FREE VIBRATION CHARACTERISTICS FOR DIFFERENT ... IJMME-IJENS.pdf · as doors, walls, and floors....

International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:10 No:03 27

I J E N S IJENS © June 2010 IJENS-IJMME 6969-034210

Free Vibration Characteristics for Different

Configurations of Sandwitch Beams

Prof. Dr. M. Abdel Salam

* & Dr. Nadia E. Bondok

**

*Design and Prod. Engin Dept Faculty Of Engineering ,Ain Shams Univ. Cairo, Egypt

** Department of Technology Development, Specified Studies Acadmy Worker’s Univ. Ras Albar, Egypt

Email: * [email protected]**[email protected]

Abstract-- A generalizes model presenting the sandwich

beams was developed to calculate the flexural rigidity and sandwich beams dynamic characteristics . Different cases such

as sandwich beams multi layer cores, sandwich beams multi

cells, sandwich beams with holes in its cores having different

shapes and different orientations were investigated. The finite

element code ANSYS 11 was used for free vibration analysis of the sandwich beams; the natural frequencies and mode

shapes and the static deflection of the sandwich beams were

calculated. The obtained results from the finite element code

ANSYS 11 such as static deflections, static rigidity and natural

frequencies were compared with that obtained from the generalized equations according the cases of investigations

which appear to be in good agreement with each others,

therefore the generalized model can be used for the best design

of the sandwich beams.

Index Term-- Sandwich beams , Static and Dynamic

characteristics, Free vibration, Finite elements, ANSYS

I. INTRODUCTION

To face the challenge of more and more complicated

loading conditions, designs of lighter, safer and efficient

structures with multi-functinality have been motivating

researchers since a long time. Light weight multi layered

structures and sandwiches also called laminate panels

belong to one such kind of advanced structures developed

up to now [1]. Lightweight material and structure designs

with elegant mechanical properties, e.g., energy absorption,

thermal isolation, anti-impact constitute always a

challenging work for aerospace, and automotive industries.

Geometrically, these structures can be considered as a result

of periodic repetitions of a basic cell in one, two or three

dimensions[2] . A sandwich material is a layered assembly

made from two thin, strong, and stiff face sheets bonded to a

lightweight compliant core material. This creates a stiff,

strong and also a very lightweight structural element. These

structures can carry both in-plane and out-of-plane loads and

exhibit good stability under compression, keeping excellent

strength to weight and stiffness to weight characteristics.

The many advantages of sandwich constructions, the

development of new materials and the need for high

performance and low-weight structures insure that sandwich

construction will continue to be in demand [3-5]. M.Cheheh

etal[6] stated that sandwich beams and plates which are

categorized as composite structures with high value of

strength to weight ratios are considered as new specific

advanced structures. Also the free vibration behavior of the

sandwich beam with FGM core material was analysed using

a meshless method, where the element free Galerkin method

as known and robust method is utilized.

As stated by [7] that, the purpose of the core is to maintain

the distance between the laminates and to sustain shear

deformations. By varying the core, the thickness and the

material of the face sheet of the sandwich structures, it is

possible to obtain various properties and desired

performance.

There are many wide varieties of core materials geometry

currently in use. Among them, honeycomb, foam, balsa and

corrugated cores are the most widely used. Usually

honeycomb cores are made of aluminum or of composite

materials: Nomex, glass thermoplastic or glass -phenolic.

The other most commonly used core materials are expanded

foams, which are often the most to achieve reasonably high

thermal tolerance, though thermoplastic foams and

aluminum foam are also used. For the bonding of laminate

and core materials, normally two types of adhesive bonding

are commonly employed in sandwich construction, i.e., co-

curing and secondary bonding. Characterization of sandwich

materials has been carried out in detail in scientific

literature. The determination of the sandwich material

behavior under crushing loads and the measurements of the

ductile fracture limits is normally done with the help of

compression tests[8,9]. Among the various honeycomb

sandwich structures, adhesively bonded aluminum

honeycomb sandwich panels are currently popular because

of their easy assembly of face sheets and cores. One

application of such structures is in the cabin sections, such

as doors, walls, and floors. Since these sandwich structures

are subjected to the dynamic service loading, the

progressive fatigue failure are frequently inspected.The

homogenized elastic properties of honeycomb sandwich

structures with skin effect were subsequently presented by

Xu and Qiao [10-12]. As shown in Fig. 1, a variety of

sandwich structures with different cell configurations and

arrangements are illustrated [1]. In ref [13] Noor etal gave

a comprehensive review of different computational models

on sandwich panels and shells. Applications were involved

in problems of heat transfer, thermal and stresses, free

vibration and damping, transient dynamic responses, design

optimization, etc. investigated the static deflection of

laminated composite beams.

Recently, Hsueh et al.[14] studied the biaxial strength of

thin multilayered disks. An analytical model of general

closed-form solutions is developed for the elastic stress

distributions subjected to biaxial flexure tests. Burgueno et

al.[15] studied the designs of hierarchical cellular sandwich

beams and plates with high structural efficiency. Based on

mailto:[email protected]**nana



experimental tests of cellular plates with different densities

of porosity and different hole arrangements over the cross -

section, the measured results of effective elastic constant

related to the bending stiffness showed a good agreement

with the shape factors and material indices. Alternatively,

Wang and McDowell [16] and Hayes et al. [17] studied

extruded metal honeycombs, i.e., the so-called linear

cellular alloys (LCA).The first work was focused on the

maximization design of elastic torsion and bending rigidities

of the circular sandwich bar structure in terms of the

triangular subcell geometry of the sandwich core while the

second work was to understand the heat transfer and

mechanical behaviors of the LCA. M. Abdel Salam & N.E.

Bondok[5] studied the effect of the sandwich beam elements

on static and dynamic characteristics of the sandwich beams

with single core where different configurations of

sandwich beams (different core and face materials and

dimensions) were presented and investigated theoretically

using developed method and experimentally using the

impulse excitation test where good agreement between

theoretical and experimental results were obtained.

The vibration characteristics of sandwich materials have

drawn much attention recently. The dynamic parameters of

a structure, i.e., natural frequency, damping and mode

shapes, are determined with the help of vibration testing

which provides the basis for rapid and inexpensive dynamic

characterization of composite structures [18]. The

importance of material damping in the design process has

increased in recent years as the control of noise and

vibration in high precision, high performance structures. In

polymeric composites, the fiber contributes to the stiffness

and the damping is enhanced owing to the internal friction

within the constituents and interfacial slip at the fiber/matrix

interfaces. [19,20] Meng-Kao etal[21] studied the dynamic

properties of sandwich beams through experimental tests

and using finite element methods. It was found out that, the

faces dominate the stiffness of the sandwich beams, the

natural frequencies were affected directly by the face

materials and decreases with the increasing fiber orientation

of the graphite /epoxy face laminates. Increasing the face of

the cores increases both frequencies and loss factors of the

sandwich beams.

The aim of the present work is to develop a generalized

model presenting the sandwich beams to calculate the

flexural rigidity , dynamic characteristics and the

characteristics of their free vibrations for different cases

such as sandwich beams multi layer cores, sandwich beams

multi cells, sandwich beams with holes in its cores having

different shapes and different orientations in wrt the

sandwich beam axis.

II. THEORETICAL ANALYSIS

Generalized flexural rigidity model for the multi layer

cores sandwich beams MLC with (nc) cores and constant

cross section.

The sandwich concept is based on two main ideas:

increasing the stiffness in bending of a beam or a panel and

doing so without adding excessive weight. Here, sandwich

beam structures may refer to multi layered structures with

symmetrical cross-sections. Suppose the considered

sandwich structure is generated by a periodic distribution of

unit cells, each of which consists of upper face, soft core

and lower face which is consider to be the upper face of the

second cell and so on as presented in fig. 1. To figure out

the problem, consider firstly the bending of a multi layered

beam with (nc) number of cores. The general term for

bending stiffness is the flexural rigidity (D), which is the

product of the material(s) elastic modulus(E), and the cross

section moments of inertia (I). For a symmetric sandwich

beam (all skins have the same thickness and material

properties also for the cores), the formula for flexural

rigidity in the case of sandwich beam with nc cores is

expressed as follows:

cc n

i

cici

n

i

fifi IiEIiEEID1

1

2

………(1) Where : Ef and Ec are the Young’s modulus of faces and

cores materials respectively.

The cross section moment of inertia for faces fiI and cores

i cI are given by:

c

22

ff

3

c ........n 0,2,4 i , )()h ( 4

bh2

12

bh )1n (

cn

i

cc

f

fi inhI

…..(2)

And

c

2

c

1n2

fc

3

ci .........n 0,2,4 i ,))1((n )(h 2

bh

12

bh n

c

ihIi

c

c

c

………(3)

Where: b denotes the beam width, h f and hc are the faces

and cores thicknesses respectively . The total beam

height(H) is therefore given by:

H = nc(hc +hf)+hf

………..(4)

Generally, flexural rigidity (D) relationship can be

expressed for (nc) –number of cores (i.e number of cells )as

follows:

c

2

c

1n22

3

33

n., 0,2,4 i, ] ))1((n )([)(2

]1)1([12

c

ih

h

E

Einhh

hbE

h

h

E

En

hbED

icf

c

f

cn

i

ccf

ff

f

c

f

c

c

ffc

….(5)



The investigation of the generalized flexural rigidity

function:

1-For sandwich beam with single core:

Let the number of cores nc =1 ie single core equation (5)

yields to:

2

3

33

)(2

)]2[(12

cf

ff

f

c

f

cffhh

hbE

h

h

E

EhbED

….(6)

Which was obtained by [5]. 2 - Comparison between sandwich beam with single core

and that with nc cores with the same height (H)

Keeping the sandwich beam height (H) constant , that can be

achieve by dividing the face height and core height by the

number of cores nc, therefore equation (5) yields to:

c

2

c

1n22

33

3

3

3

n., 0,2,4 i, ] ))1((n )([)(2

]1)1([12

c

ih

h

E

Einhh

n

hbE

h

h

E

En

n

hbED

if

c

f

c

n

i

ccf

c

ff

f

c

f

cc

c

ffc

. …(7)

3- The sandwich beams composed of multi cells having

the same dimensions.

In this case the sandwich beam can be diagrammatically

presented as shown in fig.1 and the flexural rigidity is given

as follows:

c

2

c

1n22

3

33

n., 0,2,4 i, ] ))1((n )([)(2

]1)1([12

c

ih

h

E

Einhh

hbE

h

h

E

En

hbED

icf

c

f

cn

i

ccf

ff

f

c

f

c

c

ffc

……..(8) 4- The sandwich beams with (nc ) cores and periodically

variable cross-section.

Here, another sort of sandwich beams that have periodically

variable cross-sections in the longitudinal or cross direction

of the beam axis will be investigated. This can be occur by

making hole(s) in the direction or in the cross direction of

the beam axis. The aim of this is to control the value of the

beam flexure rigidity, mass, natural frequency and modes of

vibrations (static and dynamic performance of the beam).

The beam has an upper and bottom face sheets. The core of

the unit cell of the beam has a rectangular, square or circular

holes. In order to evaluate the effective flexural rigidity, the

Boolean operation is used to subtract the contribution of

holes. With this idea in mind, the general expression of the

flexural rigidity for a cross-section can be calculated in the

following cases.

A) Sandwich beam cross-section with holes in the cores

in the direction of the beam axis:

I-Square or rectangular holes in the cores: As shown in

fig. 2, the beam has an upper and bottom face sheets. The

unit cell of the beam has an number of square or rectangular

holes (nh) in it (i.e. nh = nc). The moment of inertia of the

square or rectangular hole (with width bh , height hh and

length L ) for nc cores is expressed as follows:

c

2

c

1n2

f

3

c .........n 0,2,4 i ,))1((n )(h 2

hb

12

hb n

c

ihIi

chhhh

h …….(9)

For square hole let bh , = hh in equation (9)

II- Circular holes in the cores:

The moment of inertia of the circular holes (where hole

diameter dh i.e. bh=hh= dh and length L ) is expressed as

follows

……(10)

Generally equation ( 9 and10 ) can be given in a general form as follows:

c

2

c

1n2

f2

3

c .........n 0,2,4 i ,))1((n )(h 2

hb hb n

c

1

ihaaIi

c

hh

hhh ……(11)

Where :

in case of rectangular or square hole [a1 = 0.083 , a2 = 1]

c

2

c

1n2

f

3

c .........n 0,2,4 i 1((و((n )(h 2

hb)

4( hb

64 n

c

ihIi

chh

hhh



and in case of circular hole[a1 = 0.0156 , a2 = 0.785]

In this case, the flexure rigidity can be expressed as follows:

cc n

i

cieci

n

i

fifi IEIED1

1

2

…..(12)

Where Icie is the effective moment of inertia of the core(s)

which can be calculated as follows:

hicicie III

……(13)

And the corresponding flexural rigidity can be calculated as

follows:

c

i

chh

hhc

icf

c

f

c

n

i

ccf

ff

f

c

f

cc

ff

niiha

aEih

h

E

Einhh

hbE

h

h

E

En

hbED

c

...,6,4,2,0],))1((n )(h 2

hb

hb n [ - ] ))1((n )([)(2

]1)1([12

2

c

1n2

f2

3

c

2

c

1n22

3

33

c

1

c

…….(14)

B) Sandwich beam with holes in the cores in the across

beam axis direction:

I-Square or rectangular holes in the cores:

As shown in Fig.3, the beam has an upper and bottom face

sheets. The unit cell of the beam has nh rectangular or

square holes in the core in the direction across its axis . The

moment of inertia

of the holes can be calculated as follows:

c

2

c

1n2

f

3

ch .........n 0,2,4 i ],))1((n )(h 2

hb

12

hb [nn

c

ihIi

c

hhhh

h

….(15)

According to Fig.3, the maximum number of holes can be

given by:

)12

(2

)2(

hh

h

hL

L

L

LLn

…(16)

2-Circular holes in the cores:

As shown in Fig. 3,. The unit cell of the beam has nh

circular holes with hole diameter dh in the core in the

direction across its axis . The effective flexural rigidity, is

affected by the number and hole size, The moment of inertia

of the holes can be calculated as follows:

c

2

c

1n2

f

3

ch .........n 0,2,4 i ],))1((n )(h 2

hb b

64

1 [nn

c

ihhIi

chh

hhh ……(17)

Where the maximum number of holes can be expressed as

follows:

)12

(2

)2(

hh

hh

d

L

d

dLn

…….(18)

The corresponding flexural rigidity can be calculated as

follows

c

2

c

1n2

f2

3

c1h

2

c

1n22

3

33

.........n 0,2,4 i ],))1((n )(h 2

hba

hb n [an -] ))1((n )([)(2

]1)1([12

c

c

ih

Eih

h

E

Einhh

hbE

h

h

E

En

hbED

i

chh

hhc

icf

c

f

c

n

i

ccf

ff

f

c

f

cc

ffc

…..(19)

For nc=1 and nh=0 equation (19) can be expressed as follows

23

3

33

)(2

])2[(12

cf

ff

f

f

c

f

cffhh

hbEh

h

h

E

EhbED

…..(20)

Which was obtained by [5]

The determination of the dynamic characteristics of

multi layer cores MLC sandwich beams:

The equivalent dynamic characteristics of MLC sandwich

beams can be calculated according to its end conditions. In

case of cantilever beams supported from one end and free

from the other the dynamic characteristics can be expressed

as follows:

A-The equivalent mass ( me):

chcccffce Vbhnbhnm ])[1(

…………(21)

Where: ρf and ρc are the densities of the face core materials,

Vh is the volume of the holes in the cores which can be

calculated according to its shape, orientation w.r.t. the beam

axes and calculated as follows:

- For rectangular or square holes:



LhbnnV hhhch

…………(22)

- For circular holes:

LdnnV hhch

2785.0

…………(23)

Where: nh = 1 in case of the holes are in the direction of the

beam axes:

B-The equivalent stiffness Ke:

33 /103 LDKe

…………(24)

Where : Ke is the equivalent stiffness , N/m and the flexural

rigidity

C-The equivalent natural frequency fn:

)25.0/(2

1een mKf

…………(25)

The frequencies of different modes can be calculated

according to the following equations:

)25.0/(2

2

ee

i

i mKf

…………(26)

Where , β2

i is a constant dependent on the boundary

conditions, and i = 1, 2 . . . n are the frequency corresponds

to each mode.

D-The equivalent beam damping coefficient Ce:

Damping is an important modal parameter for the design of

structures for which vibration control and cyclic loading are

critical. In sandwich beam structures, the equivalent

damping coefficient( ec ) is affected by the damping

coefficient of the face and core materials The equivalent

sandwich beam damping coefficient is equal to the

summation of the total damping coefficient of the faces Cf

and the total damping coefficient of the Cc cores and can be

expressed as follows:

cfe CCC

…………(27)

Where : Cf : coefficient of equivalent viscous damping of

the faces beams , Ns/m Cc: coefficient of equivalent viscous damping of the core

beams, Ns/m

The coefficient of viscous damping of the faces beams

(cf)can be expressed in terms of the critical damping

coefficient of the face beam materials as follows:

…………(28)

On the other hand, the coefficient of viscous damping of the

core(cec) beams can be expressed as follows:

n

cc

cef

kC

………(29)

Where:

ηf : damping factor of the face plate material

ηc: damping factor of the core material

Therefore the equivalent coefficient of viscous damping of

the MLC sandwich beam (ce)can

be expressed as follows:

][3

3 eicccfff

n

e IEIELf

C

…………(30)

Where If and Ieic can be calculated according to MLC end

conditions and the holes orientation in the cores .

III. RESULTS AND DISCUSSIONS

The finite element code ANSYS 11 was used for free

vibration analysis of the sandwich beams; the natural

frequencies and mode shapes and the static deflection of the

sandwich beams were calculated. Element Solid45, having

eight nodes and six degrees of freedom per node, was used.

The obtained results such as static deflections, static rigidity

and natural frequencies were compared with that obtained

from the generalized equations according the cases of

investigations. In the analysis the material properties of the

core and face materials of sandwich beams are (Ef=210

GPa, Ec= 4 GPa , ρf = 7800 Kg/m3, ρc= 1200 Kg/m

3).the

sandwich beam width = 20 mm and length L= 300 mm. A

perfect bonding at the interface between the face and the

core materials was assumed. The sandwich beam, is

considered to be cantilever type, i.e fixed at one end.

1-Comparison between sandwich beam with single core

or multi cores keeping the total beam height (H)

constant.

Fig. 4. shows the static and modes shapes at response of a

sandwich beam of a single core and sandwich beams with

multi cores having the same beam height (H=16mm). For

this case the thicknesses of the faces and cores are varied

according to the number of cores (nc). It is observed that the

beams rigidity and hence the beam deflections as well as

the 1st

,2nd

,3rd

and 4th

mode shapes are the same and their

frequencies are decreased as presented in Fig. 5. These

behaviors can be explained by the influences of the faces

n

ff

ff

kC



and cores thicknesses on the( Ic and If )of the beams and

hence on the beams flexural rigidity(D).The values of

constant β2

i which depends on the boundary conditions(

fixed- free fixation), are β2

1 =1, β2

2=5.5 β2

3=13.3 β2

4=22.4.

A good agreement between the values of static deflections

and natural frequencies which obtained from ANSYS 11

with that obtained from theoretical analysis . The obtained

results indicates that both of the natural frequency and static

rigidity can be adjusted to a certain values by the best

selection of the numbers of sandwich beam cores .

2- Sandwich beams composed of multi cells having the

same dimensions.

Fig.6, shows the static and modes shapes of a sandwich

beam of a single core and sandwich beams with multi

identical cells having the same beam height (H). It is

observed that as the cell numbers increase, the beam

rigidity are increased and hence the beam deflections are

decreased as well as the 1st

,2nd

,3rd

and 4th

mode shapes are

the same and their frequencies are increased as presented

in Fig.7,. These behaviors can be explained by the

influences of the faces and cores thicknesses on the( Ic and If

)of the beams and hence on the beams flexural rigidity(D).

A good agreement between the values of static deflections

and natural frequencies which obtained from ANSYS 11

with that obtained from the theoretical analysis . The

obtained results indicate that both of the natural frequency

and static rigidity can be adjusted to a certain values by the

best selection of the numbers and dimensions of sandwich

beam cells.

3- Sandwich beams having holes in its core.

Fig. 8. shows the modes shapes of a sandwich beam of a

single core without holes in its core and another one having

a squre hole in the direction of the sandwich beams axis and

a third one having 15 holes in its core in the cross

direction of the beam axis . All the sandwich beams having

the same beam height (H). It is observed that the beam

rigidity, as well as the 1st

,2nd

,3rd

and 4th

mode shapes and

their frequencies are almost the same A good agreement was

observed between the values obtained from ANSYS 11 with

that obtained from the theoretical analysis

IV. CONCLUSIONS

From the theoretical analysis and obtained results it can be

concluded that:

1- A generalized model is developed. The model is

able to investigate the effect of the number of

cores, multi cells, and the holes shape and their

orientation w.r.t. the beam axis on the static

rigidities, natural frequencies and hence on the

dynamic properties of the sandwich beams.

2- The finite element code ANSYS 11 was used for

free vibration analysis , the natural frequencies ,

mode shapes and the static deflection of the

sandwich beams are calculated.

3- The investigation revealed that the static and

dynamic responses of the sandwich beams can be

adjusted the increasing of the number of cores or

the number of cells.

4- The values of constant β2

i which depends on the

boundary conditions( fixed- free fixation), are β2

1

=1, β2

2=5.5 β2

3=13.3 β2

4=22.4 these values are

used for calculating of the modes frequency.

5- The obtained results from the finite element code

ANSYS 11 such as static deflections, static

rigidity and natural frequencies were compared

with that obtained from the generalized model

according the cases of investigations where good

agreement between each other is obtained.

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[17] Wang, A.J., McDowell, D.L., “Optimization of a metal honeycomb sandwich beam-bar subjected to torsion and bending”.International Journal of Solids and Structures,Vol. 40, (2003).PP. 2085–2099.

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[19] Gibson, R.F., “Modal vibration response measurements for characterization of composite materials and structures”.,

Composite Science and Technology,Vol.60 ( 2000).PP. 2769–80.

[20] Wang B, Yang M., “Damping of honeycomb sandwich beams”. J. Mater Process Technol. Vol.105(2000).PP. 67–72.

[21] Li, Z., Crocker, M.J., “Effects of thickness and delamination on the damping in honeycomb-foam sandwich beams”. J. Sound

Vibr.Vol. 294 (2006).PP. 473–85. [22] Meng-Kao Yeh, Tsung- Han Hsieh,''Dynamic properties of

sandwich beams with MWNT/polymer nano-composites as core materials.'', Composites science and Technology Vol.68

(2008),PP.2930-2936.

First - cell

Second - Cell

Third - Cell

n - Cell

b L

L

b=

bh

h

h

+

h

c

hf

f

h

f

h

f Lh Lh Lh Lh

Fig. 1. Sandwich beam with nc number of cores

Fig. 3. Sandwich beam with rectangular and circular holes in the core in the

cross direction of its axes (one cell where nc = 1 for presentation)

Fig. 2. Sandwich beam with rectangular hole in the core in the direction of its axes (one cell where nc = 1 for presentation)

b

bh

hc

hf

hf

hh

L

hf

f

hf

hc



1

X

Y

Z

DEC 13 2009

09:43:30

DISPLACEMENT

STEP=1

SUB =1

FREQ=181.58

DMX =3.751

1

X

Y

Z

DEC 13 2009

09:43:48

DISPLACEMENT

STEP=1

SUB =2

FREQ=945.718

DMX =3.48

1

X

Y

Z

DEC 13 2009

09:44:02

DISPLACEMENT

STEP=1

SUB =3

FREQ=2191

DMX =3.306

1

X

Y

Z

DEC 13 2009

09:45:08

DISPLACEMENT

STEP=1

SUB =4

FREQ=3557

DMX =3.181

A) Single core

1

X

Y

Z

Two core sandwich

DEC 13 2009

09:54:54

DISPLACEMENT

STEP=1

SUB =1

TIME=1

DMX =.002997

1

X

Y

Z

Two core sandwich

DEC 13 2009

10:03:36

DISPLACEMENT

STEP=1

SUB =2

FREQ=863.632

DMX =3.561

1

X

Y

Z

Two core sandwich

DEC 13 2009

10:04:00

DISPLACEMENT

STEP=1

SUB =3

FREQ=2068

DMX =3.389

1

X

Y

Z

Two core sandwich

DEC 13 2009

10:04:15

DISPLACEMENT

STEP=1

SUB =4

FREQ=3442

DMX =3.254

B)Two cores

1

X

Y

Z

Sandwich beam with six cores

DEC 13 2009

10:33:45

DISPLACEMENT

STEP=1

SUB =1

FREQ=137.675

DMX =3.787

1

X

Y

Z


DEC 13 2009

10:33:56

DISPLACEMENT

STEP=1

SUB =2

FREQ=775.836

DMX =3.622

1

X

Y

Z


DEC 13 2009

10:34:07

DISPLACEMENT

STEP=1

SUB =3

FREQ=1908

DMX =3.469

1

X

Y

Z


DEC 13 2009

10:34:16

DISPLACEMENT

STEP=1

SUB =4

FREQ=3247

DMX =3.343

C) Six cores

Fig. 4. Mode shapes for sandwich beams having the same height H=16 mm with different number of cores.

A

B C

Fig. 5. Relation ship between number of cores and : A) Flexural regidity , B) Firest mode natural frequency, C)Mode frequency for differeent modes

1st mode

f1=181 Hz

1st mode

f1=181 Hz 2

nd mode

f2=945Hz

3rd

mode f3=2191 Hz

4th

mode f4=3557 Hz

1st mode

f1=158 Hz 1

st mode

F2=863 Hz 1

st mode

f1=137 Hz 1

st mode

f2=775 Hz

1st mode

f3=1908 Hz 1

st mode

f4=3442 Hz 1

st mode

f3=2068 Hz 1

st mode

f4=3247 Hz



1

X

Y

Z

FEB 22 2010

00:47:09

DISPLACEMENT

STEP=1

SUB =2

FREQ=945.718

DMX =3.48

1

X

Y

Z

FEB 22 2010

00:51:47

DISPLACEMENT

STEP=1

SUB =1

FREQ=273.727

DMX =2.862

1

X

Y

Z

FEB 22 2010

00:51:56

DISPLACEMENT

STEP=1

SUB =2

FREQ=1261

DMX =2.581

1

X

Y

Z

FEB 22 2010

01:08:51

DISPLACEMENT

STEP=1

SUB =1

FREQ=414.073

DMX =2.043

1

X

Y

Z

FEB 22 2010

01:09:01

DISPLACEMENT

STEP=1

SUB =2

FREQ=1573

DMX =1.853

1

X

Y

Z

FEB 22 2010

00:47:20

DISPLACEMENT

STEP=1

SUB =3

FREQ=2191

DMX =3.306

1

X

Y

Z

FEB 22 2010

00:47:31

DISPLACEMENT

STEP=1

SUB =4

FREQ=3557

DMX =3.181

1

X

Y

Z

FEB 22 2010

00:52:05

DISPLACEMENT

STEP=1

SUB =3

FREQ=2745

DMX =2.468

1

X

Y

Z

FEB 22 2010

00:52:20

DISPLACEMENT

STEP=1

SUB =5

FREQ=4256

DMX =2.389

1

X

Y

Z

FEB 22 2010

01:09:12

DISPLACEMENT

STEP=1

SUB =3

FREQ=3214

DMX =1.803

1

X

Y

Z

FEB 22 2010

01:09:26

DISPLACEMENT

STEP=1

SUB =5

FREQ=4741

DMX =1.787

A)One cell B)Two cells C)Four cells

Fig. 6. Mode shapes for sandwich beams having with different number of cells.

A

B

Fig. 7. Relation ship between sandwich beams with different number of cells and : A) Flexural regidity , B) Firest mode natur al frequency.

1st mode

f1=181 Hz 2

nd mode

f2=945Hz

3rd

mode f3=2191 Hz

4th

mode f4=3557 Hz

1st mode

f1=273 Hz 1

st mode

F2=1261 Hz 1

st mode

f1=414 Hz 1

st mode

f2=1573 Hz

1st mode

f3=3214 Hz 1

st mode

f4=4256 Hz 1

st mode

f3=2745Hz 1

st mode

f4=4741 Hz



1

X

Y

Z

JAN 13 2010

10:45:12

DISPLACEMENT

STEP=1

SUB =2

FREQ=183.271

DMX =3.749

1

X

Y

Z

JAN 13 2010

10:45:31

DISPLACEMENT

STEP=1

SUB =3

FREQ=946.894

DMX =3.488

1

X

Y

Z

JAN 13 2010

06:05:03

DISPLACEMENT

STEP=1

SUB =2

FREQ=187.846

DMX =3.859

1

X

Y

Z

JAN 13 2010

06:05:47

DISPLACEMENT

STEP=1

SUB =3

FREQ=946.108

DMX =3.566

1

X

Y

Z

JAN 14 2010

06:13:48

DISPLACEMENT

STEP=1

SUB =2

FREQ=186.142

DMX =3.842

1

X

Y

Z

JAN 14 2010

06:14:41

DISPLACEMENT

STEP=1

SUB =3

FREQ=917.004

DMX =3.515

1

X

Y

Z

JAN 13 2010

10:46:00

DISPLACEMENT

STEP=1

SUB =6

FREQ=2191

DMX =3.339

1

X

Y

Z

JAN 13 2010

10:46:21

DISPLACEMENT

STEP=1

SUB =9

FREQ=3572

DMX =3.267

1

X

Y

Z

JAN 13 2010

06:06:45

DISPLACEMENT

STEP=1

SUB =6

FREQ=2159

DMX =3.424

1

X

Y

Z

JAN 13 2010

06:07:46

DISPLACEMENT

STEP=1

SUB =9

FREQ=3487

DMX =3.365

1

X

Y

Z

JAN 14 2010

06:15:01

DISPLACEMENT

STEP=1

SUB =6

FREQ=2070

DMX =3.363

1

X

Y

Z

JAN 14 2010

06:15:46

DISPLACEMENT

STEP=1

SUB =9

FREQ=3317

DMX =3.273

A)Without holes B)With one square hole C) B)With fifteen square holes

Fig. 8. Mode shapes for sandwich beams with and without holes in the direction of beam axes.

1st mode

f1=181 Hz 2

nd mode

f2=945Hz

3rd

mode f3=2191 Hz

4th

mode f4=3557 Hz

1st mode

f1=188 Hz 1

st mode

f2=946 Hz 1

st mode

f1=186 Hz 1

st mode

f2=917 Hz

1st mode

f3=2070 Hz 1

st mode

f4=3887 Hz 1

st mode

f3=2159 1

st mode

f4=3317 Hz

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