ofan

ibei

orto,

4; ac

line 2

ite el

and

pre

excitation. The designation frequency response func-

tion could still be used to entitle a function that relates

the response of a non-linear structural system to a cer-

nated simply as response functions in non-linear sys-

tems.

Finite element methods are often used in non-linear

structural dynamics. In what the spatial model of ele-

mental structures like beams and plates is concerned, the

elements are many times derived following the so-called

82 (2* Tel.: +351-22-508-17-13; fax: +351-22-508-14-45.highly adequate to analyse the periodic, forced non-linear dynamics of beam and plate structures. An additional

purpose is to investigate the dierences in the predictions of non-linear motions when thin and thick, either beam

or plate theories are followed. To this ends response curves are derived, dening both stable and unstable solutions and

the characteristics of the motions are investigated using time plots, phase planes and Fourier spectra.

2004 Elsevier Ltd. All rights reserved.

Keywords: Non-linear vibrations; Periodic; Shooting; Thin; Thick; Beam; Plate

1. Introduction

A multi-degree of freedom linear system is charac-

terised by a set of frequency response functions (FRFs),

where each FRF relates the amplitude and phase of the

response of a determined degree of freedom to a har-

monic excitation in the same or other degree of freedom.

In linear systems, the response is linearly proportional to

the amplitude of the excitation, thus, each FRF does not

depend on the excitations amplitude and is a unique

property of the system. Therefore, the FRFs provide a

powerful way of understanding the dynamics of a linear

structure [1].

When large displacements arise and the system be-

comes geometrically non-linear, the steady-state re-

sponse is not proportional to the amplitude of the

tain harmonic excitation, in a similar way to linear

analysis. However, not only the response is a non-linear

function of the excitation amplitude, but also dierent

steady-state solutions are possible for the same fre-

quency and amplitude of excitation, depending on the

initial conditions. Other signicant points are, of course,

that in a non-linear system harmonic excitations can

cause periodic but non-harmonic, quasi-periodic or cha-

otic responses [2], and that the superposition principle

does not hold.

It is, nevertheless, still valuable to know how the non-

linear structure responds to harmonic excitations, and

dierent methods have been implemented to numerically

predict these responses. Due to the reasons explained

above, the relations between the steady-state response

and the harmonic excitations will henceforth be desig-Non-linear forced vibrationsby the nite element

P. R

IDMEC/DEMEGI, Faculty of Engineering, University of P

Received 10 March 200

Available on

Abstract

The shooting, Newton and p-version, hierarchical nlinear periodic vibrations of elastic and isotropic, beams

theories are followed. One of the main goals of the work

Computers and StructuresE-mail address: pmleal@fe.up.pt (P. Ribeiro).

0045-7949/$ - see front matter 2004 Elsevier Ltd. All rights reservdoi:10.1016/j.compstruc.2004.03.037thin/thick beams and platesd shooting methods

ro *

Rua Doutor Roberto Frias, s/n, 4200-465 Porto, Portugal

cepted 12 March 2004

0 May 2004

ement methods are applied to study geometrically non-

plates. Thin and thick or rst-order shear deformation

sented is to demonstrate that the methods suggested are

004) 14131423

www.elsevier.com/locate/compstructhin and thick, or rst-order shear deformation (FOSD),

ed.

hy w;x; 4

To employ some discretisation procedure, like the

functionsthe shape functions fNx; ygTand of time

qwtq t>>>> >>>>

: 6

1414 P. Ribeiro / Computers and Structures 82 (2004) 14131423theories [3]. One nds many discussions of the domain

of validity of these theories published, but they are most

usually in the linear vibrations realm, although some

works are also in non-linear free vibrations, as for

example reference [4].

When periodic solutions are sought, the nite ele-

ment equations of motion can be solved in the frequency

domain by the harmonic balance method [2] (HBM) or

by the incremental harmonic balance method [5], which

is similar to the HBM plus a NewtonRaphson proce-

dure [6]. In the HBM the time solution is written in the

form of a truncated Fourier series, and the coecients of

the same harmonic components are compared. In this

way, non-linear algebraic equations in the space vari-

ables and frequency are obtained. For a damped system

with n degrees of freedom, the harmonic balance methodrequires the solution of 2nk or 2nk 1 non-linear alge-braic equations, where k is the number of harmonicsused. If multi-modal and multi-frequency motions

occur, then the number of equations to solve can become

quite large [4,710], and it is cumbersome to derive the

frequency domain equations of motion. Moreover, if

the correct number and type of harmonics is not used, the

harmonic balance solution leads to incorrect data.

The numerical integration of the equations of motion

in the time domain using methods like nite dierences

or Newmarks method [11] is quite popular amongst

nite element users. Unlike the HBM, time domain

numerical integration schemes allow one to analyse non-

periodic motions. However, convergence to a steady-

state solution may take a very long time, particularly if

damping is small. Moreover, to nd periodic solutions

by numerical integration, one chooses an initial condi-

tion and integrates the system of equations until con-

vergence is achieved. With this approach it may be

dicult to ascertain that a steady-state condition was

reached, and which condition was reached when multi-

ple solutions exist. Thus, these methods are not per se

recommendable to construct the response curves.

The shooting method [1215] is a time domain tech-

nique of great potential to analyse non-linear periodic

motions and to dene response curves. Unlike in the

HBM, the original number of equations to be solved does

not depend on the number of harmonics present in the

motions Fourier spectrum. Naturally, as a time domain

procedure, the best time step to use depends on the mo-

tions Fourier spectrum, but this time step can be easily,

or automatically, changed. Moreover, the shooting

method gives as a by-product the monodromy matrix,

the eigenvalues of which dene the solutions stability.

In this paper, an algorithm based on the shooting

method is applied to solve nite element equations of

motion and study geometrically non-linear vibrations of

beams and plates. It is intended to demonstrate that

these methods constitute a valuable tool to study peri-odic motions of structures, and therefore to dene thehy

qhx t>>: >>;

The superscripts or subscripts u, v, w, hx and hy indicate,respectively, if the vectors or matrices are connected

with the membrane displacements along x or y, with thedependent generalised displacements fqtg:u0x; y; tv0x; y; tw0x; y; th0y x; y; th0xx; y; t

8>>>>>>>>>>>:

9>>>>>>=>>>>>>;

fNux; ygT 0 0 0 00 fNux; ygT 0 0 00 0 fNwx; ygT 0 00 0 0 fN hy x; ygT 00 0 0 0 fN hx x; ygT

26666664

37777775

qutqvt

8>>>>>>>>>>>=nite element or the Galerkin method, one considers

that the actual displacements are functions of spatialhx w0;y : 5In the case of a beam, only Eqs. (1), (3) and (4) apply,

dropping the y argument.response curves of elastic continua. Elastic and isotro-

pic, thin and thick, beams and plates are analysed. An-

other primary goal of the paper is to examine the

domain of validity of the thin beam and plate theories

in periodic, forced non-linear vibrations.

2. Equations of motion, shooting and Newton methods

Following the rst-order shear deformation theory

(FOSDT), as in [4], the displacement components along

the x and y directions, u, v, are functions of the mid-surface membrane translations u0, v0 and of the rota-tions of the normal to the midsurface about the x- andy-axis. The latter are denoted by h0x and h

0y . Still accord-

ing to the FOSDT, the transverse displacement w doesnot depend on the coordinate z, on the axis normal tothe plane xy. Hence, the displacements are given by:

ux; y; z; t u0x; y; t zh0yx; y; t; 1

vx; y; z; t v0x; y; t zh0xx; y; t; 2

wx; y; z; t w0x; y; t: 3For lower order modes of thin plates, the following well

known assumptions can be implemented:0displacement along z or with one of the rotations. For

The methods of solution suggested in this paper

are considered. In Eq. (10), fPg is the vector of ampli-

order dierential equations of motion (7). First this is

P. Ribeiro / Computers and Structures 82 (2004) 14131423 1415thin structures it is possible to discard the terms con-

nected with hx and hy in Eq. (6).The equations of motion can be derived by the

principle of virtual work, Hamiltons principle, or other

[3]. For geometrically non-linear problems and if

damping is included, they are of the form

M fqtg Cf _qtg Kfqtg KNLfqtg fqtg fP tg; 7

where M is the mass matrix, C the damping matrix,K the linear stiness matrix and [KNL] the non-linearstiness matrix. The latter matrix depends on the gen-

eralised transverse displacements fqtg. The dot over avariable indicates dierentiation with respect to time.

Considering stiness proportional damping, the equa-

tions of motion of thin structures are of the form [9,10]:

Mu 0 00 Mv 00 0 Mw

24

35 qutqvt

qwt

8>>>>>:

9>>>>>=>>>>>;

K1u 0 0 0 00 K1v 0 0 00 0 K1c K1c K1c0 0 K1c K1b K1c K1b0 0 K1c K1b K1b K1c

266664

377775

qutqvtqwtqhy tqhxt

8>>>>>:

9>>>=>>>;

0 0 K2 0 00 0 K2 0 0K3 K3 K4 0 00 0 0 0 00 0 0 0 0

266664

377775

qutqvtqwtqhy tqhxt

8>>>>>:

9>>>=>>>;

PutPvtPwtMhy t

8>>>>>:

9>>>=>>>;

; 9Mhxttransformed into the following system of 2n rst-orderdierential equations:

0 M M aK

_yt_qt

M 0

0 KNL

ytqt

0P t

11

the T -periodic solutions of which respect the condition

y0q0

yT

qT

f0g: 12

Thus, one needs to solve a two-point boundary value

problem. Although not an essential step, the period can

be normalised to unity, by means of transformation

s t=T , so that the integration time interval is [0,1].The system of dierential equations (11) then becomes

0 M M aK

_ys_qs

T 0

P

M 00 KNL

ysqs

:

13

The dot now indicates dierentiation with respect to s.Consider the 2n phase space vector fX sg

fys; qsg. Assigning an initial condition fsg to fX sgand rewriting (13) in a simplied manner, the followingtudes of the external forces, with period Te 2p=x,where x is the frequency of excitation. The shootingmethod [1215] is applied to nd T -periodic responses,where T is a multiple of Te, of the system of n second-should be applicable to equations of motion obtained

using any spatial discretisation procedure; but the p-version, hierarchical nite element method, which has

the major advantage of requiring fewer degrees of free-

dom than the h-version of the FEM [4,9,10], will be usedin the numerical applications.

The vector of generalised external forces, fPtg, isan explicit function of time; thus, the systems analysed

are non-autonomous. Harmonic excitations of the

following form:

fPtg fPg cosxt 10where MRy and MRx are due to the rotatory inertia,K1c is a linear stiness matrix due to shear, and fMhygand fMhxg are externally applied moments. The othersymbols are common with the ones of Eq. (8).initial value problem is obtained:

24

1416 P. Ribeiro / Computers and Structures 82 (2004) 14131423fX 0g fsg;f _X sg T M 1ff g KfX sg;fXg; fsg 2 R2n: 14One is seeking for fsg, such that the residual vector

frfsg;xg, dened asfrfsg;xg fX fsg;x; 0g fX fsg;x; 1g 15is close to zero.

frfsg;xg f0g: 16The solution of (14) and (16) is the trajectory that

starts from the vector of initial conditions fsg at s 0and arrives at the same location at s 1.

To apply the shooting algorithm an initial value is

required for fsg0. For the rst two points, the initialconditions are dened as

fsg0 0sq

0; 17

where fsqg0 is the solution of the linear problemK x2M fsqg0 fPg: 18The other initial guesses are dened by using former

periodic solutions, represented by the subscripts i andi 1:

fsg0i1 fsg0i Dfsg0i1;Dfsg0i1 fsg0i fsg0i1d: 19

The parameter d in Eq. (19) denes the increment infrequency

xi1 xi dxi xi1: 20

It will be shown later that this simple secant predictor

allows one to describe fairly complex response curves,

and to nd stable and unstable solutions.

After dening a predictor, Newtons method is ap-

plied to nd the solution of (16). Thus, fsg is correctedby using the following equation:

fsgv1 fsgv Dfsgv; 21

until convergence is achieved. The vector Dfsgv solvesthe linear system of equations

Jsv;xDfsgv frfsgv;xg: 22

The matrix J is the Jacobian of frfsg;xg with respectto fsg, which may be written as

Jfsgv;x ofrgofsg fsg

v;x

I W fsgv;x; 1: 23Thus, matrix W fsg;x; 1 is the outcome of the fol-lowing initial value problem:

_W AW ; W fsg; k; 0 I ; 25

where matrix A is

As; fsg; k oofXg T M

1ff g KfX sg: 26

The vector fXg and the matrix A are evaluated alongthe periodic solution. In this work matrix A was ana-lytically computed, with the help of a symbolic manip-

ulator, and then stored in Fortran format.

Integrating (14) and (25) one calculates W fsg;x; 1and fX fsg;x; 1g. A fourth order RungeKutta orother method [16] may be used with this purpose. Then,

the system of equations (22) is solved and fsg is updated.When Dfsg is suciently small and frg is close to zero,convergence to a periodic solution has been achieved. At

this stage, one can dene a new predictor using Eqs. (19)

and (20) and proceed to the following point on the

curve.

The monodromy matrix W fsg; k; 1 is a by-productof the shooting technique. As discussed, for example, by

Nayfeh and Balachandram [12] and by Seydel [13], the

complex eigenvalues of this matrix are the Floquet

multipliers and if a Floquet multiplier has norm greater

than one, then the solution is unstable. It is recalled that

the Floquet multipliers can leave the unit circle in three

ways. First through +1, resulting in a transcritical, a

symmetry-breaking or in a cyclic fold bifurcation. Sec-

ond, through )1, resulting in a period-doubling bifur-cation. Finally, two complex conjugate Floquet

multipliers can leave the unit circle, resulting in a sec-

ondary Hopf bifurcation.

3. Numerical applications and discussions

3.1. Beams

A clampedclamped beam, Fig. 1, with properties

given in Tables 13 is studied rst, following thin beam

theory. The meaning of the symbols used in those tables

is the following: hthickness, bwidth, Llength, Xarea of the transverse cross section, Isecond momentMatrix W ofX fsg;x;sgofsg must be evaluated at s 1.Dierentiating both sides of Eq. (14) and the vector of

initial conditions with respect to fsg, we haveoos

ofXgofsg

oofXg T M

1ff g Kfusg ofXgofsg ;

ofX 0gofsg I ; fXg; fsg 2 R

2n:

lised; thus, in the domain of validity of elastic thin

beams theory, the model is an accurate one [9].

In Fig. 2, in order to demonstrate the validity of the

procedure followed, the computed maximum displace-

ment amplitudes of the beams middle point are com-

pared with the ones experimentally measured [17]. A

point harmonic force of amplitude 0.134 N was applied

at the middle of the beam. Five transverse shape func-

tions were used in the p-version nite element. In thegure, the vertical axis gives the values of the maximum

displacement attained during a period of vibration, w,divided by the thickness h and the horizontal axis givesthe a dimensional frequency. The agreement between

the computed and experimental values is fairly good.

In the analysis of this beam, it was veried that the

maximum amplitude of vibration, where a turning point

occurs, depends heavily on the loss factor considered.

The results shown on Fig. 2 are for an undamped beam.

In order to ascertain the dierences in the dynamic

behaviour that occur due to a change in the vibration

amplitude, Fig. 3a, displays the response of the same

beam to a transverse point harmonic force with ampli-

tude 2 N, again applied at the middle of the beam.

Twelve longitudinal and eight transverse shape functions

Table 2

Thin beams material properties

Material E (N/m2) q (kg/m3) m

F

x

y

Fig. 1. Clampedclamped beam and external excitation.

Table 1

Thin beams geometric properties

h (mm) b (mm) L (mm) X (m2) I 1=12bh3(m4)

2 20 406 4 105 1.333(3) 101

P. Ribeiro / Computers and Structures 82 (2004) 14131423 1417Aluminium 7075-T6 7.172 1010 2800 0.33of area of the cross section, EYoungs modulus, qmass density and mPoissons ratio. A large number ofshape functions is employed in the HFEM model uti-

Table 3

Beams linear natural frequencies (rad/s)

Thickness Theory x1 x2

h 2L=406 Thin 396.605 1093.26h L=20 FOSDT 3960.63 10698.3

00.20.40.60.81

1.21.41.61.82

0.5 0.7 0.9 1.1

hw

Fig. 2. Transverse displacement of the beam at xwere now employed and damping is again neglected.

Unstable solutions were now found, and super-har-

monic resonances are more visible. In fact, two short

peaks due to super-harmonic resonances of order 3 and

5 appear before the main resonance; the rst peak close

to x=x1 0:2 and the second near 0.33. They are easilyvisible in logarithmic scale, which is not shown for the

x3 x4 x5

2143.26 3543.31 5375.22

20441.8 32990.9 39162.0

1.3 1.5 1.71

0: (s) numerical and (j) experimental.

sake of conciseness. Fig. 3b and c shows the time do-

main responses along three cycles of vibration and the

respective frequency spectrum of solutions close to

0:3782x1 . From these gures it becomes evident that thethird harmonic is present in the motions.

Fig. 3d shows the time series and the its coecients of

Fourier, when the excitation frequency is 970 rad/s and

the vibration amplitude is around three times the

thickness of the beam. The rst, third, fth, seventh and

ninth harmonics are now present in the motions Fourier

spectrum. It is important to notice that the seventh

harmonic is quite signicant and is greater than the third

and the fth. However, since the fth harmonic is rather

small, one would be tempted to erroneously neglect the

seventh if the HBM was employed instead of the

shooting method.

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4

hw

1

a

-0.50

-0.40

-0.30

-0.20

-0.10

0.0

0.10

0.20

0.30

0.40

0.50

t t+T t+2T t+3T

hw

b

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

t t+T t+2T t+3T

hw

c

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8 9 10

2.00

3.00

4.00

hw

d

0

0

0

0

2.00

2.50

3.00

0 1

Adimensional amplitude

Adimensional amplitude

Adimensional amplitude

Harmonics

Harmonics

tion o

t (b) 0

1418 P. Ribeiro / Computers and Structures 82 (2004) 14131423-4.00

-3.00

-2.00

-1.00

0.00

1.00

t t+T t+2T t+3T

0.0

0.5

1.0

1.5

Fig. 3. (a) Maximum transverse displacement at x 0 in funcsolutions. Displacement in function of time and Fourier series a(970 rad/s).2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Harmonics

f excitation frequency: (d) stable solutions and (r) unstable

:3530x1 (140 rad/s), (c) 0:3782x1 (150 rad/s) and (d) 2:446x1

8B

P. Ribeiro / Computers and Structures 82 (2004) 14131423 14190

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.

a

w/h

w/h

0

0.2

0.4

0.6

0.8

1

1.2

b

Super-harmonic

Super-harmonic The results from thin and thick beam theories are

compared in Fig. 4, where a beam with similar prop-

erties to the one studied before, except the thickness that

is now h L=20, is analysed. The natural frequencies ofthis beam, were computed using FOSDT and are given

in Table 3. A 2000 N point force is applied transversely

at the middle of the beam. Following any of the theo-

ries, one nds a typical main resonance, of the rst

mode, and a, also typical, super-harmonic resonance

again of the rst mode. Not so commonly found, are the

turning point that occurs in the super-harmonic branch

and the branch of solutions that bifurcates from the

main branch. The latter was found by reducing d inEq. (19).

The bifurcation from the main branch is a conse-

quence of a 1:5 internal resonance, where the rst and

third modes become coupled. The presence of these

modes was veried by plotting the shapes of the beam at

dierent instants along the vibration period (not shown).

It is curious to realise that, as one proceeds in the sec-

ondary branch of solutions, at a certain stage the max-

imum amplitude displacement at x 0 barely changes,as if it were locked. However, the smaller amplitude

waves connected to the fth harmonic, increase steadily.

Fig. 5 shows some time and phase plots of motions

before and after the bifurcation.

0.2 0.4 0.6 0.8

Fig. 4. Maximum transverse displacement at x 0 in function of excit1 1.2 1.4

Bifurcation secondary branch

ifurcation

secondary branch

/ 1

/ 1The turning point in the super-harmonic branch is

also due to due to an internal resonance and coupling

between modes, again the rst and the third mode (Fig.

6). The number of loops in the phase planeFig. 6bis

quite large, because the rst mode is linked with a super-

harmonic of order 3 and the third mode is associated

with a super-harmonic of order 15 (that gives a 1:5

internal resonance between super-harmonics). The

shooting and Newton methods accommodated this

reach dynamics rather easily, and, in the rst place, the

p-version nite element model allowed one to accuratelyconsider large order modes.

The thin beam theory predicts the main branch of

this L=h 20 beam quite reasonably, although weshould point out that the rst linear natural frequency is

3960.63 rad/s according to the thick beam theory whilst

the thin beam theory gives 4025.54 rad/s (1.64% relative

error).

Quantitatively, a quite larger dierence stems from

applying one or the other theory in what the bifurcation

and the turning points are concerned. This is natural,

since those points are due to modal interaction with

higher order modes, and, as is well known, higher order

theories provide better predictions of higher order

modes. For example the thick beam theory indicates

that the third linear natural frequency is 20441.8 rad/s,

1 1.2 1.4

ation frequency: (a) thin beam theory and (b) thick beam theory.

T-1

1

1420 P. Ribeiro / Computers and Structures 82 (2004) 14131423-0.5

0

0.5

1

T

w/ h=1. 11362-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8 w/ h =1.1160whilst the thin beam theory gives 21754.1 rad/s (relative

error: 6.4%).

Regarding the numerical procedure, because the thin

beam theory results in a lighter model, its computational

cost is much lower, not only due to the reduced number

of degrees of freedom, but also because the time step

employed in the RungeKutta method may be larger

than in the thick beam model.

3.2. Plates

The properties of the rst plate analyseda qua-

drangular steel plate with all edges immovable and

clamped (Plate 1)are given in Table 4. The letter adesignates the plates width. The rst ve linear natural

frequencies of the plate are given in Table 5. A p-version,hierarchical nite element with three out-of-plane and

six in-plane shape functions was employed to carry out

the computations, as in [10].

-1

-0.8-0.6-0.4-0.20

0.20.40.60.8

T

w/ h

-0.7

1=1.2427

Fig. 5. Transverse displacement at x 0 for points of main (x 1:11of the response curve portrayed in Fig. 4, thin beam theory.-5

-3

-1

1

3

5

-0.75 -0.25 0.25 0.75

h2

-6-4-202468

-0.5 0 0.5 1

w/ h

w/

w

h

.

h2 w

.Fig. 7 displays the plates frequency response to an

uniform harmonic, distributed force of 4000 N/m2. Due

to damping, which is taken into account by means of a

loss factor equal to 0.001, the maximum vibration

amplitude is less than 1.6 h. The displacement of the

middle point of the plate along one cycle and the

respective Fourier spectrum are shown as well, for some

frequencies of excitation. As with the beams, higher

harmonics appear.

In order to investigate the inuence of the rotatory

inertia and of the shear deformation, fully clamped

square steel plates with 500 mm width and two dier-

entthicknesses h 5 mm (Plate 2) and h 50 mm(Plate 3)were investigated. The thick plate p-versionelement employed had 3 out-of-plane, 5 membrane and

5 rotational shape functions (element with 59 DOF,

after condensation, i.e. 118 phase space co-ordinates).

Obviously, in the thin plate model there are no rota-

tional generalised coordinates, therefore the number of

DOF is only 9 (18 phase space coordinates).

-8

-15

-10

-5

0

5

10

15

5 -0.25 0.25 0.75

w/ h

h2 w

.

60x1) and of secondary branch (x 1:1362x1 and 1:2427x1)

h0.10.15

-0

w(x)

P. Ribeiro / Computers and Structures 82 (2004) 14131423 1421-8

-6

-4

-2

0

2

4

6

8

-0.4 -0.2 0 0.2 0.4

w/

0.15

0.2

0.25

0.3

0.35

a) / = 0.3416

c)w(x) = 0.3416

h2 w

.

1

/ 1Figs. 8 and 9 show the response curves of the Plates 2

and 3, due to distributed excitation forces with the

amplitudes indicated in the gures legends. Plate 2 is

thin (h=a 0:01). Therefore, the dierence between thevalues of the rst linear frequency calculated using thin

plate (1130.2187 rad/s) and thick plate theory (1129.0358

rad/s) is very small (0.1%). The non-linear response does

not dier very much as well, except when higher order

modes are excited, which is not the case of the results

given in Fig. 8.

Since Plate 3 is already a thick plate (h=a 0:1), thedierence between the values of the rst linear frequency

calculated when neglecting transverse shear and rotatory

inertia (11302.187 rad/s) and when they are considered

(10307.041 rad/s) is signicant (9.7%). For amplitudes of

vibration larger then approximately 0.25 the plates

thickness, the non-linear response is also signicantly

0

0.05

0.1

2L

2Lx

Fig. 6. Phase plots (a,b) and deformed shapes (c,d) for excitation freq

thick beam theory.

Table 4

Geometric and material properties of Plate 1

a (mm) h (mm) Material E

500 2.0833 Steel 2

Table 5

Linear natural frequencies of Plate 1 (rad/s)

x1 x2 x3

470.866 960.588 960.588-0.050

0.05

LLx0.20.25

-20

-15

-10

-5

0

5

10

15

20

.35 -0.15 0.05 0.25

w/h

b)

d) = 0.3510

= 0.3510h2 w

.

/ 1

/ 1dierent, even in what concerns the denition of the

solutions stability.

4. Conclusions

The feasibility of the nite element, shooting and

Newton methods in the determination of non-linear

periodic motions of either thin or thick, beams or plates

was demonstrated. The fact that motions with any

number of harmonics can be analysed, as long as the

time step employed in the integration of the dierential

equations of motion is small enough, is a very important

property. The procedures employed allowed namely to

derive response curves of non-linear structures, includ-

ing the denition of internal and super-harmonic reso-

nances, and the computation of stable and unstable

-0.2-0.15-0.1 2

2

uencies x=x1 0:3416 and 0.3510, super-harmonic branch,

(N/m2) q (kg/m3) m

1.0 1010 7800 0.3

x4 x5

1416.54 1724.31

1422 P. Ribeiro / Computers and Structures 82 (2004) 141314230.2

0.4

0.6

0.8

1

1.2

1.4

1.6

hwsolutions. Secondary branches were found as well. The

simple predictor used in the Newton method was quite

helpful in reducing convergence problems.

Naturally, because the fundamental frequency of

vibration was the parameter in the Newton method, it

would not be feasible to pass turning points where the

tendency of change in frequency would reverse from

increasing to decreasing, or vice-versa. Another draw-

00 0.5 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

hw

t t+T

b

00.10.20.30.40.50.60.70.8

-1-0.8-0.6-0.4-0.20

0.20.40.60.81

hw

t t+T

c

00.10.20.30.40.50.60.70.8

-1-0.8-0.6-0.4-0.20

0.20.40.60.81

hw

t t+T

d

00.10.20.30.40.50.60.70.80.9

-1.5

-1

-0.5

0

0.5

1

1.5

hw

t t+T

e

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 7. (a) Transverse displacement at x 0, y 0 in function of fre(b) 0:2124x1 (100 rad/s), (c) 0:3398x1 (160 rad/s), (d) 0:4885x1 (230aback of the procedure, in comparison with frequency

domain methods based on the harmonic balance pro-

cedure, is that the shooting method is more demanding

in computational resources.

Even for thin structures, when modal coupling occurs

the thin and thick theories give dierent results. This

occurs because modal coupling brings higher order

modes into the denition of the motion. As a result,

1.5 2

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Adimensional amplitude

Harmonics

Adimensional amplitude

Harmonics

Adimensional amplitude

Harmonics

Adimensional amplitude

Harmonics

/ 1

quency. Displacement in function of time and Fourier series at

rad/s) and (e) 1:380x1 (650 rad/s).

[2] Szemplinska-Stupnicka W. The behaviour of non-linear

vibrating systems. Dordretch: Kluwer Academic; 1990.

[3] Petyt M. Introduction to nite element vibration analysis.

Cambridge: Cambridge University Press; 1990.

[4] Ribeiro P. A hierarchical nite element for geometrically

non-linear vibration of thick plates. Meccanica 2003;

38:11530.

[5] Cheung YK, Lau SL. Incremental time-space nite strip

0.5

0.50

1.00

1.50 hw

/ 1

plate theory.

P. Ribeiro / Computers and Structures 82 (2004) 14131423 14230

0.25

0.9 1 1.1 1.2 1.3

/ 10.75

1

1.25 hw0.000.6 0.8 1 1.2 1.4 1.6

Fig. 8. Response at the center of Plate 2, to a harmonic dis-

tributed force of 2000 N/m2: (s) thick plate theory and (d) thinbeams and plates that can be studied employing thin

theories in the linear domain, may quite possibly require

that a thick theory is followed if their non-linear dy-

namic behaviour is to be accurately analysed.

Acknowledgement

The support from the Portuguese Science and

Technology Foundation, who nanced this work

under project POCTI 32641/99, FEDER, is gratefully

acknowledged.

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Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methodsIntroductionEquations of motion, shooting and Newton methodsNumerical applications and discussionsBeamsPlates

ConclusionsAcknowledgementsReferences