UNIT 13 DESIGN OF GUIDEWAYS AND SPINDLE

Transcription

1 UNIT 13 DESIGN OF GUIDEWAYS AND SPINDLE Design of Guideways and Spindle Structure 13.1 Introduction Ojectives 13. Functions of Guideways 13.3 Types of Guideways Guideways with Sliding Friction Guideways with Rolling Friction 13.4 Design of Slideways Design of Slideways for Wear Resistance Design of Slideways for Stiffness 13.5 Functions of Spindle 13.6 Design of Spindle Deflection of Spindle Axis due to Bending Deflection of Spindle Axis due to Compliance of Spindle Supports Optimum Spacing etween Spindle Supports Deflection due to Compliance of the Tapered Joint Permissile Deflection and Design for Stiffness 13.7 Summary 13.8 Key Words 13.1 INTRODUCTION Design of machine tool elements is critical in tool engineering. They must withstand against applied external load. The machine tool elements such as guideways, slideways and spindle unit are discussed in detail in the next section. Additionally, requirements, functions and types of guideways and spindle are also explained. Ojectives After studying this unit, you should e ale to understand various types and functions of guideways, the design of slideways, the design of spindle, and function of spindle. 13. FUNCTIONS OF GUIDEWAYS The Guideway is one of the important elements of machine tool. The main function of the guideway is to make sure that the cutting tool or machine tool operative element moves along predetermined path. The machine tool operative element carries workpiece along with it. The motion is generally circular for oring mills, vertical lathe, etc. while it is straight line for lathe, drilling, oring machines, etc. Requirements of guideways are : () Guideway should have high rigidity. The surface of guideways must have greater accuracy and surface finish. 17

2 Advances in Tool Engineering and Management (c) (d) (e) (f) (g) SAQ 1 () Guideways should have high accuracy of travel. It is possile only when the deviation of the actual path of travel of the operative element from the predetermined normal path is minimum. Guideways should e durale. The duraility depends upon the aility of guideways to retain the initial accuracy of manufacturing and travel. The frictional forces acting on the guideway surface must e low to avoid wear. There should e minimum possile variation of coefficient of friction. Guideways should have good damping properties. What are various functions of guideways? State requirements of guideways TYPES OF GUIDEWAYS The guideways are mainly classified according to the nature of friction etween contacting surfaces of the operative element : () Guideways with sliding friction Guideways with rolling friction Guideways with Sliding Friction The friction etween the sliding surfaces is called as guideways with sliding friction. These guideways are also called as slideways. The slideways are further classified according to the lurication at the interface of contacting surfaces. The friction etween the sliding surfaces may e dry, semi-liquid, and liquid. When the lurication is asent in etween contacting surfaces, it is called as dry friction. Dry friction is rarely occurred in machine tools. When two odies slide with respect to each other having lurication etween them, the sliding ody tends to rise or float due to hydrodynamic action of the luricant film. The principle of slider is shown in Figure F h v s θ W 18 The hydrodynamic force, Figure 13.1 : Principle of a Slider

3 F h C * v s... (13.1) Where C is constant and depends upon wedge angle θ, the geometry of sliding surfaces, viscosity of the luricant and parameter of luricant film. v s is sliding velocity. W is weight of the sliding ody. The resultant normal force acting on sliding ody, R F h W From Eq. (13.1), it is clear that the hydrodynamic force increases with increase in sliding velocity. The sliding ody rests on the stationary ody when hydrodynamic force is less than the weight of the sliding ody. Here, there are semi-liquid type friction conditions and under these conditions the two odies are partially separated y the luricant film and partially have metal to metal contact. The resultant normal force on sliding ody starts to act upwards and the ody floats as hydrodynamic force is greater than the sliding weight of the ody. The sliding surfaces are completely separated y the luricant film and liquid friction occurs at their interface. The slideways in which the sliding surfaces are separated y the permanent luricant layer are known as hydrodynamic slideways. This permanent lurication layer is due to hydrodynamic action. A permanent luricant layer etween the sliding surfaces can e otained y pumping the liquid into the interface under pressure at low sliding speed. The sliding ody is lifted y this permanent luricant layer. Such slideways are called as hydrostatic slideways Guideways with Rolling Friction These are also called as anti friction ways. The anti friction slideways may e classified according to the shape of the rolling element as : SAQ () () Roller type anti friction ways using cylindrical rollers. Ball type anti friction ways using spherical alls. Explain various types of guideways. Explain with figure the principle of slider. Design of Guideways and Spindle 13.4 DESIGN OF SLIDEWAYS Slideways are designed for wear resistance and stiffness Design of Slideways for Wear Resistance The wear resistance of slideways is mainly dependent upon maximum pressure acting on the mating surfaces. This condition may e given as p max P mp... (13.) p m maximum pressure acting on the mating surface, and p mp permissile value of the maximum pressure. It is seen during the susequent analysis that slideway designed for maximum pressure is quite complicated. Sometimes, this design is replaced y a simple procedure ased upon the average pressure acting on the mating surfaces. The condition is that: P a P ap... (13.3) p a average pressure acting on the mating surface, and 19

4 Advances in Tool Engineering and Management p ap permissile value of the average pressure. Hence from Eqs. (13.) and (13.3), the design of slideways for wear resistance requires that () (c) p m and p a to e known, p mp and p ap to e known, and The values of p mp and p ap are given for different operating conditions of slideways on the asis of experience. For determining p m and p a, the first and foremost task is to determine the forces acting on the mating surfaces. Forces acting on the mating surfaces in comination of V and flat slideways. The comination of V and flat slideways is commonly used in lathe machines. The schematic diagram of slideways and the forces acting on the system for the case of orthogonal cutting are illustrated in Figure 13.. d F y F 3 h F z γ λ W S F 1 Figure 13. : Forces Acting on Comination of V and Flat Sideways The forces acting on V and flat slideways are : () (c) Cutting force component F z (in the direction of the velocity vector) and F y (radial), Weight of carriage W, and Unknown forces F 1, F and F 3 acting on the mating surfaces. The unknown forces are calculated from following equilirium conditions : Sum of components of forces acting along Y-axis 0 i.e. Y 0 F1 F F y sin λ sin γ (13.4) Sum of components of forces acting along Z-axis 0 i.e. Z 0 F1 F F3 W F z Moment of all forces aout X-axis 0 i.e. M 0 cos λ + cos γ (13.5) x 0

5 d F1 cos λ. + F cos γ. Fz. Fy. h F3. 0 d h F 3 F1 cos λ+ F cos γ Fz Fy... (13.6) Sustituting value of F 3 in Eq. (13.5), we get, d h F1 cos λ+ F cos γ+ F1 cos λ+ F cos γ F z F y F z W 0 d h W ( F cosλ+ F cos γ ) Fz 1+ + Fy + or 1 Fz ( d + ) h W F λ+ F cos γ + Fy +... (13.7) 1 cos If the apex angle of the V is 90 o, and assume that present angle γ may change to γ 90 λ, the solution of simultaneous algeraic Eqs. (13.4) and (13.7) gives : Fz ( d + ) h F1 cos λ + Fy cos λ Fy sin λ + W cos λ... (13.8) Fz ( d + ) h F sin λ + Fy sin λ + Fy cos λ + W sin λ... (13.9) Sustituting the values of F 1 and F in Eq. (13.6) we get, F 3 F z ( ) d Fy. h + W... (13.10) Eq. (13.10) represents the forces acting on the mating surfaces in comination of two flat sideways. The schematic diagram of the slideways and the forces acting on the system under orthogonal cutting conditions are shown in Figure (13.3). Design of Guideways and Spindle d x p w F 3 h F y F z Y p F 1 S Z F 1 F z F x X F II W I Y Q Figure 13.3 : Forces Acting on Comination of Two Flat Slideways The forces acting on comination of two flat slideways are : Cutting force components, i.e. axial F x, radial F y, and F z in the direction of velocity vector. () Weight of carriage, W. (c) Unknown forces F 1, F and F 3 acting on the mating surfaces. (d) Frictional forces μf 1, μf, μf 3, μ is the coefficient of friction etween the sliding surfaces. The unknown forces F 1, F and F 3 are calculated from following equilirium conditions : 1

6 Advances in Tool Engineering and Management X 0 F + μ ( F + F + F ) R 0... (13.11) x y F F y 0... (13.1) F F y z 0 F + F F W... (13.13) 1 3 z 0 M x 0 W + Fz yp Fy h F (13.14) From Eq. (13.14) Fz yp Fy h W F3 + On sustituting the value of F 3 in Eq. (13.13) W Fz yp Fy h F1 Fz +... (13.15) The pulling force, R is calculated from Eq. (13.11) on sustituting the values of F 1, F and F 3. R F +μ ( F + F + W)... (13.16) x z y Determination of Average Pressure The average pressure can e determined as : F1 pf 1 vl F pf wl F3 pf 3 ul L length of the carriage, and v, w, u the width of slideway faces on which forces F 1, F and F 3 are acting respectively. Determination of Maximum Pressure It is necessary to estalish the points of action of the resultant normal forces F 1, F and F 3 on the respective faces for calculating the maximum pressure. The distance etween the point of action of normal force F 1 on the flat slideway I and the center of the carriage is denoted y x A. This is shown in Figure The distance etween force F acting on the vertical face of flat slideway II and the center of the carriage is denoted y xb, and the distance etween force F3 acting on horizontal face of flat slideway II and the center of the carriage is denoted y x c. For determining x, x and x, we have two equilirium conditions : A B C M y 0 F h + F x F x F x + Rz... (13.17) x z p 1 A 3 c Q 0

7 And M 0 z ( l + u) Fx yp + Fy xp RyQ F xb + μ F + μ F3 l 0... (13.18) An additional equation may e written y assuming that the moment of reactive forces F 1 and F 3 aout the Y-axis is proportional to the width of the slideway face, i.e. Fx 1 F x A c v... (13.19) u On solving Eqs. (13.17), (13.18) and (13.19), we get, the values of x, x and x. A B c Design of Guideways and Spindle The ratio of x A /L, xb /L and xc /L calculates the shape of the pressure distriution diagram and the maximum pressure on a particular face of the slideway. The procedure for determining the maximum pressure on flat slideway I is eing sujected to the normal force F1 which is descried elow. The most general case of pressure distriution along the length of contact L corresponds to a trapezoid as shown in Figure F 1 p max p min L/ y c L Figure 13.4 : Trapezoidal Pressure Distriution along Slideway Length Force F 1 acts at the center of gravity of trapezoid. The distance y c at the center of gravity from the larger arm of the trapezoid can e determined as : As a result, y c L ( pmax pmin ) L pmin L. + L. 3 pmax pmin pmin L + L L p yc 3 p max max + p + p min min... (13.0) L xa yc L pmax + pmin χ A... (13.1) 6 p + p max x 1 Now A L < 6 In this case the pressure distriution diagram represents a trapezoid. This is explained with the help of following example. min 3

8 Advances in Tool Engineering and Management Let x A 1 L < 10 From Eq. (13.18), we get, 6 pmax pmin ( pmax + pmin ) 10 p min p max p min From aove equation, it is clear that is a positive, non-zero value. Hence pressure distriution diagram must e a trapezoid. pmax + pmin p av... (13.) From Eq. (13.16), we get, 1 x A av pmax pmin p... (13.3) L On solving Eqs. (13.) and (13.3), we get p max 6x pav + L 1 A Design of Slideways for Stiffness Stiffness is one of the important parameters for designing slideways. The design of slideways for stiffness requires that the deflection of the cutting edge due to contact deformation of the slideway should not exceed certain permissile value. This value is specified from considerations of required machining accuracy. For example, in lathe machine, the vertical deflection of the tool or its horizontal deflection in the direction of feed motion has little effect on the dimensional accuracy of the machined workpiece. However, diametrical error of C takes place due to a deflection of the cutting edge y C in the radial direction. Hence the lathe slideways are designed for stiffness with a view to restrict their radial deflection. The appropriate direction for stiffness design in various machine tools can e selected in each particular case from the aove mentioned principle. In a ed using two flat slideways as shown in Figure 13.5, the radial deflection takes place due to the contact deformation C B B of the vertical face of the slideway and rotation of the saddle due to unequal contact deformation CA and C C of the horizontal flat faces. Hence radial deflection is given y C A C C C B. h Hence total radial deflection is given y C FF CA CC CB +. h... (13.4) In the lathe ed using a comination of flat and V slideways, the contact deformation of the flat slideways is shown y lowering C C. The faces of the V-slideways suffer a deformation of C A and C B. B This will result in : () Vertical lowering of the V-slideway y C C sin λ+ C cos λ v B A Horizontal displacement of the apex of the V-slideway y C C cos λ C sin λ h B A 4 C H

9 Design of Guideways and Spindle h C B Z Y C A Figure 13.5 : Radial Displacement of Cutting Edge in Comination of Two Flat Slideways This nature of deformation of flat and V-slideways results in : () radial deflection of the cutting edge y C H and rotation of the saddle due to unequal vertical lowering of the flat and V slideways which leads to radial deflection of the cutting edge y Cv Cc. h The total radial deflection then ecomes C Fv Cv Cc Ch +. h... (13.5) The contact deformation is assumed proportional to the average pressure for the purpose of stiffness design. The coefficient of proportionality d is known as contact compliance. This coefficient must e determined for each pair of slideway materials. However for approximate calculations, an average value of d mm /N may e used. Eqs. (13.4) and (13.5) can e rewritten as follows : pa pc CFF d pb + d. h... (13.6) CFv d ( pb cos λ pa sin λ ) + dh ( pb sin λ + pa cos λ p C ) (13.7) After calculating the total radial deflection of the cutting edge, the design for stiffness is carried out in accordance with C i Ci p.... SAQ 1 () What are principle parameters in designing slideways? Design the slideways for machine tool FUNCTIONS OF SPINDLE The spindle is one of the most important elements of the machine tool. The functions of the spindle of machine tool are as follows : 5

10 Advances in Tool Engineering and Management It clamps the workpiece or cutting tool in such a way that the workpiece or cutting tool is relialy held in position during the machining operation. () It imparts rotary motion or rotary cum translatory motion to the cutting tool or workpiece. (c) It is used for centering the cutting tool in drilling machine, milling machine, etc. while it centers the workpiece in lathes, turrets, oring machine, etc. Requirements of spindle are : The spindle should rotate with high degree of accuracy. The accuracy of rotation is calculated y the axial and radial run out of the spindle nose. The radial and axial run out of the spindle nose should not exceed certain permissile values. These values depend upon the required machining accuracy. The rotational accuracy is influenced mainly y the stiffness and accuracy of the spindle earings especially y the earing which is located at the front end. () The spindle unit must have high dynamic stiffness and damping. (c) The spindle earing should e selected in such a way that the initial accuracy of the unit should e maintained during the service life of the machine tool. (d) Spindle unit should have fixture which provides quick and reliale centering and clamping of the cutting tool or workpiece. (e) The spindle unit must have high static stiffness. Maximum accuracy is influenced y the ending, axial as well as torsional stiffness. (f) The wear resistance of mating surface should e as high as possile. (g) Deformation of the spindle due to heat transmitted to it y workpiece, cutting tool, earings etc. should e as low as possile. Otherwise it will affect the accuracy of the machining accuracy DESIGN OF SPINDLE Figure 13.6 shows schematic diagram of spindle. A spindle represents a shaft with length a which is acted upon y driving force F, and () cantilever of length m acted upon y external force F 1. The spindle is asically designed for ending stiffness which requires that maximum deflection of spindle nose should not exceed a prespecified value, i.e. d max d... (13.8) per F F 1 a m 6 Figure 13.6 : Principle of Working of Spindle The total deflection of spindle nose consists of deflection d 1 of the spindle axis due to ending forces F 1 and F and deflection d of the spindle axis due to compliance of the spindle supports. When the spindle has tapered hole in which a center or cutting tool is mounted, the total deflection of the center or cutting tool consists of deflections d 1, d and d 3 of the center or cutting tool due to compliance of the tapered joint.

11 Deflection of Spindle Axis due to Bending To calculate the deflection of the spindle nose due to ending, one must estalish a proper design diagram. The following guidelines may e used in this regard. () If the spindle is supported on a single anti-friction earing at each end, it may e represented as a simply supported eam, and If the spindle is supported in a sleeve earing, the supported journal is analyzed as a eam on an elastic foundation; for the purpose of the design diagram the sleeve earing is replaced y a simple hinged support and a reactive moment M r acting at the middle of the sleeve earing. The reactive moment is given as : M r C. M M ending moment at the support, and C constant 0 for small loads and 0.3 to 0.35 for heavy load. Design of Guideways and Spindle F F 1 k m M r F F 1 () M r (c) d 1 Figure 13.7 : Effect of Various Force on Spindle Figure 13.7 shows schematic diagram of spindle. Figure 13.7() depicts the design diagram of the spindle and figure 13.7(c) illustrates deflected axis of the spindle. Consider the spindle shown in Figure By replacing the rear all earing y a hinge and the front sleeve earing y a hinge and reactive moment M r, the spindle can e reduced to the design diagram as shown in Figure 13.7(). The deflection at the free end of the eam (spindle nose) can e determined y Macaulay s method and is found out to e 1 k d F m ( a + m) 0.5F km 1 M a 1 1 3EI a E is Young s modulus of the spindle material. I a is average moment of inertia of the spindle section. The deflection of the eam is shown in Figure 13.7(c). r am... (13.9) Deflection of Spindle Axis due to Compliance of Spindle Supports Let δ E and δ G represent the displacement of the rear and front support respectively. Owing to compliance support, the spindle deflects are shown in Figure From similarity of triangles OHH and OGG 7

12 Advances in Tool Engineering and Management δ m + x x d G m d 1 + δg x... (13.30) E δε G H O E (c) d G Figure 13.8 : Deflection of the Spindle due to Compliance of Support From similarity of triangles OEE and OGG, we get H δg δe x a x aδg x δ +δ E G On sustituting these values of x in Eq. (13.30), d changes to, m d 1 + δ G + δe a m a... (13.31) Hence it is clear from aove equation that displacement δ G of the front earing has greater influence upon deflection d of spindle nose than displacement δe of the rear earing. Displacement δ E R S E E and δ G R S G G Where R E and R G are the support reactions at E and G respectively. S E and S G are stiffness at E and G respectively. At equilirium, M E 0 R a Fk + M F1 ( m + a ) 0 G r R G F k M r + F1 ( m + a ) a Similarly M G 0 8 R a F M + F1m 0 E r

13 ( F + M F m) a r 1 RE Fk Mr + F1( m + a) m F + Mr F1m m Deflection d a S a a S a... (13.3) Hence total deflection d (shown in Figure 13.9) is otained as d d1 + d B A Design of Guideways and Spindle d d d 1 Figure 13.9 : Total Deflection of Spindle Axis Optimum Spacing etween Spindle Supports The ratio τ is an important parameter in spindle design. a Where, τ. The optimum value of this ratio is the one that makes sure that minimum m total deflection d can e determined from differentiating it partially with respect to τ. dd For minimum deflection d, 0 d τ. The point of the minimum of the d 1 + d curve, gives the optimum value of ratio a/m which generally lie etween 3 and 5. The value of τ opt depends upon ratio of stiffness SE SG Im of the front and rear earings, η and factor J S S I. Where S C 3EIm 3 ending stiffness of the cantilever. m I m average moment of inertia of the spindle over cantilever. I a average moment of inertia of the spindle over the supported length. G An opposite constraint on maximum span stems from the requirement that for normal functioning of the spindle driving gear, the stiffness of the span should not e less than N/µm. This constraint is expressed through the following relationship : C a a i 4/3 D a 1/3 D a average diameter of the supported length of the spindle, i 0.05 for normal accuracy machine tools, and 0.1 for precision machine tools. 9

14 Advances in Tool Engineering and Management When the spindle is supported on hydrostatic journal earings, the maximum deflection at the middle of the span should satisfy the condition: 4 d a max 10 a And the maximum span length a max should e limited y the aove constraint. This constraint is ased upon the requirement that the maximum misalignment due to deflection of the journals should not exceed one third of the earing gap Deflection due to Compliance of the Tapered Joint The spindle ends of most machine tools have a tapered hole for accommodating a center (in lathe) or cutting tool shanks (in milling and drilling machine). The deflection of center or shank at a distance d from the spindle axis the force F is acting is given y equation : d3 Δ+φ d Δ displacement of the shank or center at the edge of taper due to contact compliance, and φ angle of slope of the shank or at the edge of taper. If manufacturing the errors of the taper are ignored, Δ and φ can e calculated from following equations : 4ψ DC1 Δ ( ψ dc + C3), μm πd 4Fψ C1 φ ( ψ dc4 + C) πd C 1 coefficients of contact compliance C, C 3, C 4 coefficients that account for the diameter variation along the length of taper. 1 ψ.3c1 D D and d are expressed in cm. 4 1/4,cm Generally displacement Δ due to contact compliance can e ignored in comparison with the displacement due to ending of the shank or center Hence, d 4Fψ C1 3 dc4 ( ψ + C ) d, µm... (13.33) πd Permissile Deflection and Design for Stiffness The deflection d1, d and d3 are calculated from equations (13.9), (13.3) and (13.33) respectively. The total deflection of the spindle nose can e determined as the sum of three deflections. The design of stiffness can e carried out in accordance with equation (13.8). The permissile deflection of the spindle nose d per depends upon the machining accuracy required for the machine tool. Generally, it should e less than one-third of the maximum permissile tolerance on radial run out of the spindle nose. The machining accuracy depends not only upon the radial stiffness ut also upon its axial and torsional stiffness. The axial displacement of the spindle unit consists of the axial deformation of the spindle and the deformation of the spindle thrust earing. The torsional stiffness of machine tool spindles significantly influences the machining accuracy in metal removal operations, such as gear and thread cutting in which the feed and primary cutting motions are kinematically linked. The torsional deformation of the spindle unit consists of the

15 deformation of the spindle and deformation of drive components. As stated earlier, spindles are designed for stiffness, primarily radial. However in heavily loaded spindles the stiffness design must e proved y a strength check against fatigue failure. The strength check requires that f f min f factor of safety against fatigue failure. f min minimum value of safety factor 1.3 to 1.5 If spindles are sujected to comined ending and torsion, the factor of safety f is determined from the equation, (1 η 4 ) d 3 e σ f e... (13.34) 10 ( km ) ( T ) + di η ratio of the internal diameter to the external d e d i internal diameter of the spindle, mm d e external diameter of the spindle, mm σ Endurance limit of the spindle material, N/mm e M mean value of the ending moment acting on the spindle, N. mm T mean value of the torque acting on the spindle, N. mm k coefficient that accounts for variation of ending moment and stress concentration. coefficient that accounts for variation of torque and stress concentration. Coefficient k can e determined from the following expression : k U (1 + Q) σ dynamic stress concentration coefficient for normal stresses U σ 1.7 to.0 Ma Q ratio of amplitude of the ending moment to its average value. M Coefficient can e calculated from the following formula : σe + U Q σ τ t σ yield stress of the spindle material. y Q t y U τ dynamic stress concentration coefficient for shearing stress. 1.7 to.0 T a ratio of amplitude of torque to its average value. T... (13.35) The values of Q and Q t generally depend upon the machining conditions. For example, in a super-finishing operation there is virtually no variation of the ending moment and torque, i.e. M a T a 0 and therefore Q Q t 0. SAQ What are functions and requirements of spindle? Design of Guideways and Spindle 31

16 Advances in Tool Engineering and Management () (c) Design the spindle for machine tool. Explain the deflection of spindle due to ending and spindle support SUMMARY The guideway is used to ensure moving of the cutting tool or machine tool operative element along predetermined path. Slideways and anti friction slideways are two types of guideways. Slideway is asically designed for wear resistance and stiffness. The pressure acting on mating surfaces create wear. Hence it is essential to resist wear. The design for stiffness specifies that the deflection of cutting edge due to contact deformation of slideways should not exceed permissile value, which is required to maintain the accuracy. Spindle is generally used for centering and clamping the workpiece, and imparting rotary motion. Spindle is asically designed for ending stiffness KEY WORDS Guideways : Its function is to ensure that the cutting tool or machine tool operative element moves along predetermined path. 3