Identification of machining force model parameters from acceleration measurements
Bartosz Powalka Szczecin University of Technology, Piastow 19, 70-310 Szczecin, Poland
Jaspreet S. Dhupia*, A. Galip Ulsoyand Reuven Katz Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA E-mail: [email protected] *Corresponding author
Abstract: This paper presents a method, which does not require the use of a force dynamometer, for estimating the mechanistic cutting force model coefficients in milling. The cutting forces are reconstructed from acceleration signals measured during machining, and from experimentally evaluated Frequency Response Functions (FRFs). The quality of reconstruction is improved by the application of the Tikhonov regularisation technique, for which the measurement noise is estimated beforehand. Since the DC component of the FRF and of the acceleration signals is unknown when using piezoelectric accelerometers and impact hammers, the cutting force model estimation requires the knowledge of the cutting force pattern for a particular cutting scenario.
Keywords: cutting forces; regularisation; system identification; stability lobes.
Reference to this paper should be made as follows: Powalka, B., Dhupia, J.S., Ulsoy, A.G. and Katz, R. (2008) ‘Identification of machining force model parameters from acceleration measurements’, Int. J. Manufacturing Research, Vol. 3, No. 3, pp.265–284.
Biographical notes: Bartosz Powalka is a faculty member at Szczecin University of Technology carrying research in the area of machine tool dynamics and manufacturing. He holds a PhD in Mechanical Engineering from Szczecin University of Technology. He was awarded a research grant from the DAAD (German Academic Exchange Service) to carry research activities at Leibnitz Universitaet Hannover and a fellowship from the Polish Science foundation for a research visit at the University of Michigan. His research interests include dynamic modelling, identification and analysis of mechanical systems, e.g., machine tools.
Jaspreet S. Dhupia is a post-doctoral researcher at the University of Michigan carrying research in the area of manufacturing, dynamics and controls. He recently received his Doctorate Degree from the University of Michigan. His research interests include parametric excitation in milling, nonlinearities in mechanical structures and vehicular controls.
266 B. Powalka et al.
A. Galip Ulsoy is the William Clay Ford Professor of Manufacturing at the University of Michigan. He served as the Technical Editor of the ASMEJournal of Dynamic Systems Measurement and Control and the IEEE/ASME Transactions on Mechatronics. He is a member of the National Academy of Engineering, fellow of the American Society of Mechanical Engineers, fellow of Society of Manufacturing Engineers, a senior member of IEEE, a Corresponding Member of CIRP, and President of the American Automatic Control Council. His research interest include dynamic modelling, analysis, and control of mechanical systems; particularly manufacturing and automotive systems.
Reuven Katz is an Associate Research Scientist and a Thrust Area Leader for ‘In Process Metrology’ at the NSF Engineering Research Center for Reconfigurable Manufacturing at the University of Michigan. He holds BSc and MSc from the Technion-Israel Institute of Technology, a PhD in Mechanical Engineering from the University of Michigan and an Executive MBA from the Business School at the University of Tel Aviv. He has over 25 years of industrial experience in the fields of dynamics of machines, machine design, opto-mechanical design, manufacturing systems and R&D project management.
1 Introduction
Prediction of instability in milling is important for the selection of the cutting parameters to avoid premature tool deterioration, and maintain a satisfactory surface finish while retaining high productivity. The most common cause of instability in milling is the regenerative self-excited vibration, commonly known as machine chatter (Tlusty and Polacek, 1963). Presence of machine chatter is graphically represented in the stability lobe diagram that divides the rotational speed and depth-of-cut plane into stable and unstable regions. To evaluate these stability lobes, it is necessary to have information regarding the dynamics of the machine tool and the cutting force model. In practice, the dynamics of the machine tool is expressed in terms of the Frequency Response Function (FRF) measured at the tool tip along the feed and cross-feed directions (Altintas and Budak, 1995). The cutting process is usually described by the mechanistic cutting force model that assumes proportionality of the cutting force to the chip cross-sectional area. Summary of various mechanistic models can be found in (Smith and Tlusty, 1991). The coefficients of proportionality in the mechanistic force model are valid for a particular workpiece material and cutting tool combination.
Usually, the determination of the cutting force model parameters is done by either one of the two broad techniques. The first technique requires an orthogonal milling test in which forces are measured in the principal directions, and then milling cutting coefficients are found by transforming the forces to an oblique cutting process (Armarego and Deshpande, 1991; Budak et al., 1996). The results from such a test are also used to develop databases of cutting coefficients. However, the cutting coefficients evaluated for workpiece materials from different vendors, and even from different batches, may be inconsistent. The second technique involves cutting forces measured by dynamometer and then coefficients of the force model are estimated to fit analytically expressed cutting forces to the experimental ones (Kline et al., 1982; Sutherland and DeVor, 1986;
Identification of machining force model parameters 267
Feng and Menq, 1994). While these techniques are suitable for research laboratories, they can lead to many challenges in industry. These cutting tests can be challenging if the workpiece has complex geometry, e.g., has surface discontinuities. A method to overcome this challenge was presented in Jayaram et al. (2001). Their procedure can be applied to parts with discontinuities, and when using cutters with multiple inserts that eliminate the need for separate test part preparation. Also, the use of a dynamometer in the industrial plant during production is troublesome, as it requires changes to fixtures and modification of NC code leading to production disruption and increased cost.
The use of a dynamometer in a cutting test may be eliminated if an alternate method could effectively identify cutting forces. Several research activities have addressed this problem. Cutting forces have been identified using electric current signal of servo-drive or spindle motors (Kim et al., 1999), however, its application is limited owing to a narrow frequency bandwidth of less than 130 Hz (Jeong and Cho, 2002). Alternatively, cutting forces can be estimated from the measured response of the structure (i.e., displacement, velocity or acceleration). Capacitance sensors integrated into the spindle have been used to measure spindle deflections that are then used to estimate cutting forces (Albrecht et al., 2005; Kim et al., 2005). However, this set-up is sensitive to metal chips and dirt that are almost always present in industrial environments. Also, the capacitance sensors are sensitive to atmospheric humidity.
Piezoelectric accelerometers are widely available on most production lines and well-suited to harsh industrial environments. A method to identify forces in machines from piezoelectric ac-celerometer signals and measured frequency response functions was proposed by Okubo et al. (1985) and Spiewak (1995) proposed an application of the milling cutter instrumented with a triaxial accelerometer to determine the forces applied at the tool tip. However, in both these works, the approach was not verified using actual cutting experiments. In practice, acceleration signals during cutting are subject to noise. Estimation of cutting forces from these acceleration signals is done using the least squares method, which requires a matrix inversion operation. Thus, an ill-conditioned matrix could lead to numerical challenges in predicting cutting forces. These numerical challenges owing to ill-conditioned problems are overcome using regularisation (Vogel, 2002; Hanke and Hansen, 1993). The implementation of these numerical techniques is illustrated by simulation (Turco, 2005) or experiments on simple beam structures (Law et al., 2001; Liu and Shepard, 2005).
This paper proposes an estimation method for the mechanistic force model coefficients that does not require direct force measurements. Instead, cutting forces are reconstructed from the acceleration signals measured during machining. This method may be applied in industrial plant without interrupting production. These coefficients can be used to evaluate chatter stability lobes, which can assist in adjusting the operating parameters of the machine for higher material removal rate. Such an approach allows quick optimisation of the machining process as production is adjusted to the market demands.
This research solves the problem of estimating machine performance from measured accelerometer response by addressing two main challenges. The first challenge is the cutting force identification, which as demonstrated in this research is hard owing to numerical challenges. Therefore, the Tikhonov regularisation method (Tikhonov, 1963) is used to improve the robustness of cutting force estimation. A systematic method to apply this regularisation technique, based on estimation of noise level in the machining system, is developed. The other challenge is estimating the mechanistic cutting force parameters
268 B. Powalka et al.
when measuring response using piezoelectric accelerometers. This is because piezoelectric sensors cannot measure the DC component of a response and, therefore, the time instant of the measured signal when the cutting insert enters the workpiece is not known. This challenge is overcome by using the knowledge of the cutting force pattern to estimate this time instant.
The following section presents a brief mathematical background on the cutting force model whose parameters are identified, the frequency response function matrix for a structure that is used in force reconstruction, and the requirements regarding sensor location. Later, the force reconstruction method and results of experiments are presented.
2 Mathematical background
2.1 Mechanistic cutting force model
A mechanistic cutting force model assumes that cutting forces are proportional to the chip cross-sectional area. The constants that relate cutting forces and the chip cross-sectional area are termed specific cutting force coefficients. They depend on cutter geometry, inserts and workpiece materials, cutting conditions, etc. (Altintas, 2000). Mechanistic cutting force models define the cutting forces, Fti, Fri and Fai, acting in the tangential, radial and axial directions respectively on the ith cutting edge as a function of instantaneous chip thickness h(ϕi) and depth of cut ap (Figure 1)
enter exit
enter exit
( ( ) ) ( )1, if ( , )
( ( ) ) ( ) where ( )0, if ( , )
( ( ) ) ( )
ti tc p i i te p i ii
ri rc p i i re p i i i ii
ai ac p i i ae p i i
F K a h K aF K a h K aF K a h K a
ϕ ω ϕϕ ϕ ϕ
ϕ ω ϕ ω ϕϕ ϕ ϕ
ϕ ω ϕ
= +∈ë
= + ì ∉í= + (1)
where, ϕi is the instantaneous angular location of the ith cutting edge, the coefficients Ktc,Krc and Kac are the specific cutting force coefficients in tangential, radial and axial directions respectively. The coefficients Kte, Kre and Kae are the edge constants in the same directions. The workpiece is being cut by the cutting edge only when the ϕi takes a value between the immersion angle at entry, ϕenter and at exit, ϕexit The instantaneous chip thickness, h(ϕi), depends on the feed rate per tooth (mm/rev-tooth), ft.
Figure 1 Mechanistic cutting force model
Identification of machining force model parameters 269
Local cutting forces from working cutting edges expressed by equation (1) can be transformed to the machine coordinate system in feed (X), cross-feed (Y) and axial (Z)directions and then summed to yield the net force on the tool and the workpiece. It is noteworthy that this force is a linear function of the cutting coefficients. Therefore,
F(t) = W(t)K (2)
where, F(t) = [Fx(t) Fy(t) Fz(t)]T, K = [Ktc Kte Krc Kre Kac Kae]T and W(t) is a matrix function of spindle rotational speed, feed rate and depth of cut.
2.2 Frequency Response Function matrix
The Frequency Response Function (FRF) of a structure is defined as the ratio between its harmonic response and the applied harmonic excitation force. In machine tool applications, FRFs are conventionally evaluated using an impact hammer test, wherein, an instrumented hammer is used to provide known excitation to the machine structure, and accelerometers are used to measure the response (Altintas, 2003). The stability analysis of machine tools requires the relative workpiece-cutting tool FRF, which involves measuring the response at the cutting edge. However, in this research, the focus is on evaluating the cutting model parameters during milling and, therefore, the accelerometers cannot be mounted on the cutting edge. The solution is to mount the accelerometer at convenient locations on the spindle housing and measure the FRF matrix relating the forces acting on the cutting tool or workpiece and the response at the location of the accelerometers, prior to machining. The locations of accelerometers for measured response on the Arch-Type Reconfigurable Machine Tool (Dhupia et al., 2007), used for impact hammer test as well as the cutting experiments described later, as shown in Figure 2. The measured FRFs for response at accelerometers for input excitation at the cutting edges, also referred to as accelerances, are shown in Figure 3. Thus, during the impact test, the structure was excited by impact hammers at the cutting edge and the structural response was measured at the accelerometers.
Figure 2 Locations of accelerometer on machine tool (see online version for colours)
270 B. Powalka et al.
Figure 3 Experimentally evaluated accelerance (see online version for colours)
For the sake of generality, assume that L accelerometers are used in the experiment. Therefore the measured acceleration response at any particular frequency, , can be stacked in the acceleration vector, a( ) = [a1( ) a2( ) … aL ( )]T. The excitation force at the cutting tool can be represented in the feed, cross-feed and axial direction in the force vector, F( ) = [Fx( ) Fy( ) Fz( )]T The generated acceleration and force vectors are related via the FRF matrix, G( ), as:
( ) ( ) ( ).ω ω ω=a G F (3)
and, for L accelerometers the FRF matrix is:
1 1 1
2 2 2
( ) ( ) ( )( ) ( ) ( )
( ) .
( ) ( ) ( )
x y z
x y z
Lx Ly Lz
G G GG G G
G G G
ω ω ωω ω ω
ω
ω ω ω
è øé ùé ù=é ùé ùé ùê ú
G (4)
3 Cutting force reconstruction
The forces generated in the cutting process act on the machine structure and cause vibrations. The accelerations measured by L accelerometers are related to the cutting forces in the frequency domain through the FRF matrix equation (4). Let the spindle speed be N (RPM), and the cutting tool have n inserts. Since the frequency representation of the cutting forces during stable cutting is dominated by the tooth passage frequency
Identification of machining force model parameters 271
(Figure 4), t = 2 nN/60, and its harmonics, the response of the structure (i.e., structural vibrations) will also be dominated by these components. The response of the structure during machining is measured by the same set of accelerometers that were used for FRF measurements. Fourier coefficients of the measured accelerations at the kth harmonic of the tooth passage frequency, t, are
a(k t) = G(k t)F(k t). (5)
Thus, the estimated kth harmonic of the cutting force, ˆ ( )tkωF , can be evaluated from the measured acceleration using the least squares method as:
ˆ ( ) ( ) ( ) ( )Ht t t tk k a k kω ω ω ω= + = +F G V U aΣ (6)
where U VH = G(k t) is the Singular Value Decomposition (SVD) of the described FRF matrix G equation (4) at the kth harmonic of the tooth passage frequency. The rank of G(k t) will be three, which is equal to the number of columns, one for each Cartesian direction. Therefore, the diagonal matrix will contain only three singular values. Finally, the cutting force in the time domain can be reconstructed from the evaluated cutting force at all harmonics.
Figure 4 FFT of feed force in full-immersion cutting at 2400 RPM (see online versionfor colours)
3.1 Error analysis and regularisation
The measured acceleration is a combination of the actual acceleration and noise. During the estimation of the cutting force, this noise will be amplified if the FRF matrix, G(k t),has small singular values. Consider the noise vector, (t), when L accelerometer channels are used. Thus, the relationship between measured accelerations and actual cutting forces is:
a(k t) = G(k t)F(k t) + (k t) (7)
where (k t) is the kth harmonic of the tooth passage frequency of the noise vector. Therefore, the estimated cutting force ˆ ( )tkωF includes a component arising
owing to noise: 3
1
1
ˆ ( )Ht t i i i t i
ik k s kω ω η ω−ä
=F( ) = F( ) + U V (8)
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where Ui is the ith column vector of U, Vi is the ith column vector of V, si is the ith singular value of the FRF matrix, i(k t) is the ith element of the (k t) vector.
Equation (8) indicates that inaccuracy in estimation is influenced mostly by the smallest singular value, si. Regularisation may be employed to abate this problem. The most common regularisation technique is the Tikhonov regularisation based on the Bayesian statistical approach (Tikhonov, 1963). This technique is recommended if the system is not highly over-determined, i.e., number of measurement channel is not much larger than forces to be identified (Liu and Shepard, 2005). Because the goal of this research is to develop a method suitable for industrial application where the number of accelerometers used for response measurements should be relatively small, this technique was chosen to improve the numerical stability of the inverse problem. Applying Tikhonov regularisation, the solution is expressed as:
2
2
1ˆ ( ) = diag ( )Hireg t t
ii k
sk k
ssω ω
αë ûì ü+í ý
F V U a (9)
where, αk is the regularisation parameter for the kth harmonic component of cutting force estimation. The introduction of the regularisation parameter has the strongest effect on small singular values and, consequently, reduces the influence of the disturbances on the final estimate. However, as can be concluded from equation (9), the regularisation parameter has also similar effect on actual accelerations, and selecting a large value may lead to underestimation of the reconstructed forces. The optimal selection of the regularisation parameter is straightforward only if there is prior knowledge regarding the error distribution of the estimate. Then, the regularisation parameter can be derived on the basis of Bayesian statistics and is expressed as a ratio of measurement variance to the estimator variance. Unfortunately, this cannot be assumed in most practical problems and finding an optimal value of the regularisation parameter is still a subject of current research (Pothisiri, 2006; Choi et al., 2006; Kindermann and Leitao, 2006).
In this research work Morozov (1967) discrepancy principle was adopted to evaluate a proper regularisation parameter. According to this principle, the regularisation parameter should be chosen in such a way that the residue between the measured output and the output calculated from the regularised solution (equation (9)) is equal to the norm of the estimated noise:
ˆ( ) ( ) ( ) ( ) where ( ) = ( ) .treg kω ωω ω ω ε ω ε ω η ω=− ≈G F a (10)
Equation (10) can be used to estimate the regularisation parameter αk for each harmonic component of the cutting force if the noise vector, (t), is known.
3.2 Noise estimation
Indirect force estimation using an acceleration signal during cutting introduces noise from two main sources:
• the cooling system of the spindle
• the feed drives.
The influence of these factors along with the measurement noise is estimated by considering the portion of the measured acceleration signals that correspond to the idle
Identification of machining force model parameters 273
work of the machine tool, i.e., the spindle rotates at the speed at which cutting is performed, machine is in feed motion set for the process but the cutting tool is still not engaged into the workpiece (Figure 5). During this phase, accelerometers acquire only the disturbances and, thus, the signals delivered by the sensors represent the noise vector, (t). The noise from all channels is transformed to the frequency domain. Afterwards, its
components at the harmonics of the tooth passage frequency are selected. The norm of the noise vector at a particular harmonic from all channels is used to compute the regularisation parameter (equation (10)).
Figure 5 Selection of the acceleration signal used for noise estimation (see online version for colours)
The evaluated noise level may also be used for the selection of the number of harmonics for cutting force reconstruction. Although more harmonics theoretically give a better reconstruction of the cutting force, they also add errors and may not necessarily improve the results. The cutting force components at higher harmonics are usually much smaller (Figure 4), leading to a decreased signal to noise ratio of the acceleration signal. The decision to include the kth harmonic component into the reconstruction process is based on the comparison of the norm of the acceleration, ( )||,tkω|| a with the corresponding noise estimate, || ( ) || .k tkε η ω= If the noise estimate is of the same order as the signal norm, then this harmonic should not be included in the identification.
3.3 Phase detection
The final goal of this analysis is to determine the stability lobe diagrams for the given machine and workpiece combination, which can assist the end-user to determine the
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optimal cutting parameters for machining that achieve desired production rates. For generating the stability lobes, the mechanistic cutting model representation (equation (1)) needs to be obtained from the reconstructed cutting force signal. Thus, it is required to estimate the cutting coefficients Ktc, Krc, Kac, Kte, Kre and Kae. Since the cutting forces are linear functions of these cutting coefficients, the mechanistic cutting force model was described in matrix form in equation (2). This can also be written in the frequency domain as
ˆ ( ) ( )reg t tk kω ω=F W K (11)
where, 01
00( ) ( )e dt
T jk tt Tk t tωω −= ñW W is a matrix of Fourier series coefficients of the W(t)
matrix (equation (2)). However, the knowledge of W(t) function requires knowing the immersion angle at time instant t = 0. This is not possible when using piezoelectric accelerometers as they do not provide the DC value of the force signal, which is needed to evaluate the immersion angle. Therefore, the function W(t) is assumed to begin at immersion angle ϕ = 0 at time instant t = 0 and the force signal is considered to be delayed by an unknown time t0. Therefore,
0ˆ ( )e ( )t
tjkreg t tk kωω ω− =F W K (12)
where, k = 1, 2, … represent the first harmonic, second harmonic and so on. Using the first two harmonics, the equations may be written in matrix form as
0
02
ˆ ( )e ( ).
ˆ (2 )(2 )e
j treg t t
j ttreg t
ω
ω
ω ωωω
−
−
è ø è øé ù = é ùé ù ê úê ú
F WK
WF (13)
In a milling operation, the axial direction is assumed to be much more rigid than the feed and cross-feed directions (Altintas, 2000) and, therefore, it is not necessary to estimate the coefficients Kac and Kae. Thus, equation (13) represents four complex equations for cutting forces, two equations in both feed and cross-feed directions. The real and imaginary parts of these equations yield eight equations. However, there are four unknown cutting coefficients, i.e., Ktc, Krc, Kte and Kre, and the unknown time delay t0.
Two approaches may be used to solve the general problem in which the cutting tool may have any number of cutters. First, optimisation may be used to determine optimal values of the cutting coefficients and time delay, which minimise the residues from all equations, i.e.,
0
00 2,
ˆ ( )e ( )Min .
ˆ (2 )(2 )e
j treg t t
j tttreg t
ω
ω
ω ωωω
−
−
è ø è øé ù − é ùé ù ê úê ú
K
F WK
WF 14
Alternatively, Wang and Chang (2004) proposed an algorithm to solve for the unknown time delay, t0, while eliminating the cutting coefficients from the equations in the frequency domain. This approach requires accurate estimation of both first and second harmonics of the cutting forces.
The method used for this research uses the knowledge of the cutting force pattern for a two-insert cutting tool and estimates the time delay, t0, in the time domain to reduce the sensitivity to the lower signal to noise ratio that may be present in evaluation of
Identification of machining force model parameters 275
the cutting force component at the second harmonic. The cutting force is represented in the time domain as
ˆ ˆ( ) ( )e tjk treg t
kt k ωω=äF F (15)
where ˆ ( )tF has two rows corresponding to the feed and cross-feed forces. This evaluated cutting force has no DC component, as the DC component cannot be measured using piezoelectric accelerometers.
For full-immersion cutting with a two-insert cutting tool, there is only one insert that is cutting the material at any given time. Let the spindle speed be N (RPM). Therefore, the instant that the insert enters the workpiece, i.e., t0 must be between 0 and 60/2N. Thus, t ∈[t0, t0 + 60/2N) corresponds to one cutting cycle period. During this period, the immersion angle ϕ ∈ [0, π). For this assumption, the feed and the cross-feed forces over the entire cutting cycle are transformed to the local tangential cutting force, ˆ
tF , and local radial cutting force, ˆ
rF , using
60ˆ2 60ˆ ˆ( ) (0)
260ˆ2
t
r
NA
NN
ϕπ ϕϕ
πϕπ
è øå õæ öé ù å õç ÷ å õé ù = −æ öæ öé ù ç ÷å õ ç ÷
é ùæ öç ÷ê ú
FF F
F (16)
where, the transformation matrix
sin( ) cos( )( ) .
cos( ) sin( )ϕ ϕ
ϕϕ ϕ
−è ø= é ù−ê ú
A
Note that ˆtF and ˆ
rF must be positive over the entire cutting cycle. If ˆtF or ˆ
rF assume negative values over the cutting cycle, then this contradicts the assumption of the insert entering the workpiece at the chosen data point and this data point is neglected in further analysis.
The full-immersion cutting with two inserts is a symmetrical process around the immersion angle ϕ = 90. Therefore, | |t tF Fϕ ϕ π=∆ = −∆= and | |r rF Fϕ ϕ π=∆ = −∆= , where ∆ is some angle. Let the sampling time of the signal be ts, and let the number of samples over the cutting cycle be 2k. The error in symmetry, sym, for the measured tangential and radial forces can be represented in vector form as
sym
60 60 60ˆ ˆ ˆ ˆ ˆ ˆ(0) ( ) ( )2 2 2
.60 60 60ˆ ˆ ˆ ˆ ˆ ˆ(0) ( ) ( )2 2 2
T
t t t s t s t s t s
T
r r r s t s r s r s
F F F t F t F kt F ktN N N
F F F t F t F kt F ktN N N
η
è øè øå õ å õ å õ− − − − −é ùæ ö æ ö æ öé ùç ÷ ç ÷ ç ÷ê úé ù= é ùè øå õ å õ å õé ù− − − − −æ ö æ ö æ öé ùé ùç ÷ ç ÷ ç ÷ê úê ú
(17)
The time instant, t0, when the tool enters the workpiece is found by evaluating [0,60/2 ) symarg min || ||t N η∈ and those yielding positive values for ˆ ( )tF and ˆ ( )rF over the
entire cutting cycle. For half-immersion, the cutting forces transition between the maximum value and
zero at ϕ = 90°. Thus, this point can be easily located from the force signals.
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4 Experimental results and discussion
This section describes the application of the proposed methodology using the Arch-Type Reconfig-urable Machine Tool for experiments (Dhupia et al., 2007). The details of experimental evaluation of the FRF matrix, reconstruction of the cutting forces during end-milling of AISI 1018 steel, evaluation of the cutting coefficients and using the cutting coefficients to generate the stability lobe diagram for the Arch-Type Reconfigurable Machine Tool follows.
4.1 Transfer matrix
The transfer matrix or the FRF matrix for the Arch-Type Reconfigurable Machine Tool for the excitation at tool tip and response at the various accelerometer locations shown in Figure 2 was obtained using an impact hammer test. Four response signals were obtained using two channels from a triaxial accelerometer, and one each from two uniaxial accelerometers. The location of the accelerometers was chosen for convenience to be on the spindle and spindle housing, so it does not interfere with the cutting. Two accelerometer signals were chosen along the feed (X) and the cross-feed (Y)direction each, which are the dominant directions for machine dynamics in milling. The excitation force was provided using an instrumented hammer in the feed, cross-feed and axial direction at the tool tip. Thus, the obtained FRF matrix has a dimension of 4 × 3.
Figure 6 shows the FRF magnitude obtained when measuring the response from channel 2, i.e., the feed direction of the triaxial accelerometer when impact was provided in the feed direction at the tool tip. The frequencies 80, 89 and 110 Hz are marked in the figure and correspond to the first harmonic of the tooth passage frequency for a 2-fluted cutter at the spindle rotational speed of 2400, 2670 and 3300 RPM respectively. It may be observed that the tooth pass frequency for 2400 RPM is in the vicinity of a more damped mode when compared with the tooth pass frequency for 3300 RPM. Another spindle speed of 2670 RPM is selected to have the tooth passage frequency between the two selected modes. The transfer matrix tends to be more ill-conditioned near the lightly damped modes (Fabunmi, 1986), and may result in numerical challenges for cutting force reconstruction. The condition number for the FRF matrix,
(G(ω)) is defined as the ratio between its largest and smallest singular values, si, of the FRF matrix:
max( )( ( )) .min( )
i
i
sks
ω =G (18)
A large condition number indicates an ill-conditioned matrix and may result in numerical challenges.
Therefore, a small condition number is usually desired. Figure 7 demonstrates the variation of the condition number for the FRF matrix in the frequency range of 70–120 Hz, which corresponds to the spindle speed range of 2100–3600 RPM. The relatively small condition numbers for 2400 RPM and 2670 RPM indicate that reconstruction of the first harmonic components of the corresponding cutting forces should be less difficult when compared with the case of 3300 RPM where the condition number is larger.
Identification of machining force model parameters 277
Figure 6 Feed direction FRF with first harmonics frequencies for 2400, 2670 and 3300 RPM(see online version for colours)
Figure 7 Condition number of FRF matrix vs. spindle speed (see online version for colours)
4.2 Cutting tests
Cutting tests were performed to investigate the robustness of the proposed method to condition number of the FRF matrix and to different cutting scenarios. Thus, cutting tests were performed at 2400, 2670 and 3300 RPM, where the condition number varies as described in Section 4.1. Both half-immersion and full-immersion cutting scenarios were investigated. The operating conditions for different cutting tests are tabulated in Table 1.
In all tests, the cutting forces were reconstructed using the first two harmonics of the tooth passage frequency. In full-immersion milling, the higher harmonic components of cutting forces drop rapidly. This leads to a lower signal to noise ratio at the higher frequencies and, thus, including that these components does not improve the reconstruction quality. Also, the same first two harmonics were used for the force reconstruction in half-immersion milling. Therefore, to assess the quality of the reconstructed force, it is compared with the cutting force composed by the same first two harmonics of the forces measured using a dynamometer. The noise level at each of the two harmonic components was estimated using the portion of the acceleration signal acquired when the spindle was rotating and the machine was in a feed motion, as described in Section 3.2. This noise component was used to evaluate the regularisation parameter, α, needed for Tikhonov regularisation. Afterwards, the time instant, t0, at which the tool enters the workpiece was found using the time domain approach described in Section 3.3. Once the time instant, t0, is known, the cutting coefficients can be determined from equation (13) using the least squares approach. The evaluated cutting coefficients from reconstructed forces are tabulated in Table 2. The cutting coefficients are also determined by using the least squares approach to fit the mechanistic cutting model to the measured cutting forces using a dynamometer. The averaged cutting coefficients obtained from all cutting experiments using the dynamometer are Ktc = 1810 N/mm2 and Krc = 1085 N/mm2. The reconstruction error column in Table 2 represents the deviation in the evaluated cutting coefficients from the averaged cutting coefficients using the dynamometer.
The evaluated coefficients presented in Table 2 match reasonably well the cutting coefficients found directly from the forces measured using the dynamometer. The only exception is when cutting at 3300 RPM, where the tooth passage frequency is close to a
Identification of machining force model parameters 279
lightly damped structural frequency and, consequently, the FRF matrix has a high condition number. Figures 8–10 graphically present the results of force reconstruction, and comparison with the measured cutting forces using a dynamometer. Figure 8 considers the comparison of the reconstructed feed and cross-feed forces using the first two harmonics of the tooth passage frequency with the measured forces contributed from the same two harmonics from tests 1, 2, 4 and 5. These tests correspond to full-immersion milling at 2400 RPM, full-immersion milling at 2670 RPM, half-immersion up-milling at 2400 RPM, and half-immersion down-milling at 2400 RPM, respectively. In all these tests, good numerical stability is achieved as a result of lower condition numbers of the FRF matrix. The lowest correlation factor of 0.93 among these cases was observed for the feed force in half-immersion down-milling. The root mean square error for this case is 29 N, which is around 20% of the maximum force over the entire cutting cycle. This lower quality of reconstruction is attributed to the higher harmonic vibrations induced in the structure by initial impact inherent in the half-immersion down-milling process in each cutting cycle. Figure 9 compares the same reconstructed force with the overall measured forces from the dynamometer. This figure also indicates a good quality of reconstruction for cutting tests 1, 2, 4 and 5.
Figure 8 Comparison of accelerometer based first two harmonics of cutting forces with first two harmonics forces measured by dynamometer using a 2 flute cutter for:(a) Test 1: 2400 RPM in full-immersion; (b) Test 2: 2670 RPM in full-immersion;(c) Test 4: 2400 RPM in half-immersion up-milling and (d) Test 5: 2400 RPM in half-immersion down-milling (see online version for colours)
(a) (b)
(c) (d)
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Figure 9 Comparison of accelerometer based first two harmonics of cutting forces with forces measured by dynamometer using a 2-flute cutter for: (a) Test 1: 2400 RPMin full-immersion; (b) Test 2: 2670 RPM in full-immersion; (c) Test 4: 2400 RPM in half-immersion up-milling and (d) Test 5: 2400 RPM in half-immersion down-milling (see online version for colours)
(a) (b)
(c) (d)
Figure 10 Results of Test 3. Comparison of accelerometer based first two harmonics of cutting forces during full-immersion milling at 3300 RPM using a 2-flute cutter with:(a) the first two harmonics of the measured forces using the dynamometer and (b) the overall measured forces using the dynamometer (see online version for colours)
(a)
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Figure 10 Results of Test 3. Comparison of accelerometer based first two harmonics of cutting forces during full-immersion milling at 3300 RPM using a 2-flute cutter with:(a) the first two harmonics of the measured forces using the dynamometer and (b) the overall measured forces using the dynamometer (continued) (see online version for colours)
(b)
Figure 11 Stability lobe diagrams for cutting coefficients obtained from measured forces from dynamometer as well as reconstructed forces during (a) full-immersion milling at 2400 RPM; (b) full-immersion milling at 2670 RPM; (c) half-immersion up-milling at 2400 RPM and (d) half-immersion down-milling at 2400 RPM (see online version for colours)
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Figure 12 Stability lobe diagrams for cutting coefficients obtained from measured forces from dynamometer as well as reconstructed forces during full-immersion milling at 3300 RPM (see online version for colours)
Despite using the regularisation technique, the reconstructed cutting forces do not yield a good fit with the measured cutting forces at 3300 RPM (Figure 10). A low correlation factor of 0.68 is obtained for the reconstructed cross-feed force. This was expected because of the high condition number of the FRF matrix. As long as the cutting tests are carried out such that the tooth passage frequency does not lie in the vicinity of a lightly damped structural frequency, as in this case, the proposed regularisation technique allows for a good reconstruction of the cutting forces.
Figure 11 compares stability lobe diagrams generated using cutting coefficients estimated from the reconstructed and measured forces for tests 1, 2, 4 and 5. Stability lobes were generated using experimentally measured FRFs at the tool tip in feed and cross-feed directions and for the case of full-immersion cutting using 4-flute cutter. The experimental verification of these stability lobes has been described in Dhupia et al. (2007). Since the same FRFs were used for evaluation of stability lobes, their shape remains unaffected by the discrepancies observed in cutting coefficients. Only a low variation, i.e., about 6% is observed in the absolute stability limit, indicating that the reconstruction procedure described here is suitable for predicting machine chatter and selecting operating parameters. However, Figure 12 compares the stability lobe diagram for reconstructed cutting coefficients estimated at full-immersion milling at 3300 RPM, i.e., test 3, where poor reconstruction is obtained, therefore, a large discrepancy is observed.
5 Summary and concluding remarks
This paper presents a method for identifying a cutting force model from acceleration measurements of a machine structure. This method is suitable to harsh industrial environments because of the ease of use and ready availability of accelerometers. Such a method can be utilised to evaluate the stable operating parameters for a machine tool on an industrial line without disrupting production.
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A regularisation technique is utilised to increase the robustness of cutting force reconstruction from a variety of cutting scenarios with different operating parameters. The identified cutting forces are in good agreement with the measured forces from a dynamometer. The only exception is when cutting is carried out at a spindle speed where tooth passage frequency is in the vicinity of a lightly damped structural mode. This regularisation technique requires the estimation of the noise in the cutting process. Noise estimation is carried out using the signal when machine is in feed motion, but still not cutting the workpiece.
These reconstructed forces do not contain the DC component because of the use of piezoelectric accelerometers and, therefore, the knowledge of the cutting force pattern for a two-flute cutting tool is utilised to evaluate the full cutting force model parameters from the reconstructed cutting force. The stability lobe diagrams from the evaluated parameters matched closely to the stability lobe diagram from the cutting force model parameters obtained directly from measured forces using a dynamometer.
Acknowledgements
The authors are pleased to acknowledge the financial support of the NSF Engineering Research Center for Reconfigurable Manufacturing Systems (NSF grant# EEC-9529125) and the Foundation for Polish Science. Also, the authors are grateful for Steve Erskine’s assistance in cutting experiments.
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