JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 2, pp. 357-379, Warsaw 2004

APPLICATION OF THE TOLERANCE AVERAGING

METHOD TO ANALYSIS OF DYNAMICAL STABILITY OF

THIN PERIODIC PLATES1

Jarosław Jędrysiak

Department of Structural Mechanics, Łódź University of Technology

e-mail: jarek@p.lodz.pl

Some problems of a dynamical stability of thin periodic plates are con-
sidered. As a tool to derive the governing equations of an averaged non-
asymptotic plate model the tolerance averaging is applied, proposed for
periodic composites and structures by Woźniak and Wierzbicki (2000).
This method applied to the known Kirchhoff-type plate equation leads
to averaged models taking into account the effect of the period lengths
on the overall plate behaviour (Jędrysiak, 2001).Here, a non-asymptotic
model describing the problems of a dynamical stability of periodic plates
is formulated.Moreover, it is shown that the effect of the period lengths
plays a crucial role in some special cases of dynamical stability of such
plates, i.e. for higher oscillation frequencies of compressive forces in the
plate midplane.

Keywords: thinperiodicplate, dynamical stability, the length-scaleeffect

1. Introduction

Thin periodic plates, which are considered in this paper, are composed
of many identical small elements (Fig.1). These elements are treated as thin
plates with spans l1, l2 and called periodicity cells. Inmechanical problems of
these plates the effect of the period lengths, which will be called the length-
scale effect, on the overall platebehaviour, inparticular ondynamicsproblems,
is very interesting.

1
A part of this contribution was presented on the Xth Symposium ”Stability of Struc-

tures” in Zakopane, September 8-12, 2003.



358 J.Jędrysiak

Fig. 1. Portion of a thin periodic plate

The exact equations of the plate theory for periodic plates involve highly
oscillating, non-continuous, periodic coefficients and thus, they are too compli-
cated to apply them to investigations of engineering problems. Thus, certain
simplified models have been proposed. Two ways of formulation of averaged
2D-models of thin elastic plates having aperiodic structure along themidplane
can bementioned. Using the first of them, based on themultiscale asymptotic
expansions, the 2D-models of homogenised plates are derived from the 3D-
model of the elastic solid. Equations of these models are similar to the known
Kirchhoff-type equation of a homogeneous plate (cf. Caillerie, 1984; Kohn and
Vogelius, 1984). Using this approach, a homogenised model of a pre-stressed
periodic plate can be obtained, cf. Kolpakov (2000). However, in these avera-
ged models the length-scale effect is neglected.

In the second approach – the tolerance averaging method, presented in the
book (Woźniak andWierzbicki, 2000) for periodic composites and structures,
is applied to the 2D-model equations of periodic plates. Under different as-
sumptions of periodic plates, this approach leads to certain non-asymptotic
models of these plates, described by differential equations with constant coef-
ficients, e.g. for the Hencky-Boole-type plates by Baron (2002, dynamic sta-
bility), for wavy-type plates byMichalak (1998, stability; 2001, dynamics and
stability; 2002, dynamics), for thinplates (with the period lengths and thepla-
te thickness being of the same order) byMazur-Śniady et al. (2003, dynamics),
for Kirchhoff-type plates (with the plate thickness being small in comparison
to the period lengths) by Jędrysiak (2000, 2003b, stability; 2001, dynamics
and stability; 2003a, dynamics). Internal instability of periodic structures was
analysed by Wierzbicki and Woźniak (2002). Models of this kindmake it po-
ssible to investigate the length-scale effect on the overall plate behaviour, in
contrast to the aforementioned homogenised models.
Themain aimof this paper is to derive a non-asymptotic 2D-model of thin

periodic plates with forces acting in the midplane, which takes into account



Application of the tolerance averaging method... 359

the length-scale effect on the dynamic stability of such plates. The tolerance
averagingmethodwill be applied as a tool for themodelling.As an illustrative
example, a dynamic stability of a periodic plate band with span L along the
x1-axis will be analysed. It will be shown that in some special cases of a
dynamical stability of that plate, the length-scale effect couldnot beneglected,
e.g. for high values of oscillation frequencies of compressive forces in the plate
midplane.

2. Modelling approach

Let Ox1x2x3 be the orthogonal Cartesian co-ordinate system in the physi-
cal space, t be the time co-ordinate and indices α,β,... run over 1,2; A,B,...
run over 1, ...,N. Summation convention holds for all aforementioned indices.
Let Ω ≡ {(x,z) : −h(x)/2 < z < h(x)/2, x ∈ Π} be the region of unde-
formed plate, where x≡ (x1,x2); z≡x3; Π is the plate midplane and h(x)
is the plate thickness at the point x∈Π. The periodicity cell on the Ox1x2
plane is denoted by ∆ ≡ (−l1/2, l1/2)× (−l2/2, l2/2), where l1, l2 are the

cell length dimensions along the x1-, x2-axis. Let l ≡
√
l21 + l

2
2 be the para-

meter describing the size of the cell. Since of the parameter l is assumed to
be sufficiently small compared to the minimum characteristic length dimen-
sion of Π and sufficiently large compared to the maximum plate thickness
(hmax � l�LΠ), it will be called the mesostructure parameter. Assume that
h is a ∆-periodic function in x and all material and inertial plate properties,
e.g. mass density ρ = ρ(x,z) and elastic moduli aijkl = aijkl(x,z), are also
∆-periodic functions in x and even functions in z (cf. Jędrysiak, 2001, 2003a).
Periodic plates with the structure will be called mesoperiodic plates. Denote
by w the plate deflection andby p−, p+ loadings in the z-axis direction acting
on theupperand lower plate boundaries.Thenon-zero termsof the elasticmo-
duli tensor are aαβγδ,aαβ33,a3333; denote cαβγδ ≡ aαβγδ−aγδ33aαβ33(a3333)

−1.
Our considerations are based on the well-known Kirchhoff-type plate theory
assumptions (cf. Jędrysiak, 2001). Letus introduce ∆-periodic functions of the
mean plate properties – mass density, rotational inertia, bending stiffnesses

µ≡

h/2∫

−h/2

ρ dz ϑ≡

h/2∫

−h/2

ρz2 dz dαβγδ ≡

h/2∫

−h/2

z2cαβγδ dz



360 J.Jędrysiak

As the starting point of the modelling we assume the well known fourth
order differential equation

(dαβγδw,γδ),αβ − (nαβw,α),β +µẅ− (ϑẅ,α),α = p (2.1)

in the form known for homogeneous thin plates from the books Timoshen-
ko and Gere (1961), Volmir (1972), Kaliski (1992); here p ≡ p+ + p−; nαβ
(α,β = 1,2) are forces in the plate midplane, such that nαβ,β = 0. Howe-
ver, for mesoperiodic plates the coefficients in the above equation are highly
oscillating ∆-periodic and also non-continuous functions.
In order to derive the averaged governing equations of mesoperiodic pla-

tes, having constant coefficients and taking into account the length-scale effect,
the tolerance averagingmethod developed in the bookWoźniak andWierzbicki
(2000) for periodic composites will be applied. In thismethod some additional
concepts such as e.g. an averaging operator, a tolerance system, a slowly va-
rying function, a periodic-like function and an oscillating function explained
in detail in the above book, are used.
Let ∆(x) = ∆+x be a periodicity cell at x ∈ Π∆, Π∆ = {x : x ∈ Π,

∆(x)⊂Π}. In the analysis of periodic structures, the known averaging ope-
rator (cf. Woźniak andWierzbicki, 2000; Jędrysiak, 2001, 2003a,b) is applied

〈ϕ〉(x)≡
1
l1l2

∫

∆(x)

ϕ(y) dy x∈Π∆ (2.2)

defined on the platemidplane Π for an arbitrary integrable function ϕ. For a
periodic function ϕ in x, its averaged value from (2.2) is constant. It is tacitly
assumed that all functions under consideration satisfy the required regularity
conditions.
In order tomake the paper self-consistent, following the bookWoźniak and

Wierzbicki (2000), somemathematical notions and formulae will be used.The
tolerance averaging method is based on the concept that to every considered
physical quantity s, expressed in terms of a certain unit measure, a positive
number εs can be assigned. This number is called a tolerance parameter and
is such that for every two values s1, s2 of this quantity, if |s1−s2| ¬ εs then
s1 ∼= s2, what means that values s1, s2 can be treated as indistinguishable.
Denote by T a certain mapping (called a tolerance system), which assigns to
every quantity under consideration a tolerance parameter. Now, the following
definitions will be recalled (cf. the above book).
The continuous function Ψ, defined on Π, will be called a slowly varying

function, if for every x1,x2 ∈ Π such that ‖x1 −x2‖ ¬ l the following



Application of the tolerance averaging method... 361

condition holds |Ψ(x1)−Ψ(x2)| ¬ εΨ . We shall write Ψ ∈ SV (T) if Ψ and
all its derivatives are slowly varying functions.
The continuous function f will be called a periodic-like function if for every

x∈Π there exists a certain continuous ∆-periodic function fx such that for
every y ∈ Π and ‖x−y‖ ¬ l, we obtain |f(y)− fx(y)|εf . If derivatives
of the function f satisfy similar conditions, we will write f ∈ PL(T). It can
be shownWoźniak andWierzbicki (2000) that averaging (2.1) of periodic-like
function is a slowly varying function.
Aperiodic-like function f will be called an oscillating function if it satisfies

the condition 〈µf〉(x) ∼= 0 for every x ∈ Π∆, where µ is a positive-valued
∆-periodic function. The set of oscillating periodic-like functions with the
weight µwill be denoted by PLµ(T).
In the modelling procedure, the lemmas and assertions, formulated and

proved in the book Woźniak and Wierzbicki (2000) with the above concepts
are applied.
In the tolerance averagingmethodassumptions formulated below are used.

The Tolerance Averaging Approximation. It is assumed that for every
∆-periodic integrable function ϕ defined on Π and all integrable func-
tions Ψ ∈SV (T), f ∈PL(T), the following conditions hold

〈ϕΨ〉(x)∼= 〈ϕ〉Ψ(x) 〈ϕf〉(x)∼= 〈ϕfx〉(x) x∈Π∆ (2.3)

The Conformability Assumption. It is assumedthat thedeflection w(·, t)
of the plate midplane under consideration is a periodic-like function,
w(·, t) ∈ PL(T), i.e. the deflection is conformable to a periodic plate
structure. This conditionmay be violated only near the plate boundary.

The Midplane Forces Restriction. It is assumed that forces in the
midplane nαβ (α,β = 1,2) are also periodic-like functions,
nαβ(·, t) ∈ PL(T). Hence, they could be decomposed into
nαβ = n

0
αβ + ñαβ, where n

0
αβ ∈ SV (T) will be the averaged part de-

fined by n0αβ ≡ 〈nαβ〉, and ñαβ ∈ PL
1(T) will be the fluctuating part,

such that 〈ñαβ〉=0.

The modelling procedure of the tolerance averaging can be divided into
four steps.

1) Define the averaging part of deflection by setting W ≡〈µ〉−1〈µw〉, whe-
re µ is the platemass density. Because of w∈PL(T) it is W ∈SV (T).
Thus, the decomposition can be introduced w = W + v, where
v ∈ PLµ(T) is the fluctuating part satisfying the condition 〈µv〉 = 0.
The averaged part of deflection W will be called a macrodeflection.



362 J.Jędrysiak

2) Formulate the periodic problem (cf. Woźniak and Wierzbicki, 2000;
Jędrysiak, 2001) on ∆(x) for vx being a ∆-periodic approximation of v
on ∆(x) at x∈Π∆. Function vx satisfies the condition 〈µvx〉=0.

3) Formulate the Galerkin approximation of the above periodic pro-
blem by introducing the system of N linear-independent ∆-periodic
functions gA, A = 1, ...,N, such that 〈µgA〉 = 0, and by setting
vx(y, t) = gA(y)QA(x, t), where y ∈ ∆(x), x ∈ Π∆; QA ∈ SV (T)
are new kinematic unknowns. Functions gA are calledmode-shape func-
tions and they have to approximate the expected form of the oscillating
part of free vibration modes of the periodicity cell. Moreover, values of
these functions have to satisfy conditions l−1gA(·),gA,α(·), lg

A
,αβ(·)∈O(l).

4) After somemanipulations, the equation for the macrodeflection W and
equations for kinematic unknowns QA are obtained

(〈dαβγδ〉W,γδ + 〈dαβγδg
B
,γδ〉Q

B),αβ −n
0
αβW,αβ + 〈µ〉Ẅ −〈ϑ〉Ẅ,αα−

−〈ñαβg
B
,α〉Q

B
,β −〈ϑg

B
,α〉Q̈

B
,α = 〈p〉

(2.4)

(〈µgAgB〉+ 〈ϑgA,αg
B
,α〉)Q̈

B + 〈ϑgA,α〉Ẅ,α+ 〈dαβγδg
A
,γδ〉W,αβ +

+〈ñαβg
A
,β〉W,α+ 〈dαβγδg

A
,αβg

B
,γδ〉Q

B +n0αβ〈g
A
,αg
B
,β〉Q

B +

+〈ñαβg
A
,αg
B
,β〉Q

B = 〈pgA〉

where n0αβ and ñαβ are the averaged and the fluctuating part of in-plane
forces, respectively.

It can be shown that for plates with symmetric cells (cf. Fig.1) and
symmetric mode-shape functions gA, the following terms are equal to zero:
〈ñαβg

A
,α〉 = 〈ϑg

0
,α〉 = 0. In the subsequent considerations it will be assumed

that in the above equations the underlined terms 〈ñαβg
A
,αg
B
,β〉with the fluctu-

ating part of in-plane forces are small in comparison to terms n0αβ〈g
A
,αg
B
,β〉with

the averaged part of the forces and they will be neglected. Moreover, intro-
ducing the in-plane constant forces Nαβ (α,β =1,2) applied on the edges of
thin periodic plate, it can be assumed that the averaged part n0αβ of in-plane
forces can be replaced by Nαβ.
It should be emphasized that equations (2.4) are derived without the as-

sumption introduced inJędrysiak (2000, 2001, 2003b) that in terms (nαβw,α),β
of Eq. (2.1) the deflection w can be replaced by themacrodeflection W . Thus,
the obtained equations are more general than those presented in the above
papers.



Application of the tolerance averaging method... 363

3. Governing equations

Introducing the following notations

Dαβγδ ≡〈dαβγδ〉 D
A
αβ ≡〈dαβγδg

A
,γδ〉 D

AB ≡〈dαβγδg
A
,αβg

B
,γδ〉

GABαβ ≡ l
−2〈gA,αg

B
,β〉 m≡〈µ〉 m

AB ≡ l−4〈µgAgB〉

j ≡〈ϑ〉 jAB ≡ l−2〈ϑgA,αg
B
,α〉 P ≡〈p〉

PA ≡ l−2〈pgA〉

and neglecting in (2.4) the underlined terms, we arrive at the governing equ-
ations of the non-asymptotic model

(DαβγδW,γδ +D
B
αβQ

B),αβ −NαβW,αβ +mẄ − jẄ,αα =P
(3.1)

l2(l2mAB + jAB)Q̈B +DAαβW,αβ +D
ABQB +Nαβl

2GABαβ Q
B = l2PA

where some terms depend explicitly on the mesostructure parameter l;
Nαβ are in-plane forces.
Theabove equations are a certain generalization of the governing equations

obtained by Jędrysiak (2001) for plates with periodic structure along both the
axes in the midplane, because they involve additional terms Nαβl

2GABαβ Q
B.

These equations, having averaged constant coefficients, make it possible to
analyse the length-scale effect in dynamic processes and also in stability of
periodic plates. The basic unknowns W ,QA,A=1, ...,N, are slowly varying
functions. For a rectangular platewithmidplane Π =(0,L1)×(0,L2), twobo-
undary conditions should be defined on the edges x1 =0, L1 and x2 =0, L2
only for the macrodeflection W . Hence, functions QA are called internal va-
riables. To derive equations (3.1), themode-shape functions gA,A=1, ...,N,
for every periodic plate under consideration have to be previously obtained.
In the most cases only one (N = 1) mode-shape function g = g1, assumed
as an approximate solution to the eigenvalue problem on the cell, is sufficient
from the calculative point of view (Jędrysiak, 2001).
At the end of this section it is shown that amodelwithout the length-scale

effect is a special case of the non-asymptoticmodel. Neglecting the termswith
parameter l in equations (3.1) and substituting (3.1)2 into (3.1)1 we arrive at

[Dαβγδ −D
A
γδD

B
αβ(D

AB)−1]W,αβγδ −NαβW,αβ +mẄ − jẄ,αα =P (3.2)

The above equation of the averaged model without the length-scale effect has
form similar to the equations of homogeneous plates. Thismodelwill be called
homogenised model.



364 J.Jędrysiak

4. The problem of dynamic stability

Now, the governing equations of both the models presented in the pre-
vious section will be applied to analyse the problem of dynamic stability
of a rectangular plate. It is assumed that the plate is made of an isotropic
piece-wise periodically homogeneous material along the x1- and x2-axis and
has a periodic thickness h along both the axes. Moreover, assume that Po-
isson’s ratio ν is constant, but the plate mass density ρ and Young’s mo-
dulus E are periodically variable; loadings p are neglected and the plate
is uniformly compressed along the x1- and x2-axis in the midplane, hence
N12 =N21 =0. Let us consider a case with only one mode-shape function g
(i.e. A=N =1) assumed as the approximate solution to a certain eigenvalue
problem with periodic boundary conditions imposed on the cell in the form:
g= g1 = l2[cos(2πx1/l1)cos(2πx2/l2)+ c], where the constant c is calculated
from the condition 〈µg〉=0. For the assumed symmetric cell and symmetric
form of mode-shape function it can be shown that D112 = D

1
21 = 0. Denote

B= 〈Eh3/[12(1−ν2)]〉 and Q=Q1, x=x1,G1 ≡G1111,G2 ≡G
11
22,D1 ≡D

1
11,

D2 ≡D
1
22, D ≡D

11, and also N1 ≡−N11, N2 ≡−N22. Separating variables
x = (x1,x2) and t, the macrodeflection W and the internal variable Q can
be assumed in the form

W(x1,x2, t)=Xnk(x1,x2)Tnk(t)
(4.1)

Q(x1,x2, t)=Xnk(x1,x2)T
Q
nk(t) n,k=1,2, . . .

where functions Xnk(·) satisfy proper boundary conditions on the opposite
plate edges.
For the plate under consideration, substituting (4.1) into equations (3.1)

and after some manipulations, the equation for functions Tnk in the non-
asymptotic model is obtained

l2(l2m11+ j11){[mXnk − j(Xnk,11+Xnk,22)]
d4

dt4
Tnk+

+
d2

dt2
[(BXnk,ααββ +N1Xnk,11+N2Xnk,22)Tnk]}+

+(D− l2N1G1− l
2N2G2){[mXnk − j(Xnk,11+Xnk,22)]

d2

dt2
Tnk+ (4.2)

+(BXnk,ααββ +N1Xnk,11+N2Xnk,22)Tnk}−

−(D21Xnk,1111+2D1D2Xnk,1122+D
2
2Xnk,2222)Tnk =0



Application of the tolerance averaging method... 365

Substituting the proper functions Xnk(·) satisfying the boundary conditions
into (4.2) we obtain the frequency equation.
In the homogenised model of the considered plate, after similar manipula-

tions, from equation (3.2), we obtain

d2

dt2
[(BXnk,ααββ +N1Xnk,11+N2Xnk,22)Tnk]+

+D{[mXnk− j(Xnk,11+Xnk,22)]
d2

dt2
Tnk+

(4.3)

+(BXnk,ααββ +N1Xnk,11+N2Xnk,22)Tnk}−

−(D21Xnk,1111+2D1D2Xnk,1122+D
2
2Xnk,2222)Tnk =0

where the length-scale effect describedby termswith parameter l is neglected.
In the subsequent section an example of application of the above equations

will be shown.

5. Example – the dynamic stability of simply supported plate

strip

5.1. General analysis

As an example, let us consider a simply supported plate strip with span
L1 = L along the x-axis (x = x1), having periodic structure along both the
axes in themidplane.Theplate periodicity is causedby the periodic thickness.
However, mass density ρ andYoung’s modulus E are constant. It is assumed
that the periodicity cell is square, i.e. ∆≡ (−l/2, l/2)×(−l/2, l/2), and hence
themode-shape function is g= l2[cos(2πx1/l)cos(2πx2/l)+c]. It canbeshown
that for such a plate D1 = D2 and G1 = G2. Functions Xnk satisfying the
boundaryconditions for the simply supportedplate stripon the edges x=0,L
have the form (k=1)

Xn(x)=Xnk(x1,x2)= sin(αnx) (5.1)

where αn =nπ/L, n=1,2, .... Denote T =Tn =Tnk. Because the waveleng-
ths of Xn are sufficiently large compared to l and hence αnl � 1 and also
h/l � 1, in the sequel the simplified form of Eq. (4.2) will be applied, in
which terms l2(l2m11 + j11)d2[(BXn,1111 +N1Xn,11)T ]/dt2 can be neglected



366 J.Jędrysiak

as small compared to [D− l2G1(N1+N2)](mXn−jXn,11)d2T/dt2. Substitu-
ting (5.1) into Eq. (4.2), bearing inmind that l/L� 1, from (4.2) the explicit
asymptotic formulae can be derived

N− ≡α
2
n(B−D

2
1D
−1) N+ ≡D(G1l2)−1−N2+α2nD

2
1D
−1

Ñ+ ≡D(G1l2)−1 ω2− ≡α
4
n(B−D

2
1D
−1)(m+ jα2n)

−1

ω2+ ≡Dl
−2(l2m11+ j11)−1

(5.2)
for the ”fundamental” lower critical force N−, for the ”additional” higher
critical force N+ and its approximation Ñ+, for the lower free vibration fre-
quency ω−, for the higher free vibration frequency ω+. Using these asympto-
tic formulae, the frequency equation within the non-asymptotic model can be
written in the approximate form

d4

dt4
T +ω2+[1− (N1+N2)Ñ

−1
+ ]
d2

dt2
T +

(5.3)

+ω2+ω
2
−
(1−N1N

−1
−
)(1−N1N

−1
+ )N+Ñ

−1
+ T =0

Assuming that only N1 is a time-dependent function by N1 =Na+Nbcospt,
where p is the oscillation frequency of force N1; however, N2 is independent
of time; introducing a dimensionless time co-ordinate z = pt and denoting
T ′ = dT/dz and also

η− =ω2−p
−2 η+ =ω2+p

−2 ζ =N+Ñ
−1
+

χ− =NaN
−1
−

δ− =NbN
−1
−

χ+ =NaN
−1
+

δ+ =NbN
−1
+ χ̃+ =NaÑ

−1
+ δ̃+ =NbÑ

−1
+

χ̃=N2Ñ
−1
+ ξ= η+(1− χ̃+− χ̃) ξ− = η−(1−χ−)

ξ+ = η+(1−χ+) ϕ= δ̃+(1− χ̃+− χ̃)−1 ϕ− = δ−(1−χ−)−1

ϕ+ = δ+(1−χ+)−1

(5.4)
from (5.3) the following equation is obtained

T ′′′′+ ξ(1−ϕcosz)T ′′+ ξ−ξ+(1−ϕ− cosz)(1−ϕ+ cosz)ζT =0 (5.5)

The above equation is a starting point of the analysis of dynamic stability of
the considered plate strip in the framework of the non-asymptotic model.
It should be emphasized that, in contrast to the models presented in

Jędrysiak (2001), we obtain here not only the additional higher frequency ω+,



Application of the tolerance averaging method... 367

Eq. (5.2)5, but also the additional higher critical force N+, Eq. (5.2)2 in the
framework of the proposed non-asymptotic model.
In order to evaluate the obtained results let us consider the above problem

within the homogenised model. Substituting solutions (5.1) into Eq. (4.3) and
using formulae (5.2)1,4 we arrive at the frequency equation in the form

d2

dt2
T +ω2

−
(1−N1N

−1
−
)T =0

Assuming N1 =Na+Nb cospt, introducing a dimensionless time co-ordinate
z= pt and using (5.4), the above equation takes the form

T ′′+ξ−(1−ϕ− cosz)T =0 (5.6)

It canbeobserved that equation (5.6) for the homogenisedmodel ofperiodic
plates has a form of the known Mathieu equation, which describes dynamic
stability or parametric vibrations of different structures (e.g. bars, plates; cf.
Timoshenko (1961), Kaliski (1992)). Using this equation, regions of dynamic
instability for parameters ϕ−, ξ− can be determined.
However, in the framework of the non-asymptotic model, the fourth order

equation (5.5) is derived,whichcanbe treated asa certain generalization of the
Mathieu equation. This equation makes it possible to investigate the length-
scale effect on the dynamical stability of periodic plates. Moreover, it should
be emphasized that for this equation additional initial conditions imposed on
higher-order derivatives of function T , i.e. T ′′,T ′′′, have to be formulated.The
function T and its derivatives T ′,T ′′,T ′′′ can be treated as the dimensionless:
macrodeflection, velocity, acceleration, higher-order acceleration.

5.2. Approximate solutions

An analytical solution of the Mathieu equation (5.6) is known in the lite-
rature (cf. Timoshenko, 1961; Volmir, 1972; Kaliski, 1992). This solution can
be assumed in the following form

T(z)=
∞∑

k=0

(
Ak sin

kz

2
+Bk cos

kz

2

)
(5.7)

which is restricted in time (time t is hidden in variable z= pt).
However, in order to apply (5.7) to the generalizedMathieu equation (5.5)

some simplifications in this equation must be introduced. Thus, neglect the



368 J.Jędrysiak

constant part of the force N1 and also the force N2, i.e. Na = N2 = 0. In
this way, χ− = χ+ = χ̃+ = χ̃ = 0 and ξ = ξ+. Moreover, in most cases the
following conditions hold: Nb � N+, Nb � Ñ+, and hence it can be shown
that ϕ,ϕ+ � 1. Introduce the notations: a≡ ξ−, 2b≡ ξ−ϕ− and ε≡ω2−/ω

2
+,

hence ξ+ = a/ε. The frequency equations for both the models can be written
in the forms:

— for the non-asymptotic model (5.5)

T ′′′′+
a

ε
T ′′+

a

ε
ζ(a−2bcosz)T =0 (5.8)

— for the homogenised model (5.6)

T ′′+(a−2bcosz)T =0 (5.9)

Substituting the solution (5.7) into the above equations, after somemani-
pulations, characteristic equations of the relation between coefficients a and
b are derived in the form of continued fractions.

For the non-asymptotic model (5.8) the characteristic equations take the
form

ε

16
−
a

4
+ ζa(a− b)=

(abζ)2

81ε
16
− 9a
4
+a2ζ− (abζ)

2

125ε

16
−
25a

4
+a2ζ−...

r=1,3, . . .

ε

16
−
a

4
+ ζa(a+ b)=

(abζ)2

81ε
16
− 9a
4
+a2ζ− (abζ)

2

125ε

16
−
25a

4
+a2ζ−...

r=1,3, . . .

a

2b
=

abζ
16ε
16
− 4a
4
+a2ζ− (abζ)

2

256ε

16
−
16a

4
+a2ζ−...

r=0,2, . . .

16ε
16
−
4a
4
+ ζa2 =

(abζ)2

256ε
16
− 16a
4
+a2ζ− (abζ)

2

1296ε

16
−
36a

4
+a2ζ−...

r=0,2, . . .

(5.10)
Equations (5.10)1 and (5.10)3 are related to the part of solution (5.7) in the
form cos(rx/2), however equations (5.10)2 and (5.10)4 are referred to the part
sin(rx/2).These equations determine boundariesbetween the regions of stable
and unstable vibrations within the non-asymptotic model of periodic plate.



Application of the tolerance averaging method... 369

For the homogenisedmodel (5.9), the characteristic equations have the form

a− b−
1
4
=

b2

a− 9
4
− b

2

a−25
4
−...− b

2

a−
1

4
r2−...

r=1,3, . . .

a+ b−
1
4
=

b2

a− 9
4
− b

2

a−25
4
−...− b

2

a−
1

4
r2−...

r=1,3, . . .

1
2
a=

b2

a−4− b
2

a−16−...− b
2

a−r2−...

r=0,2, . . .

a−1=
b2

a−4− b
2

a−9−...− b
2

a−r2−...

r=0,2, . . .

(5.11)

Similar to the non-asymptotic model, equations (5.11)1 and (5.11)3 are re-
lated to the part of solution (5.7) in the form cos(rx/2), however equations
(5.11)2 and (5.11)4 are referred to sin(rx/2). The above equations determine
the boundaries between regions of stable and unstable vibrations within the
homogenised model of a periodic plate.

It can be observed that the continued fractions (5.11) for the homogenised
model have the form known in literature, being obtained from the known
Mathieu equation (5.9). Unfortunately, these relations do not describe the
length-scale effect. However, equations (5.10) for the non-asymptotic model,
derived from equation (5.8), take into account the length-scale effect. This
effect is described by terms with the parameter ε.

Diagrams of curves determining boundaries between regions of stable and
unstable vibrations by both the models are shown in the subsequent section
in Fig.3.

5.3. Numerical solutions

In order to find more exact solutions to equation (5.8) and in particular
to equation (5.5) without simplifications, it is necessary to apply numerical
methods. For this purpose, the known commercial programs for symbolic and
numerical calculations such as Mathematica or MathCad can be used. Some
diagrams of these solutions are shownanddiscussed in the subsequent section.



370 J.Jędrysiak

6. Numerical results

Numerical examples are calculated for the plate strip of periodically va-
riable thickness along both the axes in the midplane. A square periodicity
cell is assumed, i.e. ∆≡ (−l/2, l/2)× (−l/2, l/2), shown in Fig.2. The plate
thickness is defined as

h(x)=






h0 if x∈
[
−1
2
l,−1
2
γl
)
×
[
−1
2
γl, 1
2
γl
]
∪

∪
[
−1
2
γl, 1
2
γl
]
×
[
−1
2
l, 1
2
l
]
∪
(
1
2
γl, 1
2
l
]
×
[
−1
2
γl, 1
2
γl
]

h1 = ηh0 if x∈
[
−1
2
l,−1
2
γl
)
×
[
−1
2
l,−1
2
γl
)
∪

∪
[
−1
2
l,−1
2
γl
)
×
(
1
2
γl, 1
2
l
]
∪

∪
(
1
2
γl, 1
2
l
]
×
[
−1
2
l,−1
2
γl
)
∪
(
1
2
γl, 1
2
l
]
×
(
1
2
γl, 1
2
l
]

(6.1)
where γ, η ∈ [0,1], x = (x1,x2). Introduce the dimensinless parameters:
λ= l/L, η0 =h0/l, φ=N2/N−.

Fig. 2. Periodicity cell of the plate strip under consideration

6.1. Boundaries of regions of stable and unstable vibrations for an ap-

proximate solution

Some results for the approximate solution (5.7) to frequency equations
(5.8) and (5.9) are shown in Fig.3. Here, the curves of boundaries of regions
of stable and unstable vibrations are presented. These diagrams are made by
using the characteristic equations (5.10) and (5.11),whichdescribe the relation
between coefficients a−b in equations (5.8) and (5.9) in the framework of the
non-asymptotic and the homogenisedmodel, respectively. Plots shown in this
figure are made for parameters: λ = 0.1, η0 = 0.1, η = 0.7, γ = 0.5, and
n=1.



Application of the tolerance averaging method... 371

Fig. 3. Curves of relation a− b (boundaries of regions of stable and unstable
vibrations)

6.2. Solutions to the Mathieu and the generalized Mathieu equations

Fig. 4. Solutions to frequency equations for initial conditions at z=0: T =1,
T ′ =0, T ′′ =−1, T ′′′ =1, and parameter η− =0.2



372 J.Jędrysiak

Fig. 5. Solutions to frequency equations for initial conditions at z=0: T =1,
T ′ =0, T ′′ =−1, T ′′′ =1, and parameter η− =10−4

In Fig.4-Fig.8 diagrams of numerical solutions to equations (5.5) and
(5.6) are shown. These diagrams are made for different values of parameters:
η0 =0.1, η=0.7, γ =0.5 and n=1; λ=0.05, 0.1; φ=0.1; χ− =0.0, 0.1;
δ− = 0.1, 0.3, 10. In these figures we have curves of solutions for different
values of parameter η−, Eq. (5.4)1, describing the oscillation frequency of for-
ce N1, and different initial conditions at z=0, i.e. in Fig.4: η− =0.2, T =1,
T ′ = 0, T ′′ = −1, T ′′′ = 1 for Eq. (5.5) and T = 1, T ′ = 0 for Eq. (5.6); in
Fig.5: η− =10−4, T =1, T ′ =0, T ′′ =−1, T ′′′ =1 for Eq. (5.5) and T =1,



Application of the tolerance averaging method... 373

Fig. 6. Solutions to frequency equations for initial conditions at z=0: T =1,
T ′ =0, T ′′ =−1, T ′′′ =0, and parameter η− =10−4

T ′ = 0 for Eq. (5.6); in Fig.6: η− = 10−4, T = 1, T ′ = 0, T ′′ =−1, T ′′′ = 0
for Eq. (5.5) and T =1, T ′ =0 for Eq. (5.6); in Fig.7: η− =1.6·10−6,T =0,
T ′ =1,T ′′ =0,T ′′′ =1 forEq. (5.5) and T =0,T ′ =1 forEq. (5.6); inFig.8:
η− = 1.6 · 10−6, T = 1, T ′ = 0, T ′′ = −1, T ′′′ = 0 for Eq. (5.5) and T = 1,
T ′ = 0 for Eq. (5.6). In Fig.5b and Fig.6b are shown enlarged fragments of
Fig.5a and Fig.6a, respectively; however, in Fig.8c we have an enlarged frag-
ment of Fig.8a,8b. Diagrams in Fig.4 aremade for z∈ [0,200], in Fig.5a and
Fig.6a for z∈ [0,500], and in Fig.7 and Fig8a,b for z∈ [0,5000].



374 J.Jędrysiak

Fig. 7. Solutions to frequency equations for initial conditions at z=0: T =0,
T ′ =1, T ′′ =0, T ′′′ =1, and parameter η− =1.6 ·10−6

6.3. Discussion of obtained numerical results

From the obtained results of numerical examples some conclusions can be
drawn.

• The length-scale effect is negligibly small in the problem of determining
the boundaries of regions of stable and unstable vibrations and hence,
the homogenisedmodel is sufficient from the point of view of calculation
for this problem (Fig.3).

• Analysing the diagrams of numerical solutions to Eqs (5.5) and (5.6)
shown in Fig.4-Fig.8 it can be observed:

– differences between solutions from the non-asymptotic and the ho-
mogenised models are negligibly small for additional homogeneous
initial conditions of Eq. (5.5) for higher-order derivatives of func-
tion T , i.e. T ′′ =T ′′′ =0 (Fig.4);

– assuming for Eq. (5.5) additional non-homogeneous initial condi-
tions for higher-order derivatives of solution T , i.e. T ′′, T ′′′, and
high values of the oscillation frequency p of compressive force N1
(described by small values of the parameter η−), differences betwe-
en solutions from both the models are significant:

∗ there are differences between amplitudes of these solutions, in
particular for T ′′′ =1 (Fig.5 and Fig.7);



Application of the tolerance averaging method... 375

Fig. 8. Solutions to frequency equations for initial conditions at z=0: T =1,
T ′ =0, T ′′ =−1, T ′′′ =0, and parameter η− =1.6 ·10−6



376 J.Jędrysiak

∗ there are differences of fundamental periods of these solutions
(Fig.5-Fig.8), in particular for T ′′ =−1, T ′′′ =1 (Fig.5);

∗ additional oscillationswith small periods appear in solutions T
from the non-asymptotic model (Fig.5b, Fig.6b and Fig8c);

∗ solutions for the non-asymptotic model describe the phenome-
non of beating for values of the oscillation frequency p of com-
pressive force N1 closer to the higher free vibration frequency
ω+ of this model (Fig.8b);

∗ the aforementioned differences increase with increasing oscilla-
tion frequency p of compressive force N1 (i.e. for decreasing
parameter η−) and with increasing parameter λ, describing
the size of the cell;

– large values of amplitude Nb of the compressive force N1 (parame-
ter δ−) cause additional oscillations inbothof the solutions (Fig.5b
and Fig.6b – curves 5a and 5b).

7. Remarks

It has to be emphasized that the applied modelling approach, i.e. the to-
lerance averaging technique (Woźniak and Wierzbicki, 2000), different from
the knownhomogenisationmethods used for periodic plates, leads to the non-
asymptotic models, whichmake it possible to investigate the effect of the pe-
riod lengths on the overall plate behaviour (cf. Jędrysiak, 2001, 2003a,b). The
main advantage of these models is that the analysed problems are described
by relatively simple differential equations with constant coefficients. Thus, the
non-asymptotic models can be used to analyse many engineering problems.
Moreover, for the proposed non-asymptotic model of thin periodic plates, the
conditions of the physical correctness of solutions W ,QA are determined, i.e.
the macrodeflection W and the internal variables QA,A=1, ...,N, are slowly
varying functions.
In this paper, using thismodel, the effect of the period lengths on dynamic

stability problems for Kirchhoff-type plates with periodic structure is taken
into account. Applying thismodel, parametric vibrations of such plates can be
considered. This problem can be also extended to problems of loads moving
on periodic plates, which can be the equivalent dynamical compressive forces
acting in the plate midplane (cf. Szcześniak, 1992).



Application of the tolerance averaging method... 377

Summarizing our considerations, the following conclusions can be formu-
lated:

• Taking into account the effect of the period lengths on dynamic stability
for thin periodic plates leads to the fourth-order differential equation for
the unknown function of time co-ordinate, which can be treated as a
certain generalization of the Mathieu equation. On the contrary, within
the homogenised model the known Mathieu equation is obtained.

• In the framework of the non-asymptotic model proposed in this contri-
bution, the additional higher critical forces can be analysed.

• From numerical solutions to the generalized Mathieu equation it can be
observed that the effect of the period lengths plays a crucial role for high
values of the oscillation frequency p of compressive force N1, which is
manifested in:

– different fundamental periodsof the solution T in thedimensionless
time co-ordinate z, for additional non-homogeneous initial condi-
tions imposed on the higher-order derivatives of T , i.e. on T ′′, T ′′′;

– large differences between amplitudes of the function T , for non-
homogeneous additional initial conditions imposed on T ′′, T ′′′;

– additional oscillations of the solution T with very small periods
in z, for non-homogeneous additional initial conditions imposed on
T ′′, T ′′′;

– the phenomenon called the beating described by the function T
for some non-homogeneous additional initial conditions imposed on
T ′′, T ′′′ and values of the oscillation frequency p closed to higher
free vibration frequency ω+ by the non-asymptotic model.

References

1. Baron E., 2002,On themodelling of medium thickness plates with an unipe-
riodic structure, J. Theor. Appl. Mech., 40, 1, 7-22

2. CaillerieD., 1984,Thin elastic andperiodic plates,Math. Meth. in the Appl.
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3. Jędrysiak J., 2000, On stability of thin periodic plates, Eur. J. Mech.
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378 J.Jędrysiak

4. Jędrysiak J., 2001, Dispersive models of thin periodic plates. Theory and
applications, Sci. Bul. Łódź Tech. Univ., 872,series: Sci. Trans., 289, Łódź (in
Polish)

5. Jędrysiak J., 2003a, Free vibrations of thin periodic plates interacting with
an elastic periodic foundation, Int. J. Mech. Sci., 45, 8, 1411-1428

6. Jędrysiak J., 2003b, The length-scale effect in the buckling of thin periodic
plates resting on a periodicWinkler foundation,Meccanica, 38, 4, 435-451

7. Kaliski S. (edit.), 1992,Vibrations, PWN,Warsaw; Elsevier, Amsterdam

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9. Kolpakov A.G., 2000, Homogenized model for plate periodic structure with
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10. Mazur-ŚniadyK.,WoźniakC.,Wierzbicki E., 2003,On themodelling of
dynamic problems for plates with a periodic structure, Arch. Appl. Mech. (in
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11. Michalak B., 1998, Stability of elastic slightly wrinkled plates, Acta Mech.,
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12. MichalakB., 2001,Dynamics and stability of the wavy plates, Sci. Bul. Łódź
Tech. Univ., 881, series: Sci. Trans., 295, Łódź (in Polish)

13. MichalakB., 2002,On the dynamic behaviour of a uniperiodic folded plates,
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16. Volmir A.C., 1972, Non-Linear Dynamics of Plates and Shells, Nauka,
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18. Woźniak C., Wierzbicki E., 2000,Averaging Techniques in Thermomecha-
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chowa



Application of the tolerance averaging method... 379

Zastosowanie metody tolerancyjnego uśredniania do analizy stateczności

dynamicznej cienkich płyt periodycznych

Streszczenie

W pracy rozpatrzono pewne zagadnienia dotyczące stateczności dynamicznej
cienkich płyt o periodycznej budowie. Przy wyprowadzeniu równań uśrednionego
modelu nieasymptotycznego wykorzystano tolerancyjne uśrednianie, zaproponowane
przezWoźniaka iWierzbickiego (2000) domodelowania kompozytów i struktur perio-
dycznych. Zastosowanie tej metody do znanego równania teorii płyt Kirchhoffa pro-
wadzi domodeli uśrednionych, w których uwzględniony jest wpływ długości okresów
periodyczności na pracę płyty (Jędrysiak, 2001). W pracy sformułowano nieasymp-
totyczny model opisujący zagadnienia stateczności dynamicznej płyt periodycznych.
Pokazano również, że wpływ długości okresów periodyczności odgrywa znaczącą rolę
w pewnych szczególnych przypadkach stateczności dynamicznej takich płyt, np. dla
wysokich częstości oscylacji sił ściskających, przyłożonych w płaszczyźnie środkowej
płyty.

Manuscript received January 5, 2004; accepted for print January 26, 2004