Advanced Methods of Structural Analysis

14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 541

Table 14.3 Classical boundary conditions

Clamped end.yD0; D0/ Free end.QD0; MD0/

x x

yD 0 I

@y

@x

D 0

@

@x



EI

@^2 y

@x^2



D 0 I EI

@^2 y

@x^2

D 0

Pinned end.yD0; MD0/ Sliding end.QD0; D0/

x

x

yD 0 I EI@

(^2) y
@x^2
D 0
@
@x

EI@
(^2) y
@x^2

D 0 I @y
@x
D 0

14.4.2 Fourier Method...............................................

A solution of differential equation (14.14) may be presented in the form

y.x; t/DX.x/ T .t/; (14.15)

whereX.x/is the space-dependent function (shape function, mode shape function,
eigenfunction);T.t/is the time-dependent function.
The shape functionX.x/and time-dependent functionT.t/depends on the
boundary conditions and initial conditions, respectively. Plugging the form (14.15)
into the (14.14), we get

EIXIV

AX

C



T

T

D0: (14.16)

It means that both terms are equals but have opposite signs. Let


T=TD!^2 ;
then for functionsT.t/andX.x/may be written the following differential equations



TC!^2 TD0: (14.17)

XIV.x/k^4 X.x/D0; (14.18)

wherekD^4

q
m!^2
EI andmDAis mass per unit length of the beam. Thus, in-
stead of (14.14) containing two independent parameters (timetand coordinatex),
we obtained twouncoupledordinary differential equations with respect to un-
known functionsX.x/andT.t/. This procedure is called the separation of variables
method.
The solution of (14.17)isT.t/DA 1 sin!tCB 1 cos!t,where!is frequency
of vibration. This equation shows that displacement of vibrating beam obey to har-
monic law; coefficientsA 1 andB 1 should be determined frominitialconditions.