Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

2, 39, 2001

DYNAMIC RESPONSE OF A MICRO-PERIODIC BEAM

UNDER MOVING LOAD – DETERMINISTIC AND

STOCHASTIC APPROACH

Krystyna Mazur-Śniady
Paweł Śniady

Institute of Civil Engineering, Wrocław University of Technology

e-mail: kmsniady@i14odt.iil.pwr.wroc.pl; psniady@i14odt.iil.pwr.wroc.pl

In the paper, the deterministic and stochastic approach to the problem
of vibrationsof abeamwithperiodically varying geometryundermoving
load ispresented.Anewaveragedmodel for thedynamicsof theperiodic-
like beamwith a variable cross-section,Mazur-Śniady (2001), is applied.
The approach to dynamics of the periodic-like beam assumed in the
paper is based on concepts of the tolerance-averagedmodel byWoźniak
(1999). The solution obtained for a single moving force is the basis of
solutionof stochasticvibrationscausedby randomtrainofmoving forces.

Key words: dynamics, moving load, stochastic processes, periodic-like
beam

1. Introduction

The determination of vibrations caused by a moving load is one of the
classical problems of structural mechanics. The problem appears in dynamics
of bridges, railways, roads, landing strips, etc. Numerous papers in this field
were published. One of the first problemswas the determination of vibrations
of a beam under moving load. Well known papers by Krylow (1905) and Ti-
moshenko (1922) describe the vibrations of a simply supported beam caused
by the force moving along the beam with a constant velocity. For this beam
Kączkowski (1963), using the method of superposition of deflections of the
beam axis, proved that the part of solution describing aperiodic vibrations
can be written in a closed form. The closed-form solutions for beams and fra-
meswith different support conditionswere given byReipert (1969, 1970). The



324 K.Mazur-Śniady, P.Śniady

problems of finding solutions in the closed formwere presented also byMazur
and Śniady (1973) and Śniady (1976).
Thedeterministic solutionobtained for a singlemoving force canbeapplied

to the problemof stochastic vibrations of a beamunder random train of forces
travelling in the same direction with equal and constant velocity.
The problem of stochastic vibrations and reliability of the beamwas ana-

lyzed in papers by Tung (1967), Iwankiewicz and Śniady (1984), Bryja and
Śniady (1988), Śniady (1976), Sieniawska and Śniady (1990), Śniady et al.
(1998, 2001). Themodel given above can be applied to the reliability analysis
of bridge beams under traffic flow.
In this paper, the deterministic and stochastic approaches to the problem

of vibrations of beam with periodically varying geometry under moving load
are presented.
The standardmethods of analyzing the beamdynamics are effective only if

the coefficients in thewell-knowndifferential equation of thebeamare constant
or slowly varying. If the coefficients of this equation are varying rapidly then
the solution is rather difficult to obtain.
The approach presented in the paper is an application of the tolerance-

averaged model by Woźniak (1999). In this way, Mazur-Śniady (2001) has
formulated equations of the periodic-like beam in the form of a system of
averaged differential equations with slowly varying or (for the periodically
variable beam) constant coefficients which describe the length scale effect. In
contrast, in the classical homogenization theories this effect disappears, cf. for
instanceBensoussan et al. (1978), Jikov et al. (1994), Sanchez-Palencia (1980).
In this paper the vibrations of the beam with periodic structure under

moving load are analyzed. The solution obtained for a singlemoving forcewas
used to the problemof stochastic vibrations caused by randomtrain ofmoving
forces.
Since the theoretical problem of obtaining the analytical solution is very

complicated, the solution reduces to the form, which admits the numerical
analysis by using modern computational equipment (for example the Mathe-
matica package).

2. Periodic-like beam

We consider vibrations under moving load of the periodic-like straight
beam with varying cross-section. The axis of the beam coincides with the
interval [0,L] of the x-axis in 0xyz-space and the beam has the 0xy-plane



Dynamic response of a micro-periodic beam... 325

as the symmetry plane. The equation of the beam has the well known form
(under assumptions of the Euler-Bernoulli linear elastic beam theory)

[B(x)v′′(x,t)]′′+ c(x)v̇(x,t)+ρ(x)v̈(x,t)= p(x,t) (2.1)

where (·)′ = ∂(·)/∂x, ˙(·) = ∂(·)/∂t and
v(x,t) – deflection of the beam axis
B(x) – flexural beam rigidity
c(x) – damping coefficient
ρ(x) – mass density per unit length
p(x,t) – loading process.
The standard methods of analyzing the beam dynamics are effective only

if the coefficients in the equation (1.1) are constant or slowly varying. If the
coefficients B(·), c(·), ρ(·) are rapidly varying functions then the solution to
the equation (1.1) is rather difficult to obtain. We restrict our considerations
to beams for which the rapidly varying functional coefficients B(·), c(·), ρ(·)
are represented by periodic-like functions. It means that there exists a slowly
varying function l = l(x), x ∈ [0,L], max l(x)≪ L, such that in every interval
∆x =

(

x− l(x)/2,x+ l(x)/2
)

, ∆x ∈ [0,L], the functions B(·), c(·), ρ(·) can
be approximated respectively by certain l = l(x)-periodic functions Bx(ξ),
cx(ξ), ρx(ξ), ξ ∈ [x− l/2,x+ l/2]. Moreover, all cross-sectionall dimensions of
the beammust be small compared to max l(x), x ∈ [0,L]. In a special case of
l = const we consider beams with an l-periodic structure.
Functions will be averaged bymeans of the formula

〈ϕ〉=
1
l

x+ l
2
∫

x−
l

2

ϕ(ξ) dξ x ∈ Ω0 Ω0 = {x ∈ Ω : ∆(x)∈ Ω} (2.2)

where l = l(x) and ϕ(·) is an arbitrary integrable function defined on
Ω = (0,L).
If the function ϕ is l-periodic then 〈ϕ〉= const. For ϕ depending also

on the time variable, we shall also write 〈ϕ〉 instead of 〈ϕ〉(x,t).
Theproposedmodel is basedonthephysical assumption that thedeflection

of the l-periodic beam is an l-periodic-like function

v(·, t)∈ PL(l) (2.3)

It means that in every interval ∆x, ∆x ∈ [0,L], the deflection of the beam
can be approximately represented by v(ξ,t)∼= vx(ξ,t), ξ ∈ ∆x, and hence it
be treated as l = l(x) periodic in this interval.



326 K.Mazur-Śniady, P.Śniady

Let us define the averaged deflection w(x,t)

w(x,t) = 〈ρ〉−1(x)〈ρv〉(x,t) x ∈ Ω0 (2.4)

Hence the total deflection of the beam can be represented by a sum

v(x,t)= w(x,t)+d(x,t) x ∈ Ω0 (2.5)

Themodeling decomposition (2.5) is a simple consequence of the assumption
(2.3) andmakes it possible to introduce two kinds of basic unknowns, namely
a function w(·, t) which is a slowly varying function, and d(·, t) which is an
oscillating l-periodic-like function (with the weight ρ).
The deflection disturbance function d(·, t) is assumed to be in the form of

the series
d(x,t)= hA(x)ψA(x,t) x ∈ Ω0 (2.6)

(the summation convention over A = 1,2, ... holds), where hA(·) are the
a priori known oscillating l-periodic-like functions and the new unknown
amplitude of the shape functions ψA(x,t) are sufficiently regular and slowly
varying functions. It was proved by Mazur-Śniady (2001) that

〈ρhA〉=0 (2.7)

Based on concepts of the tolerance-averaged model (Woźniak, 1999), the
system of n+1 differential equations with slowly varying or (for l-periodic
beam) constant coefficients was obtained byMazur-Śniady (2001)

[

〈B〉w′′(x,t)+ 〈B(hA)′′〉ψA(x,t)
]′′

+ 〈c〉ẇ(x,t)+ 〈chA〉ψ̇A(x,t)+

+〈ρ〉ẅ(x,t) = 〈p〉(x,t)
(2.8)

〈B(hB)′′〉w′′(x,t)+ 〈B(hB)′′(hA)′′〉ψA(x,t)+ 〈ch
B〉ẇ(x,t)+

+〈chBhA〉ψ̇A(x,t)+ 〈ρh
BhA〉ψ̈A(x,t)= 〈ph

B〉(x,t)

Equations (2.8) hold for x ∈ (0,L). The boundary conditions are similar to
those formulated in the Euler-Bernoulli beam theory.
For the initial-value problem, suitable initial conditions for ψA and w

should be known.

3. Vibrations of the beam under moving force

Let us consider vibrations of a simply supported beam with periodically
varying cross-section under force Q moving along the beam axis with the



Dynamic response of a micro-periodic beam... 327

velocity u = const. In this case, in equations (2.8) the loading process is

p(x,t)= Qδ(x−ut) (3.1)

where δ(·) is the Dirac function.
For the beamwithperiodic variable cross-section, after taking A = B =1,

ψA(x,t) = ψB(x,t) = ψ(x,t), hA(x) = hB(x) = h(x), we obtain Eqs (2.8) in
the form of the system of two equations with constant coefficients

〈B〉wIV (x,t)+ 〈Bh′′〉ψ′′(x,t)+ 〈c〉ẇ(x,t)+ 〈ch〉ψ̇(x,t)+

+〈ρ〉ẅ(x,t)= 〈p〉(x,t)
(3.2)

〈Bh′′〉w′′(x,t)+ 〈B(h′′)2〉ψ(x,t)+ 〈ch〉ẇ(x,t)+

+〈ch2〉ψ̇(x,t)+ 〈ρh2〉ψ̈(x,t)= 〈ph〉(x,t)

Eqs (3.2) hold for x ∈ (0,L). For the simply supported beam we assume
functions w(x,t) and ψ(x,t) to be in the form of expansion in a sine series

w(x,t) =
∞
∑

k=1

yk(t)sin
kπx

L
ψ(x,t)=

∞
∑

k=1

qk(t)sin
kπx

L
(3.3)

In the orthogonalization process we take into account the equation (3.3) as
well as the following relations

L
∫

0

〈p(x,t)〉sin
kπx

L
dx = Qsin

kπut

L

(3.4)
L
∫

0

〈p(x,t)h(x)〉sin
kπx

L
dx =0

obtaining the set of Eqs (3.2) in the following form

ÿk(t)+
〈c〉

〈ρ〉
ẏk(t)+

〈B〉

〈ρ〉

(kπ

L

)4
yk(t)+

〈ch〉

〈ρ〉
q̇k(t)+

−
〈Bh′′〉

〈ρ〉

(kπ

L

)2
qk(t)=

2Q
L〈ρ〉
sin

kπut

L

(3.5)

q̈k(t)+
〈ch2〉

〈ρh2〉
q̇k(t)+

〈B(h′′)2〉
〈ρh2〉

qk(t)+
〈ch〉

〈ρh2〉
ẏk(t)+

−
〈Bh′′〉

〈ρh2〉

(kπ

L

)2
yk(t)= 0



328 K.Mazur-Śniady, P.Śniady

The initial conditions have the form

yk(0)= 0 ẏk(0)= 0
qk(0)= 0 q̇k(0)= 0

(3.6)

The exact analytical solution to Eqs (3.5) is very complicated and for this
reason it is better to determine the numerical results using the Mathematica
package.
Let us consider the undamped vibrations of the beam with periodically

varying cross-section under moving force (in this case c ≡ 0). We introduce
dimensionless variables

η =
x

L
T =

ut

L
(3.7)

for 0¬ η ¬ 1, 0¬ T ¬ 1.
The set of Eqs (3.2) for the loading process (3.1) after simple transforma-

tions takes the following form

∂4w(η,T)
∂η4

+
〈Bh′′〉L2

〈B〉

∂2ψ(η,T)
∂η2

+
〈ρ〉u2L2

〈B〉

∂2w(η,T)
∂T2

=
QL3

〈B〉
δ(η −T)

(3.8)

〈Bh′′〉
∂2w(η,T)

∂η2
+ 〈B(h′′)2〉L2ψ(η,T)+ 〈ρh2〉u2

∂2ψ

∂T2
=0

The solution to Eqs (3.8) being functions w(η,T) and ψ(η,T), can be
presented as the following sums (Kączkowski, 1963)

w(η,T) = wI(η,T)+wII(η,T)
(3.9)

ψ(η,T) = ψI(η,T)+ψII(η,T)

where wI(η,T), ψI(η,T) describe aperiodic vibrations and wII(η,T),
ψII(η,T) describe free vibrations of the beam. The functions wII(η,T),
ψII(η,T) enable us fulfill the apropriate initial conditions.
For the simply supported beamwe assume functions w(η,T) and ψ(η,T)

to be in the form of expansion in a sine series (similarly to the expressions
(3.3))

w(η,T) =
∞
∑

k=1

yk(T)sinkπη ψ(η,T) =
∞
∑

k=1

qk(T)sinkπη (3.10)



Dynamic response of a micro-periodic beam... 329

Taking intoaccount theabove relations andusingorthogonalization,weobtain
the set of Eqs (3.8) in the following form

〈ρ〉u2L2

〈B〉

d2yk(T)
dT2

+(kπ)4yk(T)−
〈Bh′′〉L2

〈B〉
(kπ)2qk(T)=

2QL3

〈B〉
sinkπT

(3.11)

〈ρh2〉u2
d2qk(T)
dT2

+ 〈B(h′′)2〉L2qk(T)−〈Bh
′′〉(kπ)2yk(T)= 0

Determining the particular integral of the set of Eqs (3.11), we obtain the
solution for the aperiodic vibrations of the beam

wI(η,T) =
2QL3

〈B〉

∞
∑

k=1

sinkπT sinkπη
(kπ)2C

(3.12)

ψI(η,T)=
2QL3〈Bh′′〉
〈B〉

∞
∑

k=1

(kπ)2

〈B(h′′)2〉−〈ρh2〉u2(kπ)2
sinkπT sinkπη
(kπ)2C

where

C =(kπ)2−
〈ρ〉u2L2

〈B〉
+

(kπ)2〈Bh′′〉2

〈B〉[〈ρh2〉u2(kπ)2−2〈B(h′′)2〉]

It is easy to see that the functions wI(η,T) and ψI(η,T) do not satisfy the
initial conditions, and that is why the solution to Eqs (3.11) should contain
additional functions wII(η,T) and ψII(η,T), fulfilling the initial conditions.
We expand these functions into a sine series

wII(η,T) =
∞
∑

k=1

yIIk(T)sinkπη ψII(η,T) =
∞
∑

k=1

qIIk(T)sinkπη

(3.13)
The functions yIIk(T) and qIIk(T) are obtained from the homogeneous set
of Eqs (3.2) together with the initial conditions

yIIk(0)= 0 qIIk(0)= 0

dyIIk
dT

∣

∣

∣

T=0
=
2QL3

〈B〉kπC
(3.14)

dqIIk
dT

∣

∣

∣

T=0
=

2QL3〈Bh′′〉
〈B〉[〈B(h′′)2〉−〈ρh2〉u2(kπ)2]C

It isworth to notice that functions wI(η,T) and ψI(η,T) satisfy the following
relations

∂2wI(η,T)
∂η2

=
∂2wI(η,T)

∂T2
∂2ψI(η,T)

∂η2
=
∂2ψI(η,T)

∂T2
(3.15)



330 K.Mazur-Śniady, P.Śniady

It is reason (cf Śniady, 1976; Mazur and Śniady, 1973), that the functions,
describing the aperiodic vibrations wI(η,T) and ψI(η,T), satisfy not only
the set of the partial differential equations (3.11) but also the following set of
the ordinary differential equations

d4wI(η,T)
dη4

+
〈ρ〉u2L2

〈B〉

d2wI(η,T)
dη2

+
〈Bh′′〉L2

〈B〉

d2ψI(η,T)
dη2

=
QL3

〈B〉
δ(η −T)

(3.16)

〈Bh′′〉
d2wI(η,T)

dη2
+ 〈B(h′′)2〉L2ψI(η,T)+ 〈ρh

2〉u2L2
∂2ψI(η,T)

∂η2
=0

In Eqs (3.16) the variable T is a parameter of the position of the moving
force in the basis of dimensionless variables η, T . The ordinary differential
equations (3.16) enable us obtain the aperiodic vibrations determinated by
functions wI(η,T) and ψI(η,T) to find the closed form solution (Kączkowski,
1963).
Analyzing the vibrations of the beam by means of modern computatio-

nal methods (for example the Mathematica package) is also easier to use the
set of the ordinary differential equations (3.16) instead of the set of partial
differential equations (3.8).
Finally let us find the critical velocities of themoving force.We determine

the lowest critical velocity. In the case of k =1we obtain two critical veloci-
ties. The first one is characteristic of the segment of periodicity of the beam
and is equal to

um
kr
=
1
π

√

〈B(h′′)2〉
〈ρh2〉

(3.17)

the second one is characteristic of the whole beam and fulfills the equation

π2−
〈ρ〉u2

kr
L2

〈B〉
+

π2〈Bh′′〉2

〈B〉[〈ρh2〉u2
kr
π2−〈B(h′′)2〉]

= 0 (3.18)

If the we take under consideration the following relation

π2〈Bh′′〉2

〈ρh2〉u2
kr
π2−〈B(h′′)2〉

≪ 〈ρ〉L2 (3.19)

then we obtain the aproximate value of critical velocity

ukr ∼=
π

L

√

〈B〉

〈ρ〉
(3.20)



Dynamic response of a micro-periodic beam... 331

4. Stochastic vibrations of the beam

Fig. 1. The beam loaded by a random train of forces travelling in the same
direction, all with equal, constant velocities u

Let us consider stochastic vibrations of a beam caused by a random train
of forces travelling in the same direction, all with equal, constant velocities u
(see Fig.1). The forces Qi arrive at the beam at random times ti, and this
constitutes a Poisson process N(t), and dN(t) denotes the number of forces
arrivaling within time intervals (0, t) and (t,t+dt), respectively, and P{·}
denotes the probability of the event and E[·] denotes the expected value of
the quantity in brackets. The properties of the Poisson process are as follows

P{dN(t)= 1}= λdt+o(dt)

P{dN(t)= 0}=1−λdt+o(dt) (4.1)

P{dN(t) > 1}= o(dt)

and consequently

E[dNk(t)] = λdt k =1,2, ... (4.2)

where the parameter λ is the expected arrival rate of moving forces.
The loading process assumed can be presented as follows

p(x,t)=
N(t)
∑

i=1

Qiδ[x−u(t− ti)] (4.3)

The amplitudes Qi are assumed to be random variables that are mutually
independent and independent of the times ti, hence we shall assume the
expected values E[Qr

i
] = E[Qr] = const (r =1,2, ...) to be known.

Let the dynamic influence function H(x,t− ti) denotes the response of
the beam at time t to themoving forces Qi =1, their arrival times being ti.



332 K.Mazur-Śniady, P.Śniady

The dynamic influence function depends on the velocity u and has two
different forms.
If ti ¬ t ¬ ti + L/u (i.e. the force is on the beam), H(x,t − ti) =

H1(x,t − ti), and if t > ti + L/u (i.e. the force has left the beam – free
vibrations), H(x,t− ti)= H2(x,t− ti−L/u).
The influence function H(x,t−ti)= H1(x,t− ti) is equal to the function

v(x,t) found in Section 3 for vibrations of the beamwhen Q =1, and instead
of the t we should introduce the time t− ti. The second part of the influence
function H(x,t− ti) = H2(x,t− ti−L/u) satisfies the homogeneous system
of equations (3.2) (for p(x,t)≡ 0) and the initial conditions for t = ti+L/u
respectively

H2(x,0)= H1
(

x,
L

u

)

Ḣ2(x,0)= Ḣ1
(

x,
L

u

)

(4.4)

The stochastic deflection v(x,t) of the beam is a filtered Poisson process and
can be presented in the form of the Stieltjes integral

v(x,t)=
N(t)
∑

i=1

QH(x,t− ti)=
t
∫

0

H(x,t− τ) dN(τ) =

(4.5)

=
t
∫

t−
L

u

QH1(x,t− τ) dN(τ) =

t−
L

u
∫

0

QH2(x,t− τ) dN(τ)

Taking into account relations (2.5) and (2.6) we obtains the dynamic influence
function in the form (for Q =1)

H(x,t− ti)= w(x,t− ti)+h(x)ψ(x,t− ti) (4.6)

and in view of (4.5) we have

v(x,t) =
t
∫

0

Q(τ)H(x,t− τ) dN(τ)=

(4.7)

=
t
∫

0

Q(τ)[w(x,t− τ)+h(x)ψ(x,t− τ)] dN(τ)

The expected value and variance of the deflection v(x,t) can be obtained by
taking into account equations (4.1) and (4.2). This yields



Dynamic response of a micro-periodic beam... 333

E[v(x,t)] = E[Q]λ
t
∫

0

H(x,t−τ) dN(τ)=

(4.8)

=E[Q]λ
t
∫

0

[w(x,t− τ)+h(x)ψ(x,t− τ)] d(τ)

and the variance

σ2
v
(x)= E[Q2]λ

t
∫

0

[w(x,t− τ)+h(x)ψ(x,t− τ)]2 d(τ) (4.9)

The general, the cumulants of order k have the form

κ(k)
v
(x)= E[Qk]λ

t
∫

0

[w(x,t− τ)+h(x)ψ(x,t− τ)]k d(τ) (4.10)

By analogy to equation (4.9) the variance of the velocity of the beam has the
form

σ2
v̇
(x)= E[Q2]λ

t
∫

0

[dw(x,t− τ)
dt

+h(x)
dψ(x,t− τ)

dt

]2
d(τ) (4.11)

The above formulae for the beam with periodic structure were obtained in a
similar way as for the beam with a constant cross-section, Iwankiewicz and
Śniady (1984), Sieniawska and Śniady (1990).
For the steady-state case (t → ∞) the solutions (4.8), (4.9), (4.11) have

the following forms

E[v(x,∞)] = E[Q]λ
∞
∫

0

[w(x,ξ)+h(x)ψ(x,ψ)] d(ξ)

σ2
v
(x,∞)= E[Q2]λ

∞
∫

0

[w(x,ξ)+h(x)ψ(x,ξ)]2 d(ξ) (4.12)

σ2
v̇
(x,∞)= E[Q2]λ

∞
∫

0

[dw(x,ξ)
dξ

+h(x)
dψ(x,ξ)
dξ

]2
dξ



334 K.Mazur-Śniady, P.Śniady

The deflection v(x,t) is a non-normal process as the filtered Poisson process.
For increasing parameter λ, the process v(x,t) as the sum of many indepen-
dent processes can be approximated by the normal process. For this reason,
for the steady-state case the crossing rate n+(x) of the threshold a can be
given by the Rice formula

n+(a,x)=
1
2π

σv̇(x,∞)
σv(x,∞)

exp
(

−
a2

σ2
v
(x,∞)

)

(4.13)

The reliability of the beam, as the condition of not crossing the threshold a
by the deflection of the beam, can be given by the formula

ps(x,t)= exp[n+(a,t)t] (4.14)

5. Numerical example

As an example, let us consider undamped vibrations of the beam with
l-periodic structure (in this case c(x) ≡ 0). For simplicity we restrict the
numerical analysis only to the influence of these forces which at themoment t
are on the beam (ti ∈ [t−L/u,t]), on the probabilistic characteristics of the
deflection of the beam.
The typical segment of the beam has a piece-wise constant rigidity B(·)

and a mass density ρ(·). For ξ ∈ (−a,a) we have B(ξ) = B1, ρ(ξ) = ρ1, for
ξ ∈ [−l/2,−a] and [a,l/2] we have B(ξ) = B2, ρ(ξ) = ρ2, where B1, B2,
ρ1, ρ2 are constants. For the beam of a periodic structure, the mode shape
function h(·) is l-periodic, hence this function is uniquely determined by the
function h0(ξ), ξ ∈ [−l/2, l/2], where h(x) = h(sl+ ξ) = h0(ξ), s = 1,2, , ...
with x = sl+ξ. Mazur-Śniady (2001) found themode shape functions being
the solution of the eigenproblemwith periodic boundary conditions at x±l/2
together with the corresponding jump conditions.
For the following data: a = l/4, β1/β2 =8, ρ1/ρ2 =2, the first evenmode

shape functions have the form:
— for ξ ∈ (−l/4, l/4)

h1(ξ)= l
2cos
(5.64768

l
ξ
)

−0.094638l2 cosh
(5.64768

l
ξ
)

— for ξ ∈ (l/4, l/2)

h2(ξ)=−1.70403l
2 cos
[7.98703

l

(

ξ −
l

2

)]

−0.20041cosh
[7.98703

l

(

ξ−
l

2

)]



Dynamic response of a micro-periodic beam... 335

— for ξ ∈ (−l/2,−l/4)

h2(ξ)=−h2(−ξ)

For Q = 10N, L = 20m, l = 0.4m, B2 = 8 · 106Nm, ρ2 = 500kg/m,
u = 30m/s and using the Mathematica package, we obtain the solution to
Eqs (3.5) for the general coordinate y1(t) as presented in Fig.2 and q1(t) as
presented in Fig.3. These coordinates describe the run of the beam vibrations
for a single term in the expansion (3.3).

Fig. 2. The graph of the general coordinate y1(t) [m]

For above data, the expected value and the variance of the middle point
of the beam (x = L/2) are equal for stochastic vibrations of the beam caused
by a random train of moving forces

E[v(L/2,∞)] = 3.1 ·10−8E[Q]λ

σ2
v
(L/2,∞) = 1.85 ·10−5E[Q2]λ

For the intensity λ = 0.3s−1 assuming E[Q] = 105N and
E[Q2] = 1.2 E2[Q] = 1.2 · 1010N2, we obtain the values of above expres-
sions equal to

E[v(L/2,∞)] = 0.9 ·10−3 m

σ2
v
(L/2,∞) = 0.666 ·10−5 m2



336 K.Mazur-Śniady, P.Śniady

Fig. 3. The graph of the general coordinate q1(t)[m]

6. Conclusions

In the paper the deterministic and stochastic approach to the problem of
vibrations of a beamwith periodically varying geometry undermoving load is
presented.

This approach is an application of the tolerance-averaged model (Woź-
niak, 1999). In this way,Mazur-Śniady (2001) has formulated equations of the
structured beam in the form of the system of averaged differential equations
with slowly varying (for periodic-like) or constant (for theperiodically variable
beam) coefficients which describe the length scale effect.

For the l-periodic beam we reduce the system of partial differential equ-
ations to the system of differential equations by expansion into the eigen-
functions. The solution of this system was obtained using the Mathematica
package. The solution for a single moving force was adapted to the problem
of stochastic vibrations caused by a random train of moving forces. In this
case we obtain the formulas for the probabilistic characteristics response of
the beam. The presented solutions can be applied in the analysis of dynamics
and reliability of bridges.



Dynamic response of a micro-periodic beam... 337

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338 K.Mazur-Śniady, P.Śniady

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Deterministyczne i stochastyczne drgania belki o okresowo zmiennej

geometrii wywołane ruchomym obciążeniem

Streszczenie

W pracy rozpatruje się drgania belki o okresowo zmiennej geometrii wywołane
działaniem ruchomych obciążeń. Wykorzystuje się model belki o prawie periodycz-
ne strukturze (Mazur-Śniady, 2001), otrzymany metodą uśredniania tolerancyjnego
(Woźniak, 1999).
Podano rozwiązanie zagadnienia drgań belki o okresowo zmiennej sztywności wy-

wołanych poruszającą się ze stałą prędkością siłą skupioną. Powyższe rozwiązanie
wykorzystano wyznaczając probabilistyczne charakterystyki przemieszczeń belki ob-
ciążonej losowym ciągiem ruchomych sił skupionych.

Manuscript received November 3, 2000; accepted for print February 13, 2001