Changqing Cheng Satish T.S. Bukkapatnam
Department of Industrial & Systems Engineering
Texas A & M University
Dynamics of Ultra-precision Machining and Its
Effect on Surface Roughness
Outline
Introduction
Why ultra-precision machining (UPM)
Background and literature review
Research challenge and gap
Methodology
Physics-based model: process dynamics analysis using
delayed differential equation
Statistical model: sensor-based surface roughness
estimation
Results
Summary and future work
2
Introduction
Development of sensor-based technique: especially in
advanced manufacturing processes control
Aerospace, biomedical, electronics, automotive et al. (Lee 1999,
Liu 2004)
Taniguchi curve: nano-metric manufacturing accuracy
Industry relies on ultra-precision machining (UPM) to realize
surface roughness (Ra) at 10 nm โ 30 um
3
Taniguchi 1983
Challenge in UPM 4
Quality issue: anomaly development (even in well-designed process) cannot be predicted
Physics-based models: cannot predict surface change
Cutting mechanics: cutting stresses (Marsh 2005); micro-plasticity effect (Yuan 1994, Lee 2001); tool interference (Cheung 2003);
material recovery and swelling effect (Kong 2006)
Micro-physics: crystallographic orientation of the grain (Lee
2000); metrology and process physics (Dornfeld et al. 2006)
Spectrum analysis: Ra spectrum component (Cheung and Lee
2000, Pandit and Shaw 1981, Hocheng et al. 2004)
Prahalad 2013 High-definition optical inspection
Surface roughness models
Sensor-based analytics models: applicable for real-time
Ra estimation
Vibration analysis: vibration amplitude and frequency (Lin 1998,
Abouelatta and Madl 2001, Liu 2004)
Acoustic emission (Beggan et al. 1999)
Temperature sensor (Hayashi et al. 2008)
Strain gauge sensor (Shinno et al. 1997)
Limited by nonlinear and nonstationary nature of machining
signals
5
Cheng et al. 2014
Approach
Physics domain model: consider tool radius effect,
ploughing and shearing effect, elastic material recovery; predict
system dynamic response
Sensor-based model: extract information from in situ
signals; detect change in the process
6
Process dynamics
3 types of vibrations: free vibration, forced vibration, and
self-excited vibration (chatter) (Tobias 1961)
Frictional chatter: ploughing on the work-piece surface
Regenerative chatter: overlapping cuts; source for vibration
amplification
Bring system to instability
Result in inferior part surface and increase tool wear
Most undesirable and least controllable (Quintana et al 2011)
How to model the chatter at UPM is still not well
addressed.
8
Phase difference = 0 Phase difference = ๐
Quintana et al 2011
UPM dynamics model
๐ฆ ๐ก + 2ํ๐๐๐ฆ ๐ก + ๐๐2๐ฆ ๐ก = โ
๐น ๐ฆ ๐ก ,๐ฆ ๐กโ๐
๐
๐ฆ: tool displacement
๐: period length, ๐ =1
ฮฉ
Process parameters: feed ๐0, spindle speed ฮฉ, chip width ๐ค
Thrust force model: shearing and ploughing components
9
๐0 โฅ ๐๐๐๐: material removal; both
shearing and ploughing forces exist
๐0 < ๐๐๐๐ : only elastic deformation
and ploughing force
๐0
Shearing Force
Dynamic chip thickness
๐ก๐ข = (๐0 โ ๐ฆ ๐ก + ๐ฆ(๐ก โ ๐)) โ ๐ฟ
Shearing angle (Waldorf et al 1999)
10
๐ = tanโ1๐0 โ ๐ฟ
๐ tan๐4+
๐ผ2
+๐ฟ
tan(๐2+ ๐ผ)
โ 2๐ ๐ฟ โ ๐ฟ2 + ๐ก๐/ cos ๐ผ โ ๐0 tan ๐ผ
Thrust Force
Shearing component
Shearing force parallel to shearing plane
(๐ค: chip width/ depth of cut)
Normal force on shearing plane
๐น๐ = ๐น๐ [1 + 2(๐
4โ ๐)]
Contribution to thrust force
๐น๐ก(1) = ๐น๐ cos ๐ โ ๐น๐ sin๐ = ๐๐ค 1 +๐
2โ 2๐ cot ๐ โ 1 ๐ก๐ข
Ploughing component (Waldorf 1999)
Elastic model: cylinder indentation on an elastic surface
Contribution to thrust force
11
๐น๐ = ๐๐ก๐ข๐ค/ sin๐
๐น๐ก(1)
๐= ๐ด๐๐ค๐ก๐ข
Waldorf et al. 1999 ๐น๐ก(2) =
2.375๐๐ค๐ธ
8 1 โ ๐2 ๐ฟ
๐น๐ก(2)
๐= ๐ต๐ฟ
Process dynamics
Delayed differential equation for tool dynamics
๐ฆ ๐ก + 2ํ๐๐๐ฆ ๐ก + ๐๐2๐ฆ ๐ก = โ
๐น๐ก(1) + ๐น๐ก(2)
๐= โ๐ด๐ก๐ข โ ๐ต๐ฟ
= โ๐ด๐๐ค๐0 โ ๐ด๐๐ค ๐ฆ ๐ก โ ๐ฆ ๐ก โ ๐ + ๐ฟ ๐ด๐๐ค โ ๐ต= โ๐ด๐๐ค ๐ฆ ๐ก โ ๐ฆ ๐ก โ ๐ + ๐ถ
No closed-form solution to dynamics state: ๐ฆ(๐ก)
Temporal finite element model can be used for
approximation (Bayly et al. 2003, Khasawneh et al. 2009)
The time per revolution ๐ป is divided into ๐ด elements
Approximation to the solution for the tool displacement on
each element
๐ฆ๐๐ ๐ = ๐๐๐
๐๐๐(๐) 4๐=1 ๐ฆ๐
๐โ1 ๐ = ๐๐๐๐โ1๐๐(๐)
4๐=1
๐: local time, 0 โค ๐ โค ๐ก๐; ๐ก๐: time for element ๐, ๐ก๐ =๐
๐
12
Temporal finite element model (TFEM)
Hermite basis functions (Mann et al. 2006)
13
Displacement: ๐11 ๐13/๐21 ๐23
๐1 ๐ = 1 โ 3๐
๐ก๐
2
+ 2๐
๐ก๐
3
๐2 ๐ = ๐ก๐๐
๐ก๐โ 2
๐
๐ก๐
2
+๐
๐ก๐
3
๐3 ๐ = 3๐
๐ก๐
2
โ 2๐
๐ก๐
3
๐4 ๐ = ๐ก๐ โ๐
๐ก๐
2
+๐
๐ก๐
3
โข Orthogonal; second-order
continuous
โข Coefficients represent the state
variable (displacement and
velocity) at the beginning/end of
each element
โข Boundary conditions
Velocity: ๐12 ๐14/๐22 ๐24
TFEM
Approximation leads to non-zero error
๐๐๐๐๐ ๐
4
๐=1
+ 2ํ๐๐ ๐๐๐๐๐ ๐
4
๐=1
+ ๐๐2 ๐๐๐
๐๐๐
4
๐=1
+ ๐ด๐๐ค ๐๐๐๐๐๐
4
๐=1
โ ๐๐๐๐โ1๐๐
4
๐=1
โ ๐ถ = ๐๐๐๐๐
Method of weighted residuals (Reddy 1993)
Independent trial functions: ๐1 = 1 ๐2 =2๐
๐ก๐โ 1
๐๐๐๐๐ ๐
4
๐=1
+ 2ํ๐๐ ๐๐๐๐๐ ๐
4
๐=1
+ ๐๐2 ๐๐๐
๐๐๐
4
๐=1
๐ก๐
0
+ ๐ด๐๐ค ๐๐๐๐๐๐
4
๐=1
โ ๐๐๐๐โ1๐๐
4
๐=1
โ ๐ถ ๐๐ ๐ ๐๐ = 0
14
TFEM
๐๐๐๐ ๐ ๐ + 2ํ๐๐๐ ๐ + ๐๐
2 + ๐ด๐๐ค ๐๐
๐ก๐
0
4
๐=1
๐๐ ๐ ๐๐
= ๐๐๐๐โ1 ๐ด๐๐ค๐๐
๐ก๐
0
4
๐=1
๐๐ ๐ ๐๐ + ๐ถ๐๐ ๐ ๐๐ ๐ก๐
0
15
๐๐๐๐
4
๐=1
๐๐๐๐
= ๐๐๐๐โ1๐๐๐
๐
4
๐=1
+ ๐๐๐
๐๐๐๐
= ๐ด๐๐ค๐๐
๐ก๐
0
๐๐ ๐ ๐๐
๐๐๐= ๐ถ๐๐ ๐ ๐๐
๐ก๐0
๐๐๐๐
= ๐ ๐ + 2ํ๐๐๐ ๐ + ๐๐2 + ๐ด๐๐ค ๐๐
๐ก๐
0
TFEM
๐ = 2; boundary continuous conditions
Matrix format 1 0 00 1 0
๐11 ๐12 ๐13
0 0 00 0 0
๐14 0 0๐21 ๐22 ๐23
0 0 ๐11
0 0 ๐21
๐24 0 0๐12 ๐13 ๐14
๐22 ๐23 ๐24
๐11 ๐12
๐21๐22
๐23
๐24
๐
=
0 0 00 0 0
๐11 ๐12 ๐13
0 1 00 0 1
๐14 0 0๐21 ๐22 ๐23
0 0 ๐11
0 0 ๐21
๐24 0 0๐12 ๐13 ๐14
๐22 ๐23 ๐24
๐11 ๐12
๐21๐22
๐23
๐24
๐โ1
+
00๐1
๐2
๐1
๐2
๐ต๐๐ = ๐ท๐๐โ1 + ๐ธ
16
Stability analysis
๐๐ = ๐ฎ๐๐โ1 + ๐ตโ1๐ธ (๐ฎ = ๐ตโ1๐ท Monodromy operator )
Criterion: asymptotic stability requires eigenvalues of
๐ฎ within the unit circle of the complex plane
Maximum absolute eigenvalues< ๐
Stability lobe diagram
Map the area of stability as a function of the machining
parameters (feed, depth of cut, and spindle speed)
Identify the optimum conditions that maximize the chatter-
free material removal rate and avoid inferior surface
17
Sensitivity analysis
Stability sensitivity on ๐ฟ with perturbation ๐๐ฟ
tan๐โฒ = tan๐ + โ๐ tan๐
4+
๐ผ
2โ
๐ก๐cos ๐ผ
+ ๐0 cot ๐ผ +๐0๐ + ๐ โ ๐0 ๐ฟ
2๐ ๐ฟ โ ๐ฟ2๐๐ฟ
๐ดโฒ โ1 +
๐2
tan๐+
2 tan๐ 2
3โ 3
+2 tan๐ (๐0๐ + ๐ โ ๐0 ๐ฟ)
3 2๐ ๐ฟ โ ๐ฟ2โ
(1 +๐2)(๐0๐ + ๐ โ ๐0 ๐ฟ)
2๐ ๐ฟ โ ๐ฟ2 tan๐ 2๐๐ฟ
18
๐ถ0 โช tan๐, ๐ถ0 = 0
Sensitivity analysis
2-element maximum eigenvalue analysis
๐โฒ๐๐๐ฅ โ ๐๐๐๐ฅ +tan ๐โฒ 2
tan ๐+๐0๐ + ๐ โ๐0 ๐ฟ
2๐ ๐ฟโ๐ฟ2๐๐ฟ
1+2๐
4โ๐โฒ
๐ก๐๐๐โฒโ 1 1 +
๐
2+
tan ๐ 2๐ ๐ฟโ๐ฟ2
1+๐
2๐0๐ + ๐ โ๐0 ๐ฟ
๐๐ฟ
Given ๐๐ฟ = ยฑ0.2๐ฟ, ๐๐๐๐ฅโฒ โ ๐๐๐๐ฅ ยฑ 0.08
Uncertainty for stability boundary: ๐๐๐๐ฅ close to 1
19
UPM experiment setup
Face turning of aluminum alloy disk-shaped workpiece
Cutting tools: polycrystalline diamond (๐ = 60 ๐ข๐)
Vibration sensors: Kistler 8782A500
Force sensors: 3-axis piezoelectric dynamometer Kistler
A9251A
20
Cutting Force Sensors
Marks on sample
Cutting Tool
Acoustic Emission Sensor
Vibration Sensors
Feed = 12 / 6 um per revolution 21
โข ๐ ๐ measured from MicroXAMยฎ for chatter identification
โข ๐ ๐ > 100 ๐๐: onset of the chatter
Feed = 0.75 um per revolution 23
โข ๐0 < 0.75 ๐ข๐/revolution: no material removal/chip formation;
only ploughing; high surface roughness
Summary for physics-based model
Delayed differential equation (DDE) with temporal finite
element model (TFEM)
Investigate the process dynamics for UPM
Consider the dynamic shearing and ploughing forces at
nano-scale machining
Can identify optimum conditions, tending to generate low
surface roughness
Challenges
Surface roughness Ra varies according to chip formation
process and other uncontrollable factors even under
optimum conditions
Ra variation monitoring in the incipient stages in real-time
given process parameters; vital for nano-metric range finish
assurance
24
Physics-based sensor fusion technique for Ra real-time estimation
Sensor-based model
Feature extraction
Difficult to evaluate UPM process from the raw time series
signals
Transform time series into feature space with reliable,
effective and accurate features
Identify the patterns hidden into the raw signals
25
State space reconstruction 27 Recurrence quantification analysis
๐ฅ Nonlinear vibration signal
Recurrence quantification analysis
Threshold recurrence plot
RQA extraction
Nonlinear Dynamic Characterization
Time delay ๐
Embedded dimension ๐
State space
reconstruction
๐
PCA
Total number of variables: 3 + 6 ร 9 = 57
The first 3 principal components explained over 70% of
the total variance
Contribution to the 1st principal component
28
๐๐๐_๐
๐๐๐_๐ ๐น_๐ ๐น_๐
Gaussian process (GP) model
Mapping between input ๐ โ ๐ ๐๐ and z โ ๐
z = ๐ ๐ + ํ ํ โผ ๐(0, ๐12)
Without explicit functional form, covariance structure can be used to represent the function value distribution
Z โผ ๐(0, ๐พ ๐, ๐ + ๐12๐ผ)
๐ = ๐ 1, ๐ 2, โฆ , ๐ ๐๐ and ๐ = [๐ง1, ๐ง2, โฆ , ๐ง๐๐
]
Covariance matrix ๐พ๐๐ = ๐๐(๐ ๐ , ๐ ๐), ๐ hyperparameters to be estimated
Squared exponential form
๐๐ ๐ ๐ , ๐ ๐ = ๐02 exp โ
(๐ ๐โ๐ ๐)๐๐(๐ ๐โ๐ ๐)
2+ ๐1
2
๐02: process variance
๐12: noise variance
๐ = ๐๐๐๐ ๐ โ2: length scale in each input direction
Infinitely differentiable; close points are highly correlated
Log likelihood function to optimize the hyperparameters
29
GP prediction
At new input ๐ โ โ ๐ ๐๐ , the noise-free prediction ๐โ is given
by the first two moments
Can predict a complete distribution ๐พ ๐, ๐ โ : ๐๐ ร 1 vector, each element is the covariance
between ๐ โ and one sample point
Mean: linear combination of the observation values
Covariance: difference between prior covariance and the
information explained
30
),()),((),(),()cov(
)),((),(
*
12
1****
12
1**
sSKISSKsSKssKf
ZISSKsSKf
T
T
30
Estimation result 31
Accuracy of the fitting
Over 85% of measured Ra values are within the 2-sigma
prediction band
Summary and future work
Summary
Physics-based model can predict chatter onset according
to process parameters; not applicable for real-time Ra
estimation
Physics-based statistical model can estimate the surface
roughness with accuracy over 80%
Future work
Cutting speed and thermal effects on the thrust force
Built-up edge effect: dead metal cap on tool edge
Uncertainty in the stability analysis
32