Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 1, pp. 157-169, Warsaw 2008

STEADY PERIODIC REGIME OF ROTARY MOTION

OF ROLLERS IN VIBRATING CLASSIFIERS

Victor A. Ostapenko

Dnepropetrovsk National University, Ukraine

e-mail: duo-mmf@ff.dsu.dp.ua

Vladimir P. Naduty

Vladimir F. Yagnyukov

Institute of Geotechnical Mechanics, NAS, Ukraine

e-mail: astasdn@rambler.ru

The problemof dynamics of vibrating classifierswith rollers intended for
sorting of loosematerials is considered in the paper.Themain goal of re-
search consists in substantiation of a choice of such parameters of rollers
and classifiers which provide a steady periodic mode of rollers rotation.
Differential equationsof rollers rotationunder actionof inertial forces are
essentially nonlinear. These equations are transformed to equations con-
cerning the value of delay of moving rollers concerning rotation of axles
on which these rollers are freely suspended. As values of such delays is
small, the equations of motion can be linearized with a sufficient degree
of accuracy. The linearized equations represent the inhomogeneous Hill
equations, which can be under certain conditions transformed into the
Mathieu equations. Periodic solutions to these equations are obtained,
and also stability of these solutions is investigated.

Key words: classifier, roller, rotation

1. Introduction

Classifiers with rollers are widely applied in the mountain, metallurgical and
building industries, providing high efficiency of classification and reliability
([1]; Caughey, 1960; [4]; [8]). In total, in the world, in manufacture and ope-
ration of roll screens and classifiers are engaged more than fifty firms in the



158 V.A. Ostapenko et al.

United States, the Great Britain, France, Germany, Finland, Italy, Spain, Ja-
pan, Sweden and Russia. Germany is one of the largest manufacturers and
consumers of roll screens ([1]; Caughey, 1960). FirmZemag have beenmaking
roll screens since 1992 [8]. Numerous sources specify high operational qualities
of such machines ([1]; Caughey, 1960; 4]; [8]).

At the same time, the necessity of transfering of rotary movement to each
of numerous rollers essentially complicates the design, making it bulky and
metal-consuming. For instance, every roller of screens made by Zemag has
its own individual drive gear with an electric motor. Only in small classifiers
every two rollers have a separate drive gear [8].Therefore classifierswith rollers
demand further constructive improvement.Oneof perspective directions is the
application of a vibrating drive and use of vibrating movements of working
body for rotation of rollers. In this case, in general, there is no necessity of
creation of drives for rollers. In a suggested design illustrated in Fig.1, the
rigid frame of the classifier is oscillatory driven with the help of vibrators.
Along the frame, in equal distances the axles rigidly connected to it are placed
and located in the horizontal plane.On each of axles, the roller it is freely put,
representing a ring in its cross section.

Fig. 1. Scheme of a classifier

Under the action of vibrators, the frame, and together with it the axles of
rollers, move on an elliptic or, in particular, a circular trajectory in a vertical
plane. Besides, under the action of inertial forces, also the rollers come into
movement. From the point of view of qualitative work of the classifier, it is
important to obtain conditions of periodic, synchronous and in phase rotation
of rollers.Besides, it is important to investigate conditions of stability of rollers
motion and to provide their steady rotation independent of initial conditions



Steady periodic regime of rotary motion... 159

as they have random enough character. The problem of formulation of steady
periodic solutions to the equationsmotion of rollers is considered in the paper.

2. Linearization of equation of motion

Athorough research of literature shows that there is only one attempt creating
a mathematical model for a more or less similar device. We mean the hula-
hoop, rotation of hoop around of man’s body (Caughey, 1960). In this work,
an essentially simplifiedmodel of motion of a hoop is considered. The system:
human torso-hoop is considered as a certain coulissemechanismmaking given
movements. Theanalysis of forces causing thismovement is absent.The choice
of a certain direction of influence of the human torso on the hoop, in our
opinion, obviously does not correspond to real interactions in the system.
This circumstance induces the authors to create a mathematical model

of vibrating classifiers with rollers (Ostapenko et al., 2005). The equation of
relative rotation of the rollers in such a model looks like (Ostapenko et al.,
2005)

d2α

dt2
=−R2m

Jkz

[

−(R2−R1+Rcosα)
dω

dt
+Rω2 sinα−g sin(φ−α)

]

(2.1)

where α is the angle of delay of the roller in relation to the angle φ, φ = ωt,
t – time, ω – angular velocity of rotation, R – radius of rotation of the center
of the axis, m – mass of roller, Jkz – moment of inertia of the roller with
respect to the point of its contact with the axle, and

Jkz =
m

2
(R22+R

2
3)+mR

2
2 (2.2)

Here R2 and R3 are the internal and external radii of the roller, respectively,
and R1 is the radius of the axle.
Equation (2.1) is essentially nonlinear and cannot be integrated in quadra-

tures. However, at small values of the angle α compared with its first and
second derivatives, equation (2.1), with a sufficient degree of accuracy, can be
linearized. We have for small α (sinα ≈ α, cosα ≈ 1)

sin(φ−α)= sinφcosα− cosφsinα ≈ sinφ−αcosφ (2.3)

and in the stationary mode, that is when the equality

(R2−R1+Rcosα)
dω

dt
=0 (2.4)



160 V.A. Ostapenko et al.

holds, we obtain that linearized equation (2.1) becomes

d2α

dt2
+(a1+ bcosωt)α = bsinωt (2.5)

where

a1 =
R2Rmω

2

Jkz
b =

R2mg

Jkz
(2.6)

Now it is necessary to investigate equation (2.5) from the point of view of
the existence of its steady periodic solutions. That permits one to choose pa-
rameters of the researched systemwhich provide realization of such operating
modes. It is most preferable to obtain periodic solutions with the period equal
to ω tomake one turnover of the roller correspond to one turnover of its axle.
These requires constant clearances between the rollers and, therefore, higher
quality of classification of a processable material.

Equation (2.5) can be considered as a special case of the inhomogeneous
Hill equation, or, in particular, as an inhomogeneous Mathieu equation.

3. Hill’s equation

A homogeneous equation

y′′(x)+ [Φ(x)+λ]y(x)= 0 (3.1)

with a periodic function Φ(x) and a constant λ is the Hill equation. The ba-
sic interest in the investigation of equations (2.5) and (3.1) connected to it
represents, as it was mentioned above, a question on the existence and stabi-
lity ω-periodic solutions to these equations as only at such a periodicity the
constant backlash between the rollers can be provided, and hence, qualitative
work of the classifier. To this purpose, we shall consider theHill equation in a
more general view (Smirnoff, 1969)

y′′(x)+p(x)y′(x)+q(x)y =0 (3.2)

with ω-periodic factors p(x) and q(x).

If y1(z), y2(z) is the fundamental system of solutions to equation (3.2), its
general solution can be written as

y(x)= C1y1(x)+C2y2(x) (3.3)



Steady periodic regime of rotary motion... 161

According to the Floquet theorem (McLachlan, 1947), linear homoge-
neous equation (3.2) with periodic factors has the solution in the form
y(x) = exp(µx)U(x), where U(x) is a ω-periodic function. To obtain this
solution, a condition is entered

y(x+ω)= σ[C1y1(x)+C2y2(x)] = σy(x) (3.4)

with some constant σ. Then assume σ =exp(−ωµ) and enter function

U(x)= e−ωµy(x) (3.5)

Then under condition (3.4), the function U(x) becomes ω-periodic. Num-
bers σ in equality (3.4) are determined as roots of the equation

σ2−2Aσ+1=0 (3.6)

The number A is called the characteristic Lyapunov constant. For real
q(x), A is real as well. If the roots of equation (3.6) σ1 and σ2 are simple
there are two linearly independent solutions to equation (3.2) of kind (3.4)

η1(x+ω)= σ1η1(x) η2(x+ω)= σ2η2(x) (3.7)

Then, according to the Floquet theorem, in the case of various roots of
equation (3.6), the functions η1(x) and η2(x) can be represented as

η1(x)= e
xµ1U1(x) η2(x)= e

xµ2U2(x) (3.8)

where σi = exp(ωµi), that is µi = (lnσi)/ω, Ui(x) are ω-periodic functions,
i = 1,2. If equation (3.6) has a multiple root σ, then only one function of
η1(x) of kind (3.8) exists. Any other solution to the Hill equation, linearly
independent of η1(x), cannot look like (3.8). From equation (3.6), it follows
that this equation has a multiple root only for |A| = 1. In this case, two
linearly independent solutions to equation (3.2) can be represented as

η1(x)= e
µxU(x) η2(x)= e

µx
(a21
σω

xU(x)+U3(x)
)

(3.9)

where the functions U(x) and U3(x) are ω-periodic, and the constant a21 6=0.
So, the general solution to homogeneous Hill equation (3.2)

y(x)= C1η1(x)+C2η2(x) (3.10)

will be not periodic. Moreover, if the multiple root of equation (3.6) σ = 1,
we obtain µ =0, and therefore solutions (3.9) become

η1(x)=U(x) η2(x)=
a21
ω
xU(x)+U3(x) (3.11)



162 V.A. Ostapenko et al.

From (3.11) it follows that at x → ∞ the function η2(x) unbounded
modulo grows. Therefore, in the case |A|=1, the general solution to equation
(3.2) is not periodic and stable. That is why this case does not represent
practical interest.

In general, the characteristic Lyapunov constant A essentially influence
the solutions to the Hill equation. If |A| > 1, the functions η1(x) and η2(x)
can be represented as (3.8). As the functions U1(x) and U2(x) are perio-
dic and continuous, they will be bounded at any x. In this case, roots of
equation (3.6) are different and real. Accordingly to Viet’s theorem, we have
σ1σ2 =1, with one of the roots greater and another less than 1. So, the values
exp(xµi) = exp(xω

−1 lnσi) (i = 1,2) at x → ∞ behave differently. Let, for
definiteness, |σ1| > |σ2|. Then |σ1| > 1, and |σ2| < 1. The real part of lnσ is
equal to ln |σ|. Therefore, we have ln |σ1| > 0 and ln |σ2| < 0. That means

lim
x→∞
e
x
ω
lnσ1 =∞ lim

x→0
e
x
ω
lnσ2 =0 (3.12)

Consequently, the first summand in the right part of equality (3.10) unbo-
undedmodulo grows at x →∞, whichmakes the general solution to the Hill
equation (if C1 6=0) unstable.
Hence, at |A| ­ 1 there are no two linearly independent solutions to the

Hill equation, simultaneously periodic and steady. It means that such a case,
from the point of view of the considered problem, does not represent practi-
cal interest. Therefore, it is necessary to only consider the case |A| < 1. At
|A| < 1, the roots σ1 and σ2 are determined by the equality

σ1,2 = A± i
√

1−A2 (3.13)

where σ1 and σ2 are complex conjugate and |σ1| = |σ2| = 1. For complex
σ = γ+iν, the determination of µ with the help of the equality σ =exp(ωµ)
leads to the correlation µ =(lnσ)/ω with the function ln of a complex argu-
ment. Such a function is determined by the equality

lnσ = ln(γ +iν)= ln |σ|+iargσ (3.14)

where

argσ =































arctan
ν

γ
for γ > 0

arctan
ν

γ
+π for ν ­ 0, γ < 0

arctan
ν

γ
−π for ν < 0, γ < 0

(3.15)



Steady periodic regime of rotary motion... 163

Owing to the equality of modules of σ1 and σ2 to 1, in equality (3.14) we
obtain ln |σ1|= ln |σ2|=0 for these values and therefore

µ1 = i
argσ1
ω

µ2 = i
argσ2
ω

(3.16)

Having denoted γ = A, ν =
√
1−A2, we obtain that σ1 = γ + iν;

σ2 = γ−iν. As the values σ1 and σ2 are complex conjugate, the next equality
is valid

argσ2 =−argσ1 (3.17)

Therefore µ2 = −µ1 and, hence, exp(ωµ2) = exp(−ωµ1). Thus, as in the
case of |A| 6= 1, two linearly independent solutions to the Hill equation are
represented as in (3.8), and in the considered case these solutions will become

η1(x)= e
i
β

ω
xU1(x) η2(x)= e

−i
β

ω
xU2(x) (3.18)

where β = argσ1. In equalities (3.16), the functions U1(x) and U2(x) are
ω-periodic. Therefore, in the case when 2π/β is a rational number, that is
can be expressed as 2π/β = m/n, where m and n are integers, the functions
η1(x) and η2(x) in formulas (3.12) will have the period T = mω. That is
why the general solution to Hill equation (3.10) will be also an mω-periodic
function.

If the number 2π/β is irrational, the functions exp(±βx/ω) on the one
handand the function U1(x) and U2(x) on the other handwill have not amul-
tiple period, therefore the function η1(x) and η2(x) in (3.12) and also general
solution (3.10) will be oscillation, but not periodic. Owing to the periodicity
of all functionsmaking general solution (3.10), this solution will be limited at
x →∞, and therefore steady.
Thus, it appears that theHill equation can have a steady periodic solution

only for |A| < 1 and the rational ratio 2π/β. This case represents practical
interest and should be duly addressed.

It is important here to note that the solution to the Hill equation can be
represented only in the form of Fourier series. It is well known that Fourier
series converges very slowly, and to approximate the solutionwith an essential
accuracy, it is necessary to keep a large number of terms of such a series.
Therefore, taking into account the fact that we are able to select parameters
of the system in a broad region andwe need only to provide a steady periodic
mode of its operation, to obtain a compact explicit expression for solution to
equation (2.5) we will consider this one as aMathieu equation.



164 V.A. Ostapenko et al.

4. Mathieu’s equation

In this particular case, Hill’s equation (2.5) bymeans of transformation

z =
ωt

2
(4.1)

can be represented as the Mathieu equation in one of the standard forms
(McLachlan, 1947)

d2α

dz2
+(a+16qcos2z)α =16q sin2z (4.2)

where

a =
4a1
ω2
=4

R2Rm

Jkz
q =

b

4ω2
=

R2mg

4ω2Jkz
(4.3)

It is necessary to note that if Jkz is proportional to m, then the values
of a and q do not depend on the weight of rollers, and are determined only
by geometry of the system. The parameter q, in addition, depends on the
acceleration of gravity g. Owing to presence of the factor ω2 in the denomi-
nator of the formula for q, the parameter q is small in comparison with the
parameter a. It ismuch less than unit and strongly decreases with growing ω.

Let us consider the homogeneous equation of Mathieu corresponding to
(4.2)

d2α

dz2
+(a+16qcos2z)α =0 (4.4)

As it has already been marked, from the point of view of stable work of the
classifier, it is necessary to obtain a periodic solution to equation (2.5) with
the period equal to 2π/ω. Itmeans that concerning the variable z, the period
of this solution should be equal to π. If we find periodic linearly independent
solutions to equation (4.4) in form of (3.18), it is necessary to put b/ω = mπ.
Then theperiodof the function exp(iβx/ω) = exp(imπx)will be equal to mπ.

Therefore, in view of such a transformation of the period, functions (3.18)
for equation (4.4) become

η1(z)= e
imπzU1(z) η2(z)= e

−imπzU2(z) (4.5)

In sucha representation, odd sei(z,q) and even cei(z,q)Mathieu functions
of the first kind are constructed.Thegreatest interest is put to the cases, when
the index i is odd as the solution toMathieu’s equation has the period 2π/ω.
For cases of the even index i the period of the solution is equal to 4π/ω,



Steady periodic regime of rotary motion... 165

which means that the solution will be repeated not through one but through
two turnovers of the roller.

It is necessary to note that each of the mentioned Mathieu functions will
be periodic solutions to equation (4.4) not for any value of the parameters a
and q, but only in the case when the parameter a is a certain function of the
parameter q. Such functions a(q), are called the own values ofMathieu func-
tions and are denoted by ai for functions cei(z,q) and by bi for the function
sei(z,q). Functions a(q) are analytical, and the initial terms of their expan-
sion in Taylor series are given in reference books. Thus, for small enough q
(in comparison with unit) the series converge quickly.

On the basis of the above stated, there is a following technique for selection
of parameters of the system assuming periodic movements of rollers. It is
necessary to take one of Mathieu functions to calculate am or bm and to
select such parameters of the system that equality (4.3) is valid. On the other
hand, after defining am and bm in view of (4.3) and (2.6), the parameters
should be selected so that the equality

4
R2Rm

Jkz
= cm (4.6)

holds,where cm = am or cm = bm depending on the choice of a givenMathieu
function.

5. Inhomogeneous Mathieu’s equation

Letusnowconsider inhomogeneous equation (4.2). Forbrevity,we shall denote
its right hand side by F(z). If η1(z) and η2(z) are a fundamental system of
solutions to equation (4.4), the particular solution to equation (4.2) is usually
obtained by amethod of variation of arbitrary constants. Thismeans that the
particular solution is found in the form of (3.10) considering C1 and C2 as
functions of z. Applying the method of variation of arbitrary constants, we
obtain such a particular solution to be

α(z) = C1(0)η1(z)+C2(0)η2(z)+

z
∫

0

F(ζ)

∆(ζ)
[η1(ζ)η2(z)−η1(z)η2(ζ)] dζ (5.1)

where

∆(z)= η1(z)η
′

(z)−η
′

1(z)η2(z) (5.2)



166 V.A. Ostapenko et al.

The function α(z) will be the general solution to equation (4.2) as well.
However, the solutions toMathieu’s equation are represented as

η1(z)= e
iβzU1(z) η2(z)= e

−iβzU1(−z) (5.3)

Functions (5.3) have the period equal to nπ for integer β. Nevertheless, for
integer β, functions (5.3) are dependent in the Jacobi sense and, consequently,
cannot create a fundamental systemof solutions toMathieu’s equation. In the
case when β is not integer, functions (5.3) do not have the required period.
Therefore, the obtaining of solutions toMathieu’s equation in view of (5.1)

does not contain the cases representing practical interest. In connection with
the above stated, other method for obtaining the particular solution to inho-
mogeneous equation (4.2), using not the fundamental system of solutions to
equation (4.4), but only on one solution to this equation is here applied. It is
clear that such a unique solution is necessary for the Mathieu function to be
of integer order. Thementioned method consists in the following.
Let ψ(z) bea solution to theequation (4.4).Having substituted inequation

(4.2) the following transformation

α(z) = U(z)ψ(z) (5.4)

we obtain
U ′′(z)ψ(z)+2U ′(z)ψ′(z)= F(z) (5.5)

In the domain, where ψ(z) is not equal to zero, this equation, by replacing
V (z)= U ′(z), results in a linear equation of the first order

V ′(z)+
2ψ′(z)

ψ(z)
V (z) =

F(z)

ψ(z)
(5.6)

whose general solution is the function

V (z)= exp
(

−
∫

2ψ′(z)

ψ(z)
dz
)[

C1+

∫

F(z)

ψ(z)
exp
(

∫

2ψ′(z)

ψ(z)
dz
)

dz
]

Therefore

U(z)= C1

∫

exp
(

−
∫

2ψ′(z)

ψ(z)
dz
)

+

+

∫

[

exp
(

−
∫

2ψ′(z)

ψ(z)
dz
)

∫

F(z)

ψ(z)
exp
(

∫

2ψ′(z)

ψ(z)
dz
)]

dz+C2

Taking into account that
∫

ψ′(z)

ψ(z)
dz = lnψ(z)



Steady periodic regime of rotary motion... 167

we obtain

exp
(

2

∫

ψ′(z)

ψ(z)
dz
)

=explnψ2(z)= ψ2(z)

therefore

U(z)= C1

∫

1

ψ2(z)
dz+

∫

( 1

ψ2(z)

∫

F(z)ψ(z) dz
)

dz +C2

Hence, the general solution to equation (4.2) is

α(z)= C2ψ(z)+C1ψ(z)

∫

dz

ψ2(z)
+ψ(z)

∫

( 1

ψ2(z)

∫

F(z)ψ(z) dz
)

dz (5.7)

The integral of the periodic function under certain conditions is a periodic
function of the same period as well. So, while ψ(z) and F(z) are periodic
functions of the same period and satisfy such conditions, solution (5.7) will be
a periodic function of the same period for any values of C1 and C2.

In the considered case, the state ofmatter is following. If the function ψ(z)
is assumed to be the Mathieu function of an integer order, the period of the
function ψ(z) will be equal to π or 2π. The right hand side of equation (4.2)
F(z)= 16q sin2z has the period equal to π, therefore solution (5.7) will have
a period which would be equal to the period of theMathieu function.

Concerning solution (5.7), it is necessary to make one essential remark.
This solution contains in the denominators some terms of the Mathieu func-
tions. As theMathieu functions have zero, therefore in neighborhoods of zero
of the correspondingMathieu functions, solution (5.7) loses sense, and in these
neighborhoods there will be essential features of the solution.

In these cases, it is possible to offer the following procedure. As in the
neighborhood of zero the solution to equation (4.2) linearly depends on α and
becomes small, then in such circumstances, with a high degree of accuracy, it
is possible to limit considerations to the equation

d2α

dz2
=16q sin2z (5.8)

whose general solution will be a function

α(z) = C1z +C2−4q sin2z

Function (5.8) at C1 =0 has the period equal to π. At C1 6=0, α(z) quickly
grows, α(z) no longer stays close to zero, which allow us to come back to
solution (5.7).



168 V.A. Ostapenko et al.

6. Conclusions

As a result of the carried out analysis, it was possible to show that the line-
arised inhomogeneous equation ofmotion of rollers yields periodic solutions of
periods equal to 2π/ω and 4π/ω. These solutions are obtained in an explicit
form. It was shown that periodic solutions are steady, but not asymptotic. On
the basis of these solutions, a technique of selection of parameters for design
purposes of classifiers with rollers is offered.

References

1. 1991,Allismineral systems newname formajormining andmineral processing
equipment manufacturing scope, SKII.I,Mining Review, 80, 21, 20-21

2. Caughey T.K., 1960, Hula-Hoop: an example of heteroparametric exitation,
American Journal of Physics, Mechanics, 28, 104-109

3. 1997, Good size,World Mining Equipment, 21, 10, p.3

4. 1991, Hewitt-Rolling,World Mining Equipment, 15, 2, p.5

5. McLachlanN.W., 1947,Theory and Application ofMathieu Functions, Lon-
don

6. Ostapenko V.A., Naduty V.P., Jagnyukov V.F., 2005, Mathematical
model ofmotion of classifierswith rollers of vibrating type,Vibrations in Tech-
niques and Technology, 43, 1, 97-99

7. Smirnoff V.I., 1969,Course of Higher Mathematics, Science, Moscow

8. 1999, Zemag-Waltzenrostsibe mit Einzelantriebe und Online-Steuerungsistem,
Aufbereit Techn., 40, 7, 352-353

Obszar ustalonego okresowego ruchu obrotowego rolek drgającego

sortownika

Streszczenie

W pracy zaprezentowano problem dynamiki drgającego sortownika przeznaczo-
nego do materiałów sypkich. Głównym celem analizy jest znalezienie i uzasadnienie
takiego wyboru parametrów konstrukcyjnych sortownika, które zapewnią ustalony
i okresowy stan pracy rolek. Równania ruchu są nieliniowe.W pracy dokonano prze-
kształcenia równań tak, aby opisywały opóźnienie ruchu rolek w stosunku do rotacji



Steady periodic regime of rotary motion... 169

względem osi, na których te rolki zamocowano. Przy niewielkich wartościach tego
opóźnienia, równania ruchumogą być zlinearyzowane z utrzymaniemwystarczającej
dokładności. Linearyzacja prowadzi do równań Hilla, które przy spełnieniu pewnych
warunkówmogą zostaćprzekształconedopostaciMathieu.Wartykuleprzedstawiono
rozwiązania tych równań oraz zbadano ich stabilność.

Manuscript received February 21, 2007; accepted for print April 4, 2007