Binin GENG, Ynfei ZUO,*, Zhinong JIANG, Kun FENG, Chen WANG,Jie WANG

a Key Lab of Engine Health Monitoring-Control and Networking of Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, China

b Beijing Key Laboratory of High-end Mechanical Equipment Health Monitoring and Self-Recovery, Beijing University of Chemical Technology, Beijing 100029, China

KEYWORDS Dynamic coefficients;Gas turbine engine;Measuring point response;Quantitative evaluation;Uncertainty analysis

Abstract Bearings in a gas turbine engine are the key connecting components transmitting force and motion between rotors and thin-walled flexible casing. The bearing stiffness and damping of squeeze film damper (SFD) nearby bearings are easily affected by many factors, such as assembly process,load condition and temperature variation,resulting in uncertainties.The uncertainties may influence the response of the measuring point on the casing. Hence, it is difficult to carry out the fault diagnosis, whole machine balancing and other related works. In this paper, a double integral quantitative evaluation method is proposed to simultaneously analyze the influence of two uncertain dynamic coefficients on the response amplitude and phase of casing measuring points. Meanwhile, the coupling influence of stiffness and damping accompanied by dramatic changes with rotational speeds are essentially discussed. As an example, a typical engine bearing-casing system with complex dynamic characteristics is analyzed. The impact of uncertain dynamic coefficients on the unbalance response is quantitatively evaluated.

1. Introduction

In the vibration test of gas turbine engines, it is generally impossible to measure the response of the rotors or bearings directly. Therefore, the engine fault diagnosis,1,2whole machine balancing,3,4vibration standard formulation5and other related work are based on the response analysis of measuring points on the casing. Moreover, bearings and SFD of gas turbine engine are the key connecting part between the rotors and thin-walled flexible casing system,which influence the response of measuring points greatly.Hence, the determination of the dynamic coefficients of bearings is very important and has been extensively investigated.

There are mainly two ways to address the above problem. The first one is to use the numerical method to calculate the dynamic coefficients. Hernot et al.6,7calculated the static stiffness of bearings based on the Hertz contact theory, and the change of bearing stiffness under different loads was considered. Ishfaque et al.8,9calculated the damping forces of SFD for various boundary conditions based on the Green’s function. However, the influence of the lubrication state, temperature and other factors on bearing and SFD dynamic coefficients is not fully understood. The second method is to measure the dynamic coefficients by component test. For example, Matta and Arghir10collected the vibration response under the impulse excitation at a bearing,and obtained the bearing coefficients at multiple frequencies by curve fitting based on the test data. De Santiago and Luis Andre′s11added unbalance mass to a rotor, and the equivalent stiffness and damping of the rolling bearing under working condition were identified via measuring the displacement response of the bearing shaft. According to Refs.12-14, SFD damping coefficients were evaluated by means of a mechanical impedance method with a horizontal twoway excitation tester.

However,it is difficult to measure the response at the bearing position for gas turbine engines directly.In addition,bearing and SFD dynamic coefficients are easily affected by assembly process, load condition, lubrication state and temperature variation,which results in uncertainties.15-18Whether through the numerical calculation or component test,it is challenging to obtain exact dynamic coefficients of the bearings and SFD in the working condition. Meanwhile, the structure of the engine bearing-thin-walled casing system is complex,and the dynamic characteristics are considerably influenced by rotation speeds.Many efforts have been made to investigate the impact of the supporting stiffness on the dynamic characteristics of the rotor system.19-21As such,Feng and Luo22analyzed the supporting stiffness influence on the critical speed and mode shapes of a dual-rotor engine.Zhang et al.23,24evaluated the effect of bearing stiffness on the transient and steadystate nonlinear response of a rotor system.However,the above research only qualitatively analyzed the support stiffness influence on the rotor system.In addition,most literature only considered the effects of single factor.Therefore,it is necessary to develop a quantitative method to evaluate the influence on the thin-walled casing response caused by both stiffness and damping.

Firstly, an equivalent two degree of freedom (DOF) model is used to introduce the method.To analyze the coupling influence of both bearing stiffness and SFD damping on the response of casing measuring points together with the variation under different rotational speeds, sensitivity indexes of the response amplitude and phase at a specific rotation speed are separately proposed. Accordingly, a double integral method for the quantitative evaluation is developed. Finally,a typical gas turbine engine bearing-case system is analyzed to evaluate the influence of uncertain coefficients.

2. A double integral method for quantitative evaluation

2.1. Two-degree-of-freedom bearing-casing system and its dynamic response

The engine bearing-casing coupling system is abstractly simplified as a two DOF system, and is shown in Fig. 1. Relevant studies show that the rotor system with non-linear rolling bearings or SFD exhibits complex dynamic characteristics,and the non-linear supporting forces are affected by many factors,such as preload condition, surface waviness, Hertz contact and bearing clearance.4,25-26Influenced by these factors, the vibration response on the measuring points of gas turbine casing is not uniform for different condition or different engines on the same assembly line. But it is difficult to derive the theoretical result of this engineering nonlinear problem. So, the focus of this paper is on quantitatively evaluate the response of casing measurement points by simplifying the damping of SFD as uncertain lumped damping of bearings without nonlinearity.

Then the dynamic equation of the bearing-casing coupling system can be expressed as:

where mc,kcand ccare equivalent mass,stiffness and damping of the casing system respectively, xcis the displacement response of the casing system.mb,kband cbare the equivalent mass, stiffness and damping of bearing system respectively,and xbis the displacement response of bearing system. P is the amplitude of excitation force and ω is the excitation frequency.

The response of casing and bearing can be obtained by direct solution method as follows:

For the bearing system, only the equivalent stiffness and damping of are considered, while the mass is neglected as it is usually too small. That is mb= 0, so Eq. (2) can be simplified as following form:

Fig. 1 Dynamic model of bearing-casing coupling system.

2.2. The process of the double integral method

Fig.2 shows the whole process of the proposed method.Under constant displacement excitation, set the damping being constant at one time, the amplitude-frequency response curve and the phase-frequency response curve of the casing measuring point can be obtained by changing the bearing stiffness.By changing the damping and repeat the above procedure, the response of the measuring points under a sequence(r×s)stiffness and damping can be obtained as shown in Fig.2(a).Then,the curved surface of the response amplitude and phase varying with both stiffness and damping at different rotation speeds(q steps)can be obtained,as shown in Fig.2(b).On this basis, the quantitative variation of response amplitude or phase changing with bearing stiffness or damping can be obtained by using sensitive indexes at specific speed, as shown in Fig. 2(c). Finally, a double integral calculation method in actual uncertain range is used for quantitative evaluation of the influence on the response amplitude and phase, as shown in Fig.2(d).Details of method will be further described in Sections 2.3 and 2.4.

2.3. Sensitivity index of dynamic coefficient at specific speed

It can be seen that when the bearing stiffness kbchanges,the response amplitude and phase of the casing measuring point change. When the bearing damping cbchanges, the response amplitude and phase of the casing measuring point also change.

For Eqs. (13) and (14), when kc, cc, mcand x0being constant, while kband cbchange in a larger range, as shown in Table 1. The curved surfaces of the response amplitude and phase of the casing system changing with bearing stiffness kband bearing damping cbat different rotation speeds can be plotted, as shown in Fig. 2(b).

It can be seen from Fig. 2(b) that the response amplitude and phase of the‘‘casing”measuring point are affected by both bearing stiffness and damping with coupled effect at a special range. Besides, the curved surface is changing with rotational speeds. To quantitatively evaluate the changing, four indexes of relative variation of amplitude and phase are proposed.

As shown in Fig. 3, when the bearing stiffness is K（i） and damping is C（j）,the response amplitude and phase of the measuring point under a constant displacement excitation is A（i，j）and P（i，j）. When the bearing stiffness has been changed as K（i+1） but damping kept as C（j）, the response amplitude and phase are separately denoted as A（i+1，j）, P（i+1，j）.A（i，j+1）,P（i，j+1）,A（i+1，j+1）andP（i+1，j+1） are similarly defined.

Fig. 2 Application of quantitative evaluation method for double uncertain factors.

Table 1 Details of bearing-casing coupling system.

Rkand Rcare dimensionless parameters. The physical meaning of Rkis the relative change rate of response amplitude caused by per unit change rate of bearing stiffness when bearing damping is constant. Rcis the relative change rate of response amplitude caused by per unit change rate of damping when bearing stiffness is constant.Rk＞0(or Rc＞0)indicates that the response amplitude is positively correlated with the bearing stiffness (or damping). Rk＜0 (or Rc＜0) indicates that the response amplitude is negatively correlated with the bearing stiffness (or damping).

Similarly, the relative change phase Vkto the bearing stiffness and the relative change phase Vcto the bearing damping at (K（i）,C（j）) are defined as:

Fig. 3 Schematic diagram for calculating sensitive index.

The unit of Vkand Vcis degree.The physical meaning of Vkis the relative change of response phase caused by per unit change rate of bearing stiffness when bearing damping is constant.Vcis the relative change of response phase caused by per unit change rate of damping when bearing stiffness is constant.Vk＞0 (or Vc＞0) indicates that the response phase is positively correlated with the bearing stiffness (or damping).Vk＜0 (or Vc＜0) indicates that the response phase is negatively correlated with the bearing stiffness (or damping).

In this paper, Rk、Rc、Vkand Vcare defined as the main sensitivity index, because it reflects the influence of the change of a parameter on the output response,and is a local sensitivity index.

Based on the four sensitive indexes defined above, the relative change rate of the response amplitude and phase in a large changing range of both bearing stiffness and damping can be quantitatively calculated at a sequence rotation speeds, as shown in Fig. 2c. It can be seen that the relative sensitivity denoted by colors are affected by not only bearing stiffness and damping but also rotation speed.

2.4. Quantitative evaluation method for double uncertain dynamic coefficients

where Θ denoted Rk，Rc，Vkor Vc. In Eq. (20), uncertainties of bearing stiffness and damping are considered by the uncertain boundaries [kL，kH] and [cL，cH] together with their probability density function f （kb，ω）, g （cb，ω）. Θ （ω ） seems like a weight function determined by the bearing-casing systems which can be got by a great number of pre-calculation(r×s×q) as described in Section 2.2 and sensitive analysis described in Section 2.3. Θ is the total sensitivity index defined in this paper.

As a simplified condition, it is assumed that the bearing stiffness and damping are uniformly distributed and are not affected by rotation speed in the uncertain range,the probability density function can be simplified as:

According to Eq. (20) or Eqs. (23)-(26), the sensitivity index in the variation range under different rotation speeds can be calculated, as shown in Fig. 2(d). The quantitative sensitivity of the response amplitude and phase influenced by uncertain bearing stiffness and damping under different rotation speeds can be got. The results of two degrees of freedom model show that the sensitivity is highest at the natural frequency of the system.

3. Impact of uncertain dynamic coefficients on unbalance response

Section 2 simplifies the bearing-casing coupling system as a two-degree-of-freedom system,and proposes a double integral quantitative method for evaluating the influence of uncertain bearing stiffness and damping on the response amplitude and phase. In this section, a typical engine bearing-casing system is analyzed as a typical application, and the impact of uncertain bearing stiffness and damping on the response of measuring points is analyzed.

3.1. A typical engine bearing-casing model

A typical gas turbine engine structure is shown in Fig. 4. The engine has six bearings,of which No.1,No.2,No.3 and No.6 are low-pressure rotor bearings,and No.4 and No.5 are highpressure rotor bearings. According to the actual installation status of the engine, the constraint conditions are applied to the casing model. For the bearing-casing coupling model of the engine, the casing is modelled by 3-D solid finite element and the bearing are modelled by a kind of 2-D finite element as Ref. 27.

The harmonic analysis is carried out by applying a constant rotating displacement excitation at the position of the bearing No.2 of the gas turbine engine.The amplitude of displacement excitation is x0=10-4m. The rated speed of high-pressure rotor is 13300 r/min, and that of low-pressure rotor is 10200 r/min. The frequency of the harmonic displacement excitation is set as 0-230 Hz by 1 Hz per step, covering the rotation frequency of the working range.

Fig. 4 Schematic diagram of the typical engine structure.

It will take a large amount of computing resources if the whole casing model is used, because of the large number of sample calculations together with the huge number of DOFs.Actually,The reduction method proposed by Ref.27 was used here to reduce the models, which can ensure accuracy and improve computational efficiency greatly.

3.2. Sensitive analysis at specific speed

Fig. 5 shows the amplitude-frequency response curve of the casing measuring point by changing the stiffness of the spring element at bearing No.2 with constant damping.It can be seen that when the bearing stiffness changes, not only the response amplitude changes but also some frequency positions of the response peaks change, which makes it very hard to conclude in word. Similar to the amplitude, the response phase changes complicated too.

Fig.5 shows the response change with only stiffness,which already being complicated to analyze. In order to analyze the influence with both stiffness and damping, the variations of response amplitude and phase under different rotation speeds are analyzed. Fig. 6 shows the curved surface of response amplitude varying with bearing stiffness and damping at three different rotation speeds corresponding to the speeds signed by blue dashed lines in Fig. 5. It can be seen that the response amplitude is affected by both bearing stiffness and damping with coupled effect at a special range. Besides, the curved surface is changing with rotational speeds and the response shows complex phenomenon to exactly describe.

Fig. 5 Amplitude-frequency response curves of casing measuring points under different bearing stiffness.

Fig. 6 Curved surface of response amplitude with bearing stiffness and damping at different rotational speeds.

Fig. 7 Relative change rate of response amplitude of casing measuring point with bearing stiffness.

Fig. 8 Relative change rate of response amplitude of casing measuring point with bearing damping.

In order to quantitatively analyze the influences, the indexes proposed in Section 2.3 are used. According to Eqs.(15) and (16), the relative change rate of response amplitude with bearing stiffness and damping are calculated under different rotation speeds.Figs.7 and 8 show the relative change rate of the response amplitude with bearing stiffness and bearing damping at 51 Hz, which is the idle speed frequency of the engine. Similarly, the relative changes of response phase are calculated according to Eqs.(17)and(18),as shown in Figs.9 and 10.

The boundaries of sensitive areas are showed by red lines in Figs. 7-10. They are roughly determined by the measured vibration, and can be adjusted according to the specific problems in the project,which is not universal or standard.Base on the sensitive analysis results and the requirement of engineering we suggest, if |Rk|≤0.5 (or |Rc|≤0.5), the response amplitude changes slowly with the bearing stiffness (or damping),and is not sensitive.If|Vk|≤10°(or|Vc|≤10°),we think that the response phase changes slowly with the bearing stiffness (or damping), and is not sensitive.

Fig. 9 Relative change of response phase of casing measuring point with bearing stiffness.

From Fig. 7, it can be seen that the sensitive area of response amplitude to bearing stiffness varying with damping.With the increase of bearing damping, the sensitive area of response amplitude to bearing stiffness decreases gradually.When the bearing damping increases to a certain value, the bearing stiffness has no sensitive area. Similar conclusions can be obtained from Fig. 8 for the sensitivity of response amplitude to damping.

From Fig. 9, it can be seen that the sensitive area of response phase to bearing stiffness varying with damping.With the increase of bearing damping, the sensitive area of response phase to bearing stiffness decreases gradually. When the bearing damping increases to a certain value, the bearing stiffness has no sensitive area.From Fig.10,it can be seen that the sensitive area of response phase to bearing damping varying with stiffness. When the bearing stiffness is small, there is only a negative correlation sensitive area. When the bearing stiffness increases,a positive correlation sensitive area appears simultaneously and changes towards to the negative one.When the bearing stiffness increases to a certain value, the bearing damping has no sensitive area.

In conclusion,the bearing dynamic coefficients have a coupling effect on the casing-bearing system.Only when the bearing stiffness and damping are in a certain range, can the response amplitude and phase of the system be affected.

3.3. Quantitative evaluation of double uncertain dynamic coefficients on casing response

Fig. 10 Relative change of response phase of casing measuring point with bearing damping.

Fig. 11 Variation of R*k with rotation speed.

The quantitative sensitive analysis in Section 3.2 give a roughly configuration of both stiffness and damping changing in a large range. But in a specified situation, the stiffness and damping are uncertain in a smaller range and obey a probability density function. Based on the results of Section 3.2, the quantitative evaluation method proposed in Section 2.4 is used.

For the bearing No.2 of the gas turbine engine,the empirical stiffness is about 1.25×108N/m based on test and numerical calculation. According to the engineering experience, the uncertain range of bearing stiffness is about ±30%of the empirical stiffness:

Fig. 12 Variation of V*k with rotation speed.

Fig. 13 Variation of R*c with rotation speed.

Fig. 14 Variation of V*c with rotation speed.

According to the above calculation results,it can be used as a reference for the selection of the engine balance speed region.It is suggested that the dynamic balance should be carried out in such a speed region, where the response of the casing measuring point is insensitive to the bearing stiffness and damping.At this time,the probability of successful dynamic balancing is higher.On the contrary,if dynamic balancing is carried out in the region sensitive to bearing stiffness and damping,the probability of successful dynamic balancing may be lower because the amplitude and phase are easily affected by slight changes in bearing dynamic coefficients.

4. Conclusion

In order to analyze the influences of uncertain bearing dynamic coefficients on bearing-casing system, a double integral method for quantitative evaluation of influence on the response amplitude and phase caused by bearing coefficient uncertainties is proposed. Taking a typical gas turbine engine bearing-casing model as the research object, the application analysis is carried out by using the proposed evaluation method, which can be summarized as follows:

(1) Based on the two-degree-of-freedom lumped parameter model of the bearing-casing coupling system, the coupling influence of both bearing stiffness and SFD damping on the response of casing measuring point together with the variation under different rotational speeds are analyzed. The sensitivity indexes of response amplitude and phase to the change of bearing dynamic coefficients at a specific speed are proposed. Accordingly, a double integral quantitative evaluation method is proposed.

(2) A typical engine bearing-casing complex system is analyzed as a typical application,the influences of uncertain bearing stiffness and damping on the amplitude and phase of engine casing measuring points at different rotation speeds are analyzed by the above-mentioned evaluation method. The rotation speed region in which the response amplitude and phase of the measured points are less affected by the uncertainty of the bearing dynamic coefficients is obtained, which can provide reference for the selection of the balance rotation speed of the engine.

In addition, the sensitivity analysis method of multiparameter coupling can also provide reference for the optimization of dynamic characteristics of actual bearing-casing system, which helps to predict the uncertainty of dynamic characteristics of the casing system and even the whole machine system at the design stage, and then select a reasonable parameter intervals to improve the reliability of the structural design of the whole engine system.

Acknowledgements

This study was co-supported by the Young Scientists Fund of the National Natural Science Foundation of China (No.51905025), the Joint Funds of the National Natural Science Foundation of China (No. U1708257) and the Fundamental Research Funds for the Central Universities (No. JD1911).