BENDING STIFFNESS AND DYNAMIC CHARACTERISTICS OF A ROTOR WITH SPLINE JOINTS

Transcription

1 Proeedings of ASME 0 International Mehanial Engineering Congress & Exposition IMECE0 November 5-, 0, San Diego, CA IMECE BENDING STIFFNESS AND DYNAMIC CHARACTERISTICS OF A ROTOR WITH SPLINE JOINTS ABSTRACT Wu Fayong Shenyang Aeroengine Researh Institute Shenyang, Liaoning, China, 005 Ma Yanhong Shool of energy and power engineering, Beihang University Beijing, China, 009 Spline is a ind of typial joint strutures widely used in the aero-engine rotor system. The stiffness distribution of the spool is affeted by the spline joint s stiffness diretly. Therefore, it is signifiant to investigate the spline joint s stiffness harateristis and its influenes on the dynami harateristis of the rotor system. To figure out the fators that affet the bending stiffness of the spline joint, a mehanial model whih taes ontat state into aount was built based on the struture analysis and fore state analysis. In addition, an atual rotor with a spline joint was established to measure the stiffness under different loads, and the results were ompared with the analysis by three modeling methods. Furthermore, the natural frequenies of the rotor were tested. Obtained results indiate that the strutural disontinuity, the surfae stiffness and the learane fit ause the loss of loal angular stiffness and linear stiffness. Meanwhile, the stiffness of the rotor with the spline joint is weaer along with the inrease of the load/deformation, and beome stable when undergoing the huge load/deformation. Besides, the loal blending stiffness is presented as nonlinear and unertainty subjeted to an interval, whih further affeted the dynami harateristis of rotor system. INTRODUCTION Joints are employed to hold two or more parts together to form an integrated rotor frame in aero-engines due to the requirements of assembly. Four types of joints are ommonly used: bolt, pin, end-tooth and spline. Among these, spline is usually used to onnet two spools with disassemble property. However, the mating ontat of the joint hanges with the loadings, espeially the bending moments. It an result in the bending stiffness variation, affeting the dynami harateristis of the rotor system. In the traditional Liang Zhihao Shool of energy and power engineering, Beihang University Beijing, China, 009 Zhang Dayi Shool of energy and power engineering, Beihang University Beijing, China, 009 rotordynami analysis, the omplex geometri features and the ontat harateristis are usually ignored while the joint is modeled as a whole struture. As a result, some fators, suh as ontat, rub, fretting wear, whih may lead to inaurate alulations annot be desribed. The main reason is the disontinuous of the joint strutures. In reent years, studies on the missile [, ], a non-rotating mahine, found that the stiffness of joints influened the dynami haraters. The stiffness of the missile would derease 0-0% due to the joint. Consequently the modal frequeny and the modal shape would be hanged as well. And the gap between the surfaes may ause the non-probabilisti distribution of the mehanis parameters. It should be noted that numerous researhes have been done onerning about the spline joint in the rigid rotor of the ground rotating mahinery. Nevertheless, these studies were mainly about the stability and exiting fore due to the misalignment. While for the high speed flexible rotor whih is usually woring above the bending ritial speed, the influene aused by loal stiffness is more signifiant. Yin et al. [], proposed a onept of urvi oupling beam element to model the urvi joint in the turboshaft engine. Aording to his researh, the stiffness of urvi oupling beam element drops 70%, whih an hange the modal shape and frequeny. Mei and Li [4], and Kang and Cao [5] found the fit looseness of spline may inrease the flexibility of the oupling, and studied the damping and stability. Li and Ma [6], and Liu et al. [7] studied the dynami effets of the stiffness and ontat state of the rotor joints, mainly about the bolted joint, based on nonlinear finite-element method and experiments. The sensitivity analysis of ritial speeds and vibration modes show that the stiffness and ontat state vary with external loads and geometry strutures, and affet the rotor system operating. Reently, researhes [8-] about the analysis method and mehanial harateristi of spline joint have made a lot of progress, and they are mainly foused on the ontat state, fretting wear and the fretting fatigue. However, Copyright ASME 0

2 these studies do not provide muh attention to stiffness haraters of spline joint and its influene to the rotor system. The sope of the urrent researh lies in the mehanism of stiffness loss and how does it affet the dynami haraters of the rotor system. In this paper, the main purpose is to investigate the stiffness mehanism of the spline joint and its influenes on the rotor system. A stiffness analytial model of spline joint was built, explaining the mehanism of the stiffness loss. The influenes on the angular stiffness and the linear stiffness due to the strutural parameters and the external load were analyzed. A atual low pressure spool with the spline joint was employed to measure the stiffness and the nature harateristis. A parametri modeling method based on ANSYS software was proposed to model the spline joint. Then the test results of spool s stiffness were ompared with two types of finite element models, ontat element model and parametri model. At last, the distribution harateristis of the stiffness and the modal frequenies were analyzed. TYPICAL STRUCTURES OF SPLINE JOINTS A typial rotor system, whih onsists of omponents with distint funtions and different materials, ommonly used in high thrust ratio turbo fan engine is shown in Figure. Bolted joints and spline joint are used to assemble the omponents, meanwhile, they transmit torque and energy from the turbine to ompressor. It is important to point out that the basi differene between the joint and the integral struture is the ontinuity of the interfae. The interfae ould merely endure pressure stress, but tensile stress. And the ontat stress is losely related to the ontat state, whih may be far open, near ontat, sliding and stiy. Figure Cross setion diagram of dual rotor system For different design requirements, many types of spline joints are proposed. Based on the number of the teeth, they an be divided into two ategories: single set of teeth and double sets of teeth, as shown in Figure. The two sets of teeth loates one behind the other in Figure (b), whih is named tandem teeth spline joint, while the joint in Figure () is named parallel teeth spline joint. a. single set of teeth In the single set of teeth shown in the Figure (a), fan shaft and LP shaft are diretly onneted. A bearing loated at the position of the spline is designed to redue the deformation. But for the reason of misalignment and bending vibration, there is still angular deformation. Figure (a) also shows the bending moment is supported by two pairs of ylindrial entering surfae: A and B. In the tandem teeth spline joint shown in Figure (b), the fan shaft and LP shaft are onneted by a onnetion shaft. The left spline onnets with the fan shaft, entering by surfae A and B. The right spline onnets with the LP shaft, entering by surfae C and D. In the parallel teeth spline joint of Figure (), the outer spline onnets with the fan shaft, while the right spline onnets with the LP shaft. As it an be observed, when the struture is omplex, the mehanial harateristis are omplex along. This paper studies the harateristis of the spline joint with single set of teeth, and fous on the bending stiffness. BENDING STIFFNESS OF SPLINE JOINT b. double sets of tandem teeth. double sets of parallel teeth Figure Different types of spline joints Based on the mehanial analysis, the strutural harateristis of joint strutures were simplified, the model of mehanial harateristis of the bending stiffness was established, and the loal stiffness harateristis of joint strutures were analyzed in this hapter. Sine there are ontat interfaes between joint strutures, rather than ontinuous bodies, the law of strutural stiffness variation annot be desribed by the setional harateristi-ei, while the deformations in the radial diretion and the angular diretion are unoordinated. Therefore, the resistane ability to angular deformation and linear deformation of joint strutures should be desribed by linear stiffness and l Copyright ASME 0

3 angular stiffness. They are defined as the fore (torque) needed by a unit displaement (angle). l F / x () M / In order to ompare the stiffness of joint strutures and the stiffness of integral onsolidation strutures, dimensionless parameters - the equivalent linear stiffness l and equivalent angular stiffness were used for analysis, whih is defined as the ratio of stiffness of joint strutures and stiffness of integral strutures. Consequently the equivalent stiffness ould express the degree of the impat of strutural ontat state on loss of stiffness. l l / l0 () / 0 Fore state analysis A LP flexible rotor system in an aero-engine, often wors above the bending ritial speed, hene when the struture is bending and deforming, the fore state of the joint struture withstanding the bending load as it is shown in the Figure (a) below. The left end of the LP rotor is fored by axial pre-tightening fore F of lonut, and the right end of the LP shaft is fored by the bending moment M. When the engine is running, the fan and turbine omponents withstand the forward and baward enormous aerodynami loads, respetively, thus the tightening fore is heavier, and the ontat state is basially stable. The ontat state of the entering surfae B has a main influene on the bending stiffness of the struture, under the lateral (bending moment) loads of the different degrees, as shown in (b). The ontat area of the entering surfaes is different, so is the ontat pressure, hene the bending stiffness is also different. bending stiffness analysis, stiffness analysis model of the spline joint struture was built, with the basi assumptions as following: () Beause of the learane between the teeth, the effet of tooth to struture bending stiffness is negleted. () The lo nut and shaft are assumed as an entirety. () The left entering surfaes are displaement oordination in radial diretion. (4) The right entering surfaes are ontat surfaes, and there are both elasti deformation and axial sliding due to strutural disontinuity. Based on these, we establish the mehanial model of the spline joint struture whih is shown in the Figure 4. The deformation of the struture is omposed of three parts: ) the bending deformation of the inner shaft; ) the bending deformation of the outer shaft; ) deformation of the ontat interfae. Thus the stiffness also onsists of three parts: the outer shaft stiffness, the inner shaft stiffness and the ontat stiffness. Then and an be desribed by EI and length L, while the an be desribed based on the ontat mehanis. Figure 4 Mehanial model of the spline joint First, in the mehanial harateristis analysis model, to analyze displaement inoordination of the inner and outer spools, the left end of the interfae was fixed, as shown in Figure 5. a. fore ondition of spline joint b. stress ondition of surfae B Figure Seth of analysis model for the spline stiffness Stiffness analysis model Based on the fore analysis and the requirements of the Figure 5 Displaement ondition of spline joint When the right end of the inner shaft withstood the moment M, the orresponding linear displaement and angular displaement are: FL w EI () F x ML FL w EI EI ML FL EI EI (4) where w is the radial deformation of the outer spool, w is the radial deformation of the inner spool, x is the Copyright ASME 0

4 relative deformation of the right ends of the outer and inner spools, is the angular deformation of the inner spool, and is the ontat stiffness. By the radial deformation oordination onditions of the inner and outer shafts and ontat interfaes w w x, it an be derived that M L w EI EI L L L E I M 4L L 4 E I E I L (5) In the same way using the definitions of linear stiffness and angular stiffness, they an be written as M EI l Lw L L EI (6) M EI 4L L 4 L EI EI L By substitution of the linear and angular stiffness of inner and outer shafts into abovementioned equations, respetively, it an be obtained that M l l Lw l (7) M 4 4 L where l and represent the stiffness of the outer spool, l and represent the stiffness of the inner spool respetively. These stiffness an be derived through the formulas of mehanis of materials. Stiffness loss In order to validate derived formulae, the equivalent stiffness of the three states: integral onsolidation, the stiff ontat surfae and the elasti ontat surfae are analyzed, as it is shown in Table. Contat of surfae is not onsidered in integral onsolidation, and the stiffness an be derived through the formulas of mehanis of materials. The axial slide of ontat surfae is onsidered in the stiff ontat surfae whose ontat stiffness is muh higher than the shaft stiffness. In alulation, the ontat stiffness is more than two orders of magnitude from. In Elasti ontat surfae, both slide and deformation in radial diretion are onsidered. The value of ontat stiffness is similar to the shaft stiffness. is given the same value with in alulation to analyze the effet of stiffness loss qualitatively. Table Equivalent stiffness of three states States Stiffness l Integral onsolidation Stiff ontat surfae l ; 00l Elasti ontat surfae l It an be seen that, due to the disontinuity of the joint struture interfae, the deformation of the inner and outer spools annot oordinate in eah diretion, he angular stiffness loss is 0% in the ideal state. Further onsider the ontat stiffness of the ontat surfae, the redutions of linear and angular stiffness result in stiffness loss. It is revealed that both the disontinuity of the joint struture interfae and the ontat state variation may weaen the struture stiffness. Influene by learane The spline struture is onsidered non-ontinuity whih auses loss of stiffness in the above analysis, while at the same time the atual struture of the mating surfaes presents a ertain learane. In this setion influene of the fitted state on the stiffness was studied and examined. When the struture is exposed by the bending deformation, there are two stages: the first stage for the entering surfaes are not in ontat, then the stiffness is equivalent to the bending stiffness of the inner shaft. The seond stage when the two entering surfaes ontat with the load inreasing, the stiffness need to be onsidered as the stiffness of two spools and the gap between them. The total stiffness is determined by the inner and outer spools stiffness together with the ontat stiffness. The stiffness of the spline struture is defined as follows: l w< l l w (8) l w< w 4 4 L (9) Based on pratial experiene, the ontat surfae of the joint struture usually wears beause of the effet of alternating load,. Fretting damage of the entering surfaes ould generate or enlarge gaps, that in turn further weaen the stiffness of the joint struture. Therefore the stiffness harateristis of the same struture with same external onditions are not stati, with ertain variation ranges, whih illustrates that the stiffness of the joint struture also has the harateristis of interval distribution. When the gap is suffiiently great, the rotor will exhibit the harateristis of the different enters, leading to addition of the misalignment inentive loads on the system. 4 Copyright ASME 0

5 Stiffness[0 6 N/m] Load Low Pressure Spool Figure 5 TEST OF ROTOR WITH A SPLINE JOINT In the last setion we proposed how the loal stiffness of spline joint was hanging. A real rotor with the spline joint was used to test the stati stiffness and modal harateristi, and the results were ompared with the result of parametri model built based on the FEM method. Test rotor Support Support Spline Joint Assembly Test rig of a LP rotor with spline joint () The range of stiffness and divergene beomes wider with a maximum of stiffness,.9%, when there is a small load/deformation. () The stiffness of the spline joint seem to be more stable at big load/deformation, for both the range and the deviations of the stiffness are narrow when the fore is bigger than 900N Experimental Data Fitting Curve The stati stiffness and the modal harateristis of an aero-engine LP rotor (without blades) were measured, as shown in Figure 5. It ould be seen, that the supporting points were loated at the bearings near two ends of the rotor. And the load point was set at the bearing near the spline joint. The load is loaded by weight, onneting to the rotor by steel wire. 0 dial indiators are arranged along the axial diretion of the rotor to measure the lateral displaements. Experiments on stati stiffness The displaement measuring point was set at the load point sine the stiffness harateristis of the whole rotor an be desripted here. Fourteen sets data are shown in Table. The load was varied in range N. The results of stiffness test are plotted out in Figure 6 (The urve is the fitting result). Table Test results of stati stiffness Fore [N] Displaement [ μm ] Stiffness [0 6 N/m] Standard deviation [0 6 N/m] From the test result, we an onlude that: () The bending stiffness of the rotor dereases for the load s inreasing Nonlinearity Derease Steady Figure 6 Fore[N] N/m.9% of max stiffness Test results of stati stiffness Comparison between simulation and experiments Three modeling methods for entering surfae were proposed to simulate it based on the FEM software, as shown in Figure 7. () Integral onsolidation model: the outer and inner ylindrial entering surfae is fixed. () Detailed ontat model: based on nonlinear ontat elements. Beause the surfae is fae-fae ontat, ontat elements, TARGE70 and CONTA74 were seleted. The oeffiient between the metal material was seleted for the oeffiient of sliding frition, the value is 0.. () Parametri model: parameters of onneting part materials were optimized. The modulus of elastiity was assumed to be 70% ompared with analysis above. Contat 7 stiffness: 0 N/m. The parameters of materials used in the FEM models are listed in Table. Table Material parameters Part Material [ g / m ] E [ GPa ] fan TC turbine steel Copyright ASME 0

6 Displaement[mm] Figure 7 Modeling methods for spline joint Test Data Fixed Model Contat Model Parametri Model Loation(mm) Figure 8 Comparison with simulation and experiments Spring elements were used to simulate the bearing stiffness of the rotor at both ends, and the value of 8 stiffness: 0 N / m. Then stati analysis of rotor was Model Frequeny[Hz] performed. Figure 8 shows the omparison between the order defletion urves from above three models under 000N load and experiments. Based on the ahieved results, it is lear that the integral onsolidation model has the highest disagreement with experiments. Beause effets on stiffness from the joint struture were ignored. For omparison, the other two models are highly aordant with the tests. Anyway it should be noted, that the ontat model is diffiult to build and time-onsuming to alulate. However, parametri model ould solve this problem and be more easily appliable in simulation analysis of rotordynami problems. -000N 0 000N EXPERIMENTS ON MODAL CHARACTERISTICS In the same way, to understand the effets of the ontat state of onnetion parts to the rotordynami properties, various deformations were simulated by applying different transverse loads (none, 000N up, 000N down). The modal harateristis of the rotor was alulated by ANSYS. The transverse vibration frequeny up to 4th order and the divergene distribution are presented in Table 4. Table 4 Modal results and divergene distribution f [Hz] Divergene [%] Obtained results have shown that the distribution of divergene of st order is the widest, up to 8.90%. The divergenes of the other orders are smaller, but still more than %. It should be noted, that these divergenes originate from the hanging stiffness, whih is from unstable ontat state of onneting part. It is evident, and it will play an important role in the rotordynami analysis. CONCLUSION AND PROSPECT In this paper we explored the mehanism of the stiffness loss by building a loal bending stiffness analysis model. Based on omparison of the experiments and finite elements simulation results, it ould be onluded that: () There is a signifiant effet of the spline joint on the loal stiffness of the rotor, the loss of loal angular stiffness 6 Copyright ASME 0

7 and linear stiffness reah 0% and 6% respetively, whih is aused by the strutural disontinuity and the surfae stiffness. Besides, the learane fit will further inrease the stiffness loss. () The stiffness of rotor with spline joint strutural will be weaer along with inrease of load/deformation. The flutuation of stiffness of a rotor will exeed 0% for a small load/deformation. However, the stiffness of the rotor will beome stable when the rotor undergoes the huge load/deformation. () There are also some divergene distributions for modal harateristis leading by the unertainty of spline struture of loal ontat states. The divergene of lower order modeling frequeny reahes 0%. This was verified by the aero-engine low pressure rotor modal test. (4) The parametri model of the spline struture loal stiffness ould desript the real stiffness features more effetively and will be more effiient and easily appliable in the dynami simulation of the rotor system with the spline joint as a result. In onlusion, we also suggest that the ontat stiffness, frition damping and sliding deformation would affet the preision of stiffness analysis model of spline joint struture. It is still an open topi about the dependeny between the spline struture stiffness and its nonlinear variation relation from the oordinate parameters and load parameters. In the further study, we will mae an aent on the influenes of rotordynami harateristis due to the spline joint struture. As a onsequene, the determination of the distribution interval dispersion also needs further exploration. The model we ve proposed here is also far from optimization. And at the same time, it is believed that this researh will help to improve the reliability and stability of the rotor systems. NOMENCLATURE E F I M l l, elasti modulus fore moment of inertia bending moment linear stiffness angular stiffness ontat stiffness equivalent stiffness l0, 0 stiffness of integral strutures l, stiffness of outer spool l, stiffness of inner spool w radial deformation of outer spool w radial deformation of inner spool x relative deformation of two spools angular deformation of inner spool poisson ratio ACKNOWLEDGEMENTS The paper is supported by Aeronautial Siene Foundation of China (05). REFERENCES [] Gao J. H., 005, The study for the model tehnology of engine asing and imitative alulation, Master thesis, Beihang University, Beijing, China. [] Wang J. M., Li G. D., Huang W. Y., 006, The researh for the dynami harateristis of the missile with the jointed struture by the test method, Struture and Environment Engineering, (), pp [] Yin Z. Y., Li G. D., Hung W. Y., 99, Stiffness of a shaft setion with urvi ouplings and its effet on dynami harateristi of a rotor, Journal of Vibration Engineering, 6(), pp [4] Mei Q., Li N., 008, Dynami harateristis analysis and test of an elasti oupling, Journal of Vibration and Sho, 7(6), pp. 8-. [5] Kang L. X., Cao Y. H, 00, Dynami instability of heliopter transmission rotating shafts with spline oupling, Journal of Beijing University of Aeronautis and Astronautis 6(6), pp [6] Li J. H., Ma Y. H., 009, Dynami design method of spline joint struture for rotor system, Aeroengine, 5(4), pp [7] Liu S. G., Ma Y. H., Zhang D. Y., 0, Studies on dynami harateristis of the joint in the aero-engine rotor system, Mehanial Systems and Signal Proessing, 9, pp [8] Leen S. B., Hyde T. H., Ratsimba C. H., 00, An investigation of the fatigue and fretting performane of a representative aero-engine spline oupling, The Journal of Strain Analysis for Engineering Design, 7(6), pp [9] Ratsimba C. H., MColl I. R., Williams E. J., 004, Measurement, analysis and predition of fretting wear damage in a representative aeroengine spline oupling, Wear, 57(), pp [0] Ding J., Sum W. S., Sabesan R., 007, Fretting fatigue preditions in a omplex oupling, International Journal of Fatigue, 9(7), pp [] Limmer L., Nowell D., Hills D. A., 00, A ombined testing and modelling approah to the predition of the fretting fatigue performane of splined shafts, Proeedings of the Institution of Mehanial Engineers, Part G: Journal of Aerospae Engineering, 5(), pp [] Bannister R. H., 980, Methods for modeling flanged and urvi ouplings for dynami analysis of omplex rotor onstrutions, Journal of Mehanial Design, 0(), pp Copyright ASME 0