Error Modeling Error Budgets Low cost method to help evaluate concepts before detailed solid models, FEA (which will not catch geometric errors ), because: Nothing is perfect Need to estimate accuracy and repeatability of concepts Need to better predict loads/life of bearings! Start with basics Stick figures Structural loop Error budget spreadsheets Simple error models If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics Roger Bacon Bumpy path as measured by sensor Ideal path Y Circle traced out by X and Y axes moving with sine and cosine paths X Non-perfect rollers Non-perfect surface High frequency straightness error (smoothness) Straightness error Bumps caused by axis reversal 2016 Alexander Slocum 2-1
Stick figure: The sticks join at centers of stiffness, mass, friction, and help to: Define the sensitive directions in a machine Locate coordinate systems Error Modeling: Stick Figures Set the stage for error budgeting The designer is no longer encumbered by cross section size or bearing size It helps to prevent the designer from locking in too early on a concept Error budget and preliminary load analysis can then indicate the required stiffness/load capacity required for each stick and joint Appropriate cross sections and bearings can then be deterministically selected 2016 Alexander Slocum 2-2
Error Modeling: Structural Loops The Structural Loop is the path that a load takes from the tool to the work It contains joints and structural elements that locate the tool with respect to the workpiece It can be represented as a stick-figure to enable a design engineer to create a concept Subtle differences can have a HUGE effect on the performance of a machine The structural loop gives an indication of machine stiffness and accuracy The product of the length of the structural loop and the characteristic manufacturing and component accuracy (e.g., parts per million) is indicative of machine accuracy (ppm) Long-open structural loops have less stiffness and less accuracy 2016 Alexander Slocum 2-3
Structural Loops: Equivalent Springs Structural stiffness between two coordinate systems is modeled as stiffness of the structure between the two (6x6 matrix) & stiffness of the attachment (6x1 vector) 2016 Alexander Slocum 2-4
Bearings Error Motions Bearings are not perfect, and when they move, errors occur in their motion Accuracy standards are known as ABEC (Annular Bearing Engineers Committee) or RBEC (Roller Bearing Engineers Committee) of the American Bearing Manufacturers Association (ABMA) ABEC 3 & RBEC 3 rotary motion ball and roller bearings are common and low cost ABEC 9 & RBEC 9 rotary motion ball and roller bearings are used in high precision machines The International Standards organization (ISO) has a similar standard (ISO 492) An error budget is used to keep track of all the error motions in a machine Remember Abbe and sine errors and how they can amplify bearing angular errors! 50 Parallelism P (mm) 40 30 20 10 0 0 1000 2000 3000 4000 5000 6000 Rail length (mm) Normal (N) High (H) Precision (P) Super precision (SP) Ultra Precision (UP) Amplitude (m) 2.50E-07 2.00E-07 1.50E-07 1.00E-07 // P Peaks likely due to rolling elements (ball and cam roller surface errors) Surface finish effects // P // P Overall bow in rail 5.00E-08 2016 Alexander Slocum 2-5 0.00E+00 1.00E-03 1.00E-02 1.00E-01 Wavelength (m)
Error Motions: Linear Bearings Error motions of a carriage supported by a kinematic arrangement of bearings (exact constraint) can be determined "exactly" Error motions of a carriage supported by an elastically averaged set of bearings can be estimated by assuming the bearings act in pairs Calculations are done using the running parallelism error information from the bearing supplier Running parallelism number is usually a systematic (repeatable) error Roll: εxparallelism Assume all Random error motion may typically be 10% of the running bearings on each rail Roll: εx Assume all move vertically in an bearings on each Z rail x opposite direction Roll: ε Assume all move vertically in an x bearings direction on each rail opposite move vertically in an opposite direction ε = 2δ/L Horizontal Straightness: δy Assume all bearings move horizontally Horizontal Straightness: δ Assume Y y δh move horizontally all bearings δh Z Y Z X YL X w w L Vertical Straightness: δz Assume all bearings move vertically Vertical Straightness: δ Assume all z qz (Pitch) δv bearings move vertically δv Z 2016 Alexander Slocum qy (Yaw) LZ LX ε = 2δ/L Pitch: εz Assume Z and rear bearing X front Pitch: εz Assume pairs move in opposite front and rear bearing vertical directions pairs in opposite Pitch:move εz Assume vertical directions front and rear bearing pairs move in opposite vertical directions εy = 2δ/LX qx (Roll) Yaw: ε Assume y front and rear bearing Yaw: y Assume pairs εmove in opposite front and bearing horizontalrear directions X pairs in opposite Yaw:move εy Assume horizontal directions front and rear bearing 2-6 pairs move in opposite
Error Motions: Linear Bearing Rotation Analysis When all the bearings are in a plane defined by the CS axes, angular error motions are simple to model: ε x = 2δ/L Z ε Y = 2δ/L X ε Z = 2δ/L X When the bearings are not in a plane, a simple algorithm can be fooled Conditional IF statements can be used to avoid issues Roll about X axis in this example is Y ε x = 2δ/L Y REMEMBER Calculations are done using the running parallelism error information from the bearing supplier Running parallelism error is a systematic (repeatable) error Random error motion is typically 10% of the running parallelism Parallelism P (mm) 50 40 30 20 10 0 0 1000 2000 3000 4000 5000 6000 Rail length (mm) Normal (N) High (H) Precision (P) Super precision (SP) Ultra Precision (UP) 2016 Alexander Slocum 2-7 // P // P // P X -Z
Error Motions: Rotary Bearings Standards exist for describing and measuring the errors of an axis of rotation: Axis of Rotation: Methods for Specifying and Testing, ANSI Standard B89.3.4M-1985 Radial, Axial, and Tilt error motions are of concern Upper bound: Radial error motion equals bore roundness Lower Bound: Radial bearings act as elastic averaging elements and radial error motion is bore roundness/averaging factor (3-5 for rolling elements, 10-20 for hydrostatic or aerostatic) Precision Machine Designers measure error motions and use Fourier transforms to determine what is causing the errors MRS center error motion value PC center error motion value PC center 500 Inner motion Displacement (nanometers) 400 300 200 Displacement due to machine deformation MRS center Total Error Motion Outer motion 100 0 0 50 100 150 Frequency (Hz) 200 250 2016 Alexander Slocum 2-8 Average Error Motion Fundamental Error Motion
Error Motions: Rotary Motion Estimates Rotary bearings usually only come with an overall quality rating (e.g., ABEC 9, ISO 5) The rating indicates ID and OD tolerance of the bearing The accuracy of the supported element (e.g., shaft) axis of rotation is usually dominated by the accuracy of the bore, shaft, alignment, and clamping method. Mel Liebers at Professional Instruments [MLiebers@airbearings.com] has tremendous insight on bearing measurement and mounting As he points out, screw-actuated locknuts can also be used to preload a bearing and deform a shaft to correct for errors and thus achieve greater accuracy» E.g. http://www.ame.com/ As a first order estimate, assume the root square sum of the bore and shaft roundness are representative of the radial accuracy of the supported shaft. Similar for axial accuracy Tilt accuracy can be estimated by radial accuracy divided by spacing between bearing sets If just a single bearing set is used, tilt accuracy can be estimated by the flatness of the bearing mount (bore) divided by the bearing pitch diameter 2016 Alexander Slocum 2-9
Sensor Placement Effects If a linear encoder is used to measure an axis position, it can reduce the Abbe error To model the effect of sensor placement, scale the angular error that affects error motions in the direction of sensor action Add sensor location effect as an input in the Error Budget spreadsheet a b δ Abbe = aα δ Abbe with sensor = (a-b)α α is scaled by (a-b)/a 2016 Alexander Slocum 2-10
Thermal Growth Errors: Heat sources and paths There are many different types of thermal errors and paths Thermal effects in manufacturing and metrology (After Bryan.): Heat source/ sinks Room environment Heat added or removed by coolant systems Coolants Electronic Hydraulic Frame Cutting Lubricating systems oil stabilizing fluid oil People Electrical and electronic Friction Heat created by the machine Frame stabilization Motors, transducers Amplifiers, control cabnets Spindle bearings Other Hydraulic Miscellaneous Heat created by the cutting process Heat flow paths Conduction Convection Radiation Conduction Convection Radiatio Conduction Convection Radiatio Temperature gradiants or static effects Temperature variations or dynamic effects Temperature field Uniform temperature other than 20 degrees C Nonuniform temperatures Memory of previous environment Affected Structure Part Master Frame Station-change effect Error components Form error Total thermal error 2016 Alexander Slocum 2-11 Size error
Very troublesome They are always changing Thermal Growth Errors Time constants can be from seconds to many hours Very troublesome because components' heat transfer coefficients can vary from machine to machine Surface finish and joint preload can have an effect Design strategies to minimize effects: Isolate heat sources and temperature control the system Maximize conductivity, OR insulate Combine one of above with mapping and real time error correction May be difficult for thermal errors because of changing boundary conditions. Combine two of the above with a metrology frame 2016 Alexander Slocum 2-12
Thermal Growth Errors: Linear Expansion Simple to estimate Axial expansion of tools, spindles and columns, caused by bulk temperature change ΔT, is often a significant error At least it does not contribute to Abbe errors δ = αlδt Axial expansion in a gradient (one end stays at temperature, while the other end changes) ( 1 2) αl T T δ = 2 For a meter tall cast iron structure in a 1 C o /m gradient, δ= 5.5 µm This is a very conservative estimate, because the column will diffuse the heat to lessen the gradient 2016 Alexander Slocum 2-13
Thermal Growth Errors: Bimaterial Effect Deformation of a bimaterial plate moved from one uniform temperature to another: δ = α = t t 1 2 1 2 ( α1 α 2) ΔT( L ) 2 4( EI 1 1+ EI 2 2) 1 1 + t + + t1 EA 1 1 EA 2 2 ( α1 α 2) ΔT( L ) 2 + t 2( EI 1 1+ EI 2 2) 1 1 + 2 + t1 EA 1 1 EA 2 2 Example: 1m x 1m x 0.3m with 0.03 m wall thickness surface plate If not properly annealed, after top is machined and the bottom retains a 0.5 cm layer of white iron: δ = 0.10 µm/c o, α = 0.41 µrad Similar effects are incurred by steel bearing rails grouted to epoxy granite structures Consider using a symmetrical design (steel on the bottom) to offset this effect Two materials may have similar expansion coefficients, but very different conduction coefficients and density! For a quick estimate of transient effect, assume that the coefficient of expansion of one member is scaled by the ratio of the conduction coefficients Beware placing a precision surface on top of a cabinet that contains electronics! 2 2016 Alexander Slocum 2-14
Thermal Growth Errors: Bimaterial Effect Example: Two size 55 linear guides bolted to a granite bed, later used at a different temperature (e.g., in the summer) How can these errors be counteracted? How can symmetry be used? Does segmenting steel members reduce the effect? Example: steel bearing rails attached to granite beam Two materials may have similar expansion coefficients, but very different conduction coefficients and density! For a quick estimate of transient effect, assume that the coefficient of expansion of one member is scaled by the ratio of the conduction coefficients BiMat.xls Determine thermal errors in a bi-material beam Written by Alex Slocum. Last modified 2016.10.21 by AS Enter numbers in BLACK outputs in RED Be consistent with units Beam and environment Units Value Length of beam: L mm 1000 Change in temperature: DT C 4.0 Cross section 1 (e.g., bearing rails) properties Modulus of Elasticity: E_1 mm 2.00E+05 Coefficient of thermal expansion: alpha_1 1/C 1.20E-05 Enter YES if entering just moment of inertia and X section area, else NO NO User entered values for I, A TOTAL (e.g., if 2 bearing rails) Moment of inertia, I_1ue mm^4 25000 TOTAL (e.g., if 2 bearing rails) Cross section area, A_1ue mm^2 750 TOTAL flange thickness (top + bottom) (0 for rect. beam): ft_1 mm 0 Height: h_1 mm 20 Width: b_1 mm 40 TOTAL web thickness (left + right) (bi=bo for rect. beam): wt_1 mm 0 Moment of inertia: I1 mm^4 26667 Area: Ar1 mm^2 800 Cross section 2 (e.g., structure) properties Modulus of Elasticity: E_2 N/mm^2 6.67E+04 Coefficient of thermal expansion: alpha_2 2.40E-05 Enter YES if entering just moment of inertia and X section area, else NO NO User entered values for I, A TOTAL (e.g., if 2 bearing rails) Moment of inertia, I_2ue mm^4 1800000 TOTAL (e.g., if 2 bearing rails) Cross section area, A_2ue mm^2 2200 TOTAL flange thickness (top + bottom) (0 for rect. beam): ft_2 mm 6 Height: h_2 mm 100 Width: b_2 mm 100 TOTAL web thickness (left + right) (bi=bo for rect. beam): wt_2 mm 6 Moment of inertia: I_2 mm^4 1827092 Area: A_2 mm^2 800 Results Max. displacement error mm -0.148 Max. slope error radians -0.000594 2016 Alexander Slocum 2-15
Thermal Growth Errors: Bimaterial Effect Note its not the ratio of CTEs but the difference! How can these errors be counteracted? How can symmetry be used? Does segmenting steel reduce the effect? What would size 20 linear guides do to a 100mm square aluminum tube? What if there were an axis mounted orthogonal to the tube? 2016 Alexander Slocum 2-16
Thermal Growth Errors: Thermal Gradients One of the most common and insidious thermal errors Beam length = L, height = h, section I, gradient ΔT, straightness error: ε T M y α yδt = = ρ h EI = ρ ( L ) 2 2 M 2 L αδt δ T = = 2EI 2h Slope error at the ends of the beam (α=m(l/2)/ei): αδtl θ T = 2h For a 1x1x0.3 m cast iron surface plate with ΔT=1/3 C o (1 C o /m), δ = 1.5 µm and θt = 6.1 µrad This is a very conservative estimate, because the plate will diffuse the heat to lessen the gradient In a machine tool with coolant on the bed, thermal warping errors can be significant Angular errors are amplified by the height of components attached to the bed 2016 Alexander Slocum 2-17
Thermal Growth Errors: Thermal Gradient Causes of gradients A machine tool structure may be subjected to a flood of temperature controlled fluid Evaporative cooling (common on large grinders and milling machines) Room temperature may vary wildly during the day Overhead lights can create gradients in sensitive structures Plastic PVC curtains are extremely effective at reducing infrared heat transmission A large machine on a deep foundation (relies on the concrete for support), can have problems: Several meters under the ground, the concrete is at constant temperature The top of the machine and the concrete are at room temperature Internal heat sources (motors, spindles, ballscrews, process) 2016 Alexander Slocum 2-18
Thermal Growth Errors: Design Strategies Example Conduction: Use thermal breaks (insulators) Keep the temperature the same in the building all year! Channel heat-carrying fluids (coolant coming off the process) away Convection: Use sheet metal or plastic cowlings to control heat flow Radiation: Plastic PVC curtains (used in supermarkets too!) are very effective at blocking infrared radiation Use indirect lighting outside the curtains, & never turn the lights off! Always ask yourself if symmetry can be used to minimize problems 62.5 grams of prevention is worth a kilo of cure! Workpiece zone Wheel zone Flood coolant Flood coolant Insulation layer (5 mm foam) Sheet metal trough 2016 Alexander Slocum 2-19
Thermal Growth Errors: Symmetry and Thermocentric Design Symmetry: Avoid inducing angular displacements Cause differential expansion to cancel Beware aluminum structure and steel bearing rails Steel on granite can also be an issue if not careful Thermocentric design: Expansion in one direction cancels effect of expansion in another Always ask yourself if symmetry can be used to minimize problems Contact angle Effective width CL Effective width Face-to-face Back-to-back 2016 Alexander Slocum 2-20
Dynamic Errors from Rotating Components Rotating components can cause errors Out-of-balance induced forces (e.g., motors, leadscrews ) Bowed-components (e.g., leadscrews) Modal analysis and frequency analysis can be used to help determine the source of the error The spindle speed was 1680 rpm (28 Hz) Was the 2x speed error from out-of-round bore or shaft, or is it a balancing problem? 500 Displacement (nanometers) 400 300 200 100 Displacement due to machine deformation 0 0 50 100 150 Frequency (Hz) 200 250 2016 Alexander Slocum 2-21
Dynamic Errors from Accelerating Componennts Accelerating components can cause errors Force from the acceleration Impulse from the starting or stopping of an acceleration (jerk) Round those motion profiles! 2016 Alexander Slocum 2-22
Which Error is it? Temperatures of different principle components and locations need to be plotted along side a quality control parameter (e.g., part diameter) In addition, all other functions on the machine should also be plotted E.g., lubricators that squirt oil to bearings every N minutes can cause a sudden temporary expansion of the machine Predictions can be made using fundamental theory or finite element models However, nothing beats real data from a real system The problem lies in interpolating the data Constant adjustment (via SPC) does not address the problem Temperature 0 0 Δ T environment Part error. Δ T top and bottom structure. etc.. Part error Time Sliding bearing lubricator cycle 2016 Alexander Slocum 2-23
The Fourier Transform: Your Error Hunting Companion The Fourier transform, when plotted as error amplitude as a function of wavelength, is an invaluable diagnostic tool It can help identify the dominant sources of error, so design attention can be properly allocated For a system with rolling elements, the center of the rolling element moves πd, while the element that rolls upon it moves 2πD! Amplitude (m) 2.50E-07 2.00E-07 1.50E-07 1.00E-07 5.00E-08 Peaks likely due to rolling elements (ball and cam roller surface errors) Surface finish effects Overall bow in rail 0.00E+00 1.00E-03 1.00E-02 1.00E-01 Wavelength (m) 2016 Alexander Slocum 2-24