Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 4, pp. 1095-1108, Warsaw 2016
DOI: 10.15632/jtam-pl.54.4.1095

NONLINEAR VIBRATIONS OF PERIODIC BEAMS

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

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

Geometrically nonlinear vibrations of beams with properties periodically varying along the
axis are investigated. The tolerance method of averaging differential operators with highly
oscillating coefficients is applied to obtain governing equations with constant coefficients.
The proposedmodel describes dynamics of the beamwith the effect of microstructure size.
In an example, an analysis of undamped forced nonlinear vibrations of the periodic beam
is shown. Moreover, the results obtained for undamped free vibrations of periodic beams
by the tolerance model are justified by those results from the finite element method. These
results can be used as a benchmark in similar problems.

Keywords: nonlinear vibrations, periodic beams, tolerancemodelling

1. Introduction

The paper concerns with geometrically nonlinear vibrations of beamswith geometric andmate-
rial properties periodically varying along the x-axis. Moreover, such beams can interact with a
periodically inhomogeneous viscoelastic subsoil. A fragment of such a beam is shown in Fig. 1.

Fig. 1. A fragment of a periodic beam

Equations of motion of such structures have usually highly oscillating, periodic, non-
-continuous functional coefficients. For this reason, emphasis is placed on the formulation of
continuous models of the considered structures. In the proposed method of modelling, this is
performed by substituting the original equations with an effective model with constant coeffi-
cients. This makes it possible to avoid full discretization of the problem.
Structures with physical properties regularly arranged in the body domain are commonly

found innature andarewidely used in engineering.The continuous interest in suchobjects is due
to their specific properties. Properly designed composite structures are characterized by, among
others, favourable ratio of stiffness toweight.This, due to the trends inmodern technology for the
design of lightweight, high-strength structures, indicates actuality of the problem. In particular,
periodic structures exhibit some interesting and desirable dynamic properties, namely, theymay
serve as filters for some specific vibration frequency bands, cf. Banakh and Kempner (2010).
Analysis of the so-called locally resonant beams were published in numerous papers, e.g. by
Olhoff et al. (2012), where the optimization of beam geometry in order to obtain the maximum
width of the frequency band-gap leads to a periodic structure.



1096 Ł. Domagalski, J. Jędrysiak

Direct numerical modelling of structures of this kind, such as finite element discretization,
is one of the possible ways of analysis the considered problems. The computational cost of the
parametric analysis within the discretization approach is, however, proportional to the varia-
tion range of the parameters under consideration. Thus, it is advisable to strive to formulate
alternative continuous models in order to reduce the computational cost.
Among the analytical methods applied in stationary problems of periodic structures, the

most widespread are those based on the rigorous mathematical theory of asymptotic homoge-
nization of differential operators. In this approach, the actual periodic structure is modelled
as a homogeneous anisotropic structure with some effective properties. The above-mentioned
effective properties are obtained through analysis of the so-called periodicity cell problem. The
fundamentals of this theory are described byBensoussan et al. (1978), Sanchez-Palencia (1980),
Bakhvalov and Panasenko (1984), Jikov et al. (1994), Lewiński and Telega (2000), Krysko et
al. (2008). Some various works are devoted to derivation of micro-periodic beam equilibrium
equations in the frame of homogenization theory, wherein the starting point of analysis are
three-dimensional elasticity theory equations, see Kolpakov (1991, 1995, 1998, 1999), Syerko et
al. (2013). Certain analytical approaches and the finite element method are also used to evalu-
ate strength and buckling of sandwich beams having corrugated cores, e.g. by Magnucki et al.
(2013).
The literature on the problems of linear vibrations of periodic beams is extensive. In most

of the research papers, attention is focused on local resonance properties of such structures.
The two-scale asymptotic expansions are applied by He et al. (2013) in analysis of beams with
periodically variable stiffness. A common approach making use of the theory of Floquet-Bloch
waves in the analysis. This was applied in the analysis of the Timoshenko (Chen and Wang,
2013) and Euler-Bernoulli (Chen, 2013) beam vibrations. The problem of wave propagation in
a periodic elastically supported beam was considered by Yu et al. (2012) using the transfer
matrix method. The direct approach with use of the Heaviside step function in the case of the
forced oscillation of the plate band on a periodic elastic substrate was applied by Sylvia and
Hull (2013), where the investigations were brought to a one-dimensional problem.
Description of dynamic problems within geometrically linear theories imposes severe re-

strictions on deformations, limiting the displacements order to the smallest dimension of the
structural element considered. Since the considered structures are slender at the macro level,
this limitationmakes it impossible to correctly analyze thewhole spectrumof their applicability.
In addition, some physical phenomena that occur in vibrations of nonlinear systems which have
a significant impact on motion characteristics are impossible to investigate in terms of lineari-
zed theories. Nonlinear vibrations of homogeneous nano-beams cooperatingwith a homogeneous
visco-elastic substratewere considered byWang andLi (2014), vibrations of the sandwichbeams
by Krysko et al. (2008). The paper by Awrejcewicz et al. (2011) contains comparison of nonli-
near vibrationmodels of the Euler-Bernoulli beam derived through FEMdiscretization and the
finite differencesmethod.Despite the large number of studies dealing with non-linear vibrations
(e.g. Sedighi et al., 2013; Hryniewicz and Kozioł, 2013), the majority of publications relates to
systems with a relatively small number of degrees of freedom.
In this contribution, in order to replace differential equations with highly oscillating co-

efficients by equations with constant coefficients, the tolerance modelling, see Woźniak and
Wierzbicki (2000),Woźniak et al. (2008, 2010) is applied. This approach was introduced for the
purpose of analysis of various thermomechanical problems of periodic elastic composites in a
series of papers, e.g. for thin periodic plates on a foundation by Jędrysiak (1999), for micro-
-periodic beams under moving load byMazur-Śniady and Śniady (2001),for periodic beams for
plates with the microstructure size of an order of the plate thickness – for periodic thin by
Mazur-Śniady et al. (2004), for periodicmedium-thickness by Baron (2006), for thin functional-
ly graded by Jędrysiak (2013), for multiperiodic fibre reinforced composites by Jędrysiak and



Nonlinear vibrations of periodic beams 1097

Woźniak (2006), for periodic shells by Tomczyk (2007), for functionally graded plates by Wi-
rowski (2012). This technique was also used in vibration analysis of periodic beams within the
linear theory byMazur-Śniady (1993), where the equations ofmotion and their generalization by
including the influence of the axial force, elastic subsoil and viscous dampingwere derived. The
books byWoźniak andWierzbicki (2000),Woźniak et al. (2008, 2010) contain the fundamentals
of this theory and numerous examples of application.
Themainaimof this note is toderive averaged governing equations of thenonlinear tolerance

model of dynamics of periodic beams on a viscoelastic foundation and showa certain application
of this model to a special problem.Moreover, some justifications of the results by the proposed
model are presented by the results obtained from the finite element method for a benchmark
problem of free vibrations of a linear periodic beam.

2. Formulation of the problem

The object under consideration is a linearly elastic prismatic beam, bilaterally interacting with
a periodic viscoelastic foundation. Let Oxyz be an orthogonal Cartesian coordinate system in
which theOx axis coincideswith the axis of thebeam, the cross section of thebeam is symmetric
with respect to the plane of the load Oxz, the load acts in the direction of the axis Oz. The
problem can be treated as one-dimensional, so that we define the region occupied by the beam
as Ω≡ [0,L], whereL stands for the beam length.
The beam is assumed to bemade of many repetitive small elements, called periodicity cells,

defined as ∆ ≡ [−l/2, l/2], where l ≪ L is length of the cell and named the microstructure
parameter.
Our considerations are based on the Rayleigh theory of beams with von Kármán type non-

linearity. Since we are interested in the transverse vibrations only, the effect of axial inertia is
neglected in further considerations. Let ∂k = ∂k/∂xk be thek-th derivative of a functionwith re-
spect to thex coordinate, overdot stands for the derivativewith respect to time. Letw=w(x,t)
be the transverse deflection, u0 = u0(x,t) longitudinal displacement, EA = E(x)A(x) and
E(x)J = EJ(x) tensile and flexural stiffness, k = k(x) and c = c(x) – elasticity and damping
coefficients of the foundation, µ = µ(x) and ϑ = ϑ(x) mass and rotational moment of inertia
per unit length and q= q(x,t) – transverse load. The strain and kinetic energy density per unit
length of the beam are

W =
1
2
EA
(
∂u0+

1
2
∂w∂w

)2
+
1
2
EJ(∂2w)2 K=

1
2
µẇẇ+

1
2
ϑ∂ẇ∂ẇ (2.1)

For the subsoil, we apply the Kelvin-Voight model, so that the dissipative force is assumed in
the form

p= p(x,t)= c(x)ẇ(x,t) (2.2)

The equations of motion can be obtain from the extended (Woźniak et al., 2010) principle
of stationary action A=A(u0,w) formulated as

δA= δ
1∫

0

L∫

0

L dxdt=
1∫

0

L∫

0

δL dxdt=
1∫

0

L∫

0

[(∂L
∂u0
−∂

∂L

∂(∂u0)

)
δu0

+
(∂L
∂w
−∂

∂L

∂(∂w)
+∂2

∂L

∂(∂2w)
−
d

dt

∂L

∂ẇ
+
d

dt
∂
∂L

∂(∂ẇ)

)
δw
]
dxdt=0

(2.3)



1098 Ł. Domagalski, J. Jędrysiak

where the Lagrangian is

L(x,t,w,ẇ,∂w,∂2w,∂u0)=W−K+pw+
1
2
kww−qw

=
1
2
EA
(
∂u0+

1
2
∂w∂w

)2
+
1
2
∂2wEJ∂2w−

1
2
µẇẇ−

1
2
ϑ∂ẇ∂ẇ+pw+

1
2
kww− qw

(2.4)

The system of nonlinear coupled differential equations for the longitudinal displacements u0
and the transverse deflection w resulting from (2.3) can be written as

∂
[
EA
(
∂u0+

1
2
∂w∂w

)]
=0

µẅ−∂(ϑ∂ẅ)+ cẇ+kw+∂2(EJ∂2w)−∂
[
EA
(
∂u0+

1
2
∂w∂w

)
∂w
]
= q

(2.5)

The coefficients EA,EJ, k, µ, ϑ, c are highly oscillating, often non-continuous functions of
the x-coordinate. The main aim of this note is to derive an approximately equivalent model,
which describes geometrically nonlinear vibrations of periodic beams bilaterally interactingwith
a periodic viscoelastic foundation, taking into account the effect of microstructure size.

3. Introductory concepts and basic assumptions of the tolerance modelling

The averaged equations of periodic beams with large deflections are derived using the concepts
and assumptions of the tolerance modelling technique, see Woźniak et al. (2010). The funda-
mental concepts are: the tolerance system, averaging operation and certain classes of functions
such as the tolerance-periodic (TP), slowly-varying (SV ), highly oscillating (HO) and fluctu-
ation shape (FS) functions. The tolerance parameter, associated with the tolerance relation, is
denoted by d, 0<d≪ 1. The highest order of function derivative that can be included into a
certain function class is denoted by α.
Let ∆(x) = x+∆, Ω∆ = {x ∈ Ω : ∆(x) ⊂ Ω} be a cell with its center at x ∈ Ω∆. The

averaging operator for an arbitrary integrable function f is defined by

〈f〉(x)= l−1
∫

∆(x)

f(y) dy x∈Ω∆ y∈∆(x) (3.1)

It can be shown (Woźniak et al., 2010) that for a periodic function f ofx, its average is constant.
The first of the basic assumptions is the micro-macro decomposition of the unknown trans-

verse deflection and longitudinal displacement

w(x,t) =W(x,t)+hA(x)VA(x,t)

A=1, . . . ,N W(·, t),VA(·, t)∈SV 2
d
(Ω,∆) hA(·)∈FS2

d
(Ω,∆)

(3.2)

and

u0(x,t)=U(x,t)+g
K(x)TK(x,t)

K =1, . . . ,M U(·, t),TK(·, t)∈SV 1
d
(Ω,∆) gK(·)∈FS1

d
(Ω,∆)

(3.3)

Here and hereafter, the summation convention holds.
The new basic kinematic unknowns W(·) and U(·) are called the transverse and the axial

macrodisplacements, respectively; VA(·) and TK(·) are additional kinematic unknowns, called
thefluctuation amplitudes.Theunknown functionsare assumed tobe slowly-varying.Thehighly
oscillating fluctuation shape functions hA and gK are postulated a priori in every problem



Nonlinear vibrations of periodic beams 1099

under consideration and are assumed to describe unknownfields oscillations caused by structure
inhomogeneity. These functions have to satisfy the following conditions

〈µhA〉=0 〈µgK〉=0〈µhAhB〉=0 for A 6=B

〈µgKgL〉=0 for K 6=L ∂mhA ∈O(l2−m)

∂ngK ∈O(l1−m) A,B =1, . . . ,N K,L=1, . . . ,M

(3.4)

The second assumption is the tolerance averaging approximation

〈f〉(x)= 〈fx〉(x)+O(d) 〈f∂
α(φAΨ)〉(x)= 〈f∂αφA〉(x)Ψ(x)+O(d)

〈fΨ〉(x)= 〈f〉(x)Ψ(x)+O(d) 〈f∂(γKΞ)〉(x) = 〈f∂γK〉(x)Ξ(x)+O(d)

〈fΞ〉(x) = 〈f〉(x)Ξ(x)+O(d) f ∈TPα
d
(Ω,∆) Ψ ∈SV 2

d
(Ω,∆)

φA ∈FS2
d
(Ω,∆) Ξ ∈SV 1

d
(Ω,∆) γK ∈FS1

d
(Ω,∆)

x∈Ω α=1,2 A=1, . . . ,N K =1, . . . ,M 0<d≪ 1

(3.5)

in which the terms of the order of the tolerance parameter O(d) are assumed to be negligibly
small.

4. The governing equations of the proposed models

4.1. The equations of the tolerance model

After substitution of micro-macro decompositions (3.2) and (3.3) into Lagrangian (2.4), the
next step of modelling is averaging (3.1) over an arbitrary periodicity cell with approximations
(3.5).
The averaged action functional has the following form

δAh = δ
1∫

0

L∫

0

〈Lh〉 dxdt=
1∫

0

L∫

0

δ〈Lh〉 dxdt=0 (4.1)

where the averaged Lagrangian is formulated as follows

〈Lh〉=
1
2
D∂2W∂2W +DA∂2WVA+(P −Q)W +

1
2
KWW +

1
2
B
(
∂U+

1
2
∂W∂W

)2

+ l2
(
PA−QA+KAW +

1
2
l2KABVB

)
VA−

1
2
MẆẆ −

1
2
J∂Ẇ∂Ẇ

− (l2MAẆ + lJA∂Ẇ)V̇A+
[
BK
(
∂U+

1
2
∂W∂W

)
+
1
2
BKLTL

]
TK

−
1
2
l2(l2MAB +JAB)V̇BV̇A+ l

[
BA
(
∂U+

1
2
∂W∂W

)
+BAKTK

]
VA∂W

+
1
2
l2
[
BAB
(
∂U+

3
2
∂W∂W

)
+BABKTK

]
VBVA

+
1
2
l3
(
BABC∂W +

1
4
lBABCDVD

)
VAVBVC +

1
2
DABVAVB

(4.2)

Under essential boundary conditions it leads to a system of Euler-Lagrange equations

∂〈Lh〉

∂U
−∂
∂〈Lh〉

∂(∂U)
= 0

∂〈Lh〉

∂W
−∂
∂〈Lh〉

∂(∂W)
+∂2

∂〈Lh〉

∂(∂2W)
−
d

dt

∂〈Lh〉

∂(Ẇ)
+
d

dt

(
∂
∂〈Lh〉

∂(∂Ẇ)

)
=0

∂〈Lh〉

∂TK
=0

∂〈Lh〉

∂VA
−
d

dt

∂〈Lh〉

∂(ẆA)
= 0

(4.3)



1100 Ł. Domagalski, J. Jędrysiak

After somemanipulations, we arrive at the following system of equations

∂(B∂U +BKTK)+∂
(1
2
B∂W∂W + lBAVA∂W +

1
2
l2BABVBVA

)
=0

D∂4W +DA∂2VA+KW + l2KAVA+CẆ +MẄ −J∂2Ẅ

+ l2CAẆA+ l2MAV̈A− lJA∂V̈A−Q

−∂
[(
B
(
∂U +

1
2
∂W∂W

)
+BKTK + lBAVA∂W +

1
2
l2BABVBVA

)
∂W
]

−∂
[
lBAKVATK +

(
∂U +

1
2
∂W∂W

)
lBAVA+ l2BAB∂WVBVA

+
1
2
l3BABCVAVBVC

]
=0

(DAB + l4KAB)VB + l4CABẆB + l2(l2MAB +JAB)V̈B +DA∂2W + l2KAW

+ l(lMAẄ +JA∂Ẅ)+ l2CAẆ − l2QA+ l
[
BA
(
∂U +

1
2
∂W∂W

)
+BAKTK

]
∂W

+ l2
(
BAB∂U +

3
2
BAB∂W∂W +BABKTK +

3
2
lBABCVC∂W

+
1
2
l2BABCDVCVD

)
VB =0

BKLTL+BK∂U +
1
2
BK∂W∂W + lBAKVA∂W +

1
2
l2BABKVBVA =0

(4.4)

with constant coefficients related to the beam properties

〈EA〉 ≡B 〈EA∂hA〉≡ lBA 〈µ〉≡M

〈EA∂gK〉≡BK 〈EA∂hA∂gK〉≡ lBAK 〈µhA〉≡ l2MA

〈EA∂gK∂gL〉≡BKL 〈EA∂hA∂hB∂gK〉≡ l2BABK 〈µhAhB〉≡ l4MAB

〈EJ〉 ≡D 〈EA∂hA∂hB〉≡ l2BAB 〈ϑ〉≡ J

〈EJ∂2hA〉≡DA 〈EA∂hA∂hB∂hC〉≡ l3BABC 〈ϑ∂hA〉≡ lJA

〈EJ∂2hB∂2hA〉≡DAB 〈EA∂hA∂hB∂hC∂hD〉≡ l4BABCD 〈ϑ∂hA∂hB〉≡ l2JAB

(4.5)

to the subsoil properties and to the transverse load

〈c〉 ≡C 〈k〉≡K 〈q〉≡Q

〈chA〉≡ l2CA 〈khA〉≡ l2KA 〈qhA〉≡ l2QA

〈chAhB〉≡ l4CAB 〈khAhB〉≡ l4KAB
(4.6)

It is a system of 2+N+M differential equations for themacrodisplacementsU(·),W(·) and
for the fluctuation amplitudes of the axial displacement TK(·) and of the deflection VA(·). The
coefficients of these equations are constant, some of themdepend on the size l of the periodicity
cell.
Equations(4.4) can be simplified to the following form

D∂4W +DA∂2VA+KW + l2KAVA−N∂2W − lÑA∂VA

+CẆ +MẄ −J∂2Ẅ + l2CAẆA+ l2MAV̈A− lJA∂V̈A−Q=0

(DAB + l4KAB)VB + l4CABẆB + l2(l2MAB +JAB)V̈B + lÑA∂W + l2ÑABVB

+DA∂2W + l2KAW + l(lMAẄ +JA∂Ẅ)+ l2CAẆ − l2QA =0

(4.7)

The nonlinear terms (underlined) involve the axial force

N =EAε0 =EA
(
∂u0+

1
2
∂w∂w

)
(4.8)



Nonlinear vibrations of periodic beams 1101

averaged with the certain gradients of the fluctuation shape function as weights




N

lÑA

l2ÑAB




=





〈N〉
〈N∂hA〉
〈N∂hA∂hB〉





=




B0 lB
C

0 l
2B
CD

0

lB
A

0 l
2B̃AC0 l

3B
ACD

0

l2B
AB

0 l
3B
ABC

0 l
4B
ABCD

0






1
L

L∫

0






1
2
∂W∂W

VC∂W
1
2
VCVD





dx+






δ0
0
0









(4.9)

where

δ0 =L
−1

L∫

0

∂U dx=L−1[U(L)−U(0)] (4.10)

stands for the relative elongation of the beam axis. The coefficients of (4.9) are

B0 =B−BL(B−1)LKBK B̃AB0 =B
AB −BAK(B−1)LKBLB

B
A

0 =B
A−BAL(B−1)LKBK B

ABC

0 =B
ABC −BABL(B−1)LKBKC

B
AB

0 =B
AB −BABL(B−1)LKBK B

ABCD

0 =B
ABCD−BABK(B−1)KLBLCD

(4.11)

It can be seen that the axial displacement can be eliminated in a way common in the
conventional Euler-Bernoulli or Timoshenko theories of uniform beams.

4.2. The equations of the tolerance-asymptotic model

Neglecting in equations (4.4) or (4.7) the terms with the microstructure parameter l and
introducing the effective bending stiffness of the beam

D0 ≡D−D
A(D−1)ABDB (4.12)

we arrive at the equations

B0∂
(
∂U +

1
2
∂W∂W

)
=0

MẄ −J∂2Ẅ +CẆ +KW +D0∂
4W −B0

(
∂U +

1
2
∂W∂W

)
∂2W =Q

(4.13)

The above equations do not describe the effect of the cell size on the behaviour of periodic
beamsunder consideration.Hence, the asymptoticmodelmakes it possible to analyse vibrations
on themacrolevel only. In the framework of this approximation, certain higher eigenfrequencies
and eigenforms of vibrations cannot be obtained.

5. Examples of application

Let us consider a hinged-hinged beam with immovable ends. The beam has a constant cross
section and is provided by a system of periodically distributed system of concentrated masses
M1,M2 with rotational inertia I1, I2, as it is shown in Fig. 2.
A single cell is shown in Fig. 3. Themass distribution in a periodicity cell is given by

µ(y)=µ0+M1δ(y)+M2δ
(
y+
l

2

)

ϑ(y)=ϑ0+ I1δ(y)+ I2δ
(
y+
l

2

)
y∈ (x)

(5.1)



1102 Ł. Domagalski, J. Jędrysiak

Fig. 2. The considered beam and its fragment

Fig. 3. A single periodicity cell

In this Section, the free undamped and forced vibrations will be analyzed. In the case of
forced vibrations, we assume that the transverse load is given by

q(x,t)= q0 sin
(πx
L

)
cos(Ωt) (5.2)

6. The method of solution

6.1. Fluctuation shape functions

The fundamental assumption of the tolerance approach is the macro-micro decomposition
(3.2), (3.3) of the unknown displacements. It can be seen that the fluctuation shape functions
(FSF) play a crucial role in the analysis. As it has been mentioned, these functions represent
the oscillations of displacements in a periodicity cell. The common practise is to use approxi-
mate functions, usually the sine and cosine that are infinitely differentiable. Another way is to
utilize the periodic eigenproblem solutions of a periodicity cell, which can be obtained through
numerical analysis.
Here, this is done throughboth the abovementionedways. For the symmetric periodicity cell

(Fig. 3), two transverse andone longitudinal approximatemodes of cell vibrations are considered

h1(y)= l2
[
cos
(2Aπy
l

)
+ c
]

h2(y)= l2 sin
(2Aπy
l

)

g1(y)= l
[
cos
(2Kπy
l

)
+c
]

c=
M2−M1

µ0l+M1+M2

(6.1)

The constant c is calculated from condition (3.4)1.

Fig. 4. FE-based fluctuation shape functions: transverse (a), (b), and longitudinal (c) modes of the
periodicity cell vibrations

The refined fluctuation shape functions (Fig. 4) are obtained from a finite element analysis
of the cell. The calculations are performed in the environment of Maple. The periodicity cell is
divided into two elements, and the periodic boundary conditions are assumed.



Nonlinear vibrations of periodic beams 1103

Since the fluctuation shape functions satisfy the orthogonality conditions, there is possibi-
lity to obtain cell vibrations mode shapes taking more approximate FSFs and observing the
convergence of total oscillation on a periodicity cell. In the case of the finite element based cell
solution, a similar strategy can be adopted.

6.2. Solutions of the tolerance model

The solutions of the tolerancemodel equations, as well as the loads, are assumed in the form
of truncated Fourier series

{
W(x,t)
VA(x,t)

}
=

n∑

m=1

{
wm(t)
vA
m
(t)

}
Xm(x)

{
Q(x,t)
QA(x,t)

}
=

n∑

m=1

{
qm(t)
qA
m
(t)

}
Xm(x) A=1, . . . ,N

(6.2)

For the hinged-hinged boundary conditions, the linear natural vibration modes are

Xm(x)= sin(ξm)x ξm =
mπ

L
(6.3)

Application of the Galerkin method leads to a system of m× (1+N) ordinary differential
equations

Ky+Mÿ=q K=K0+KNL(y) (6.4)

where

y=y(t) =
{
w1(t) w2(t) · · · v

1
1(t) v

1
2(t) · · · v

2
1(t) v

2
2(t) · · ·

}T
(6.5)

The linear natural frequencies andmode shapes are sought for as solutions to the linearized
eigenproblem

|K0−ω
2
M|=0 (6.6)

In the case of free and forced nonlinear vibrations, equations (6.4)1can be converted into a
system of the first order ordinary differential equations

ÿ=M−1(q−Ky) ⇔

{
ẏ=v

v̇=M−1(q−Ky)
(6.7)

and solved by forward numerical integration. The calculations have been performed in Maple,
using own procedure based on the Runge-Kutta-Fehlberg (RKF45) method.

7. Calculational results

The length of the beam is L = 1.0m, Young’s modulus E = 205GPa, mass density ρ =
7850kg/m3. The other dimensionless parameters are as follows

b

h
=5

h

l
=0.1

l

L
=0.1

M1
ρAl
=5.1

M2
M1
=0.5

√
I1
M1l2

=10
I2
I1
=0.5

(7.1)



1104 Ł. Domagalski, J. Jędrysiak

Let us introduce a dimensionless central macrodeflection, deflection fluctuation and load ampli-
tude

w=
Wcenter
h

v=
Vcenter
h

p=
1
h

q0L
4

π2EJ
(7.2)

In the forced vibrations analysis, the load frequency is 5/4 times the lowest free undamped
frequency of the beam

Ω=
5
4
ω1 (7.3)

7.1. Free undamped vibrations

We restrict ourselves to considering only the first two (m=2) terms of Fourier series (6.2)
and two FSFs (N =2) so that the model hasm(1+N)= 6 degrees of freedom.
In order to validate the results, a finite element method procedure for beam dynamics ana-

lysis has been written in Maple. The Rayleigh beam elements with Hermitian polynomials and
consistent mass matrix have been applied.
The results of comparative analysis of calculations obtained for the finite element (40 ele-

ments)model and the tolerancemodel, using the approximate (trigonometric) and refined (finite
element based) fluctuation shape functions (FSF), are shown inTable 1 (linear eigenfrequencies)
and Fig. 5 (linear eigenvectors).

Table 1.Comparison of linear eigenfrequencies of the considered beam

Finite Tolerance model
Mode element approximate FSFs FE-based FSFs

ωFE [rad/s] ωTA [rad/s] ∆ [%] ωTA [rad/s] ∆ [%]

1.w1 15.861 15.876 0.094 15.872 0.071
2.w2 32.884 33.050 0.505 33.041 0.477
3. v22 223.491 239.454 7.143 212.200 5.052
4. v21 224.244 246.109 9.751 224.088 0.069
5. v12 14855.007 15129.460 1.848 15034.591 1.209
6. v11 14989.493 15129.460 0.934 15034.591 0.301

Figure 6 presents the total central deflection versus the quotient of nonlinear frequency to
linear frequency of freemacro- andmicro-vibrations (backbone curves) corresponding to thefirst
macro-mode and symmetric cell vibrations (m=1,N =1). Parameters (7.1) are kept constant
except the dimensionless microstructure parameter λ= l/L that is equal to 1/10, 1/12 or 1/15.
Studying the forced vibrations, the first 200 load periods are considered. The bifurcation

diagram of the total dimensionless deflection w at x = 0.5L versus the dimensionless load
amplitude is shown inFig. 7. Figure 8 presents a close-up of the bifurcation diagramof deflection
together with the diagram of its velocity dw/dt.

8. Discussion of results

From the results of linear free vibrations analysis, it can be seen that the finite element ba-
sed fluctuation shape functions indicate a better performance than the first-term trigonometric
ones, although the significant differences appear only for the antisymmetric mode shapes of the
periodicity cell. This applies both to the frequencies (Table 1) and the mode shapes (Fig. 5).
It should be noted that approximation of the lowest order possible is applied in the numerical



Nonlinear vibrations of periodic beams 1105

Fig. 5. Comparison of linear eigenforms of the considered beam: (a) finite element model, (b) tolerance
model with trigonometric FSFs, (c) tolerancemodel with FE-based FSFs

Fig. 6. Backbone curves for dimensionless macrodeflection (a) and fluctuation (b) at x=0.5L for
λ=1/10 (solid line), λ=1/12 (dashed line), λ=1/15 (dotted line)

analysis on the macro- andmicrolevel. As it is mentioned above, these results can be improved
through amore accurate analysis of the cell problem.
When it comes to nonlinear free vibrations the backbone curves presented in Fig. 6 indicate

that the lower order (macro-) nonlinear vibrations frequency is practically not affected by the
periodicity cell length, while the higher order frequency is much more sensitive to its variation.
It is caused by the fact that in the geometrically nonlinear formulation the strain terms that
include the displacement fluctuations are dependent on the microstructure parameter l, what
does not take place in the linear model.
Studying the results of the forced vibrations case (Figs. 7 and 8), the most characteristic

feature is that thebifurcationdiagram isvery scattered, although therearemanynarrowperiodic
windows. This is due to the fact that free vibrations are not damped, so they do not fade with
time. For the values of p higher than ∼ 3, it can be seen that the envelope of the diagram
suddenly loses its regularity and becomes jagged, which may indicate irregular vibrations. This
requires a more detailed analysis.



1106 Ł. Domagalski, J. Jędrysiak

Fig. 7. The bifurcation diagram of the deflectionw vs. the load amplitude p

Fig. 8. A blow-up of the bifurcation diagram: the deflectionw and the velocity dw/dt vs. the load
amplitude p

9. Closing remarks

In this contribution, the tolerancemodel is shown,whichdescribes geometrically nonlinearvibra-
tions of a periodically inhomogeneous beam. Themodel is developed by applying the tolerance
averaging method directly to the 1D beam theory equations. Hence, the fundamental equations
with highly oscillating, periodic, non-continuous functional coefficients are replaced by the equ-
ations with constant coefficients. It should be stressed that the aim is to develop a low degree
of freedommodel that would be able to provide results that are not available for the first order
asymptotic models. Some applications of the proposedmodel including free and forced undam-
ped vibrations are presented. A comparison of linear eigenfrequencies and mode shapes with a
finite element model is shown.
The following general remarks can be formulated.

1. It can be observed that the proposed toleranc emodel makes it possible to investigate the
effect of the microstructure size on dynamic problems of periodic beams under considera-
tion, e.g. the “higher order” vibrations related to the beammicrostructure.

2. The governing equations of the tolerance model have a physical sense for the unknowns
W ,U, VA,A=1, . . . ,N, TK,K =1, . . . ,M, being slowly-varying functions.



Nonlinear vibrations of periodic beams 1107

3. The asymptotic model of periodic beamsmakes it possible to investigate only lower order
(fundamental) vibrations.

The issues anticipated to be addressed in the future work are:

• taking into account the structural and material heterogeneity of the beam and the visco-
elastic subsoil,

• more detailed analysis of the solutions to the cell problem,

• detailed analysis of the properties distribution in a periodicity cell,

• analysis of the initial shortening/elongation of the beam axis.

These problems will be investigated in forthcoming papers.

Acknowledgement

The authors are grateful for the support provided by the National Science Centre, Poland (Grant
No. 2014/15/B/ST8/03155).

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Manuscript received October 23, 2015; accepted for print January 24, 2016