JOURNAL OF THEORETICAL
AND APPLIED MECHANICS
45, 1, pp. 61-72, Warsaw 2007
REGULAR AND CHAOTIC VIBRATIONS OF
A VIBRATION-ISOLATED HAND GRINDER
Institute of Applied Mechanics, Cracow University of Technology
The paper is concerned with qualitative analysis of a non-linear model
describing vibration of a vibration-isolated hand grinder. A discontinu-
ous description of grinding forces is introduced, which accounts for the
possible separation of the grinding wheel from the object during the pro-
cess. Eight non-linear ordinary di?erential equations are obtained which
describe dynamics of the system. Numerical analysis is done using me-
thods of numerical integration and the Fast Fourier Transform. The in-
?uence of selected parameters on the character of vibration is studied
and some measures are calculated which characterize the quality of the
vibration isolation system.
Key words: vibrations, chaos, grinding, vibration isolation, non-linear
Harmful vibrations during mechanical processing (such as grinding or milling)
have to be avoided since they deteriorate the quality of the product as well
as have a negative e?ect on the human-operator. The main sources of the-
se vibrations are kinematic and inertial excitations (Alfares and Elsharkawy,
2002; Gradi?sek et al., 2001; Karube et al., 2002; ?uczko and Markiewicz, 1986;
Suh et al., 2002). In order to reduce vibration levels transmitted to the ope-
rator, vibration isolation systems are mounted between the tool body and the
handle. In the case of passive vibration isolation systems their parameters are
usually selected using linear models.
However, for large vibration amplitudes it is necessary to account for non-
linear phenomena, such as brought about e.g. by the loss of contact of the
grinding wheel with the object being worked (?uczko et al., 2003). Moreover,
J. ?uczko et al.
it is desirable to determine the in?uence of parameters that undergo changes
during the grinding process, such as rotational speed or pressure on the tool
handle, on dynamic characteristics of the system.
Model of the system
Figure 1 shows a schematic view of a hand grinder equipped with a vibration
isolation system. The following elements have been taken into account in the
model: tool body (1) (with the subassembly motor-spindle-grinding wheel),
handle (2) and object being processed (base) (3). The resilient connections
between the tool body and the handle represent the passive vibration isolation
system, whereas the ?exible elements which link the body to its surroundings
are a simpli?ed model of the operator interaction. The ?exible connections
consist of extension-compression springs of sti?nesses cj (j = 1, 2) and torsion
springs of sti?nesses kj, which can be considered as a result of the reduction
of any resilient connections to the point at a distance lnj from the respective
centre of mass Sn (n = 1, 2). In a similar way, one can de?ne parameters which
describe energy dissipation. Assuming that the damping matrix is proportional
to the sti?ness matrix, the damping of a given connector is given by a single
coe?cient ?j. In order to limit the displacements of the handle relative to the
body, a motion limiter (4) is introduced into the model. The properties and
geometry of the motion limiter are determined by the parameters c4, ?4, ?,
Fig. 1. Model of the system: 1 – body, 2 – handle, 3 – workpiece, 4 – limiter
It has been assumed that the tool body and the handle can undergo a ge-
neral motion in space, with the exclusion of longitudinal and torsional degrees
of freedom. Therefore, longitudinal and torsional vibrations are excluded from
Regular and chaotic vibrations...
the analysis. The kinematic excitation is accounted for by de?ning parame-
ters m0 (the unbalanced mass) and e0 (eccentricity), and assuming that the
rotational speed remains constant. Introducing moving co-ordinate systems
with origins at the centres of mass Sn of subassemblies: the body with the
rotating elements (n = 1) and the handle (n = 2), motion of the system can
be described by specifying the co-ordinates xn and yn of points Sn and the
angles ?n and ?n, which for small vibrations describe rotations of the local
axes with respect to the ?xed ones.
Fig. 2. Kinematic relations: (a) grinding forces, (b) reaction of the limiter
The case of processing of a ?at surface is being considered. It has been
assumed that when the grinding wheel remains in contact with the base (for
xA = x1 + l?1 > 0), a linear relationship T = f N holds between the tan-
gential and normal components of the grinding force (Fig. 2a). Additionally,
it has been assumed that the normal reaction force N is described by the
Voigt-Kelvin model, but it cannot take on negative values. By introducing the
notation s = d/dt, this reaction force is given by the following formula
N (xA, sxA) = c3(1 + ?3s)xAH(xA)H[(1 + ?3s)xA]
where H is the Heaviside step function. In a similar way, the normal reaction
force of the limiter (Fig. 2b) is written as
R(rB, srB) = c4(1 + ?4s)rBH(rB ? ?)H[(1 + ?4s)rB]
Here, it has been assumed that the impact takes place when rB =
x2 + y2 >
> ?, where
xB = (x1 + l14?1) ? (x2 + l24?2)
yB = (y1 ? l14?1) ? (y2 ? l24?2) (2.3)
The Cartesian components of the limiter reactions are calculated as
x = R
y = R
J. ?uczko et al.
Using the laws of change of momentum and angular momentum about the
centres of mass Sn, motion of the system can be described by the following
set of eight second-order ordinary di?erential equations
m1s2x1 + c1(1 + ?1s)(x1 ? x2 ? l11?1 + l21?2) + Rx + N = m0e0?2 cos ?t
m1s2y1 + c1(1 + ?1s)(y1 ? y2 + l11?1 ? l21?2) + Ry + T = m0e0?2 sin ?t
I1s2?1 ? I0?s?1 + (1 + ?1s)[k11?1 ? k21?2 ? c1l11(x1 ? x2)] +
+Rxl14 + N l = m0e0l?2 cos ?t
I1s2?1 + I0?s?1 + (1 + ?1s)[k11?1 ? k21?2 + c1l11(y1 ? y2)] +
?Ry l14 ? T l = ?m0e0l?2 sin ?t
m2s2x2 ? c1(1 + ?1s)(x1 ? x2 ? l11?1 + l21?2) + c2(1 + ?2s)(x2 ? l22?2) +
?Rx = Qx
m2s2y2 ? c1(1 + ?1s)(y1 ? y2 + l11?1 ? l21?2) + c2(1 + ?2s)(y2 + l22?2) +
?Ry = Qy
I2s2?2 ? (1 + ?1s)[k21?1 ? (k1 + c1l221)?2 ? c1l21(x1 ? x2)] +
+(1 + ?2s)(k22?2 ? c2l22x2) ? Rxl24 = M?
I2s2?2 ? (1 + ?1s)[k21?1 ? (k1 + c1l221)?2 + c1l21(y1 ? y2)] +
+(1 + ?2s)(k22?2 + c2l22y2) + Ryl24 = ?M?
k11 = k1 + c1l2
21 = k1 + c1l11l21
k22 = k2 + c2l222
In equations (2.5), m1 and m2 are respectively the masses of subassemblies 1
and 2, I1 and I2 are moments of inertia of the subassemblies with respect to
the x-axis (or the y-axis, thanks to axial symmetry) passing through points
S1 and S2, and I0 is the moment of inertia of the rotating elements about the
axis of symmetry. The generalised forces Qx, Qy, M? and M? represent the
operator action on the system. When the grinding wheel remains in contact
with the object, one can assume that these forces remain constant. The case
when the grinding wheel loses contact with the object is more complex, espe-
cially as far as the generalized forces Qy and M? are concerned. The human
operator is an active system since he reacts to changes in the working condi-
tions. During grinding, the operator tries to adjust the magnitudes of forces so
as to equilibrate the corresponding components of the grinding forces. When
the loss of contact occurs, the operator tries on one hand to bring the tool back
into contact with the object (without changing the values of Qx, M?), and,
on the other hand, counteracts sudden movements of the tool in the direction
tangent to the surface being processed by a sudden change (in the model an
Regular and chaotic vibrations...
instant change) of the force Qy and the moment M?. To account for this
behaviour, the following simpli?ed relations are used in the model
Qy = f QxH(N )
M? = Qxd
M? = Qyd
where d is the distance of the resultant operator’s action from the centre of
The analysis is done in a dimensionless form, and non-dimensional quan-
tities are used where the amplitudes are calculated relative to the e?ective
amplitude of inertial excitation e = m0e0/(m1 + m2), angles are taken rela-
tive to e/l (l = l13) and the non-dimensional time ? = ?0t is referred to the
circular frequency ?0 =
c2/(m1 + m2) of the simpli?ed linear model.
The equations, when written in the dimensionless form, depend on the
m1 + m2
0 = l
and on the coe?cient of dry friction f . In equations (2.8), the index n is
the number of the subsystem (n = 1, 2), and the index j corresponds to
the number of the ?exible element (j = 1, 2, 3, 4). Moreover, the following
conditions hold: ?2 = 1, ?21 = ?0 ? ?11, ?24 = ?0 + ?14 and µ1 + µ2 = 1. By
introducing the state vector
x1 y1 l?1 l?1 x2 y2 l?2 l?2 ?
the system of equations (2.5) can be written in a compact matrix form
Mu” + (G + 2? C
1 + 2?2
2)u? + (C1 + C2)u = p(? ) + q + r + s
Here, the matrices M, G, C1 and C2 are respectively the mass-, gyroscopic-
and sti?ness matrices, and p(? ) is the vector of inertial excitation
p(t) = ?2[cos ??, sin ??, cos ??, ? sin ??, 0, 0, 0, 0]?
The vectors q, r and s describe non-linear terms, respectively related to the
model of the operator
q = [0, 0, 0, 0, q, f qH(n), ?q, ??f qH(n)]?
J. ?uczko et al.
the model of the limiter
r = [?rx, ?ry, ??14rx, ?14ry, rx, ry, ?24rx, ??24ry]?
and the model of the grinding forces
s = [?n, ?f n, ?n, f n, 0, 0, 0, 0]?
In order to calculate n, rx and ry one makes use of formulae (2.1)-(2.4),
where dimensionless components of the state vector (2.9) and the respective
non-dimensional parameters (e.g. parameters ?3, ?4, 2?3, 2?4 and ? in lieu of
c3, c4, ?3, ?4 and ?) are introduced. The matrix M is diagonal and has the
M = diag [µ1, µ1, µ1?1, µ1?1, µ2, µ2, µ2?2, µ2?2]
The only non-zero terms of the gyroscopic matrix G are given by G43 =
= ?G34 = ?µ1?1?.
In order to de?ne the sti?ness matrix, we introduce an auxiliary matrix
j (?nj , ?mj ) = ?
???j?nj 0 ?j +?j?nj?mj
?j + ?j?nj?mj?
The matrices C1 and C2, which represent respective ?exible links can be
written as the following block matrices
?? 1(?11, ?21)
?? 1(?21, ?11)
? 1(?21, ?21)
? 2(?22, ?22)
Results of the qualitative analysis of the vibration-isolated hand grinder will
be described below, with emphasis put on the selection of some of the para-
meters of the vibration isolation system and on the explanation of physical
phenomena brought about by the percussive nature of the grinding forces.
The results have been obtained using methods of numerical integration and
the Fast Fourier Transform, which have been used in studying non-linear oscil-
lations, e.g. by Awrejcewicz and Lamarque (2003). More details about the use
of spectrum analysis to determine the character of vibrations have been di-
scussed in Ferdek and ?uczko (2003). In discussion of the results, the criterion
Regular and chaotic vibrations...
index J1 (or J2) of the e?ciency of the vibration isolation system will be
used. It is de?ned as the ratio of the rms values of accelerations (respectively
velocities) at the point B on the handle (front grip – see Fig. 1), calculated
for the tool with- and without the vibration isolation system. By analysing
the in?uence of the parameters ?1, ?1 and ?11 on the value of the criterion
indices Jk, estimates of the optimum parameter values from the point of view
of minimising vibration levels have been found. The character of vibration has
also been studied, depending on the values of these parameters and the values
of parameters (?, q) which characterize the grinding process. The following set
of values of parameters have been used in the numerical calculations: µ1 = 0.8,
µ2 = 0.2, ?1 = 1.5, ?2 = 0.5, ? = 0.1, f = 0.5, ?1 = 1.5, ?2 = 1, ?3 = 400,
?4 = 100, ?1 = 1.5, ?2 = 1, ?1 = 0.1, ?2 = 0.5, ?3 = 0.05, ?4 = 0.05,
?0 = 0.5, ?11 = 0.25, ?21 = 0.25, ?22 = 0, ?13 = 0.5, ?23 = 1.5, ?14 = 0.5,
?24 = 1, ? = 10, ? = 0.75, q = 10, ? = 5.
Figure 3a shows the dependence of the criterion index J1 on the parameters
?1, ?1 (for ? = 5 and q = 10), and Fig. 3b illustrates the zones of di?erent
vibration types in the (?1, ?1) plane. The minimum of J1 (and also of J2) is
achieved in the neighbourhood of the point ?1 = 1.5, ?1 = 1.5. As seen in
Fig. 3b, in the neighbourhood of this point, sub-harmonic vibrations of type
1:2 take place.
Fig. 3. In?uence of parameters c1 and ?1 (? = 5, q = 10, ?11 = 0.25) on:
(a) criterion index J1, (b) zones of di?erent vibration types
In a similar way, Figs. 4a and 4b illustrate the in?uence of the parameters
? and ?11. In a relatively wide neighbourhood of ?11 = 0.25, the index J2
(Fig. 4a) assumes a value close to the minimum, regardless of the rotational
speed ?. This holds true, even though for high values of ? the vibrations are
chaotic, as is seen in Fig. 4b.
The type of vibrations and the value of the criterion index depend also on
the remaining parameters, characterizing both the model of the tool and the
J. ?uczko et al.
Fig. 4. In?uence of ? and ?11 (c1 = 1.5, ?11 = 1.5, q = 10) on: (a) criterion
index J2, (b) zones of vibration types
grinding process. A proper choice of the vibration isolation system requires
evaluation of the sensitivity of the solution to changes in these parameters.
Below, the discussion is limited to the in?uence of the rotational speed and
the operator pressure on the tool handle, since these two parameters have
been found to have the biggest e?ect on the dynamic behaviour of the sys-
tem operator-tool-base. The operator controls the grinding process mainly by
changing the force q. The rotational speed ? also undergoes changes as a
result of the limited power of the motor.
Fig. 5. In?uence of ? and q (c1 = 1.5, ?11 = 1.5, ?11 = 0.25) on: (a) e?ciency
index J2, (b) vibration zones
Figures 5a and 5b illustrate, in the same format as in the previous ?gures,
the in?uence of the parameters ? and q on the criterion index J2 and on the
character of vibrations. By studying the zones in the (?, q) plane in Fig. 5b,
one can note that only for small values of the non-dimensional rotational spe-
ed ? the grinding process takes place without separation of the grinding wheel
Regular and chaotic vibrations...
from the motion limiter. This zone becomes a little wider as the dimensionless
pressure q increases. For higher values of ?, which is the case of most prac-
tical applications, there appear zones of sub-harmonic vibrations of an order
increasing with ? separated by narrow zones of sub-harmonic (mostly of ty-
pe 1:4) or chaotic oscillations. The criterion index (Fig. 5a) decreases with the
rotational speed, which signi?es that the selected vibration isolation system is
e?cient also in the case of irregular vibrations.
Fig. 6. Bifurcation diagram (c1 = 1.5, ?11 = 1.5, ?11 = 0.25, q = 5)
Figure 6 shows a bifurcation diagram obtained using the stroboscopic me-
thod by taking the displacement u1 at selected time instants (every excitation
period). This diagram corresponds to the section of the (?, q) plane shown
in Fig. 5b taken for q = 5. In the bifurcation diagram, one can distinguish
alternate regions of sub-harmonic and chaotic oscillations, where the order of
the sub-harmonic vibrations increases with the rotational speed.
Fig. 7. Time history, spectrum, phase portrait and trajectory of the point B on the
handle (c1 = 1.5, ?11 = 1.5, ?11 = 0.25, q = 5): (a) 4T -periodic vibrations, ? = 4.6,
(b) chaotic vibrations, ? = 5.0
Figure 7 shows time histories of the displacement xB(t), phase portraits
(xB, x? ) and frequency spectra S(?) of the signal x
B for two values of the
excitation frequency ?. For the ?rst value of ?, the period of oscillations is
four times of that of the excitation (sub-harmonic oscillations of type 1:4),
J. ?uczko et al.
the spectrum has pronounced maxima at points ?/? = k/4, where k is a
natural number, and the corresponding curve in the phase plane is closed.
For the second value of ?, the oscillations are chaotic, which is manifested
in the irregular time history by the continuous spectrum and irregular phase
For chaotic vibrations, the phase portraits are di?cult to interpret. Much
more information is gained by making stroboscopic portraits or Poincar´e maps.
For the subsequent regions of chaotic vibrations shown in the bifurcation dia-
gram in Fig. 6, one obtains di?erent shapes of fractals shown in Fig. 8.
Fig. 8. Stroboscopic portraits of chaotic oscillations (c1 = 1.5, ?11 = 1.5, ?11 = 0.25,
q = 5): (a) ? = 5.0, (b) ? = 6.5, (c) ? = 7.5, (d) ? = 9.0
In the paper an approach to the modelling and to the qualitative analysis
of vibrations of a non-linear system in the presence of unilateral constraints
resulting both from the grinding process and the presence of the motion li-
miter is discussed. The adopted model of the grinding process can predict
sub-harmonic and chaotic vibrations brought about by the percussive charac-
ter of grinding forces. The numerical analysis has shown that the parameters
? and q have a pronounced qualitative e?ect on motion of the system. A little
less important, but not negligible, is the e?ect of the parameters f , ?3, ?3.
The regions of di?erent vibration types shown in Fig. 3-Fig. 5 as well as the
bifurcation diagram (Fig. 6) do not undergo qualitative changes for di?erent
values of the parameters ?3, ?3, f that describe the model of the grinding
forces. By increasing the values of the parameters f and ?3 or by reducing
the value of ?3, the zones of chaotic vibrations tend to get bigger.
The inclusion of the motion limiter in the model prevents excessive static
displacements and allows an e?cient choice of the parameters of the passive
vibration isolation system without a need of imposing additional constraints.
With a badly designed vibration isolation system, the parameters of the motion
limiter can have an important e?ect on the type of vibration in the system.