Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

44, 4, pp. 867-880, Warsaw 2006

INFLUENCE OF ADDITIONAL MASS RINGS ON

FREQUENCIES OF AXI-SYMMETRICAL VIBRATIONS OF

CLAMPED CIRCULAR PLATES OF LINEARLY VARIABLE

THICKNESS

Jerzy Jaroszewicz

Faculty of Mechanical Engineering in Suwalki, Technical University of Bialystok

e-mail: jerzyj@pb.edu.pl

Longin Zoryj

Mechanical Engineering Faculty Technical University of Lviv, Ukraine

Andrzej Katunin

Faculty of Mechanical Engineering in Suwalki, Technical University of Bialystok

The aim of this paper is to analyze the influence of values and radius of an
additional mass ring on the continuous distribution of mass of a clamped
circular plate of linearly variable thickness. The linear theory of thin plates
is used for description of small buckling vibrations. The authors applied the
partial discretizationmethod which is based on the discretization of the con-
tinuousmass and continuous buckling rigidity function. It is also based on the
method of Cauchy’s influence function, which gives particularly exact effects
for distributed-continuous systems such as that presented in this paper. It is
shown that an approximate result leads to the exact value with the discreti-
zation degree of less than 5, and it is not dependent on the value and radius
of the concentratedmass. Exact results of calculations lead to accurate valu-
es discovered by Conway for plates of linearly variable thickness without an
additional mass and to accurate values discovered by Roberson for plates of
constant thickness with the mass concentrated in the center.

Keywords: circularplates,variable thickness,boundary-valueproblem,partial
discretizationmethod

1. Introduction

The model presented in this paper can be applied to numerous structures
such as diaprahrgms, bottom parts of boilers and cylindrical containers, etc.
Boundary-value problems of axi-symmetrical vibrations of circular plates of



868 J. Jaroszewicz et al.

variable thickness carrying an additional mass distributed in concentric rings
are investigated in this paper. Well known papers by Roberson and Conway
are devoted to investigations of a concentratedmass in the centre of a plate of
constant thickness or of a plate of variable thickness without any discrete inc-
lusions.The investigation concerns an analogy between the lateral vibration of
a conical bar and axi-symmetric vibration of a circular disc of linearly varying
thickness (cf. Conway, 1957). Conway analyzed the basic natural frequency of
axi-symmetric vibrations of circular plates with clamped edge, whose flexural
rigidity D varieswith the radius r according to the law D=D0r

m, where D0
and m are constants (cf. Conway, 1958). Free, flexural, axi-symmetric vibra-
tions of a clamped circular disc were investigated, solutions in terms of Bessel
functionswere found for certain values of m andPoisson’s ratio ν. Vibrations
of a circular plate clamped at its edge and carrying a concentratedmass at its
center were considered in Roberson (1951). The plate was excited by motion
of the rigid framing, to which it had been clamped. The first four natural
frequencies were displayed graphically as functions of themass ratio andwere
calculated more precisely for µ = 0, µ = 0.05 and µ = 0.10 (cf. Roberson,
1951). An auxiliary mass µ0 was concentrated on a concentric radius of the
plate, which did not have a thickness dimension and exerted no effect upon
the flexibility of the plate.

A method of partial discretization was applied to the investigation of the
influence of mass on vibrations of a circular plate of constant thickness (Zoryj
and Jaroszewicz, 2000). The effectiveness of that method was evaluated. The
paperwas focused on analysis of the influence of themass concentrated at the
centre of the plate on themain frequency. In the paper, a method of spectral
functions, proposed byBernštein andKieropian (1960) was used solely for the
analysis of systemscharacterisedbyconstantparameterswithnoconsideration
given to friction.

In this paper, we propose double sided Bernštein-Kieropian estimators for
the frequency which should be calculated with different degrees of accuracy.
Therefore, the method of characteristic series, which makes use of the Cau-
chy influence function to solve these problems, appears to be attractive. The
direction of the theorem of vibrations of continually-discrete, linearly elastic
systems (based on utilizing theCauchy influence functions and the characteri-
stic seriesmethod) is useful for constructing and studying universal frequency
equations (cf. Zoryj, 1982; Jaroszewicz and Zoryj, 1994, 1996). Jaroszewicz
and Zoryj (2005) proved that FEM is very effective in vibration analysis but
in comparison with the Cauchy function method, the functional dependence
of mass rigidity is characteristic. It gives directional optimality of such plates.

As it is well known, equations of motion can be presented by forces or
displacements which are applied in the partial discretization method. Using
the method of partial discretization for a plate with a continuous or discrete-



Influence of additional mass rings... 869

continuousdistributionofmass is replacedbydiscrete systemswithone, twoor
n degrees of freedom dependending on the discretization degree. The rigidity
distribution of created discrete systems is the same as in the input system –
themasses are concentrated on chosen radii, their radii have no thickness and
they do not influence the flexural rigidity of the plate.
This investigation method introduces an additional mass where the di-

stribution depends on the radial coordinate. Jaroszewicz (2000) considered a
constant thickness plate only.
Figure 1 shows amodel of a clamped circular plate with variable thickness

and with a mass inclusion. R denotes radius of the plate, h0 – thickness of
the plate on clamping, ri – radius of the mass ring, mi, ci – values of the
concentrated masses and elastic supports.

Fig. 1. Model of the plate

2. Formulation of the problem

Weconsider circular plate of the radius Rwithdiscrete inclusions, the location
of which depends on the radial coordinate ri and, its flexural rigidity D is a
power function in the following form

D=D0
( r
R

)m
0<r¬R (2.1)

There are two possible variants of changes of the plate thickness:

• m> 0 – a diaphragmwith thickness decreasing toward the axial center

• m< 0 – a disc with thickness increasing toward the axial center.



870 J. Jaroszewicz et al.

If m= 0. the plate thickness is constant.
Investigation of free axi-symmetrical vibrations of such a plate leads to the

following boundary problem (cf. Zoryj and Jaroszewicz, 2002)

L0[u]−
ρh0
(
r
R

)m
3

D0
(
r
R

)m ω
2u−

K∑

i=1

αiu(ri)δ(r−ri)= 0

(2.2)

0<r1 <r2 < ... < rK <R

for which the following boundary conditions exist

u(R)= 0 u′(R)= 0 (2.3)

These conditions correspond to the case of a circumferentially fixed circular
plate.
The differential operator has the form (cf. Hondkiewic, 1964)

L0[u]≡u
IV+
2

r
(m+1)uIII+

1

r2
(m2+m+νm+1)uII+

1

r2
(m−1)(νm−1)uI

The following nomenclature is used in (2.1)-(2.4): m – coefficient of the power
flexural rigidity function, whose value may assume arbitrary real numbers;
D0 = Eh

3
0/[12(1− ν

2)] – flexural rigidity at r = R; u = u(r) – amplitude
of deflection; h= h0(r/R)

m/3 – variable thickness; h0 – thickness at r =R;
ω – frequency parameter; δ(r) – Dirac’s delta function; ρ – mass density of
the plate; ν –Poisson’s ratio. The inclusion parameters in discretemasses mi
and rigidity of elastic supports ci, which are located on the radius circle ri,
are described by the following formula

αi =
1

D0
(
ri
R

)m(miω
2
− ci) (i=1,K) (2.4)

In this case, solutions to equation L0[u] = 0 in expressions (2.2) and (2.3) as
well as their first derivates at r = 0 should be limited. For circular plates of
the diaphragm type,we have h(r)­ 0 because 0<r¬R,m> 0 and h0 > 0.
The substitutemodel of the analyzed plate (Fig.1) is defined by following

assumptions:

• The additional mass ring is distributed on the circle with the radius r0
from the range 0 to R.

• The linear theory of thin plates and small deflections is used (cf. Timo-
shenko, 1940).

• The rigidity of elastic supports ci is neglected.



Influence of additional mass rings... 871

The assumption of the partial discretization method is the substitution
of continuous distribution of the plate mass by a sequence of concentrated
masses mi located on circles with radii ri, which can be determined by the
following formula (cf. Jaroszewicz and Zoryj, 2006)

ri =
R

2K
(2i−1) (i=1,K) (2.5)

where K denotes the degree of discretization, which is equal to the amo-
unt of concentrated masses that substitute the continuous mass of the plate.
The number of distributedmasses depends on the thickness variability coeffi-
cient (m).
For a circular plate of linearly variable thickness (m=3), the concentrated

masses equivalent to the degree of discretization K have following form

Mi =
mi
2π
=
K3− (K−1)3

3K3
ρh0R

2 (i=1,K) (2.6)

The sum of masses received from the discretization is equal to the total mass
of the plate

M =
2

3
πR2ρh0

For a plate with one additional mass ring m0 of the radius r0, the relative
value of the additional mass with respect to the total plate mass µ0 and its
relative radius χ0 are defined by the following formulas

µ0 =
m0
πR2ρh0

χ0 =
r0
R

(2.7)

3. Determination of the influence matrix for plates with linearly

variable thickness

The elements of the flexibility influence matrix are found by considering an
appropriate static problem (cf. Jaroszewicz and Zoryj, 2005)

L0[u]≡u
IV+
8

r
uIII+

12

r2
uII =

FjR
3

r4jD0
δ(r−rj)

(3.1)

u(R)= 0 u′(R)=0

The differential operator L0[u] in (3.1) has been obtained for the coefficient
of the power flexural rigidity function m = 3, which corresponds to linearly
variable thickness and Poisson’s ratio ν = 1/3, which corresponds to steel.



872 J. Jaroszewicz et al.

The value of Fj characterises a flexural force concentrated on a circle with the
radius rj(rj ­ ri).

A limited solution to equation (3.1) is determined by the formula

u=C0+C1r+
FjR

3

r4jD0
K0(r,rj)Θ(r−rj) (3.2)

where: C0, C1 are arbitrary constants; K0(r,rj) =K0(r,α)
∣∣
α=rj
; K0(r,α) –

fundamental function of the operator L0[u]; Θ(r) – Heaviside’s function.

From (3.2) we obtain

u′ =C1+
FjR

3

r4jD0
K′0(r,rj)Θ(r−rj) (3.3)

Substituting the right-hand sides of (3.2) and (3.3) to the boundary conditions
of (3.1), we determine the constants C0, C1 after which, from formula (3.2),
we obtain a solution to the mentioned problem in the form

uj =
FjR

3

r4jD0
[K′Rj(R−r)−KRj +KrjΘrj] (3.4)

where the following denotes

KRj ≡K0(R,rj) K
′
Rj ≡K

′
0(R,rj)

Krj ≡K0(r,rj) ΘRj ≡Θ(R−rj)
(3.5)

Hence, considerin that βij =uj(ri) for Fj ≡ 1 (Zoryj and Jaroszewicz, 1982),
we arrive from (3.4) at the following formula

βij =
R3

r4jD0
[K′Rj(R−ri)−KRj] (3.6)

since ri ¬ rj, then θrj =0. It is know (cf. Zoryj and Jaroszewicz, 2002) that
the fundamental function of the operator L0[u] in equation (3.1) has the form

K0(r,α) =
1

6
(rα2−α5r−2)+

1

2
(α4r−1−α3) (3.7)

Substituting dependencies (3.4), (3.5) to (3.6), we determine the elements of
the flexibility influence matrix β= [βij] (cf. Zoryj and Jaroszewicz, 2005)

βij =
R2

3D0χi

[3
2
χi(χi+χj)+

3

2

χi
χj
−
1

2

(χi
χj

)2
−3χi−χ

2
iχj
]

χi =
ri
R
(3.8)



Influence of additional mass rings... 873

In particular, the diagonal monomial elements

βij =
R2

3D0χi
(1−χi)

3 (i=1,K) (3.9)

are consistent with the formula obtained by Zoryj and Jaroszewicz (2002).
It should be noticed from (3.6) that the multinominal elements βij could be
arrived at equally similarly, and it might be proved that βji = βij, which
confirms symmetry of the matrix β.

It is not difficult to find formulas for βji for plates with constant thickness

βij =
(
1−
r2j −r

2
i

R2
−
r2jr
2
i

R2
+2
r2j +r

2
i

R2
ln
rj
R

)R2

8
(i¬ j) (3.10)

and βij =βji, 0<r1 <r2 < ... < rk <R.

4. The characteristic equation and the double estimator for the

base frequency

Formulas (3.8) and (3.9) allow us to write down an equation for small vibra-
tions of a given model for arbitrarily limited number K, in an reverse form
that is expressed through dislocations (Zoryj, 1982)

K∑

j=1

Mjβij
d2qj
dt2
+qi =0 (i=1,K) (4.1)

where qj and Mj represent generalised coordinates and concentrated masses
defined in (2.7), respectively.

The natural frequencies and forms of the system described by equation
(4.1) can be calculated by well known methods (cf. Jaroszewicz and Zoryj,
2005). For the formulated influence matrix and equation of motion of the
discretizated system for K = 2, a characteristic equation for plates without
additional masses has been obtained in a power series form

∆=1−a1Λ+a2Λ
2
− . . .=0 (4.2)

where

Λ=ω2
R4ρh

D
a0 =1

a1 =
m1
2π
β11+

m2
2π
β22 a2 =

m1m2
4π2
(β11β22−β

2
12)



874 J. Jaroszewicz et al.

The estimators of the base frequency ω1 can be calculated by Bernstein’s
double estimators for frequency coeffcients in equation (4.2) (cf. Jaroszewicz
et al., 2004)

ω1 = γ±
1

R2

√
D0
ρh0

(4.3)

where

γ− =

√√√√
a0√

a21−2a0a2
γ+ =

√√√√
2a0

a1+
√
a21−4a0a2

γ± =
1

2
(γ−+γ+)

By taking into consideration (4.2) and replacing in it a with ã, it is easy to
get an equation for the plate with an additional mass ring

ã0− ã1Λ+ ã2Λ
2 =0 (4.4)

where

ã0 =1 ã1 =
m0
2π
β00+a1

ã2 =
m0m1
4π2
(β00β11−β

2
01)+

m0m2
4π2
(β00β22−β

2
02)+a2

5. Results of calculations

As an example of calculation, a linearly variable thickness plate with one
additional mass ring is given. Bellow, calculations for a simple case of two-
degree discretization with one additional mass, which corresponds to K+1,
arepresented.This single additionalmass increases thedegree of discretization
by 1. Using formulas (2.5)-(2.7), (3.8), (4.2)-(4.4), the following values have
been obtained

χ0 =0.2 χ1 =0.25 χ2 =0.75

m0 =0.1πρh0R
2 m1 =

1

12
πρh0R

2 m2 =
7

12
πρh0R

2



Influence of additional mass rings... 875

ã0 =1 ã1Λ=
1

2π
(m0β00+m1β11+m2β22)= 0.068129

R4ρh0
D0

ã2Λ=
1

4π2

[
m0m1

∣∣∣∣∣
β00 β01
β01 β11

∣∣∣∣∣+m0m2

∣∣∣∣∣
β00 β02
β02 β22

∣∣∣∣∣+m1m2

∣∣∣∣∣
β11 β12
β12 β22

∣∣∣∣∣
]
=

=2.35492 ·10−5
(R4ρh0
D0

)2

β00 =
R2(1−χ0)

3

3D0χ0
=0.8533

R2

D0

β11 =
R2(1−χ1)

3

3D0χ1
=0.5625

R2

D0

β22 =
R2(1−χ2)

3

3D0χ2
=0.00694

R2

D0

β01 =
R2

3D0χ0

[3
2
χ0(χ0+χ1)+

3

2

χ0
χ1
−
1

2

(χ0
χ1

)2
−3χ0−χ

2
0χ1
]
=0.675

R2

D0

β02 =
R2

3D0χ0

[3
2
χ0(χ0+χ2)+

3

2

χ0
χ2
−
1

2

(χ0
χ2

)2
−3χ0−χ

2
0χ2
]
=0.0324

R2

D0

β12 =
R2

3D0χ1

[3
2
χ1(χ1+χ2)+

3

2

χ1
χ2
−
1

2

(χ1
χ2

)2
−3χ1−χ

2
1χ2
]
=0.03009

R2

D0

ω1 =3.841
1

R2

√
D0
ρh0

(γ1)− =3.8409557

(γ1)+ =3.8409809

Calculations were carried out for the above-mentioned example for many va-
lues of coefficients µ0 and χ0. The results are presented in tables and figures
below.Thenatural frequency is displayed graphically as a function of themass
ring coefficients µ0 and χ0 (Fig.2) and as a function of the discretization de-
gree K (Fig.3-Fig.5). The results of calculations are presented in Table 1 to
Table 5. Table 1 andTable 2, and in Fig.2 give results for plates with linearly
variable thickness (m=3, ν =1/3).

In order to check themethodwithRoberson’s results, particular caseswith
constant thickness of different values and radii of the additionalmass ringhave
been investigated. The results are presented inTable 3 toTable 5 and inFig.3
to Fig.5.



876 J. Jaroszewicz et al.

Fig. 2. Double estimators for the natural frequency parameter (γ±) of the plate with
linearly variable thickness with one additional mass versus coefficients µ0 and χ0

Table 1. Calculations of γ±(µ0) for χ0 =0.02, K =2 (when µ0 =0.001,
the discretization degree was K =10)

µ0 0.001 0.5 0.99 2 2.5

γ± 7.7918 0.5035 0.3583 0.2523 0.2257

µ0 3 3.5 4 4.5 5

γ± 0.2061 0.1908 0.1785 0.1683 0.1597

Table 2. Calculations of γ±(χ0) for µ0 =0.1,K =2

µ0 0.025 0.05 0.075 0.1 0.125

γ± 1.2471 1.7936 2.2308 2.6119 2.9564

µ0 0.15 0.175 0.2 0.225 0.25

γ± 3.2732 3.5672 3.841 4.0961 4.3336

Table 3. Calculations for χ0 =0.001; µ=0

K 2 3 4 5

a1 0.0118 0.011 0.0107 0.0106

a2 9.16097 ·10
−6 9.2951 ·10−6 8.8804 ·10−6 8.6301 ·10−6

γ− 9.5127 9.9196 10.0501 10.1074

γ+ 9.5262 9.94 10.071 10.1282

γ±/∆ 9.5194/6.8005% 9.9298/2.7824% 10.0605/1.5028% 10.1178/0.9418%

γs/∆ 9.1860/10.0646% 9.5175/6.8191% 9.6403/5.6168% 9.6975/5.0568%



Influence of additional mass rings... 877

Fig. 3. Natural frequency parameter (γ) and terms of the characteristic series
(a1,a2) of the constant thickness plate without the additional mass versus

discretization degree (K) for χ0 =0.001;µ=0

Table 4. Calculations for χ0 =0.001; µ=0.05

K 2 3 4 5

a1 0.015 0.0142 0.0139 0.0137

a2 1.5778 ·10
−5 1.6373 ·10−5 1.6061 ·10−5 1.5841 ·10−5

γ− 8.4873 8.7851 8.882 8.9248

γ+ 8.5017 8.8063 8.9046 8.9477

γ±/∆ 8.4945/5.7423% 8.7957/2.4001% 8.8933/1.3171% 8.9363/0.84%

γs/∆ 8.1716/9.3253% 8.4023/6.7654% 8.4864/5.8322% 8.5254/5.3995%

Fig. 4. Natural frequency parameter (γ) and terms of the characteristic series
(a1,a2) of the constant thickness plate with one additional mass versus

discretization degree (K) for χ0 =0.001;µ=0.05

Table 5. Calculations for χ0 =0.001; µ=0.1

K 2 3 4 5

a1 0.0181 0.0173 0.017 0.0169

a2 2.239 ·10
−5 2.345 ·10−5 2.3242 ·10−5 2.3053 ·10−5

γ− 7.71107 7.9367 8.0105 8.0431

γ+ 7.7233 7.9541 8.0292 8.0621

γ±/∆ 7.7172/4.8563% 7.9454/2.0429% 8.0198/1.1256% 8.0526/0.7212%

γs/∆ 7.4328/8.3626% 7.6052/6.2371% 7.6674/5.4703% 7.6961/5.1164%



878 J. Jaroszewicz et al.

Fig. 5. Natural frequency parameter (γ) and terms of the characteristic series
(a1,a2) of the constant thickness plate with one additional mass versus

discretization degree (K) for χ0 =0.001; µ=0.1

6. Conclusions

The suggested method takes into consideration not only the mass but also
the radius of an additional mass ring. It is shown that if the radii of additio-
nal masses increase, the main frequency increases as well. The radius of the
mass ring is not taken into consideration in themethod applied byRoberson.
However, the presented method gives a deviation less than 1% for a small di-
scretization degree (K =5) in comparison to the precise solution obtained by
Roberson.As themass value grows from0 to 0.1, the deviation decreases from
0.94% to 0.72% (Tables 3-5).We discovered that solutions for themass values
illustrated in Fig.4. and Fig.5 coincide with Roberson’s solutions. Roberson
did not take into account the change of thickness of the plate, therefore the
obtained results cannot be compared. However, the results for cases without
additional masses correspond to Conway’s results. As shown in Table 1, the
result for the case when µ = 0.001 and χ0 = 0.02 (7.7918) is different from
the exact value obtained by Conway (8.75) by 1.1% for the degree of discreti-
zation K =10. Simultaneous analysis of the variable thickness and additional
mass by analytical methods has not been encountered in the literature on the
subject.

Acknowledgement

This work was financially supported within a scientific grant No.W/ZWM/1/05

sponsored by the Technical University of Bialystok.

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Influence of additional mass rings... 879

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Engineering, Lviv, 5, 37-38 (in Ukrainian)



880 J. Jaroszewicz et al.

Wpływ dodatkowych pierścieni masowych na częstości drgań

osiowosymetrycznych płyty o liniowo-zmiennej grubości utwierdzonej na

obwodzie

Streszczenie

Wpracyzbadanowpływwartości i promieniadodatkowejmasypierścieniowejwy-
stępującej w ciągłym rozkładzie masy płyty o liniowo-zmiennej grubości. Pominięto
przy tymwpływpierścieniamasowegona sztywnośćgiętnąpłyty.Zastosowano liniową
teorię cienkich płyt do opisu małych drgań giętnych. Zastosowanametoda dyskrety-
zacji polega na dyskretyzacjimasy ciągłej płyty i pozostawieniu ciągłej funkcji sztyw-
ności. Bazuje ona na metodzie funkcji wpływu Cauchy’ego, która daje szczególnie
dobre efekty dla układów dyskretno-ciągłych. Pokazano, że rozwiązanie przybliżone
zbiega do wartości ścisłej już przy stopniu dyskretyzacji mniejszym niż 5, niezależnie
od wartości i promienia rozłożenia masy skupionej. Uzyskane wyniki obliczeń zmie-
rzają do ścisłych wartości uzyskanych przez Conway’a dla płyty o liniowo-zmiennej
grubości bez dodatkowej masy i do wyników Robersona dla płyty o stałej grubości
z dodatkowąmasą skupioną w środku symetrii.

Manuscript received June 9, 2006; accepted for print October 16, 2006