DEFORMATION OF SHAFTS AND AXLES LOADED WITH TRANSVERSE FORCE APPLIED IN THEIR CONSOLE PARTS

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1 INNOVATION IN WOODWORKING INDUSTRY AND ENGINEERING DESIGN, /4 (5): 7 77 DEFORMATION OF SHAFTS AND AXLES LOADED WITH TRANSVERSE FORCE APPLIED IN THEIR CONSOLE PARTS Slavcho Sokolovski, Evguenia Slavtcheva, Nencho Deliiski University of Forestry, Kliment Ohridski blvd, 797 Sofia, Bulgaria slav_sokolovski@yahoo.com; deliiski@netbg.com Ministry of Agriculture and Food Sofia, Bulgaria, evslavtcheva9@yahoo.fr ABSTRACT The methodology for calculation of the shafts and the axles subjected to deformation taking into account the radial clearance in the bearings is presented in the paper. Deflection of the shafts and axles is determined for to cases: the first one is hen the antifriction bearings are assumed to be perfectly solid supports and the second one is hen the above-mentioned are assumed to be elastic. The deflection of the antifriction bearings is a sum of the initial clearance and the elastic deformations of the antifriction bodies and their rings. The aim of the present ork is to determine the elastic deformations of the antifriction bodies and the rings of the bearings. Key ords: shafts, axles, antifriction bearings, deflection, radial clearance INTRODUCTION Deformations (elastic deflections) of the shafts and the axles cause a harmful impact on the joints related to them (keyjoints, splines, etc.), as ell as on the bearings, gear-drives and other details and junctions of the machines. The aforecited increase the stress concentration, decrease the fatigue resistance of the parts, accelerate the ear-out, and orsen the ork precision of the mechanisms and machines. Considerable deflections of the shafts and axles could bring to edging of the bearings, to aggravation of the frequency characteristics of the machines, to formation of vibrations, etc. [S. Sokolovski, N. Deliiski ; N. Staneva, V. Vlasev, ].. DIFFERENT TYPE OF DEFLECTIONS OF THE SHAFTS AND AXLES Bending of the shafts and axles in their console part here the operating bodies of the ood-orking machines are situated (circular sas, band-sa belts, etc.), has a negative influence on their ork. The resistance of the shafts against vibrations and the rotation precision are decreased. The unfavorable influence of the deflections of the shafts and axles requires their exact determination and seeking for their diminution. On fig. is shon deflection of shafts and axles loaded ith a force exercising an effort outside the supports hich is the most common case about ood-orking machines. In the cases hen the shafts are loaded on both their endings ith forces (band-sas, circular blades, etc.) the deflections are determined on both endings of the shaft, in the sections here the external forces are applied. The total load of the shafts and axles f in their console part is shon on fig. в as a sum of the deflection f in perfectly solid supports (fig.а) and the deflection f in elastic supports, i.e. f = f + f. () On fig. a are shon the deflection f and the type of the elastic line of the shaft or the axle after their bending at perfectly solid

2 7 SLAVCHO SOKOLOVSKI, EVGUENIA SLAVTCHEVA, NENCHO DELIISKI supports. Generally, in these cases, the antifriction bearings are admitted to be perfectly solid supports. The deflection f is defined by the relation [S. Sokolovski, 7]: Fa f a b, () E I Where f is the deflection of the shaft or the axle under the action of the external force (in the console part), m; F external force of the shaft or the axle, N; I moment of inertia of the section of the equivalent shaft or axle ith a constant diameter, m 4. It is determined by the ellknon formula πd 4 ekv I ; () 64 d ekv diamenter of the equivalent shaft or axle ith a constant diameter, m; E modulus of linear deformation of the shaft or axle material, Pa; a, b distances according to fig., bearing is of a considerable significance as the yield (displacement) in the bearing changes the type of the elastic line, the theoretically calculated deflections and thence the accuracy of the elaborated detail. The rigidity of the bearing depends on its construction, on the form of the rolling bodies etc. In this report the factors on hich depend the displacements of the bearing are examined. On fig. б is shon the deflection f of perfectly rigid shafts and axles depending on the displacements in the bearings f A and f B. This deflection is determined geometrically by the similarity of the to right triangles ΔВАА and ΔВСС (fig. ), for hich could be ritten the folloing equality b f A f B, (4) a b f f B Whence it follos that a f f A fb f A, (5) b Where f is the deflection of the perfectly rigid shafts and axles under the action of the external force depending on the displacements of the bearings, m; f A, f B displacements of the bearings in the supports, Figure : Shafts and axles deflection loaded ith a force outside the supports: а at perfectly solid supports; б at elastic supports; в total deflection In fact the supports ith antifriction bearings are elastic. The bigger the elastic yield (displacement) of one bearing subjected to a single load is, the smaller its rigidity is. When putting bearings on mandrels and operating shafts of oodorking machines the rigidity of the Figure : Scheme of geometrical determination of the deflection of shafts and axles in accordance ith the displacements in the shafts The displacement of the shafts is a sum of their initial clearance G r and the elastic deformations of the rolling bodies and rings δ: f A = G ra + δ A - for the bearing in support А, (6)

3 DEFORMATION OF SHAFTS AND AXLES LOADED WITH TRANSVERSE 7 f B = G rb + δ B - for the bearing in support В, (7) here G ra and G rb are the initial radial clearances respectively for the bearing in support А and for the bearing in support В. Their values could be found in the catalogues for rolling bearings; δ A, δ B elastic deformations of the most loaded rolling bodies and rings of the bearings, respectively for the bearing in support А and for the bearing in support В.. DETERMINATION OF DISPLACEMENT IN BEARINGS, PROVOKED BY THE ELASTIC DEFORMATIONS OF THE ROLLING BODIES AND RINGS It is knon that in the case of the radial bearings, the rolling bodies hich are disposed in the loaded zone of the rings, they receive different parts of the total force. The most loaded is the loest rolling body hich is situated in the plane of action of the radial force R. The load could be defined by taking into consideration the clearance in the bearing according to the folloing relation [К. Кolesnikov and others, 4]: Q 5R, (8) z here Q is the force acting on the loest rolling body of the bearing, N; R the radial load of the bearing (support reactions of the shaft), N; (respectively R А - for the bearing in support А; R B for the bearing in support В); z the number of the bearing balls hich is taken by the catalogues for antifriction bearings. Under the action of the force Q in the zones of contact of the rolling body ith the rings (fig. а) occur elastic contact deformations of the rolling body and the rings shon on the scheme on fig. б. The rolling body changes its form from a circle to an ellipse, and hollos (small pits). For the determination of the joint displacement of the most loaded rolling body and the rings of the bearing, let us consider their elastic deformations beteen the internal ring and the rolling body (point on fig. а) and the deformations beteen the external ring and the rolling body (point on fig. а). After the elastic deformations occur the center of the bearing, consequently the center of the shaft ill be translocated on a distance δ hich is the sum of the deformations of the contact in point - δ and the contact in point - δ, i.e. δ = δ + δ. (9) Figure : Scheme of the deformations in an antifriction bearing: а before loading; б after loading The shift in the contact of the point (fig. a) beteen the internal ring and the rolling body δ is a sum of the deformations of the rolling body (a ball or a roller) δ s and of the internal ring δ r, i.e. δ = δ s + δ r. () The shift in the contact of the point (fig. a) beteen the external ring and the rolling body δ is a sum of the deformations of the rolling body (a ball or a roller) δ s and of the external ring δ r, i.e. δ = δ s + δ r. () To determine the deformations δ of the contact in point (fig. а) e could use the relations at the contact of a sphere or a cylinder ith convex surfaces, hereas for the deformations δ of the contact in point

4 74 SLAVCHO SOKOLOVSKI, EVGUENIA SLAVTCHEVA, NENCHO DELIISKI (fig. а) e can use the relations at the contact of a sphere or a cylinder ith concave surfaces. When specifying these deformations (displacements) it is necessary to be expressed in dependence ith the applied force in the bearing. The applied force of the bearing is perceived by a very small surface of contact of the rolling bodies and the rings and therefore the strains in these contacts are considerable hich provokes elastic deformations. On the basis of the theory of the German physicist Herz [H. Herz, 88] are obtained formulae for determination of the contact surface (hich has the form of an ellipse in the ball bearings and a rectangle in the roller bearings), the normal strains and deformations beteen sphere and plane, sphere and sphere, cylinder and plane and cylinder and cylinder. The deformation δ in ball bearings (point contact) of the most loaded ball ith a force Q is equal to [L. Perel, 98] δ.79 6 Q ρ K, () here δ is the deformation of the most loaded ball in the case of ball-bearings, m; Q force exercising an effort on the loest rolling body of the bearing, N. It is determined ith the help of formula (8); K parameter characterizing the curve of the contacting surface. It is determined by table [L. Perel, 98] in accordance to the subsidiary parameter τ, characterizing the difference of curvedness of the contact surface. Table : Values of the subsidiary parameter τ and the parameter K τ K τ K τ K τ K τ K τ K,999,7,989,9,979,465,968,58,946,588,9,68,998,49,988,4,978,47,966,56,94,598,88,79,997,79,987,4,977,476,964,5,98,68,84,755,996,,986,4,976,48,96,54,94,68,8,79,995,,985,47,975,486,96,546,9,66,7,859,994,6,984,4,974,49,958,55,96,64,6,94,99,5,98,44,97,495,956,599,9,64,5,98,99,6,98,447,97,5,954,565,98,65,4,96,99,7,98 54,97,55,95,57,94,657.,99,99,84,98,459,97,59,95,577,9,664 Note. Intermediate values are obtained through interpolation The subsidiary parameter τ is determined in dependence ith the type of contact (sphere or cylinder) and of the form of the contact surfaces (convex or concave) [L. Perel, 98]: - for radial ball single-ro and radial ball spherical bearings: in contact of a ball ith the internal ring τ.9d τ ; (). D D D cos α in contact of the ball ith the external ring τ.9d τ.d D D ; (4) - for radial roller spherical bearings ith barrel-shaped rollers: in contact of a barrel-shaped roller ith the internal ring τ

5 DEFORMATION OF SHAFTS AND AXLES LOADED WITH TRANSVERSE 75 τ D 4 D ; (5) D 4 D in contact of a barrel-shaped roller ith the external ring τ D 4 D τ. (6) D 4 D In the formulas () (6) D is the diameter of the rolling bodies (balls or rollers), m; α is the angle of contact, ; D is the diameter of the circumference on hich the rolling bodies are disposed, It is determined as the average diameter of the internal d and external D diameters of the bearing, i.e. D =.5(d +D). (7) The values of D, α, d and D are given in the catalogues for antifriction bearings. ρ Σ - the sum of the curve, m -. It is determined in accordance to the type of the contact (sphere or cylinder) and to the form of the contact surfaces (convex or concave) [L. Perel, 98]: - for radial ball single-ro and radial ball spherical bearings: in contact of a ball ith the internal ring ρ Σ D ρ. ; (8) D in contact of a ball ith the external ring ρ Σ D ρ. ; (9) D - radial roller spherical bearings ith barrel-shaped rollers: in contact of a barrel-shaped roller ith the internal ring ρ Σ. D 4 ρ ;() D in contact of a barrel-shaped roller ith the external ring ρ Σ. D 4 ρ ;() D - radial roller bearings ith cylindrical rollers: in contact of the cylindrical roller ith the internal ring ρ Σ D ρ ; () D D in contact of the cylindrical roller ith the external ring ρ Σ D ρ ; () D D here D the diameter of the rolling bodies (balls or rollers), The values can be found in the catalogues for antifriction bearings; D diameter of the circumference on hich the rolling bodies are disposed, m, determined ith the help of formula (7); α contact angle,. It is determined due to the catalogues for antifriction bearings. Deformation δ in roller bearings (linear contact) of the most loaded roller ith force Q is [L. Perel, 98],95 - Q δ., (4),85 l here δ is the deformation of the most loaded roller in roller bearings, m; Q the force charging the loest rolling body of the bearing, N. It is described ith the help of formula (8); l the length of the rollers, m, given in the catalogues for antifriction bearings. Verification of the proposed methodology is made by the folloing example: Determination of the total deflection of shaft (fig. ), loaded in its console part ith a force F = 5 N. Diameters of the shaft

6 76 SLAVCHO SOKOLOVSKI, EVGUENIA SLAVTCHEVA, NENCHO DELIISKI are: in the console part, hich is equal to the equivalent d С = d ekv =. m, diameters of the bearing necks are d А = d В =.4 The shaft is elaborated from quality steel 45. The shaft is mounted in bearings on to similar radial ball single-ro bearings 68 ith d =.4 m, D =.9 m, D =.65 m, D =, 58 m, z = 8 and α =. The bearing body is shared by the to bearings hich provide the coaxiality of the bearings. The distances are: a =. m, b =.4 Support reactions are (fig. ): Fa b 5..4 R A 5 N; b.4 Fa 5. R B 75 N. b.4 Total deflection of the shaft f is f = f + f = = 54-6 Deflection f of the shaft under the action of the external force is f Fa E I 6 a b The moment of inertia of the equivalent shaft section ith constant diameter is 4 4 πd I ekv.4 m ; E =. Ра modulus of linear deformation of steel. he deflection f of the shaft under the action of the external force depending on displacements in the bearings is a f f A f B f A b The displacement of the bearings is: - for the bearing in support А f A = G ra + δ A = = The initial clearance of the antifriction bearing is G ra = -6 The elastic deformation δ A of the bearing in the support А is δ A = δ A + δ A = = --6 The elastic deformations in the contact in point (the ball ith the internal ring) is δ A K A Q A.6(46 ρ A.78).8 The force exercising an effort on the loest ball 5RA 5 5 QA 46 N. z 8 Parameter K А from table depending on the subsidiary parameter τ А.9D τ A.D За τ А =.94; K А =.6. The sum of the curve is ρ A. D m. The elastic deformations in the contact of point (the ball ith the external ring) δ A K A. Parameter K А from table depending on the subsidiary parameter τ А Q.695(46 A ρ A.)

7 DEFORMATION OF SHAFTS AND AXLES LOADED WITH TRANSVERSE 77.9 D τ A. D За τ А =.89; K А =.695. The sum of the curve is ρ A. D m. - for the bearing in support В f B = G rb + δ B = = 4-6 The initial clearance of the antifriction bearing is G rb = -6 The elastic deformations δ В for the bearing in the support В is δ В = δ В + δ В = = -6 The elastic deformations in the contact of point (the ball ith the internal ring) δ B K B 5.6 The force exercising an effort on the loest ball 5RB 5 75 QB N. z 8 Parameter K В from table depending on the subsidiary parameter τ В.9D τ B. D D D Q B.6(468.8 ρ B.94. За τ В =.94; K В =.6. The sum of the curve is.78) ρ B. D m. The elastic deformations in the contact of point (the ball ith the external ring) δ B K B 5.4 Parameter K В from table depending on the subsidiary parameter τ В.9D τ B. D D D Q.695(468.8 A ρ.89. За τ В =.89; K В =.695. The sum of the curve is B.) ρ B. D m. CONCLUSION The results of the example sho that the most important deflection of the shafts and the axles is in the console part from the action of the external load. It could be reduced by decreasing the console part, if it is possible, or by increasing the diameter of the shaft. The deflection of shafts and axles caused by the deformations in the bearings is considerably smaller than the total deflection (for the case considered above it is about 6 %). The deflection of shafts and axles caused by the deformations in the bearings is necessary to be done hen mounting bearings of precise metal and oodorking machines, centrifuges,

8 78 SLAVCHO SOKOLOVSKI, EVGUENIA SLAVTCHEVA, NENCHO DELIISKI ventilators, electro motors and etc. It is imperative to determine the deflection of the shafts and axles caused by the deformations in the bearings hen mounted on bearings of exact metal and oodorking machines, centrifuges, ventilators, electro motors and etc. In conventional machines it is advisable. REFERENCES. Kolesnikov K., N. Alfutov and al. (4). Machine parts. Publisher MGTU N.E. Baumana. Mosco, p. 55 (in russian). Perel L. (98). Rolling bearings. Calculation, design and service of the supports. Reference book. Publisher Mechanical engineering. Mosco, p. 54 (in russian). Sokolovski S. (7). Machine elements. Publisher house at LTU- Sofia, 8 p. (in bulgarian) 4. Sokolovski S., N. Deliiski (). Calculation of fatigue, deformation and vibration. Part : calculation of fatigue, deformation and vibration (in russian) 5. Staneva, N., Vlasev, V. (). Study of static strenght of a milling oodorking machine spindle by FEM, Innovation in oodorking industry and engineering design, / (): 7 78.