1、94FTMI Fatigue Analysis of Shafts for Marine Gearboxes by: E. William Jones and Anying Shen, Mississippi State University and Robert E. Brown, Caterpillar, Inc. American Gear Manufacturers Association TECHNICAL PAPER Fatigue Analysis of Shafts for Marine Gearboxes E. William Jones and Anying Shen, M
2、ississippi State University and Robert E. Brown, Caterpillar, Inc. The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract The design of shafts for marine gearboxes. whi
3、ch may include the effects of torsional vibration, is presented. The influence of the vibratory torque on the values of shaft diameter and safety factor is discussed. Use of the Finite Element Method to evaluate unknown stress concentration factors is illustrated. A program for the design of shafts,
4、 which are subjected to fatigue, has been developed. Copyright 1994 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 1994 ISBN: 1-55589-635-9 FATIGUE ANALYSIS OF SHAFTS FOR MARINE GEARBOXES E. William Jones, Ph.D., P.E. Anying Shen, Graduate St
5、udent Department of Mechanical Engineering Mississippi state University, Mississippi State, MS 39762 Robert E. Brown, Project Engineer-Gear Design caterpillar, Inc. NOMENCLATURE Ma alternating reversed-bending moment, lb in mean bending moment, Ib in alternating shaft torque, Ib in mean shaft torque
6、, Ib in alternating axial force, Ib mean axial force, Ib alternating shear force, Ib mean shear force, Ib alternating normal stress, psi mean normal stress, psi alternating shear stress, psi mean shear stress, psi von Mises equivalent stress, psi cross section area, in2 moment of inertia of shaft, i
7、n4 polar moment of inertia of shaft, in4 radius of the shaft, in shaft outside diameter, in shaft inside diameter, in fatigue stress concentration factor in bending fatigue stress concentration factor in torsion fatigue stress concentration factor in tension modifying factor for stress concen tratio
8、n theoretical stress concentration factor in bending theoretical stress concentration factor in torsion theoretical stress factor in tension concentration notch sensitivity factor for bending 1 qt notch sensitivity factor for torque qp notch sensitivity factor for tension k shaft fatigue limit modif
9、ication ka shaft surface finish factor kb shaft size factor kc shaft reliability factor kd shaft temperature factor ke shaft life factor kg shaft miscellaneous factor Sfe shaft fatigue (endurance) limit of polished, unnotched test specimen in reverse bending, psi Stl shaft ultimate tensile strength,
10、 psi Sf shaft fatigue limit, psi Sy shaft tensile yield strength, psi Fs shaft factor of safety for fatigue Fs shaft factor of safety for yielding Ws shaft rotation frequency Wv vibration frequency phase angle INTRODUCTION The traditional method for designing shafts for rine gearboxes is to use the
11、ABS formulaJ. Although it is easy to use, it has some drawbackse First, the vibrato ry torque is not included in the calcula tion even though it is often evaluated by torsional analysis and its magnitude is the major contributor to the value of the AGMA application factors, Ka and Ca“ Second, the st
12、ress concentration factors, surface finish factor and size factor, which may vary significantly from shaft to shaft, are not treated as independent design variables_ This paper presents shaft design formula which explicitly includes vibra tory torque and stress concentration factors. A sensitivity s
13、tudy of the influence of the vibratory torque on the shaft diameter and safety factor is pre sented. A computer program, which is written in C, is developed to facilitate the use of this method of shaft fatigue analysis. An example of stress concen tration factor evaluation by the Finite Element Met
14、hod is given to illustrate the modeling procedure. SHAFT LOADING The shaft load includes the bending moment (M), shaft torque (T), axial force (P), and horizontal shear force (V). If the forces due to longitudinal and trans verse vibrations are neglected, the M, P, and V are linear functions of T. T
15、he maximum shaft torque may be ex pressed as T=Tm+Ta (1 ) The amplitude of the shaft torque, T(t), at any time tis: where Tm = mean torque, Ta = alternating torque, Wv = vibration frequency, = phase angle. (2 ) The value Ta may be primarily due to torsional vibrations, which are functions of the pri
16、me mover and driven machinery. The amplitude of the vibrating torque may be computed by torsional vibratin analy sis. The Military standard 167 J gives limiting values for vibratory torques, which suggests values for Ta for this class of vessels. The application factors Ca and Ka used in calculation
17、 of compres sive and bending stresses for gear teeth are load modifiers, which account for the increase in tooth load due to external effects including torsional vbrations. According to Det Norske Veritas 1, “Gener ally, the torsional vibrations shall not cause the maximum cyclic torques to exceed t
18、he approved Ka times the approved rated torque. Further, the highest permissible application factor for gears is Ka = 1.5. Note the application factor used in this connection only refers to the influence of torsional vibrations.)“. Hence, when the gear designers select the values of Ca and 2 Ka the
19、values suggest magnitudes for the alternating torque Ta. The torque due to torsional vibra tions depends on system parameters and is not uniformly distributed across the system, but the system is usually ill defined when the gearbox is being de signed. For a stationary shaft, the bending moment is a
20、 linear function of torque (3 ) where Cl is a constant. strictly speaking, Equation(3) is true only for a constant torque T. If the torque T includes vibratory torque, the value of the bending moment M is no longer the simple function of torque T, then C1 will depend on the system characteristics an
21、d the frequency of the torque Ta. For a rotating shaft, the moment may be expressed as (4) where Ws = shaft rotation frequency. Substituting (2) into (4) yields: M = Cl(Tm+Tasin(“vt+) )sin(“st) (5 ) The vibration frequency Wy is usually 0.5 to 12 times that of shaft rotation fre quency ws. is not al
22、ways O. M varies with frequency and phase angle. The fol lowing formula gives the bending moment for the worst possible combination. For the rotating shaft the mean component of moment, Mm is usually zero. Mm=O (7) Hence, the alternating moment, Ma is: Ma = CITm+CITa (8) Equation(8) implies that if
23、vibratory torque Ta is 25% of mean torque Tm then Ma will increase to 125% of the value produced by the mean torque acting alone. If axial vibration is neglected, the axial force P is also a linear function of torque T (9 ) where C2 is a constant. Therefore (10) If vibratory torque Ta increases 25%,
24、 then Pa increases 25%. DISTRIBUTION OF STRESSES IN SHAFTS Figure 1 shows the distribution of four types of stresses over a cross sec tion of the shaft. From left to right these stresses are: normal stress due to bending, normal stress due to axial force, shear stress due to torsion, shear stress du
25、e to bending. CT= o I 0“:- .E rA VQ 1b=jb Neutral Axis Figure 1. stress Distribution in the Shaft The value of Q and b for a solid circular-section shaft per Figure 2 are: Q = (R2_y;)3/2 b = 2VR2-y; (11 ) For a hollow shaft per Figure 2 with Yl r, Q (R2-yf)3/2_ (r2_y;)3f2j (12) b = 2 (VR2-y; -Vr2-y
26、) For a hollow shaft per Figure 2 with Yl r, 3 (13) where R = shaft outside radius, r = shaft inside radius, Yl = distance from the center to the point where the shear stress, Tbl is evaluated. (see Figure 2). B r R Figure 2. Shaft Cross Section For this complex stress state, the distortion-energy t
27、heory will be used to obtain the equivalent von Mises stress. Generally, the maximum equivalent stress occurs at the point A in Figure 2 because the bending stress is dominant. However, for a very short shaft, the maximum equiv alent stress may occur between point A and B. An example is given in App
28、endix A. DERIVATION OF SHAFT DESIGN FORMULA Given the shaft alternating bending moment Ma, mean bending moment Mm alter nating torque Ta and mean torque Tm the stress components for a solid round shaft are: Moc 32 Mo a=-=-x _ I rr d 3 os Toc 16 To Ta=-=-x _ J rr d 3 os _ 32 x Mm “m- - rr d 3 os 16 T
29、m Tm=-X- rr d 3 os (14) Instead of using krl/Kf = 1/ (1+q(K,-1) to reduce the materials strength3, which ignores the differences in stress concentration factors between bending, torsion, and tension, the fatigue concen tration factors, Kfu and Kftl are applied in the calculation of alternating stres
30、ses: where Ga=Kjb 32 Ma 1f -3 dos Ta=Kjt 16 Ta 1f -3 dos Kjb = 1+qb(Kb-1) Kjt = l+qr(K,-l) (15 ) (16) Complex stress components are combined by using von Mises equivalent stress: G = 4 222 (oxm-Oym) + (Oym-ozm) + (ozm-oxm) 2 2 + 6 (Txym + Tyzm 2 .5 + Tzxm) J (17) The von Mises stress due to alternat
31、ing moment and alternating torque is: When applying the von Mises equation for equivalent alternating stress, it is assumed that the alternating moment and torque are at the same frequency. This assumption is usually not true. But for infinite life shaft design, the above formula is assumed to be va
32、lid since the alternating torque occurs at a frequency which is a multiple of the critical shaft speed. The von Mises mean stress is 4 The following ASHE elliptic equation will be used to predict fatigue: (20) If the factor of safety, Fs is based on a linear relationship between Mal Tal Mm and Tml t
33、he variable Fs“may be separated per Equation(21) A more convenient form of this equation may be in terms of shaft outside diameter: (22) For a hollow shaft, (23 ) Hence I 16 1 a=-3- 4 1fdos (l-(dis/dos) ) 4 (KfoHa) 2+3 (KjtTa)2J0.5 (24) Substituting into ASME elliptic equation yields the following e
34、quation which may be solved by iteration for the outside shaft diameter. where the fatigue strength, Sf is esti mated by Equations (22) and (25) give shaft diameters based on fatigue failure mode. However, the shaft may fail by yielding when the full load is first applied. This static failure mode i
35、s discussed in the following section. ALGORITHM FOR SHAFT DESIGN PROGRAM WITH AXIAL FORCE, MOMENT AND TOROUE (1) Factor of safety evaluation for the fatigue mode of failure: For a given load and shaft dimen sions, use the following formulas to get the factor of safety. Alternating normal stress: whe
36、re Alternating shear stress: where Von Mises alternating stress: 5 Mean stresses: von Mises mean stress: The factor of safety for fatigue is: (2) Factor of safety evaluation for the general yielding mode of failure (static failure mode): Maximum normal stress: Maximum shear stress: Tmax = von Mises
37、maximum stress: I max ( 02 + 3 -r2 ) 112 max max The factor of safety for yielding is: FY = Sy s -1-max (3) Design procedure based on fatigue failure mode: Figure 3 shows the steps for shaft diameter design. (4) Design procedure based on ABS shaft diameter equation4: where b=0.073 + (27,800/Sy) m=17
38、2,000/(60,000 + Sy) alculate Fs yes Fs (T + T ) = 31 (1610 + m bm m 2529) = 7169 psi. Substitute Ga 2904, A-4 Gm = 7169.8, and Sf = 28661, Sy = 124000 into Eq.(A-5) to obtain the safety factor at point B Fs = 8.6. The maximum horizontal shear stress is Tbmax (A-ll ) Substituting Vm = 51670 1b, Va =
39、12915 lb, A = 42.78 in2 into Eq. (A-II) yields Tbmax = 013 p.si. var; iSB maximum stres 21 pont B s 0max - 3 (T max + Tbmax) - 3 (3161 + 2013) = 8962 psi. Hence the factor of safety for yielding at point B is FsY = Sy / Gmax = 124000/8962 = 13.8. The above results clearly indicates that the stress at point A is worse than that at point B for this case. For a very short shaft, the bending moment may become negligible and the contribution due to the horizontal shear stress would become significant.
