Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

57, 1, pp. 141-154, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.141

FREE VIBRATIONS SPECTRUM OF PERIODICALLY INHOMOGENEOUS

RAYLEIGH BEAMS USING THE TOLERANCE AVERAGING TECHNIQUE

Marcin Świątek, Łukasz Domagalski, Jarosław Jędrysiak
Lodz University of Technology, Department of Structural Mechanics, Łódź, Poland

e-mail: marcin.swiatek@dokt.p.lodz.pl; lukasz.domagalski@p.lodz.pl; jarek@p.lodz.pl

In this paper, linear-elastic Rayleigh beams with a periodic structure are considered. Dy-
namics of such beams is described by partial differential equations with non-continuous
highly oscillating coefficients. The analysis of dynamic problems using the aforementioned
equations is very often problematic to perform. Thus, other simplified models of Rayleigh
beams are proposed. Some of thesemodels are based on the concept of the effective stiffness.
Among them, one candistinguish the theoryof asymptotic homogenization.However, in the-
se models, the size of the mesostructure parameter (the size of a periodicity cell) is often
neglected. Therefore, a non-asymptotic averagedmodel of the periodic beam is introduced,
called the tolerance model, which is derived by applying the tolerance averaging technique
(TA). The obtained tolerancemodel equations have constant coefficients, and in contrast to
other averagedmodels, some of them depend on the size of the periodicity cell.

Keywords: periodicity cell, Rayleigh beam, tolerance averaging technique

1. Preface

Beamsare the simplest representations of periodic structures.Numerous examples of engineering
applications, for instance in acoustic isolations, are themain reason for interest in such objects.
In such beams, one can distinguish a small repetitive element called the periodicity cell. Periodic
objects can represent approximate models of some complex systems.
Propagation of the elastic wave and linear vibrations in periodic beams are considered in

many papers. Vibration band gaps were investigated by Xiang and Shi (2009) by the diffe-
rential quadrature method. A comprehensive research on inhomogeneous beams vibrations was
presented by Hajianmaleki and Qatu (2013). The transfer matrix method, adapted in analysis
of flexural wave propagation in a beam on an elastic foundation and in investigating natural
frequencies of non-uniform beams, can be found in Yu et al. (2012) and Xu et al. (2016), re-
spectively. Wave propagation in beams with periodically varying stiffness is considered in Chen
(2013) by the use of themultireflectionmethod. In this paper, linear-elasticRayleigh beamswith
a periodic structure are considered. Dynamics of such beams is described by partial differential
equations with non-continuous highly oscillating coefficients. The analysis of dynamic problems
using the aforementioned equations is very often problematic to perform.Thus, other simplified
models of Rayleigh beams are proposed. Some of these models are based on the concept of the
effective stiffness. Among them, one can distinguish the theory of asymptotic homogenization
introduced in works by Kohn and Vogelius (1984), Papanicolau et al. (1978), Bakhvalov and
Panasenko (1989), Sánchez-Palencia (1980) and Zhikov et al. (1994). The microperiodic beam
equilibrium equations in frames of the homogenization theory were studied by Kolpakov (1991,
1998, 1999). However, in governing equations of these models, the size of the mesostructure
parameter (the size of the periodicity cell) is often neglected. Therefore, a non-asymptotic ave-
ragedmodel of the periodic beam is introduced. Thismodel is called the tolerancemodel and is



142 M. Świątek et al.

derived by applying the tolerance modelling technique, c.f. Woźniak et al. (2008), Awrejcewicz
(2010), Woźniak andWierzbicki (2000). The obtained tolerance model equations have constant
coefficients and, in contrast to other averaged models, some of them depend on the size of the
periodicity cell. The proposedmethod can be adopted to any differential equations with highly
oscillating coefficients. The suggested approach, in contrast to the asymptotic homogenization,
enables analysis of themesostructure size. Themethod found numerous applications in structu-
ral mechanics. Macro-dynamics of microperiodic elastic beams was analysed by Mazur-Śniady
(1993). Geometrically nonlinear vibrations of slender mesoperiodic beams were investigated in
the paper by Domagalski and Jędrysiak (2016). Themethod was widely applied in the analysis
of microstructured plates: thin plates with an elastic periodic foundation, Jędrysiak (2003), ho-
neycomb lattice-type plates, Cielecka and Jędrysiak (2006), geometrically nonlinear thin plates,
Domagalski and Jędrysiak (2015) and thin functionally graded plates, Jędrysiak (2013, 2014)
and Kaźmierczak and Jędrysiak (2011). The TA technique was also applied in plates stabili-
ty problems, cf. Jędrysiak (2000) and Jędrysiak and Michalak (2011). The tolerance averaging
technique was also applied in the analysis of wavy type platesMichalak (2001) andmany other
engineering problems.
In this paper, anewtolerancemodel of aRayleighbeamwithweakly slowly-varying functions

is proposed Tomczyk (2013), Jędrysiak (2017). Natural boundary conditions are also obtained
and presented for a newly derived tolerance model. The presented tolerance model equations
are used to determine natural vibration frequencies and natural forms of vibrations. Solutions
obtained from the proposedmodel are compared with those corresponding to the finite element
model. The paper is arranged as follows: basic assumptions of the inhomogeneous Rayleigh
beams are presented in Section 2. The elemental and essential basis of the tolerance averaging
technique are quoted in Section 3. The main model equations for examples considered in this
paper are derived in Section 4. The numericalmethods of solution, validation of themodel, final
results and comparison with the finite element method are presented in Section 5. Finally, the
discussion and conclusions are given in Section 6.

2. Formulation of the problem

A beam made of a linear-elastic material, associated with a three-dimensional Cartesian co-
ordinate system Oxyz is considered. The beam axis is collinear with the x-axis of the local
coordinate system.The problem can be treated as one-dimensional, so that there is defined a re-
gionΩ≡ [0,L] occupiedby the beam,whereL is the beam length.The consideredbeamconsists
of many repetitive elements, called periodicity cells. The basic cell is defined as∆≡ [−l/2, l/2],

Fig. 1. A periodicity cell

where l≪L is length of the cell and is named the mesostructure parameter. The following de-
notations are introduced: lateral deflection w =w(x,t), lateral stiffness EJ =E(x)J(x), mass
per unit length µ=µ(x), rotational moment of inertia per unit length ϑ=ϑ(x) and transverse
load q = q(x,t). Furthermore, let ∂k = ∂k/∂xk be the k-th derivative of a function taken with
respect to thex coordinate, and the overdot stands for the derivative takenwith respect to time.



Free vibrations spectrum of periodically inhomogeneous Rayleigh beams... 143

Thus, the strain and kinetic energy of the beam can be described in the following form

W =
1
2
EJ∂2w∂2w K=

1
2
µẇẇ+

1
2
ϑ∂ẇ∂ẇ (2.1)

The Lagrangian functionL=L(x,t,w,ẇ,∂ẇ,∂2w) is defined as

L=W−K−qw (2.2)

The equations of motion are given by Hamilton’s principle

δA= δ

t1
∫

t0

L
∫

0

L dxdt=

t1
∫

t0

L
∫

0

δL dxdt=0 (2.3)

After some common variation calculus operations, the equation of motion of the Rayleigh beam
with highly oscillating non-continuous coefficients is obtained

∂2(EJ∂2w)+µẅ−∂(ϑ∂ẅ)= q (2.4)

3. Tolerance modelling

3.1. Preliminary notions

Themain objective of this paper is to propose a new averaged model of the Rayleigh beam.
This new approach is based on the concept of weakly slowly-varying functions. The averaged
equations of the periodic beam are derived using the tolerance modelling technique. The fun-
damental concepts of the tolerance modelling approach – tolerance relations, slowly-varying
functions (SV ), tolerance periodic functions (TP), fluctuation shape functions (FSFs) and ave-
raging operation, are outlined in the monographs by Woźniak and Wierzbicki (2000), Woźniak
et al. (2008), Awrejcewicz (2010). There are introduced the following denotations:∆(x)≡x+∆,
Ω∆ ≡ {x∈Ω : ∆(x)∈Ω}, x∈Rm. Subsequently, a subset ∆ of Rm is called the periodicity
cell with l as a cell dimension. Every cell ∆(x), x∈Ω∆, refers to the cell in Ω with the center
at x. The averaging operator for an arbitrary integrable function f is defined by

〈f〉(x)=
1
l

∫

∆(x)

f(y) dy x∈Ω∆ y∈∆(x) (3.1)

Themicro-macro decomposition is a fundamental operation of the tolerance averaging tech-
nique. It states that the transverse deflection of the beamw(x,t) (unknown of the partial diffe-
rential equations describing behavior of the microheterogeneous structure) can be decomposed
into: the unknown averaged displacement W(x,t) (a weakly slowly-varying function in the pe-
riodicity direction) and the highly oscillating fluctuation of the displacement, represented by the
knownhighly oscillating∆-periodic fluctuation shape functionhA(x)multiplied by the unknown
fluctuation amplitude VA(x,t) – weakly slowly-varying (WSV ) in the periodicity direction. In
this case, the micro-macro decomposition becomes

w(x,t) =W(x,t)+hA(x)VA(x,t) A=1, . . . ,N W(·),VA(·)∈WSV 2
d
(Ω,∆)

(3.2)

From now, W(x,t) is a new basic kinematic unknown and VA(x,t) is an additional kinematic
unknown. The uppercase integer states that the unknown functions are assumed to be weakly
slowly-varying up to the second derivative order. The function F(·) will be referred to as the



144 M. Świątek et al.

weakly slowly-varyingwith respect to the cell∆andthe tolerance givenby δ≡ (α,δ0,δ1, . . . ,δR),
if and only if the following condition is satisfied

∃(x,y)∈Ω2
[

(x
α
≈ y) ⇒ F(x)

δ0
≈F(y) ∧ ∂kF(x)

δk
≈ ∂kF(y), k=1,2, . . . ,R

]

(3.3)

where ∂0F(·)≡F(·).
Under the above conditions, it can be written F ∈WSVR

δ
(Ω,∆). In the applications of the

tolerance modelling, the tolerance parameterα= l is known a priori as a certainmesostructure
length, whereas values of the tolerance parameters δ0,δ1, . . . ,δR can be determined only a poste-
riori, i.e. after obtaining a solution to the considered initial-boundary value problem.The highly
oscillating fluctuation shape functions hA are postulated a priori in every problem under con-
sideration and describe the unknown fields oscillations caused by the structure inhomogeneity.
Apart from the restriction of l-periodicity, the FSFs have to satisfy the following conditions

〈µhA〉=0 〈µhAhB〉=0 forA 6=B

∂mhA ∈O(l2−m) A,B =1, . . . ,N
(3.4)

Another assumption is the tolerance averaging approximation. For the purposes of this article,
the following denotations are introduced. Let e,f ∈ L2

loc
(R) be the known l-periodic func-

tions and let F ∈ WSV 1
d
(0,L), d ≡ (l,δ0,δ1). By the tolerance averaging of eF + f∂1F is

meant 〈eF + f∂1F〉T(x) ≡ 〈e〉F(x) + 〈f〉∂1F(x) for every x ∈ (l/2,L− l/2). The tolerance
averaging approximation is an approximation of 〈eF +f∂1F〉(x) by 〈eF +f∂1F〉T(x) for every
x∈ (l/2,L− l/2). Thus, the tolerance averaging approximation has the form

〈eF +f∂1F〉(x) = 〈eF +f∂1F〉T(x)+O(l) d≡ (l,δ0,δ1) (3.5)

where e(·), f(·) are the known functions and F(·) is unknown in the initial-boundary value
problem under consideration.

3.2. The averaged model equations

The averaging operation is performed, after substituting micro-macro decomposition (3.2)
into Lagrangian (2.2). Thus, the variation of the averaged action functional can be written as

δA= δ

t1
∫

t0

L
∫

0

〈Lh〉 dxdt=

t1
∫

t0

L
∫

0

δ〈Lh〉 dxdt=0 (3.6)

Knowing that

−κ= ∂2w= ∂2W +∂2(hAVA)= ∂2W +∂(∂hAVA+hA∂VA)

= ∂2W +∂2hAVA+2∂hA∂VA+hA∂2VA

− δκ= ∂2δW +∂2hAδVA+2∂hA∂δVA+hA∂2δVA

−M =EJ∂2w=EJ(∂2W +∂2hBVB +2∂hB∂VB +hB∂2VB)t

(3.7)

the Lagrangian variation is

δL= δW− δK−qδw=Mδκ−µẇδẇ−ϑ∂ẇ∂δẇ−qδw (3.8)

Finally

t1
∫

t0

L
∫

0

δK dxdt=

t1
∫

t0

L
∫

0

(µẅδw+ϑ∂ẅ∂δw) dxdt (3.9)



Free vibrations spectrum of periodically inhomogeneous Rayleigh beams... 145

and

δL=Mδκ+(µẅ−q)δw+ϑ∂ẅ∂δw (3.10)

Let micro-macro decomposition (3.2) be substituted into the components of the Lagrangian
and averaged over a periodicity cell. It should be noted that

δw= δW +hAδVA (3.11)

The variation of the averaged bending energy gives

〈δL〉= 〈Mδκ〉= 〈M〉∂2δW + 〈M∂2hA〉δVA+2〈M∂hA〉∂δVA+ 〈MhA〉∂2δVA (3.12)

where

〈M〉= 〈EJ〉∂2W + 〈EJ∂2hB〉VB +2〈EJ∂hB〉∂VB + 〈EJhB〉∂2VB

〈M∂2hA〉= 〈EJ∂2hA〉∂2W + 〈EJ∂2hA∂2hB〉VB +2〈EJ∂2hA∂hB〉∂VB

+ 〈EJ∂2hAhB〉∂2VB

〈M∂hA〉= 〈EJ∂hA〉∂2W + 〈EJ∂hA∂2hB〉VB +2〈EJ∂hA∂hB〉∂VB + 〈EJ∂hAhB〉∂2VB

〈MhA〉= 〈EJhA〉∂2W + 〈EJhA∂2hB〉VB +2〈EJhA∂hB〉∂VB + 〈EJhAhB〉∂2VB

(3.13)

The total variation of the Lagrangian is

δL= 〈M〉∂2δW +
(

〈ϑ〉∂Ẅ + 〈ϑ∂hA〉V̈A
)

∂δW +
(

〈µ〉Ẅ + 〈µhA〉V̈A−〈q〉
)

δW

+
(

〈M∂2hA〉+ 〈µhA〉Ẅ + 〈µhAhB〉V̈B + 〈ϑ∂hA〉∂Ẅ

+ 〈ϑ∂hA∂hB〉V̈B −〈qhA〉
)

δVA+2〈M∂hA〉∂δVA+ 〈MhA〉∂2δVA

(3.14)

After some transformations

δL=
[

∂2〈M〉−∂
(

〈ϑ〉∂Ẅ + 〈ϑ∂hA〉V̈A
)

+ 〈µ〉Ẅ + 〈µhA〉V̈A−〈q〉
]

δW

+
(

〈M∂2hA〉−2∂〈M∂hA〉+ 〈µhA〉Ẅ + 〈µhAhB〉V̈B + 〈ϑ∂hA〉∂Ẅ

+ 〈ϑ∂hA∂hB〉V̈B +∂2〈MhA〉−〈qhA〉
]

δVA+∂
(

〈M〉∂δW
)

+∂
[(

〈ϑ〉∂Ẅ + 〈ϑ∂hA〉V̈A−∂〈M〉
)

δW
]

+∂
(

〈MhA〉∂δVA
)

−∂
[(

∂〈MhA〉−2〈M∂hA〉
)

δVA
]

(3.15)

This leads to a system of differential equations

δW : ∂2〈M〉−∂
(

〈ϑ〉∂Ẅ + 〈ϑ∂hA〉V̈A
)

+ 〈µ〉Ẅ + 〈µhA〉V̈A−〈q〉=0

δVA : 〈M∂2hA〉−2∂〈M∂hA〉+ 〈µhA〉Ẅ + 〈µhAhB〉V̈B

+ 〈ϑ∂hA〉∂Ẅ + 〈ϑ∂hA∂hB〉V̈B +∂2〈MhA〉−〈qhA〉=0

(3.16)

and natural boundary conditions

(

〈ϑ〉∂Ẅ + 〈ϑ∂hA〉V̈A−∂〈M〉
)

δW
∣

∣

∣

L

0
+ 〈M〉∂δW

∣

∣

∣

L

0
+ 〈MhA〉∂δVA

∣

∣

∣

L

0

+
(

∂〈MhA〉−2〈M∂hA〉
)

δVA
∣

∣

∣

L

0
=0

(3.17)



146 M. Świątek et al.

TheN+1 differential equations for the macro-deflection and its fluctuation amplitudes are

∂2〈M〉−〈ϑ〉∂2Ẅ −〈ϑ∂hA〉∂V̈A+ 〈µ〉Ẅ + 〈µhA〉V̈A−〈q〉=0

〈M∂2hA〉−2∂〈M∂hA〉+ 〈µhA〉Ẅ + 〈µhAhB〉V̈B + 〈ϑ∂hA〉∂Ẅ

+ 〈ϑ∂hA∂hB〉V̈B +∂2〈MhA〉−〈qhA〉=0

(3.18)

The weight-averaged bendingmoments have the following form


















〈M〉
〈M∂2hA〉
〈M∂hA〉
〈MhA〉



















=











〈EJ〉 〈EJ∂2hB〉 〈EJ∂hB〉 〈EJhB〉

〈EJ∂2hA〉 〈EJ∂2hA∂2hB〉 〈EJ∂2hA∂hB〉 〈EJ∂2hAhB〉

〈EJ∂hA〉 〈EJ∂hA∂hB〉 〈EJ∂hA∂hB〉 〈EJ∂hAhB〉

〈EJhA〉 〈EJhA∂2hB〉 〈EJhA∂hB〉 〈EJhAhB〉





























∂2W
VB

2∂VB

∂2VB



















(3.19)

where W(x,t), VA(x,t) and their derivatives are the new kinematic unknowns. Together with
the averaged equation of motion, the following natural boundary conditions (for x=0,L) with
averaged coefficients are obtained

〈ϑ〉∂Ẅ + 〈ϑ∂hA〉V̈A−∂〈M〉=0 or δW =0

〈M〉=0 or ∂δW =0

∂〈MhA〉−2〈M∂hA〉=0 or δVA =0

〈MhA〉=0 or ∂δVA =0

(3.20)

It is worthmentioning that expressions (3.20) reduce to classic natural boundary conditions for
a homogeneous beam (Fung, 1965). Moreover, the underlined coefficients are dependent on the
mesostructure size l. The external load is assumed tobe zero in the analysis of natural vibrations
of the beam.

4. Examples of applications

In this Section, the derived averaged model is adapted in a study of some special problems.
The object under consideration is a simply supported beam with length L. The beam has a
rectangular cross section and is made of some small repetitive elements. The periodicity cell,
presented in Fig. 1, has a symmetrical shape and is divided into three segments. The segment
material and geometrical properties may vary depending on each case.
One of the most significant components of the tolerance modelling is determination of the

fluctuation shape functions. Thefluctuation shape functions can be assumed as forms of eigenvi-
brations one the periodicity cell. In this model, FSFs are obtained from finite element analysis
of the periodicity cell, although the common practice is to use approximate solutions such as
l-periodic trigonometric functions.
In order to obtain a systemof algebraic equations ofmotion, theGalerkinmethod is applied.

The trial solutions are assumed in the form of truncated trigonometric series

W(x,t)=
Mw
∑

m=1

Xm(x)Wm(t) V
A(x,t)=

M
A

V
∑

n=1

YA
m
(x)VA

m
(t) A=1, . . . ,N (4.1)

where theweight functionsXm andYAm are chosen to satisfy the boundary conditions of a simply
supported beam

Xm(x)= sin
mπx

L
YA
m
=















sin
nπx

L
for A∈ESF

cos
(n−1)πx
L

for A∈OSF
(4.2)



Free vibrations spectrum of periodically inhomogeneous Rayleigh beams... 147

The functionsXm andYAm satisfy the assumed boundary conditions atx=0,L, whereESF and
OSF stands for an even and odd fluctuation shape function respectively. The known relation
is solved with respect to the unknown trial function coefficients (Zienkiewicz et al., 2013). The
number of terms in the expansion results from the condition of convergence of the solution. In
order to obtain the natural frequencies of the beam, the eigenproblem of the dynamic stiffness
matrix is solved. In the numerical solutions, the size of the matrix is limited to the finite value

Fig. 2. The considered beam

5. Results and discussion

This Section is dedicated to the analysis of free vibrations of a Rayleigh beam. The beam has
length L = 1.0m and is composed of 10 periodicity cells with length l = 0.1m and cross
section width b=0.01m. Three different beams with variable cross section height hM, Young’s
modulus EM and mass density ρM are analyzed. For each beam, there are considered three
individual cases. The properties of the central periodicity cell segment – height hR = 0.008m,
Young’s modulus ER = 205GPa and density ρR = 7850kg/m3 are constant in all analyzed
cases.
As an example, 3 cases: A, B and C are analyzed. For each case, one of the material or

geometrical parameter of the periodicity cell has an individual value. The values of hM, EM
and ρM parameters for all cases are presented in Table 1.

Table 1.Analyzed-cases

Case hM [m] EM [GPa] ρM [kg/m3]

A1 0.004

205.000

7850.00A2 0.005
A3 0.006
B1

0.008

3925.00
B2 1962.50
B3 981.25
C1 102.500

7850.00C2 51.250
C3 25.625

In order to validate the tolerance model, a finite element method procedure is applied in
Maple software. The finite element model is assembled with 30 Rayleigh beam elements with
Hermitian polynomials and the consistent mass matrix. As a result, the model has 31 nodes
with 62 degrees of freedom.
The natural frequencies, which are obtained using the tolerance averaging technique (TA)

and the finite elementmethod (FE), are compared in Table 2. The validation of first 22 natural
frequencies for cases A2, B2 andC2 for themesostructureα=0.5 is performed. TheTAmodel
results are presented as gray dots, and the FE model as black rings. The received values of
frequencies are given in [Hz]. The first five natural frequencies for cases A2, B2, C2 are listed in
Table 2. 80.878Hz, 147.330Hz, and 73.581Hz are the least derived values of natural frequencies
in cases A2, B2, and C2, respectively. The relative error does not exceed 2% for the first five



148 M. Świątek et al.

frequencies. It is noticeable that thepresentedbandwidth is not entirely continuous anduniform.
Among all obtained frequencies in Fig. 3, separated bands of frequencies can be observed –
the chains preceded and followed by some intervals. These interruptions in the bandwidth,
highlighted with gray backgrounds, are called band gaps. In case A, the first band interval
reveals between the 9th and 10th natural frequency. The difference between these frequencies
arrives at 3133Hz. Another gap appears between the 20th and 21st natural frequency. In this
case, the difference rises to 8454Hz. In case B, the gaps occur between the 10th, 11th and 20th,
21st free vibration frequencies, and the intervals in the bandwidth reach 7657 and 14164Hz,
respectively. In case C, the band gaps reveal at the same frequencies as in case A, and the
magnitudes of the interludes are 3663Hz and 7171Hz, respectively.

Table 2.Natural frequencies for case A2, B2 and C2

ωi
A2 B2 C2

TA FE |∆ω|
|ωFE|

TA FE |∆ω|
|ωFE|

TA FE |∆ω|
|ωFE|

n [Hz] [Hz] [%] [Hz] [Hz] [%] [Hz] [Hz] [%]
1 80.878 80.971 0.115 147.330 147.314 0.011 73.581 73.659 0.105
2 322.607 324.108 0.463 589.309 589.054 0.043 293.269 294.534 0.430
3 722.358 730.114 1.062 1325.846 1324.555 0.097 655.683 662.294 0.998
4 1274.964 1300.213 1.942 2356.475 2352.406 0.173 1154.338 1176.231 1.861
5 2043.391 2036.098 0.358 3679.510 3669.598 0.270 1843.340 1834.854 0.462

Fig. 3. Comparison of natural frequencies bandwidth of the considered beam for α=1/2,
case A2, B2, C2

In Figs. 4-6, the band gaps neighborhood is shown. The tolerance solutions are represented
by solid lines, and the finite element solutions are represented by dashed lines. In this case, the
presented frequencies are functionsof the saturationparameterα.All frequencies arepresented in
relation to the constant value – the natural frequency obtained from the finite element method.
As a result, the solutions are presented in the dimensionless form. The first two gaps in the
observedbandwidthrangeareanalyzed. IncasesAandC, theanalyzed frequencies are increasing
with the argument of a function. In case B, a decreasing relationship can be observed. In the
enclosed figures, two types of band gaps can be noticed. The first type of the gap is between
the same frequencies in the entire domain of the mesostructure parameter. The second type of
the gap changes its character along with the α parameter. The following relationship can be
observed: the compared models have the best convergence for low natural frequencies and low
values of the α parameter.



Free vibrations spectrum of periodically inhomogeneous Rayleigh beams... 149

Fig. 4. Band-gap neighboring eigenfrequencies as a function of the α parameter, case A

In Table 3, the eigenmodes of the considered beam model for α = 1/2 are compared. It
can be noticed that in B2 case the band gaps occur in different places in comparison with A2,
B2 and C2 cases (Fig. 3). What is more, there is a difference in the order of symmetrical and
antisymmetrical eigenmodes (cf. Table 3).

6. Final remarks

In this paper, the authors present a new averaged model of a linear-elastic periodic Rayleigh
beam. Dynamics of the beam is described by partial differential equations with non-continuous



150 M. Świątek et al.

Fig. 5. Band-gap neighboring eigenfrequencies as a function of the α parameter, case B

highly oscillating coefficients. The exact model equations are transformed into a form that can
be solved numerically.

The new model implements the notion of weakly slowly-varying functions. The proposed
equations are derived using the tolerance averaging approach. In contradiction to other homo-
genizedmodels, the tolerance averaging technique allows one to observe some averaged effective
properties of a structure. Despite the inhomogeneity of the structure, this newmodel introduces
somenewunknowns– averaged deflection. It also allows one to observe somedynamic properties
of the beam, depending on the size of the periodicity cell.



Free vibrations spectrum of periodically inhomogeneous Rayleigh beams... 151

Fig. 6. Band-gap neighboring eigenfrequencies as a function of the α parameter, case C

The solutions derived from tolerance averaging have been compared with the finite element
model solutions. The finite element model has 31 nodes with 62 degrees of freedom.

In this paper, 3 cases have been compared:A,B andC.Agood agreement has been obtained
between the two methods in all analyzed cases. What is more, there is an evident dependency
between the occurrence of band gaps and the shape of eigenmodes. Therefore, the proposed
solution enables one to formulate model equations which can be solved with known numerical
methods (e.g. Galerkin method). That is why the suggested technique can be used in the para-
metric analysis of the structures under consideration. The problems that can be considered in



152 M. Świątek et al.

Table 3.Comparison of natural frequency bandwidths of the considered beam for α=1/2

Case ω9 ω10 ω11 ω20 ω21

A2

TA

FE

B2

TA

FE

C2

TA

FE

futureworks are: forced vibrations of inhomogeneousRayleigh beams, greater diversity of boun-
dary conditions, analysis of structural andmaterial heterogeneity of the beamand a viscoelastic
subsoil.

Acknowledgements

This contribution was supported by the National Science Centre of Poland under grant
No. 2014/15/B/ST8/03155

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Manuscript received March 24, 2018; accepted for print August 8, 2018