Time-varying Stiffness Characteristics of Shaft with Slant Crack

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1 Internationa Conferene on Modeing, Simuation and Appied Mathematis (MSAM 05) Time-varying Stiffness Charateristis of Shaft with Sant Cra Hengheng Xia Shoo of Aeronautia Manufaturing Engineering Nanhang Hangong University Nanhang, China Yaoxian Xiao Shoo of Aeronautia Manufaturing Engineering Nanhang Hangong University Nanhang, China Zhinong Li * Shoo of Aeronautia Manufaturing Engineering Nanhang Hangong University Nanhang, China Abstrat The stiffness matrix of a shaft with sant transverse ra under the ommon ation of axia fore, moment and torque was derived based on materia mehanis, oa fexibiity and frature mehanis theory and appy strain energy reease rate method. The opening and osing reguarity of the ra and the shaft rigidity varying harateristis was simuated in a steady state rotating by appy the stress intensity fator is zero method, and the time-varying harateristis of inination of 5 raed shaft stiffness was detaied study. The studies have shown that, the stiffness of shaft has not been redued, but has some breathing harateristis of futuations with inreasing ra depth in the steady state ra axis of rotation ye. And the ouped vibration was easiy aused so that dynami harateristis of shaft is more ompex. eywords-sant-raed shaft; shaft stiffness; time-variant harateristis I. INTRODUCTION Noninear dynami harateristis of a raed shaft is mainy affeted by the time-varying stiffness of the shaft, us estabishing reasonabe mode for auating ra shaft stiffness is one of the important ontent to study the dynami harateristis of the shaft ra. In fat, sine many of the rotor systems are used to transmit torque, so sant ra produe inevitabe. And how to auratey and preisey desribe the rigidity of sant ra rotor system is the ey to studying the harateristis of sant ra rotor system dynamis and the foundation. The study of sant ra rotor system began in twentieth Century 0 time, and have made some progress [-9. Lin Yani [5- did the researh based on stress intensity fator is zero method and made out the infuening of oad and ra depth on shaft stiffness.darpe[9 anaysis additiona fexibiity and stiffness matrix of arbitrary ange ra rotor under the effet of inds of oads, but he did not wor out the infuene of ra opening and osing of a shaft stiffness when the rotor is rotate.however, the iterature [- did not onsider the open-ose reguarity of ra and the time-varying harateristis of rotor stiffness in a steady rotation ye. Obviousy, the hange rue of ra shaft stiffness has the very big differene in the stationary state and rotating ondition. Though the iterature [9 has been onsidered the ra opening and osing aw and time varying stiffness harateristis in steady rotation ye, however, this is ony a diagnosis of sant ra rotor of 5, and ignore the infuene of ra ange of sant ra rigidity, has obvious shortage. This paper derivates stiffness matrix by the way of strain energy reease rates whih was proposed by Dimarogonas, using the fator that Stress intensity fator is zero,whih was proposed by Darpe to simuate the ra breathing properties, and on this basis, study the ra opening and osing aw and time varying stiffness harateristis of the 5 sant ra shaft in steady rotation ye. II. SLANT CRAC MODEL O THE ROTOR SYSTEM Consider a shaft of radius R with a sant ra having a depth of α oriented at ange of β reative to the axis of the shaft (ig.(a)).ig.(b) shows the detais of the ross-setion showing open and osed area to the ra above the ra edge,the shaft is oaded with axia fore p, bending moment p and p and torque p. The opening mode stress intensity fators are given as beow: P P : Ⅰ P ( + os β ) πa sin β πa P Y P : Ⅰ ( + os β ) πa () () 05. The authors - Pubished by Atantis Press 5

2 P h P : Ⅰ ( + os β ) πa P h sin ( β ) πa h P Y + Ⅰ sin ( β ) πa P : () () Δ ij W q q i j () Where s is no ra shaft fexibiity and is expressed as: s diag EI EI EA exibiity matrix an be expressed as foows: EA () Therefore, the tota stress intensity fator is given by foowing expression: Ⅰ Ⅰ Ⅰ Ⅰ Ⅰ + πa Among them, assuming, aording to the h iterature [0, three infuene oeffiients is given by foowing expression: ( ( α / ) + 0.7( sin ) ) tan h [ ( sin ) tan tan os os Where J is strain energy density funtion and is expressed as: [ + ( v J + Ⅰ ) (5) () (7) () (9) (0) Suppose A for ra opening area (The first stress intensity fator Ⅰ > 0 ), the ra rotor reease of strain energy an be expressed as: [ αsin + ( + ν ) α sin β da R Ι os π [ y αsin Ι + y ( + ν ) α sin β os E [ h αsin + ( + ν ) h α sin β os A sin d [ h α + y α sin β os ΙΙ + α ( + ν ) os da yαsin Ι + hsin βα ysin β os βα Ι + hsin hsin β sin βα β os βα Ι + hsin β os β osβα ysin βαh Ι + ysin βα sin yhsin β os βα Ι + sin βα sin βh hyα sin β os β os + h α sin β osβ os Therefore, ra shaft fexibiity matrix as foows: ( ) W Ⅰ dαdy + A A + v E dady () Using Castigiano theorem, the additiona fexibiity an be expressed as: (5) The stiffness matrix for inverse matrix fexibiity matrix:

3 () By no ra shaft fexibiity of s an get no ra shaft stiffness matrix s, Then the matrix an be obtained by dividing the s matrix with whih is expressed as: III. s diag EI EI s EA EA OPENING AND CLOSING CHARACTERISTICS O SLANT CRAC SECTION AND STINESS ANALYSIS (7) () igure shows that, when the ra depth is shaow, the raed shaft stiffness variation of the main diretion is not obvious, with the inrease of ra depth, the stiffness have a ertain deine, when the ra depth is deeper, the axia diretion ra stiffness derease maximum, axia and torsion diretion stiffness hanges are symmetria about ange 0, whie the two bending stiffness is not symmetria about ange 0, and two bending stiffness variation range is about the same, ra torsion diretion stiffness hange trend is more ose to the axia stiffness; the stiffness hange of axia,torsion and transverse bending diretion were dereased first and then inreased, whie the stiffness( ) of bending diretion with the above three diretion stiffness hange trend is different, it is first dereased and then have a modest rise then hit bottom, finay returned to the origin of the trend. Due to the rotation of the rotor wi generate a ertain degree of vibration, whih produes the ouping stiffness; it is onuded that the bending ouping stiffness ( ) and the ouping stiffness ( ) is amost zero through auation and anaysis, so did not give the two diretion stiffness variation map; an be seen from the graph,, the ouping stiffness hange is not obvious when the ra depth is shaow, with the inrease of ra depth, the hanges graduay beome obvious; axia and bending diretion of the ouping stiffness ( ) as the rotation of the rotor ange hange no onger has been positive, but futuate around zero, simiar to sine urve hanges, axia stiffness and bending stiffness ouping ( ) between ange of 0 and 0 trend simiar to the main diretion of and hange, but in the 0 to 0 is a sma bump. the ouped stiffness ( ) hanges trend of axia and torsion diretion are simiar to the main diretion of and with the hange of ange in a ompete ye, just the vaue for the negative, ouping stiffness of bending and torsion diretion ( ) aong with the rotation Ange hanges are simiar to the sine urve, and ie the ouping stiffness ( ), but the hange range is smaer; Can aso see that the ouping stiffness of a the diretion are zero when the rotation of the rotor ange is 0 degree in the initia moments, the above anaysis an now that more ouping vibration wi mae ra rotor dynami performane more ompiated. IV. CONCLUSION This paper is anayzed from the time-varying harateristis of sant ra: The raed shaft stiffness of the main diretion aong with ra ange is not obvious under the ra depth is shaow when the ra shaft to rotate, the stiffness of the main diretion have a degree of deine with the inrease of ra depth, the stiffness of axia ra diretion deine the argest when the ra depth is deeper; the stiffness hange in the diretion of axia and transverse bending and torsion were dereased first and then inreased; for ouping stiffness, the stiffness hange is not obvious when the ra depth is shaow, but the hanges graduay beome obvious with the inrease of ra depth, the ouped stiffness( )hange of axia and bending diretion is simiar to sine urve with shaft ange hanges, the ouped stiffness( )hange of bending and torsion diretion is simiar to sine urve with shaft ange hanges, a the ouping diretion stiffness at the initia moment stiffness is zero. ACNOWLEDGEMENT This wor was finaniay supported by the Natura Siene oundation of China (5509, 50, 50757, ) and the Graduate Innovation ound of Jiangxi Provine (YC0-S9). REERENCE [ Ihimonji M and Watanabe S. The dynamis of a rotor system with a shaft having a sant ra. JSME, Internationa Journa, Series III 9,(), 7-7. [ Ihimonji M, azao Y, Watanabe S and Nonaa S. The dynamis of a rotor system with a sant ra under torsiona vibration. Noninear and Stohasti Dynamis, 99,7: -9. [ Prabhaar S, Sehar A S and Mohanty A R. Transient anaysis of a sant-raed rotor passing through its fexura the ritia speed. Mehanism and Mahine Theory, 00,7(9): [ Sehar A S, Mohanty A R and Prabhaar S. Vibrations of raed rotor system: transverse ra versus sant ra. Journa of Sound and Vibration, 005,79:0-7. [5 Lin Y, Chu. The dynami behavior of a rotor system with a sant ra on the shaft[j. Mehania Systems and Signa Proessing, 00, (): [ Yani Lin, uei Chu, Numeria and experimenta investigations of fexura vibrations of a rotor system with transverse or sant ra, Journa of Sound and Vibration, Vo., Nos.-, 0 Juy 009. [7 Lin Yani. Dynami harateristis of the rotor system with a sant ra on the shaft[d.beijing: Tsinghua University,00. [ Lin Yani, Chu uei, Hao Rujiang, et. Dynami Anaysis of a Jeffott Rotor System with an Opening Sant Cra on Its Shaft[J. Journa of Vibration and Sho, 00, 7(): 5-9. [9 Darpe A. Dynamis of a Jeffott rotor with sant ra[j. Journa of Sound and Vibration, 007, 0(): -. 7

4 [0 A Darpe, Gupta, A Chawaa. Couped bending, ongitudina and torsiona vibrations of a ra rotor [J. Journa of Sound and Vibration, 00,9(-):-0. (a) IGURE I SLANT CRAC SHAT AND SLANT CRAC SECTION (b) (a) Ange of rotation (b) Ange of rotation () Ange of rotation (d) Ange of rotation

5 (e) Ange of rotation (f) Ange of rotation (g) Ange of rotation (h) Ange of rotation IGURE II CRAC SHAT STINESS CHANGES WITH THE ROTATION ANGLE AND THE DEPTH O THE CRAC 9

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