Mechanical Design. Design of Shaft

Transcription

1 Mechanical Design Design of Shaft

2 Outline Practical information Shaft design

3 Definition of shaft? It is a rotating member, in general, has a circular cross-section and is used to transmit power. The shaft may be solid or hollow. It is supported on bearings and it rotates a set of gears or pulleys for the purpose p of power transmission. The shaft is generally acted upon by bending moments, torsion and axial forces.

4 Shaft versus axle and spindle Axle is a non-rotating member used for supporting rotating wheels, etc., and do not transmit any torque. Spindle is simply defined as a short shaft. However, design method remains the same for axle and spindle as that for a shaft.

5 Whatdoesitmean shaft shaft design? Material selection Geometric layout Stress and strength: static and fatigue Deflection and rigidity: bending defl., torsional twisting, slope at bearings and shaft- supported elements, and shear deflection due to transverse loading on short shafts. Vibration: critical speed

6 Material selection Many shafts are made from low carbon, colddrawn or hot-rolled steel. Alloy steel: Nickel, chromium and vanadium are some of the common alloying materials. However, alloy steel is expensive. Shafts usually don t need to be surface hardened unless they serve as the actual journal of a bearing surface. Hardening of surface (wear resistant): case hardening and carburizing ; cyaniding and nitriding.

7 Geometric layout

8 Geometric layout The geometry of shaft is generally that of stepped cylinder. There is no magic formula to give the shaft geometry for any given design situation. ti

9 Geometric layout The best approach his to learn from similar il problems that have been solved and combining the best to solve your own problem. A general layout to accommodate shaft elements, e.g. gears, bearings, and pulleys, must be specified early in the design process. Shoulders are used for axially locating shaft elements and to carry any thrust loads. Common Torque Transfer Elements: keys, set screws, pins, press or shrink fits, tapered fits.

10 Geometric layout Small pinions are often machined onto shafts. Sequence of assembly should be thought. Use chamfers to ease assembly and avoid interferences. Consider stress risers due to grooves and sharp steps in shafts. What can fail and how will it happen?

11 Shaft design based on strength Design is carried out so that stress at any location of the shaft should not exceed material yielding. Stress due to torsion: τ xy T r 16 T = = J π d 1 c ( ) 3 4 o τ τxy : Shear stress due to torsion T : Torque on the shaft Note: T 704 hp 9549 kw N ( R P M ) N ( R P M )

12 Shaft design based on strength Bending stress: σ b M y 3M = = I π d (1 c) 3 o M : Bending moment at the point of interest do : Outer diameter of the shaft c: di/do

13 Shaft design based on strength Axial stress: Fa σ a Fa 4 α Fa = = A π d (1 c ) o Fa: Axial force (tensile or compressive) α: Column-action factor(= 1.0 for tensile load) α arises due to the phenomenon of buckling of long slender members which h are acted upon by axial compressive loads. Fa

14 Shaft design based on strength Ail Axial stress (continue): α α 1 =, ( λ = L / r ) λ λ s yc =, λ > 115 π ne n = 1.0 for hinged end; n =.5 for fixed end n = 1.6 for ends partly restrained, as in bearing g, L = shaft length syc = yield stress in compression

15 Shaft design based on strength Maximum shear stress theory (ductile mat.): Failure occurs when the maximum shear stress at a point exceeds the maximum allowable shear stress for the material. Therefore, σ x τmax τ = allowable = + τ xy τ allowable ( 1+ c) 16 α Fa do = M + + π d 3 ( 1 4 ) 8 o c T

16 Shaft design based on strength Maximum normal stress theory (brittle mat.): σ x σ x σmax σ = allowable = + + τ σ allowable ( 16 α Fd 1 ) a o + c = M π d ( 1 ) 8 o c ( ) α Fa do 1+ c + M + + T 8 xy

17 Shaft design based on strength Von Mises/ Distortion-Energy theory: σ = σ = σ + 3τ max allowable x xy σ allowable ( 1+ c) 16 α Fa do = 3 ( 4 ) M T π do c

18 Shaft design based on strength τ ASME design code (ductile material): ( 16 α F 1 ) a do + c k M ( 1 c ) 8 = + + π ( ) allowable 3 4 m t π do kt where, km and kt are bending and torsion factors accounts for shock and fatigue. The values of these factors are given in ASME design code for shaft.

19 Shaft design based on strength ASME design code (brittle material): σ allowable ( 16 α F 1 ) a do + c = k mm + π d 3 ( 1 4 ) 8 o c ( ) α Fa do 1+ c + kmm + + ( kt t ) 8

20 Shaft design based on strength ASME design code: Combined shock and fatigue factors Type of load Stationary shaft Rotating shaft km kt km kt Gradualy applied load Suddenly applied load, minor shock Suddenly applied load, heavy shock

21 Shaft design based on strength ASME design code: Commercial steel shafting τallowable = 55 MPa for shaft without keyway τallowable = 40 MPa for shaft with keyway y Steel under definite specifications τallowable = 30% of the yield strength but not over 18% of the ultimate strength in tension for shafts without keyways. These values are to be reduced by 5% for the presence of keyways.

22 Standard sizes of shafts Typical sizes of solid shaft that are available in the market are: diameter increments up to 5 mm 0.5 mm 5 to 50 mm mm 50 to 100 mm.0 mm 100 to 00 mm 5.0 mm

23 Example: problem A pulley drive is transmitting power to a pinion, i which in turn is transmitting power to some other machine element. Pulley and pinion diameters are 400 mm and 00 mm respectively. Shaft has to be designed for minor to heavy shock. A C. D B m m m

24 Shaft design based on fatigue Any rotating shaft loaded by stationary bending and torsional moments will be stressed by completely reversed bending stress while the torsional stress will remain steady (i.e., Ma = M; Mm = 0; Ta = 0; Tm = T). h f bl l (f ) A design method for variable load (fatigue), like Soderberg, Goodman or Gerber criteria can be followed.

25 Shaft design based on fatigue Design under variable normal load (fatigue)

26 Shaft design based on fatigue A is the design point for which the stress amplitude is σa and the mean stress is σm. In the Soderberg criterion the mean stress material property is the yield point Sy, whereas in the the Goodman and Gerber criteria the material property is the ultimate strength Sut. For the fatigue loading, material property is the endurance limit Se in reverse bending.

27 Shaft design based on fatigue where Soderberg criterion i mod-goodman d criterion i FS σ FS σ FS σ FS σ a m a m + = 1 + = 1 Se S y Se Sut FS Gerber criterion σ FSσ S e a m + = 1 Sut σa = stress amplitude (alternating stress); Se= endurance limit (fatigue limit for completely reversed loading); σm = mean stress; Sy = yield strength; σut = ultimate tensile strength and FS= factor of safety.

28 Shaft design based on fatigue Design under variable shear load (fatigue)

29 Shaft design based on fatigue It is most common to use the Soderberg criterion. FS K S e S K σ FS σ + = 1 S Kf = fatigue stress f a m σ y + σ = S y f a y m Se FS concentration factor Sy K f σ a S e Sys K fs τ a + σm = σeq + τm = τeq S es Normal Stress Equation Shear Stress Equation

30 Shaft design based on fatigue Sy K f σ a Sys K fs τ a + σm = σeq S S e es + τ = τ m eq σeq and τeq are equivalent to allowable stresses (Sy/FS) and (Sys/FS), respectively. Effect of variable stress has been effectively defined as an equivalent static stress. C ti l f il th i b d t Conventional failure theories can be used to complete the design.

31 Shaft design based on fatigue Max. shear stress theory + Soderberg line (Westinghouse Code Formula) d σ eq τmax = τallowable = + τ eq S y S y K f σ a S y K fsτ a = + σm + + τ m FS Se Se FS K M M K T T = π Se Sy Se Sy 3 3 f a m fs a m

32 Shaft design based on rigidity Deflection is often the more demanding constraint. Many shafts are well within specification for stress but would exhibit too much deflection to be appropriate. Deflection analysis at even a single point of interest requires complete geometry information for the entire shaft.

33 Shaft design based on rigidity It is desirable to design the dimensions at critical locations to handle the stresses, and fill in reasonable estimates for all other dimensions, before performing a deflection analysis. Deflection of the shaft, both linear and angular, should be checked at gears and bearings.

34 Shaft design based on rigidity Slopes, lateral l deflection of the shaft, and/or angle of twist t of the shaft should be within some prescribed limits. Crowned tooth Diametralpitch, P = number of teeth/pitch diameter. 1 in = 5.4 mm.

35 Shaft design based on rigidity In case of sleeve bearings, shaft deflection across the bearing length should be less than the oil-film thickness. θ rad = Twist angle: ( ) T N. mm L mm ( ) ( ) ( ) ( 4 / ) G N mm J mm G: shear modulus; J: polar moment of inertia The limiting value of θ varies from 0.3 deg/m to 3 deg/m for machine tool shaft to line shaft respectively.

36 Shaft design based on rigidity Lateral ldeflection: Double integration Moment-area Energy (Castigliano Theorem) δ= f (applied load, material property, moment of inertia and given dimension of the beam). From the expression of moment of inertia, and known design parameters, including δ, shaft dimension may be obtained.

37 Double Integration Method dy M x ( ) d y M = θ () x = = dx EI dx EI () x dx y ( x) = M x ( ) EI ( x ) dx Use boundary conditions to obtain integration constants

38 Conjugate beam method Was developed by Otto Mohr in 1860 Slope (real beam) = Shear (conj. beam) Deflection (real beam) = Moment (conj. beam) Length of conj. beam = Length of real beam The load on the conjugate beam is the M/EI The load on the conjugate beam is the M/EI diagram of the loads on the actual beam

39 Conjugate beam method Real lbeam Conjugate beam Real beam Conjugate beam

40 Critical speed of rotating shaft Critical speed of a rotating shaft is the speed where it becomes dynamically y unstable. The frequency of free vibration of a non- rotating shaft is same as its critical speed. 60 wδ + w δ w δ N ( ) critical RPM = g π δ δ... n δn ( ) 1 1 n n ( w ) 1 1+ w + + w

41 Critical speed of rotating shaft W1, W. : weights of the rotating bodies (N) δ1, δ. : deflections of the respective bodies (m) For a simply supported shaft, half of its weight may be lumped at the center for better accuracy. For a cantilever shaft quarter of its weight For a cantilever shaft, quarter of its weight may be lumped at the free end.

42 Shaft design: general considerations Axial thrust loads should be taken to ground through a single thrust bearing per load direction. Do not split axial loads between thrust bearings as thermal expansion of the shaft can overload the bearings. Shaft length should be kept as short as possible, to minimize both deflections and stresses.

43 Shaft design: general considerations A cantilever beam will have a larger deflection than a simply pysupported (straddle mounted) one for the same length, load, and cross section. Hollow shafts have better stiffness/mass ratio and higher h natural frequencies than solid shafts, but will be more expensive and larger in diameter.

44 Shaft design: general considerations Slopes, lateral deflection of the shaft, and/or angle of twist of the shaft should be within some prescribed limits. First natural frequency of the shaft should be at least three times the highest forcing frequency. (A factor of ten times or more is preferred, but this is often difficult to achieve).

45 Example 3 Determine the diameter of a shaft of length L =1m, carrying a load of 5 kn at the center if the maximum allowable shaft deflection is 1mm. What is the value of the slope at the bearings. Calculate the critical speed of this shaft if a disc weighting 45 kg is placed at the center. E=09 GPa. ρst = 8740 kg/m^3.

46 Shaft design: summary Shaft design means material selection, geometric layout, stress and strength (static and fatigue), deflection and slope at bearings. Conjugate beam method for slope and deflection calculations. Some design considerations: (axial load, shaft length, support layout, hollow shafts, slopes and deflection, operating speed)