Error Modeling - MIT

2/21/17 © 2016 Alexander Slocum 2-1

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

Ideal path

Non-perfect rollersNon-perfect surface

Stra

ight

ness

err

or

“Bumpy path” as measured by sensorH

igh

freq

uenc

y st

raig

htne

ss e

rror

(s

moo

thne

ss)

Y

X

“Bumps” caused by axis reversal

Circle traced out by X and Y axes moving with

sine and cosine paths

“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


Error Modeling: Stick Figures

•  Stick figure: –  The sticks join at centers of stiffness,

mass, friction, and help to: •  Define the sensitive directions in a

machine •  Locate coordinate systems•  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


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

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)


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!

0.00E+00

5.00E-08

1.00E-07

1.50E-07

2.00E-07

2.50E-07

1.00E-03 1.00E-02 1.00E-01Wavelength (m)

Am

plitu

de (m

)Overall bow in railSurface finish effects

Peaks likely due to rolling elements (ball and cam roller surface errors)

50

40

30

20

10

00 1000 2000 3000 4000 5000 6000

Rail length (mm)

Para

llelis

m P

(mm

) Normal (N)

High (H)

Precision (P)

Super precision (SP)Ultra Precision (UP)

// P

// P

// P

Roll: εx Assume all bearings on each rail move vertically in an opposite direction

Yaw: εy Assume front and rear bearing pairs move in opposite horizontal directions

Pitch: εz Assume front and rear bearing pairs move in opposite vertical directions

2/21/17 © 2016 Alexander Slocum 2-6 X

qX (Roll)qZ (Pitch)

qY (Yaw)

Z

Y

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 •  Random error motion may typically be 10% of the running parallelism

XY

Z

XY

Z

Horizontal Straightness: δy Assume all bearings move horizontally

Vertical Straightness: δz Assume all bearings move vertically

wL

δh

δv







XY

Z

XY

Z

Horizontal Straightness: δy Assume all bearings move horizontally

Vertical Straightness: δz Assume all bearings move vertically

wL

δh

δv

LX LZ εx = 2δ/LZ

εZ = 2δ/LX

εY = 2δ/LX

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: –  εx = 2δ/LZ

–  εY = 2δ/LX

–  εZ = 2δ/LX

•  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 –  εx = 2δ/LY

•  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


-Z

X

Y

50

40

30

20

10

00 1000 2000 3000 4000 5000 6000

Rail length (mm)

Para

llelis

m P

(mm

) Normal (N)

High (H)

Precision (P)

Super precision (SP)Ultra Precision (UP)

// P

// P

// P


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…

250200150100500

0

100

200

300

400

500

Frequency (Hz)

Dis

plac

emen

t (na

nom

eter

s)

Displacement due to machine deformation

Average Error Motion Fundamental Error Motion

Total Error Motion

PC center

MRS center

Inner motion

MRS center error motion value

PC center error motion value

Outer 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 [[email protected]] 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

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


δAbbe = aαδAbbe with sensor = (a-b)α

α is “scaled” by (a-b)/a

a b


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 added or removed by coolant systems

CoolantsElectronic Hydraulic Frame Cutting Lubricating systems oil stabilizing oilfluid

Roomenvironment

PeopleHeat created by the machine

Electricaland

electronic

Frame stabilizationMotors, transducersAmplifiers, control cabnetsSpindle bearingsOther

HydraulicMiscellaneous

Friction

Heat created by the cutting process

Conduction Convection Radiation Conduction Convection Radiation Conduction Convection Radiation

Temperature gradiantsor static effects

Temperature variationsor dynamic effects

Nonuniform temperatures Memory of previousenvironment

Uniform temperatureother than 20 degrees C

Part Master Frame

Form error Size error

Total thermal error

Station-changeeffect

Heatsource/sinks

Heatflowpaths

Temperaturefield

Affected Structure

Error components


Thermal Growth Errors •  Very troublesome

–  They are always changing –  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


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

–  Axial expansion in a gradient (one end stays at temperature, while the other end changes)

–  For a meter tall cast iron structure in a 1 Co/m gradient, δ= 5.5 µm•  This is a very conservative estimate, because the column will diffuse the heat to lessen the

gradient

L Tδ α= Δ

( )1 2

2L T Tα

δ−

=


Thermal Growth Errors: Bimaterial Effect •  Deformation of a bimaterial plate moved from one uniform temperature to another:

•  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/Co, 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

1 2

1 1 2 21 2

1 1 2 21

1 2

1 1 2 21 2

1 1 2 21

24 1 1

22 1 1

2

LT

E I E It t

t E A E A

LT

E I E It tt E A E A

α αδ

α αα

− Δ=

+ ⎛ ⎞+ + +⎜ ⎟

⎝ ⎠

− Δ=

++ ⎛ ⎞+ +⎜ ⎟

⎝ ⎠


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


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?


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:

–  Slope error at the ends of the beam (α=M(l/2)/EI):

–  For a 1x1x0.3 m cast iron surface plate with ΔT=1/3 Co (1 Co/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

( )2

222 2

T

T

y y Th

EIM

LM TLEI h

αε

ρ

ρ

αδ

Δ= =

=

Δ= =

2TTLh

αθ

Δ=


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)


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!

Workpiecezone

Wheelzone

Insulation layer (5 mm foam)Sheet metal trough

Flood coolantFlood coolant


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

Effectivewidth

Face-to-face

Effective width

CL

Contact angle

Back-to-back


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?

250200150100500

0

100

200

300

400

500

Frequency (Hz)

Dis

plac

emen

t (na

nom

eter

s)

Displacement due to machine deformation


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!


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

0 0

Tem

pera

ture

Part

err

or

Time

Δ T environmentPart errorΔ T top and bottom structureetc.

.

.

.

Sliding bearing lubricator cycle


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!

0.00E+00

5.00E-08

1.00E-07

1.50E-07

2.00E-07

2.50E-07

1.00E-03 1.00E-02 1.00E-01Wavelength (m)

Am

plitu

de (m

)

Overall bow in railSurface finish effects

Peaks likely due to rolling elements (ball and cam roller surface errors)

Error Modeling - MIT

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