ultimo trabajot

7/31/2019 Ultimo Trabajot

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K. V. Avramov

O. Tyshkovets

Department of Nonstationary Vibrations,

National Academy of Sciences of Ukraine,

2/10 Dm. Pozharskoho Street,

61046 Kharkiv, Ukraine

K. V. Maksymenko-SheykoDepartment of Applied Mathematics and

Numerical Methods,

A.N. Podgorny Institute for Mechanical

Engineering Problems,

National Academy of Sciences of Ukraine,

2/10 Dm. Pozharskoho Street,

61046 Kharkiv, Ukraine

e-mail: [email protected]

Analysis of Nonlinear FreeVibration of Circular Plates WithCut-Outs Using R-FunctionMethodGeometrically nonlinear vibrations of circular plate with two cut-outs are simulated bythe von Karman equations with respect to displacements. The combination of the

RayleighRitz method and the R-function method, which allows satisfying all boundaryconditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrationsare expanded using these vibrations modes. The dynamical system with three degrees of

freedom is derived by Galerkin method. The influence of cut-outs size on linear andnonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, dif-

ferent internal resonances occur in the plate. The nonlinear vibrations of the system fordifferent internal resonances are analyzed by multiple scale method.

DOI: 10.1115/1.4001496

Keywords: circular plates with cut-outs, R-function method, conjugate modes, multiplescale method, internal resonances

1 Introduction

Plates are bearing elements of many structures and machines.

They widely used in gas-turbine engines, water turbines, brakes

systems, and water tanks. If the plate is bended more than half

thickness, the geometrical nonlinear model of the plate deforma-

tion must be considered 1 . Parametric vibrations of circularplates are treated in the book 2 , where the system of ordinary

differential equations with time-periodic coefficients of plate vi-

brations is derived. Farnsworth and Evan-Iwanowski 3 consid-ered axisymmetric vibrations of circular clamped plate. They used

harmonic balance method to analyze the nonlinear vibrations.

Axisymmetric vibrations of circular clamped plate are described

by the Duffing equation in Ref. 4 . Goloskokow and Filippow 5considered passage through a resonance of a circular disk with

constant thickness assuming linear time dependence of excitation

frequency. The vibrations of the circular disks are described by the

von Karman equations in Ref. 6 . It is shown that the internalresonances frequently occur in the circular plates. The vibrations

of clamped elliptic plates carrying concentrated masses are treated

in Ref. 7 . The frequency response of such system is hard. Axi-symmetric vibrations of circular plate are described by one Duff-

ing equation in Refs. 8,9 . Hadian and Nayfeh 10 analyzed theinteraction of three axisymmetric vibration modes using the mul-

tiple scale method.

The interactions of two conjugate vibration modes are consid-

ered mainly in the modern research of nonlinear dynamics of cir-

cular plates

11

. The system of three partial differential equations

describing radial, tangential, and bending vibrations of rotatingdisk is derived in Ref. 12 . The nonlinear vibrations of the diskrotating with constant angular velocity are considered in Ref. 13 .

The internal resonance between the modes, which are moved in

forward and backward directions, is analyzed. Luo and Tan 14considered the traveling waves of the computer disk rotated with

constant velocity. The nonlinear interaction of two conjugate

modes of circular disk studied in Refs. 15,16 . Nayfeh et al. 16modeled this interaction by the system of two ordinary differential

equations, which is studied by the multiple scale method. The

free-edge circular plate under the action of time-periodic lateral

force is considered in the paper 17 . It is shown that the interac-tion of two conjugate modes is described by hard frequency re-

sponse. Lee et al. 18 analyzed circular plate on an elastic foun-dation. The bifurcation diagram of periodic and chaotic vibrations

is treated. Arafat and Nayfeh 19 considered axisymmetric dy-namical behavior of circular plate under the action of thermal

loads in plane. The internal resonance 1:3 between the first and

the third axisymmetric vibration modes is treated.

The R-function method is suggested to obtain analytically theequations of complex domain boundaries 20 . Mainly, thismethod is used to solve the problems of mathematical physics for

the domains with complex boundaries. Kurpa 2123 developedthe R-functions theory to analyze the vibrations of plates and shal-

low shells.

In this paper, R-function method is used to obtain the vibration

modes of circular plates with two cut-outs. The nonlinear vibra-

tions of circular plate are expanded using these vibration modes

with R-functions. The nonlinear interaction of two conjugate vi-

bration modes and axisymmetric one is investigated by the mul-

tiple scale method.

2 Problem Formulation

The circular plate with two cut-outs Fig. 1 is considered. It isassumed that the deformation-displacement relations are nonlinear

and strain-deformation relations are linear. The vibrations are

treated in cylindrical coordinates r,,z . Then the displacementsof plates material points along r,,z are denoted by ur, u, uz,respectively. The equations of nonlinear shell in curvilinear coor-

dinates are used to derive the equations of plate vibrations 24 .The following equations of the plate nonlinear vibrations in polar

coordinates are derived:

Contributed by the Technical Committee on Vibration and Sound of ASME for

publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August5, 2008; final manuscript received February 23, 2010; published online August 18,

2010. Assoc. Editor: Sotirios Natsiavas.

Journal of Vibration and Acoustics OCTOBER 2010, Vol. 132 / 051001-1Copyright 2010 by ASME

Downloaded 07 Feb 2011 to 200.119.3.209. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm


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ur,rr 3 u,

2r2+

1 +

2ru,r +

1

2r2ur,+

ur,r

r

ur

r2+ uz,ruz,rr

1 +

2r3uz,

2++ 1

2r2uz,uz,r +

1

2r2uz,ruz,+

1

2ruz,r

2

= 1 2

Eh ur

1

r2u,+

3

2r2ur,+

1 +

2rur,r+

1

2ru,r

1

2r2u+

1

2u,rr

+1

r3uz,uz,+

1 +

2ruz,ruz,r+

1

2r2uz,ruz,+

1

2ruz,rruz,

= u1 2

Eh

h2

124uz =

1 2

Ehuz +

1

r ruz,r ,r ur,r + 1

2uz,r

2+

rur +

ru,

+

2r2 uz,2

+1

r uz,r u,r +1

rur,

1

ru+

1

ruz,ruz,+

1

r2uz,

1

rur +

1

ru,+

1

2r2uz,

2 + ur,r +

2uz,r

2 1where u=

2u/ t2, uz,r= uz / r, is mass per unit area, E and

are the Young modulus and the Poisson ratio, h is the thickness ofthe plate, and

2 =

2

r2+

1

r r+

1

r2

2

2

More general equations of plate deformations, then the system 1are derived in Ref. 12 .

The membrane forces and moments are determined in the fol-lowing forms:

Nr =Eh

1 2ur,r +

1

2uz,r

2 +

rur +

ru,+

2r2uz,

2N=

Eh

1 21

rur +

1

ru,+

1

2r2uz,

2 + ur,r +

2uz,r

2Nr=

Eh

2 1 + u,r +

1

rur,

1

ru+

1

ruz,ruz,

Mr = D uz,rr +

ruz,r +

r2uz,

M= D1

ruz,r +

1

r2uz,+ uz,rr

Mr= D 1 1

ruz,r

1

r2uz, 2

3 Linear-Free Vibrations of Plate

The Galerkin method is used to obtain the ordinary differential

equations of the plate vibrations 1 and the nonlinear vibrationsof plate with cut-outs are expanded using eigenmodes of linearvibrations. The RayleighRitz method 25 is used to obtain theeigenmodes of vibrations. Now this method is considered. The

elastic strain energy of the plate in the polar coordinates has thefollowing form:

=Eh

2 1 2

ur,r2 +

2

r u,+ ur ur,r +

1

r2u,+ ur

2

+1

2 u,r u

r+

1

rur,

2 rdrd+ D2

uz,rr2

+ 2uz,rr1

r2uz,+

1

ruz,r

+

1

r2uz,+

1

ruz,r

2

+ 2 1

1

ruz,r

1

r2uz,

2

rdrd 3

where is the domain of the plate. The kinetic energy of the platehas the following form:

T=

2

ur2 + u

2 + uz2 rdrd 4

where is mass per unit area.The linear vibrations of the plate with cut-outs are presented in

the following form:

ur r,,t

u

r,,t

uz r,,t =

ur r,

u

r,

uz r, sin

pt+

5

Equation 5 is substituted into the Hamiltonian action and theintegration is carried out. As a result the following functional isderived:

S=Eh

2 1 2

ur,r2 +

2

r u,+ ur ur,r +

1

r2u,+ ur

2

+1

2 u,r u

r+

ur,

r

2 rdrd+ D2

uz,rr2

+ 2uz,rr1

r2uz,+

1

ruz,r +

1

r2uz,+

1

ruz,r

2

+ 2 1

1

ruz,r

1

r2uz,

2 rdrd 2p2

ur2 + u

2 + uz2 rdrd

6

In future analysis the clamped plate is considered. The boundary

of the plate with cut-outs is denoted by . The boundary condi-tions are as follows:

uz =uz

n = u = ur = 0 7

where n is normal to the boundary .

Fig. 1 Circular plate with two cut-outs

051001-2 / Vol. 132, OCTOBER 2010 Transactions of the ASME



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The equation of the plate boundary Fig. 1 is obtainedanalytically using the R-function method

2022

. The R-function

satisfies the following conditions:

r, = 0, r,

r, 0, r, 8

The known approach 20 is used to construct the R-function. Inthis paper, two functions i r, ; i =1,2 are considered. For the

circular plates with two cut-outs Fig. 1 , the function 1 in Car-tesian coordinates has the following form:

1 x,y = 203 01

1 =1

2R R2 x2 y2

2 =1

a1

a12

4 x2

3 =1

b1y2

b12

4 9

where 0 and 0 are the operations of R-conjunction and

R-disjunction 20 , which can be presented in the following form:

203 = 2 + 3 + 22 + 32

401 = 4 + 1 42 + 12 10where 4 =203. The functions 9 have the following form in

polar coordinates:

1 =1

2R R2 r2 , 2 =

1

a1

a12

4 r2 cos2 11

The steps of R-function derivation are presented on Fig. 2.

R-function in the form 9 with small magnitude a1 has two localextreme. The following function 2 removes these extremes:

2 x,y = 203 01 0 12

where =max 1 r, ; 01.The eigenmodes of circular plate with two cut-outs can be pre-

sented in the following forms:

uz r, = k2 r,

k=0

m1

Zk c r cos k + Zk

s r sin k

u r, = k r,k=0

m2

k c r cos k +k

s r sin k

ur r, = k r,k=0

m3

Rk c r cos k + Rk

s r sin k , k= 1 or 2

13

where Zk

c r ,Zk

s r , . . . ,Rk

s r are the following polynomials:

Zk c r =

j=0

M1,k

Ak,j c

rj, Zk s r =

j=0

M2,k

Ak,j s

rj, k= 0,1, . .. , m1

k c r =

j=0

M3,k

Bk,j c

rj, k s r =

j=0

M4,k

Bk,j s

rj, k= 0,1, ... , m2

Rk c r =

j=0

M5,k

Ck,j c

rj, Rk s r =

j=0

M6,k

Ck,j s

rj, k= k= 0,1, ... ,m3

14

In order to satisfy the boundary conditions 7 , the expansion foruz contains the function k

2 r, and the expansions for u and urhave the function k r, .

Equations 13 and 14 are substituted into 6 and the integra-tion over the area of the plate is performed. As a result thefollowing functional is derived:

S= S a0, . . . , al

l =j=1

3

k=1

mj

M2j1,k + M2j,k 15

where a0 , . . . , al are the coefficients of the polynomials 14 : X= a0 , . . . , al = A0,0

c,A

0,1

c, . . . , C

m3,M6,k

s . Following the RayleighRitz method, the following system of linear algebraic equations isderived to minimize the functional 15 :

S

aj= 0, j = 0, ... ,l 16

Equation 16 can be rewritten in the form of eigenvalue problem:

K p2MX= 0 17

where K= kijj=1 , l + 1i=1 , l + 1

, M= mijj=1 , l + 1i=1 , l + 1

.

4 Finite Degree of Freedom Model of Nonlinear Vibra-tions

The nonlinear vibrations of the plate are expanded by using theobtained eigenmodes:

uz =i=1

L

qi t uz i r, 18a

u =i=1

L

i t u i r, 18b

Fig. 2 The steps of R-function derivation

Journal of Vibration and Acoustics OCTOBER 2010, Vol. 132 / 051001-3



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ur =i=1

L

i t ur i r, 18c

where q1 t , q2 t , . . . ,L t are the generalized coordinates. Usingthe data of the numerical analysis of linear vibrations, which aretreated in Sec. 7, the number of terms of expansion 18 is taken

L = 3.Equation 18 is substituted into the first two equations of the

system 1 and the inertial terms are annihilated as the eigenfre-quencies of vibrations in plane are much higher than the eigenfre-quencies of the bending vibrations. The Galerkin projections are

carried out. As a result, the system of linear algebraic equationswith respect to 1 , . . . ,3 is derived:

B 1

2

3

1

2

3

= l=13=13 qlqAl

1

Al 2

Al 3

Dl 1

Dl 2

Dl 3

19The parameters of the system 19 are presented at Appendix A.

Using the matrix R = rij j=1,6i= 1 , 6

, R =B1, the displacements of

the middle surface u, ur can be presented as

u =i=1

3

l=1

3

=1

3

l i

u i r, qlq

ur =i=1

3

l=1

3

=1

3

l i

ur i r, qlq 20

where

l j

==1

3

rj,Al

+ rj,+3Dl

l j

=

=1

3

rj+3,Al

+ rj+3,+3Dl , l,,j = 1,2,3

Equation 20 is substituted into the third equation of the system 1 and the Galerkin method is applied. As a result, the followingthree degree of freedom dynamical system is derived:

qk + pk2

qk =i=1

3

l=1

3

=1

3

Gil k

qiqlq, k= 1,2,3 21

The parameters Gil

kare presented in Appendix B.

The following dimensionless variables and parameters are used:

= p1t, i =qi

h, i = 1,3, Hil

k=

h2

p12

Gil k

, l, = 1,2,3

22

where 01. Then the dynamical system 21 can be presentedas

k + k2k =

i,l,

Hil k il, k= 1,2,3 23

5 Nonlinear Vibrations in the Case of Internal Reso-

nance 212

In this section the plate with one internal resonance is consid-ered:

21 = 2 + 24

where is detuning parameter. It is shown in Sec. 7 that forcertain plate parameters, the vibrations with internal resonance

24 are taken place.Following the multiple scale method 6 , the dynamics of the

system 23 is presented as

k = k,0 T0,T1, . . . + k,1 T0,T1, . . . + . . . 25

where T0 = and T1 =. Now the expansion 25 is substitutedinto the system 23 and the following equations are derived:

k,0 = Ak exp ikT0 + A

k exp ikT0 26

2k,1

T02

+ k2k,1 = 2ikAk exp ikT0 + 2Ak exp ikT0

j=1

3

Di k Aj

+ IN. E. T, k= 1,2,3 27

where IN.E. T is inessential for future analysis terms, Ai=AiA

i ;A

i is the complex conjugate value, and Di k

=Hiki

k+H

iik

k

+Hkii

k. There are no summands in Eq. 27 , which are determined

by the internal resonance 24 . Excluding secular terms from Eq. 27 , the following system of modulation equations with respect tocomplex variables is derived:

ikAk = Akfk A1,A2,A3 , k= 1,2,3 28

where

fk A1,A2,A3 =i=1

3

Di k Ai , k= 1,2,3

The following change of the variables is applied to the system

28 : Ak=0.5ak exp ik . As a result, the system of modulationequations is derived:

ak = 0

ak kk + fk a1, a2, a3 = 0 29

where

fk a1, a2, a3 =1

4i=1

3

Di k

ai2

The general coordinates of the dynamical system 23 and thevariables of the modulation equations 29 are connected as

k = ak cos kt+ k + O 30

The fixed points of the modulation equations correspond to theperiodic vibrations of the mechanical system 23 , which have thefollowing form:

k = ak cos kt + O 31

The fixed points are described by the system of nonlinear alge-braic equations, which are derived from Eq. 29 . Table 1 showsall types of fixed points. Properties of fixed points are representedin rows of table. Thus, nine types of fixed points are obtained. Thefirst column of Table 1 presents the properties of modulation vari-ables. The systems of nonlinear algebraic equations with respectto the fixed points are presented in the second column. The inde-pendent variables, which are set with some step for the backbonecurve calculations, are shown in the third column. The system ofnonlinear algebraic equations or analytical solutions of these sys-tems are shown on the fourth column. The frequencies of nonlin-ear vibrations are demonstrated on the fifth column.




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6 Nonlinear Vibrations Analysis in the Case of Inter-

nal Resonances 321 ; 32

In this section, the plate with the following two internal reso-nances is considered:

3 = 21 +

3 = 2 + 32

where and are the detuning parameters. As shown in Sec. 7,two internal resonances are observed for the certain parameters ofthe plate.

The multiple scales method is used to study the dynamics of thesystem 23 . Then the solution is presented in the form of Eq.

25 . Equation 25 is substituted into Eq. 23 and the coefficientsof 0 and 1 are equated. As a result, the following is derived:

k,0 = Ak exp ikT0 + A

k exp ikT0 , k= 1,3 33

2k,1

T02

+ k2k,1 + 2

2k,0

T0 T1

= 2j=1

3

Dj k

AkAjA

j exp ikT0 + GkA3AkA

2 exp iT0 k +

+ GkA

3AkA2 exp iT0 k + 3Hkkk k

Ak2Ak exp ikT0

+ PkA22A3 exp i 2 + FkA3

2A2 exp iT0 3 +

+ IN. E. T, k= 1,2,3 34

where

Gk = H32k k + H23k

k + Hk23 k + H2k3

k + H3k2 k + Hk32

k

Pk = H232 k

+ H322 k

+ H223 k

Fk = H323 k

+ H233 k

+ H332 k

, k= 1,2,3

Equation 33 is substituted into Eq. 34 and the secular terms areequated to zero. As a result, the system of modulation equations

with respect to the complex variables A1 ,A2 ,A3 is derived:

2i1A1

T1= 2A1

j=1

3

Dj 1 Aj + G1A3A2A1 exp iT1

+ G1A1A2A

3 exp iT1 + 3H111 1

A12A1

2i2A2

T1= 2A2

j=1

3

Dj 2 Aj + G2A3 A2 exp iT1 + G2A2

2A3 exp

i

T1 + P2A22A

3exp i

T1 + F2A3

2A2

exp i2

T1

+ 3H222 2 A2

2A2

2i3A3

T1= 2A3

j=1

3

Dj 3 Aj + G3A3

2A2 exp iT1 + G3A2 A3 exp

iT1 + P3A22A3 exp i2T1 + F3A3

2A2 exp iT1

+ 3H333 3 A3

2A3 35

The change of the variables Aj =0.5aj exp ij , j =1,2,3 is ap-

Table 1 The fixed points of the modulation equations

1 2 3 4 5

1 a10, a20, a30 fj a1 , a2 , a3 = 0,

j = 1,2, f3 a1 , a2 , a3 0

a3 D1 j

a12 +D2

ja2

2 = D3 j

a32, j = 1 , 2 j =j ; j = 1,2,

3 = 3

3f3 a1,a2,a3

2 a10, a20, a30 f2 a1 , a2 , a3 =0 , f3 a1 , a2 , a3 = 0,

f1 a1 , a2 , a3 0

a3 D1 j

a12 +D2

ja2

2 = D3 j

a32, j = 2 , 3

j =j ; j = 2,3,

1 = 1

1f1 a1,a2,a3

3 a1

0, a2

0, a3

0 f1

a1

, a2

, a3

=0 , f3

a1

, a2

, a3

= 0,

f2 a1 , a2 , a3 0

a3

D1

ja

1

2 +D2

ja

2

2 = D3

ja

3

2, j = 1 , 3 j =j ; j = 1,3,

2 = 2

2f2 a1,a2,a3

4 a1 =0 , a20, a30 f2 a1 , a2 , a3 = 0,

f3 a1 , a2 , a3 0

a3a2

2 = D3

2

D2 2 a3

21 = 0 ;2 =2,

3 = 3

3f3 a1,a2,a3

5 a1 =0 , a20, a30 f3 a1 , a2 , a3 = 0,

f2 a1 , a2 , a3 0

a3a2

2 = D3

3

D2 3 a3

21 = 0; 3 =3,

2 = 2

2f2 a1,a2,a3

6 a2 =0 , a10, a30 f1 a1 , a2 , a3 = 0,

f3 a1 , a2 , a3 0

a3a1

2 = D3

1

D1 1 a3

22 =0 ; 1 =1,

3 = 3

3f3 a1,a2,a3

7 a2 =0 , a10, a30 f1 a1 , a2 , a3 0,

f3 a1 , a2 , a3 = 0

a3a1

2 = D3

3

D1 3 a3

22 =0 ; 3 =3,

1 = 1

1f1

a1,a2,a3

8 a3 =0 , a10, a20 f1 a1 , a2 , a3 = 0,

f2 a1 , a2 , a3 0

a2a1

2 = D2

1

D1 1 a2

23 =0 ; 1 =1,

2 = 2

2f2 a1,a2,a3

9 a3 =0 , a10, a20 f1 a1 , a2 , a3 0,

f2 a1 , a2 , a3 = 0

a2a1

2 = D2

2

D1 2 a2

23 =0 ; 2 =2,

1 = 1

1f1 a1,a2,a3




6/11

plied to Eq. 35 and the system of six modulation equations withrespect to the variables aj ,j ; j =1,2,3 is obtained. One morechange of the variables

a1,a2, a3,1,2,3 = a1, a2, a3,

= 3 2 + T1 36

is applied to the system of modulation equations. As a result, thefollowing dynamical system is derived:

a1 = 0

a2 = P2

82a2

2a3 sin +

F2

82a3

2a2 sin 2

a3 =F3

83a3

2a2 sin

P3

83a2

2a3 sin 2

= +1

823 jjaj

2 +

823a2a3 cos +

F2

82a3

2

P3

83a2

2 cos 2 + 382

H222 2 a2

2 3

83H333

3 a32 37

where

= 3 2G2 + P2 2 2G3 + F3

j = 2Dj 2 3 2Dj

3 2, j = 1,2,3

The fixed points of the system of the modulation equations 37describe the vibrations 31 of the system 23 with the frequen-cies k:

k = k k, k= 1,2,3

811 = 2j=1

3

Dj 1

aj2 + 3H111

1a1

2 + 2G1a2a3 cos

822 = 2j=1

3

Dj 2

aj2

+ 2G2 + P2 a2a3 cos + F2a32

cos 2

+ 3H222 2 a

22

833 = 2j=1

3

Dj 3

aj2 + 2G3 + F3 a3a2 cos + P3a2

2 cos 2

+ 3H333 3

a32

The following groups of the fixed points exist in the system of theautonomous equations 37 :

a1 = 0, 1 = 0, a2 0, a3 0 case A

a1 = 0, 2 = , a2 0, a3 0 case B

a1 = 0, a2 = 0, a3 0, 0 case C

a

1 = 0, a

3 = 0, a

2 0, 0 case D

a1 0, a2 0, a3 0, 0 case E 38

Now, case A is considered. The parameter a2 is prescribed withsome step to analyze the backbone curve of vibrations. For every

value of a2, the parameter a3 is determined from the quadraticequation:

a32 3 + F23 32H333

3 + a2a3 + 823+ a22 2 P32

+ 33H222 2 = 0 39

These fixed points correspond to the vibrations of the system 23 ,

which is determined by the following relation: 1 = 0 ; 20 ; 3 =0. If the change of the variables a3a3 is applied tothe system of equations with respect to the fixed points case B ,then the system of equations for the fixed points case A is ob-tained.

Case C is treated. The variable a3 is preset with some step to

analyze the motions. For every value a3, the parameter is ob-tained from the equation

cos 2 =2

F2

3

3H333

3 3

23

8

a32

40

The fixed points case S correspond to the following vibrationsof the system 23 :

1 = 2 = 0, 3 = a3 cos 3t

3 = 3 2D3

3+ 3H333

3

83a3

2 41

Now case D is considered. The value a2 is preset with some step

to analyze the vibrations. For every value a2, the angle is ob-tained from the equation

cos 2 =3

P3

2

23+

3

2H222

2+

8

a22

42

The fixed points D correspond to the following vibrations of the

system 23 :1 = 3 = 0, 2 = a2 cos 2t

2 = 2 D2

2+ 3H222

2

82a2

2 43

The value a2 is preset with some step to analyze the vibrations,

which correspond to the fixed point case E . For every value a2,the parameters , a3 , a1 are obtained from the equations:

cos =P2a2

2F2a3

a3 = a2P2P3F

2F

3

a12 = 1

1 823 a22 2 + 33H222

2 + a32 32H333

3 3

a2a3 cos F23a32 P32a2

2 cos 2 44

The fixed points case E correspond to the system vibrations 31 .

7 Numerical Analysis of Linear Vibrations

The steel plate with the following parameters is considered:

R = 0.25 m, E= 2 1011 Pa, = 0.3,

= 7800 kg/m3, h = 5 103 m 45

The linear bending vibrations of the plate are analyzed. As follows

from the functional 6 , bending linear vibrations uz and longitu-dinal motions u

r

, u

are uncoupled. Therefore, these motions canbe analyzed numerically independently. The calculations of linearvibrations are performed by two methods. The first one is the

above-presented combination of R-function method andRayleighRitz approach and the second one is the finite elementcalculations by the software ANSYS. The effect of the cut-out sizeon the plate linear vibrations is analyzed. A parametric study by

altering the values of a1 and b1 are performed. The eigenfrequen-

cies of the plate for different values of the parameters a1 and b1are shown on Table 2. These results are obtained by finite element

method. The values of a1 and b1, and the first three vibrationeigenfrequencies are presented in the columns of Table 2. The

results of the calculations for the same values of a1 and b1, which




7/11

are obtained at different densities of finite element mesh, are pre-sented at the first and the second rows of Table 2. The results,which are obtained at rougher mesh, are presented at the secondrow. The values of the third eigenfrequency strongly depend onthe parameters of the cut-outs, as crest of the third eigenmode is

close to the cut-outs. The plates without cut-outs have p2 =p3. Ifthe area of cut-outs is increased, the distance of conjugate modeeigenfrequencies is increased too Table 2 .

The numerical calculations by means of R-function method arecarried out for the plate with the following parameters of cut-outs:

a1 =0.4 m and b1 =0.1 m. The R-function 2 for these plate pa-rameters is presented in Fig. 3. Figure 4 shows the first threeeigenmodes of the plate bending vibrations. Figure 4 a shows thefirst axisymmetric umbrella-shaped mode. Two conjugate modesare presented in Figs. 4 b and 4 c .

The results of the eigenfrequency calculations, which are per-

formed with the function 1 r, , are shown on Table 3. Thevalues of the first three eigenfrequencies obtained with differentdegrees of polynomials 13 and 14 are presented in Table 3.As follows from this table, the convergence of the results is ob-

served. For comparison the data obtained by the finite element

calculations are presented on the fifth row. The relative differences

of the results, which are presented at fourth and fifth rows, are

shown on the sixth row.

The results of eigenfrequency calculations with the function

2 r, are presented at Table 4. The eigenfrequencies of vibra-tions obtained with different degrees of polynomials 14 areshown at the first three rows of Table 4. The data obtained by

finite element method are shown at the fourth row of the table.

The relative differences of the results, which are shown at the

third and the fourth rows, are presented at the fifth row.

The results obtained using the function 2 is closer to the data

of finite element method, and then the results obtained with 1.

High accuracy of the results Table 4 is achieved at low degree of

polynomials 14 in comparison with the data of Table 3.Note that the second and the third eigenfrequencies for per-

fectly circular plate are multiple. However, if the shape of the

plate boundary is not perfectly circular, the multiple frequencies

are fibered. This phenomenon is observed in the considered sys-

tem. The second and the third eigenfrequencies of the plate are

shown at the second and the third row of Table 2, respectively. As

Table 2 The eigenfrequency, obtained by finite element method

a1 m

b1 m

p1 Hz

p2 Hz

p3 Hz

1 0.46 0.02 203.76 414.77 431.392 0.46 0.02 203.85 414.51 431.723 0.45 0.02 206.01 414.63 439.224 0.40 0.02 223.23 416.49 493.695 0.36 0.02 245.68 420.15 556.206 0.30 0.02 299.67 432.42 677.197 0.46 0.05 204.86 414.73 435.418 0.45 0.05 207.58 415.1 444.689 0.40 0.05 226.52 418.26 504.99

10 0.30 0.05 313.12 445.71 719.5511 0.46 0.1 206.26 415.12 440.7112 0.45 0.1 209.27 415.99 451.0913 0.40 0.1 231.51 422.66 522.7714 0.30 0.1 330.98 468.06 759.4915 0.45 0.2 210.10 416.60 454.3116 0.30 0.2 348.98 503.31 779.58

Fig. 3 Function 2




8/11

follows from Table 2, the multiple eigenfrequencies of the per-fectly circular plate and eigenfrequencies of the plate with twocut-outs differ significantly.

The eigenfrequencies of longitudinal and circumferential vibra-

tions of the plate ur and u are analyzed. The RayleighRitzmethod and approximation of the vibrations 13 are used forcalculations. The analysis of eigenfrequencies is performed withboth functions 1 and 2. Table 5 shows the results of the eigen-frequency calculations with function 1 for different degrees ofpolynomials 13 . The eigenfrequencies are presented in hertz.The eigenfrequencies of pair of conjugate eigenmodes are pre-sented in the second and third columns and axisymmetric mode isshown on the fourth column of Table 5. The eigenfrequencies

obtained with the function 2 are presented in Table 6. The eigen-frequencies calculated using the function 1 are presented in thistable too. The relative differences of the results obtained using

functions 1 and 2 are presented in the last row of Table 6. Westress that the degrees of polynomials to achieve the accuracy are

less, if the function 2 is used. The data obtained with the func-tion 2 are used in future analysis. These eigenmodes are denoted

by u

i r, and ur

i r, .

8 Numerical Analysis of Nonlinear Vibrations

The nonlinear vibrations, which are considered in Sec. 5 Table1 , are analyzed numerically. In cases 2 and 3 Table 1 , the sys-tem of algebraic equations with respect to the fixed point coordi-

nates for the plate parameters 45 has no solutions. AsD

3

3/D

2

30, there are no vibrations with the parameters 45

corresponding to case 5 Table 1 .Figure 5 shows the results of the backbone curve calculations.

The backbone curves for cases 1 and 6 Table 1 are shown onFig. 5 a . The numbers of backbone curves correspond to the dataof Table 1. Figures 5 b 5 d show the backbone curves corre-sponding to cases 79, respectively. Two backbone curves 6 and 1

Fig. 5 a describe the different types of motions a2 = 0, a10,a30 and a10, a20, a30, respectively.

The free vibrations Fig. 5 are connected to the plate vibra-tions. The case Fig. 5 a corresponds to the interaction ofumbrella-shaped mode and two conjugate modes. Other backbone

curves, which are presented on Fig. 5, describe the interaction ofumbrella-shaped mode and one of conjugate modes.

Now the nonlinear vibrations of the plate with the parameters

45 and a1 =0.46 m and b1 = 0.02 m are considered. As followsfrom Table 2, two internal resonances 32 occur for these param-eters.

Now the nonlinear vibrations of the system for the cases 38are analyzed. The vibrations corresponding to case D are not ob-

served, as the condition 1cos 21 is violated for the con-sidered plate parameters in Eq. 42 . The results of the calcula-tions for the rest cases of Eq. 38 are shown on Fig. 6. Thebackbone curve corresponding to cases A and C is shown in Fig.6 a by the solid and dotted lines, respectively. Figure 6 b showsthe vibrations corresponding to case E. Thus, the backbone curvescomplying with cases A, C, and E are hard and soft, respectively.

Now the free vibrations Fig. 6 are connected to the platevibrations. The case Fig. 6 a corresponds to the interaction oftwo conjugate vibration modes of the plate and case B corre-sponds to vibrations according to one of the conjugation modes.Thus, the backbone curve Fig. 6 a describes the transfer fromthe plate vibrations with two conjugate modes to the vibrations

Fig. 4 The eigenmodes of bending vibrations

Table 3 The eigenfrequencies, which are calculated using function 1r,

p1, Hz p2, Hz p3, Hz

1 M1,0 = 4 ; M1,2 =4 ; M1,4 =2 ;M1,1 =13; M1,3 =0 ; M2,1 = 1 273.37 456.08 616.94

2 M1,0 =7 ; M1,2 =7 ; M1,1 = 8;M1,3 =2 ; M2,1 = 2 258.59 450.05 605.64

3 M1,0 =10; M1,2 =10; M1,1 = 8 ;M1,3 =5 ; M2,1 = 5 254.76 444.97 596.95

4 M1,0 =13; M1,2 =13; M1,1 = 8 ;M1,3 =8 ; M2,1 = 8 254.24 444.18 595.73

5 FEM 231.51 422.66 522.776 Relative difference 8.9% 4.8% 12.2%

Table 4 The eigenfrequencies, which are calculated with function 2r,

p1, Hz p2, Hz p3, Hz

1 M1,0 =6 ; M1,2 = 0; M1,1 = 4 ; M1,3 = 0 ; M2,1 = 2 ; M2,3 = 0 256.76 449.41 566.28

2 M1,0 =6 ; M1,2 = 2; M1,1 = 4 ; M1,3 = 2 ; M2,1 = 3 ; M2,3 = 0 250.64 449.01 545.95

3 M1,0 =6 ; M1,2 = 4; M1,1 = 4 ; M1,3 = 4 ; M2,1 = 2 ; M2,3 = 2 249.99 448.33 543.074 FEM 231.51 422.66 522.775 Relative difference 7.3% 5.7% 3.7%

Table 5 Eigenfrequencies of longitudinal and circumferentialvibrations, obtained using 1

1 2, Hz 3, Hz 4, Hz

M3,1 =3 ; M4,1 =3 ; M5,1 = 3; M6,1 = 3;

M6,2 =7 ; M3,0 =7 ; M3,2 = 7 7233.33 7644.83 8593.35

M3,1 =5 ; M4,1 =5 ; M5,1 = 5; M6,1 = 5;

M6,2 =9 ; M3,0 =9 ; M3,2 = 9 7228.21 7641.44 8579.8

M3,1 =6 ; M4,1 =9 ; M5,1 = 9; M6,1 = 6;M6,2 =10; M3,0 =10; M3,2 = 10 7228.58 7643.78 8588.19




9/11

Fig. 5 Backbone curves for the case of internal resonance 212

Fig. 6 Backbone curves. a Cases A and C of Eq. 38. b Case E of Eq. 38.




10/11

with one of the two conjugate vibrations modes. The backbonecurve Fig. 6 b describes the vibrations with both umbrella-shaped mode and two conjugate modes.

9 Conclusions

The R-function method, which allows obtaining approximatelythe vibration modes and accurately satisfies the boundary condi-tions of the plate, is suggested to analyze the circular plate withtwo cut-outs in this paper. The nonlinear vibrations of the plate are

expanded into these modes containing R-functions. Both axisym-metric umbrella-shaped mode and two conjugate modes are usedin this expansion.

To discretize the system of the nonlinear partial differentialequations of the plate, the Galerkin method is applied to everyequations of the system. As a result, the nonlinear dynamical sys-

tem with three degrees of freedom is derived. This system de-scribes the nonlinear interaction between umbrella-shaped modeand two conjugate modes.

The following plate vibrations are analyzed:

1. the vibrations with umbrella-shaped mode and two conjuga-tion modes

2. the vibrations with umbrella-shaped mode and one of con-jugation modes

3. the vibrations according two conjugation modes4. the vibrations with one of conjugation modes

The motions according to umbrella-shaped mode are not ob-served. The backbone curves presented in this paper are both softand hard types.

Acknowledgment

This research is particularly supported by the Ukraine Found ofFundamental Research Grant No. F25.1/042 . The first authorexpresses thanks to Ph.D. student I.D. Breslavsky for his help.

Appendix A

B = bij j=1,6i=1,6

bl,i =

R i ur l

rdrd;bl,i+3 =

G i ur l

rdrd

bl+3,i =

A i r, u l

rdrd;bl+3,i+3 =

B i u l

rdrd, i

= 1,2,3, l = 1,2,3

R i = ur,rr i

+1

2r2ur, i

+1

rur,r i

1

r2ur i

G i =1 +

2ru,r i

3

2r2u, i

A i =3

2r2ur, i +

1 +

2rur,r i

B i = 1

r2u, i + 1

2ru,r i 1

2r2u i + 1

2u,rr i , i = 1,2,3

Al j

=

Ul r, ur j

rdrd

Ul= uz,r l

uz,rr +

1 +

2r3uz, l

uz,

1 +

2r2uz, l

uz,r

1

2r2uz,r l

uz,

1

2ruz,r l

uz,r

Dl j

=

Clu j

rdrd

Cl= 1

r3uz, l

uz,

1 +

2ruz,r l

uz,r

1

2r2uz,r l

uz,

1

2ruz,rr l

uz, j,l, = 1,2,3

Appendix B

Gilm k =

Eh

k 1 2

Filmuz k

rdrd

k = u

z

k 2

rdrd

Film =1

r ruz,r

i ,rk=1

3

ur,r klm k + r

ur klm

k+

ru, k lm

k

+1

ruz,r i

k=1

3

u,r k lm k + 1r

ur, k lm

k

1

ru k lm

k

+1

r2uz, i

k=1

3

1r

ur k lm

k+

1

ru, k lm

k+ ur,r

klm k

+1

2r ruz,r

i ,r uz,r l

uz,r m

+

r2uz, l

uz, m + 1

r2uz,r i

uz,r l

uz, m

+1

r2uz, i 1

2r2uz, l

uz, m +

2uz,r l

uz,r m

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