# Finite Element Modeling and Stability Analysis of End Milling - PDF

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```					                     Finite Element Modeling and Stability Analysis of End Milling Chatter

T. Qu 1 K. Behdinan2 B. Lin2 A. Khajepour1
1
Department of Mechanical Engineering, University of Waterloo, Waterloo, ON, Canada, N2l 3G1
2
Department of Mechanical Engineering, Ryerson Polytechnic University, Toronto, ON, Canada, M5B 2K3

1. Introduction                                                                                                                   uj
End milling is one of the mostly used cutting processes for
machining surfaces. The cutting force and structure
F rj        vj
deflections vary according to the variation of the radial and
tool
axial depth of cut in surface machining. The milling system                                                                        j
θ        Ftj
can become unstable due to the inherent feedback existing
between the cutting forces and the structural deflections.
This self-excited vibration is known as chatter. Chatter is                                                            Ω               j-1
the fundamental reason, which limits the efficiency of the

a
z         y
machining process. It will result in undesirable finishing
workpiece
surface, decreases tool life, and produces excessive loads
x
on the structure of the machine. Fundamental chatter                                                (0,0,0)
theory was developed half a century ago. Now,                     Figure 1. Dynamic beam model of end milling
regeneration of chip thickness and mode coupling are
regarded as the main mechanism, which associated with             constant coefficients determined by the experiment,
chatter.                                                          h (φ j ) is the chip thickness and can be expressed in the
following form:
For milling chatter, the first detailed mathematical model              h (φ j ) = [V sin φ j + R j 0 (φ j ) − R j (φ j )] g (φ j )          (2)
with time varying cutting force coefficients was presented        where V is the feed per tooth, R j 0 and R j represent the
by Sridhar et al. [1]. Later, a comprehensive analytical
method to solve that model was developed by Minis and             radial displacement of the cutter at the previous and
Yanushevsky [2]. Budak and Altintas [3] also derived the          present cuts at the position φ j , respectively, g (φ j ) is a
finite order characteristic equation for the stability analysis   unit step function which determines whether the jth tooth
in milling. Fourier series components are used to                 is in or out of the cut. Namely, g (φ j ) = 1 when
approximate the time varying dynamic cutting force
φ st ≤ φ j ≤ φ ex where φ st and φ ex are the start and exit angular
coefficients, and chatter free axial depth of cuts and spindle
speeds are derived analytically. In the past, the milling         immersions           of   the     cutter,     respectively.           Otherwise
process was modeled as a lumped mass system. They are              g (φ j ) = 0 .
valid for some kinds of machining processes, such as face
milling, etc. For end milling, the cutter is in the shape of a    The total dynamic cutting force on the cutter can be
cantilever beam, and the cutting forces are distributed           resolved into two components in the x and y directions in
loads. In order to account for the flexibility, a cantilever      the stationary coordinates (x,y,z):
beam model is considered for the better understanding of                Fx  1        a11 a12   x( t) − x (t − τ ) 
the chatter mechanism.                                                   = a Kt                                    (3)
 Fy  2       a21 a22   y( t) − y( t − τ )
In this paper, an end milling machine tool has been modeled       where
as a finite element (FEM) Timoshenko cantilever beam. The             a11          − g j [sin 2φ j + K r (1 − cos 2φ j )]
element mass and stiffness matrix were developed by                   a  N t −1 − g [1 + cos 2φ + K sin 2φ ] 
 12                                             j 
 =           ∑
j            j      r
Nelson [4]. The cutting forces model is used in [3]. The                                                                                     (4)
 a 21  j = 0  g j [1 − cos 2φ j − K r sin 2φ j ] 
stability analysis is made based on the characteristic
 a22 
              g j [sin 2φ j − K r (1 + cos 2φ j )] 
equation derived in frequency domain.                                                                                      

2. Milling Force Model                                            where x(t) and y(t) represent the displacements of the
The cutting force model is capable of describing the              spindle in the x and y directions, N t , the tooth number of
primary characteristics of dynamic milling force. An end
a    a12 
milling cutter and its cross section are shown in Figure 1.       the cutter, and a (t ) =  11       is referred to as the
The tangential and radial cutting forces on the jth tooth are                               a21 a22 
given as:                                                         directional dynamic milling coefficient which is periodic
Ft ( j) = K t a h(φ j ), Fr ( j ) = K r K t h(φ j ) (1)      with the tooth passing frequency. When the time varying
where, a is the axial depth of cut , K t and K r are cutting      dynamic cutting forces are approximated by the average
component of the Fourier series expansion, α (t) can be                                    cutter has one natural frequency (348Hz), which is the same
expressed as:                                                                              as the dominated frequency in [3].

 [cos2φ − 2K φ + K sin 2φ )]φ                                         For the assumed end-milling cutter, we obtained two sets of
α11                                             φ
ex

α                        r         r
φ
st
       stability lobes. One is obtained by applying the
 12  Nt  [ − sin 2φ − 2φ + K r cos2φ ]φ                                
ex
Timoshenko cantilever beam model and the other
α (t) =   =      
st
 (5)
α21  4π  [− sin 2φ + 2φ + K r cos2φ ]φ
φ                                 considering an Euler-Bernoulli cantilever beam for the

ex
st

α 22 
        [ − cos 2φ − 2K φ − K sin 2φ )]φ                          ex          cutter, based on the method described by J. W. Sutherland
                   r         r          φ                 st          [5]. After calculating the modal mass and stiffness for the
Hence the cutting force in (3) can be reduced to:                                          cutter, we apply Altintas’ stability analysis method [3]
 Fx  N 1          α     α 12   x (t) − x(t − τ )                                 around the dominated mode (348Hz). The stability lobes are
 =     ∑    a K t  11                            
α 21 α 22   y (t) − y (t − τ ) 
(6)    shown in figure 2. It is obvious that the Timoshenko
 Fy  k =1 2                                                                          cantilever beam modal results in more conservative results.

3. FEM Timoshenko Cantilever Beam Model
We model the end-milling cutter using Timoshenko’s beam
theory. There are two coordinate reference systems used to
describe the motion of an element: a fixed coordinate (x,y,z)
and a rotating coordinate (u,v,z) fixed at the shaft.
According to [4], each element of this Timoshenko beam is
modeled with eight degrees of freedom. For each node, the
element has four degrees of freedom: two translational
displacements in x and y directions, and two rotational
displacements about the x and y axes, respectively.                                        Figure 2. The stability limit predictions

Applying the extended Hamilton’s principle, the element                                    5. Conclusion
translational and rotational mass matrices, element stiffness                              A dynamic milling model has been developed for end
matrix and shape function etc. can be obtained. The total                                  milling chatter. The cutter is modeled as a FEM
M                                 K
mass matrix [ ] and stiffness matrix [ ] can be easily                                     Timoshenko cantilever beam. Analytic stability analysis
assembled from the single element matrix. The damping in                                   has been performed. Stability lobes are compared for
each degree of freedom is representing [C]. So, the dynamic                                Timoshenko beam finite element analysis and Euler-
model for the end-milling tool can be expressed in the                                     Bernoulli beam modal analysis. The results show that
following form:                                                                            Timoshenko beam model predict more conservative stable
[M ]q&(t) + [C ]q(t) + [K ]q(t ) = (1 − e −τD )[B (t )]q(t) (7)
&           &                                                                      region.
where q(t) is the nodal displacement vector, [B(t)], a time-
varying matrix associated with the cutting forces, and e −τD                               The future work will be on increasing the number of
elements for the Timoshenko and Euler-Bernoulli beam for
is a time delay operator ( e −τD q(t)=q(t-τ)).
deterring the stability lobes more accurately. The study will
show the effect of the number of elements in the
4. Stability Analysis                                                                      Timoshenko and Euler-Bernoulli beam in the stability lobes.
Assuming the chatter frequency as ω c , the system
characteristic equation can be obtained as:                                                6. References
− i ωτ
[ K ] − ω [ M ] + iω c [ C] − (1 − e
2
c                                      )[ B( iω c )] = 0               (8)    [1] R.Sridhar, R.E.Hohn and G.W.Long: “ A general
formulation of the milling processing equation.
By introducing the transfer function between the cutting                                   Contribution to machine tool chatter research-5” , ASME J.
force and cutter tip displacement, the critical depth of cut                               of Engr. for Industry, 90, 317-324, (1968)
, alim , and the free spindle speed, Ω , can be derived as a                               [2] I.Minis and T.Yanushevsky: “A new theoretical
function of ω c .                                                                          approach for the prediction of machine tool chatter”, ASME
J. of Engr. for Industry, 115, 1-8, (1993)
To address the flexibility of the cutting tool,we constructed                              [3] Y.Altintas and E.Budak: “Analytical prediction of
the stability diagram and compared with the modal analysis                                 stability lobes in milling”, Annals of the CIRP, 44, 357-362,
results. The dominated mode in [3] was selected to use.                                    (1995)
Assume a 8-teeth end milling cutter with 19mm diameter,                                    [4] H.D.Nelson: “A finite rotating shaft element using
241mm in length, in a half immersion-up milling of aluminum                                Timoshenko beam theory”, J. of Mech. Design, 102, 793-
workpiece. Cutting constants are assumed as K t =1500                                      803, (1980)
MPa , K r =0.3 and damping ratio is 0.1. The end-milling                                   [5] J.W.Sutherland: “A dynamic model of the cutting force
system in the end milling process”, ASME Bound vol.-PED,
33 ,53-62, (1988)

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