The authors have declared that no competing interests exist.
In the present document, the dynamic analysis of isotropic materials applied to the laminated composite beam has been studied to improve the simplification of the homogenization process of heterogeneous composite materials.
This study is based on the conventional stratified theory of Von Karman's formulation and the kinetic energy of Euler Bernoulli developed through displacement relationships.
The model used in this study to explain the problem of nonlinear geometry in the free vibration applied of laminated beam is based on the Hamilton's principle for solving the motion equation and determining the frequency of the reaction of the rolled beam.
The numerical of the symmetric an asymmetric material gives a good idea of the behaviour of the orthotropic materials, and to minimize the problem of many areas used these materials such as: dental medicine, aeronautical, aerospace, because they have a good strength, high rigidity.
The comparison between the different lay up may give greater importance to this study and open the search to simplify the problem of non linear geometry and give a good result about the older other published.
For instance the application of medicine,the researcher has to decode the compatibility of the human body with new materials,and to minimize the problem of deterioration,The comparison with different lay up give more interesting by the mechanical engineering and the researcher's to give the solution of no linear problem and optimize the composite material’s to adaptation with problem study.
The laminated composite material became important in many field such as: mechanical, electronic, aerospace, aeronautic, dental medicine, medicine, nanotechnology.
They have good properties physics and mechanic can be give more interesting for researcher's and ingenieur’s of different area.
for example the used of composite beam in nanotechnology to concentrate of this properties for search in the development of laminated composite material’s in biology and medicine.
Further more, the application of orthotropic composite beam is increasing in different area because they have more advantages as : aeronautic used the material’s because they have a good soupless,high rigidity.
Therefore this study is based on the application of orthotropic laminated composite material’s in different area and offered the occasion the analyze the procedure of homogenization for composite beam .
The model used in this paper based in the theory of Von Karman for the strain displacement relationships and the equation of Euler Bernoulli which developed the equation of total energy.
This study is based on theory of geometry nonlinear in different area and the problem became very important in many field because ,more the users of composite material’s .such us the aeronautic and aerospace search in properties to improve the rigidity and soupless of material for effect of temperature .
The part is based in the equation of laminated composite material’s of case the orthotropic and isotropic material’s written about the theory of Von  Karman for non linear strain displacement relationships:
The Total elastic energy VT developed
Where;
Total potential energy V_{T} developed in







0,24  1,01  0,24  1,009  237  1,05 
0,32  1,017  0,3  1,016  316  1,087 
0,4  1,028  0,39  1,026  0,39  1,13 
0,5  1,04  0,47  1,03  0,47  1,18 
554  1,054  0,55  1,05  0,54  1,24 
0,6  1,07  0,63  1,06  0,62  1,30 
0,7  1,09  0,71  1,08  0,7  1,37 
0,8  1,1  0,78  1,09  774  1,44 
0,88  1,12  0,86  1,11  0,85  1,5 
944  1,15  0,94  1,14  922  1,59 
The study based in the calculation of nonlinear frequency of orthotropic composite material’s and the test realized about the orthotropic laminated composite beam ,they have the properties perfect: high rigidity, and good stiffness such us: ET=225 GPA , EL=210 GPA ; ρ = 2389Kg/m^{3} ; h=0,01m
The work supported by Mohammed V university in Rabat