Identification of dynamic rigidity for high speed

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 02 Issue: 07 | Jul-2013, Available @ http://www.ijret.org 146
IDENTIFICATION OF DYNAMIC RIGIDITY FOR HIGH SPEED
SPINDLES SUPPORTED ON BALL BEARINGS
Jakeer Hussain Shaik1
, Srinivas J 2
1
Research Scholar, 2
Associate Professor, Department of Mechanical Engineering, National Institute of Technology,
Rourkela, Odisha, India, jakeershaik786@yahoo.co.in
Abstract
The widespread use of high-speed machining in recent decades has led to a significant area of research on issues that limit its
productivity. Regenerative chatter is a well‐known machining problem that results in unstable cutting process, leads to the poor
surface quality, reduced material removal rate and damage on the machine tool itself. The main requirement for the stability of system
dynamics is the information of tool tip frequency response functions (FRF’s).The present work considered a coupled model of spindle-
bearing system by using the angular contact ball bearing forces on stability of machining. Using Timoshenko beam element
formulation, the spindle unit is analyzed by including the gyroscopic and centrifugal terms and the bearing contact forces are arrived
from Hertzian contact theory. Then, the model is used for studying the effects of viscous damping to obtain the tool point FRF for the
dynamic spindle.
Index Terms: Spindle dynamics, Stability, High speed effects, Bearing contact forces, Hertzian contact theory, Finite
element modeling.
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1. INTRODUCTION
In recent years there has been keen interest by manufacturers
to increase Productivity and Surface quality of part by using
High speed milling machines. This has direct impact on cost
of machining and Quality. Dymamic rigidity is one of the
most critical characteristics of machine tools, especially for
high precision and high performance machining applications.
This technology is mainly limited by the performance of the
spindle which has a significant influence on the machining
accuracy. Self-excited vibrations of the tool result in unstable
cutting process which leads to the chatter on the work surface
and it reduces the productivity. At large rates of material
removal, tool chatter creates the unavoidable flexibility
between the cutting tool and the work piece; if it is
uncontrolled the chatter causes a rough surface finish and
dimensional inaccuracy of the work piece along with
unacceptable loud noise levels and accelerated tool wear. The
chatter stability of the tool is dependent on the dynamic
behavior of the spindle system, which is often expressed as the
frequency response function (FRF) at the tool tip. In other
words, tool-tip FRF is a key variable in determination of
stability limit [1]. The objective of the design engineer is to
predict the cutting performances of the spindle during the
design stage by relying on engineering model of the process
and system dynamics. Early spindle research has focused
mainly on static analysis, where as current research is
extended to the dynamics for optimal design.
Over the last two decades, numerous approaches have been
addressed to elaborate the tool-tip FRF through modeling and
experiments. With introduction of receptance coupling
substructure analysis by Schmitz in early 2000, several authors
[2-7] have illustrated the approach to predict the tool point
FRFs. Erturk et al.[8] presented analytical models to obtain the
tool-tip FRF. Faassen et al. [9] predicted chatter stability lobes
of high speed milling on the basis of experimental FRFs at
different spindle speeds. Abele and Fiedler [10] introduced a
sub-space-state-space-identification method to measure and
calculate the dynamic behavior of spindle-tool systems during
high speed machining, consequently, stability lobe diagrams
were determined based on the identified FRFs. Zaghbani and
Songmene [11] used operational modal analysis to estimate
the machine tool dynamic parameters during high machining
operations, and the dynamic stability lobes were calculated
using the extracted modal parameters. The experimental
method is used to identify speed-varying dynamics of spindles
in the above works, which is direct and feasible. However, it is
time-consuming to repeat the modal tests for every spindle
speed and prone to errors. With known details of the spindle
geometry, drawbar force, bearing parameters and preload, etc.,
an alternative method to obtain the dynamic behavior of
spindle system is the finite element (FE) method. With the
accurate spindle model, the FRFs at the tool tip are simulated
and then chatter stability lobes can be predicted. Chen and
Wang [12] modeled the integrated spindle-bearing system and
they also found that significant errors occurred in predicting
stability lobes if the load and speed effects on the

__________________________________________________________________________________________
shaft/bearing dynamics were neglected. Tian and Hutton [13]
considered the gyroscopic effects of the rotating spindle and
proposed a chatter model in milling systems. They found that
gyroscopic effects reduced the critical axial depth of cut.
Similarly, Movahhedy and Mosaddegh [14] predicted the
chatter in high speed milling including gyroscopic effects on
the basis of FE spindle model. Gagnol et al. [15] developed a
dynamic model of a high-speed spindle system and then
proposed a dynamic stability lobe diagram by integrating the
speed-dependent FRFs of spindles into analytical approach of
Altintas and Budak [16]. Angular contact ball bearings are
commonly used in high-speed spindles due to their low-
friction properties and ability to withstand external loads in
both radial and axial directions. Although the ball bearings
appear to be a simple mechanism, their internal geometry is
quite complex and often display very complicated nonlinear
dynamic behavior. It is obvious that the spindle machining
system supported by ball bearings present interesting stability
characteristics and nonlinear responses. Several recent studies
[17-21] illustrated the dynamics of high speed spindles under
different preload mechanisms and operating speeds.
Present paper describes a coupled model of spindle-bearing
system by considering the effects of ball bearing Hertz contact
forces on the spindle dynamics. The governing differential
equations of motion of spindle system are employed and
numerical simulations are carried-out using Timoshenko beam
element model.
2. MODELING OF SPINDLE-BEARING SYSTEM
The spindle system generally consists of spindle housing
carrying spindle shaft over the front and rear bearings, a tool
holder and tool as shown in Fig.1.
Fig-1: Schematic of end-mill spindle unit
The dynamic behavior of the spindle-tool unit can be well
established through the modeling of the restricted spindle
rotating system as shown in Fig.2. The spindle has six
elements, seven nodes and total 28 degrees of freedom. Both
the fifth and seventh nodes of the shaft are supported by two
angular contact bearings. The x and y directional cutting
forces act on the tool tip. Tool is assumed to be rigidly
connected to the tool holder which is fixed to the spindle shaft.
Fig-2: Equivalent analysis model
The equations of motion for the spindle shaft due to rotation
are given by [22-23]:
x
2
zs2
2
FAv
x
v
P
x
v
AGk
xdt
vd
A =ρΩ−





∂
∂
−





θ−
∂
∂
∂
∂
−ρ
(1)
y
2
ys2
2
FAw
x
w
P
x
w
AGk
xdt
wd
A =ρΩ−





∂
∂
−





θ+
∂
∂
∂
∂
−ρ
(2) (2)
xys2
y
2
z
2
y
2
M
x
w
AGk
x
EI
dt
d
J
dt
d
I =





θ+
∂
∂
+
∂
θ∂
−
θ
ρΩ+
θ
ρ
(3)
yzs2
z
2
y
2
z
2
M
x
v
AGk
x
EI
dt
d
J
dt
d
I =





θ−
∂
∂
−
∂
θ∂
−
θ
ρΩ−
θ
ρ
(4)
Here, v and w are the bending deformation in two
perpendicular directions of the spindle unit. ρA is mass per
unit length of spindle shaft, ks is shear deformation factor, EI
is flexural rigidity, AG is shear rigidity of the shaft material ,
ρJ is polar modulus , Fx and Fy are the components of
external forces, Mx and My are the components of transverse
moments and Ω is the speed of rotation of spindle shaft.
A rotating shaft with eight degrees of freedom per node finite
beam element with gyroscopic terms based on Timoshenko
beam theory formulated by Nelson [24] is employed to model
the rotating spindle. The element translational and rotational
mass matrices, the stiffness matrix and gyroscopic matrix are
1
2
3
4
5
6
7
Front bearings
Rear bearings

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obtained by applying Hamilton’s principle. The governing
equation in matrix form obtained using FE method is:
( ) F}q{]M[]K[}q]{C[}q]{M[ c
2
=Ω−++ &&&
(5)
Where the system element matrices are given by [18]:
[M] = [MT] + [MR] (6)
[MT] = [M0] + φ[M1] +φ2[M2] (7)
[MR]= [N0] + φ[N1] +φ2[N2] (8)
[K]= [K0] + φ[K1] (9)
[C] =-Ω[G] + α[M] + β[K] (10)
φ=
2
12
s
EI
k AGl (11)
The front and rear positions of the spindle shaft are supported
by the similar angular contact ball bearings. Without
considering the axial forces and moment loadings of the
spindle system, the angular contact ball bearings can be
assumed as deep-groove ball bearings and the dynamic
problem of the angular contact ball bearing is made as one of
the two degrees of freedom. The local Hertz contact forces and
deflection relationships for a ball with the inner and outer
races may be written as a following set of restoring forces:
Fx1=
∑ ϕδ−ϕ+ϕ−−
=
bN
1i
i
5.1
i1bi1b1b1b1b cos)sinycosx(kxc &
(12)
Fy1=
=
bN
1i
i
5.1
i1bi1b1b1b1b sin)sinycosx(kyc &
(13)
Fx2=
=
bN
1i
i
5.1
i2bi2b2b2b2b cos)sinycosx(kxc &
(14)
Fy2=
=
bN
1i
i
5.1
i2bi2b2b2b2b sin)sinycosx(kyc &
(15)
Here δ refers to the initial clearance of the bearings, Nb is
number of balls, xb1 and yb1, xb2 and yb2 are the
displacements of mass elements distributed at the front and
rear bearing nodes along x and y directions respectively, kb1
and cb1, kb2 and cb2 are the stiffness and damping of front
and rear bearings respectively, and the
angle
2
( 1)
( )
i
b
r
t i
R r N
π
φ
  Ω
= + −  +    (where i=1,2,…Nb)
is the angular location of the ith ball. Here the term: xbcosφi+
ybsinφi-δ is accounted only when it is positive, otherwise it is
taken as zero and the bearing races are at loss of contact with
bearing balls. An idealized condition is considered in present
work with δ=-0.05 µm (considering practical conditions, the
interference fit generally adopted to chatter problems where
negative clearance is justified) and negligible value of bearing
damping, inner radius r=40 mm, outer radius R=60mm,
number of ball Nb=8, kb1=kb2=13.34×109 N/m3/2. The
spindle FRF consisting of real and imaginary parts can be
given expressed as:
[H(jω)]=[Re(ω)]+j[Im(ω)]=[-[M]ω2 +jω[C]+([K]-Ω2[Mc])]-1
(16)
Here, Re and Im are, respectively, the real and imaginary part
of the transfer function of the spindle tool tip.
3. RESULTS
The parameters of the finite element model of the spindle are
illustrated in Table 1. Except the element-1 (Silicon carbide
tool) and all other elements are taken as steel. Densities and
shear modulus are considered for the problem is obtained from
tables.
Table -1: Parameters for finite element model of the spindle
[18].
Parameter Elements of the spindle
1 2 3 4 5 6
Length(mm) 45 40 50 20 200 20
Outer
dia.(mm)
19 74.5 40 60 60 60
Inner
dia.(mm)
0 0 0 0 40 40
E (GPa) 450 210 210 210 210 210
Parameter 45 40 50 20 200 20
Computer program is developed in MATLAB to analyze the
spindle system. The assembly matrices and static condensation
approach to eliminate the rotational degrees of freedom are
included in the program. Initially, by assuming the bearings to
be perfectly rigid, Campbell diagram is obtained as shown in
Fig.3. Using undamped natural frequencies, the coefficients of
Rayleigh’s damping α and β are obtained as 17.32 and 3.87e-6
respectively for 1% damping ratio.

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500
1500
2500
3500
4500
5500
6500
7500
8500
0 10000 20000 30000 40000 50000
naturalfrequency(Hz)
rotational speed (rpm)
CAMPBELL DIAGRAM
BW1
FW 1
BW 2
FW 2
BW 3
FW 3
Fig-3: First three natural frequencies of spindle rotor
Here the viscous damping forces and centrifugal stiffening are
not accounted. The forward and backward whirl modes are
obvious due to the gyroscopic effect at three first natural
frequencies 1098.8 Hz, 4079.1 Hz and 7590.57 Hz
respectively. In order illustrate the centrifugal stiffening effect
and bearing dynamics, direct frequency response function at
the tool tip hxx(jω) is illustrated at different spindle speeds
(Ω). Fig.4. depicts the FRF without considering Rayleigh’s
damping for the spindle mounted on rigid bearings at two
different speeds. The effect of centrifugal stiffening is
observed at higher speeds only as seen in Fig.5.
Fig-4: Frequency response at 7000 rpm
Fig-5: Frequency response at 15000 rpm
Unlike direct FRF, the cross FRF is a smooth curve without
showing backward (BW) and forward (FW) whirl modes as
seen from the Fig.6. obtained at 15000 rpm.
Fig-6: Cross frequency response at 15000 rpm
The effect of Rayleigh’s (viscous) damping on the tool tip
FRF is shown at 12000 rpm as shown in Fig.7. It is seen that
damping affects considerably the whirl modes.
Fig-7: Effect of viscous damping at a speed 12000 rpm
For studying the bearing effect, the solution is obtained as a
transient analysis problem first in time domain and then the
frequency spectrum is obtained from fast Fourier
transformation (FFT) algorithm. The reduced coupled
differential equations (14 in number) were solved by explicit
Runge Kutta solver time integration schmes. Fig.8. and Fig.9.
shows the time histories and FFT diagram at the tool-tip node.
Fig-8: Time histories
Fig-9: Frequency responses
Backward
Forward

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It is has been observed clearly from the frequency responses
that the natural frequencies of the spindle have been affected
by the bearing contact forces.
CONCLUSIONS
In this work a coupled model of spindle bearing system is
considered for illustration and it was analyzed using finite
element analysis by considering the effect of shear
deformation and rotary inertia of spindle shaft. It was
observed that the Centrifugal and gyroscopic forces on the
spindle rotor affect respectively the stiffness and damping of
the system. The viscous damping was also accounted and the
bearings were treated first as a rigid supports. Bearing forces
expressed as functions of corresponding nodal displacements
using Hertz contact theory have also been included in the final
model in order to investigate effect of the angular contact
bearings. The resulting FRF obtained by the spindle bearing
model can be used in the existing analytical and numerical
models for constructing the accurate stability lobe diagrams
for the high speed machining process.
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Identification of dynamic rigidity for high speed

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