· pdf file · 2016-05-24title: microsoft word - eccomas2016-gamzi_s paper.docx

ECCOMAS Congress 2016 VII European Congress on Computational Methods in Applied Sciences and Engineering

M. Papadrakakis, V. Papadopoulos, G. Stefanou, V. Plevris (eds.) Crete Island, Greece, 5–10 June 2016

Keywords: Non-linear vibration, rectangular membrane, discrete system, Hamilton’s Principle, Spectral analysis, axial tension, shear stress. ABSTRACT. This paper investigates the non-linear vibration of a rectangular membrane subjected to axial tensions Tx and Ty, in addition to shear stress resultant force Txy applied to the membrane’s four edges. The studied membrane is isotropic and simply supported, the spatial and time variables of the displacement function are separable and the motion is harmonic. The approach used in this work is the discrete model developed by BENAMAR and coauthors for non-linear vibrations of thin plates and shells. The method consists on expanding the displacement function in the form of a finite series and applying Hamilton’s principle to obtain the non-linear amplitude equation, the solution of which leads to the non-linear mode shapes and resonance frequencies of the membrane at large vibration amplitudes. First, we compared the linear results found by the present method to the nonlinear results computed for small values of the amplitude and to linear resulats found by the the approximate Rayleigh Ritz method , for different values of the tension ratio Ty/Tx. Then we examined the amplitude dependence of the fundamental mode shape of the membrane in the middle section and the section near the edge. Afterwords we studied the impact of the shearing stress on the frequency-amplitude dependence and finally we examined the effect of the tensions in the x and y directions on the vibration frequency and for differet values of the maximum displacement. The numerical results obtained here compare well with the ones obtained previously using large deflection theory of membranes and solving the corresponding non-linear partial differential equations and they showed that the response of the membrane is highly affected by the shearing stress resultant value and that the nonlinear effect is amplified by high axial tensions. They also showed that the mode shape depends increasingly on the amplitude in the nonlinear case and that the nonlinear vibration frequency corresponds to the linear one in the case of small maximum displacement values while the ratio of the two frequencies increases with the decreasing value of the tension ratio, for important values of maximum displacement . 1 INTRODUCTION: Vibration analysis of membranes is a topic of a high interest since they intervene in many applications in engineering such as aerospace, acoustics, biotechnology… These structures

Non-linear vibration of a rectangular membrane subjected to axial tensions and shear stress resultant force, applied to the membrane’s four edges

A.GAMZI (1) and R. BENAMAR (2)

(1) Engineer researcher at Ecole Mohammadia d’Ingénieurs, Laboratoire d’Etudes et de Recherches en Simulation, Instrumentation et Mesures, LERSIM, BP. 765 Agdal, Rabat, Morocco,

Email :[email protected] (2) Université Med V Agdal, Ecole Mohammadia d’Ingénieurs, Département des E.G.T, Laboratoire

d’Etudes et de Recherches en Simulation, Instrumentation et Mesures, LERSIM, BP. 765 Agdal, Rabat, Morocco, Email : [email protected]

A. GAMZI, R. BENAMAR

can, in a severe environment, be the subject of high deflections that may largely exceed their thickness. In such cases, linear models do not describe properly the membrane dynamic behavior. Hence, the importance of models that can take into account the geometric nonlinearities of the membrane under large vibration amplitudes, and their effects on the response of the structure has led many researchers to work on the nonlinear vibrations of membranes using different methods. For the linear case, various approaches were adopted. J. L. Ferrela and L.A. Medeiros [1] used the model of Kirchhoff and Carrier to deduce and analyze a mathematical model for small deformations of a circular membrane with moving boundaries. C.Shin, J.Chung and H.H.Yoo [2] studied the dynamic responses of the in-plane and out-of-plane vibrations for an axially moving membrane using Hamilton’s principle. The membrane considered is supported by two pairs of rollers, one is driving and the other one is driven and subjected to different tensions in the right and the left edges. Applying the extended Hamilton’s principle, the equations of motion have been derived for the moving membrane and discretized by the Galerkin method, and then a time integration method was applied to compute the dynamic responses of the membrane. The results found show that the in-plane responses have shorter periods than the out-of-plane responses. Also, dynamic instability occurs in the out-of-plane motion rather than in the in-plane motion because of the presence of a lower critical speed.In another paper about free vibration analysis of rectangular orthotropic membranes at large deflections, Z. Lian, L.Chang-Jiang, H. Xiao-Ting, and C.Shan-Lin [3] applied the large deflection theory of membranes and D’Alembert’s principle, derived the governing vibration equations of rectangular membrane with four edges fixed, and obtained the power series formula of nonlinear vibration frequency that was found to be dependent on the initial conditions. Vibration of arbitrary shaped membranes was analyzed by W.X. Wu, C. Shu and C.M. Wang [4] using the local radial basis function-based on differential quadrature method which allows a numerical discretization of Helmholtz equation. A system of algebraic equations was then solved to obtain wavenumbers and mode shapes of the membrane. In ref [5] H. Ersoy, Ö. Civalek and L. Özpolat used the discrete singular convolution approach in the free vibration analysis of rectangular membranes with variable density, which allowed them to reduce the partial differential equations of motion, into an eigenvalue problem. For the nonlinear case, only a few research works are available in the literature. In a paper by S. Kukathasan and S. Pellegrino [6], the finite element simulation technique was applied to predict the natural frequencies and mode shapes of wrinkled membrane structures, both in vacuum and in air. Several conclusions were reached such as the increase of damping in all modes as the amplitude of excitation increases, the difference in the frequency variation for each mode and the presence of the second and third harmonics of the excitation frequency in the response of the wrinkled membrane, excited in the second mode. Also Renata M. Soares and Paulo B. Gonçalves [7] studied the linear and non-linear free vibrations of a pre-stretched circular hyperelastic membrane with varying the density or the thickness. The membrane material was assumed to be isotropic and incompressible and its behavior was described by the neo-Hookean constitutive law. Taking into account finite deformations, the governing equations of motion of the membrane under a uniform radial stretch, were derived using a variationnal formulation. Then, the linearized equations of motion were analytically solved to obtain the vibration modes and frequencies of the hyperelastic membrane and these normal modes were used, together with the Galerkin method, to obtain low-order approximations of the nonlinear dynamic response. The results showed the strong influence of the radial stretching ratio on the natural frequency and the nonlinear frequency-amplitude dependence. They also analyzed the linear and nonlinear vibration response of a pre-stretched hyperelastic rectangular membrane under time varying lateral pressure and finite initial deformations [8]


using the same method used earlier for circular membranes. The results found show the influence of the stretching ratio on the vibration frequencies and nonlinear frequency-amplitude relation. In this paper, the membrane considered is rectangular and isotropic, the time and the space are supposed to be separable in the displacement function and the motion is assumed to be harmonic. The space displacement function is then expanded in the form of a finite series of functions using the discrete model developed by BENAMAR and co-authors ref [9 to 11] Hamilton’s principle is applied to obtain the nonlinear algebraic amplitude equation which is solved using a MATLAB program developed to that matter. Some of the results obtained are presented and compared with the ones found in the literature.

2 GENERAL FORMULATION: Consider the transverse vibration of a rectangular membrane with a length a and a width b, subjected to axial tensions Tx and Ty , along with shear stress resultant Txy, as shown in Figure 1.

Fig.1 The membrane examined with a length a and a width b, subjected to axial tensions Tx and Ty , along with shear stress resultant Txy

2.1 The membrane strain and kinetic energy expressions at large vibration amplitudes: 2.1.1 The strain energy due to the axial tensions Tx and Ty , and the shear stress resultant Txy: According to reference [12] , the strain energy due to the axial tensions Tx and Ty , and the

shear stress resultant Txy is given as :

𝑉! =!!

𝑇! .!"!"

!+ 𝑇! .

!"!"

!+ 2𝑇!"

!"!". !"!"

.𝑑𝑆! (1)

In which W(x,y,t) is the transverse displacement function and dS=dx.dy is the elementary surface. 2.1.2 The axial strain energy:


The axial strain energy induced by large deflections is given in [13] as a combination of terms involving the first partial derivatives of the in-plane displacements U and V with respect to x and y, and the partial derivative of the transverse displacement W with respect to either x or y. However, according to [2], the effect of the in-plane displacements on the nonlinear strain

energy is insignificant compared to that of the terms containing !"!"

! and !"

!"

!. Therefore,

it was assumed in this work that the in-plane displacements can be neglected in the expression of the axial strain energy induced by large deflections, which is given in [13] as follows:

𝑉! =!!

!"!"

!+ !"

!"

! !.𝑑𝑆! (2)

In which 𝐶 = !"

!!!! is the membrane axial stiffness.

2.1.3 The kinetic energy: The kinetic energy of the membrane is given by:

𝑇 = !!

𝜌𝐻 !"!"

!.𝑑𝑆 ! (3)

2.2 Discretization of the membrane strain and kinetic energy expressions: It is supposed that the time and space are separable and that the motion is harmonic. Thus, the transverse displacement can be written as:

𝑊 𝑥,𝑦, 𝑡 = 𝑤 𝑥,𝑦 . sin (𝜔𝑡) (4)

Using Einstein summation convention for repeated indices, the displacement function W(x, y) can be written as a finite series of n basic functions wi(x) as follows:

𝑤 𝑥,𝑦 = 𝑎! .𝑤!(𝑥,𝑦) (5) With : 𝑎! : The contribution coefficient; 𝑤!(𝑥,𝑦) : The ith basic function of the transverse displacement. Substituting (5) in (1) , (2) and (3) leads to the following discrete formulations: § The strain energy due to the axial tensions Tx and Ty , and the shear stress resultant Txy

𝑉! = !

! 𝑎!𝑎!𝑘!"sin !(𝜔𝑡) (6)

In which : 𝑘!" is the linear stiffness tensor.

𝑘!" = 𝑇!!!!!". !!!!"

+ 𝛽 !!!!". !!!!"

+ 2 𝛾 !!!!". !!!!" .𝑑𝑆! (7)


With : β = !!!!

and 𝛾 = !!"!!

§ Axial strain energy due to the large deflections: 𝑉! = !

!𝑎!𝑎!𝑎!𝑎!𝑏!"#$sin !(𝜔𝑡) (8)

In which : 𝑏!"#$ is the nonlinear stiffness tensor. 𝑏!"#$ =

!!

!!!!". !!!!"

+ !!!!". !!!!"

. !!!!"

. !!!!"+ !!!

!". !!!!"

.𝑑𝑆! (9) Kinetic energy: 𝑇 = !

!𝜔!𝑎!𝑎!𝑚!"cos! (𝜔𝑡) (10)

In which : 𝑚!" = 𝜌𝐻.𝑤!𝑤! .𝑑𝑆 ! (11) is the mass tensor.

3 NON-DIMENSIONAL FORMULATION: The non-dimensional parameters are defined as follows: 𝑤! 𝑥,𝑦 = 𝐻.𝑤!∗

!!, !!= 𝐻.𝑤!∗(𝑥∗,𝑦∗) (12)

Replacing 𝑤! 𝑥,𝑦 with the expression given above in (7), (9) and (11), and putting 𝛼 = !

! ,

the following non-dimensional expressions are obtained:

𝑘!"∗ = !!!∗

!!∗.!!!

∗

!!∗+ 𝛽.𝛼! !!!

∗

!!∗.!!!

∗

!!∗+ 2. 𝛾.𝛼 !!!

∗

!!∗.!!!

∗

!!∗.𝑑𝑆∗!∗ (13)

With:

!!"!!"∗ = !!

!!!!

(14)

And:

𝑏!"#$∗ = 3 !!!∗

!!∗.!!!

∗

!!∗+ 𝛼² !!!

∗

!!∗.!!!

∗

!!∗!!!

∗

!!∗. !!!

∗

!!∗+ 𝛼² !!!

∗

!!∗. !!!

∗

!!∗.𝑑𝑆∗!∗ (15)

With: !!"#$!!"#$∗ = !"!

!

!! (16)

And 𝐷 = !.!!

!".(!!!!) (17)

Then 𝑚!"

∗ = 𝑤!∗𝑤!∗𝑑𝑆∗!∗ (18)


!!"

!!"∗ = 𝑎. 𝑏.𝜌𝐻! (19)

And then, using the law of conservation of energy which is : 𝑇!"# = 𝑉!"# (20) The following expression of the nonlinear frequency is obtained:

𝜔! = !!!!!!"!

!!!!!!!!!!!!"#$!!!!!!"

(21)

Replacing 𝑘!" , 𝑏!"#$ and 𝑚!" by their correspondent non-dimensional parameters, we find that:

𝜔! = !!!"!!

!!!!!!"∗!

!!!!!

!!!!!!!!!!"#$∗

!!!!!!"∗ = !!

!"!! 𝜔∗! (22)

So, the non-dimensional frequency parameter is given by:

𝜔∗! = !!!!!!"

∗ ! !!!!!

!!!!!!!!!!"#$∗

!!!!!!"∗ (23)

And !

!

!∗!= !!

!"!! (24)

It can be noticed that the non-dimensional parameter of the tensor 𝑘!"∗ depends on the membrane’s aspect ratio α, the chosen basic function 𝑤!∗, along with the ratios of the axial tensions and shear stress resultant γ and β, while the non-dimensional parameters of the nonlinear stiffness tensor and the mass tensor depend only on the aspect ratio of the membrane and the basic function chosen. It should also be highlighted that the expression of the non-dimensional parameter of the frequency depends on the axial tension in the x direction, besides the other non-dimensional parameters. This is slightly different from what was found for plates in references [9 to 11]. Applying Hamilton’s principle for a conservative system leads to :

𝛿 𝑉 − 𝑇 .𝑑𝑡 = 0!!!! (25)

Substituting T and 𝑉 = 𝑉! + 𝑉! by their expressions found previously, we obtain:

𝑉 − 𝑇 .𝑑𝑡 = !!!(𝑎!𝑎!𝑘!"∗ +

!!!!!!!

𝑎!𝑎!𝑎!𝑎!𝑏!"#$∗ − 𝑎!𝑎!𝜔∗!𝑚!"∗

!!!! ) (26)

Solving (20) consists on finding the minimum of (26) which leads to the calculation of its partial derivatives with respect to the 𝑎!’s (for r =1, .. ,n) and stating that they vanish. In other words, the following set of coupled algebraic equations has to be solved: 𝑎!𝑘!"∗ +

!!!!!!!

𝑎!𝑎!𝑎!𝑏!"#$∗ − 𝑎!𝜔∗!𝑚!"∗ = 0; (27)


𝑓𝑜𝑟 𝑟 = 1, . . ,𝑛 The above non-linear algebraic system can be written in a matrix form as: 3 !

!!!!𝐵∗ 𝐴 𝐴 + 2 𝐾∗ 𝐴 − 2𝜔∗! 𝑀∗ 𝐴 = 0 (28)

4 NUMERICAL RESULTS: 4.1. Choice of the basic functions: The membrane considered in this paper is simply supported on the four edges; therefore the basic functions chosen to describe the motion of the membrane should satisfy the corresponding boundary conditions. This leads us to adopt the following basic function proven to be accurate for our purpose in ref [12]: 𝑤!"∗ 𝑥,𝑦 = sin !"#

!. sin !"#

! 𝑛,𝑚 𝑎𝑟𝑒 𝑒𝑛𝑡𝑖𝑟𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 (29)

In which: m,n=(1,2,3…) are integer numbers. In the reminder of this paper, and for facility, the double index was converted in to a simple one via the relation: 𝑘 = 3 𝑖 − 1 + 𝑗 (30) These functions constitute an orthogonal basis, they were normalized in such manner that: 𝑚!"

∗ = 𝑤!∗ 𝑥∗,𝑦∗ .𝑤!∗ 𝑥∗,𝑦∗ .!∗ 𝑑𝑆∗ = 𝛿!" (31) 4.2. Linear analysis: The linear response of the system can be obtained by omitting the non-linear term of equation (26). This leads to solution of the classical eigenvalues problem: K∗ A − ω∗! M∗ A = 0. (32) In order to be able to compare with the linear results given by W. Leissa in [12], a square membrane is considered, subjected to uniform tensions in the x an y directions. Also, Tx is taken equal to !

!

! and the poison’s ratio is taken equal to 0,3.

§ 𝛼 = 1 § 𝛽 = !!

!!= 1


For different values of 𝛾 = !!"

!! , the results obtained are summarized in Table 1.

𝛾 =𝑇!"𝑇!

Present linear results

(8x8 functions)

Present nonlinear results for small displacement

a11=0,005

Results from reference [12] obtained by Rayleigh Ritz

method (8x8 functions) 0 4,4384 4,4399 4,443

0,1 4.4340 4,4361 4,438 0,2 4.4206 4,4243 4,425 0,5 4.3180 4,3311 4,322 0,75 4.1217 4,1488 4,126 0,85 3.9818 4,0292 3,989 0,98 3.5869 3,8089 3,591

1 3.3729 3,7666 3,376

Table 1: Non-dimensional frequencies for Biaxial Membrane Tension with Shear stress resultant

In Table 1, a comparison was made between linear results found by the present method, nonlinear results computed for small values of the amplitude (a11=0.005) and linear results found in Ref [12] by the approximate Rayleigh Ritz method, for different values of the coefficient 𝛾 = !!"

!!.

A good agreement can be seen between all the values found in the linear case by the present method and those found by Rayleigh Ritz method in ref [12]. The nonlinear results for small amplitudes show a good agreement especially for values of 𝛾 lower than 0,5. However, they start to slightly diverge from the linear ones, once 𝛾 exceeds 0,5 , which leads to conclude that when the shearing stress resultant value reaches half of the tension in the x direction and starts exceeding it, the frequency estimate predicted via the linear model does no longer correspond to the one calculated using the non-linear model for small amplitudes, which would mean that the non-linear effect is more significant in such cases. In order to validate the results found by the present approach, other satisfactory comparisons were made with the linear solutions found in the literature, corresponding to the variation of α . These are presented in Table2.

α Present results (8x8 functions)

DQ Method Laura et al.

[15]

Galerkin Method Laura

[15] DSC results [6]

0.4 8.4504 8,45901 8,45901 8,45906 0.6 6.1000 6,10617 6,10618 6,1062 1 4.4384 4,44289 4,44291 4,44305

Table 2 : Comparison of the fundamental frequency of various homogeneous rectangular membranes for different values of the aspect ratio (For β = 1 and γ = 0 ):


4.3. Nonlinear analysis: 4.3.1. The first nonlinear mode shape of a rectangular membrane: The numerical results presented in table 3 correspond to values of the contribution coefficient a11 of the first basic function varying from 0,05 to 1 for three values of the coefficient𝜷.The aspect ratio was taken equal to 1. These results present on one hand the maximum non dimensional amplitude and on the other hand the corresponding values of the frequency ratio!!"!!.

𝜷=0,25 𝜷 = 𝟎,𝟓 𝜷=1

𝑾𝒎𝒂𝒙 𝑾𝒏𝒍

𝑾𝒍 𝑾𝒎𝒂𝒙

𝑾𝒏𝒍

𝑾𝒍 𝑾𝒎𝒂𝒙

𝑾𝒏𝒍

𝑾𝒍

0 1 0 1 0 1 0,0509 1,0531 0,0507 1,0446 0,0505 1,0337 0,1576 1,3968 0,1571 1,3395 0,1564 1,2641 0,2652 1,9028 0,2648 1,7849 0,2641 1,6256 0,5334 3,3824 0,5332 3,115 0,5327 2,7442 0,801 4,9471 0,8009 4,5348 0,8006 3,9594 1,0685 6,5361 1,0684 5,9808 1,0681 5,2039

Table 3:Frequency ratio !!"!!

for various vibration amplitudes for the fundamental mode of a square membrane: (α=1)

Fig. 2 : The amplitude-frequency dependence with the variation of the coefficient 𝜷 The amplitude frequency dependence versus the variation of the tension ratio 𝛽 is plotted in figure 2, Table 3 summarizes most of these results. It can be seen on the curves that the amplitude frequency dependence is more pronounced with the increasing values of the tension ratio 𝛽. It can also be observed, that for higher values of 𝛽 the maximum displacement amplitude increases. This leads to the conclusion that when the tension in the y direction increases with respect to the one in the x direction, the deflections are more important.

0

1

2

3

4

5

6

7

0 0,2 0,4 0,6 0,8 1 1,2

wnl*/w

l*

Wmax*Ty/Tx=0,25 Ty/Tx=0,5 Ty/Tx=1


4.3.2. The amplitude dependence of the fundamental nonlinear mode shape of a square membrane subjected to uniform axial tensions 𝐓𝐱 = 𝐓𝐲 and 𝐓𝐱𝐲 = 𝟎 In order to investigate the amplitude dependence of the membrane fundamental nonlinear mode shape (uniform tension) at large vibrations amplitude, for the section close to the edge x*=0,05 and the middle section x*= 0,5 , normalized sections of the fundamental mode shape were plotted in figures 3 and 4. Fig. 3 : Amplitude dependence of the membrane fundamental mode shape at large vibration amplitude. Section very close to the edge x*=0.05 Fig. 4 : Amplitude dependence of the membrane fundamental mode shape at large vibration amplitude, in the middle section: It can be observed that the mode shape is amplitude dependent and that the non-linear effect is more pronounced in the vicinity of the edges.

00,20,40,60,81

1,2

00,075

0,15

0,225

0,3

0,375

0,45

0,525

0,6

0,675

0,75

0,825

0,9

0,975

Wno

r

Y*

x*=0,05

Wmax=0,05 Wmax=0,25

Wmax=0,75 Wmax=1

0

0,2

0,4

0,6

0,8

1

1,2

00,075

0,15

0,225

0,3

0,375

0,45

0,525

0,6

0,675

0,75

0,825

0,9

0,975

Wno

r

Y*

X*=0,5

Wmax=0,05 Wmax=0,25

Wmax=0,75 Wmax=1


4.3.3. The impact of the shearing stress on the frequency-amplitude dependence: In order to examine the effect of the shear stress resultant on the frequency-amplitude dependence, the frequency ratios and the corresponding non dimensional maximum amplitudes have been computed for different values of the contribution coefficient a11 of the first basic function (from 0,05 to 1) and different values of the coefficient γ = !!"

!!.

The numerical results are plotted in figure 5 given below. It can be seen from the graphs in fig.5 that for values of γ below 0.5, the frequency-amplitude dependence is almost identical and very pronounced. Once γ reaches and exceeds 0,5 the frequency-amplitude dependence is less pronounced but remains identical for all the values of γ ≥ 0,5. It can also be observed that for higher values of γ (γ ≥ 0,5), the value of the maximum displacement amplitude increases and the frequency ratio decreases. This can be explained by the fact that when the shearing stress resultant is important, compared to the other tensions, the nonlinear effect becomes more important.

Fig. 5 : The impact of shear stress resultant on the frequency-amplitude dependence

4.3.4. The effect of the tensions in the x and y directions on the nonlinear vibration frequency: To be able to examine the effect of the tensions in the x and y directions on the nonlinear vibration frequency, the frequency ratio was calculated for different values of β and each time a fixed value of the maximum displacement amplitude was taken. The results are plotted in fig 6. .

0

1

2

3

4

5

6

0 0,5 1 1,5

Wnl*/W

l*

Wmax*

γ=0 γ=0,2 γ=0,4γ=0,5 γ=0,6 γ=0,7


Fig.6 : The effect of tensions in the x and y directions of the vibration frequency: It appears from the graph that for small maximum displacement amplitude values, the frequency ratio is almost the same for all the different values of the tension ratio β and they all approach 1, which means the nonlinear vibration frequency converges to the linear one when the vibration amplitude tends to 0. For higher values of the maximum displacement amplitude, the frequency ratio increases even for lower values of β.

5. CONCLUSION: In this paper, the nonlinear vibration amplitudes of a simply supported rectangular isotropic membrane, subjected to axial tensions and shearing stress resultant has been examined. The theoretical model developed previously by R. Benamar and Co-authors [9 to 11] for thin elastic structures subjected to large deflections, in particular beams and plates was adapted to the membrane case. The approach is based on expanding the displacement function in the form of finite series and applying Hamilton’s principle in order to determine the nonlinear amplitude equations. The results found show that:

§ The shearing stress resultant value has an important effect on the response of the membrane. It appeared that when this value exceeds significantly half the value of the axial tensions in the x or y directions, the vibration frequency increases and the linear model seems to be no longer appropriate for obtaining accurate results;

§ Increasing the axial tensions ratio induce a more significant nonlinear effect; § In the nonlinear case, the mode shape depends increasingly on the amplitude; § For small maximum displacement amplitudes, the nonlinear vibration frequency

corresponds to the linear one no matter how the tensions in the x and y directions are varied. For more important values of the maximum displacement amplitude, the frequency ratio increases with decreasing the values of β.

REFERENCES [1] Vibrations of elastic membranes with moving boundaries, Juan Limaco Ferrela, Luis Adauto Medeiros, Nonlinear Analysis 45 (2001) 363 – 382; [2] Dynamic responses of the in-plane and out-of-plane vibrations for an axially moving membrane, Changho Shin, Jintai Chung, , Hong Hee Yoo, Journal of Sound and Vibration 297 (2006) 794–809; [3] Free Vibration Analysis of Rectangular Orthotropic Membranes in Large Deflection, Zheng Zhou-Lian, Liu Chang-Jiang, He Xiao-Ting, and Chen Shan-Lin, Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2009, Article ID 634362, 9 pages doi:10.1155/2009/634362;

0

1

2

3

4

5

6

7

0 0,5 1 1,5 2Wnl*/Wl*

β

Wmax*=0,05 Wmax*=0,25 Wmax*=1


[4] Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method, W.X. Wu, C. Shu, , C.M. Wang, Journal of Sound and Vibration 306 (2007) 252–270; [5] Free vibration analysis of rectangular membranes with variable density using the discrete singular convolution approach, H. Ersoy, Ö. Civalek and L. Özpolat, Asian Journal Of Civil Engineering (Building And Housing) Vol. 11, no. 1 (2010) pages 83-94. [6] Nonlinear Vibration Of Wrinkled Membranes, S. Kukathasan and S. Pellegrino, University of Cambridge, Cambridge, CB2 1PZ, UK 44th, AIAA 2003-1747; [7] Membranes With Continuously Varying Thickness, Renata M. Soares and Paulo B. Gonçalves, ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 11, NO. 1 (2010) PAGES 83-94 [8] Nonlinear Vibrations of a Rectangular Hyperelastic Membrane; H. Ersoy, Ö. Civalek and L. Özpolat, ENOC 2011, 24-29 July 2011, Rome, Italy. [9] The effect of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part I: Simply supported and Clamped Clamped Beams, R. Benamar , M.M.K Bennouna and R.G. White. Journal of sound and vibration 1991, 149 (2), 179-195; [10] The effect of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part II: Fully Clamped Isotropic Rectangular plate, R. Benamar , M.M.K Bennouna and R.G. White. Journal of sound and vibration 1993, 164 (2), 295-316. [11] The effect of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures. Part III: Fully Clamped Isotropic Rectangular plate. Measurements of the mode shape amplitude dependence of the spactial distribution of harmonic distortion,R. Benamar , M.M.K Bennouna and R.G. White. Journal of sound and vibration 1994, 175 (3), 377-395; [12] Membrane Vibrations,W. Leissa and S. Qatu, Vibrations of continuous Systems, Chapter 5, pages 181 – 216. [13] Plates and Shells, Tomaz Wierzbicki , Pages 14-16.

· pdf file · 2016-05-24title: microsoft word - eccomas2016-gamzi_s paper.docx

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