ABSTRACT: In this work, large amplitude vibrations of a heated Timoshenko composite beam having delamination is studied. The model of delamination considers the contact interaction between sublaminates including normal forces, shear forces and additional damping due to the sublaminate interaction. Numerical calculations are performed in order to estimate the influence of the delamination, the geometrically nonlinear terms and elevated temperature on the response of the beam.

KEY WORDS: Timoshenko beam, delamination, large amplitude vibration, contact interaction, damping, temperature influence.

1 INTRODUCTION Delamination is a major problem in multilayer composite structures. Due to this reason the development of adequate models describing the phenomena arising in the delaminated part of the structure is a very important topic in real engineering applications. Models which describe the dynamic behaviour of the delaminated structures could be very useful in the development of the vibration based methods for delamination detection.

There are many models in the literature which may predict the dynamic behaviour of beams with delamination. In [1] Wang et al. introduced the so-called “free model” where sublaminates were free to vibrate, which resulted in interpenetration of the delaminated sublaminates. In [2] the so-called “constrained model” was developed assuming that delaminated sublaminates were constrained to have identical transverse displacements. This model excluded the possible opening vibration mode of sublaminates, which is observed in experimental studies. Lou and Hanagud, [3] proposed a model according to which the contact separation phenomenon between sublaminates is considered by using a piecewise linear spring model. Then, in [4] an extended model was suggested where the sublaminate interaction was presented by a nonlinear interpenetration model. A detailed review of the dynamics of earlier works on beams with delamination can be found in Luo’s and Hanagud’s works [3] and in the recently published article for free vibrations of beams with delamination [5]. In most of the models of dynamic behaviour of beams with delamination, the shear forces during the sublaminate interaction and the additional damping arising due to friction between sublaminates are neglected. A model of the dynamic response of a composite Timoshenko beam which takes into account the above mentioned phenomena was recently developed by the authors of this study in [6]. In this paper a numerical analysis of the dynamic behaviour of a composite beam was performed and some comparisons with experimental results were provided. The experimental tests, however, were performed only for the case of composite beams without delamination. In [7] an experimental study of

the dynamic behaviour of beams with delamination was performed. In order to model the delamination in layered composite beams small inclusions of different materials were inserted, modelling the delamination. In this way the model of dynamic behaviour of the contact interaction introduced in [6] was verified and the importance of the additional damping due to sublaminate interactions was proved.

In all of the above mentioned papers the beam vibrations were modelled by using the small deflection beam theory. In order to take into account the longitudinal deflections in the delaminated parts of the beam in the previous studies [ 2, 4, 6 and 7 ] these parts of the beam were assumed to be subjected to artificially introduced axial force. An attempt to consider the large deflections of a beam with delamination was done in [8]. The present work is an extension of the work presented in [8]. The influence of the temperature changes are taken into account in the equation of motion.

In most of the analysis of the dynamic behaviour of a beam with delamination the environmental conditions are neglected. A very important condition which has to be considered is the temperature, which may vary in high ranges in real applications. Temperature changes can and do affect substantially the vibration response of a structure. Thermal loads introduce stresses due to a thermal expansion, which lead to changes in the modal properties. Although the temperature and elastic behaviours are in fact coupled [9, 10], for thin structures it is often reasonable to assume that the temperature distribution is independent of the deformation of the structure or that the structure gets the elevated temperature instantly. This approach is widely used to model the thermoelastic behaviour of structures. The geometrically nonlinear vibrations of structures at elevated temperature are studied by many authors ( see, for example [10] - [13]) . It is shown in [14] that on a lot of occasions the presence of a temperature field can either mask the effect of damage (delamination in our case) or increase it, which will render a vibration based damage detection method ineffective - it might give no alarm when a fault is present or might give a false alarm.

LARGE AMPLITUDE VIBRATIONS OF TIMOSHENKO BEAMS WITH DELAMINATION IN TEMPERATURE ENVIROMENT

Emil Manoach1, Jerzy Warminski2, Anna Warminska3

1Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria , 2Department of Applied Mechanics, 3Department of Thermodynamics, Fluid Mechanics and Aviation Propulsion Systems, Lublin University of Technology, Lublin, Poland

email: e.manoach@imbm.bas.bg, j.warminski@pollub.pl, a.warminska@pollub.pl

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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In the present work the model of dynamic behaviour of beam with delamination suggested in [6] and [7] is extended by introducing the geometrically nonlinear terms in the equations of motion considering large amplitude vibrations. The beam is considered to be subjected to temperature loading as well. A numerical procedure and a computer code for solution of the problem of the large amplitude vibrations of heated Timoshenko beams with delamination are developed. Numerical results of the large amplitude vibration of intact and damaged beams were provided and comparisons with the case of small amplitude vibrations were performed. The influence of the elevated temperature on the response of the damaged beam is also studied.

2. BASIC EQUATIONS

2.1. Equations of motion and boundary conditions

The considered composite laminated beam was conditionally considered to consist of four sections (Fig. 1). Section 1 and section 4 are without defects. The cross-section between these two parts contains a delamination. Sections 2 and 3 denote the parts of the beam’s cross-section divided by the delamination.

Figure 1. Geometry of the beam. x1 and x2 denote the beginning and the end of delaminated area.

By hi (i = 2, 3) the thicknesses of the delaminated parts of the cross-section are denoted and zd is the coordinate of the delamination.

According to the model suggested in [6] for section 2 and 3

we assume that normal and shear forces Nσ and Sσ act when these two sections are in contact. An additional damping due to delamination, which is experimentally observed, is also included by adding rate dependent terms in the expressions for the contact forces:

2 3 2 3( , ) ( ) ( )N N Nx t K w w K w wσ = − + −ɺ ɺ ɺ (1)

[ ]322 2 3 3 3 3 2 2( , )S S Sww

x t K h h K h hx x

σ ψ ψ ψ ψ ∂ ∂ = − − − + − ∂ ∂

ɺ ɺ ɺ (2)

Here by wi the transverse displacements of ith section of the beam is denoted, ψi is the angle of rotation of the normal of the cross-section to the beam mid-axes. In (1) and (2) NK ,

SK , NKɺ and SKɺ are coefficients. They are connected with

the mechanical properties of the material. NKɺ and SKɺ are related directly with the coefficients of the dry friction of the material. The contact phenomenon between section 2 and 3 is modelled according to this model by considering each contacting sublaminate acting as a two parametric elastic foundation (with spring constants KN and KS) on which the other contacting sublaminate is resting. The possibility of separation (opening of the delamination) and friction between delaminated layers is taken into account. The spring constants KN and KS are presented by the following formulas ([4] and [6]) :

2 32 2 3 3

2( )

/ /N b

K H w wh E h E

≈ −+

(3)

[ ]2 2 3 3 2 3( / ) ( / ) ( )sK b h h G h h G H w w≈ + − (4)

where by H(.) is denoted the Heaviside step function ,

0 for 0

( )1 for 0

Hδ

δδ

≤= >

(5)

In Eqs. (3) Gi (i=2,3) denote the shear modulus of the cross section i.

In the following considerations the longitudinal inertia effect is neglected. Thus, the governing equations for the large amplitude thermoelastic vibration of the beam could be written as follows:

2 2

2 2i i i T

i i i i i iT

u w wE F E F E b

x xx x

γα

∂ ∂ ∂ ∂= − +

∂ ∂∂ ∂,

(6 a-c)

( ) ( )

2 2

22 2

2 2 3 3 2 2 3 3 2 2 3 3

2 2

12 2

2

( 1) ( ) ( 1) ,

1

2

i i i i Ti i i i i i i i iT i

i S i S

i i i ii i i i i

i ii i iT

wI E I c kG F bE

t x xt x

K h w h w h h K h hx

w w wF kG F c

x tt x

u wE F

x x

ψ ψ ψ χρ ψ α

γ ψ ψ γ ψ ψ

ψρ

α

∂ ∂ ∂ ∂ ∂ = − − − + ∂ ∂ ∂∂ ∂

∂ − − − − − − − ∂

∂ ∂ ∂ ∂= − − ∂ ∂∂ ∂

∂ ∂ + + − ∂ ∂

ɺ ɺ ɺ

2

2

2 3 2 3( 1) ( ) ( 1) ( ) ( , )

iT

i N i N

w

x

K w w K w w q x t

γ

γ γ

∂

∂

+ − − + − − −ɺ ɺ ɺ

( )2

2

22

hh

T Thh

T x z t zdz T x z t dzχ γ−

−

= =∫ ∫/

/

//

, , , ( , , )

Here ui denotes the longitudinal displacement of i-th beam section. The parameter γ has been introduced to take into

account the fact, that for section 1 and 4 the terms containing

, , ,N S N SK K K Kɺ ɺ disappear, i.e.:

1 for =2 and 3

0 for =1 and 4

i

iγ

=

(7)

The notations of the geometrical parameters in (6) are

defined by the following expressions:

Section 4 Section 2

Section 3

Section 1

b

h

x

z

l

x1 x 2

h2

h3 zd

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3 3

3 3

2 3

2 3

2 2, , ,

3 3 3 3

/ 2 , / 2

d di i

d d

h hz z

F bh I b I b

h h z h z h

− = = − = −

= − = −

(8)

Equations (6) are solved at the following boundary

conditions (clamped-clamped beam):

1 1(0, ) 0, (0, ) 0,w t tψ= = 4 4( , ) 0, ( , ) 0,w l t l tψ= = (9)

and the continuity conditions:

1 1 2 1 1 1 3 1

1 1 2 1 1 1 3 1

1 2 3 1

1 2 3

( , ) ( , ), ( , ) ( , ),

( , ) ( , ), ( , ) ( , )

w x t w x t w x t w x t

x t x t x t x t

M M M x x

Q Q Q

ψ ψ ψ ψ= == =

= + == +

(10) (10)

4 2 2 2 4 2 3 2

4 2 2 2 4 2 3 2

4 2 3 2

4 2 3

( , ) ( , ), ( , ) ( , ),

( , ) ( , ), ( , ) ( , )

w x t w x t w x t w x t

x t x t x t x t

M M M x x

Q Q Q

ψ ψ ψ ψ= == =

= + == +

where Mi and Qi are the bending moments and the shear forces of i-th beam’s section.

2.2. Solution of the problem

In order to consider the longitudinal displacements in a delaminated beam it could be assumed that it consists of two parallel beams - I and II. Beam I consists of section 1,2 and 4 and beam II consists of sections 1, 3 and 4. From Eq. (6 a), it can be seen that

2

0.5I II

u wC

x x

∂ ∂ = − ∂ ∂ I=1,2 (11)

where index I denotes a number of the beams defined above. This means that the geometrically nonlinear term GL in Eqn. (6 c) could be presented as:

2 2

2

1

2iI I

L Iwu w

G Cx x x

∂∂ ∂ = + = ∂ ∂ ∂ , I=1,2 (12)

The constants CI can be determined from the boundary condition uI(1,t)=0. After integration it could be written in the form

21

20

I II

w wC dξ ξ

ξ ξ∂ ∂

= −∂ ∂∫ . (13)

Thus, the geometrical nonlinear term for sections 1 and 4 of

the beam is 2

22I II

LC C w

Gx

+ ∂ = ∂ - for section 2 it is

2

2L Iw

G Cx

∂=∂

and for section 3 it is 2

2L IIw

G Cx

∂=∂

.

Using this approach the possibility of arising different longitudinal deflections in the part of the beam having delamination (in the opening mode of the delamination) is taken into account. Thus, the inclusion in the governing equations of the axial force P , to which section 2 and 3 of the beam are assumed to be subjected in [2], [4] and [6], is avoided.

The governing equations of motion are solved numerically by the finite difference method. The following dimensionless variables are introduced for the numerical solution:

w w l= , x x l= , t tc l= , 2 /c E ρ==== (14a-d)

The governing equations (6), the boundary conditions (9) and the continuity conditions (10) are discretized along x by the finite difference method using the central difference

formulas with order 2( )O s (where s is the discretization

step):

( ) ( ) ( )( )211

2 2

( ) ( ) ( )( )21 1

2 2

( ) ( )( )1 1

2,

21, 2,3, 4

, etc.2

j j jji ii i

j j jjii i i

j jji i i

w w ww

x s

jx s

x s

ψ ψ ψψ

ψ ψψ

−+

+ −

− −

− +∂=

∂− +∂

= =∂

−∂=

∂

(15 a,b)

In the applied notation in Eq. (12) the upper index in brackets denotes a number of the beam sections and the subscript i is the number of the discretization node in the space domain. A doubled numeration of the nodes in sections 2 and 3 is introduced. If section 1 is discretized by N1 nodes, section 2 and 3 are discretized by N2 nodes (they coincide) and section 4 – by N4 nodes, then the total number of the second order ordinary differential equations (ODE) obtained after the discretization is N=N1+N4+2N2 . By a simple substitution they are transformed in 4N first order ODE with respect to the variables:

{ }, , ,T

w wψ ψ=y ɺ ɺ ,

y = [A]y + qɺ (16 a,b)

The ordinary differential equations (16) are solved numerically by an implicit method using the backward differentiation formulas (well known as the Gear’s method (see [15] ) and the IMSL Fortran Library software package. An iteration procedure is used to recalculate the reaction forces in the sublaminates when a contact interaction arises. First, the problem is solved by using the small deflection beam theory (SDBT). Then the derivative ( ) /j

iw x∂ ∂ , 2 ( ) 2/j

iw x∂ ∂ , the longitudinal displacements (according to

Eqn (12) ) and LG were calculated numerically, and the

problem was solved again by using the large deflection beam theory (LDBT). At each time step the difference between the deflections of the nodes with a doubled numeration was checked and if penetration appeared the

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recalculation of the equation for the same time step was performed introducing the normal and shear contact forces. In this way the interpenetration between sublaminates was prevented. When the corresponding nodes from the sublaminates are not in contact the reaction forces and the corresponding damping terms are set to zero.

3. NUMERICAL RESULTS AND DISSCUSSION

A numerical study was performed in order to test the created algorithms and the computer programmes and to study the influence of the geometrically nonlinear terms and the elevated temperature on the response of the beam with delamination.

The first task was to repeat the calculations performed in [7] and to compare them with the experimental results published in the same article. In the above mentioned article inclusions were embedded in small parts of two adjacent layers of 12-ply unidirectional composite layered beam, in order to model delamination. Then, the beams with and without inclusions were subjected to short pulses and the numerical and experimental results were compared. In the present work this calculations were repeated by using the large deflections beams theory. Obtained results practically coincide with the experimentally obtained results in [7]. As far as the deflections are small the results obtained by SDBT and LDBT are practically indistinguishable. That is why these results are not shown here.

The tested beam had the following dimension l=0.23 m, h=0.00325 m b=0.01815 m. The material properties of the beam without delamination (inclusions) were measured as: E=47065.19 MPa, ρ=1900 kg/m3, ν=0.3189. The delamination was considered to have a length of 0.03 m located at zd =-0.00054m at the central part of the beam along its length.

The beams were discretized by 139 nodes along the beam length. In order to describe the delamination in the numerical model 21 nodes with doubled numeration were used in the middle of the beam. This means, that for beams with delamination, section 1 and section 4 had 59 nodes each, whereas section 2 and 3 had 21 nodes each. Each node from section 2 had a corresponding node from section 3 with the same x coordinate. The total number of nodes for the beam with delamination was 160. Thus, the total number of solved ODE was equal to 640. The calculations were performed with different numbers of nodes. The chosen number of nodes guaranteed the obtaining of the mesh independent results. Тhe dynamic responses of the beams with delamination

subjected to a harmonic loading was studied by using SDBT and LDBT. The case of a beam without delamination was also considered. In order to avoid the influence of the inclusions which has different density and rigidity in the next example the mechanical properties of sections 1,2,3 and 4 are accepted to be the ones for a glass-epoxy material. The beam was subjected to harmonic loading with amplitude p0=2.4 kN at the beam centre with excitation frequency ωe=1360 rad/s. The first eigen-frequency of the intact beam was obtained to be ω1=1366.8 rad/s. The results from the numerical calculations are plotted in Figure 2

Figure 2. Time history diagram of the response of the beam

subjected to a harmonic loading with P0=2.4 kN and ωe=1360 rad.s-1 . 1 (blue line) – SDBT with delamination; 2 (Red line) – large deflections with delamination; 3 (black line) – LDBT

without delamination.

Due to the fact that the excitation frequency is very close to the first natural frequency of the beam, in the results computed according to SDBT strong beating can be observed. The deflections in this case are much larger than the ones computed by LDBT. As can be expected the deflections of the beam with delamination are larger than the one of the intact beam (see curve 2 and 3). This case shows the importance of the consideration of the large deflections in the governing equations of beams with delamination.

The next examples were computed for a beam with the following physical properties and geometrical dimensions: E = 7000 MPa ρ= 2778 kg/m3 ν=0.34 l=0.5 m h=0.005m b=0.05m . The delamination was larger – 0.07 m at the centre of the beam and located at zd = -0.000325 m. The discretization of the beam was the same as in the previous example. The loading is distributed along the whole beam length according to the formula

0( , ) sin( / ) sin( )ep x t P x l tπ ω=

The exciatition frequency for this case was ωe=1360 s-1 and P0=300N/m.

Results from the response of the beam’s centre for the selected period of time are shown in Fig 3. It is seen that the deflections computed according to LDBT differ from the one obtained by SDBT. The deflections computed by LDBT lead to an amplitude smaller than the ones obtained by SDBT. The presence of delamination increases the amplitude of the vibrations.

In the next figures (4 and 5) the opening and closing modes of the delamination for a short period of time can be oserved. It is interesting to note that the opening of the delamination is larger in the case of consideration of the beam according to LDBT. The normal reaction force during the contact between the sublaminates for the same period of time shows the opposite trend – the reaction force for the case of SDBT is larger than the one in the case of LDBT. It is seen that the

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simple consideration of the longitudinal displacements by introducing an artifically longitudinal force which acts at both ends of delamination (with oposite signs) cannot describe adequately the influence of the large deflections on the response and especially on the contact interaction between sublaminates.

Figure 3. Time history diagram of the beam center subjected

to harmonic loading. 1 – (black colour) – SDBT , no delamination; 2 (red colour) – SDBT beam with delamination;

3 (green colour) -LDBT , delamination; 4 – LDBT, no delamination.

Figure 4. The differences between the deflections from section 2 and section 3 in the centre of delamination (w70-w91) in time.

Red colour – SDBT; Green colour – LDBT.

Figure 5. The normal reaction forces (dimensionless) between the sublaminates at the centre of the delamination. Red colour

– SDBT; Green colour – LDBT.

In the next examples the influence of the temperature on the vibration of the beam was studied.The same beam like the one from the previous example was subjected to the same mechanical loading but was subjected to the temperature changes. A more complicated case when the delamination was located close to the left beam edge was considered. The delamination , placed at zd = -0.000325 m was located between nodes 24 and 45, i.e. it had a length of 15 % of the beam’s length.

The responces of the centre of the delamination of the beam subjected to uniformly distributed along beam’s length load p= 150sin( )etω N/m are plotted in Fig.5. The excitation

frequency was 324eω = rad/s ( 1 647.95ω = rad/s). The beam

are subjected to 3 different temperature changes ∆T =10, 20 and 39 K. Also the response of the beam without delamination at node 35 was plotted. Inspecting the curves plotted there it is clearly seen that the elevated temperatures lead to larger amplitudes. After a long transition regime the vibrations in all cases become periodic. The Poincaré maps of the responses are plotted in Fig. 7 for 5000t > . Comparisons between the responses of the beams with and without delamination can be made in Figs. 8 and 9. As can be expected the amplitude of vibration is larger for the beam having delamination. The differences between the responses, however, increase with the increasing temperature, i.e. the elevated temperature enhances the influence of the delamination on the response of the beam.

4. CONCLUSIONS.

An extension of the model (proposed by the authors of this study) of the dynamic behaviour of composite beams having delamination has been developed. The proposed model, considering a contact-separation of the sublaminates during forced response, has been extended for the case of large amplitude vibrations of thermally loaded beams. The influence of the geometrically nonlinear terms and the delamination on the response of the beam have been studied

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numerically in the case of the harmonic loading of heated and unheated beams. The numerical studies have shown that the results of the extended model considering large deflections

Fig. 6. Responses of the beam (at the centre of the

delamination) subjected to harmonic loading . P0=150 N/m , 324eω = rad/s. 1 (black color) – beam without delamination

∆=0.; 2 (green color) - ∆T =10; 3 (blue color) - ∆T =20; 4 (red color) - ∆T =30.

Fig. 7. Poincare maps of the responses of the beam (at the center of the delamination) subjected to harmonic loading.

P0=150 N/m , 324eω = rad/s. 1 (black colour) – beam without

delamination, ∆Τ=0; 2 (green colour) - ∆T =10; 3 (blue colour) - ∆T =20; 4 (red colour) - ∆T =30.

Fig. 8. . Responses of the beam at node 35 for ∆T =0. 1 –

black solid line - beam without delamination; 2 red dashed line beam with delamination .

Fig. 9. Responses of the beam at node 35 for ∆T =30. 1 (black solid line) - beam without delamination; 2 (red dashed line) -

beam with delamination .

differ from the ones obtained by using SDBT beam theory. The differences between the responses of the beam according to LDBT model and the SDBT model can be essential when the excitation frequency is close to the first natural frequency of the beam. The increased temperature leads to vibration with larger amplitude. The influence of the delamination on the response of the beam becomes more essential when the temperature increases.

ACKNOWLEDGMENTS.

The first author wishes to acknowledge the support received from the Bulgarian NSF Grant DCVP-02/1. The second and third authors would like to acknowledge the financial support

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of Structural Funds in the Operational Programme – Innovative Economy (IE OP) financed from the European Regional Development Fund –Project “Modern material technologies in aerospace industry”, Nr POIG.01.01.02-00-015/08-00.

REFERENCES [1] J.T.S Wang, Y.Y Liu, J.A. Gibby, Vibrations of split beams. J. of Sound

and Vibr. 84, (1982), 491–502. [2] P.M Mujumdar, S. Suryanarayan, Flexural vibration of beams with

delaminations. J. Sound Vib., (125) , 1988, 441-461. [3] H. Luo, S. Hanagud, Dynamics of delaminated beams. Int. J. of Solids

and Structures 37 (2000 ), 1501-1519. [4] J. Wang, L. Tong, A study of the vibration of delaminated beams using

a nonlinear anti-interpenetration constraint model. Composite structures 57 (2002), 483-488.

[5] P. Qiao, F. Chen, On the improved dynamic analysis of delaminated beams. J. Sound and Vibrations, 331 (2012), 1143-1163

[6] E. Manoach, J. Warminski, A. Mitura, S. Samborski, Dynamics of a composite Timoshenko beam with delamination. Mechanics Research Communications 46 (2012) , 47-53

[7] E. Manoach, J. Warminski, A. Mitura, S. Samborski, Dynamics of a laminated composite beam with delamination and inclusions. Eur. Phys. J. Special Topics 222 (2013), 1649–1664

[8] Manoach, E., Warminski, J., Mitura, A., Samborski, S. in Z. Dimitrovová, J.R. de Almeida, R. Gonçalves (eds.) Large amplitude vibrations of Timoshenko beams with delamination, in Proceedings of the 11th International Conference on Vibration Problems (ICOVP-2013), Lisbon, Portugal, 9-12 September, 2013, AMPTAC, ISBN 978-989-96264-4-7, 10 pages. paper : 78

[9] E.A Thorton., Thermal Structures for Aerospace Applications, AIAA Education Series, Reston, Virginia (1996)

[10] E. Manoach., P. Ribeiro, Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams, Int. Journal of Mechanical Sciences, 46 (2004), 1589–1606

[11] 14. R.C Zhou., Y Xue, C.Mei, Finite Element Time Domain-Modal Formulation for Nonlinear Flutter of Composite Panels, AIAA Journal 32, (1994), 2044–2052

[12] 15. P. Ribeiro, E. Manoach, The effect of temperature on the large amplitude vibrations of curved beams, Journal of Sound and Vibration 285, (2005), 1093–1107

[13] M. Amabili, S Carra. Thermal effects on geometrically nonlinear vibrations of rectangular plates with fixed edges, Journal of Sound and Vibration 321 (2009), 936-954

[14] Manoach E., Samborski S., Mitura A., Warminski J., Vibration based damage detection in composite beams under temperature variations using Poincaré maps, Int. Journal of Mechanical Sciences, 62, (2012), 120–132

[15] C.W. Gear, The automatic integration of ordinary differential

Equations. Comm. ACM. 14 (1971), 176-179.

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