Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 2, pp. 305-321, Warsaw 2003

ON DYNAMIC STABILITY OF AN UNIPERIODIC MEDIUM

THICKNESS PLATE BAND

Eugeniusz Baron

Department of Building Structures Theory, Silesian University of Technology

e-mail: ckibud@polsl.gliwice.pl

The aim of this contribution is to apply equations derived by Baron
(2002) to the analysis of the dynamic stability of a periodically ribbed
simply supportedplate band.Thegeneral equationofmotion for thepla-
te band subjected to the time-dependent axial force was obtained. The
considerationsare relatedto thedynamicanalysis forarbitraryboundary
conditions. The obtained frequency equation can be treated as a certain
generalization of the known Mathieu equation. By applying the proce-
dure used for investigation of the Mathieu equation, two fundamental
regions of the dynamic instability are determined. The obtained results
are similar to those derived from known solutions, but also depend on
the period l.We also deal with a new higher free vibration frequency.

Key words: modelling, dynamic stability, medium thickness plate,
periodic structure

1. Introduction

The main aim of this contribution is to apply general equations of uni-
periodic plates formulated by Baron (2002) to detect the effect of repetitive
cell size on the dynamic plate behaviour in the case of dynamic instability. In
the aforementioned paper a newmodelling approach to themedium thickness
elastic periodic plates was presented in the framework of the Hencky-Bolle
theory. A new averaged 2D-model of Hencky-Bolle elastic plates with a one-
directional periodic (uniperiodic) structure was obtained by using the tole-
rance averaging of partial differential equations with periodic coefficients, cf.
Woźniak andWierzbicki (2000). This model describes the influence of the re-
petitive (periodic) cell size on the overall plate behaviour (the microstructure
length-scale effect).



306 E.Baron

In the present contributionwe analyse the problemof dynamic plate insta-
bility related to the parametric resonance of the plate band, cf. Bolotin (1956).
The plate bandunder consideration is simply supported on the opposite edges
and subjected to a time-dependent compressive axial force acting in the di-
rection normal to the edges, see Fig.1. Hence, the governing equations of the
medium thickness uniperiodic plate derived by Baron (2002) will be specified
for a plate band made of an orthotropic homogeneous material and having
a periodically variable thickness in the x1-axis direction and constant in the
x2-axis direction, see Fig.1. On these assumptions the frequency equation for
the uniperiodic plate band will be derived by the tolerance averaging of the
2D-plate equations. Assuming that the compressive axial force is constant,
the free vibration frequencies and static critical force will be calculated. For
the time-dependent compressive force given by N(t) = N0 + N1cos(pt), it
will be shown that instead of the knownMathieu equation we obtain a forth-
order differential equation. By applying the procedure similar to that used
for investigations of the Mathieu equation, two fundamental dynamic insta-
bility regions are determined. The dynamic stability of plates periodic in two
dimensions was investigated by Wierzbicki andWoźniak (2002).

Fig. 1. Uniperiodically ribbed plate band

The obtained solutions related to the dynamic stability are similar to the
known solutions but also depend on the periodic cell size, and hence they
constitute a certain generalization of the results derived from the Mathieu
equation. The results derived on the basis of the tolerance averaging model



On dynamic stability of an uniperiodic... 307

are compared to those derived in the framework of the asymptotic model, cf.
Baron (2002).

Throughout the paper the subscripts α,β,... run over 1,2, subscripts
i,j, ... over 1,2,3 and superscripts A,B,... over 1,2, ...,N; summation co-
nvention holds for all aforementioned indices.

2. Preliminaries

To make the paper self-consistent we recall some fundamental concepts
and denotations following Baron (2002).

By x=(x1,x2) we denote theCartesian orthogonal coordinates of a point
on the plate midplane Π = (0,L1)× (0,L2) and by z the Cartesian coordi-
nate in the direction normal to the midplane. By z = ±δ(x), x ∈ Π, we
denote functions representing the upper and lower plate boundary, respecti-
vely; hence 2δ(x) is the plate thickness at a point x ∈ Π. By ρ = ρ(x,z)
and Aijkl(x,z) we denote the mass density and elastic moduli tensor of the
plate, and assume that every plane z = const is a plane of elastic symmetry.
We also define Cαβγδ := Aαβγδ − Aαβ33A33γδ(A3333)

−1, Bαβ := Aα3β3. We
assume that δ(·), ρ(·), Aijkl(·) are periodic functions with the period l with
respect to the x1-coordinate, ρ(·), Aijkl(·) are even functions of z, and all afo-
rementioned functions are sufficiently regular ones of the x2-coordinate. We
also assume that l ≫ maxδ(x). Let p+ and p− denote tractions (along the
z-axis) situated on the upper and lower plate boundaries, respectively, Nαβ
be the prestressing tensor of the plate midplane and b be the z-coordinate of
a constant body force acting along the z-axis direction. Furthermore, let t be
the time coordinate.

The averaged value of an arbitrary integrable function ϕ(x1,x2, t) in the
periodicity interval (x1− l/2,x1+ l/2) will be denoted by

〈ϕ〉(x, t)=
1

l

x1+l/2∫

x1−l/2

ϕ(ξ,x2, t) dξ x=(x1,x2)

In the special case when ϕ(·) is an uniperiodic function, i.e. ϕ(·) is periodic
only in the x1-direction, the above averaged value is independent of x1, and
will be denoted by 〈ϕ〉.



308 E.Baron

Under denotations (in the formulae for Dαβ there is no summation over
α and β!)

µ :=

δ∫

−δ

ρ dz J :=

δ∫

−δ

z2ρ dz p := p++p−+ b〈µ〉

Gαβγδ :=

δ∫

−δ

z2Cαβγδ dz Dαβ :=

δ∫

−δ

KαβBαβ dz

where Kαβ are the shear coefficients, we recall the system of equations for
unknown displacements w and rotations ϑα

(Gαβγδϑ(γ,δ)),β −Dαβϑβ −Jϑ̈α =0

Noαβw,αβ +[Dαβ(ϑβ +w,β)],α−µẅ+p =0

representing the Hencky-Bolle theory of the medium thickness plates, cf. Je-
mielita (2001). For the periodic plates under consideration this is a system
of equations with functional coefficients depending on x1, which are highly
oscillating and can be non-continuous. Direct solutions to the boundary va-
lue problems related to these equations are very complicated. That is why,
some simplified models of the plates have been proposed, cf. Baron (2002).
We canmention here homogenized models based on the asymptotic approach
which leads to differential equations with constant coefficients. However, the
asymptotic homogenization neglects the effect of periodicity cell size on the
macrodynamic plate behaviour (length-scale effect). The new modelling ap-
proach, which describes the above length-scale effect was presented by Baron
(2002). This approach applies the tolerance averaging technique to the pla-
te equations, cf. Woźniak and Wierzbicki (2000). According to the tolerance
averaging technique, we introduce the decompositions

ϑα = ϑ
o
α+ϑ

∗

α w = w
o+w∗

where the slowly varying functions wo(·), ϑoα(·) are theaveraged displacements
and rotations, respectively, and the oscillating functions w∗(·), ϑ∗α(·) are fluc-
tuations of these fields, cf.Woźniak andWierzbicki (2000). Using the procedu-
re proposed by Baron (2002), the functions w∗(·), ϑ∗α(·) can be approximated
bymeans of the formulae

ϑ∗xα(x, t)
∼= ha(x1)Θ

a
α(x, t) a =1,2, ...,n

w∗x(x, t)
∼= gA(x1)W

A(x, t) A =1,2, ...,N
(2.1)



On dynamic stability of an uniperiodic... 309

where Θaα(x, t), W
A(x, t) are new unknown functions which are slowly vary-

ing with respect to the x1-coordinate, and h
a(x1), g

A(x1) are certain linear
independentmode-shape functions satisfying the conditions

〈Jha〉=0 〈µgA〉=0 ha(x1)∈ O(l)

gA(x1)∈ O(l) lh
a
,1(x1), lg

A
,1(x1)∈ O(l)

Taking into account the aforementioned conditions we shall also introduce the
functions

h
a
= l−1ha gA = l−1gA

which are of order O(1) when l → 0. In most cases, the mode-shape func-
tions are postulated a priori, and hence formulae (2.1) can be interpreted as
a certain kinematic hypothesis related to the form of expected displacement
disturbances caused by the plate periodic structure. The degree of approxi-
mation ∼= in (2.1) depends on the number of terms on the right-hand sides in
(2.1).

The governing equations of the averaged 2D-model of nonhomogeneous,
medium thickness elastic plates with a one-directional periodic structure,
which have been derived by Baron (2002), consist of:

— equations of motion

Mαβ,β −Qα−〈J〉ϑ̈
o
α =0

(2.2)

Noαβw
o
,αβ +Qα,α−〈µ〉ẅ

o+p =0

— equations for Θa, WA

l2〈Jh
a
h
b
〉Θ̈bα+M

a
α −M̃

a
α,2 =0

(2.3)

l2〈µgAgB〉ẄB +QA− lQ̃A,2−N
o
αβ〈g

A〉wo,αβ − l〈g
Ap〉=0

— constitutive equation

Mαβ = 〈Gαβγδ〉ϑ
o
(γ,δ)+ 〈h

a
,1Gαβ1δ〉Θ

a
δ + l〈h

a
Gαβ2δ〉Θ

a
δ,2

Qα = 〈Dαβ〉(ϑ
o
β +w

o
,β)+ l〈h

a
Dαβ〉Θ

a
β + 〈g

A
,1Dα1〉W

A+ l〈gADα2〉W
A
,2

Maα = 〈h
a
,1h
b
,1Gα11δ〉Θ

b
δ + 〈h,1Gα1γδ〉ϑ

o
(γ,δ)+ l〈h

a
,1h
b
Gα12δ〉Θ

b
δ,2+

+l2〈h
a
h
b
Dαβ〉Θ

b
β + l〈h

a
Dαβ〉(ϑ

o
β +w

o
,β)+

+l〈h
a
gA,1Dα1〉W

A+ l2〈h
a
gADα2〉W

A
,2 (2.4)



310 E.Baron

M̃aα = 〈h
a
hb,1Gα21δ〉Θ

b
δ + 〈h

a
Gα2γδ〉ϑ

o
(γ,δ)+ l〈h

a
h
b
Gα22δ〉Θ

b
δ,2

QA = 〈gA,1g
B
,1D11〉W

B + 〈gA,1D1β〉(ϑ
o
β +w

o
,β)+ l〈g

A
,1h
a
D1β〉Θ

a
β +

+l〈gA,1g
BD12〉W

B
,2

Q̃A = 〈gAgB,1D21〉W
B + 〈gAD2β〉(ϑ

o
β +w

o
,β)+ l〈g

AhaD2β〉Θ
a
β +

+l〈gAgBD22〉W
B
,2

Equations (2.2)-(2.4) constitute the starting point for the analysis of para-
metric vibrations and dynamic stability of the medium thickness periodically
ribbed plate band. In a general case, the coefficients in (2.2)-(2.4) can depend
on the x2-coordinate.

3. Equations of the uniperiodic plate band

In this section we specify equations (2.2)-(2.4) for the plate band made
of an orthotropic homogeneous material, having the uniperiodic (with the
period l) variable thickness δ(x1) along the x1-axis, see Fig.1. In this case,
the coefficients in (2.2)-(2.4) are constant. Taking into account the orthotropy
of the plate we denote

G11 = G1111 G22 = G2222

G12 = G1122 = G2211 G = G1212 = G1221 = G2112 = G2121

D1 = D11 D2 = D22

In the subsequent considerations only twomode-shape functions will be taken
into account. Hence, we define

h = h1(x1)= lh(x1) g = g
1(x1)= lg(x1)

where the values of h(x1) and g(x1) are of order O(1) when l → 0. We also
denote

Θ1(x, t)= Θ
1
1(x, t) Θ2(x, t)= Θ

1
2(x, t) W(x, t)= W

1(x, t)

ϑ1(x, t)= ϑ
o
1(x, t) ϑ2(x, t)= ϑ

o
2(x, t) w(x, t)= w

o(x, t)



On dynamic stability of an uniperiodic... 311

For the cylindrical bending of the plate band in the Ox2z-plane, all unk-
nowns are independent on x1. Setting x = x2, equations (2.2)-(2.4) reduce to
the form

〈G〉ϑ′′1 −〈D1〉ϑ1−〈J〉ϑ̈1+ l〈hD1〉Θ1+ 〈h,1G〉Θ
′

21−〈g,1D1〉W =0

〈G22〉ϑ
′′

2 −〈D2〉ϑ2−〈J〉ϑ̈2+ 〈h,1G12〉Θ
′

1− l〈hD2〉Θ2−〈D2〉w
′ =0

(N22+ 〈D2〉)w
′′−〈µ〉ẅ+ 〈D2〉ϑ

′

2+ l〈hD2〉Θ
′

2+p =0

−l2〈h
2
G〉Θ′′1 + l

2〈h
2
D1〉Θ1+ 〈h

2
,1G11〉Θ1+ l

2〈h
2
J〉Θ̈1+

+l2〈hD1〉ϑ1+ 〈h,1G12〉ϑ
′

2+ l〈hg,1D1〉W =0 (3.1)

−l2〈h
2
G22〉Θ

′′

2 +(l
2〈h
2
D2〉+ 〈h

2
,1G〉)Θ2 + l

2〈h
2
J〉Θ̈2+

+〈h,1G〉ϑ
′

1+ l〈hD2〉(ϑ2+w
′)+ l2〈hgD2〉W

′ =0

−l2〈g2D2〉W
′′+ 〈g2,1D1〉W + l

2〈g2µ〉Ẅ + 〈g,1D1〉ϑ1+

+l〈hg,1D1〉Θ1− l
2〈ghD2〉Θ

′

2 =0

where the denotations ∂f/∂x = f ′(x), ∂2f/∂x2 = f ′′(x) for an arbitrary dif-
ferentiable function f are introduced. Thus, the problem under consideration
is governed by the system of six partial differential equations (3.1) for six unk-
nown functions: theaveraged rotations ϑα(x,t), averaged displacement w(x,t)
and extra unknowns Θα(x,t), W(x,t). Equations (3.1) have a physical sense
provided that Θα(·, t) and W(·, t) are slowly varying functions, cf. Woźniak
andWierzbicki (2000).

Taking into account the symmetric form of the periodic cell, cf. Fig.1,
we assume that the mode-shape function h(x1) is odd while the mode-shape
function g(x1) is even. In this case, the system of equations (3.1) can be
separated into two independent systems. The first of these systems is the
following system of equations describing evolutions of the functions ϑ1(x, t)
and Θ2(x, t)

〈G〉ϑ′′1 −〈D1〉ϑ1−〈J〉ϑ̈1+ 〈h,1G〉Θ
′

2 =0
(3.2)

−l2〈h
2
G22〉Θ

′′

1 +(l
2〈h
2
D1〉+ 〈h

2
,1G〉)Θ2+ l

2〈h
2
J〉Θ̈2+ 〈h,1G〉ϑ

′

1 =0

The secondonedescribes evolutions of the functions ϑ2(x, t),w(x, t),Θ1(x, t),
W(x, t)



312 E.Baron

〈G22〉ϑ
′′

2 −〈D2〉ϑ2−〈J〉ϑ̈2+ 〈h,1G12〉Θ
′

1−〈D2〉w
′ =0

(N22+ 〈D2〉)w
′′−〈µ〉ẅ+ 〈D2〉ϑ

′

2+p =0

−l2〈h
2
G〉Θ′′1 +(l

2〈h
2
D1〉+ 〈h

2
,1G11〉)Θ1+ l

2〈h
2
J〉Θ̈1+ (3.3)

+〈h,1G12〉ϑ
′

2+ l〈hg,1D1〉W =0

−l2〈g2D2〉W
′′+ 〈g2,1D1〉W + l

2〈g2µ〉Ẅ + l〈hg,1D1〉Θ1 =0

Neglecting the rotational inertia terms and assuming the homogeneous initial
conditions, equations (3.2) yields ϑ1 = 0 and Θ2 = 0. Bearing in mind that
N22 =−N, N = N(t), together with p = p(x, t), neglecting terms involving
J and taking into account denotations

ϕ1[w] = [l
2〈h
2
G〉〈G22〉w

′′− (l2〈h
2
D1〉+ 〈h

2
,1G11〉)How]

′′

ϕ2[w] = [l
2〈h
2
G〉〈G22〉w

′′−〈h2,1G11〉H1w]
′′

ψ1[w] = ϕ1[w]−〈D2〉[l
2〈h
2
G〉w′′− (l2〈h

2
D1〉+ 〈h

2
,1G11〉)w]

ψ2 = ϕ2[w]−〈D2〉(l
2〈h
2
G〉w′′−〈h2,1G11〉w)

ζ1[p] = l
2〈h
2
G〉(〈G22〉p

′′−〈D2〉p)
′′− (l2〈h

2
D1〉+ 〈h

2
,1G11〉)(Hop

′′−〈D2〉p)

ζ2[p] = l
2〈h
2
G〉(〈G22〉p

′′−〈D2〉p)
′′−〈h2,1G11〉(H1p

′′−〈D2〉p)

where

Ho = 〈G22〉−
〈h,1G12〉

2

l2〈h
2
D1〉+ 〈h

2
,1G11〉

H1 = 〈G22〉−
〈h,1G12〉

2

〈h2,1G11〉

we obtain from equations (3.3) a single equation for the averaged deflection
w(x,t) in the form

−N(l2〈g2D2〉ψ
′′

1 −〈g
2
,1D1〉ψ2)

′′+ 〈D2〉(l
2〈g2D2〉ϕ

′′

1 −〈g
2
,1D1〉ϕ2)

′′−

−l2〈g2µ〉(−N̈ψ1+ 〈D2〉ϕ̈1+Nψ̈1)
′′−〈µ〉(l2〈g2D2〉ψ̈

′′

1 −〈g
2
,1D1〉ψ̈2)+(3.4)

+l2〈g2µ〉
∂4

∂t4
ψ1 = l

2〈g2D2〉ζ
′′

1 −〈g
2
,1D1〉ζ2− l

2〈g2µ〉ζ̈1

Equation (3.4)willbe treatedasageneral equationofmotionof anuniperiodic,
medium thickness plate band.



On dynamic stability of an uniperiodic... 313

4. Frequency equation

By separating the variables, the unknown function w(x,t) in (3.4) will be
assumed in the form

w(x,t) = wo(x)T(t) (4.1)

Let use assume that the plate band is simply supported on the edges x =0,
x = L. In this case, the unknown function wo(x) will be assumed in the form

wo(x)=
∞∑

n=1

wn sin(knx) kn =
nπ

L
n =1,2, ... (4.2)

In the special case, when p =0, taking into account (4.1), (4.2), and denoting

Hn = 〈G22〉
(
1+k2nl

2 〈h
2
G〉

l2〈h
2
D1〉+ 〈h

2
,1G11〉

)
−

〈h,1G12〉
2

l2〈h
2
D1〉+ 〈h

2
,1G11〉

Dn = 〈D2〉
(
1+k2nl

2 〈h
2
G〉

l2〈h
2
D1〉+ 〈h

2
,1G11〉

)

κn =
〈D2〉

k2nHn+Dn

◦

κ=
(
1+
〈h2,1G11〉

l2〈h
2
D1〉

)−1
(4.3)

c1 =1−κn
◦

κ
〈G22〉

Hn
+k2nl

2 〈g
2D2〉

〈g2,1D1〉

c2 =1−κn
◦

κ
(
1+

k2n〈G22〉

〈D2〉

)
+k2nl

2 〈g
2D2〉

〈g2,1D1〉

weobtain from(3.4) the following frequency equation for the simply supported
uniperiodic plate band

l2
d4T

dt4
+
[c2〈g2,1D1〉
〈g2µ〉

+
k2nl
2

〈µ〉
(−N +k2nHnκn)

]d2T
dt2
+

(4.4)

+k2n

[ 〈g2,1D1〉
〈g2µ〉〈µ〉

(−c2N + c1k
2
nHnκn)− l

2d
2N

dt2
1

〈µ〉

]
T =0

If the plate satisfy the condition d ≪ l ≪ L, where d =2maxδ(x), we obtain
c2 ≈ c1, 〈Dn〉≈ 〈D2〉 and then

Hn ≈ H = 〈G22〉−
〈h,1G12〉

2

l2〈h
2
D1〉+ 〈h

2
,1G11〉

κn ≈
〈D2〉

k2nH + 〈D2〉
(4.5)



314 E.Baron

Taking into account (4.5) and assuming that the axial force is compressive
(N < 0) and constant, we obtain from (4.4) the frequency equation

l2
d4T

dt4
+
[c2〈g2,1D1〉
〈g2µ〉

+
k2nl
2

〈µ〉
(−N +k2nHnκn)

]d2T
dt2
+

(4.6)

+k2n
c2〈g

2
,1D1〉

〈g2µ〉〈µ〉
(−N +k2nHκn)T =0

From the above equation, after some transformations, we derive the following
formulae for the higher ω1 and lower ω2 free vibration frequencies

ω21 =
c2〈g

2
,1D1〉

l2〈g2µ〉
ω22 =

k2n
〈µ〉
(−N +k2nHκn) (4.7)

For the static critical force we obtain

Nkr,n = k
2
nHκn (4.8)

It must be emphasised that in the first approximation the lower frequency
ω2 and the static critical force Nkr,n coincide with those derived from the
asymptotic model which will be presented in Section 6. The higher frequency
ω1 will be obtained by using the tolerance averaging model.

5. Dynamic stability

In this section we shall investigate the dynamic instability for the unipe-
riodic, simply supported plate band, described by equation (3.4). We shall
examine the case of dynamic instability caused by the parametric resonance,
cf. Bolotin (1956). Using the standard procedure, we assume that the com-
pressive axial force in the plate midplane is time dependent and governed by
the relation

N(t)= N0+N1cos(pt)

where N0, N1 are constant. The solution to equation (3.4) will be assumed in
the form

w(x,t) =
∞∑

n=1

Tn(t)sin(knx) kn =
nπ

L
n =1,2, ... (5.1)



On dynamic stability of an uniperiodic... 315

Now, the nth free vibration frequency have the form

Ω2n =
k4nHκn
〈µ〉

(
1−

N0
Nkr,n

)

At the same time, the modulation factor (depth of the modulation) has the
form

2µn =
N1

Nkr,n−N0

(
1−

p2

ω21

)

Taking into account denotations (4.3) and assumptions (4.5), we finally obtain

l2
d4Tn
dt4
+
c2〈g

2
,1D1〉

〈g2µ〉

{d2Tn
dt2
+Ω2n[1−2µncos(pt)]Tn

}
=0 (5.2)

It must be emphasised that equation (5.2) is a certain generalization of the
knownMathieu equation; it takes the form of theMathieu equation provided
that in (5.2) the length-scale effect is neglected.
The analysis of dynamic stability leads to the determination of the insta-

bility regions on the (p/Ωn,µn)-plane. Hence, we ask whether, for the given
quotient of the exciting frequency of the axial force p and the free vibra-
tion frequency Ωn and for the given modulation factor µn, the plate band
vibrations are stable or instable. Thus, we have to determine the instability
regions (resonance regions) for solutions to equation (5.2). Within the reso-
nance regions vibrations grow up in an unlimited way as t →∞. Outside and
at the boundaries of the resonance regions there exist periodic solutions to
equation (5.2) with the parametric excitation periods Tp = 2π/p and 2Tp.
The considerations will be restricted to the parametric vibrations for the first
harmonic components of series (5.1). The subsequent resonance regions for
3Tp, 4Tp, ... and for higher harmonic components (n = 2,3, ...) do not play
an important role in most engineering problems. That is why we are looking
for the instability solutions for the equation

d4T

dt4
+ω2

[d2T
dt2
+Ω2

(
1−2µcos(pt)

)
T
]
=0 (5.3)

where, for the sake of simplicity, we neglected the index 1 and denoted

ω2 = ω21 =
c2〈g

2
,1D1〉

l2〈g2µ〉

Looking for the solutionswith the period 2Tp related to the boundaries of the
first instability region we substitute

T(t)=
∞∑

i=1,3,...

(
ai sin

ipt

2
+ bicos

ipt

2

)



316 E.Baron

into (5.3).After comparing the coefficients of pertinent trigonometric functions
to zero,weobtain the following infinite systemof the linear algebraic equations

[
1+µ−

( p
2Ω

)2
+ε2
( p
2Ω

)4]
a1−µa3 =0

(5.4)
[
1−
( ip
2Ω

)2
+ε2
( ip
2Ω

)4]
ai−µ(ai−2+ai+2)= 0 i =3,5, ...

[
1−µ−

( p
2Ω

)2
+ε2
( p
2Ω

)4]
b1−µb3 =0

(5.5)
[
1−
( ip
2Ω

)2
+ε2
( ip
2Ω

)4]
bi−µ(bi−2+ bi+2)= 0 i =3,5, ...

where ε = Ω/ω. For sufficiently small values of the modulation factor, the
characteristic determinants of systems (5.4) and (5.5) can be approximated
by the first components of relations (5.4)1 and (5.5)1. In this case, in the
(p/Ω,µ)-plane, we obtain the boundaries of the first instability region given
by

( p
Ω

)2
≈ 1+µ

( p
Ω

)2
≈ 1−µ (5.6)

and an extra condition for the excitation force frequency p 6=2ω.

Solutions (5.6) are similar to theknown solutions,Bolotin (1956).However,
the free vibration frequency Ω and the modulation factor µ depend on the
period l. We have also obtained the extra resonance frequency for the axial
excitation force.

Analogously, we look for the solution with the period Tp. Setting

F(t)= bo+
∞∑

i=2,4,...

(
ai sin

ipt

2
+ bicos

ipt

2

)

we obtain two homogeneous, infinite systems of linear algebraic equations

bo−µb2 =0
[
1−
( p
Ω

)2
+ε2
( p
Ω

)4]
b2−µ(2bo+ b4)= 0 (5.7)

[
1−
( ip
2Ω

)2
+ε2
( ip
2Ω

)4]
bi−µ(bi+2+ bi−2)= 0 i =4,6, ...



On dynamic stability of an uniperiodic... 317

[
1−
( p
Ω

)2
+ε2
( p
Ω

)4]
a2−µa4 =0

(5.8)
[
1−
( ip
2Ω

)2
+ε2
( ip
2Ω

)4]
ai−µ(ai+2+ai−2)= 0 i =4,6, ...

For small values of themodulation factor, the dimensions of the characteristic
determinants related to systems (5.7) and (5.8) can be restricted to two rows
and two columns.Hence, by applying to (5.7) and (5.8) a similar procedure to
that in the analysis of the classicalMathieu equation, we obtain the boundary
of the second instability region

( p
Ω

)2
=1−2µ2+ε2(1−2µ2)

(5.9)
( p
Ω

)2
≈ 1+

1

3
µ2+ε2

9−14µ2

9+8µ2

and the extra resonance frequency for the axial excitation force p 6= ω and
p 6=0.5ω. In (5.9)weunderlined the termswhichare qualitatively conformable
to the known solutions, Bolotin (1956). The length-scale effect described by
the terms including coefficients depending on ε, yields a correction of the
boundaries of the instability regions.

The above method of analysis of the parametric resonance problem seems
to be the simplest possible. It yields sufficiently good results provided that the
modulation factor satisfies the condition µ < 0.6. The obtained relations con-
firmthe correctness of thepresentedaveraged 2D-model of amediumthickness
uniperiodic elastic plate. By applying this model to the analysis of dynamic
instability of the considered plate, we obtained a certain generalization of the
classical results, which are being found when ε tends to zero.

6. Asymptotic model

In this section we recall the procedure presented in Sections 3-5 for equ-
ations of the asymptoticmodel derived byBaron (2002).We take into account
the assumptions and denotations given in Section 3. In the framework of the
asymptotic model, obtained for l → 0, the following system of equations of
the plate band can be derived



318 E.Baron

〈G〉ϑ′′1 −〈D1〉ϑ1−〈J〉ϑ̈1+ 〈h,1G〉Θ
′

2 =0

〈G22〉ϑ
′′

2 −〈D2〉ϑ2−〈J〉ϑ̈2+ 〈h,1G12〉Θ
′

1−〈D2〉W
′ =0

(N22+ 〈D2〉)w
′′−〈µ〉ẅ+ 〈D2〉ϑ

′

2+p =0

〈h2,1G11〉Θ1+ 〈h,1G12〉ϑ
′

2 =0 (6.1)

〈h2,1G〉Θ2+ 〈h,1G〉ϑ
′

1 =0

〈g2,1D1〉W + 〈g,1D1〉ϑ1 =0

On assumption that the mode-shape function g(x1) is even, after neglecting
the rotational inertial terms, and for the homogeneous initial conditions, we
obtain ϑ1 =0,Θ2 =0and W =0. In this case, introducing, as in theprevious
section, the denotations N22 =−N, N = N(t) the system of equations (6.1)
reduces to the following equation of motion

[How
′′−N(How

′′−〈D2〉w)]
′′−〈µ〉(Hoẅ

′′−〈D2〉ẅ)= Hop
′′−〈D2〉p (6.2)

where

Ho = 〈G22〉−
〈h,1G12〉

2

〈h2,1G11〉

Assuming that the plate band is simply supported on the edges x =0, x = L
and for p =0, by separating the variables, we obtain from (6.2) the following
frequency equation

d2T

dt2
+
k2n
〈µ〉
(−N +k2nHoκn)T =0 (6.3)

where

κn =
〈D2〉

k2nHo+ 〈D2〉
kn =

nπ

L
n =1,2, ...

If the axial force N is constant and compressive, then from (6.3) we obtain
the free vibration frequency given by

ω2o =
k2n
〈µ〉
(−N +k2nHoκn)

and the static critical force

Nkr,n = k
2
nHoκn

In the framework of the asymptotic model described above it is not possible
to determine the structural (higher) free vibration frequency.



On dynamic stability of an uniperiodic... 319

When investigating the plate dynamic stability (i.e. an instability caused
by the parametric resonance), we assume that the compressive axial force is
given by

N(t)= N0+N1cos(pt)

Ifwe look for a solution to (6.3), in the form (5.1), thenwe arrive at the known
Mathieu equation

d2Tn
dt2
+Ω2n(1−2µcos(pt))Tn =0

where

2µn =
N1

Nkr,n−N0

is a modulation factor and

Ω2n =
k2nHoκn
〈µ〉

(
1−

N0
Nkr,n

)

is the nth free vibration frequency.Using the procedure for the determination
of the instability region boundaries described in Section 5, we obtain:
— for the first instability region (vibrations with period 2Tp)

( p
2Ω

)2
≈ 1+µ

( p
2Ω

)2
≈ 1−µ

— for the second instability region (vibrations with period Tp)

( p
Ω

)2
≈ 1−2µ2

( p
Ω

)2
≈ 1+

1

3
µ2

After comparing the results obtained from the tolerance averaging model
with those derived from the asymptotic one, the following conclusions can be
formulated:

• the square of the lower resonance frequency ω2o calculated from the
asymptotic model is an approximation of order O(l2) of the frequen-
cy ω22 derived from the tolerance averagingmodel, i.e., ω

2
2 = ω

2
o+O(l

2);

• the higher resonance frequency ω21, determined from equation (4.7)1,
cannot be derived from the tolerance averaging model;

• in the asymptotic model, the analysis of the dynamic stability of the
plate band leads to the knownMathieu equation.



320 E.Baron

7. Conclusions

This contribution represents both a supplement and continuation of the
paper byBaron (2002). By applying the tolerance averaging procedure formu-
lated in the above paper, the general equation of a new averaged 2D-model
of themedium thickness elastic plates with a one-directional periodic structu-
re has been obtained. This model takes into account the effect of the period
length on the overall plate behaviour.

The aim of this contribution was to apply the general equations of uni-
periodic plates formulated by Baron (2002) for detection of the influence of
the effect of repetitive cell size on the dynamic plate behaviour in the case of
dynamic instability. It was assumed that the plate band under consideration
is homogeneous and orthotropic. The uniperiodic structure of the plate band
was related to the periodically spaced system of ribbs, the axes of which are
normal to the edges. The main result of this paper is that the equations of
motion for the uniperiodic plate band reduce to a certain single equation (3.4)
for the plate band subjected to a time-dependent axial force provided that the
plate is homogeneous and orthotropic.

Equation of motion (3.4) can be applied to the analysis of free vibrations,
forced vibrations and dynamic stability (parametric resonance) of the plate
band with arbitrary boundary conditions.

The frequency equation for a simply supported plate band (4.6), derived
from (3.4), makes it possible to determine the static critical force and the fun-
damental (lower) free vibration frequency. The obtained results are a certain
generalization of known solutions, cf. Bolotin (1956), and take into account
the influence of the period length l on the overall plate behaviour.

On the basis of equation (4.6) a higher free vibration frequency, explicity
depending on the period l, has been obtained as well. This frequency is called
the structural one, and cannot be derived from the asymptotic model.

For the axial force given by N(t)= N0+N1cos(pt) we obtained equation
(5.2), which can be treated as a certain generalization of the known Mathieu
equation. It is a fourth-order ordinary differential equation, which reduces to
the knownMathieu equation provided that the period length l is neglected.

By applying a procedure similar to that used for the investigation of the
Mathieu equation, cf. Bolotin (1956), two fundamental regions of dynamic
instability have been determined. The obtained results are a certain generali-
zation of those derived from the known solution, i.e. in the proposed approach
we have dealt with a certain new parameter, namely the structuralfree vibra-



On dynamic stability of an uniperiodic... 321

tion frequency. In the determination of this frequency the definition of the
modulation factor was taken into account and an extra condition on the axial
force frequency was imposed on this frequency.

References

1. Baron E., 2002, Onmodelling of medium thickness plates with a uniperiodic
structure, J. of Theor. and Appl. Mech., 40, 7-22

2. Bolotin B.B., 1956,Dinamicheskaya ustǒıchivost uprugikh sistem, Gos. Izd.
Tekh.-Teor. Lit., Moskwa

3. Jemielita G., 2001, Meandry teorii płyt, in: Mechanika Sprężystych Płyt i
Powłok, edit. C.Woźniak, PWNWarszawa

4. Wierzbicki E., Woźniak C., 2002, Continuum modelling and the internal
instability of certain periodic structures,Arch. of Appl. Mech., 72, 451-457

5. Woźniak C., Wierzbicki E., 2000,Averaging Techniques in Thermomecha-
nics of Composite Solids, Wydawn. Pol. Częstochowskiej

Stateczność dynamiczna uniperiodycznego pasma płytowego średniej

grubości

Streszczenie

Celempracy jest zastosowanie równańuzyskanychprzezBarona (2002)do analizy
stateczności dynamicznej periodycznie użebrowanego, swobodnie podpartego pasma
płytowego. Wyprowadzono ogólne równanie ruchu takiego pasma płytowego obcią-
żonego zależną od czasu siłą osiową. Otrzymane wyniki zastosowano do analizy za-
gadnień dynamiki przy dowolnych warunkach brzegowych.Wyprowadzono równanie
częstości dla pasmapłytowego swobodnie podpartego. Stanowi onopewneuogólnienie
znanego równaniaMathieu. Stosując trybpostępowania, jakprzy rozwiązywaniu rów-
nania Mathieu, wyznaczono dwa podstawowe obszary niestateczności dynamicznej.
Uzyskane wyniki są zgodne z rozwiązaniami znanymi, uwzględniają jednak zależność
rozważanego problemu od wymiaru powtarzalnego segmentu płyty. W rozważaniach
pojawia się dodatkowa wysoka częstość drgań własnych, której nie obejmują wyniki
uzyskane przez Bolotina (1956).

Manuscript received December 19, 2002, accepted for print January 23, 2003