Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 1, pp. 109-121, Warsaw 2008

LIMITATIONS IN APPLICATION OF BASIC FREQUENCY

SIMPLEST LOWER ESTIMATORS IN INVESTIGATION OF

NATURAL VIBRATIONS OF CIRCULAR PLATES WITH

VARIABLE THICKNESS AND CLAMPED EDGES

Jerzy Jaroszewicz

Mariusz Misiukiewicz

Wojciech Puchalski

Technical University of Bialystok, Mechanical Faculty in Suvalki, Poland

e-mail: jerzyj@pb.bialystok.pl

In this paper,we discussed the effects of application theBernstein-Keropian
simplest lower estimators for calculation of basic natural vibration frequen-
cies of variable-thickness circular plates. Following a thorough analysis, we
showed a significant role of the operators of the 4th orderEuler equations of
motion describing vibrations of circular plates with exponentially variable
thickness. The dependence of solutions on values of the variable thickness
power index of the plate and Poisson’s ratio of the plate material was in-
vestigated. The variable power thickness diaphragmmodel was verified by
FEMwhich confirmed the usefulness of the simplest estimator for following
materials: titanium, steel, zinc. Comparisonof the results obtained after ap-
plying the simplest lower estimators for the basic frequencywith the results
of exact solutions for particular cases found in the literature, confirmedhigh
accuracy of the appliedmethod.

Key words: circular plates, variable thickness, boundary-value problem,
Cauchy function method

1. Introduction

In his paper, Conway (1958b) analyzed the basic natural frequency of axi-
symmetric vibrations of clamped edge circular plateswhen the flexural rigidity
D varied with the radius r according to the law: D = D0r

m, where D0 and
m are constants. The author obtained characteristic equations using special
Bessel’s functions for particular caseswhen the plate thickness changed lineary



110 J. Jaroszewicz et al.

andparabolic. He chose a few combinations of variable thickness parameter m
and Poisson’s ratio ν according to the formula: ν =(2m−3)/9, which led to
exact solutions. Although these solutions have a limited practical value, they
are unique and may be used to estimate the accuracy rate of approximate
solutions (Vasylenko, 1992; Kovalenko, 1959).

We propose a method of characteristic series using the Cauchy influence
function to solve these problems. The method seems to be attractive, becau-
se it gives terms of characteristic series in an analytical form. The theory of
vibrations of discrete-continuous linearly elastic systems,whichhas beendeve-
loped in this work, is useful for constructing and studying universal frequency
equations (Jaroszewicz and Zoryj, 2005). Thismethod is based onmaking use
of the Cauchy influence functions and a characteristic series (Jaroszewicz and
Zoryj, 1997). Properties of the Cauchy influence function are applied in this
method. One property of the Cauchy influence function is especially signifi-
cant – not only the function itself but also its derivatives in respect to the
parameter α always create a basic system for solutions (Haščuk and Zoryj,
1999).

It was shown that the proposedmethod allows us to analyse the influence
of additional mass rings on the frequencies of natural vibrations of clamped
circural plates with linearly variable thickness by means of the mass par-
tial discretization methods (Jaroszewicz and Zoryj, 2006; Jaroszewicz et al.,
2006). By comparison, the spectral function method, proposed by Bernštein
in 1960, was used solely for the analysis of systems characterised by constant
parameters with no consideration given to friction (Bernštein and Kieropian,
1960). In the paper by by Jaroszewicz and Zoryj (2000) the characteristic se-
ries method was successfully applied to solve the boundary-value problem of
free transversal vibrations of an axially loaded vertical cantilever with variable
parameters. Main frequencies of axi-symmetric vibrations of thin plates with
variable distribution of parameters were analyzed by Jaroszewicz et al. (2004)
bymeans of the characteristic series method. The influence of Young’s modu-
lus, Poisson’s ratio andmass density of the material on the base frequency of
circular plates of the diaphragm typewith variable thickness was discussed by
Domoradzki et al. (2005). The paper addresses the problems concerning pro-
perties of the fundamental system of Euler’s operators as well as limitations
and singularities of the above mentioned methods for solving the boundary
value problems of variable thickness circular plates.

The authors examine circular plates with variable thickness by means of
the influence function method. It is worth pointing out that Conway did not
apply the exact solution for the considered case (Conway, 1958a,b).



Limitations in application of basic frequency... 111

2. Formulation of the problem

Weconsider an R-radius circular plate having a clamped edge. Its thickness h
and flexural rigidity D change in the following way

h=h0
( r

R

)

m

3

D=D0
( r

R

)m
0<r¬R (2.1)

and

D0 =
Eh30

12(1−ν2)

where D0, h0, m ­ 0 are constants, r denotes the radial coordinate, E –
Young’s modulus, and ν – Poisson’s ratio.
Investigation of free, axi-symmetric vibrations of such a plate is reduced to

analysis of the boundary problem (Conway, 1958b, Hondkiewǐc, 1964; Zoryj
and Jaroszewicz, 2005)

L0[u]−pr−
2

3
mu=0 u(R)= 0 u′(R)= 0 (2.2)

where

L0[u]≡uIV +
2

r
(m+1)uIII +

1

r2
(m2+m+νm−1)uII +

+
1

r3
(m−1)(νm−1)uI

(2.3)

p=
ρh0
D0
R
2

3
mω2

u= u(r) denotes the amplitude of flexural vibrations, ρ – density of the ma-
terial of the plate, ω – parameter of frequency (angular velocity). Boundary
value conditions corresponding to clamped edge (2.2)2,3 are defined as zero
deflection and zero angle of deflection for r = R. Additional conditions per-
taining to the centre of symmetry of the plate (r=0) have limited values of
deflection u(0)<∞ and zero values of the angle of deflection u′(0)= 0.
The value m = 0 refers to the plate with constant thickness; m > 0 –

to plates of the diaphragm type with thickness decreasing toward the center;
m< 0 – to plates of the disc type with thickness increasing toward the center
(Hondkiewič, 1964; Jaroszewicz and Zoryj, 2005). We determine the border
of variation of the power index m­ 0 for which the most simple estimators
of the basic frequency ω1 exist and, therefore, they can be calculated, i.e. we
search for the lowest proper value of boundary problem (2.2).



112 J. Jaroszewicz et al.

In problem (2.2), a limitation of solutions growing r and their first three
derivatives with respect to the independent variable r is required (Conway,
1958b).

3. Derivation of the frequency equation

The necessary limited solution of equation (2.2)1 will be derived according to
the known formula (Haščuk and Zoryj, 1999; Jaroszewicz and Zoryj, 2005)

Sj = sj0+psj1+p
2sj2+ . . . j=1,2 (3.1)

where

Sjk =

r
∫

0

K(r,τ)τ−
2

3
msj,k−1(τ) dτ k=1,2, . . . (3.2)

S10, S20 are the solutions (limited for r = 0) to Euler’s equation L0[u] = 0.
Therefore, K(r,τ) – its Cauchy’s function, is the solution to the equation
L0[u] = 0, which satisfies the conditions

K(r,τ)=K′(r,τ) =K′′(r,τ) = 0 K′′′(r,τ) = 1 (3.3)

In order to build the above mentioned function K(r,τ), we need solutions to
the given equation, which correspond to operator (2.3)1. Substituting u= r

s

for p=0 in (2.2)1, we obtain an appropriate algebraic equation with respect
to the parameter s (Hondkiewič, 1964)

s
{

s3+2(m−2)s2+[4−5m+m(m+ν)]s+m(2−m)(1−ν)
}

=0 (3.4)

The roots of this equation and the corresponding Cauchy function were pre-
viously determined for a few values of mwhere m¬ 3 (Conway, 1958b). That
is why in this paper, we examine the case

2<m<∞ ν =
1

m
(3.5)

Theparameter m is the samewhichwas used in operator (2.3)1. In the case of
a constant thickness plate m= 0, the coefficient of natural frequency γ(m),
describedby formula (4.3), is notdependenton ν,whichwas shownbyConway
(1958b), Hondkiewič (1964), Jaroszewicz and Zoryj (2005).



Limitations in application of basic frequency... 113

Obviously, for operator (2.3)1, the coefficient of u
′ equals zero and the

roots of equation (3.4) are as follows

s1 =0 s2 =1 s3 =1−m s4 =2−m (3.6)

where s3 and s4 are determined fromthe square equation,whichwas obtained
from (3.4) for s1 =0 and s2 =1.
Hence, corresponding fundamental system(3.6) of solutions to theequation

L0[u] = 0 takes the following form

u1 =1 u2 = r u3 = r
1−m u4 = r

2−m (3.7)

The application of the above system, leads to the formula

K(r,τ) =
1

1−m

[ 1

m
(−rτ2+r1−mτ2+m)+

1

2−m
(r2−mτ1+m− τ3)

]

(3.8)

Now solutions (3.1) canbe constructed.After calculating thefirst two integrals
of (3.2) (for j=1,2 and k=1), we obtain the following relations

S1(r)=1+A1pr
a S2(r)= r+B1pr

a+1 (3.9)

and

A1 =
1

ab(b+1)(a−1)
B1 =

1

a(a+1)(b+1)(b+2)
(3.10)

where

a=4−
2

3
m b=2+

m

3
(3.11)

After taking into consideration the formulas for u3 and u4 (3.7) for conditions
(3.5), we can draw a conclusion that the limited solutions to equation (2.2)1
are determined using the formula

u(r)=C1S1(r)+C2S2(r) (3.12)

where C1 and C2 are arbitrary constants. After inserting this solution into
condition (2.2)2,3, we obtain homogenous systemof two algebraic equations for
the constants C1 and C2, whose determinant is the left part of the frequency
equation of problem (2.2). Therefore

∣

∣

∣

∣

∣

S1(r) S2(r)
S′1(r) S

′

2(r)

∣

∣

∣

∣

∣

=0 (3.13)



114 J. Jaroszewicz et al.

Considering formulas (3.9)-(3.11), we obtain the first approximation (accurate
to p, hence to the square of the frequency), which takes the following form

1−a1pRa =0 (3.14)

where

a1 =−[A1(1−a)+B1(1+a)] (3.15)

Taking into consideration the equation

pRa =
ρh0
D0
R
2

3
mR4−

2

3
m =
ρh0
D0
R4ω2 (3.16)

we transform (3.16) to the following form

1−a1
ρh0
D0
R4ω2 =0 (3.17)

Remark: for m= 3, ν = 1/3, we obtain A1 = 1/24, B1 = 1/120 (Jarosze-
wicz and Zoryj, 2005). Formulas (3.10), (3.11) are in accordance with
the values quoted in the above remark.

4. The simplest lower estimator for the basic frequency

All roots of the frequency equation of the problem defined by (2.2), (2.3)
ought to be real, positive and inmost cases, simple numbers according to the
general characteristics of the frequency spectra of elastic systems (Vasylenko,
1992). Therefore, we obtain the simplest estimator of the basic (the smallest)
frequency from (3.17)

ω1 >γ(m)
1

R2

√

D0
ρh0

(4.1)

or

ω1 >γ(m)
h0
R2

√

E

12ρ(1−ν2)
(4.2)

where

γ(m)=
1
√
a1

(4.3)



Limitations in application of basic frequency... 115

Examples:

1. For the constant thickness plate (m=0), the coefficient a1 =1/96 (see
Jaroszewicz et al., 2004). So it follows that γ(0) =

√
96≈ 9.798, which

consists 95.92% of the exact value 10.214 (3.1962) (Vasylenko, 1992).

2. For the diaphragm m = 3, ν = 1/3 (linearly variable thickness) a1 =
=1/60 (seeJaroszewicz andZoryj, 2005). So, γ(3)

∣

∣

ν=1/3
=
√
60≈ 7.746,

which consists 88.56% of the exact value 8.747 (Conway, 1958b).

Examples of constant thickness plates (m=0) and the law of linearly variable
thickness (m=3) are presented in this paper as the standard solutions.

For cases (3.5) using (3.10) and (3.11), we transform (3.15) to the following
form

a1 =
2

ab(b+1)(b+2)
(4.4)

hence for m = 3 with a = 2, b = 3, on the base of equations (3.10), (3.15),
a1 equals 1/60.

Taking into consideration relationships (3.11), it is possible to transform
formula (4.4) into

a1 =
34

(6−m)(6+m)(9+m)(12+m)
(4.5)

On the base of expression (4.5) for m > 2 and ν = 1/m, we formulate the
conclusion that the simplest lower estimator can be applied when a1 > 0 un-
der the necessary condition that m< 6. The term a1 has negative values for
m > 6 and it is indefinite for m = 6. These two cases (m > 6 and m = 6)
cannot be accepted. It should be pointed out that construction materials ha-
ving ν ¬ 1/6 do not exist in reality. The border value of ν ¬ 1/6 corresponds
to a concrete plate. However, manufacturing a concrete plate for m > 6 is
technologically impossible and, therefore, such a case has only theoretical va-
lue.

We introduce a change of the coefficient γ(m) for m > 3 applying (4.5)
(ν = 1/m). The results of calculations are shown in Fig.1. Precise values of
γ(m) are displayed in Table 1.

The results of calculations of the base frequency for selectedmaterials and
values of m are presented in Table 2.



116 J. Jaroszewicz et al.

Fig. 1. The curve showing the influence of the plate thickness index on the simplest
estimator of the basic frequency coefficient

Table 1.Results of calculations of the variable-rigidity plate for m=3.25 to
m=5.999

m 3.25 3.5 3.75 4 4.25 4.5

γ 7.659451488 7.537201975 7.374735166 7.166451332 6.905149243 6.581223292
m 4.75 5 5.25 5.5 5.75 5.999

γ 6.181238257 5.685155024 5.060184657 4.244186290 3.081354161 0.199979495

Table 2. Results of calculations of the base frequency coefficient

Coeffi- Poisson’s First term of Simplest Value of
No. cient ratio of plate characteristic estimator exact

m material ν series a1 γ solution

1 0
arbitrary from 1

96
9.798

10.214
values 0-0.5 (Conway, 1958a)

2 3
1

3

1

60
7.746

8.730
(Conway, 1958a)

3 4
1

4
2
4

3

10

3

13

3

16

3
7.166

Exact value is
unknown4 5

1

5
2
2

3

11

3

14

3

17

3
5.685

5 5.9 0.169 0.255 1.979
6 5.99 0.16(7) 2.505 0.632

In the case of the constant thickness plate m= 0 (No. 1), the coefficient
of natural frequency γ(0) is not dependent on ν, which has been already
described.



Limitations in application of basic frequency... 117

5. FEM verification of the results of calculations of the base

frequency simplest estimators

The SolidWorks software was used in themodeling of diaphragm type plates
made of the selected materials: titanium, zinc, steel, concrete, which satisfy
the condition of the following relation between the thickness indexmandPois-
son’s ratio ν =1/m. Calculationswere performed for thin orthotropic circular
plates attached on perimeter. The plates were characterized by exponentially
changing thickness and the following main dimensions: radius R = 250mm,
thickness in the area of attachment h0 =10mm.Ahalf of the radial section of
the plate is presented inFig.2. Shaping the geometry of a platewith exponen-
tially changing thickness (described by formula (2.1)) requires selection of the
minimum thickness in the centre of symmetry equal to 0.1mm, which amo-
unts to 1% of the plate thickness in the area of attachment. The values of the
radius r0 for which the thickness is in reality lower than 0.1mmare displayed
in Fig.2. Dimensions of applied FEM elements allow the smallest thickness
to be 0.1mm on the plate surface limited by the radius r0 in relation to the
symmetry axis. The case of concrete considered in a given point in time has
only a theoretical value.

Fig. 2. Lengthwise section along the radial coordinate r of the plate characterised
by geometrical dimensions r0,R, h0

The mesh with changing size of elements was generated automatically.
The numerical analysis was conducted using the CosmosWork software. Two
solvers: theDirect sparse solver and the FFEPlus were applied to perform the
numerical analysis. TheFEManalysis gives convergent results with analytical
calculations by means of the Cauchy function method. The first five modes



118 J. Jaroszewicz et al.

and corresponding natural frequencies were investigated. The results of the
analysis are presented in Fig3.

Fig. 3. The base mode of natural vibrations of a titanium diaphragm (330.19Hz,
scale of deformation: 0.025)

The differences between the results of calculations using FEM and the
Cauchy function method are set together in Table 3.

Table 3. The base frequency of the plate with nonlinearly variable thickness
h(r→ 0)= 0.01mm

Material ν
Coeffi-
cient
m

D0
[Nm]

Mass
[kg]

Simplest
estima-
tor
γ

Frequency Frequency
Diffe-
rence
∆ [%]

of base by FEM
estimator analysis
f [Hz] f [Hz]

Titanium 0.36 2.78 9778 6.093 7.795 288.67 330.19 12.6
Steel 0.27 3.7 18560 9.834 7.412 283.12 347.15 18.4
Zinc 0.25 4.0 10531 8.410 7.166 219.61 267.35 17.9

Concrete 0.17 5.9 1716 1.983 1.979 46.26 211.76
Not

acceptable

The results of calculations of the base frequency using the simplest estima-
tor, presented in Table 3, are accepted for titanium, steel and zinc, whereas
for concrete they are not acceptable due to the fact that the difference exceeds
18%. It is worth to notice that the case of the concrete plate is only theoreti-
cal because such a structure cannot bemanufactured in practice. The results
of numerical analysis show that the simplest estimator can be used within
2.78¬m¬ 4.



Limitations in application of basic frequency... 119

6. Conclusions

Deriving the abovementioned formulas for theCauchy functions aswell as the
fundamental systems of the function operator L0[u], it allows one to study
the convergence problem (velocity of convergence) of solutions to equation
(2.2)1 in the form of power series with respect to frequencies depending on the
parameters m and ν.

Having the influence functions of the operator L0[u], we can determine
corresponding solutions such as (3.12)-(3.17) and use them for any given and
physically justified parameters m and ν (2 < m < ∞; ν = 1/m), when the
exact solutions are unknown. On the basis of the quoted solutions, simple
engineering formulas for the frequencies estimators of circular plates, which
are characterised by variable parameter distribution, can be derived, and li-
mitations on their application can be identified. The simplest lower estimator
calculated using the first element of series (4.4) allows ont to observe with
considerable credibility the effect of the material constants: Young’s modu-
lus E, Poisson’s ratio ν, density ρ on frequencies of axi-symmetric vibrations
of circular plates. Thickness or rigidity of the tested plates change along the
radius according to an exponential function.

The coefficient of the base frequency γ(m)= 1/sqrta1 can be determined
for 3 < m < 6 (Fig.1), because γ(m) → 0 for m → 6 it corresponds to
a=4−2m/3→ 0, 1/

√
a1 → 0. In this case, the simplest lower estimator gives

strongly underestimated values, therefore it cannot be even used in approxi-
mate calculations. However, the simplest lower estimators can be applied to
preliminaryengineering calculations for constant andvariable-thickness plates,
when 0¬m¬ 4.

Acknowledgement

This work was financially supported by through a scientific grant sponso-

red by the Technical University of Bialystok, Mechanical Faculty of Suwalki,

No.W/ZWM/1/07.

References

1. Bernstein S.A., Kieropian K.K., 1960, Opredelenije častot koleba-
nij ster(nevych system metodom spektralnoi funkcii, Gosstroiizdat, Moskva,
p.281



120 J. Jaroszewicz et al.

2. Conway H.D., 1958a, An analogy between the flexural vibrations of a cone
and a disc of linearly varying thickness, Z. Angew. Math. Mech., 37, 9/10,
406-407

3. ConwayH.D., 1958b,Some special solutions for theflexural vibrationsof discs
of varying thickness. Ing. Arch., 26, 6, 408-410

4. Domoradzki M., Jaroszewicz J., Zoryj L., 2005, Anslysis of influence of
elastisity constants andmaterial density on base frequency of axi-symmetrical
vibrations with variable thickness plates, Journal of Theoretical and Applied
Mechanics, 43, 4, 763-775

5. Haščuk P., Zoryj L.M., 1999, Linijni modeli diskretno-neperervnyh mekha-
nichnych system, Ukrainski technologii, Lviv, p.372

6. Hondkiewič W.S., 1964, Sobstviennyje kolebanija plastin i obolochek Kiev,
Nukova Dumka, p.288

7. Jaroszewicz J., Zoryj L., 1997, Metody analizy drgań i stateczności
kontynualno-dyskretnych układów mechanicznych, Rozprawy Naukowe Poli-
techniki Białostockiej, 54, Białystok, p.126

8. Jaroszewicz J., Zoryj L., 2000, Investigation of the effect of axial loads
on the transverse vibrations of a vertical cantilever with variables parameters,
International Applied Mechanics, 36, 9, 1242-1251

9. Jaroszewicz J., Zoryj L., 2005,Metody analizy drgańosiowosymetrycznych
płyt kołowych z zastosowaniem metody funkcji wpływu Cauche’go, Rozprawy
Naukowe Politechniki Białostockiej, 124, Białystok, p.120

10. Jaroszewicz J., Zoryj L., 2006, Themethod of partial discretization in free
vibration problems of circular plates with variable distribution of parameters,
International Applied Mechanics, 42, 3, 364-373

11. Jaroszewicz J., Zoryj L., Katunin A., 2004, Dwustronne estymatory czę-
stościwłasnychdrgańosiowosymetrycznychpłytkołowycho zmiennej grubości,
Materiały III Konferencji naukowo-praktycznej „Energia w Nauce i Technice”,
Suwałki, 45-56

12. Jaroszewicz J., Zoryj L., Katunin A., 2006, Influence of additional mass
rings on frequencies of axi-symmetrical vibrations of linear variable thickness
clamped circular plates, Journal of Theoretical and Applied Mechanics, 44, 4,
867-880

13. Kovalenko A.D., 1959, Kruglyje plastiny peremenntoj tolshchiny, Gosudar-
stvennoje Izdanie Fiziko-Matematicheskoj Literatury,Moskva, p.294

14. Vasylenko N.V., 1992,Teoriya kolebanij, Vyshcha Shkola, Kiev, p.429



Limitations in application of basic frequency... 121

Ograniczenia w stosowaniu najprostszych dolnych estymatorów częstości

podstawowej w badaniu drgań własnych płyt kołowych typu diafragma

Streszczenie

W pracy omówiono efekt zastosowania najprostszych estymatorów Bernštejna-
Kieropianado obliczania podstawowej częstości osiowosymetrycznychdrgańwłasnych
płyt o zmiennej grubości typu diafragma. Przeanalizowano i uwypuklono istotną rolę
operatorów równań Eulera czwartego rzędu oraz ich fundamentalnej funkcji w osio-
wosymetrycznych zagadnieniach drgań płyt kołowych o potęgowo zmiennej grubości.
Zbadano zależność rozwiązania od wartości wskaźnika zmiany grubości płyty i liczby
Poissona. Weryfikacja modelu diafragm o potęgowo zmiennej grubości przy pomo-
cyMES potwierdziłamożliwość zastosowania najprostszego estymatoraw przypadku
metali: tytanu, stali, cynku.Porównaniewynikówobliczeńnajprostszychestymatorów
z dołu częstości podstawowej uzyskanych przy wykorzystaniu zaproponowanejmeto-
dy oraz przy zastosowaniu znanych z literatury rozwiązań ścisłych dla szczególnych
przypadków zmiany grubości płyty potwierdziło wysoką dokładność proponowanej
metody.

Manuscript received April 18, 2007; accepted for print July 5, 2007