We have investigated possible motions of the holder on top of the polishing pad during the process of dielectric plate grinding taking into account the forces of dry friction about its axis. A mathematical model of the mechanical device has been elaborated to describe the process of dielectric wafer grinding. The model is in the form of a non-autonomous nonlinear system with a variable structure. The structure of the phase space of the dynamical system was investigated, the qualitative studies of the possible motion modes were carried out. The values of the geometrical and dynamic parameters that qualitatively and quantitatively influence the modes of the holder motion were obtained. It was found that the inclusion of dry friction forces on the axis of the free holder result in a periodic motion of the mechanism with long stops. We present the calculations of the parameters for the type 3PD-320 machine

A. G. Panasenko

A. V. Grezina

L. A. Igumnov

V. S. Metrikin

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 501 
The effect of dry friction forces on the process of 
dielectric wafer grinding 
L. A. Igumnov1, V. S. Metrikin2, A. V. Grezina3, A. G. Panasenko4 
1, 2Research Institute for mechanics, National Research Lobachevsky State University of Nizhni Novgorod, 
Nizhny Novgorod, Russia 
3, 4National Research Lobachevsky State University of Nizhni Novgorod, Nizhny Novgorod, Russia 
2Corresponding author 
E-mail: 1igumnov@mech.unn.ru, 2v.s.metrikin@mail.ru, 3alekcandra-grezina@yandex.ru, 
4a.g.panasenko@yandex.ru 
(Received 25 May 2016; accepted 3 September 2016) 
Abstract. We have investigated possible motions of the holder on top of the polishing pad during 
the process of dielectric plate grinding taking into account the forces of dry friction about its axis. 
A mathematical model of the mechanical device has been elaborated to describe the process of 
dielectric wafer grinding. The model is in the form of a non-autonomous nonlinear system with a 
variable structure. The structure of the phase space of the dynamical system was investigated, the 
qualitative studies of the possible motion modes were carried out. The values of the geometrical 
and dynamic parameters that qualitatively and quantitatively influence the modes of the holder 
motion were obtained. It was found that the inclusion of dry friction forces on the axis of the free 
holder result in a periodic motion of the mechanism with long stops. We present the calculations 
of the parameters for the type 3PD-320 machine. 
Keywords: grinding, friction, mathematical model, phase space, holder, regimes with a long stop. 
1. Introduction 
The authors in [1, 2] focus on the study of the kinematics of the process and the quality of the 
abrasive machining of wafer surfaces. Here, we investigate the dynamics of the mechanical 
grinding of dielectric wafers pasted to a free holder, whose axes are subjected to frictional forces. 
Schematic diagram of the mechanism is shown in Fig. 1. The wafers to be ground are pasted to 
the bottom of the holder (a small circle with the center at ସܱ), which is placed on the polishing 
pad (a large circle with its center at ܱଷ ). The grinding is carried out by a complex relative motion 
of the holder and the polishing pad. The polishing pad rotates at angular velocity ߱଴. The motion 
of the holder’s axis (the center of ସܱ ) is determined by the operation of a crank-and-rod 
mechanism in which the crank ଵܱܤ rotates at angular velocity ߱. The crank ଵܱܤ through the rod 
AB, which is hinged at points A and B, acts on the rigid structure ܣܱଶ ସܱ, so that the axis of the 
holder ( ସܱ) oscillates. The rotation of the holder is determined by the effect of the friction forces 
between the polishing pad and the wafers pasted to the holder, and by the friction torque on the 
axis of the holder. 
 
Fig. 1. Schematic diagram of the grinding mechanism 
THE EFFECT OF DRY FRICTION FORCES ON THE PROCESS OF DIELECTRIC WAFER GRINDING.  
L. A. IGUMNOV, V. S. METRIKIN, A. V. GREZINA, A. G. PANASENKO 
502 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
1.1. Mathematical model  
In the development of the mathematical model it is assumed that the relative material removal 
rate is proportional to the relative velocity of the piece being ground and the polishing pad 
(Preston’s law [3]).  
Introducing the notation: 
ଵܱܤ = ܾ,     ܣܤ = ܿ,    ܣܱଶ = ݀, ܱଶܱଷ = ܴ଴, ܱଶ ସܱ = ܴଵ, ܱଶ ଵܱ = ݂, ߰ = ߱ݐ, 
and making a series of simple transformations, we obtain the formula for calculating the velocity 
௥ܸ of the point ܯ on the holder relative to the polishing pad in the form: 
௥ܸ = ܴ଴ඥ[ߩଶ(߱଴ − ሶ߮ )sinߛ + ߩଵ߱଴sinߤ]ଶ + [ߩ ሶ߮ + ߩଶ( ሶ߮ − ߱଴)cosߛ − ߩଵ߱଴cosߤ]ଶ, (1)
where: 
ሶ߮ = ߱[݂ିଵܾcos(߮ − ߰) − ݀ିଵܾcos߰][݂ିଵܾcos(߮ − ߰) − sin߮]ିଵ,
Φ(ݐ) = (G − Φଵ(ݐ))/2݀ඥ݂ଶ + ܾଶ − 2݂ܾsin߰ = sin(߮ − ߴ),
ܩ = ݂ଶ + ݀ଶ + ܾଶ − ܿଶ, Φଵ(ݐ) = 2݂ܾsin߰, cos߮଴ =
݂ଶ − ܿଶ + ݀ଶ
2݂݀ ,
ߩ = ܴܴ଴ ,    ߩଵ = ඥ1 + ߩ(ߩ − 2cos(߮ − ߮଴)), ߩଶ =
ݎ
ܴ଴,
tgߴ = ݂ − ܾsinܾ߰cos߰ ,     tgߤ = −sin߮଴(ߩ − cos(߮ − ߮଴))
ିଵ.
It follows from Preston’s law [3] that the relative rate of material removal at different points 
of the wafers on the holder ܵ(ݎ, ߛ) = ܵ(ߩଶ, ߛ) is determined from the following Eq. (2): 
ܵ = න ௥ܸ
ଶగ/ఠ
଴
(ݐ)݀ݐ. (2)
Fig. 2 shows for ݂ = 22.7; ܿ = 22.5; ݀ = 3.5; ܴ଴ = 40; ܾ = 0.4. in three-dimensional space 
[݃ܽ݉݉ܽ(ߛ), ݎ݋2(ߩ), ܸ/ܴ଴(ܵ)] the results of the calculations of the relative value of material 
removal on the wafers being ground depending on the coordinates of the wafer points: the angle 
ߛ, the radius ߩ, ߱ at ߱଴ = 0.5ܿିଵ for the mode of the holder fixed on its axis, using the MATLAB 
package. Let us now consider in more detail the process of grinding in the free mode of the holder. 
In this case, the holder’s rotation is determined by the action of the moment ܯଵ of the frictional 
forces on the holder from the pad and the friction torque on the axis of the holder. 
To determine ߊଵ, we choose a coordinate system ݔ′ݕ′ parallel to the system ݔݕ with the origin 
that coincides with the center of the holder. In this system, the inertial forces acting on small parts 
of the holder are equal to Δ۴ = −Δ݉܅଴  where Δ݉  is a part of the mass, and ܅଴  is the 
acceleration of the center of the holder. These forces do not generate torque, since the holder is 
disk-shaped and the wafers being ground are fixed, as a rule, symmetrically without changing the 
position of the center of gravity that coincides with the center of the disk. Thus we conclude that 
the moment ߊଵ is only determined by the action of the friction forces between the holder (the 
wafers on the holder) and the polishing pad.  
A detailed study of the friction forces associated with the process of polishing is a separate 
large task. Therefore, to simplify, we assume that the frictional force acting on the element of the 
holder with the area ݀ݏ is equal to: 
THE EFFECT OF DRY FRICTION FORCES ON THE PROCESS OF DIELECTRIC WAFER GRINDING.  
L. A. IGUMNOV, V. S. METRIKIN, A. V. GREZINA, A. G. PANASENKO 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 503 
݀۴௓ = −߯ܞ௥݀ݏ. (3)
The admissibility of this representation of the frictional forces is based on the similarity of the 
abrasive with lubricant to a high viscosity liquid. Constant ߯ is proportional to its viscosity. Then 
we can write Eq. (4): 
݀ۻଵ = [ܚ ⋅ ܌۴௓] = −߯[ܚ × ߥ௥]݀ݏ. (4)
 
Fig. 2. The dependence of the relative rate of material removal ܵ on the coordinate points of the wafer 
Going to the coordinate system ߦ, ߞ, ߟ fixed relative to the holder (the axis ߦ is directed along 
ସܱܹ, the axis ߞ – along ସܱܷ, the axes ߦ, ߞ, ߟ form a right-handed space) and calculating the 
vector value [ܚ × ߥ௥] as: 
[ܚν௥] = ݎ൫cos(ߛ)ݒ௥కൣ܍క܍఍൧ + sin(ߛ)ݒ௥఍ൣ܍క܍఍൧൯ = ݎ൫cos(ߛ)ݒ௥఍ − sin(ߛ)ݒ௥క൯܍ఎ, 
it is possible to compute the moment relative to the axis ߟ in the following way: 
ߊఎ = −߯ න
௥బ
଴
න ݎଶ
ଶగ
଴
[cosଶ(ߛ)(߱଴ܴ + (߱ଷ − ߱଴)ݎcos(ߛ) − ߱଴ݎcos(ߤ))
     −sin(ߛ) ቀ൫(߱ଷ − ߱଴)ݎsin(ߛ) − ߱଴ݎsin(ߤ)൯ቃ ݀ݎ݀ߛ = −
߯ߨݎ଴ସ(߱ଷ − ߱଴)
2 ,
(5)
where ߱ଷ is the angular speed of the holder in the coordinate system ݔ and ݕ. Here, ݎ଴ is the radius 
of the holder. Thus, in the chosen coordinate system we have: 
ܬ ݀߱ଷ݀ݐ = −
߯ߨݎ଴ସ(߱ଷ − ߱଴)
2 − ߊ௓, (6)
where ܬ is the moment of inertia of the holder with the wafers. Assume further that the force of 
friction on the axis is the Coulomb-Amontons friction and represent the moment in the following 
form: 
ߊ௓ = ܤsign(߱ଶ),    ߱ଶ ≠ 0, ߊ௓ ∈ [−ܤ, ܤ], ߱ଶ = 0.
Since ߱ଷ = ߱ଵ + ߱ଶ, where ߱ଶ is the velocity of the holder relative to its axis, and ߱ଵ is the 
angular velocity of the holder drive piece, which can be considered for the above scheme 
approximately equal to ߱ଵ = ܣsin(߱ݐ + ߮଴), ߱଴ is the polishing pad velocity, the Eq. (6) can be 
rewritten as Eq. (7): 
THE EFFECT OF DRY FRICTION FORCES ON THE PROCESS OF DIELECTRIC WAFER GRINDING.  
L. A. IGUMNOV, V. S. METRIKIN, A. V. GREZINA, A. G. PANASENKO 
504 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
݀߱ଶ
݀ݐ + ߣ߱ଶ = −ܣ൫߱cos(߱ݐ + ߮଴) + ߣsin(߱ݐ + ߮଴)൯ + ߣ߱଴ −
ߊ௓(߱ଶ)
ܬ , ߣ =
߯ߨݎ଴ସ
2ܬ .  (7)
Introducing the dimensionless time ߬ = ߱ݐ + ߮଴ − Φ , coordinate ݔ = (߱ଶ − ߱଴)/߱଴  and 
parameters ݇ = ߣ/߱ , ܨ = ܣ√ߣଶ + ߱ଶ/(߱߱଴) , ܽ = ܤ/(ܬ߱଴߱) , Eq. (7) can be rewritten as 
follows Eq. (8): 
ݔሶ + ݇ݔ = −ܨcos߬ − ܽsign(ݔ + 1), tgΦ = ߣ߱. (8)
The phase space of Eq. (8) is two-dimensional in the coordinates (߬, ݔ). It has a straight line Г 
( ݔ = −1 ), which divides the plane into three subspaces ܺା(ݔ > −1) , ܺି(ݔ < −1)  and Г  
(ݔ = −1). The motion of the image point in said subspaces is described by the Eq. (9): 
ݔሶ + ݇ݔ = −ܨcos߬ − ܽsign(ݔ + 1), ݔ ≠ −1, |݇ + ܨcos߬| ≤ ܽ, ݔ = −1. (9)
Note that the motion of the image point in the subspace Г (ݔ = −1) takes place in time 
intervals Δ௜ ∈ [߬௜, ߬௜ାଵ] corresponding to the junction of the phase trajectories coming from the 
subspaces ܺି, ܺା.  
Fig. 3 (a, b) shows the phase trajectories for different parameter values ܽ = 9 (Fig. 3(a)) and 
ܽ = 7 (Fig. 3(b)), respectively, at ݇ = 0.71, ܨ = 3.2. 
 
a) 
 
b) 
Fig. 3. Phase trajectories in the plane ߬, ݔ 
What all the parameters have in common is the presence of parameter intervals Δ௜ of the holder 
stops, which increase with a rise of the coefficient of static friction. The rotation with the stopping 
of the holder begins only from some values of ܽ (see Fig. 3(b)). With the increase of the parameter 
݇  the intervals are shifted to the region of the shorter times, and the oscillation amplitude  
decreases. In the ideal case, when we can neglect friction on the axis of the holder (ߊଶ = 0), it 
follows from Eq. (6) that with the obvious relation (߯ݎ଴ସߨ/2)/(ܬ߱଴) > 0  we will obtain 
lim௧→ஶ߱ଷ(ݐ) = ߱଴ . This means that the angular velocity of the holder ߱ଶ  relative to its axis is  
߱ଶ = ߱଴ − ߱ଵ(ݐ). Note that for the chosen values of the parameters of the mechanism the relation 
߱ଵ = ሶ߮ ≈ 0.25sin(߱ݐ + ߮଴) holds true and hence ߱ଶ > 0. Assuming ߊଶ ≠ 0 and that it can be 
approximated by the force of friction obeying the Coulomb-Amontons law, it follows from Eq. (6) 
that lim௧→ஶ߱ଷ(ݐ) = ߱଴ − Δ߱,  where Δ߱  is a small constant, which is equal to  
Δ߱ = 2|ܯଶ|୫ୟ୶/(߯ߨݎ଴ସ). In this case, for the steady state mode, instead of the relationship Eq. (1) 
we will have the following Eq. (10):  
THE EFFECT OF DRY FRICTION FORCES ON THE PROCESS OF DIELECTRIC WAFER GRINDING.  
L. A. IGUMNOV, V. S. METRIKIN, A. V. GREZINA, A. G. PANASENKO 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 505 
௥ܸ = ܴ଴ඥ[ߩଶΔ߱sinߛ + ߩଵ߱଴sinߤ]ଶ + [ߩ ሶ߮ + ߩଶΔ߱cosߛ − ߩଵ߱଴cosߤ]ଶ. (10)
2. Conclusions 
Note that according to Eq. (10):  
– In the ideal case (ߊଶ = 0) there is no dependence of the relative velocity ௥ܸ from the position 
of the point on the holder ߩଶ, that is, the removal of the material on the grinding disc will be 
uniform across the surface;  
– The friction on the axis of the holder leads to a weak dependence ௥ܸ from ߩଶ and thus to the 
non-uniform removal of the material; 
– The dynamics of the mechanism is determined by the ratio of the friction torque on the axis 
and in the contact of the wafers with the polishing pad;  
– There are three modes of the mechanism motion: free rotation, rotation with periodic stops 
of the holder; the holder does not rotate on its axis.  
Acknowledgements 
Development of mathematical model is financially supported by Russian Science Foundation, 
project No. 16-19-10237. Investigation of numerical results was financed within the framework 
of the base part of State Task of the Ministry of Education and Science Project,  
No. 2014/134 2226. 
References 
[1] Khomich N. S., Lugovik А. Yu., Fedortsev Р. V., Korzun A. E., Kukhto P. V. Simulation of the 
kinematics of the process of magnetic abrasive polishing of wafers. BSTU Herald, Vol. 1, 2009, 
p. 33-38, (in Russian). 
[2] Gavrishuk Е. М., Komarovv N., Metrikin V. S., Panasenko А. G. Mathematical of wafer grinding 
on the machines of type 4PD-200 and 3PD-320. Bulletin of the Samara SC RАS, Vol. 13, Issue 1, 
2011, p. 992-995, (in Russian). 
[3] Nanz G., Camille L. E. Modeling of chemical-mechanical polishing: a review. IEEE Transactions on 
Semiconductor Manufacturing, Vol. 8, Issue 4, 1995, p. 382-389. 


The effect of dry friction forces on the process of dielectric wafer grinding

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The effect of dry friction forces on the process of dielectric wafer grinding

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