High frequency bandwidth cutting force measurement in milling using capacitance displacement sensors

High frequency bandwidth cutting force measurement in milling using

capacitance displacement sensors

Andreas Albrechta, Simon S. Parkb,*, Yusuf Altintasc, Gunter Pritschowa

aInstitut fur Steuerungstechnik der Werkzeugmaschinen und Fertigungseinrichtungen, Universitat Stuttgart, 70174 Stuttgart, GermanybDepartment of Mechanical and Manufacturing Engineering, The University of Calgary, Calgary, AB, Canada T2N 1N4

cManufacturing Automation Laboratory, Department of Mechanical Engineering, The University of British Columbia, Vancouver, BC, Canada V6T 1Z4

Received 25 October 2004; accepted 30 November 2004

Abstract

This article presents a method of measuring cutting forces from the displacements of rotating spindle shafts. A capacitance displacement

sensor is integrated into the spindle and measures static and dynamic variations of the gap between the sensor head and the rotating spindle

shaft under cutting load. To calibrate the sensing system, the tool is loaded statically while the deflection of the tool is measured with the

capacitance probe. With this calibration, the displacement sensor can be used as an indirect force sensor. However, the measurement

bandwidth is limited by the natural modes of the spindle structure. If cutting force frequency contents are within the range of the natural

modes of the spindle structure or higher, the measurements are distorted due to the dynamic characteristics of the spindle system. In order to

increase the bandwidth of the indirect force sensor by compensating for the spindle dynamics, the design of a Kalman filter scheme, which is

based on the frequency response function (FRF) of the displacement sensor system to the cutting force, is presented in this paper. With the

suggested sensing and signal processing method, the frequency bandwidth of the sensor system is increased significantly, from 350 to

approximately 1000 Hz. The proposed indirect force sensor system is tested experimentally by conducting cutting tests up to 12,000 rpm

with a five-fluted end mill. Besides cutting forces, the measured displacements can also be affected by factors such as roundness errors,

unbalance at different speeds, or dilatation of the spindle shaft due to temperature variations. Methods to compensate for these disturbing

effects are also described in the paper.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Displacement sensor; Dynamic compensation; Indirect force sensor; Kalman filter; Milling; Modal curve fitting; Monitoring

1. Introduction

Machining of large and complex integral parts like

aircraft ribs, dies, molds, turbine rotors and vanes often

requires the removal of large amounts of metal. In order to

reduce production time and cost, there is an increasing

demand for higher metal removal rates. To achieve high

productivity and accuracy, process disturbances like self-

excited chatter vibrations, forced vibrations due to unba-

lance, overload, collision and tool breakage, or excessive

tool wear need to be monitored and suppressed. The

measurement of cutting forces is the key information needed

to monitor, troubleshoot, or control the machining

0890-6955/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijmachtools.2004.11.028

* Corresponding author. Tel.: C1 403 220 6959; fax: C1 403 282 8406.

E-mail address: [email protected] (S.S. Park).

operations. Process optimization can be achieved either by

offline simulation or online process monitoring and

diagnosis [1,2]. Offline simulation helps to optimize the

operation in the planning stage before the actual machining

takes place in the shop, or serves as a tool for troubleshoot-

ing [3]; whereas online monitoring of the machining process

is essential in recognizing disturbances like collision, tool

breakage, tool wear, or unstable process conditions such as

chatter or tool failure [4]. Adaptive control of machining

operations also requires accurate measurement of cutting

forces during production [5–8]. In short, a reliable cutting

force measurement system, which has a high bandwidth to

cover a wide range of cutting speeds, is required.

The most common method to measure cutting forces in

machining operations is through table dynamometers.

Typical table dynamometers consist of piezoelectric

International Journal of Machine Tools & Manufacture 45 (2005) 993–1008

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Nomenclature

dAir_Cutting displacement profile measured while air

cutting at cutting speed (mm)

dCutting displacements measured while cutting including

disturbing effects (mm)

dF resultant displacement signal caused by cutting

forces (mm)

Fd force measured indirectly from the displacement

sensor (N)

Fa actual force applied to the tool tip (N)

KS static stiffness of the spindle sensor system

(N/mm)

GS static compliance of the spindle sensor system

(mm/N)

F(s) dynamic compliance of the spindle sensor

system (mm/N)

fn natural frequencies of the spindle structure (Hz)

un,k natural frequencies of the spindle structure

(rad/s)

zk damping ratio of the spindle structure (l)

ak compliance equivalent term of the spindle

structure (mm/Ns2)

bi numerator coefficient of the transfer function

ai denominator coefficient of the transfer function

Ai system matrix

Bi input matrix

Ci measurement or output matrix

x state vector

u input vector (here: uZFa)

z output vector (here: zZdF)

T similarity transformation matrix

W observability matrix

G system noise matrix

w process noise

v measurement noise

x estimate for the system state vector from the

Kalman filter

z estimate for the system output from the Kalman

filter (here: zZ dF)

dF estimate for the spindle flange displacement

(mm)

K Kalman filter gain matrix

Co observer or Kalman filter output matrix

zo Kalman filter output (here: zoZ Fa)

Fa estimate for the actual cutting force from the

Kalman filter (N)

GFa=dFcontinuous Kalman filter transfer function

(N/mm)

td discrete sampling time (s)

P state estimation error covariance matrix

Q system noise covariance matrix

R measurement noise covariance matrix

Dw temperature variation (K)

aw thermal expansion coefficient (1/K)

L geometric size/length (m)

U output voltage of the capacitance displacement

sensor (mV)

KU sensitivity factor of the capacitance displace-

ment sensor (mV/mm)

A. Albrecht et al. / International Journal of Machine Tools & Manufacture 45 (2005) 993–1008994

sensors that are clamped between two plates [9]. Although

table dynamometers provide accurate and effective force

measurement, they are more suitable for laboratory or

experimental use rather than for practical application on

production machines, due to the limitation of workpiece

size, mounting constraints, high sensitivity to overload,

and high costs. Furthermore, the dynamic characteristics

of table dynamometers are strongly dependent on the

workpiece mass, which may change during machine

operation. To overcome limitations of workpiece mass

and size, a force sensor can be integrated to the spindle

itself instead of installing it on the machine table. For

example, Kistler AG [10], Aoyama et al. [11], and Smith

et al. [12] proposed rotating force and torque dynam-

ometers. They are attached between the spindle and tool

as an adapter, and measure cutting forces very close to the

tool. However, the rotating force sensor has an additional

mass and overhang, which reduce the dynamic stiffness of

the spindle system. A more rigid solution has been

proposed, in the form of a force ring sensor integrated to

the spindle housing, by Kistler AG in cooperation with

several laboratories [13,14]. The ring sensor consists of

six piezo quartz elements measuring cutting forces in X-,

Y-, and Z-directions. Since the sensors are located away

from the tool tip, they are affected by the dynamics of the

spindle, which can distort the measurements. Although the

spindle integrated force ring sensor provides accurate

cutting force sensing when using dynamic compensation,

it has high capital and installation costs. A more rugged

and cost effective solution is desirable for use in the

production environment.

In this study, the radial displacements of the rotating

spindle shaft are used to measure cutting forces indirectly

via a capacitance sensor installed in the spindle housing.

The measurement system is insensitive to overload and not

subject to wear because the sensors are not in contact with

the rotating spindle. The design satisfies the following

criteria for ideal force measurement systems [15,16]:

†
No reduction in the static and dynamic stiffness of the
machine tool;

†
No restriction of working space and cutting parameters;

A. Albrecht et al. / International Journal of Machine Tools & Manufacture 45 (2005) 993–1008 995

†

Fig

side

Wear and maintenance free, easy to replace, and cost

effective;

†
Function is independent of workpiece mass, size, and
geometry; and,

†
Reliable signal transmission.
Since the displacement sensor is at a distance from the

actual force applied at the tool tip, the dynamics of the

spindle (including tool and tool holder) limit the frequency

bandwidth of the indirect force sensor system. To

compensate for the effects of the spindle dynamics, a

disturbance Kalman filter compensation technique [14] is

. 1. The spindle integrated displacement sensor system: (a) schematic

view; (b) sensor and target, top view.

used to recover the cutting force signals from the distorted

displacement measurements obtained from the capacitance

probe.

2. Spindle integrated displacement sensor setup

A capacitance type displacement sensor is installed on

the main spindle of a three-axis CNC vertical machining

center. For the experiments, the sensor is attached externally

to the front of the spindle system using a bracket, which is

clamped around the spindle housing (see Fig. 1). The

outstanding cylindrical flange of the spindle shaft, with

88.9 mm (3.5 00) in diameter and 9.525 mm (3/8 00) in width,

serves as a target for the capacitance sensor. In this study,

cutting force measurement in X-direction is used as an

example case, which can be extended to Y- and Z-directions

using an identical procedure.

The selected displacement sensor is a high-resolution

capacitive sensor (LION PRECISION C1-A/B) for non-

contact displacement measurement. Amplification and

offset voltage can be adjusted on the sensor’s circuit board

(LION PRECISION PM755D). The sensor is calibrated in

combination with the original target, the spindle flange. The

sensitivity factor of the sensor is determined to be

21.4 mV/mm with a gap of 1.016 mm (0.04 00) between

sensor and target. At the actual standoff, resolution and

range of the displacement sensor are 30 nm and 1000 mm,

respectively.

Additionally, an optical position encoder is mounted on

the rear of the spindle shaft. The encoder is required by a

preprocessor algorithm compensating for spindle run out,

roundness errors of the spindle flange, and unbalance of

spindle, tool, and tool holder. The algorithm uses the

displacement profile recorded over one spindle revolution,

while air cutting at cutting speed, and subtracts the profile

Fig. 2. Transfer function of the table dynamometer (reference force sensor)

in X-direction with the workpiece clamped on top.

Fig. 3. Static calibration test setup.

Fig. 4. Static calibration test results for the X-direction (the remaining static

offset is caused by dilatation as only one displacement sensor was used

here).


from the displacement signals measured while cutting. The

encoder serves as a synchronization clock. Due to the

processing time required for synchronization and subtrac-

tion of the air cutting profile, dAir_Cutting, from the

displacement signal measured while cutting (dCutting), the

algorithm works intermittently. It reads both signals in

sequences of a certain number of revolutions from a buffer

and returns the recovered cutting forces, Fd, as

Fd Z dFKS Z ðdCutting KdAir_CuttingÞKS (1)

where dF are the displacements caused by cutting forces;

and, KS is the static calibration factor in N/mm or the static

stiffness. The length of the signal sequences is dependent on

the available buffer size and the processor speed.

The cutting experiments are conducted with a five-fluted,

19.05 mm diameter end mill which is clamped around a

hydraulic tool holder. A block of an aluminum Al7075-T6

alloy is used as work material during the experiments. The

test workpiece is mounted on top of a piezo quartz table

dynamometer (KISTLER 6255 B), which serves as a

reference force sensor. The bandwidth of the table

dynamometer, including the test workpiece, is measured

to be about 650 Hz when tolerating a measurement error of

at a maximum 10% in magnitude (see Fig. 2). Between 650

and 1000 Hz, the magnitude error increases up to 70%,

whereas phase shift remains almost invariant up to 1000 Hz.

Beyond 1000 Hz, phase shift drops significantly. Hence, it is

possible to use the Fourier spectrum of the dynamometer

signal as reference for force components at frequencies

between 650 and 1000 Hz when considering this magnitude

error.

Displacement, encoder and force signals are all passed

through an anti-aliasing filter (Krohn Hite 3905B) and then

captured using the data acquisition system developed in-

house.

The indirect force sensor system is first calibrated

statically when the spindle is at standstill by applying

a gradually increasing load on the tool tip while measuring

both reference force and displacement of the spindle flange,

as is shown in Figs. 3 and 4. Displacement over force yields

the static compliance, GS, of the spindle system. Static

stiffness, KS, is the inverse value

GS ZdF

Fa

Z1

KS

z0:02 ðmm=NÞ (2)

To obtain the dynamic compliance, F(s), between the tool

tip and the spindle flange, which is the sensor target, the

spindle structure is excited on the tool tip by applying a

short impact force, Fa, with an instrumented hammer. The

displacement response signal, dF, is measured with the

spindle integrated capacitance sensor at the spindle flange.

Both impact force and displacement response of the spindle

structure are recorded synchronously and processed using a

Fourier analyzer system (CutPro-MalTFe) developed in-

house.

Fig. 5. Measured and modeled dynamic compliance function of the spindle integrated displacement sensor system FðuÞZdFðuÞ=FaðuÞ in X-direction.


The obtained frequency response or dynamic compli-

ance function, F(s), is shown in Fig. 5. The first major

resonance peak is at approximately 490 Hz. The second

one is at 709 Hz, which originates from a bending mode

of spindle and tool holder at the tapered interface. The

third peak is at 929 Hz, which is due to the tool holder

FðsÞ ZdFðsÞ

FaðsÞZ

b5s5 Cb4s4 Cb3s3 Cb2s2 Cb1s Cb0

s6 Ca5s5 Ca4s4 Ca3s3 Ca2s2 Ca1s Ca

Z5:46!105s4 C2:93!108s3 C1:77!1

s6 C9:95!102s5 C6:37!107s4 C3:87!1010s3 C

Table 1

Modal parameters obtained by curve fitting the measured dynamic

compliance function

k fn (Hz) x a (mm/Ns2)

1 490 0.042 1.69!104

2 709 0.025 8.85!104

3 929 0.044 4.41!105

assembly bending at the CAT 40 spindle taper interface.

The transfer function measurements indicate that the

uncompensated sensor system can only measure cutting

forces reliably that have harmonics of less than 350 Hz,

assuming an error of maximum 10% in magnitude. Due to

the large target diameter, static and dynamic cross-talks

are found to be negligible for the displacement sensor

system.

The experimentally measured dynamic compliance

function is approximated to be a three degrees-of-freedom

system. The modal parameters are identified using a modal

curve fitting technique as

FðsÞ ZdFðsÞ

FaðsÞZ

X3

kZ1

ak

s2 C2xkun;ks Cu2n;k

ZX3

kZ1

ak

s2 C2xkð2pfn;kÞs C ð2pfn;kÞ2

(3)

where k is the number of modes; dF are the displacements

measured by the spindle integrated capacitance sensor; and,

Fa is the actual force acting on the tool tip. The identified

modal parameters are given in Table 1. The measured and

curve fitted transfer functions are in good agreement, as can

be seen in Fig. 5. The modal equation (Eq. (3)) can be

expanded into polynomial form as:

0

013s2 C4:71!1015s C1:23!1020

1:19!1015s2 C3:43!1017s C6:41!1021(4)

3. Dynamic compensation

The objective of dynamic compensation is to reduce the

influence of spindle dynamics through a disturbance

Kalman filter. The Kalman filter compensates for the

influence of the structural dynamics while attenuating high

frequency noise and thus overcoming problems posed by the

simple transfer function inversion [17,18].


In order to design the Kalman filter, the transfer function

of the spindle dynamics given in Eq. (4) is mapped into state

space form that yields

_x Z Asx CBsu

z Z Csx:(5)

where x is the state vector; uZFa is the input vector or the

actual force applied to the tool; and, zZdF is the

measurement vector or the displacement sensor reading.

The state space equation (Eq. (5)) representing the

spindle dynamics can be expressed in the following

scheme [19]

_x1

_x2

_x3

_x4

_x5

_x6

26666666666664

37777777777775

Z

Ka5 Ka4 Ka3 Ka2 Ka1 Ka0

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

26666666666664

37777777777775

x1

x2

x3

x4

x5

x6

266666666664

377777777775

C

1

0

0

0

0

0

26666666666664

37777777777775

Fa

½dF�Z½b5 b4 b3 b2 b1 b0�

x1

x2

x3

x4

x5

x6

266666666664

377777777775

ð6Þ

where the system polynomial parameters ai and bi are

given in Eq. (4). The matrices As and Cs contain both

very large and very small numbers, which result in poor

conditioning with respect to inversion and eigenvalue

analysis. In order to cope with the problem, the system is

transformed into an equivalent system

_xnZAnxnCBnu

zZCnxn

(7)

using a similarity transformation

AnZTAsTK1;BnZTBs;CnZCsT

K1 (8)

such that An has, as nearly as possible, equal row and

column norms. The similarity transformation matrix, T,

which is found based on the algorithm in [20], is a

permutation of a diagonal matrix whose elements are

integer powers of two to prevent the introduction of

round-off errors:

TZdiagð24;218;231;243;254;265Þ

The identified equivalent balanced system is:

AnZ

K99:50K388:79K28:86K21:70K30:50K277:95

16384 0 0 0 0 0

0 8192 0 0 0 0

0 0 4096 0 0 0

0 0 0 2048 0 0

0 0 0 0 2048 0

266666666664

377777777775

BnZ½16 0 0 0 0 0�T

CnZ½0 2:084 0:136 2:016 0:261 3:333�

The observability matrix, W, is found to be full rank,

which guarantees the observability of the system:

WTZ½CTn AT

n CTn /ðAnK1

n ÞTCTn � (9)

3.1. Disturbance model expansion

The aim of dynamic compensation is to reconstruct the

actual force, Fa, exerted on the tool, which is the system

input, u, in Eqs. (6) and (7). Since the Kalman filter only

yields estimates for state vector, x, and output, zZ dF, the

balanced system in Eq. (7) is expanded with the actual force,

Fa, as an additional unknown state in the state vector. It is

assumed that the cutting force signals are piece-wise

constant and both the actual force and the displacement

signal measured by the capacitive sensor are contaminated

with system noise, w, and measurement noise, v, yielding

_xeð7!1Þ Z Aeð7!7Þxeð7!1Þ CGð7!1Þwð1!1Þ

zð1!1Þ Z Ceð1!7Þxeð7!1Þ Cvð1!1Þ

(10)

where G is the system noise matrix. The expanded state

vector is depicted as below; the former input vector has

disappeared

xe Z ½ xTnð1!6Þ Fað1!1Þ �

T; ue Z ½0� (11)

The expanded and noise contaminated state space model

(denoted by e) given by Eq. (10) can be rewritten as:

_xnð6!1Þ

_Fað1!1Þ

" #Z

Anð6!6Þ Bnð6!1Þ

0ð1!6Þ 0ð1!1Þ

" #xnð6!1Þ

Fað1!1Þ

" #CGð7!1Þwð1!1Þ

zð1!1Þ Z ½Cn 0ð1!1Þ �xnð6!1Þ

Fað1!1Þ

" #Cnð1!1Þ

(12)

In this form, the cutting force can be estimated through a

disturbance Kalman filter designed for the expanded model


of the spindle integrated force sensor system as

_xe ¼ Aexe þ �Kðz K zÞ ¼ Aexe þ Kðz KCexeÞ

¼ ðAe KKCeÞxe þ Kz

zo ¼ Coxe ¼ Fa; with Co ¼ ½0ð1!7Þ 1�

(13)

where K is the Kalman filter gain matrix; and, Fa is an

estimate for the actual cutting force, Fa. The continuous

Kalman filter transfer function can be derived from Eq. (13)

as:

Fa ZC0 adj½sI K ðAe KKCeÞ�

det½sI K ðAe KKCeÞ�K

�dF Z GFa=dF

dF (14)

For digital signal processing, Eq. (13) is transformed into an

equivalent discrete transfer function using continuous-to-

discrete-transformation and zero-order-hold

xeðkC1ÞZexpfðAeKKCeÞtdgxeðkÞ

C

ðtd

0expfðAeKKCeÞtgKdt

�zðkÞ

FaðkÞZCoxeðkÞ

(15)

GFa=dFZ

5:67!105s6 C5:64!108s5 C3:61!1013s4 C2:20

s7 C1:11!104s6 C1:25!108s5 C7:09!1011s4 C3:2

where in this application, the discrete sampling time is

tdZ0.1 ms.

3.2. Kalman filter gain matrix

A Kalman filter is a time variant observer designed to

suppress state estimation errors, ~xZ xKx, due to system

and measurement noise. The Kalman filter gain matrix

is identified by minimizing the state estimation error

covariance matrix, PZE½ ~x ~xT �. Based on the assumption

that system and measurement noise are uncorrelated

zero-mean white noise signals with covariance matrices

QZE½w wT �O0, RZE½ v vT �R0 and E½w vT �Z0,

the minimum state estimation error covariance matrix P can

be evaluated by solving the following time variant Riccati

equation [21,22,24]:

_P Z AeP CPATe CGQGT KPCT

e RK1CeP (16)

Using the solution of Eq. (16), the optimal Kalman filter

gain matrix is obtained as:

K Z PCTe RK1 (17)

The measurement noise covariance matrix, R, is determined

from the average electrical RMS reading of the displace-

ment sensor when the machine is stationary and from the

average differences in air cutting fluctuations. Assuming

that the unknown cutting force is the only state variable

affected by the system noise, w, one obtains the structure of

the system noise matrix, G. The system noise covariance

matrix, Q, is tuned to accommodate the compensations. For

this case R, Q, and G are:

R Z ½ 56:59 �; Q Z ½1:82!1013�; G Z ½ 0ð1!6Þ 1 �T

(18)

With respect to real-time signal processing, the stationary

solution of the above Riccati equation (Eq. (16)), PN

ð _PZ0Þ, and the corresponding static Kalman filter gain

matrix, KN, are used in this application, which yields a sub-

optimal time-invariant Kalman filter—also called a Wiener

filter [24]. When transient oscillations have disappeared, the

time-invariant Kalman filter yields the same results as the

time-variant one [24]. In this example, the Kalman filter

gain matrix, KN, is:

KNZ½0:0046 0:0386 0:0373 0:0070 K0:0008 0:0004 5:6711�T

!1!105

Substituting KN for K in Eq. (14) yields the time-invariant

Kalman filter transfer function between the reconstructed

force at the tool tip, Fa, and the displacement of the spindle

flange measured by the capacitive sensor, dF:

!1016s3 C6:77!1020s2 C1:95!1023s C3:64!1027

8!1015s3 C1:32!1019s2 C2:21!1022s C6:97!1025

The actual force applied at the tool tip is reconstructed from

the preprocessed measurements, dF, of the displacement

sensor (see Eq. (1)) by applying the time-invariant Kalman

filter as a discrete time domain recursive filter at each

sampling instant according to Eq. (15). In continuous time,

the reconstructed cutting force is obtained as:

Fa Z GFa=dFdF Z GFa=dF

ðdCutting KdAir_CuttingÞ (19)

The modeled frequency response function (FRF) of the

uncompensated force sensor system ½FðsÞZdF=Fa�Ks

(i.e. normalized model FRF), the FRF of the Kalman

filter ½GFa=dFðsÞZ Fa=dF�K

K1s (i.e. normalized KF FRF), and

the cascaded FRF of the compensated force sensor system

FCðsÞZFðsÞGFa=dFare illustrated in Fig. 6. Depending on

the ratio of R and Q, the cascaded FRF of the compensated

force sensor system has a more or less perfect unity gain at

the regions of modal frequencies. Deviations may occur,

especially at anti-mode locations. However, compared to

the uncompensated system, the distortions of indirect force

measurements are significantly reduced with the Kalman

filter. In this case, the compensated system has nearly linear

phase shift up to 1000 Hz with a phase lag of K1098 at

1000 Hz. When measuring cutting forces synchronously in

X-, Y-, and Z-directions, the Kalman filters for Y- and Z-

directions can be designed with approximately the same

phase delays as for X-direction, up to a specified frequency.

This way, the true orientation of the original force vector

can be maintained in the indirect force measurements.

Fig. 6. Normalized FRFs of the uncompensated indirect force sensor system, the Kalman filter, and the compensated indirect force sensor system (cascaded

FRF) in X-direction.


4. Results and discussion

Several cutting tests conducted with a five-fluted end mill

are presented here as an example. With the five-fluted cutter,

the spindle system is excited at tooth passing frequency,

which is five times the spindle frequency. The workpiece

material used was an Al7050-T6 alloy. The tests were

performed at 1.5 mm depth of cut and 0.1 mm feed per tooth

in full immersion mode without any lubricant. The cutting

forces measured with the reference force sensor (table

dynamometer), the uncompensated spindle sensor, and the

Kalman filter compensated spindle sensor are shown in

Figs. 7–11. The top graph of each of these figures shows the

dynamic components of the cutting forces in time domain.

The bottom graph of each of these figures depicts a cutout of

the magnitude spectrum of the force signal focused on tooth

passing frequency, ftp. Since only one displacement sensor

was used in this experiment, the sensor system was not

temperature compensated. This leads to a static offset in the

reconstructed forces when the temperature of the spindle

flange varies between the measurements taken when cutting

air and when cutting metal. A proposal to compensate for

this effect is described later in this paper.

The table dynamometer reliably measures cutting

forces below 650 Hz with less than a 10% error in

magnitude. This corresponds to a spindle speed of

7800 rpm (ftpZ650 Hz) when using a five-fluted cutter.

Forces with higher frequencies are increasingly distorted

due to the dynamic characteristics of the dynamometer.

However, forces between 650 and 1000 Hz are almost

only distorted in magnitude with a maximum error of 70%

at 1000 Hz. Phase shift remains nearly invariant for all

frequencies below 1000 Hz (see Fig. 2). Therefore, it is

possible to still use the dynamometer as reference force

sensor for spindle speeds between 7800 and 12,000 rpm

(ftpZ650–1000 Hz) when focusing on single frequencies

in the spectrum and considering the corresponding

magnitude error according to the dynamometer transfer

function (Fig. 2). In the following, the force component at

tooth passing frequency is used for comparison and

assessment of the quality of force reconstruction at

different spindle speeds.

The cutting test results for a spindle speed of 1000 rpm

(ftpZ83.33 Hz) are given in Fig. 7. The bandwidth of the

uncompensated spindle sensor (350 Hz) is large enough to

capture the first four harmonics of the tooth passing

frequency. Therefore, the uncompensated spindle sensor

effectively measures cutting forces and yields almost

identical measurements as the dynamometer.

When the spindle speed is increased to 5700 rpm, tooth

passing frequency (ftpZ475 Hz) becomes very close to the

first mode (486 Hz); therefore, the dynamics of the first

mode distort the displacement measurements. The Kalman

filter compensates for the distortions and brings the force to

the level provided by the dynamometer (see Fig. 8).

At a spindle speed of 7800 rpm, tooth passing frequency

(ftpZ650 Hz) reaches the bandwidth limit of the dynam-

ometer when allowing only a 10% error in reference force

measurement. The frequency spectrum in Fig. 9 shows that

the uncompensated spindle sensor cannot measure the force

Fig. 8. Cutting force measurement at 5700 rpm (spindle frequency fSpZ95 Hz, tooth passing frequency ftpZ475 Hz). Dry cutting aluminum Al7075-T6 with a

five-fluted end mill at 0.1 mm feed per tooth and 1.5 mm depth of cut in full immersion mode. All cutting force signals are displayed in both time and frequency

domain. In the frequency spectra, the focus is always on the force component at tooth passing frequency, which is used to assess the quality of indirect force

measurement with and without dynamic compensation. In the diagrams, ‘Ref’ denotes the reference cutting force from the table dynamometer; ‘Disp’ denotes

the force measured by the uncompensated displacement sensor system; and, ‘KF’ denotes the force from the Kalman filter compensated displacement sensor

system.

Fig. 7. Cutting force measurement at 1000 rpm (spindle frequency fSpZ16.7 Hz, tooth passing frequency ftpZ83.3 Hz). Dry cutting aluminum Al7075-T6 with

a five-fluted end mill at 0.1 mm feed per tooth and 1.5 mm depth of cut in full immersion mode. All cutting force signals are displayed in both time and

frequency domain. In the frequency spectra, the focus is always on the force component at tooth passing frequency, which is used to assess the quality of

indirect force measurement with and without dynamic compensation. In the diagrams, ‘Ref’ denotes the reference cutting force from the table dynamometer;

‘Disp’ denotes the force measured by the uncompensated displacement sensor system; and, ‘KF’ denotes the force from the Kalman filter compensated

displacement sensor system.


Fig. 10. Cutting force measurement at 9000 rpm (spindle frequency fSpZ150 Hz, tooth passing frequency ftpZ750 Hz, magnitude error of reference force is

20%). Dry cutting aluminum Al7075-T6 with a five-fluted end mill at 0.1 mm feed per tooth and 1.5 mm depth of cut in full immersion mode. All cutting force

signals are displayed in both time and frequency domain. In the frequency spectra, the focus is always on the force component at tooth passing frequency,

which is used to assess the quality of indirect force measurement with and without dynamic compensation. In the diagrams, ‘Ref’ denotes the reference cutting

force from the table dynamometer; ‘Disp’ denotes the force measured by the uncompensated displacement sensor system; and, ‘KF’ denotes the force from the

Kalman filter compensated displacement sensor system.

Fig. 9. Cutting force measurement at 7800 rpm (spindle frequency fSpZ130 Hz, tooth passing frequency ftpZ650 Hz). Dry cutting aluminum Al7075-T6 with a

five-fluted end mill at 0.1 mm feed per tooth and 1.5 mm depth of cut in full immersion mode. All cutting force signals are displayed in both time and frequency

domain. In the frequency spectra, the focus is always on the force component at tooth passing frequency, which is used to assess the quality of indirect force

measurement with and without dynamic compensation. In the diagrams, ‘Ref’ denotes the reference cutting force from the table dynamometer; ‘Disp’ denotes the

force measured by the uncompensated displacement sensor system; and, ‘KF’ denotes the force from the Kalman filter compensated displacement sensor system.


Fig. 11. Cutting force measurement at 12,000 rpm (spindle frequency fSpZ200 Hz, tooth passing frequency ftpZ1000 Hz, magnitude error of reference force is

70%). Dry cutting aluminum Al7075-T6 with a five-fluted end mill at 0.1 mm feed per tooth and 1.5 mm depth of cut in full immersion mode. All cutting force

signals are displayed in both time and frequency domain. In the frequency spectra, the focus is always on the force component at tooth passing frequency,

which is used to assess the quality of indirect force measurement with and without dynamic compensation. In the diagrams, ‘Ref’ denotes the reference cutting

force from the table dynamometer; ‘Disp’ denotes the force measured by the uncompensated displacement sensor system; and, ‘KF’ denotes the force from the

Kalman filter compensated displacement sensor system.


component at tooth passing frequency any more (165%

deviation!); whereas the Kalman filter compensated spindle

sensor reaches the quality of the dynamometer.

At 9000 rpm, tooth passing frequency is 750 Hz, which is

between the second and the third mode (709 and 929 Hz),

and the magnitude amplification of the dynamic compliance

of the spindle system becomes close to the static value

again. Hence, the uncompensated spindle sensor produces

a smaller error than before. The reference force has a

magnitude error of about 20% itself at 750 Hz; the

uncompensated indirect force measurement deviates around

37% (62% absolute); whereas the Kalman filter compen-

sated measurement differs by only 7% (29% absolute) (see

Fig. 10). The overall quality of force measurement is still

quite good at this speed.

When spindle speed is increased to 12,000 rpm, tooth

passing frequency becomes 1000 Hz, which is on the falling

flank of the third mode of the structural dynamics of the

spindle system. At 1000 Hz, the transfer function of the

dynamometer measurement has an error of 70% in

magnitude. Phase shift is still very small. Nevertheless,

the dynamometer is excited at its limits here. In the

frequency spectrum of Fig. 11, the uncompensated indirect

force sensor and the dynamometer measure similar signal

amplitudes, whereas the Kalman filter compensated

measurement is strongly dampened. Considering the

magnitude error of the reference signal of 70% at

1000 Hz, the Kalman filter compensated indirect force

measurement is approximately 52% too small, whereas the

uncompensated measurement is roughly 97% too high.

Since magnitude amplification of the cascaded transfer

function decreases steadily for higher frequencies, the

Kalman filter compensated indirect force sensor system

seems to have reached its bandwidth here (see Fig. 6).

Cutting tests, ranging from 50 to 1000 Hz tooth passing

frequency, have been conducted; and, all showed quite good

results. This proves that the suggested dynamic compen-

sation technique is a good and feasible way to compensate

for the dynamic characteristics of a sensor system and, thus,

to increase its bandwidth while suppressing high frequency

noise at the same time.

4.1. Run out, roundness errors and unbalance

(air cutting measurements)

The displacement sensor measures the variation of the

gap size between the sensor head and spindle flange. Hence,

it also records displacements due to run out and roundness

errors of the spindle flange, where run out may change with

the spindle speed due to unbalance. These errors are

compensated for by subtracting the spindle flange profile,

which is measured while air cutting, from the displacement

signals measured while cutting at the same speed (see

Eq. (1)). The two signals are synchronized using the encoder


mounted on the spindle. The encoder has a reference

channel, which gives one impulse per revolution that can be

triggered at exactly the same spindle position. Alternatively,

a spindle integrated angular position encoder can be used for

more precise spindle position measurement. However, when

spindle speed fluctuations are low, the encoder principle

works accurately enough. Nevertheless, any fluctuation in

spindle speed (e.g. due to load variations) leads to slightly

different spindle periods in air cutting and cutting signal;

and thus, the numbers of samples per revolution may differ.

In order to overcome the problem, the two signals are

synchronized by splitting the cutting signal in sequences

corresponding to one revolution and adapting the air cutting

profile through re-sampling. For online applications, the

processing time required for synchronization and subtrac-

tion of the air cutting profile has to be considered.

4.2. Thermal effects

During the machining operation, the temperature of the

spindle system may fluctuate. Mechanical energy is

dissipated in the cutting process or, for example, in the

spindle bearings, which causes an increase of temperature.

Cooling is provided by an in-built cooling system in the

spindle housing or by natural convection of the surrounding

air. When machining, the dynamic balance of heat flux

stabilizes after a while, and so do temperatures. However,

temperature fluctuations of a few degrees, plus or minus, are

immanent. The spindle flange is, as with any metal object,

subject to dilatation. When the temperature increases, the

spindle flange diameter enlarges in consequence; and vice

versa, when temperatures, drop the spindle flange diameter

shrinks. This leads to variations in the gap size between the

displacement sensor and spindle flange, which can be

falsely interpreted as forces by the spindle sensor. The same

applies if the sensor clamp or the spindle housing enlarges

or shrinks due to temperature variations.

Assuming an equal temperature distribution, the dilata-

tion, DL, of a metal object with an overall length, L, can be

Fig. 12. Variation of the gap between sensor head and spindle flange wh

calculated as:

DL Z awLDw (20)

where aw is the thermal expansion coefficient; and, Dw is the

temperature variation. Since the spindle flange, which has a

diameter of 88.9 mm, is made out of steel (thermal

expansion coefficient of steel [25]: aw;Steel Z10:5K13:0!10K6 1=K), an increase in temperature of 1 K theoretically

expands the spindle flange approximately 0.93–1.16 mm,

provided an equal temperature distribution. Thus, the gap

between the displacement sensor and spindle flange

decreases 0.47–0.58 mm, accordingly. Since temperature

fluctuations are comparably slow, this change in gap size

is interpreted as a quasi static force offset of about

23.3–28.9 N, when using the static stiffness of the spindle

sensor system (KSz50 N/mm) as calibration factor. So,

even very small temperature fluctuations lead to quite large

quasi static offsets in indirect force measurement.

In an experiment that is supposed to resemble a cutting

process where heat is generated at the tool tip, the tool is

heated with a fan while measuring the temperature on the

spindle flange surface with a thermocouple and recording

the displacement signal from the spindle sensor. Displace-

ment and reconstructed force are plotted versus temperature,

as shown in Figs. 12 and 13, respectively. Both quantities

show a linear dependency on temperature variation.

According to the measurements, an increase in temperature

of only 1 K (measured at the surface of the flange) leads to a

reduction in gap size of approximately 0.8 mm, which is

interpreted as a quasi static force offset of around 40 N. The

measured value is higher than the theoretical value from

above, which is most likely caused by an unequal

temperature distribution. For instance, it is possible that

temperature accumulates in the tool holder, which is

clamped on the spindle, due to a slower heat transfer at

the tapered interface that leads to additional mechanical

strain and, thus, to a larger force offset.

The effect of dilatation does not affect the measurement

of dynamic force components. However, it produces large

en warming the spindle system at the tool tip using a heater fan.

Fig. 13. Static force offset caused by dilatation of the spindle flange when warming the spindle system at the tool tip using a fan.


quasi static offsets, which cannot be neglected. Hence, the

adverse effect of dilatation must be compensated for, in

order to obtain dependable cutting force measurement. This

is especially important when large targets are used, since

dilatation is proportional to the geometric size, L, of the

target object (see Eq. (21)). On the other hand, a large target

diameter is recommendable to avoid cross-talk.

A very simple way to automatically compensate for

dilation is to use multiple displacement sensors. Dilatation

leads to an overall increase or decrease of the spindle flange

or the spindle housing diameter; whereas cutting forces

deflect the spindle flange in one direction only (see Fig. 14).

By using pairs of two equally calibrated displacement

sensors in a face-to-face-arrangement and coupling their

Fig. 14. Compensation for the effect of dila

outputs with a negative sign, changes in the spindle flange or

the spindle housing diameter due to temperature variation

are no longer detected. When the temperature varies, both

sensors measure the same positive or negative change in gap

size, which is compensated for, due to the negative coupling

of the sensor outputs. However, when a force deflects the

spindle shaft, one sensor registers a decrease and the other

one an increase in gap size according to the amount of

deflection. Provided that there is equal calibration, one

obtains a voltage proportional to twice the amount of

deflection when subtracting the two signals. When perform-

ing the coupling of the two sensors on a digital computer or

the NC-controller, different calibration factors can also be

considered.

tation on indirect force measurement.

Fig. 15. Cartesian arrangement of displacement sensors for two-dimensional force measurement and automatic temperature compensation.


In case of force measurement in X- and Y-directions, the

above-described principle requires four displacement sen-

sors in a Cartesian arrangement, as is shown in Fig. 15.

Below are the coupling equations, in order to compensate

for dilation and to extract the deflection due to cutting forces

in X- or Y-direction, respectively

dF;x Z1

2ðd1 Kd3Þ Z

1

2

U1

KU;1

KU3

KU;3

�(21)

dF;y Z1

2ðd2 Kd4Þ Z

1

2

U2

KU;2

KU4

KU;4

�(22)

where Ui are the output voltages; and, KU,i are the sensitivity

factors for the displacement sensors in mV/mm. In order to

further reduce the costs for the sensor hardware, it is

possible to achieve the same with only three sensors, which

are arranged at angles of 1208 along the circumference of

the spindle flange like a Mercedes-star, as shown in Fig. 16

dF;x Z1

2ðd1 Kd2Þsinð608Þ Z

1

2

U1

KU;1

KU2

KU;2

�sinð608Þ

(23)

dF;y Z1

2½ðd1 Cd2Þcosð608ÞKd3�

Z1

2

U1

KU;1

CU2

KU;2

�cosð608ÞK

U3

KU;3

� �(24)

In order to obtain optimal temperature compensation

performance, equal temperature distribution should be

guaranteed in the spindle system. Therefore, the spindle

and sensor ring should be designed rotation-symmetrically.

4.3. Play in the spindle bearings

Play in the spindle bearings may also cause a static offset

in indirect force measurement, which changes sign with the

cutting direction. This offset cannot be compensated for

with the above-described sensor arrangement. The easiest

way to compensate for play is to provide sufficient pre-load

for the spindle bearings.

4.4. Changing dynamics with respect to rotational speeds

Since dynamic stiffness may change due to variation in

bearing stiffness as a function of spindle speed, the

experimental modal analyses are performed to acquire the

FRFs with respect to the spindle speeds from 0 to

Fig. 16. Mercedes-star arrangement of displacement sensors for two-dimensional force measurement and automatic temperature compensation.


12,000 rpm. A small ball bearing is, therefore, attached to a

cylindrical blank tool, which is clamped on the tool holder

(see Fig. 17). The outer bearing ring is kept stationary while

the spindle rotates; and, an impact force is applied on the

stationary outer bearing ring to measure the transfer

functions of the spindle sensor system at different speeds.

Additionally, dynamic compliance is measured using a laser

sensor in order to verify the variation of natural frequencies,

damping and stiffness of the spindle at different speeds.

Based on observations, the first two modes are almost

insensitive to variation of spindle speed; whereas the third

modal frequency decreases up to 12% at 12,000 rpm. To

take account of the variation in bearing stiffness, it is

possible to update the parameters of the discrete Kalman

filter when spindle speed approaches very high levels.

However, these parameters must be determined offline and

in advance. They can be stored in the signal processing unit

until they are needed.

Fig. 17. Experimental setup for dynamic frequency response function

measurements while the spindle rotates.

4.4.1. Receptance coupling

Whenever the length or mass of a tool is changed, the

spindle dynamics may change, too. Thus, a new FRF

measurement and an update of the filter parameters are

required. In addition, when the depth of cut is very large, the

dynamic characteristics may also change, because the

model is based on the measurement from the tool tip.

When using the indirect force sensor system in production,

the Kalman filter parameters must be determined in advance

for each tool in the magazine. The parameters can be stored

in the signal processing unit (e.g. the NC-controller) to

enable an automatic update of the Kalman filter when a tool

is exchanged. As proposed by the authors in [23], a

receptance coupling approach, with joint identifications to

couple an arbitrary tool to the spindle, can be used to predict

the FRF of the spindle sensor system using the new tool to

take FRF measurements for each tool.

5. Conclusions

This paper presents an indirect cutting force sensing

method using the displacements of the rotating spindle

shaft measured by a spindle mounted capacitive sensor.


The bandwidth of the uncompensated indirect force sensor

system is limited to 350 Hz, due to vibration modes of the

spindle structure. Using a Kalman filter, which quasi inverts

a model of the dynamic compliance between tool tip and

displacement sensor reading point, the bandwidth of the

proposed sensor system is increased from 350 to almost

1000 Hz. The method has been verified through several

milling experiments.

Acknowledgements

The research is supported by NSERC, Boeing, and Pratt

and Whitney Canada. The work has been completed at the

Manufacturing Automation Laboratory, University of

British Columbia.

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Documents

High frequency bandwidth cutting force measurement in milling using capacitance displacement sensors