Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 3, pp. 789-812, Warsaw 2010

A NEW 2D SINGLE SERIES MODEL OF TRANSVERSE
VIBRATION OF A MULTI-LAYERED SANDWICH BEAM

WITH PERFECTLY CLAMPED EDGES

Stanisław Karczmarzyk

Warsaw University of Technology, Institute of Machine Design Fundamentals, Warszawa, Poland

e-mail: karczmarzyk st@poczta.onet.pl

A new two-dimmensional, single series local model of the transverse vi-
bration of amulti-layer, one-span sandwich beam composed of isotropic
layers with ideally (perfectly) clamped ends is proposed in the paper.
The model is derived within the local theory of linear elastodynamics
and it is composed of two two-dimmensional fields and of two approxi-
mations of three-dimmensional fields satisfying exactly the equations of
motion as well as the Saint-Venant compatibility equations of the the-
ory. All through-the-thickness boundary conditions of the local theory
of elastodynamics as well as all local compatibility equations (for the
displacements and stresses) between adjoining layers are fulfiled in the
model. Both the cross-sectional warping and the transverse complian-
ce(s) in each layer of the beam are taken into account, thus themodel is
applicable to the classical three-layer sandwichbeamand toamulti-layer
sandwich or laminated narrow structure.

Key words: sandwichbeam, perfect clamping, transverse vibration, local
model

Notations

E – Young’s modulus
hj – thickness of jth layer of beam
L – length of beam
Ux,Uy,Uz,ux,uy,uz – displacements in directions x, y, z, respectively
ux(j),uy(j),uz(j) – displacements within jth layer

t – time
x,y,z – space variables

X(T) – trigonometric function of variable x

X(H) – hyperbolic function of variable x



790 S. Karczmarzyk

zj – coordinate of one (upper) surface of jth layer
εqr – strain tensor
λL,µL – Lame’s parameters
µ=µL,µ(j) – shear modulus and shear modulus of jth layer,

respectively
ν – Poisson’s ratio
ρ,ρ(j) – density and density of jth layer, respectively

σzz,σzx,σzz(j),σzx(j) – stresses and stresses in jth layer, respectively

1. Introduction

Many papers have been published lately on vibration analysis of sandwich
structures and, in particular of sandwich beams. Unfortunately, most of them
are devoted to presentation of general analyticalmodels and are limited to nu-
merical investigation of the simply supported structures – see e.g. Frostig and
Baruch (1994), Cabańska-Płaczkiewicz (1999), Kapuria et al. (2004). It is no-
ted that thepaperbyLewiński (1991) contains theoretical considerationswhile
papers by Lewiński (1991), Cupiał and Nizioł (1995), Szabelski and Kaźmir
(1995) refer to rectangular plates. In papers by Chen and Sheu (1994), Fasa-
na andMarchesiello (2001), Nilsson andNilsson (2002), Backstom andNilson
(2006, 2007), some numerical results for clamped-clamped, clamped-free and
free-free beams are also presented. In some of the papers, the numerical re-
sults are not tabulated and, therefore, are not useful for detail comparisons.
Some of the above papers contain comparisons of numerical results for diffe-
rent theories, see Kapuria et al. (2004), Backstom and Nilson (2006), Hu et
al. (2006, 2008), Wu and Chen (2008). In a paper by Backstom and Nilson
(2006) the numerical results (amplitudes) are comparedwithmeasured values
for the beamwith both ends free.

Majority of analytical beammodels were derived following the variational
procedure and the same path as in the case of laminated composites – see e.g.
Kapuria et al. (2004), Hu et al. (2008), Wu and Chen (2008). After looking
through the analytical and numerical results for the simply supported beams,
one may notice that the models of the eigenvalue problem of sandwich struc-
tures have got some deficiencies. Some of them are shown e.g. in a paper by
Hu et al. (2006), where the evaluation of kinematic assumptions applied by
different authors is proposed.

Some other deficiencies can be easily noticed. For example, in paper of
Frostig andBaruch (1994) the in-plane normal stresses in the core are omited



A new 2D single series model of transverse vibration... 791

and the equilibrium equation instead of the equation of motion for the core is
applied. Despite of the simplifications, the model is not compared (in Frostig
and Baruch, 1994) with other models. In paper of Kapuria et al. (2004), high
inaccuracy of eigenfrequencies of a sandwich beam predicted by the FSDT is
shown. In Backstom and Nilson (2006, 2007), the compatibility equations of
stresses between adjoining layers are not satisfied. In Wu and Chen (2009),
highpercentagedifferencesbetweenpredictions of eigenfrequencies bydifferent
analytical models are given and commented.

Because of various assumptions and simplifications introduced into the
models of vibration of sandwich structures, the comparisons limited to sim-
ply supported members can imply misunderstandings since the comparative
results for any two models may be dependent on boundary conditions of the
structure(s). To some extent, it is suggested e.g. in Fasana and Marchesiello
(2001), where the percentage differences between the eigenfrequencies predic-
tedby the twomodels arewithin the range (3.86-0.56) for the simply supported
structure andwithin the range (5.85-4.22) for the free-free structure.Thus, in-
stead of investigating the simply supported beams, a direct investigation of
the clamped-clamped (C-C) sandwich structures is much more desired since
it can be useful because of their practical importance.

There is much less papers devoted to vibration analysis of clamped-
clamped unidirectional three-layer sandwich structures. Here, a few are col-
lected (Nilsson and Nilsson, 2002; Raville et al., 1961; Sakiyama et al., 1996;
Sokolinsky and Nutt, 2002; Howson and Zare, 2005). It is noticed that the
experimental data given in Raville et al. (1961) are compared in Sakiyama et
al. (1996), Sokolinsky and Nutt (2002), Howson and Zare (2005). Vibrational
models presented in Nilsson and Nilsson (2002), Raville et al. (1961), Soko-
linsky and Nutt (2002) were obtained according to the variational procedure.
In Sakiyama et al. (1996), the Green functions approach is used, and in How-
son and Zare (2005) a direct approach is employed to obtain the equations of
motion.

It is an aim of the paper to present and discuss the new two-dimmensional
(2D)model of transverse vibration of aC-C sandwichmulti-layered beamwith
perfectly clamped edges, that is to show both its mathematical details and
some comparison of numerical results. Thismodel is a next result of investiga-
tions of sandwich structures by the present author within the local theory of
linear elastodynamics. Several vibrational models for the unidirectional, both
cantilever (Karczmarzyk, 1995, 1996) and clamped-clamped (C-C) (Karczma-
rzyk, 1999, 2005), sandwich structures have been elaboratedwithin the appro-
ach. The former models and the new local model of the present author were



792 S. Karczmarzyk

obtainedwithout the a priori expandingdisplacement and stress fields (within
the structures) into series. However, the stress and displacment fields in the
new model are finally expanded into the single series of the eigenfunctions of
the classical Bernoulli-Euler theory of beams.All through-the-thickness boun-
dary conditions and the compatibility equations of the local theory of linear
elastodynamics as well as some specific edge boundary conditions have been
satisfied in the former models (Karczmarzyk, 1995, 1996, 1999, 2005) and in
the newmodel. However, as far as the present author knows, the perfect clam-
ping edge boundary conditions for the sandwich beam are fulfiled for the first
time within the local elastodynamic approach in the present paper.

The newmodel is directly applicable to the beams consisting of any num-
ber of layers. This is its important feature since the multi-layered sandwich
structures are rarely investigated in the literature but they occur frequently
in the modern composite constructions (see e.g. Wu et al., 2003).

There are many formal differences between the new model and the mo-
dels presented by other authors above mentioned in particular for the C-C
sandwich beam. First, displacements and stresses within the new local model
satisfy thewell knowndifferential equations ofmotion of the local theory of li-
near elastodynamics – expressed in stresses and displacements. The equations
of motion of the other authors were derived for an assumed number of layers
(usually equal to three) usuallywithin the variational procedure or within the
Bolle-Mindlin procedure. Thus, to apply (eventually) the variational theories,
e.g. for a five-layer sandwich beam one needs to derive first new equations of
motion. Secondly, the kinematic assumptions in the present new model and
in the former models are quite different. The functions of space variable in
the direction perpendicular to the interfaces appearing in the present model
are unknown while their counterparts in the former models are assumed as
known (linear or nonlinear) functions of the variable. On the other hand, the
formof functions of space variable in the direction parallel to the interfaces (to
length) of the beam is assumed in the presentmodelwhereas the functions are
derived from the equations of motion in the former models. Thirdly, the final
(computational) formof the problemwithin the present localmodel is derived
after satisfying both the local edge boundary conditions and all through-the-
thickness local boundary conditions and compatibility equations. In fact, the
final form of the problem consist of two transcendental uncoupled equations.
The computational form of the problem within the former models is derived
byusing only the edge boundary conditions since through-the-thickness condi-
tions have been satisfied (more or less exactly) in the procedure(s) of deriving
the equations of motion.



A new 2D single series model of transverse vibration... 793

There aremore formal distinctionswhich imply somemerit differences, not
discussed here, between the local newmodel and the former models however,
dispite of them the numerical results showmerit compatibility of the models.
Themain advantage of the newmodel, stressed here, is its direct applicability
to the analysis of multi-layered (eg. five-layered) sandwich structures that is
when the adjoining layers in such structures are of incomparable stiffnesses. It
is also emphasized that the eigenfunctions for the C-C beam within the new
local model are the same as in the classical Bernoulli-Euler beam theory.
Theexemplary structures considered in thepaper are shown inFig.1.They

are composedofhomogeneous, isotropic layers.The layers areperfectlybonded
one to another. Each layer is perfectly clamped at the edges i.e., in Fig.1a at
x=±L/2. Any parameter of the structure(s) is not formally limited.

Fig. 1. Multi-layered sandwich C-C beams: (a) three-layer, (b) five-layer. Thickness
of jth layer hj = zj −zj+1

In the case of the beam symmetric about itsmiddle plane (mid-plane) it is
desired to impose the following assumptions on location of the origin of coor-
dinate system. It is convenient to place it in themiddle plane of the structure
and in themiddle of the span (mid-span) – as shown inFig.1a. Location of the
origin in the mid-plane enables us to split the boundary problem in two sub-
problems – the transverse flexural problemand the transverse breath problem.
Location of the origin in the mid-span enables decoupling of the symmetric
and anti-symmetric modes of vibration.
Thenewmodel is presented in the further text as follows.All the equations

and conditions of the local theory of linear elastodynamics, however without
the well known Hooke law, are listed in the 2nd section. Two 2D solutions
to the local 2D equations of motion of the theory of linear elastodynamics,
derived here by the present author in an original way, are described in the 3rd
section. Two 3D solutions to the local 3D equations ofmotion of the theory of
linear elastodynamics are given in the 4th section. The 2D and 3D solutions



794 S. Karczmarzyk

are presentedwidely in order to facilitate understanding the content of the 5th
section.Theessential new ideas of deriving thenewmodel (after combining the
2D and 3D fields) are presented in the 5th section. Exact formulas necessary
to create the final numerical form of the boundary problem (i.e. to create the
matrix of the problem) and some details on the final form are given in the
6th section. Numerical results and comparisons as well as some comments are
given in the 7th section. Section 8th contains a few conclusions.

2. Statement of the problem

Theboundaryproblem is formulated and solved entirelywithin the local linear
theory of elastodynamics. The new solution (model) is composed of two 2D
(plane) components and two 3D components.

The following 2D local equations of motion, containing the plane stress
state components σxx, σzz, σzx and the corresponding displacements ux, uz,
are satisfied by the 2D components of the model within each layer of the
structure separately

∂σxx
∂x
+
∂σzx
∂z
= ρ
∂2ux
∂t2

∂σzx
∂x
+
∂σzz
∂z
= ρ
∂2uz
∂t2

(2.1)

Equations (2.1) can be expressed entirely in terms of the field ux, uz
(Karczmarzyk, 1996) that is

µ∇2ux+(λ+µ)
(∂2ux
∂x2
+
∂2uz
∂x∂z

)
= ρ
∂2ux
∂t2

(2.2)

µ∇2uz +(λ+µ)
(∂2ux
∂x∂z

+
∂2uz
∂z2

)
= ρ
∂2uz
∂t2

The parameters λ, µ in Eqs (2.2) are defined as follows

λ=λL
1−2ν
1−ν

=2µL
ν

1−ν
λL =2µL

ν

1−2ν
µ=µL (2.3)

where λL, µL are the Lame material parameters and ν denotes the Poisson
ratio of a particular homogeneous layer of the structure. Symbols ρ, t in (2.1)
and (2.2) stand for the layer density and time, respectively.



A new 2D single series model of transverse vibration... 795

The 3D components of the new solution satisfy the full 3D equations of
motion of the local linear theory of elastodynamics, i.e.

µ∇2ux+(λL+µ)
(∂2ux
∂x2
+
∂2uy
∂y∂x

+
∂2uz
∂z∂x

)
= ρ
∂2ux
∂t2

µ∇2uy+(λL+µ)
(∂2ux
∂x∂y

+
∂2uy
∂y2
+
∂2uz
∂z∂y

)
= ρ
∂2uy
∂t2

(2.4)

µ∇2uz +(λL+µ)
(∂2ux
∂x∂z

+
∂2uy
∂y∂z

+
∂2uz
∂z2

)
= ρ
∂2uz
∂t2

The 2D and 3D fields, satisfying the above equations of motion, fulfil the
Saint-Venant compatibility equations expressed in terms of strains εqr in the
following well known abbreviated form

εkl,mn+εmn,kl−εkm,ln−εln,km =0 (2.5)

The following through-the-thickness local boundary conditions, (2.6), and
compatibility equations (2.7) for the whole structure are satisfied by the total
stress and displacemnnt fields within the newmodel

σ̃zz(1)(x,z=z1)= σ̃zx(1)(x,z=z1)= σ̃zz(p)(x,z=zp+1)=
(2.6)

= σ̃zx(p)(x,z=zp+1)= 0

and

σ̃zz(j)(x,z= zj+1)= σ̃zz(j+1)(x,z= zj+1)

σ̃zx(j)(x,z= zj+1)= σ̃zx(j+1)(x,z= zj+1)

ũz(j)(x,z= zj+1)= ũz(j+1)(x,z= zj+1)

ũx(j)(x,z= zj+1)= ũx(j+1)(x,z= zj+1)

(2.7)

where j=1,2, . . . ,p−1.
The symbolswith the sign”∼”denote the total stresses anddisplacements,

the subscript p means the number of layers, subscripts, 1, j, j+1 identify
the 1st, jth and (j+1)th layer, respectively. The coordinates z1, zj, etc. are
explained in Fig.1.

It is noticed that assuming inEqs (2.6) the normal stresses as non-equal to
zero, we have the boundary conditions for the forced vibration. It is explained
that the stresses result from the Hooke law applied in the paper.



796 S. Karczmarzyk

The following edge boundary conditions are satisfiedwithin the newmodel
(j=1,2, . . . ,p)

ũx(j)(x=±L/2,z)= 0 ũz(j)(x=±L/2,z) = 0
(2.8)

∂ũz(j)
∂x

∣∣∣∣
(x=±L/2,z)

=0

As far as the present author knows, local edge boundary conditions (2.8)
for the perfect clamping of the edges for all layers of the sandwich structure
have been fulfiled for the first time within the local elastodynamic approach.

3. Solutions to the 2D (plane) local equations of motion of the
linear elastodynamics

In order to derive 2D solutions for an isotropic continuous layer, the following
kinematic assumptions are used

ux =−g(z)T(t)
dX(T)

dx
uz = f(z)X

(T)T(t)

d2X(T)

dx2
=−α2X(T) α2 > 0

(3.1)

The functions g, f of the space variable z are unknown, the function X(T)

of the space variable x will be defined later. The function T(t) = exp(iωt),
where i2 = −1 and ω, t are the vibration frequency and time, respectively.
Due to (3.1), Eqs (2.2) can be transformed to the following form

−µ
d2g

dz2
+[(λ+2µ)α2−ρω2]g+(λ+µ)

df

dz
=0

(3.2)

(λ+2µ)
d2f

dz2
− (µα2−ρω2)f+(λ+µ)α2

dg

dz
=0

Equations (3.2) canbe solved inmanyways, andone of them,which is very
convenient, is shown below. It is noticed that Eqs (3.2) may be rearranged as
follows

−µ
d2g

dz2
+(µα2−ρω2)g+(λ+µ)

(
α2g+

df

dz

)
=0

(3.3)

µ
d2f

dz2
− (µα2−ρω2)f+(λ+µ)

(
α2
dg

dz
+
d2f

dz2

)
=0



A new 2D single series model of transverse vibration... 797

The underlined term occurs in each of Eqs (3.3). It is seen form Eqs (3.3)
that,

d2g

dz2
=
(
α2−

ρω2

µ

)
g≡β21g β

2
1 =α

2−
ρω2

µ

d2f

dz2
=
(
α2−

ρω2

µ

)
f ≡β21f g=−

1

α2
df

dz

(3.4)

Thus, thefirst rearrangementofEqs (3.2) leads to thefirst solution, expres-
sed in the following matrix form:
— for β21 > 0

[
f1
g1

]
=



cosh(β1z) sinh(β1z)

−
β1
α2
sinh(β1z) −

β1
α2
cosh(β1z)



[
C1
C2

]
(3.5)

— for β21 < 0, β
2
1 =−β21
[
f1
g1

]
=



cos(β1z) sin(β1z)

β1
α2
sin(β1z) −

β1
α2
cos(β1z)



[
C1
C2

]
(3.6)

Equations (3.2) can be also rearranged in a second manner

−(λ+2µ)
d2g

dz2
+[(λ+2µ)α2−ρω2]g+(λ+µ)

(d2g
dz2
+
df

dz

)
=0

(3.7)

(λ+2µ)
d2f

dz2
− [(λ+2µ)α2−ρω2]f+(λ+µ)α2

(dg
dz
+f
)
=0

Again, there is a term (underlined) occuring in both Eqs (3.7). It is seen
directly from Eqs (3.7) that the functions g,f are now defined as follows

d2g

dz2
=
(
α2−

ρω2

λ+2µ

)
g≡β22g β

2
2 =α

2−
ρω2

λ+2µ

d2f

dz2
=
(
α2−

ρω2

λ+2µ

)
f ≡β22f f =−

dg

dz

(3.8)

Thus, the second rearrangement of Eqs (3.2) leads to the second solution,
expressed in the following matrix form:— for β22 > 0

[
f2
g2

]
=




cosh(β2z) sinh(β2z)

−
1

β2
sinh(β2z) −

1

β2
cosh(β2z)




[
C3
C4

]
(3.9)



798 S. Karczmarzyk

— for β22 < 0, β
2
2 =−β22
[
f2
g2

]
=




cos(β2z) sin(β2z)

−
1

β2
sin(β2z)

1

β2
cos(β2z)




[
C3
C4

]
(3.10)

The constants Cl, l=1,2,3,4, in (3.5), (3.6) and (3.9), (3.10) are unknown.

4. Solutions to the 3D (plate) local equations of motion of the
linear elastodynamics

In order to derive 3D solutions for an isotropic continuous layer, the following
kinematic assumptions are used

Ux =−G(z)Y (y)
dX(H)

dx
T(t) Uy =−G(z)

dY

dy
X(H)T(t)

Uz =F(z)Y (y)X
(H)T(t)

d2X(H)

dx2
=α2X(H) α2 > 0

d2Y

dy2
=−β2Y β2 > 0

(4.1)

Ux, Uy and Uz are displacements dependent on three space variables x,y,z.
The functions G,F of the variable z are unknown.The function X(H) of the
variable x as well as Y (y) will be defined later. T(t) is the same function of
timewhichappears in (3.1).NowEqs (2.4) canbe transformed to the following
(two) ordinary differential equations

−µ
d2G

dz2
− [(λL+2µ)(α2−β2)+ρω2]G+(λL+µ)

dF

dz
=0

(4.2)

(λL+2µ)
d2F

dz2
+[µ(α2−β2)+ρω2]F − (λL+µ)(α2−β2)

dG

dz
=0

It is noticed that a first rearrangement of Eqs (4.2) is as follows

−µ
d2G

dz2
− [µ(α2−β2)+ρω2]G− (λL+µ)

[
(α2−β2)G−

dF

dz

]
=0

(4.3)

µ
d2F

dz2
+[µ(α2−β2)+ρω2]F − (λL+µ)

[
(α2−β2)

dG

dz
−
d2F

dz2

]
=0



A new 2D single series model of transverse vibration... 799

The underlined term occurs in each of Eqs (4.3). It is seen formEqs (4.3) that

d2G

dz2
=−
(
α2−β2+

ρω2

µ

)
G≡−R21G R

2
1 =α

2−β2+
ρω2

µ

d2F

dz2
=−
(
α2−β2+

ρω2

µ

)
F ≡−R21F G=

1

α2−β2
dF

dz

(4.4)

Now it is seen that the first solution to Eqs (4.3) can be expressed in the
matrix form:
— for R21 < 0,R

2
1 =−R21

[
F1
G1

]
=




cosh(R1z) sinh(R1z)
R1
α2−β2

sinh(R1z)
R1
α2−β2

cosh(R1z)



[
D1
D2

]
(4.5)

— for R21 > 0

[
F1
G1

]
=




cos(R1z) sin(R1z)

−
R1
α2−β2

sin(R1z)
R1
α2−β2

cos(R1z)



[
D1
D2

]
(4.6)

The second rearrangement of Eqs (4.2) is as follows

−(λL+2µ)
d2G

dz2
−[(λL+2µ)(α2−β2)+ρω2]G+(λL+µ)

(d2G
dz2
+
dF

dz

)
=0

(4.7)

(λL+2µ)
d2F

dz2
+[(λL+2µ)(α

2−β2)+ρω2]F +

−(λL+µ)(α2−β2)
(dG
dz
+F
)
=0

Directly from Eqs (4.7), one obtains

d2G

dz2
=−
(
α2−β2+

ρω2

λL+2µ

)
G≡−R22G

R22 =α
2−β2+

ρω2

λL+2µ
(4.8)

d2F

dz2
=−
(
α2−β2+

ρω2

λL+2µ

)
F ≡−R22F F =−

dG

dz

Finally, it is seen that the second solution to Eqs (4.2) can be expressed in
the following matrix form:



800 S. Karczmarzyk

— for R22 < 0,R
2
2 =−R22

[
F2
G2

]
=



cosh(R2z) sinh(R2z)

−
1

R2
sinh(R2z) −

1

R2
cosh(R2z)



[
D3
D4

]
(4.9)

— for R22 > 0
[
F2
G2

]
=




cos(R2z) sin(R2z)

−
1

R2
sin(R2z)

1

R2
cos(R2z)




[
D3
D4

]
(4.10)

The constants Dl, l=1,2,3,4, in (4.5), (4.6) and (4.9), (4.10) are unknown.

5. Idea of the new solution to the boundary problem –
combination of the 2D and 3D fields

Let us assume the following relationships

d2gi
dz2
=
d2Gi
dz2
⇔β2igi =−R

2
iGi

(5.1)
d2fi
dz2
=
d2Fi
dz2
⇔β2ifi =−R

2
iFi i=1,2

The above assumption is one of new ideas in the paper. Equations (5.1) will
be satisfied if the following equalities are valid

β2i =−R
2
i ∧ gi ≡Gi ∧ fi ≡Fi ⇒

(5.2)

⇒
d2gi
dz2
=
d2Gi
dz2

∧
d2fi
dz2
=
d2Fi
dz2

i=1,2

It is evident that Eqs (5.1) will be satified if β2 is defined as follows

β21 =−R
2
1 ⇔ β

2 =2α2

(5.3)

β22 =−R
2
2 ⇔ β

2 =2α2+ρω2
( 1
λL+2µ

−
1

λ+2µ

)

If λL ≈λ, as assumed in the further text, we can write down

β21 =−R
2
1 ∧ β

2
2 =−R

2
2 ⇔ β

2 =2α2 (5.4)

It is noted however that the final form of the solution proposed here is the
same irrespective of applying or omiting the assumption λL ∼=λ.



A new 2D single series model of transverse vibration... 801

Due to Eqs (5.1)-(5.4), one can write displacements (4.1) by replacing
Gi(z) with gi(z) and Fi(z) with fi(z), for i=1,2, respectively

Uxi =−gi(z)Y (y)
dX(H)

dx
T(t) Uyi =−gi(z)

dY

dy
X(H)T(t)

Uzi = fi(z)Y (y)X
(H)T(t) i=1,2

d2X(H)

dx2
=α2X(H) α2 > 0

d2Y

dy2
=−2α2Y

(5.5)

It is explained here that the function Y is assumed to be even, i.e.

Y (
√
2αy)=Y (−

√
2αy) Y =cos(

√
2αy)= cos[αL(

√
2y/L)] (5.6)

It is obvious that for a sufficiently small y, the following approximations
are valid

Y ∼=1
dY

dy
∼=0 (5.7)

Approximations (5.7) imply a limitation of the model proposed here to a
narrow structure. After taking into account (5.7), one obtains approximations
of diplacements (5.5)

Uxi ∼=−gi(z)
dX(H)

dx
T(t) Uyi ∼=0 Uzi ∼= fi(z)X(H)T(t)

(5.8)

i=1,2
d2X(H)

dx2
=α2X(H) α2 > 0

Let us assume the following (total) displacement field within the isotropic
layer

ũx =
∑

i

ũxi =
∑

i

(uxi−Uxi)∼=−
∑

i

gi(z)
(dX(T)

dx
−
dX(H)

dx

)
T(t) ũyi ∼=0

(5.9)
ũz =

∑

i

ũzi =
∑

i

(uzi−Uzi)∼=
∑

i

fi(z)(X
(T)−X(H))T(t) i=1,2

Assumption (5.9) is the next new idea of the solution presented here. It is
stated that the 3D components of the solutions (of the equations of motion)
derived in Section 4 do not occur in the final, total displacement field (5.9).
They have ”disapeared” due to assumptions (5.1) and (5.7). The only trace
of including the 3D components into field (5.9) is the function X(H) and its
derivative. We further assume, as in Karczmarzyk (1999), that the functions



802 S. Karczmarzyk

of the variable x for the symmetric (about the mid-span) vibration are as
follows

X(T) =
cos(αx)

cos αL
2

X(H) =
cosh(αx)

cosh αL
2

(5.10)

When the anti-symmetric (about themid-span) vibrations are considered, the
functions of the variable x are defined as follows

X(T) =
sin(αx)

sin αL
2

X(H) =
sinh(αx)

sinh αL
2

(5.11)

It is noted that functions (5.10), (5.11) are the eigenfunctions within the
classical Bernoulli-Euler theory of beam. It is seen that irrespective of the type
of vibration, the following equalities are satisfied, for x=±L/2

X(T)(x=±L/2)−X(H)(x=±L/2)= 0 ⇔ ũzi(x=±L/2)= 0 (5.12)

The right-hand side of Eq. (5.12) is one of the edge(s) boundary conditions
for the perfect clamping of the edge(s). It means that irrespective of value
of the variable z, the transversal (out-of-plane) vibrational displacement at
the edges x =±L/2 is equal to zero. It is noticed that the derivative of the
transverse displacement equals to zero at x=±L/2 for any value of z.
The second edge boundary condition for the perfect clamping solution is

as follows

ũxi(x=±L/2)= 0 ⇔
dX(T)

dx

∣∣∣∣
x=±L/2

−
dX(H)

dx

∣∣∣∣
x=±L/2

=0 (5.13)

After substituting functions (5.10) into the right-hand side Eq. (5.13), one
obtains the following transcendental equation for the symmetric modes of vi-
bration enabling us to calculate α

sin αL
2

cos αL
2

+
sinh αL

2

cosh αL
2

=0 (5.14)

When functions (5.11) are used, the right-hand side Eq. (5.13) is transformed
to the form

cos αL
2

sin αL
2

−
cosh αL

2

sinh αL
2

=0 (5.15)

In the literature, there are the following approximate values of α satisfying
Eqs (5.14) and (5.15), i.e.:



A new 2D single series model of transverse vibration... 803

— for symmetric modes

α1L

2
∼=2.365

αkL

2
∼=
(4k−1)π
4

k=2,3,4, . . .

α1L

2
∼=2.365

αmL

2
∼=
(2m+1)π

4
m=3,5,7, . . .

(5.16)

— for anti-symmetric modes

α1L

2
∼=3.927

αlL

2
∼=
(4l+1)π

4
l=2,3,4, . . .

α2L

2
∼=3.927

αmL

2
∼=
(2m+1)π

4
m=4,6,8, . . .

(5.17)

6. Through-the-thickness boundary and compatibility equations
and a numerical form of the boundary value problem

Uponthebasis of thedisplacementfield,definedby (5.9), (3.5), (3.6), (3.9) and
(3.10), we are able to derive the total strain field and, after its substitution
to the Hooke law, we obtain the following expressions for the total stresses
within the layer (i=1,2)

σ̃zx =
∑

i

σ̃zxi =
∑

i

µ
(
fi−
dgi
dz

)(dX(T)

dx
−
dX(H)

dx

)
T(t)=

=
∑

i

Szxi
(dX(T)

dx
−
dX(H)

dx

)
T(t)

(6.1)

σ̃zz =
∑

i

σ̃zzi =
∑

i

(
λα2gi+(λ+2µ)

dfi
dz

)
(X(T)−X(H))T(t)=

=
∑

i

Szzi(X
(T)−X(H))T(t)

It is seen fromEqs (6.1) that irrespective of value of the variable z (inclu-
ded in the functions gi, fi), the total shear stresses and normal (out-of-plane)
stresses are equal to zero at the edges x=±L/2. In order towrite the explicit
expressions for the stresses, we use the following relationships for the stress
components, i.e.

σ̃zx1 =µ
(
1+
β21
α2

)
f1
(dX(T)

dx
−
dX(H)

dx

)
T(t)

σ̃zz1 =2µ
df1
dz
(X(T)−X(H))T(t)



804 S. Karczmarzyk

σ̃zx2 =2µf2
(dX(T)

dx
−
dX(H)

dx

)
T(t)

σ̃zz2 =
[
2µ+λ

(
1−
α2

β22

)]df2
dz
(X(T)−X(H))T(t) (6.2)

σ̃zx1 =Szx1
(dX(T)

dx
−
dX(H)

dx

)
T(t) Szx1 =µ

(
1+
β21
α2

)
f1

σ̃zz1 =Szz1(X
(T)−X(H))T(t) Szz1 =2µ

df1
dz

σ̃zx2 =Szx2
(dX(T)

dx
−
dX(H)

dx

)
T(t) Szx2 =2µf2

σ̃zz2 =Szz2(X
(T)−X(H))T(t)

Szz2 =
[
2µ+λ

(
1−
α2

β22

)]df2
dz

If we use (3.5), (3.6), (3.9), (3.10) and (6.2), we obtain Szx =Szx1+Szx2
and Szz =Szz1+Szz2 in the explicit form:

— for β21 > 0, β
2
2 > 0

[
Szz
Szx

]
=

[
2µβ1 sinh(β1z) 2µβ1cosh(β1z) Asinh(β2z) Acosh(β2z)
Bcosh(β1z) B sinh(β1z) 2µcosh(β2z) 2µsinh(β2z)

]



C1
C2
C3
C4




(6.3)
where

A=
[
2µ+λ

(
1−
α2

β22

)]
β2 B=µ

(
1+
β21
α2

)

— for β
2
1 =−β21 > 0, β

2
2 =−β22 > 0

[
Szz
Szx

]
=

[
−2µβ1 sin(β1z) 2µβ1cos(β1z) −Asin(β2z) Acos(β2z)
Bcos(β1z) B sin(β1z) 2µcos(β2z) 2µsin(β2z)

]




C1
C2
C3
C4





(6.4)
where

A=
[
2µ+λ

(
1+
α2

β
2
2

)]
β2 B=µ

(
1−
β
2
1

α2

)

Analogously, g= g1+g2, f = f1+f2 in the explicit form are as follows:



A new 2D single series model of transverse vibration... 805

— for β21 > 0, β
2
2 > 0

[
−g
f

]
=




β1
α2
sinh(β1z)

β1
α2
cosh(β1z)

1

β2
sinh(β2z)

1

β2
cosh(β2z)

cosh(β1z) sinh(β1z) cosh(β2z) sinh(β2z)








C1
C2
C3
C4




(6.5)

— for β
2
1 =−β21 > 0, β

2
2 =−β22 > 0

[
−g
f

]
=



−
β1
α2
sin(β1z)

β
1

α2
cos(β1z)

1

β2
sin(β2z) −

1

β2
cos(β2z)

cos(β1z) sin(β1z) cos(β2z) sin(β2z)







C1
C2
C3
C4




(6.6)
For a particularmode of free vibration, the symbol α in expressions (6.3)-

-(6.6)must be replacedwith αm,m=1,2,3 – see (5.16)2, while the symbol ω
(frequency) appearing in β1, β2 must be replaced with ωm (mth eigenfrequ-
ency).
It is noticed that any limitation on parameters of the beam (such as thick-

nesses, densities etc.) as well as any restriction on the ratios h(j)/h(j+1),
ρ(j)/ρ(j+1), µ(j)/µ(j+1), L/ht, ht = h1 + h2 + . . .+ hp, etc., have not been
introduced into the model. Therefore, it can be applied for the vibration ana-
lysis of both themulti-layered, slender and thickset, sandwich beams and the
classical laminated beams consisting of stiffness-comparable layers. Obviously,
this statement is true provided that the edge fixing of the structure assures
perfect edge clamping boundary conditions (2.8).
It is noted (repeated) that the stress and displacement fields derived in

Sections 3-6 occur in an isotropic, homogeneous (let us say in jth) layer of the
multi-layered structure. If we want to have such fields for any (j+k)th layer,
we have to substitute the material parameters ρ, λ and µ for this particular
layer into theall above expressions– inparticular into formulas (6.3)-(6.6).The
distinction between the stress and displacement fields for two different layers,
let us say jth and (j+k)th, is seen in the following exemplary expressions

d2g(j)
dz2

=
(
α2−

ρ(j)ω
2

µ(j)

)
g(j) ≡β21(j)g(j) β

2
1(j) =α

2−
ρ(j)ω

2

µ(j)

d2g(j+k)

dz2
=
(
α2−

ρ(j+k)ω
2

µ(j+k)

)
g(j+k) ≡β21(j+k)g(j+k) (6.7)

β21(j+k) =α
2−
ρ(j+k)ω

2

µ(j+k)



806 S. Karczmarzyk

In order to obtain a numerical form of the boundary problem, we have to
use the derived expressions for the displacements and stresses and substitu-
te it into through-the-thickness boundary conditions (2.6) and compatibility
equations (2.7). The creation and the structure of the resulting matrix of the
eigenvalue problem for a three-layered sandwich beam is illustrated in the
following formal expression (scheme)




z= z1 σ̃zz(1) =0

σ̃zx(1) =0

z= z2 σ̃zz(1) = σ̃zz(2)
σ̃zx(1) = σ̃zx(2)
ũz(1) = ũz(2)
ũx(1) = ũx(2)

z= z3 σ̃zz(2) = σ̃zz(3)
σ̃zx(2) = σ̃zx(3)
ũz(2) = ũz(3)
ũx(2) = ũx(3)

z= z4 σ̃zz(3) =0

σ̃zx(3) =0




≡




++++
++++
++++ ++++
++++ ++++
++++ ++++
++++ ++++

++++ ++++
++++ ++++
++++ ++++
++++ ++++

++++
++++







C1(1)
C2(1)
C3(1)
C4(1)
C1(2)
C2(2)
C3(2)
C4(2)
C1(3)
C2(3)
C3(3)
C4(3)




=0≡

(6.8)
≡AC=0

Thepluses in thematrix Adenote, in a general case, the non-zero elements
of the matrix. After solving the equation detA = 0, one obtains the eigen-
frequencies ωm. There are many ways for numerical solving of the eigenvalue
equation. One of them is using a standard software module for evaluation
the determinants. The other way may be transformation of thematrix to the
smallest dimension and then obtaining a computational code. It is noted that
the whole eigenvalue problem is expressed by one of Eqs (5.14), (5.15) and
Eq. (6.8).
If the matrix 0 in Eq. (6.8) is replaced with a non-zero matrix containing

components of sinusoidally varying loads of the structure, Eq. (6.8) together
with one ofEqs (5.14) and (5.15)will be thefinal,matrix formof theboundary
problem of the forced vibration (in this case, the loads will be expanded into
series (5.14) and (5.15)). Anyway, the boundary problem in its final form
consists of two uncoupled Eqs: (5.14) or (5.15) and (6.8).
Thedimension of the squarematrix A for the structure consisting of p lay-

ers is equal to 4p× 4p. It is easy to show that for the structure symmetric
about themiddle plane, thematrix dimension can bedecreased two times. For
the symmetric structure, the boundary problem (6.8) splits into two subpro-
blems: one for the transverse flexural vibration and the other for the transverse



A new 2D single series model of transverse vibration... 807

breath problem. Thus, for the classical sandwich beam symmetric about the
mid-plane (as shown in Fig.1a), whose outer layers are of the same thickness
and the samematerials, the matrix A dimension is equal to 6×6≡ 2p×2p,
p=3.

Let us finally note that after substituting into (3.5), (3.6), (3.9), (3.10),
(6.3)-(6.6) and (6.8) αm = mπ/L for m = 1,2,3, . . ., after replacing
the function X(T) − X(H) in (5.9) by the sinus Fourier series function
X(x)= sin(mπx/L) and the function d(X(T)−X(H))/dx in (5.9) by the func-
tion αmcos(mπx/L), we obtain a local 2D solution to the sinusoidal vibration
problem of the simply supported multi-layer sandwich beam (Karczmarzyk,
1999). This advantageous property of the model proposed here shows its ef-
ficiency and its (limited) similarity to the classical Bernoulli-Euler theory of
homogeneous beam based on the assumption of plane cross-sections.

The opposite idea of replacing the sinus Fourier series functions with the
Bernoull-Euler eigenfunctions was first proposed and numerically verified by
thepresent author inKarczmarzyk (2005). Itwas onlyan intuitive proposition.
In thepresentpaper, the idea checked inKarczmarzyk (2005)hasbeen justified
for the first time mathematically. Due to the present paper we know, among
other things, that themodel is exact and accurate only for sufficiently narrow
sandwich structures (beams) and not for wide rectangular plates with two
parallel edges clamped and the other edges free.

7. Numerical results and comparisons

In order to check the newmodel, some computations have beenmade for the
input data given in Table 1. The results are listed and compared in Table 2.
Eight eigenfrequencies ωSK obtained after numerical solving of eigenvalue
problem (5.14), (5.15), (6.8) for the C-C beam, for the input data given in
Raville et al. (1961), Sakiyama et al. (1996), Sokolinsky and Nutt (2002),
Howson andZare (2005), are presented.Apart fromthenewresults, the reader
will find in Table 2 the eigenfrequencies presented (for the structure) in the
literature, i.e., ωExExp – obtained experimentally (Raville et al., 1961) and
listed in Sakiyama et al. (1996), Sokolinsky and Nutt (2002), Howson and
Zare (2005) and ωRAV , ωSAK, ωVSS, ωHZ computed according to themodels
by Raville et al. (1961), Sakiyama et al. (1996), Sokolinsky and Nutt (2002),
Howson and Zare (2005), respectively. The percentage differences between the
results predicted by the different models are shown in Fig.2.



808 S. Karczmarzyk

Table 1. Parameters of the classical three-layered sandwich beam of length
L=1.21872m

Para- h E ν ρ µ λ
meter [mm] [Pa] [–] [kg·m−3] [Pa] [Pa]
Layer 1 0.40624 0.6890 ·1010 0.33 2687.3 0.2590 ·1010 0.2551 ·1010
Layer 2 6.34750 0.1833 ·109 0.33 119.69 0.6891 ·108 0.6788 ·108
Layer 3 0.40624 0.6890 ·1010 0.33 2687.3 0.2590 ·1010 0.2551 ·1010

Table 2. Flexural eigenfrequencies of the sandwich C-C beam according to
different models

Vibr. Mode (m)
[rad·s−1] 1(s) 2(a) 3(s) 4(a) 5(s) 6(a) 7(s) 8(a)
ωSK 220.50 597.91 1144.0 1834.7 2645.1 3551.4 4537.4 5568.3
ωExExp – – 1165.5 1761.2 2509.5 3362.8 4277.0 5448.8
ωRAV 229.88 617.81 1173.8 1872.0 2685.6 3596.0 4575.3 5618.7
ωVSS 217.40 584.96 1113.4 1776.9 2552.9 3419.9 4358.6 5353.3
ωSAK 210.88 567.77 1081.2 1727.3 2484.5 3332.2 4252.7 5230.3
ωHZ 217.38 584.96 1113.1 1776.8 2553.0 3420.1 4359.2 5354.2

Fig. 2. Percentage differences between eigenfrequencies listed in Table 2

It is explained that the notation m = [1(s),3(s),5(s),7(s)] is used in
Fig.2 to denote vibration symmetric about themiddle of the beam span (mid-
span), while the notation m= [2(a),4(a),6(a),6(a)] refers to vibration anti-
symmetric about the mid-span. The abbreviations used in Fig.2 are defined
as follows: RAV −SK =100(ωRAV −ωSK)/ωRAV , VSS−SK =100(ωVSS−
ωSK)/ωVSS, SAK − SK = 100(ωSAK − ωSK)/ωSAK, SK − ExExp =
100(ωSK −ωExExp)/ωExExp, VSS−ExExp=100(ωVSS −ωExExp)/ωExExp.



A new 2D single series model of transverse vibration... 809

The abbreviations SK − Exp and VSS − Exp do not stand for de-
finitions analogous to the above outlined, but they are approximate, po-
tential elongations of the curves SK − ExExp and VSS − ExExp, re-
spectively. Unfortunately, the eigenfrequencies of the first symmetric mo-
de and first unsymmetric mode of vibration are not explicitely given (ta-
bulated) in the literature (Raville et al., 1961; Sakiyama et al., 1996;
Sokolinsky and Nutt, 2002; Howson and Zare, 2005) and, therefore, the
present author was not able to calculate SK − Exp = 100(ωSK −
ωExExp)/ωExExp, VSS −Exp = 100(ωVSS − ωExExp)/ωExExp for the two
lower modes.

It is seen from Table 2 and in Fig.2 that the following relationships, con-
cerning the computational eigenfrequencies, are observed, ωRAV > ωSK >
ωVSS > ωSAK. It is noted that the eigenfrequencies ωVSS are almost equ-
al to the eigenfrequencies ωHZ. This means that models presented in So-
kolinsky and Nutt (2002), Howson and Zare (2005) are compatible. The re-
sults predicted by the new model and models by Raville et al. (1961), So-
kolinsky and Nutt (2002), Howson and Zare (2005) are close. The curves
RAV −SK and VSS−SK are parallel and distant approximately by 6%.
The model by Sakiyama et al. (1996) gives lower eigenfrequencies than the
new model and the models by Sokolinsky and Nutt (2002), Howson and
Zare (2005).

However, the comparisons of the computational results and the existing
experimental results (Raville et al., 1961), see the curves SK−ExExp and
VSS−ExExp, suggest high inaccuracy of all the models (see Raville et al.,
1961; Sakiyama et al., 1996; Sokolinsky and Nutt, 2002; Howson and Zare,
2005; and the new one) for the lower modes of vibration. This is suggested
by the elongations SK−Exp and VSS−Exp. The computed eigenfrequ-
encies for the lower modes of vibration are probably much lower than the
corresponding measured values. The first mode eigenfrequency according to
the model by Sokolinsky and Nutt (2002) seems to be some 16% lower than
the expected experimental value. It is difficult to explain this phenomenon
exactly, but one of potential explanations is suggested here. Most proba-
bly, the existing experimental eigenfrequencies ωExExp, presented in Ravil-
le et al. (1961) and listed in Sakiyama et al. (1996), Sokolinsky and Nutt
(2002), Howson and Zare (2005), were measured for the vibrating sandwich
beam with fixed (or free – see e.g. Nilsson and Nilsson, 2002) edges (ends),
which in any case were not perfectly clamped – see definition of boundary
conditions (2.8).



810 S. Karczmarzyk

8. Conclusions

A new two-dimmensional, single series local model of transverse vibration of
a multi-layered one-span sandwich beam with perfectly clamped edges has
been presented in the present paper. It is derived in the local theory of linear
elastodynamics after satisying all the rigorous requirements of the theory.
The eigenfunctions for the C-C sandwich multi-layered beam within the

newmodel are the same as in the classical theory of homogeneous beam, based
on the assumption of plane cross-sections.
The model is applicable to beams composed of any number of layers ir-

respective of their parameters. It is applicable to structures with both edges
clamped or simply supported, after replacing (if it is necessary) theBernoulli-
Euler eigenfunctions with the sinus Fourier series functions.
In the case of a beam symmetric about its middle plane, the model splits

into two submodels: one for the transverse flexural anti-symmetric vibration
and the second for the transverse symmetric (breathing) vibration.
Themodel predicts the eigenfrequencies close to the counterparts predicted

by different former models published by other authors.

References

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A new 2D single series model of transverse vibration... 811

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812 S. Karczmarzyk

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Nowy dwuwymiarowy pojedyńczo szeregowy model drgań poprzecznych
wielowarstwowej belki sandwiczowej z idealnie utwierdzonymi

krawędziami

Streszczenie

W tej pracy jest przedstawiony nowy dwuwymiarowy, pojedyńczo szeregowy, lo-
kalny model drgań poprzecznych wielowarstwowej, jednoprzęsłowej belki sandwiczo-
wej, złożonej z warstw izotropowch, z idealnie utwierdzonymi końcami. Model ten,
otrzymany w ramach lokalnej teorii liniowej elastodynamiki, składa się z dwóch pól
dwuwymiarowych i dwóch aproksymacji pól trójwymiarowych spełniających ściśle
równania ruchu oraz warunki zgodności Saint-Venanta. W modelu zostały spełnio-
ne wszystkie warunki brzegowe po grubości, jak również lokalne warunki ciągłości
(przemieszczeń i naprężeń) między przylegającymi warstwami. Uwzględniono depla-
nacje przekrojowe, jak też poprzeczne podatności każdej warstwy i dlategomodel ten
jest stosowalny zarówno do klasycznej trójwarstwowej belki sandwiczowej, jak i do
wielowarstwowej struktury sandwiczowej czy laminatowej.

Manuscript received October 8, 2009; accepted for print March 4, 2010