting stiffness values are within a spe-cific, feasible range. If they are outside the range, the mountingstiffness will no longer affect the rotor and stator coupling.

Acknowledgment

We would like to thank the engineer G. Q. Feng of theShenyang Aero-engine Institute for his valuable suggestions. Inaddition, we would like to thank the graduate students H. F.Wang, P. P. Song, and B. B. Liu for their help in conducting themodal experiments.

Funding Data

� Funding for Outstanding Doctoral Dissertation in NUAA(Grant No. BCXJ17-10).

� Funding of Jiangsu Innovation Program for Graduate Educa-tion (Grant No. KYLX16_0387).

� National Natural Science Foundation of China (Grant No.51675263).

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effect of the aero-engine mounting stiffness on the whole

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