Nonlinear vibration of plates by the hierarchical finite element and continuation methods

International Journal of Mechanical Sciences 41 (1999) 437—459

Nonlinear vibration of plates by the hierarchicalfinite element and continuation methods

P. Ribeiro, M. PetytInstitute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, UK

Received 30 June 1998

Abstract

The hierarchical finite element (HFEM) and the harmonic balance methods are applied to analyse thegeometrically nonlinear vibration of thin, isotropic plates. The von Karman type of nonlinearstrain—displacement relationships are used. Symbolic computation is employed in the derivation of themodel. The equations of motion are solved by the Newton and continuation methods. Free and steady-stateforced vibration are analysed. The excitations considered are harmonic plane waves at both normal andgrazing incidence. The stability of the obtained solutions is investigated by studying the evolution ofperturbations to the solutions. The convergence properties of the HFEM and the influence of the middleplane in-plane displacements are discussed and results compared with published experimental and numericalresults. ( 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Hierarchical finite element method; Nonlinear; Plates; Vibration

Notation

a length of the plateb width of the plateE Young’s modulus[E] elastic matrixfj

out-of-plane shape functionsMFN vector of dynamic forcesgj

in-plane shape functionsh length of the finite elements or thickness of the platei J!1[I] identity matrix[J] Jacobian of MFNDJ D determinant of [J]

0020-7403/99/$— see front matter ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 9 8 ) 0 0 0 7 6 - 9

[K1b], [K1

p] linear bending and stretching stiffness matrices

[K2], [K3] and [K4] components of nonlinear stiffness matrix[Knl] nonlinear stiffness matrix[M

b], [M

p] bending and in-plane mass matrices

xMy"xMx

My

Mxyy couple resultants, moment per unit length (Nm/m)

xNw(x)y row matrix of out-of-plane shape functionsxNw(x)y , xNv(x)y row matrices of in-plane shape functionsP amplitude of applied excitationMPN vector of generalized external forces (amplitudes)MPM N vector of generalized external forces (time-dependent)pi

number of in-plane shape functionspo

number of out-of-plane shape functionsMq

pN in-plane generalized displacements

MqwN transverse generalized displacements

Re(cn) real part of the complex number cnt timex¹y"x¹

x¹

y¹

xyy stress resultants, force per unit length (N/m)

u in-plane displacement in the longitudinal directionv in-plane displacement in the width directionw transverse displacementw.!9

amplitude of vibration displacementMw

cN, Mw

sN coefficients of cosine terms and sine terms, respectively

Greek lettersa loss factorb damping factorMep

0N, Meb

0N linear membrane and bending strains

MepLN geometrically nonlinear membrane strain

j characteristic or Floquet exponentl Poisson’s ratioo mass densityp characteristic or Floquet multiplierx angular frequencyxj

natural linear frequencies[u2

j] diagonal matrix of squares of natural linear frequencies

) area of the platef viscous damping ratio

1. Introduction

Aircraft skin-panels, particularly those near the exhaust of jet engines, are subjected to highlevels of acoustic pressure. On the other hand, there is a frequent use of thin-walled structuralcomponents in the design of aircraft [1]. As a result, large amplitude, geometrically non-linearvibration of aircraft panels, with consequent reduced fatigue life, often occurs [2, 3].

Geometric nonlinearity in plates with fixed ends causes an increase of the resonance fre-quency with the amplitude of vibration. The nonlinear mode shape changes during the period of

438 P. Ribeiro, M. Petyt / International Journal of Mechanical Sciences 41 (1999) 437—459

vibration and is amplitude-dependent [4—6]. As this behaviour cannot be explained bylinear models, nonlinear mathematical models have been developed. For thin plates, thesemodels are usually based on von Karman large deflection theory [7], which provides a goodapproximation.

Commonly, the h-version of the finite element method (FEM) is used to carry out a discretiza-tion in the spatial domain, the time variation of the solution is expressed by means of a Fourierseries and the harmonic balance method (HBM) is applied. Lau et al. [8, 9] developed a triangu-lar element, including middle plane in-plane displacements, analysed free and forced vibrationand detected internal resonances. Mei and Decha-Umphai [10], presented a finite element methodfor plates subjected to harmonic loading, in which the strain energy was linearized. The equa-tions of motion were transformed into an eigenvalue problem by using a harmonic forcematrix. Rao et al. [11] applied an isoparametric element to study the free vibration of plates andstiffened plates; middle plane in-plane deformation and inertia were taken into account. Instead ofapplying the FEM, Benamar et al. [4] expanded the transverse displacement in the form of a finiteseries of functions, to derive the strain and kinetic energies. The middle plane in-plane displace-ments were neglected. The same authors [12], presented a set of experiments which were carriedout in order to investigate the dynamic response of fully clamped plates. It was verified that theslope of the mode shape near the clamps and the frequency of vibration increase with the amplitudeof vibration.

The solutions of systems of nonlinear equations are obtained by application of iterative methods.In each iteration, the nonlinear matrices are reformulated. Therefore, the number of degrees offreedom (DOF) has a substantial influence on the time needed to solve the model. Han and Petyt[5, 6, 13] applied the hierarchical finite element method (HFEM) to study the free vibration ofisotropic and laminated plates with geometrical non-linearity. The use of very high-order poly-nomial displacement shape functions made it possible to model the whole plate with only oneelement, to obtain accurate results with far fewer DOF than using the h-version of the finite elementmethod and, consequently, to analyse modes of order higher then one. The distortion of the modeshapes with amplitude was illustrated.

In nonlinear steady-state vibrations, the frequency response function (FRF) curve can havemulti-valued regions, linked by turning points [14]. In free vibration, particularly for largeamplitudes of vibration displacement, more than one mode may exist for a given frequency. TheNewton method or other simpler iterative methods usually applied, do not always provide a meansof describing these multi-valued regions [13]. On the other hand, with continuation methods [14]these regions can be automatically described. Continuation methods are computationally heavierthan other simpler methods, which makes the reduction of the number of DOF even moreimportant.

In this paper the HFEM method is used to construct the spatial model of thin, rectangu-lar, isotropic, fully clamped plates and applied to study their geometrically nonlinear freeand forced vibration. The harmonic balance method is applied to derive the equations ofmotion in the frequency domain. These equations are solved by a continuation method. Theexternal applied force is represented by waves at normal and grazing incidence. Although theformer is more commonly found in the literature, the latter is a more realistic representationof acoustic excitation. The stability of the obtained solutions is studied by applying Floquet’stheory [15].

P. Ribeiro, M. Petyt / International Journal of Mechanical Sciences 41 (1999) 437—459 439

2. Mathematical model

2.1. Equations of motion

For each element, the middle plane displacements (Fig. 1) are expressed in the form:

Gu0

v0

w0H"[N] G

qp

qwH (1)

[N]"CxNuy 0 0

0 xNuy 0

0 0 xNwyD , (2a)

xNuy " xg1(m)g

1(g) g

1(m)g

2(g) 2gp

i(m)gp

i(g)y , (2b)

xNwy " xf1(m) f

1(g) f

1(m) f

2(g) 2f

po(m) f

po(g)y (2c)

where piand p

oare the number of in-plane and out-of-plane shape functions used in the model; MgN

and M f N are the vectors of in- and out-of-plane shape functions; MqpN and Mq

wN are the generalized

in- and out-of-plane displacements and [N] the matrix of shape functions. The set of shapefunctions used is derived from Rodrigues’ form of Legendre polynomials [13, 16]. The in-planeshape functions are given by

gr"

INT(r@2)+n/0

(!1)n(2r!2n!5)!!2nn! (r!2n!1)!

mr~2n~1, r'2, (3)

and the out-of-plane shape functions are given by

fr"

INT(r@2)+n/0

(!1)n(2r!2n!7)!!2nn!(r!2n!1)!

mr~2n~1, r'4, (4)

Fig. 1 (a) Rectangular plate: x, y and z—global coordinate system; u0, v

0and w

0—middle-plane displacements; a, b and

h—plate dimensions. (b) m, g—local coordinate system.


where r!!"r(r!2)2(2 or 1), 0!!"(!1)!!"1 and INT(r/2) denotes the integer part of r/2. Theseshape functions satisfy fully clamped boundary conditions. To analyse plates with differentboundary conditions, other shape functions—for example third-order polynomials—would beadded to the model.

Since only one element is going to be used to model the whole plate,

m"2x/a, g"2y/b. (5)

The equations of motion, without damping, are derived by equating the sum of the virtual workof the inertia forces and of the elastic restoring forces to the virtual work of the external forces.Considering only transversely applied external forces, using von Karman nonlinear strain displace-ment relationships and neglecting shear deformation and rotatory inertia, one obtains

P)

(Mdep0NT#Mdep

LNT) [A] (Mep

0N#Mep

LN) d)#P

)Mdeb

0NT [D] Meb

0Nd)

#P)

oh(du0u(0#dv

0v(0#dw

0w(

0) d)"xdu

0dv

0dw

0y P

)[N]TG

0

0

PMd(x, y, t)Hd), (6)

[A]"Eh

(1!l2) C1 l 0

l 1 0

0 0 12(1!l) D , (7)

[D]"h2[A]. (8)

where Mep0N and Meb

0N are the linear membrane and bending strains; Mep

LN is the geometrically

nonlinear membrane strain; E, l, o denote Young’s modulus, Poisson’s ratio and density; andPMd(x, y, t) is the distributed applied force (N/m2).The strain—displacement relationships are

MepoN"C

u0,x

v0,y

u0,y

#v0,xD, Mep

LN"C

(w0,x

)2/2

(w0,y

)2/2

w0,x

w0,yD, Meb

oN"C

!w0,xx

!w0,yy

!2w0,xyD, (9)

where, x denotes differentiation with respect to x. Substituting Eqs. (9) into Eq. (6) and allowing thevirtual generalised displacements to be arbitrary gives:

CM

p0

0 MbD G

qp

qwH#AC

K1p

0

0 K1bD#C

0 K2

0 0 D#C0 0

K3 0D#C0 0

0 K4DBGqp

qwH"G

0PM H. (10)

[Mp] and [M

b] are the in-plane and bending inertia matrices; [K1

p] and [K1

b] the in-plane and

bending linear stiffness matrices; [K2], [K3] and [K4] the nonlinear stiffness matrices and MPM N isthe vector of generalised external forces.


With the introduction of mass proportional hysteretic damping—which depends on the damp-ing factors b

p, b—these equations become

CM

p0

0 MbDG

qp

qwH#

bp

uM

p0

0bu

MbGqRp

qRwH#AC

K1p

0

0 K1bD#C

0 K2

K3 K4DB Gqp

qwH"G

0PM H .

(11)

All submatrices in Eqn (11) are symmetric except [K2] and [K3], which are related by[K3]"2[K2]T [13].

Neglecting middle plane in-plane inertia and damping the following equations result fromEqn (11)

[Mb] Mq

wN#

bu

[Mb] MqR

wN#[K1

b] Mq

wN#[Knl] Mq

wN"MPM N, (12)

where [Knl ]"[K4]!2[K2]T [K1p]~1[K2]. The integrals involved in calculating the inertia and

stiffness matrices in Eqn (12) were evaluated using symbolic computation [17].The excitations considered will be of the form MPM N" MPNcos(ut). The steady-state response

Mqw(t)N will be expressed as:

Mqw(t)N"

n+i/1

MwciNcos(iut)#Mw

siNsin(iut) (13)

This equation is inserted into the equations of motion (12) and the HBM is applied. This methodcan be easily implemented in a program produced with the symbolic manipulator Maple [17, 18],using two commands. The first, trign, is defined as trign :"readlib(‘trig/reduce’):, where readlib isa Maple library function. ¹rign replaces all nonlinear trigonometric functions by linear ones; forexample cos3(ut) is replaced by 3

4cos(ut)#1

4cos(3ut). The second is the command coeff, which is

used to select the harmonics terms, that is the coefficients of cos(xt), sin(ut), cos(3ut), sin(3ut), etc.Most of the work will be carried out using one harmonic, then, for the sake of simplicity, the

equations of motion will be presented for this case only (i"1 in Eqn. (13)). If more harmonics areused, the derivation of the equations is similar and merely results in more algebraic terms. Theequations of motion obtained are of the following form:

MFN"A!u2CM

b0

0 MbD#C

0 bMb

!bMb

0 D#CK1

b0

0 K1bDBG

wc

wsH#G

F1

F2H!MPN"M0N

(14)

where the nonlinear terms are

MF1N"

2¹ P

T

0

[Knl] MqwN cos(ut) dt"(3

4[KN¸1]#1

4[KN¸3]) Mw

cN#1

4[KN¸2] Mw

sN, (15)

MF2N"

2¹ P

T

0

[Knl] MqwN sin(ut)dt"1

4[KN¸2] Mw

cN#(1

4[KN¸1]#3

4[KN¸3]) Mw

sN, (16)


[KN¸1]"[K4(MwcN, Mw

cN)]!2[K2(Mw

cN)]T[K1

p]~1[K2(Mw

cN)], (17)

[KN¸2]"[K4(MwcN, Mw

sN)]!2[K2(Mw

cN)]T[K1

p]~1[K2(Mw

sN)]*

!2[K2(MwsN)]T[K1

p]~1[K2(Mw

cN)], (18)

[KN¸3]"[K4(MwsN, Mw

sN)]!2[K2(Mw

sN)]T[K1

p]~1[K2(Mw

sN)]. (19)

[K4(MwcN, Mw

cN)] means that [K4] is a quadratic function of Mw

cN; [KN¸1], [KN¸2], [KN¸3],

[Mb] and [K1

b] are symmetric matrices. The vector of generalized displacements is defined by

MwN" Gw

cw

sH. (20)

The total number of degrees of freedom is n"2 ip2o, for a damped model, and n"ip2

o2, for an

undamped model, where i represents the number of harmonics.

2.2. Generalized external forces

When the excitation considered is an harmonic plane wave at normal incidence, the force perunit area at an arbitrary point is given by P

dcos(ut), where P

dis the pressure amplitude. In this case

the vector of generalized forces of equation (14) is

MPN"G P1

~1

PdMNw (x)Nd)

0 H. (21)

When the excitation considered is an acoustic harmonic plane wave at grazing incidence, theforce per unit area at an arbitrary point is given by P

gcos(ut!kx), with k"u/c and where P

gis

the pressure amplitude, k is the wave number and c is the speed of sound in air. In this case thevector of generalized forces is

MPN"GP)Pgcos(k(x cosa#y sina)) MNwNd)

P)Pgsin(k (x cosa#y sina)) MNwN d)H

"GP

gab4 P

1

~1P

1

~1

cosAkam2

cosa#kbg2

sinaB MNwNdgdm

Pgab

4 P1

~1P

1

~1

sinAkam2

cosa#kbg2

sinaB MNwNdg dm H (22)

*With this formulation, [K4] in [KN¸2] must be calculated using 2xNw,xyMw

cNxNw

,xyMw

sN.


where a is the angle between the direction of propagation of the wave and the positive x-axis. Thelast expression is simplified to

MPN"GP)Pgcos(kx) MNwNd)

P)Pgsin(kx) MNwNd)H"G

Pgab4 P

1

~1P

1

~1

cosAkam2 B MNwNdgdm

Pgab

4 P1

~1P

1

~1

sinAkam2 B MNwNdgdm H , (23)

if the wave is travelling in the direction of the x axis of the plate (a"0°).The integrations present in vector (22) were carried out symbolically [17]. As they were very

affected by numerical round-off errors, it was necessary to use a large number of digits.

3. The continuation method

In forced vibration, Newton’s method is used in the nonresonant region. For each frequency, thefirst approximation of MwN is the MwN from the last point of the FRF curve. By solving the system ofequations

[J] MdwN"!MFN, (24)

MdwN is obtained and MwN is corrected. [J] is the Jacobian of MFN defined by

[J]"LMFN/LMwN. (25)

The process is repeated until convergence is achieved. The frequency is then changed to anotherfixed value and the same method applied.

In forced vibration there are multiple solutions in the vicinity of resonance frequencies, which aredifficult to obtain with the Newton method alone. In these regions and in free vibration a continua-tion method is utilized [14, 19—21].

The continuation method is composed of two main loops. In the external loop a predictor to thesolution is defined. For that, the two last determined points of the curve—(MwN

i, u2

i) and (MwN

i~1,

u2i~1

)—are used. The prediction of MwNi`1

is thus obtained in the following way:

MwNi`1

"MwNi#*MwN

i`1, *MwN

i`1"(MwN

i!MwN

i~1)dwaux

wm

. (26)

dwaux is the amplitude of the first increment vector, *MwNi#1, and wm

is the amplitude of the vector(MwN

i!MwN

i~1). A prediction for u2

i`1must also be calculated. This results from the equation

u2i`1

"u2i`1

#*u20, (27)

*u20"$s/(MdvNT

1MdwN

1)1@2. (28)

s and MdwN1

will be defined afterwards. The sign in Eq. (28) is chosen following that of the previousincrement, unless the determinant of [J] has changed sign. In the last case a sign reversal is applied.To calculate MdwN

1in Eq. (28), the last known frequency of the FRF curve is used.


Now, the approximated solution must be corrected. This correction is carried out in an internalloop. Applying Newton method to Eq. (14),

[J]MdwN![M] MwNi`1

du2"!MFN. (29)

Note that, unlike Eq. (24), variations in the frequency are also considered. The fact that both thegeneralized displacements and the frequency are unknowns, allows one to pass the turning pointsof the curve. However, there is one extra unknown: the frequency of vibration. Consequently,another equation is needed. This is obtained by constraining the distance between the twosuccessive points of the FRF curve, the arc-length s, to a fixed value, by the following constraintequation

s2"E*MwNi`1

E2. (30)

From Eq. (29), one has

MdwN"du2MdwN1#MdwN

2, (31)

MdwN1

and MdwN2

result from the equations

[J]MdwN1"[M] MwN

i`1, (32)

[J]MdwN2"!MFN. (33)

Then the corrected value of MwN will be

MwNi`1

"MwNi#*MwN

i`1(34)

with

*MwNi`1

"(*MwNi`1

)13%7*064

#MdwN. (35)

Substituting *MwNi`1

from Eq. (35) into the constraint equation (30) gives the relation for du2 ,

a1(du2)2#a

2du2#a

3"0, (36)

where

a1"MdwNT

1MdwN

1, a

2"2((*MwN

i`1)13%7*064

#MdwN2)TMdwN

1,

a3"((*MwN

i`1)13%7*064

#MdwN2)T ((*MwN

i`1)13%7*064

#MdwN2)!s2. (37)

Equation (36) has two solutions. To avoid a return to the known part of the curve, the anglebetween the incremental amplitude vector of the previous iteration and the one of the presentiteration should be positive. If both angles are positive the appropriate root is the one that is closerto the linear solution of Eq (36).

The corrected value of the natural frequency is given by

*u2i`1

"(*u2i`1

)13%7*064

#du2, (38)

u2i`1

"u2i#*u2

i`1. (39)


The iterations are repeated until the inequalities

D du2/u2i`1

DSerror 1, (40)

EMwNi`1

!(MwNi`1

)13%7*064

E/EMwNi`1

ESerror 2, (41)

EMFNESerror 3, (42)

are satisfied.If the roots of Eq. (36) are complex, if too many iterations are necessary in order to achieve

convergence or if (ui`1

!ui) is greater than the value desired by the user, then the arc-length is

reduced and the process restarted.

4. Stability of the solutions

The study of the local stability of an equilibrium solution is concerned with what happens ifa system is slightly disturbed from it. A perturbation near an unstable equilibrium condition leadsto a departure from this condition and the inverse occurs near a stable equilibrium condition. Ina nonlinear system more than a single equilibrium solution may appear and the question ofstability is particularly important, because only stable equilibrium solutions exist actually, whereasan unstable one cannot be maintained [15, 22]. To investigate the local stability of the harmonicsolution a small disturbance is added to the steady-state solution

MqJ N"MqwN#Mdq

wN (43)

and its evolution is studied. If MdqwN dies out with time then Mq

wN is stable, if it grows then Mq

wN is

unstable.Inserting the disturbed solution (43) into eq. (12), expanding the nonlinear terms by means of

Taylor series around MqwN and ignoring terms of order higher than Mdq

wN, the following equations

are obtained:

[Mb] Mdq(

wN#

bu

[Mb] MdqR

wN#[K1

b] Mdq

wN#

L ([Knl] MqwN)

LMqwN

MdqwN"M0N. (44)

The coefficients L([Knl] MqwN)/LMq

wN are periodic functions of time. With symbolic manipulation,

using the Maple commands presented in Section 2, they can easily be expanded in a Fourier series.If Mq

wN is of the form Eq. (13) with i"1, then

L ([Knl] MqwN)

LMqwN

"[[p1]#[p

2] cos(2ut)#[p

3] sin(2ut)], (45)

[p1]"

1¹ P

T

0

LLMq

wN([Knl]Mq

wN) dt, (46)

[p2]"

2¹ P

T

0

LLMq

wN([Knl]Mq

wN) cos(2ut) dt, (47)

[p3]"

2¹ P

T

0

LLMq

wN([Knl]Mq

wN) sin(2ut) dt. (48)


Multiplying Eqs. (44) by the transpose of the modal matrix [B] and using modal coordinates MmN,one arrives at

Mdm1® N#bu

[I] Mdm5 N#[u2j] MdmN#[B]T([p

1]#[p

2] cos(2ut)#[p

3] sin(2ut))[B] MdmN"M0N,

(49)

where [u2j] is the diagonal matrix of linear natural frequencies. This is a system of extended

coupled Hill’s equations [22]. The first derivative term can be eliminated by introducing a newvector of variables

MdmN"e~(b@2u) *I +t Mdm1 N, (50)

obtaining, because matrix e~(b@u) *I +t commutes with any other matrix and is non-singular,

Mdm1® ]#A[u2j]!

14 A

buB

2[I]#[B]T ([p

1]#[p

2] cos(2ut)#[p

3] sin(2ut)) [B]BMdm1 N"M0N

(51)

Following Ref. [15], the solution of Eq. (51) is expressed in the form

Mdm1 N"ejt (Mb1N cos(ut)#Ma

1N sin(ut) ) (52)

which should allow one to determine, in a first approximation, the first-order simple unstableregion.

Inserting Eq. (52) into Eq. (51) and applying the HBM results in Ref. [23],

C0 [I]

![M0] ![M

1]D G

X

!H"j GX

!H, (53)

where

MXN"Gb1

a1H, (54)

[M1]"C

0 2u[I]

!2u[I] 0 D, (55)

[M0]"C

[B]T [J11

] [B]!Au2#A12

buB

2

B [I]#[u20j

]

[B]T[J21

] [B]

[B]T[J12

] [B]

[B]T [J22

] [B]!Au2#A12

buB

2

B [I]#[u20j

] D , (56)

[J11

]"LMF

1N

LMwcN, [J

12]"

LMF1N

LMwsN, [J

21]"

LMF2N

LMwcN, [J

22]"

LMF2N

LMwsN. (57)


The characteristic exponents j are the eigenvalues of the double size matrix in Eq. (53). If the realpart of j!b/(2u) is positive for any j, then the solution is unstable, otherwise it is stable.

For large DOF systems, the solution of Eq. (53) takes a long time, even if only the eigenvalueswith major real part are determined. A simplification of the stability study, in which the determina-tion of j is avoided for most points of the curve, is possible [19—21], as is explained in the followingparagraphs.

The Floquet multipliers p are related with the characteristic or Floquet exponents by

j"1¹

ln(p) (58)

Stability of MmM N is lost when EpE"1, which can happen in three ways [14]:

p"1 Q j"0, (59a)

p"!1 Q j is purely imaginary, (59b)

Im(p)O0Nj is a complex number. (59c)

Mechanism (59a) of losing stability is characteristic of turning points and will be the only oneconsidered here. Thus, due to transformation (50), the stability limit is defined by

j"12

bu

. (60)

Inserting Eq. (60) in Eq. (53) one arrives at

C[B]T[J

11] [B]!u2[I]#[u2

0j] [B]T[J

12] [B]#b

[B]T[J21

] [B]!b [B]T[J22

] [B]!u2[I]#[u20j

]DGb1

a1H"G

0

0H. (61a)

Eq. (61a) is equivalent to

C[B]T 0

0 [B]TD [J]C[B] 0

0 [B]D"G0

0H, (61b)

A non-trivial solution of Eq. (61b) exists if

det AC[B]T 0

0 [B]TD [J]C[B] 0

0 [B]DB"0 Q DB D4 DJ D"0 Q DJ D"0, (62)

Thus, in the stability limit, the determinant of the Jacobian of MFN, DJ D, is zero. DJ D is a polynomialin Mw

cN, Mw

sN and u; therefore, it is a continuous function in those coefficients. All the experimental

and numerical analysis of nonlinear vibration of plates, indicate that the shape of vibration, definedin this model by Mw

cN and Mw

sN, is a continuous function of the amplitude and the frequency of

vibration. Hence, DJ D varies in a continuous way through the FRF curve and if there is a change inits sign between two consecutive points of the FRF curve, then DJ D"0 for a particular pointbetween these two. In that particular point, the stability limit might have been crossed.


So, a complete study of the first-order solution’s stability is carried out in the following way:

1. Determination of the stability of the first solution by finding the characteristic exponents. Forlow amplitudes, in the nonresonant area, this is not necessary. In fact, using a perturbation method[22], it can be demonstrated that in these conditions the solution is always stable.

2. Calculation of DJ D, which is needed in the continuation method and, when the Newton methodis applied, can be easily calculated from [J]. If DJ D changes sign or if DJ D is approximately zero thencalculate the characteristic exponents to verify if the stability of the solution changed.*

In all the numerical cases studied the characteristic exponents were always either purely real orpurely imaginary, as in Ref. [19]. This suggests that stability of the present system can never be lostthrough processes (59b) or (59c) and would completely justify the former simplification of thestability study. However, no demonstration that the characteristic exponents assume that form wasfound in the literature. A discussion on the form of this exponents is presented in the appendix.

5. Applications

5.1. Plates analysed

The methods described were applied to two isotropic plates, with all edges immovable andclamped. Both are steel plates, with material properties E"21.0]1010 N/m2, l"0.3, o"7800kg/m3 and geometric properties as defined in Table 1. Plate 1 is a rectangular plate, which is usedfor comparison of the present solution with experimental [12, 26] and numerical [5] results for freevibration. Steady state forced vibration of Plate 1 is as well analysed. Plate 2 is a square plate whichis used for comparison of the present solution with numerical results for free [5, 11] and forced [10]vibration.

5.2. Convergence properties of the HBM and the HFEM

Convergence studies of the HBM and HFEM were carried out by analysing the variation of thesolution with the number of harmonics, number of out-of-plane shape functions and number ofin-plane shape functions. Plate 1 was used and free and forced vibration were analysed. Theexternal excitation is a harmonic plane wave at normal incidence, with an amplitude of 10 N/m2.Since the plate, the external excitation and the boundary conditions are symmetric with respect toboth axes x and y, only modes for which the transverse displacement is symmetric with respect tox and y (doubly symmetric) are excited. Thus, only symmetric out-of-plane shape functions need tobe included in the model. However, both symmetric and antisymmetric in-plane shape functions

*It would seem logical that if DJ D changes sign, then due to continuity (j!b/(2u)) changes sign as well; consequently,a determination of the characteristic exponents does not seem to be necessary [19—21]. However, one should rememberthat DJ D depends not only on j, but also on MwN and that what was demonstrated was (j!b/(2u))"0ND J D"0, nothingelse.


Table 1Geometric properties of plates 1 and 2

Plate a (mm) b (mm) h (mm)

1 486 322.9 1.22 500 500 2.0833

Fig. 2. Convergence with number of harmonics (pi"6, p

o"6). Plate 1. Displacements calculated at point x"0, y"0

and at t"0: (—) 1 harmonic, (e) 2 harmonics, (#) 3 harmonics.

must be used. In fact, the in-plane displacement components are anti-symmetric with respect to oneaxis and symmetric with respect to the other one, that is: u(x, y)"u (x, !y)"!u(!x, y) andv(x, y)"v (!x, y)"!v(x, !y).

The equations of motion are of the Duffing type. In free vibration there is no forcing term. Inforced vibration, the excitation is purely harmonic, with no constant term. Consequently, the timesolution will only contain odd harmonics [22]. In Fig. 2 the results obtained in free vibration withone, two and three harmonics are displayed. The approximation with one harmonic provides quitean accurate solution and will be used in the rest of this paper.

In Table 2 the convergence of the linear natural frequency values with the number, po, of

out-of-plane shape functions is shown. In Fig. 3 the backbone curves (which describe the relationbetween amplitude of vibration and frequency) and the frequency response function curves are


Table 2Convergence of linear natural frequencies(rad/s) of Plate 1 withpo. Doubly-symmetric modes.

po

2 3 4 5

u1

487.343 487.283 487.276 487.276u

21233.04 1197.30 1195.92 1195.91

u3

2378.31 2267.64 2263.73 2263.69

Fig. 3. Convergence with poin the vicinity of: (a) first mode, (b) second mode, (c) third mode. FRFs: h p

o"2, s p

o"3,

#po"4; Backbone curves: ——. Values calculated at (x, y)"(0, 0)

displayed for different values of poand for p

i"6. Both in the linear and in the nonlinear cases, the

results obtained with po"2 are accurate around the first mode; for the second and third modes

three out-of-plane shape functions provide a very reasonable approximation to the solution. Thismeans that with 9 DOF in free undamped vibration, and 18 DOF in forced vibration withdamping, the first three modes can be analysed.

In Fig. 4 the backbone curves and the FRF curves are displayed for different values of piand

po"3. For the amplitudes considered, p

i"4 provides results with a good accuracy. The exclusion

of the middle plane in-plane displacements (pi"0) increases the stiffness of the model. The

influence of the middle plane in-plane displacements is particularly visible around the first mode,due to the larger amplitudes of vibration attained.

The rate of convergence with po

and piis the same in free and forced vibration.

5.3. Free vibration analysis

In order to validate the HFEM model, results were compared with experimental and othertheoretical results.


Fig. 4. Convergence with piin the vicinity of: (a) first mode, (b) second mode, (c) third mode. FRFs: h p

i"0, # p

i"4,

e pi"5, s p

i"6; Backbone curves: ——. (x, y)"(0, 0).

Fig. 5. Comparison between the first resonance frequency predicted by the HFEM (po"3 and p

i"6); (—), and the

measured one, (#) [12, 26]. Plate 1. (x, y)"(0,0).

In Fig. 5 the first resonance frequencies obtained with the HFEM, for Plate 1, are compared withthe experimental ones [12, 26]. The agreement is quite good. With the exception of the lastexperimental point, the difference between the experimental and the HFEM values is almostindependent of the amplitude of vibration. Due to the finite stiffness of any clamping structure,exact clamped conditions are not possible in practice, which might account for most of thedifference between the HFEM and the experimental values. In Fig. 6, the backbone curves for thefirst three doubly-symmetric modes of vibration are shown for large amplitudes of vibration.Because the continuation method was utilised, no difficulties were encountered in achievingconvergence.


Fig. 6. Backbone curves of first three double-symmetric modes of vibration of plate 1. po"5, p

i"6. (x, y)"(0, 0).

Table 3Comparison of frequency ratios u/u

1of immovable fully clamped square isotropic plates

w.!9h

Ref. [5]

49 DOF

Ref [11];

425 DOF

HFEM

9 DOF (po"3, p

i"6)

HFEM

6 DOF (po"4, p

i"7)

w.!9

/h u/u1

w.!9

/h u/u1

0.2 1.0068 1.0095 0.2099 1.0079 0.21377 1.00820.6 1.0600 1.0825 0.6007 1.0632 0.60780 1.06471 1.1599 1.2149 1.0011 1.1670 1.0012 1.1668

For plate 2, comparison is made in Table 3 between the HFEM results and results from theliterature. The number of DOF of the HFEM model is much lower than the number of DOF of theh-version finite element method used in [11]. The results in this paper agree closely with the onesfrom Ref. [5], in which the HFEM was as well applied, but the equations of motion were solved byanother iterative procedure. The symmetries of the problem were used in the present work,substantially reducing the number of degrees of freedom.


Fig. 7. FRF in the vicinity of the first, (a), second, (b), and third, (c), double-symmetric modes due to excitation byharmonic plane wave at normal incidence with P"10 N/m2. (s) stable solutions, (#) unstable solutions. Plate 1.(x, y)"(0, 0).

5.4. Forced vibration analysis

Forced vibration due to harmonic plane waves was carried out. For plate 1, plane waves atnormal and grazing incidence with a"0° were considered. For similar excitation amplitudes, themaximum amplitudes of vibration caused by the two different excitations occurred around the firstmode and were similar—Figs. 7a and 8. However, the wave at grazing incidence excites modes thatare not doubly-symmetric, explaining the second resonance in Fig. 9, at u/u

1+1.54. Therefore, for

a wave at grazing incidence both symmetric and anti-symmetric shape functions were used.With the continuation method, stable and unstable solutions were obtained. The stability of the

solutions was studied by using the sign of the determinant as an indicator of a possible change inadjacent solution’s stability status, as explained in Section 4. This approach was confirmed bya complete determination of the characteristic exponents, which is not very time demanding whenusing the HFEM model, due to its small number of DOF.

The results for Plate 2 are shown in Table 4. Nine DOF were sufficient to achieve convergence.The calculated values are, with the exception of the value at w

.!9/h"#0.2, close to the values

calculated in Ref. [10], in which 54 DOF were necessary to achieve convergence.

6. Conclusions

In this work, the geometrically nonlinear free and forced vibration of isotropic plates has beenstudied by applying the hierarchical finite element method. The model was validated by compari-son with numerical and experimental results from the literature.

The HFEM model is characterized by its small number of DOF and of elements. This results invery significant time savings when constructing the model, iteratively solving the equations ofmotion and calculating the characteristic exponents by solving an eigenvalue problem. With thecontinuation method, convergence was achieved where other methods fail: multi-mode regions in


Fig. 8. FRF in the vicinity of the first mode due to excitation by harmonic plane wave at grazing incidence withPg"10 N/m2. (s) stable solutions, (#) unstable solutions. Plate 1. (x, y)"(0, 0), p

o"5, p

i"6.

Fig. 9. FRF due to excitation by harmonic plane wave at grazing incidence with Pg"10 N/m2: (——) stable solutions,

(— — — —) unstable solutions. (x, y)"(a/4, b/4), po"5, p

i"6.


Table 4Frequency ratio u/u

1of immovable fully clamped isotropic square plates under uniform

harmonic distributed force P0"0.2* (P

d"873.82 N/m2)

w.!9h

Finiteelement*

HFEMpo"3, p

i"6 (9 DOF)

HFEMp0"4, p

i"7 (16 DOF)

54 DOFw.!9

/h u/u1

w.!9

/h u/u1

$0.2 0.1180 #0.2001 0.2442 #0.2000 0.24321.4195 !0.2005 1.4399 !0.2072 1.4275

$0.6 0.8905 #0.5992 0.8962 #0.6008 0.89711.2083 !0.5997 1.2114 !0.59011 1.2120

$1 1.0700 #1.000 1.0800 1.0013 1.08031.2429 !1.001 1.2491 !0.9952 1.2475

*From Ref. [10]: P0"cP

d/oh2u2

1c"::/dxdy/::/2dx dy, /, normalised mode shape.

Pd, amplitude of external applied force (N/m2).

free vibration and unstable solutions in forced vibration. Since this method is computationallyheavier then other simpler iterative methods, the advantage of modelling with a small number ofDOF is increased.

The convergence with the number of shape functions used in the model is particularly quick. Thenumber of out-of-plane displacement shape functions necessary to obtain convergence in nonlinearanalysis is roughly the same as in linear analysis. For significant nonlinearities, more in-planeshape functions then out-of-plane shape functions are necessary. This is explained by the facts thatfor each out-of-plane half-wave (visualizing the first mode as one half-wave, the second astwo-halfwaves, etc.) two in-plane half-waves exist and that middle plane in-plane displacementsonly are important for large amplitudes. The same number of shape functions is needed to achieveconvergence in harmonic free and forced vibration.

It was shown that the middle plane in-plane displacements have a softening spring effect, in thesense that their exclusion makes the stiffness increase more with amplitude.

More modes are excited when the wave impinges tangentially than normally. The former isa more realistic representation of acoustic excitation and should be preferred to waves at normalincidence in the analysis of aircraft panels.

Symbolic manipulation was employed to accurately integrate products of shape functions ofhigh order, which define the mass and stiffness matrices, to derive the Jacobian matrix and washelpful in the separation of the coefficients of each harmonic when applying the HBM.

Acknowledgement

P. Ribeiro acknowledges the scholarship PRAXIS XXI/BD/3868/94 from the Science andTechnology Foundation, Portugal


Appendix: The characteristic exponents’ form

The characteristic exponents that determine the stability of the solutions are defined by thefollowing eigenvalue problem

(j2[I]#j[M1]#[M

0]) MXN"G

0

0H . (A1)

This type of eigenvalue problem is typical of a conservative gyroscopic system, because matrices[I] and [M

0] are symmetric and matrix [M

1] is skew-symmetric [24]. According to Ref. [19] the

resulting eigenvalues are purely real or purely imaginary. If the former is true, stability can only belost by process (59a) which would fully justify the simplification of the stability study sugested insection 4. In fact, for imaginary j

ReAj!12

buB"!

12

bu

, (A2)

and for real j

ReAj!12

buB"j!

12

bu

. (A3)

Consequently, if j is imaginary the solution is always stable, if j is real the stability limit isdefined by

ReAj!12

buB"0Qj"

12

bu

. (A4)

and conclusions similar to the ones made in Section 4 after Eq. (60) can be taken.However, no proof was found in the literature that when [I] and [M

0] are symmetric and [M

1]

is skew-symmetric, j is either purely imaginary or purely real. To verify the form of j, let usintroduce the notation

MXN"M½N#iMZN, MXM N"M½N!iMZN, (A5)

where the real vectors M½N and MZN are the real and imaginary parts of MXN. Introducing Eq. (A5)into Eq. (A1) one obtains the following quadratic equation

j2(x½y[I]M½N#xZy[I] MZN)#2j(x½y[M1] MZN)i#( x½y[M

0] M½N#xZy[M

0] MZN)"0.

(A6)

Defining

a"x½y[I] M½N#xZy [I] MZN, b"2x½y [M1] MZN and

c"x½y [M0] M½N#xZy [M

0] MZN, (A7)


where a, b and c are real numbers, the following equation is obtained:

j"!2bi$J!b2!4ac

2a. (A8)

If matrix [M0] is positive definite, it is easy to verify that j is purely imaginary, agreeing with Ref.

[25]. However, if [M0] is not positive definite, then !4ac can be positive and it can happen that

!b2!4ac'0. In this case j will be complex, with real and imaginary parts both different fromzero, as long as bO0; if b"0, j will be purely real.

In all numerical cases tried in this work (with a special attention to the cases in which !4ac waspositive) it was verified that b"0 and consequently j was either purely real or purely imaginary.The vectors M½N and MZN, and the matrix [M

1], are very sparse, which explains why it is likely to

happen that b"0, but it does not prove that it always happens. Consequently, it is not overriddenthat a loss of stability of the types (59b) or (59c) occurs.

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Nonlinear vibration of plates by the hierarchical finite element and continuation methods

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