Yongfeng WANG, Ynhong MA, Jie HONG,,*

a School of Energy and Power Engineering, Beihang University, Beijing 100083, China

b Research Institute of Aero-Engine, Beijing 100083, China

KEYWORDS Coupling vibration;Critical speed;Dynamic stiffness;Rotor dynamic;Rotor-supports system

Abstract The dynamic response of the rotor depends on not only itself but also the dynamical characteristics of the structures that support it.In this paper,the coupling vibration characteristics of the rotor and supporting structure are studied using one simple rotor-supports model firstly,and then the dynamic stiffness of the typical supporting structure of an aero-engine is investigated in use of both numerical and experimental methods.While,one simulation strategy is developed to include dynamic stiffness of realistic supports in the dynamical analysis of the rotor system. The simulated and tested results show that the dynamic stiffness of the supporting structure not only depends on the structural parameters but also is related to the frequency of the excitation force. The dynamic stiffness is affected by the damping and inertia effect when the excitation frequency is high and closed to the resonance frequency of the support,which may decrease the dynamic stiffness sharply.More resonance frequencies may be induced and the critical speed could be reduced or increased.While higher vibration response peak and overload of the bearing may also be caused by the varied dynamic stiffness, which needs to be avoided in the design of the rotor-supports system.

1. Introduction

In rotating machinery, the rotor operates around the supporting structures, the features of which play of vital important role in the dynamic characteristics of the rotor.1,2Studies have shown that the correlation between analytical calculations and experimental results can be poor in some cases if just constant static stiffness and damping are adopted but the mass of supports is excluded from the analysis.3There are usually underor unmodeled dynamic components considered to be frequency-dependent in a rotor-support system, especially for the machinery with lightweight but heave loads like aeroengine.4,5Engineering practice shows that the stiffness of the structure varies under the dynamic excitation instead of keeping at a constant value, which is named the dynamic stiffness.6,7The dynamic stiffness theory combines the static stiffness of the structure and the dynamic effects of its mass and damping,which comprehensively reflect the characteristics of the structure under dynamical excitation.

To obtain the accurate dynamic characteristics,such as the critical speed and the unbalanced response, and design the rotor properly, the accurate dynamic stiffness of the support is necessary.8,9Finite Element Analysis (FEA) is probably the most widely used method to obtain the dynamic stiffness of complex structures. Three-dimensional elements model of the supporting structures for different kinds of machines have been established by Kubany10and Ding11et al., and the dynamic stiffness curves were calculated based on the displacement response acquired by harmonic response analysis. However, the accuracy of the simulation results often depends on the correctness of the model. To guarantee the accuracy of results, the dynamic stiffness tests will be the other choice in the study of factual engineering structures.12Muszynska et al.13outline the sweep frequency rotating force perturbation method for identifying the dynamic stiffness, and the results are presented in the direct and quadrature dynamic stiffness formats for identifying more easily.Gade and Herlufsen14presented the measurement strategy using a shaker for the dynamic stiffness of an isolator, based on both resonant and non-resonant methods, while the hammer exciting method is used by more researchers because of its practicability and facilitation. The experimental stiffness tests have been carried out in past years for different kinds of structures, such as bearing supports of gas turbine,15,16the engine rubber mount,17wind turbine,18support hydraulic system,19and so on.

There is no doubt that the dynamic characteristics of supporting structures must be included in the analysis of the dynamical response of the rotor.20The procedure for the coupling supports system has been developed,21,22which allows the mass, damping, and static stiffness of the supports to be added beyond the bearing. The component mode synthesis method,23the transfer matrix method,24or the impedance coupling method25can be used to build the rotor-supports model considering the stiffness and damping of supports,but it’s still not convenient to be used for the practical engine with complex rotors and supporting structures. In recent years, the FEA method is widely used, and the dynamic models of the whole engine consisting of both rotors and supporting structures have been established.26–28The dynamic supporting stiffness is included automatically since the model consists of both the rotor and the supporting structures. However, it’s tedious work to build the dynamic model that consists of not only the rotor but also the complex supporting structures using finite element methods, and it’s also time-consuming to acquire the vibration response of entire rotor-support systems for factual rotating machinery.

Meanwhile,the mathematical model does not always accurately represent the physical system as expected, especially for some characteristics that are extremely difficult to be simulated,such as the complex contact interfaces.Thus,the testing data of the practical structures are generally welcomed, which can be compared with the analytical results for verification.29,30Using the measured data to update the finite element matrices of rotor systems is an effective way to get more accurate results for rotor dynamics analysis,and some exploratory research have been carried out.31But it’s still a great challenge to use the measured supporting stiffness data directly in the dynamic analysis of the realistic structures, which need to be deeply investigated to include the dynamical characteristics of supporting structures correctly in the analysis of rotorsupport systems.

This paper concentrates on the dynamic stiffness of the supporting structure and its influence on the vibration of rotors. The non-negligible influence of supporting structures on the dynamic response of the rotor is revealed, and both simulation and experimental methods were used to investigate the dynamic stiffness characteristics of the typical flexible supporting structure in the aero-engine. One computation strategy was developed to include the simulated or tested dynamic stiffness of realistic supports directly in the dynamical analysis of the rotor system,and the updated simulation method was given to study the critical speed and the frequency response of the rotorsupports system. The research in this paper provides an accurate and efficient method to include the accurate dynamical characteristics of supporting structure in the analysis of the rotor-support system, which is valuable for the design work of some lightweight rotating machines like aero-engine.

2. Coupling vibration of the rotor and supporting structure

2.1. Mathematical model of the rotor with supporting structures

One dynamic model of the rotor,considering not only the stiffness of the supports but also the dynamic effects of mass and damping, was developed to investigate the coupling vibration characteristics of the rotor and supporting structure,as shown in Fig.1.The typical Jeffcott rotor with a single disk located at the one-half span of the shaft is adopted,while the masses(ms),the stiffnesses(ks),and the viscous damping(cs)of the supporting structures are also modeled instead of with rigid supports.The stiffness and the viscous damping of the massless elastic shafts at the location of the disk are recorded as krand cr,while the mass of the rigid disk is mdwith the eccentricity e.The stiffness of the rolling bearings(kb),as well as the damping(cb), are considered, while the mass of which is ignored. ω is the rotating speed, t is the time, r is the off-centre distance,and θ is the whirling angle. Fkand Fcis forces caused by the elastic deformation of rotor and damping separately.

The differential equations for the vibration of the rotor coupled with supporting structures(Fig.1)are obtained based on Lagrange’s method, which can be represented as:

The differential equation for the vibration of the system in the x-direction is:

Fig. 1 Schematic of dynamical model for rotor-supports system.

The vibrations in the x and y directions are the same for the centrosymmetric rotor-supports system, ignoring the coupling in the x and y directions.Thus,Eq.(3)meets the need to investigate the typical dynamic characteristics of the rotor coupled with supporting structures.

2.2. Coupled vibration characteristics

The vibration response of the rotor-supports system in the xdirection can be assumed to be xd=Xdcosωt and xs=Xscosωt under unbalanced excitation, where Xdand Xsare the vibration amplitudes.The effect of damping on the resonance frequencies is little, which can be ignored in the analysis of rotor’s critical speed, and the differential Eq. (3) can be further simplified as:

Fig. 2 shows the typical dynamical response of the rotorsupports system with the mass of supports,which is compared with the classical Jeffcott rotor, using the physical parameters listed in Table 1. The results show the great effect of the supporting structures on the dynamical response of the rotor.More critical speeds are induced as the mass effect of the supports is taken into consideration,one of which comes from the rotor (similar to the Jeffcott rotor with rigid supports), while the other one is caused by the resonance of the supporting structures. When the mass of supports is taken into consideration, the resonance frequency will also be lower, which is due to the inertia effect of the mass.Under the dynamical excitation,an effective force that acts in the direction of motion of the mass is induced,which is opposite to the stiffness force and acts to reduce the stiffness of the system.

In a word, the supports of the rotor can’t be regarded as a spring with constant stiffness, for the inertia effect caused by the mass of supports is also play a nonnegligible part.To study both the static stiffness and the dynamic effects of mass and damping,the dynamic stiffness of the structures must be used.

3. Dynamic stiffness of supporting structures

3.1. Mechanism of the dynamic stiffness

Fig. 2 Response curves of rotor systems with or without mass effect of supports.

Table 1 Physical parameters of rotor-supports system.

Dynamic stiffness combines the static effects of spring(K)and the dynamic effects of mass(M)and damping(C).33As shown in Fig. 3. The simple spring-mass system is given with no applied force,a static force,and a dynamic force.When a static force FSis applied (middle), the positive spring stiffness produces a force proportional to the displacement that opposes the applied force.When a dynamic force FDis applied(Right), the inertia of the mass creates an effective force that acts in the direction of motion of the mass, opposite to the spring stiffness force.This inertia effect is equivalent to a negative stiffness and acts to reduce the spring stiffness of the system, which is called mass stiffness. Similarly, the dynamic stiffness originates in the damping force is called the damping stiffness, which is in the same direction as the spring stiffness force.

Assume that the dynamic excitation load is a simple harmonic force expressed as FD= F0sin(ωt), followed by a displacement in the same direction y = Y sin(ωt). Then, the dynamic equation of the system can be formulated based on the force equivalent method.

3.2. Simulation results of the dynamical supporting stiffness

The dynamic stiffness of the supporting structure can be expressed as the ratio of the amplitude of the applied excitation load to the amplitude of the dynamic response at the same point. The widely used FEA method is adopted for the simulation, considering complex geometric features of the aeroengine supporting structure shown in Fig. 5. The supporting structure consists of the outer casing, eight evenly distributed frames, the inner casing, and the housing of bearings, which will hold the rolling bearing and transfer the loads from the rotor to stators. The FEA model was built in the ANSYS/Workbench 18.0 using SOLID186 elements, and the material parameters are given in Table 2. The structure is treated as a linear system with all the contact interfaces bonded, and the damping ratio of the supporting structures is assumed to be 0.01, considering both structural damping and the damping induced at the contact interfaces of joints.35The harmonic response analysis is carried out, and the Frequency Response Functions (FRFs) are obtained and analyzed. Only part of the ring in the direction of the excitation load contacts with the housing, as shown in Fig. 5(c), which is simulated using the bonded contact element in simulation.The nonlinear characteristics induced at contact interfaces of the bearing are ignored in this manuscript.

The excitation load is lumped on a node of bearings in the vertical direction with a constant value(FD=1000×sin(ωt)),and the flanges of the outer casing are fully restricted,as shown in Fig. 5(b). The dynamic responses of the supporting structures are acquired in ANSYS Firstly, using the harmonic response analysis method. And then, the frequency response functions, as well as the amplitude of excitation force, are loaded in MATLAB to calculate the dynamic supporting stiffnesses, which are shown in Fig. 6 and Fig. 7.

Fig. 3 Schematic diagram of static effects of spring and dynamic effects of mass and damping.

Fig. 4 Amplitude-frequency curve of spring-mass system.

Fig. 5 Diagram of one typical supporting structure model.

Table 2 Material parameters of the structure.

The simulation results show that the dynamic stiffness of the supporting structure is approximately equal to the static stiffness when the frequency of the excitation load is much lower than the resonance frequency, while the dynamic response of the structure is small as well. But once the excitation frequency closes to the resonance frequencies, the response amplitude increases sharply, and the dynamic stiffness can decrease to just one percent of the static stiffness.While,the dynamic stiffness may also be higher than the static stiffness at some specific frequency over the first-order resonance, which is mainly caused by the inertia effect.

The dynamic stiffness characteristics at the position bearing M are different from that of bearing N,though bearing M and bearing N are on the same supporting structure, which is mainly affected by the modal shape of the structure.As shown in Fig.8,the stiffness of bearing N is more sensitive to the firstorder swing mode compared to the bearing M,but less affected by the swing mode when close to the node of mode shape.

Fig. 6 Numerical results at the position bearing M.

Fig. 7 Numerical results at the position bearing N.

Fig. 8 Modal shapes of supporting structure.

3.3. Experimental investigations

The most widely used hammering test method was adopted to obtain the dynamic stiffness of the supporting structure, and assuming that the supporting structure is approximately linear,thus the frequency of the response will be the same as the frequency of the excitation load.The supporting structure is fixed on the amount during the testing work,the stiffness of which is assumed to be similar to that of the structure connected with the supporting structure. The excitation load is applied by one hammer, and the magnitude of the load is measured by the force sensor. Meanwhile, the acceleration sensor is attached to the structure,which is used to obtain the acceleration response of the structure consistent with the excitation direction, as shown in Fig. 9.

According to the measured excitation force FD(ω) and acceleration response YA(ω) of the structure at the bearing position, the acceleration admittance function HA(ω) can be obtained.

Fig. 9 Method of measurement and test device.

Based on the relationship between the dynamic stiffness KD（ω ) and the acceleration admittance function HA（ω ), the dynamic stiffness values of the structure at each bearing position can be expressed as:

The testing results of the dynamic stiffness curves at different bearing positions of the supporting structure are shown in Fig.10,and three repeated tests were carried out to guarantee the accuracy. The testing results are also compared with the simulation results, which are basically in agreement with each other, confirming the accuracy of the simulation method. The results verify that the resonances of the supporting structure have a great influence on the dynamic characteristics, which may cause a sharp reduction in the dynamic stiffness.

There are still some differences between the testing and simulation results:

Firstly,the tested dynamic stiffness is smaller than the simulation results at low frequencies far from the resonance. The reason is that when the frequency is low,the dynamic stiffness mainly depends on the static stiffness. In the simulation, the displacements of the flanges of the outer casing are fully restricted.But the supporting structure is placed on the mounting structures in testing, the elasticity of which decreases the stiffness at low frequencies.

Secondly, the tested dynamic stiffnesses at the resonance frequency are larger than the simulation results, and the resonant frequency bands of the tests are much wider as well. It is due to the difference of the damping, which has an important effect on the frequency bandwidth and the dynamic stiffnesses near the resonance frequency. In the simulation, some damping sources are ignored,like the contact interfaces.The damping induced at the contact interfaces increases the total damping of the realistic structure, resulting in the wider frequency band and bigger dynamic stiffness at the resonant frequency for the testing.

4. Influence of the dynamic stiffness on the rotor

4.1. Strategy to include the dynamic stiffness in the rotor dynamic analysis

The supporting structure is used to mount the rotor system in the aero-engine, as shown in Fig. 11(a). The bearing M of the structure studied in Section 3 is used to support the back-end of the high-pressure rotor, while the low-pressure rotor supported by bearing N is not shown here. The front-end of the rotor is supported by another supporting structure (Fig. 12),the dynamic stiffness of which can be studied in the same method.

Fig. 10 Testing results of dynamic stiffness,

Fig. 11 Schematic of dynamic model of aero-engine rotor.

Fig. 12 Dynamic stiffness of Support 1.

The typical high-pressure rotor-supports systems of aeroengine can be simplified as a rotor system with two offset disks and supported at both ends,as shown in Fig.11(b).The structural characteristics of the rotor are ignored,but the mass and the stiffness of the rotor are modeled using the Euler-Bernoulli beam element,mass element.The supporting structure and the rolling bearing are modeled by the spring-damper elements,while the damping ratio induced in the structure is assumed to be constant (0.01) and the dynamic stiffness is included for the spring, but the mass of the rolling bearing will be ignored. The detailed deterministic geometric and physical parameters of the model are given in Table 3.

To model the dynamic characteristics of the supporting structure accurately and efficiently, the dynamic stiffnesses were adopted in the dynamic model of the rotor-supports system shown in Fig. 11. The dynamic stiffness of support 2(bearing M) has been acquired in Section 2, as shown in Fig. 10, and the testing results are used for the rotorsupports model.It should be noted that the stiffness at low frequency was modified by the simulation result to be consistent with the static stiffness value.The FEA model of support 1 has also been built,the dynamic stiffness of which was acquired by simulation method same as in Section 3, and the results are shown in Fig. 12.

Table 3 Values of physical parameters of double disk rotorsupports system.

In most classical FEM commercial tools, like ANSYS, the modal analysis or harmonic response analysis method can’t be directly used to study the dynamic response of the rotorsupports system with varied supporting stiffness. One developed computation strategy is given out to calculate the frequency response of the rotor-supports system considering the dynamic stiffness, which is shown in Fig. 13. The structural and mechanical parameters of both rotor and supports (such as M, K, C, F) are determined by the practical structure in analysis, while the frequency interval and the step length δ can be chosen according to the accuracy and time limit. In the simulation of this manuscript, the frequency interval is[0, 500 Hz], with a step length (δ) 1 Hz. The ω1represents the frequency in the calculation, and the ωendis the end frequency.

Firstly, the dynamic stiffness of supporting structures can be obtained through the harmonic response analysis method based on the structural and mechanical parameters, or based on the experimental methods. Then, a narrow starting frequency range is chosen, in which the supporting stiffnesses can be assumed to be constant,and the dynamic characteristics of the rotor are studied using the classical dynamical method.The obtained response will be stored and recorded. Next, the frequency range will be judged and a new narrow range will be chosen unless it’s the end. The stiffness of the supports will be updated based on the dynamic stiffness results. Based on the loop statement used in the computation procedure, the dynamic response of the rotor within the whole frequency range is finally investigated and given out.

Fig. 13 Computation strategy for the rotor-supports system considering the dynamic stiffness.

4.2. Simulation results and discussion

Both the critical speed and the dynamic response of the rotorsupports system have been calculated using the dynamic model and method developed in the front section.Modal analyses are firstly carried out to learn about the natural characteristics of the rotor-supports system,the first two of which(with constant static stiffness) are shown in Fig. 14. The first-order is the bending vibration of the whole rotor, and the nodes of the vibration are close to the supports. While, the second-order is the translation motion of the disks together with the ends of the rotor, and the vibration node is at the center of the rotor.

Fig. 15 is the Campbell diagram of the rotor-supports system, which shows the variation of the eigenfrequencies of the rotor, both with the dynamic stiffness and with the constant static stiffness 1.04 × 108N/m and 6.2 × 107N/m. When the stiffness is constant, the eigenfrequencies of the rotor may increase(or decrease)monotonously as the rotating speed increases, due to the gyroscopic effect of the rotating rotor.But the dynamic stiffness may have a much greater influence on the eigenfrequencies, and the nonmonotonic drastic variation of which is not surprising at high frequency due to the resonance of supporting structure. Thus the order of critical speed for the rotor-supports system may be increased,which may further lead to the more complex vibration response of the rotor.

Fig. 15 Critical speeds of rotor-supports systems considering dynamic stiffnesses.

The results show that there are two critical speeds below 500 Hz for the rotor with constant static support stiffness,which is 162 Hz and 418 Hz separately. However, once the dynamic stiffnesses of the supports were taken into consideration, more critical speeds (162 Hz, 320 Hz, 392 Hz) appear,which is caused by the vibration of supports. The dynamic stiffnesses have almost no influence on the first-order critical speed (162 Hz) because the support stiffness mainly depends on the static stiffness at low frequency. However, when the rotating speed is close to the resonance of the supports, the critical speeds decrease because of the decreases of stiffnesses,and more critical speeds are induced for the nonmonotonic variation of supporting stiffness.

The mode shapes of the rotor-support systems are deeply influenced by the dynamic stiffness of the supports, as shown in Fig. 16. The first-order mode is the bending vibration of the whole rotor,but the deformation of the rotor close to support 2 will increase as the dynamic stiffness is reduced. At the resonance frequency of support 2, the constraint provided by the supporting structure is weak, and the mode shape change to the swing of the rotor due to the vibration of supports.The second-order is the translation motion of the disks together with the ends of the rotor, which is sensitive to the dynamic stiffness of both supporting structures. The variation of the stiffness at support 2 is much larger, thus the deformation close to support 2 changes more obviously. The coupling vibration of the rotor and supporting structure causes a more complex vibration mode shape, which induces the complex vibration response at different excitation frequencies.

Fig. 14 Mode shapes of the rotor with static stiffness.

Fig. 16 Influence of supporting dynamic stiffness on rotor mode shape.

The dynamic response of the rotor-supports system was also studied based on the developed simulation strategy, with the unbalance (10 g.cm) applied on disk 2 (Fig. 11), and the dynamical responses of which are shown in Fig. 17 and Fig. 18. Similar to the results of critical speeds, the dynamic stiffnesses have very little influence on the displacements and the reaction forces of the rotor at low frequency. It is mainly because the variation of the stiffness is very small when the frequency is low,which could be ignored in the dynamic analysis of the rotor-support systems. But both the displacements of the disks and the reaction forces of the bearings are changed at a higher frequency close to the resonance frequency of the support. For the rotor system with dynamic supporting stiffness, two response peaks at lower frequencies replace the single peak of the second-order critical at 416 Hz.

The variations of the amplitudes at disk 1 and disk 2 are different. For disk1, the two response peaks of the rotor with dynamic stiffness are all smaller than the single peak, but the response peak of disk 2 at 394 Hz is much larger than the original one if the dynamic stiffness is taken into consideration.The variation characteristics of the reaction forces on the supports are just similar to that of vibration responses, and a much larger reaction force may appear considering the dynamic supporting stiffnesses.

Fig. 17 Response of rotor under unbalance force.

Fig. 18 Reaction forces of supports.

It is worth mentioning that serious defects may be covered if the supporting stiffness of the rotor is treated as the constant static value instead of the dynamic stiffness. Lots of aeroengine rotors operate between the first two critical speeds,and the decrease of the second-order critical speeds and more critical speeds caused by the dynamic stiffness may have a great influence on the safety and reliability of the aero-engine.For example, the rotor seems to be able to operate safely around 300 Hz if the supporting stiffnesses are treated as the constant static stiffness, and both the displacements and reaction forces of the rotor are small. But actually, the features of the dynamic stiffness of the supporting structure will cause a large vibration response,and the reaction force at the bearings may also be over limits,which may lead to serious damage and even disastrous accidents for the aero-engine in operation.

5. Conclusions

This paper focuses on the effect of the dynamical supporting stiffness on the vibration of rotors. The vibration characteristics of the rotor coupled with supports were investigated firstly,based on the simple rotor-support system, which reveals the non-negligible influence of supporting structures. Then both the numerical and the experimental methods were adopted to obtain the dynamic supporting stiffness of the typical supporting structure in the aero-engine, which were used to build one practical rotor-supports model. At last, one simulation strategy was developed to update the dynamic stiffness in the simulation of the rotor-support system, based on which the influence of the dynamic stiffness on the critical speed and vibration response was fully investigated. The following conclusions could be drawn:

(1) More critical speeds can be induced and the response amplitude may be changed dramatically in some frequency as the mass effect of the supports is taken into consideration, which is caused by the resonance of the supporting structures.And the resonance frequency will also be lower due to effective force opposite to the stiffness force induced by the inertia effect of the mass.

(2) The dynamic stiffness depends on not only the structural parameters of the system but also the frequency of the excitation force.The testing results of the dynamic stiffness are basically in agreement with the simulation results for the supporting structure of the aero-engine,which confirmed the accuracy of the method. The dynamic stiffnesses have almost no influence on the first-order critical speed (162 Hz), because the support stiffness mainly depends on the static stiffness at low frequency. But once the rotating speed comes close to the resonance of the supports,the critical speeds will change due to the variation of the supporting stiffness caused by the vibration, and more critical speeds may appear.

(3) Similar to the results of critical speeds,the dynamic stiffnesses have very little effect on the vibration response and the reaction forces of the rotor at low frequency.But both the vibration of the disks and the reaction forces of the bearings are changed at the frequency close to the resonance frequency of the supporting structure.The features of the dynamical supporting stiffness may cause large vibration responses and over-limit reaction force at the bearings,which may lead to serious damage and even disastrous accidents for the aero-engine in operation.

There are still some defects in the paper, such as the supporting structure being placed on the mounting frame instead of on the realistic engine, which has some influence on the tested stiffness value especially at low frequency. Thus, the dynamic stiffness used in the rotor-supports system needs to be modified using the simulation results. But actually, it’s better to measure the dynamical supporting stiffness in the assembled state, which can be directly used in the dynamic simulation of the rotor-supports system.Additionally,the contact interfaces were all ignored in this study, but the stiffness loss and the damping induced on the interfaces may also have a great influence on the dynamical supporting stiffness, especially when the excitation frequency is close to the resonant frequency. The method to study the influence of the bolted interfaces on the dynamic characteristics of the rotor-bearing system may be another interesting topic.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to acknowledge the financial support from the National Natural Science Foundation of China(No. 52075018), and the National Science and Technology Major Project of the Ministry of Science and Technology of China (Nos. 2017-IV-0011-0048 and 2017-I-0008-0009).