1. 4. MECHANICAL ERRORS

2. 4.1 Introduction Shirley and Jaikumar identified error sources with respect to part and machine contributions to systematic and random/dynamic errors. Mechanical errors can be associated with the part, the machine or the process. Static and Measurable errors compliance; the most pervasive source of error / elastic distortion motion error ; angular, positional errors and squareness, straightness and parallelism of axis motion due to machine element behavior. setup errors ; incorrectly position the tool tip in the tool holder, or fail to square the fixture with the machine axes are mechanical errors. programming errors measurement errors Dynamic errors are due primarily to motion and the variations of forces displacements and acceleration due to that motion. Vibration / distortion of the cutting tool or fixtures / work material variation burr formation / built up edge (BUE) property variation process characteristics

3. 4.2 Errors due to machine elements Machine structural components and their orientation and relative motion are major contributors to mechanical errors.

4. 4.2 Errors due to machine elements

5. Figure 4.3 illustrates the effect of non-parallel motion caused by straight but non-parallel guideways positioning a spindle above a worktable. A positional error of O 1 O‘ occurs for a vertical column motion of h at a radius r from the column. Clearly, increasing h will increase the error. And, the displacement measuring hardware is invariably mounted on the column of the machine introducing a substantial Abbé error. 4.2 Errors due to machine elements

6. “Crosswind” is defined as being the vertical displacement of one end of the front slideway with respect to the rear slideway. The effect of this, as the saddle is traversed along the slideway, is to move the cutting tool radially with respect to the workpiece.

7. 4.2 Errors due to machine elements It can be seen that the relationship between δh and z is nonlinear and the effect of crosswind on the geometric shape of the workpiece is to produce a non-linear taper.

8. 4.3 Kinematic design When machine base is over constrained (a non-kinematic mounting), distortion of the floor (due to swelling of concrete during a “wet” season) will cause the base to distort. This distortion is a part form error due to the Abbé offset error.

9. 4.3 Kinematic design 4.3.1 Connectivity “connectivity between structural elements” - for example between a spindle and the frame or bed of a lathe. kinematic design - deterministic, less reliance on manufacturing of the components, limit to the load capacity and stiffness. - has concentrated loads at points and can experience local deformation. - is preferable to replace excessive surface and line contact with point constraints. elastically averaged design - nondeterministic, heavier reliance on the particulars of the manufacturing process, no limit on the load capacity and stiffness. - over-constrained as in Figure 4.6 - susceptible to warping due to uneven thermal expansion.

10. 4.3 Kinematic design Kinematic three point support eliminates the problem of machine base distortion due to uneven foundation support. While the machine may not stay level, it will not see any distortion of its frame due to the foundation support.

11. 4.3.2 Kinematic elements Lord Kelvin / Clark Maxwell One point holds the body with respect to translation in x; two points hold the body with respect to translation in y and rotation about y; three points hold the body with respect to translation in z and rotation about x and z. “when a rigid body is constrained by more than the necessary minimum number of constraints, the redundant constraints will cause strain which results in distortion, wear, and higher than necessary cost to achieve a specified precision.”

12. Good kinematic design seeks to utilize only the minimum number of degrees of freedom to constrain the motion of a body. Thus only six constraints in the correct positions are necessary to define and fully locate a rigid body positionally with respect to a fixed frame of reference. The ball and groove in Figure 4.9d is an example of a common machine element. 4.3.2 Kinematic elements

13. Kelvin coupling or clamp ; a device for locating one machine element with respect to another in all degrees of freedom. It is comprised of the ball on plate (one degree of freedom), ball in vee groove (two degrees of freedom), and ball in trihedral nest (three degrees of freedom) elements. 4.3.2 Kinematic elements

14. Three machined intersecting radial grooves, or a trihedral hollow (as by electrical discharge machining or a coining operation). 4.3.2 Kinematic elements

15. The “Stewart platform”; A parallel robot that incorporates six prismatic actuators, commonly hydraulic jacks. > a machine with six degrees of freedom based in six parallel actuators.

16. 4.3.3 Contact and complex support Construct the ways and axes from kinematic design principles. Any clamping or loading of one element to another to provide positive contact and insure stability should be done through kinematic contacts. Clamp design with point contact  This adjustable kinematic ball seat with clamp maintains the kinematic ball in cone design and has the clamping mechanism through the contact.

17. 4.3.3 Contact and complex support Axes can be constructed of kinematic elements as well. The vee slot and balls used to kinematically support the table. It is necessary to carefully fix the center of gravity of the load on the table to insure that the assembly is stable. Also note the axis of reaction necessary to insure smooth linear translation.

18. 4.3.3 Contact and complex support Although Figure 4.14 illustrates an ideal but usable mechanism, Traditional machine tool design relies on more robust designs that tend to relax the kinematic configuration. >> Semi-kinematic. This shows the use of conventional bearing elements in place of the balls.

19. 4.3.3 Contact and complex support These elements are shown in more detail in Figure 4.16. The error due to ball inaccuracy is illustrated along with relative velocities of table and ball carriage. 0.00006in >> 1.5um

20. 4.3.3 Contact and complex support Figure shows a configuration of semi-kinematic slide with bearings

21. 4.3.3 Contact and complex support

22. Figure 4.19 shows a configuration of semi-kinematic slide with a table “hung” for mounting a machine element with a vertical spindle or axis of rotation. Exact kinematic vertical supports are possible although complicated.

23. 4.3.3 Contact and complex support Generally, the most stable is as described here, from Slocum.

24. 4.3.3 Contact and complex support Other less stable configurations are shown here.

25. 4.3.3 Contact and complex support Large planes can be supported kinematically to maintain their insensitivity to disturbances.

26. 4.3.3 Contact and complex support

27. This concept can be applied to more complex assemblies of planar surfaces as for mirror segments for the Keck telescope, Figure 4.24. Figure 4.25 shows this concept applied to the support of a large flat surface. Check your windshield wiper mechanism next time you are driving and notice how this approach insures that the blade maintains contact even with a changing windshield curvature.

28. 4.3.4 Summary of kinematic design The key concerns are: which approach gives a more repeatable motion ? which approach provides the highest stiffness for ideal one degree of freedom motion ? which approach maintains the desired performance and characteristics best over the typical range of usage and wear conditions? what is the relative cost of manufacture for the same level of performance? Kinematic design will improve kinematic precision. -If the center of gravity and axis of reaction concerns are met, kinematic design will reduce the possibility of undesirable motion. -Long ways will also help minimize this. -The point contacts are ideal in a kinematic design, large loads transmitted through these contacts can result in substantial compliance and wear and, in this case, elastic averaging by the use of larger contacts is advantageous. -when ever possible, the number of constraints should be reduced to the minimum achievable.

29. 4.4 Structural compliance 4.4.1 Microscale compliance Hertz contact mechanics, 1896, Germany Equivalent radius of contact R e = (1/R 1 + 1/R 2 ) -1 Equivalent modulus of elasticity E e = (4/3) [(1− μ 1 2 )/E 1 + (1− μ 2 2 )/E 2 ] -1 Equivalent circular contact radius a = (F R e / E e ) 1/3 Deflection δ = a 2 / R e = (F 2 R e / E e 2 ) 1/3 Contact pressure, q = F / πa 2 = F / πδR e or aE e /πR e Example) F=1N, R1=1.5mm, R2=∞, Re=1.5mm, Ee=110 GPa a=25um, δ = 0.4um, if the load is increased by 10%, this results in a 6% increase in deflection. The stiffness k of the system for a 1N load and 0.4μm deformation, is 2.5N/um.

30. 4.4 Structural compliance To avoid surface damage, the contact pressure, q, should be less than τ max for the surface. >> Brinnelling will occur for contact pressures greater than this. If τ max is one half the maximum tensile stress (σ max / 2) and τ max = q/3 then we should insure that the maximum Hertzian contact stress, qmax = (3/2) σ allowable−tensile. This is most serious for contact probes for measurement instruments but can also affect the precision of fixtures and other machine elements. This was the basis of the earlier comment under the discussion of metrology that non-contact techniques are preferred.

31. 4.4.2 Macroscale compliance Bodies supported in either kinematic or non-kinematic fashion will all exhibit distortion due to their compliance (elastic) behavior. A symmetrical body will, ideally, have the same symmetric distortions. Overconstraint of mounts (i.e. non-kinematic) can result in significant nonsymmetrical distortion due to “forced geometrical congruence.” And, if a kinematic mounting is used, its deterministic nature enables the computation of the amount of distortion, and then to design the system to maintain the distortion below a threshold. The problem of machine distortion is enhanced by the physical size of most machines, the mass of their moving elements and the unfortunate fact that they are rarely supported kinematically.

32. 4.4.2 Macroscale compliance Any movement in the concrete due to thermal expansion or absorption of water and subsequent expansion, will cause differential movement of the machine tool and distortion of the ways, etc. >>> One example

33. The major source of static deflection of the machine bed is due to movement of the mass of the workpiece and table over the bed of the machine tool. 4.4.2 Macroscale compliance

34. The two diagrams show the machine deformation under the static load of the table at two locations. The graph shows compliance (inverse stiffness) as a function of position in μm/kN. 4.4.2 Macroscale compliance

35. To effectively design for minimization of compliance, one must consider the stiffness of various machine elements. 4.4.2 Macroscale compliance

36. Unfortunately, most machine elements behave nonlinearly with some process variables. For example, we have already seen that spindle runout varies with rotational speed. A change in a factor of two can be seen over the range of 400 to 1400rpm. 4.4.2 Macroscale compliance

39. When the machine elements are assembled into a machine tool their individual behavior contributes to the overall behavior of the machine tool. This is not the primary sensitive direction with respect to surface finish but does affect part accuracy. Notice the significant change in compliance with load.

40. 4.4.2 Macroscale compliance In some cases, counterweights can be used to offset this change in compliance with position or load.

41. 4.5 Bearings and spindles 4.5.1 Bearings The purpose of bearings is to insure smooth motion of one element to another.

42. 4.5 Bearings and spindles Contact; Sliding and rolling Noncontact; Hydrostatic, aerostatic and magnetic Hydrodynamic COF μ sliding bearings with little of no fluid film (i.e. contact) 0.01 - 0.1 Hydrostatic bearings (as well as hydrodynamic) 0.001 - 0.006 for oil 10 -7 to 10 -8 for air Rolling contact bearings 0.02 - 0.04.

43. 4.5 Bearings and spindles At very slow speed in the high friction a “stick slip” phenomenon can occur. This is often a problem with machine tool axes during complex motion where one axis may slow to zero velocity, change direction and then accelerates (as in a two axis circular motion). This can be avoided, or at least minimized, by using material pairs in bearings that have an increasing frictional resistance (force due to friction) with increasing speed rather than the inverse.

44. 4.5 Bearings and spindles Bearings (in ways and in spindles) are significant sources of error due to compliance. Considerations; accuracy accuracy of the motion of the components supported lateral deviation/repeatability as part of a system under servo control (backlash, etc.) repeatability similar to accuracy (component and system) resolution friction level/smoothness of motion surface finish, shape accuracy short travel ranges (where flexural bearings are generally preferred) Often, bearing are pre-loaded to eliminate problems with backlash of excessive compliance with the following effects: nonlinear deformation higher stiffness reduced backlash, increased repeatability to a point higher the preload, greater component deformation (Hertzian stresses), higher friction, lower resolution and repeatability

45. 4.5 Bearings and spindles An additional row of balls in the bearing reduces the amplitude of compliance but increases the frequency of variation.

46. 4.5 Bearings and spindles i.heavy/medium — which is good for shock and vibration loading conditions, overhanging offset loads in heavy machining conditions generating large cutting forces; ii.medium/light — for lower shock and vibration conditions and machining applications; iii.light — for small vibration environment; iv.very light — precision machines and no overhanging loading environments; v.very light but with clearance — for no load conditions where thermal effects are not important.

47. 4.5 Bearings and spindles Bearings can introduce damping in a structure depending on the type of bearing: sliding contact and fluid bearings best because of viscous nature of film bearing maybe only damping in the system friction stick-slip friction causes tracking errors about zero velocity and limit cycling in servos sliding or dynamic friction helps damp out motion sliding friction causes heat Bearing performance thus further aggravating the thermal problems: influence of temperature on friction properties and, thus, dynamic behavior of machine influence of temperature on accuracy, repeatability and resolution; effect of differential expansion heat transfer characteristics across the bearing; does it act as an insulator or conductor?

48. 4.5 Bearings and spindles General concerns span the range of sensitivity to operating environment: influence of dirt, moisture, etc. on bearing’s performance sealability can environment be effectively sealed out size and configuration size/strength ratios or size/performance ratios weight weight/strength ratios or weight/performance ratios support equipment (to keep the bearing operating) upkeep, lubrication, dry air, pressure regulation maintenance ease of service frequency of service what are MTBF and MTTR values; availability material compatibility interaction/reaction of bearing materials to other materials mounting requirements bearing mounts/supports required life MTBF(Mean time between failure), MTTR(mean time to repair)

49. 4.5 Bearings and spindles These are especially useful at high rotational speeds as utilized increasingly in high speed machining applications.

50. 4.5 Bearings and spindles Hydrostatic bearings have a traditional wide application in both linear and rotational motions. Figures shows a variety of hydrostatic bearing configurations.

51. 4.5 Bearings and spindles Figure shows several practical configurations of hydrostatic guide bearings for machine axes.

52. 4.5.2 Aerostatic bearings and spindles Figure illustrates the construction of a hydrostatic annular thrust bearing. The fluid restrictors insure a balanced and uniform flow of fluid into the pads in the bearings. Balance is needed to insure equal and opposite pad pressures for true running and stiffness.

53. As the gas enters the bearing clearance and eventually exhausts it reduces further to atmospheric pressure Pa at the bearing outlet. Changes in the restriction or clearance between the bearing elements affect the downstream pressure Pd for a given restriction and thus allowing a higher load (for a smaller clearance) to be carried by the bearing. A higher clearance has the opposite effect. There is always a trade off between load capacity and stiffness in the design of the bearing. 4.5.2 Aerostatic bearings and spindles

55. Consideration must be given to driving the table so as not to induce unnecessary load on the bearings and, importantly, for both axes and spindles consideration must be given to the rest positions of the elements when the air pressure is reduced or shut off.