Mechanics of Composite Materials And Vibration

8/16/2019 Mechanics of Composite Materials And Vibration

1/23

Jordan University of science and technology

Department of Civil Engineering

SEMINAR

CE7!

"#rgans Donation$

Anas Adeeb Mohammed Elsharo.

20143023010

Date% May& ' ()*+

S,-mitted to% .rof+ /ahia A-del0Ja1ad


2/23

A2S3RAC3%

Laminated composite materials are widely used in the range of application according to theirhigh strength high durability and low weight compared to other materials used in structural

application. !he composite laminas are the main material used in manufacturing the body of the

space roc"ets airplanes and cars. Accordingly the aim of this pro#ect is to study the natural

fre$uencies and mode shapes for the laminated composite plates and the e%ect of increasing the

number of laminas in the natural fre$uency. !he pro#ect then will be e&tended to study the

'ibration of special orthotropic plates rested on elastic foundation and the case where the plate

is immersed in a medium.

IN3R#DUC3I#N%

(omposite Materials consist of two or more materials which are together producedesirable properties that cannot be achie'ed with any of its constituents alone. )iberreinforced composite materials for e&ample contains high strength and high modulus*bers in a matri&.

+n the composites *bers are the principal load,carrying capacity and the matri& "eepsthe *bers together and acts as a load transfer medium between *bers and protects the*bers from being e&posed to the en'ironmental and e&ternal conditions. )ibers ha'enear crystal,si-ed diameter and ha'e a 'ery high length to diameter ratio.

)iber,reinforced composite materials for structural applications are often of a thin layercalled lamina. A lamina is a macro unit of material whose material properties aredetermined using proper laboratory tests.

tructural members such as bars beams and plates are then formed by stac"ingnumber of layers with each other to reach the desirable properties or strength re$uiredby this structural element.


3/23

!he *ber orientation in each lamina and the stac"ing se$uence are chosen by thedesign engineer in order for the laminated member to reach the re$uired sti%ness andstrength and to be of a lightweight based on the function of the structural member inconsideration.

!he structural application of the composite materials and *ber reinforced materialsincluding the use of these materials in aeronautics and mo'ing 'ehicles where highstrength material is needed but with a lightweight.

!he *gure below shows the use of carbon *ber reinforced polymers in the body of A,3/0airplane the () plates and sheets ha'e a high strength with lightweight which helpsthe airplane body to ha'e the desired strength without any loss in the functionality ofeciency.

!he below table shows some materials used as composite material and their mechanicaproperties


4/23

According to the use of composite plates in 'ibrating mechanical and structuralmembers then there is a need to study the 'ibration natural fre$uencies and modeshapes for the composite plates. !his study is intended to show how lamination angleand how the number of layers will a%ect the natural fre$uencies and mode shape forspecial orthotropic simply supported plates.

ibration analysis has its beginnings with 5alilei 6178491842: who sol'ed by geometricameans the dependence of the natural fre$uency of a simple pendulum on the pendulumlength.

!he mathematics of di%erential calculus prospered and pa'ed the way for Le ond;


5/23

0De4nitions%

efore deri'ing the static and constituti'e relations for composite plates we will ma"e

some de*nitions related to composite plates and their types

5aminate, A material consisting of layers 6laminae: bonded together.

! ransversely isotropic0 materials are special orthotropic materials that ha'e one a&is

of symmetry 6any other pair of a&es that are perpendicular to the main one and

orthogonal among them are also a&es of symmetry:

#rthotropic, A material that has di%erent mechanical properties in three mutually

perpendicular planes.

?ote that the properties of the material are direction speci*c in this case. Al

unidirectional laminae are indi'idually orthotropic. Most laminated composites fall into

this category.

Homogeneous-

material or system has the same properties at e'ery point@ it is uniform

without irregularities.

?ote that since the laminated composite plates consist of *bers connected into a matri&

they are ne'er truly homogenous. ?e'ertheless while studying the elastic response of

plates in a macroscopic scale the plate could be considered homogeneous.

Angle ply Laminates- laminates containing plies that are oriented on angles other than 0

and 0 degree.

Balanced Laminates- )or each positi'e angle ply laminate in the laminae there is negati'e

angle ply. !his does not include 0 and 0 degree laminas.

Symmetric Laminas- !he plies of the laminate are mirror image about the geometrica

midplane.

Principal Material Direction- ;irections parallel and perpendicular to the direction of the *bers

in a lamina.6!hey are not the direction of the principal stress:.

https://en.wikipedia.org/wiki/Transverse_isotropyhttps://en.wikipedia.org/wiki/Transverse_isotropyhttps://en.wikipedia.org/wiki/Transverse_isotropy


6/23


7/23

Mechanics of Composite .lates%

0Constit,tive Relations%

+n the analysis of laminates the lamina is always assumed to be in a Bare gi'en as

following

6here %

i8 are the components of the lamina sti%ness matri& which depend on both modulus of

elasticity and poission


8/23

3ransformation of lamina sti9ness matri: %

+f the laminated composite plate is an angle ply the lamina sti%ness matri& elements

are transformed by the following relations

Strain and c,rvat,re relations %

Although the laminate is made up of multiple laminae its assumed that the indi'idual

laminae are perfectly bonded together so it beha'es as a unity ?onhomogeneous and

anisotropic Clate.

Strain0displacement relations for the in plane strains %


9/23

3he strains at the middle s,rface %

3he c,rvat,re at middle s,rface %

Res,ltant ;orces and Moments %

;orce per ,nit length % Moment per

,nit length %


10/23

After using the stress strain relations in the pre'ious integrations the e$uations ofresultant forces and moments can be written in the following form

!he pre'ious e$uations can be rewritten also by introducing some constants that depend

on sti%nesses as following

Dhere

!he complete set of e$uations can be written in a matri& form as following

Ai# is the laminate e:tensional sti9nesses and given

2i# is the laminate co,pling sti9nesses and given -y

Di# is the laminate -ending sti9nesses and given -y


11/23

+t can be easily shown that the coecient i# is related to geometry and material

symmetry about the middle surface. +n other words i# is non-ero for asymmetry plates

about its mid surface.

!he isotropic lamina properties lead to the condition

A18 A28 ;18 ;28 0 .

Analysis of Small 3ransverse De


12/23

S,mmation of the forces along the >0direction %

S,mmation of the moments a-o,t the :0direction %

S,mmation of the moments a-o,t the y0direction %

After s,-stit,tion 1e get the moment e?,ili-ri,m e?,ation %

Special case %

!rans'erse deFection of the rectangular specially orthotropic plate which is simply

supported in all edges and carrying a load of $ 6&y: .

!he general iharmonic e$uation is


13/23

!he iharmonic e$uation for this special case will be as following

Moments in terms of displacement 1 %

>ere ?a'ier solution is acceptable here because all the re$uirements are met . !he boundary conditions are

!he fourier series e&pansions for both e&ternal loading and displacement are

!he boundary conditions and the di%erential e$uation are satis*ed and the constants

are


14/23

As a result we can conclude that the same deri'ation techni$ues ha'e been used to

deri'e the displacement e&pression in both the con'entional medium thin plates and

laminated composite plate . >owe'er the main di%erence appears in the Fe&ural

sti%ness of plate ; . !he laminated composite plate has di%erent Fe&ural sti%nesses

depending on the material properties for each layer . !herefore the displacement

formula for the two types of plates will be the same but with di%erent constants in their

fourier e&pansions . )rom which moments normal and shear forces slopes and normal

and shear stresses can be found easily by applying the same methodologies of

con'entional plate .

3ransverse @i-ration of 5aminated .late

+n this section the trans'erse free 'ibration of laminated composite plate is studied in

order to *ned the natural fre$uencies and the mode shapes . !his study is 'ery important

because this type of plates is mainly used in locations that e&posed to dynamic loadings

and its dynamic beha'iors must be concerned .

!he same method of deri'ation is ta"en into account to deri'e the displacement formula

for free 'ibrated plate . Gne term is added to the *nal biharmonic e$uation . !his term is

the trans'erse inertial force that acts in the -,direction . !he inertial force depends on

second deri'ati'e of displacement with respect to time . !herefore natural fre$uencies

and mode shapes are deri'ed directly from displacement e$uation .

!he series of e$uation will lead to the general moment e$uilibrium e$uation


15/23

+n this we the free trans'erse 'ibration of a specially orthotropic rectangular and s.s

plate is considered .

!he moment e$uilibrium e$uation in terms of displacement will be as following %

)or this case the solution is assumed to be

3he -o,ndary conditions are satis4ed -y sol,tion in the form of %

3he nat,ral fre?,ency e?,ation %


16/23


17/23

!o deri'e the e$uation of motion the following y,direction forces must be added to the

pre'ious *gure

1, +nertia forces which is e$ual to

2, pring force 6elastic bed force: which is e$ual to HID@ where H is the sti%ness of the

bed.

umming the force in the y direction and based on the deri'ation of the plate 'ibration

we will end up with the following result

!o sol'e the e$uation of motion for the case of simply supported plates we will also

consider a solution of the type

6here Dmn is the mode shape of the 'ibration and J1 is the &,a&is and J2 is the y,

a&is.

ubstituting the proposed solution into the e$uation of motion we will end up with

relationship


18/23

3he overall sol,tion 1ill tae the follo1ing form %

@i-ration of .lates Immersed in a medi,m% Damping of

@i-ration Amplit,de

!he resistance force will depend upon the 'elocity of the plate and directly multiplied by

a constant ( called the drag coecient. Accordingly the e$uation of motion will be of theform

!he solution is assumed to be of the type

It is sho1n from the previo,s e?,ationthe e9ect of the elastic -ed onincreasing the fre?,ency and lo1eringthe period of vi-ration+

;or the vi-ration of plates immersed in avisco,s


19/23

ubstituting the proposed solution in the e$uation of motion we will get

implifying the abo'e e$uation the abo'e e$uation will be of the form

!he last e$uation is a $uadratic e$uation in which it is solution depends upon the 'alue

of the damping or drag coecient (

3he e?,ation 1ill have t1o roots of the form%

ubstituting the abo'e solutions in the proposed solution for the plate deFection we will

end up with the e&ponential of real part plus an imaginary part. !he e&ponential of theimaginary part will be treated using Euler formula in order to transform it to a harmonic

form. !he solution will be of the form


20/23

Dhere Amn and mn are constants that can be found from the initial conditions for e'ery

'alue of m and n.

Matla- E:amples%

ecause we didnKt ha'e any reference and all the deri'ations here ha'e been carried out

by our own calculations we decided to ma"e some e&amples on matlab to 'alidate the

results and formulas we get .

!herefore for the *rst case we sol'e a two layer laminated composite plate in free

'ibration . !he material properties are

E1 13/ 5Ca. E2 5Ca. 512 8. 5Ca.

12 0.3.

;ensity1.8&10,3 gmm3

Dith thic"ness of 2mm.

!he ; Matri& will be as follow

; 2.7432 1./108 0

1./108 8.0374 0

0 0 4.8000


21/23

Nat,ral ;re?,ency 1ill -e as follo1%

w11 8=3.21= radsec.

w12w211020 radsec.

3he 4rst t1o mode shapes are %

;or a plate on elastic the follo1ing e:ample has -een solved %

Assume that the pre'ious plate resting on an elastic foundation where the sti%ness is" 2 ?mm.

wmn80.3 rad sec.

!he olution )or a plate with arbitrary properties and conditions will ha'e the following

shape


22/23

;or this case the period and damped nat,ral fre?,ency are %

! 0.82 sec .Dd 10.3 radsec .

Recommendations%

)or the case where the plate has other boundary conditions other thansimply supported for the cases discussed later A. D. Leissa in a ?AA

publication listed many shape function for plates with di%erent boundary

conditions including cantile'er and free plates@ the substitution of these

shape functions into the pre'iously deri'ed e$uations will gi'e a solution for

those plates whether they are supported on an elastic foundation or

'ibrating inside a medium. !he following shows the shape functions

proposed by Leissa


23/23

Date post:	05-Jul-2018
Category:	Documents
Upload:	sameralawneh
View:	238 times
Download:	1 times

Mechanics of Composite Materials And Vibration

Documents