Introduction to Acoustics

922 Part G Structural Acoustics and Noise
Part G 22.4
which yields the equation of the eigenfrequencies
(Fig. 22.11)
cos ωL ωL
cosh = 1 . (22.162)
c c
The numerical resolution of (22.162) shows that the
eigenfrequencies fn (in Hz) are
fn ≈
�
EI π
ρS 8L 2
�
3.011 2 , 5 2 , 7 2 ,... ,(2n + 1) 2�
.
(22.163)
The eigenfrequencies are not harmonically related so
that the impulse response of the beam is not periodic.
Beam with a Variable Cross Section
As another example, we consider the flexural vibrations
for a beam with variable cross section. Here, we describe
the Galerkin method [22.12]. It has the advantage
that it remains valid even in the case of nonconservative
systems. In this method, the eigenfunctions Φ(x)of
(22.154) are approximated by a finite sum of p terms
Φ (p) p�
(x) = a jφ j(x) , (22.164)
j=1
where φ j(x) are arbitrary functions that satisfy the
boundary conditions at both ends of the beam. Inserting
(22.164) in(22.154), and defining λ (p) = (ω2 ) (p) as
the approximate eigenvalues of order p, one can define
the Galerkin residue
�
R Φ (p) �
(x) = d2
dx2 �
EI(x) d2Φ (p)
dx2 �
− λ (p) ρ(x)S(x)Φ (p)
following (22.164), this can be written as
�
R Φ (p) � p�
(x) = a j ×
j=1
(22.165)
�
d2 dx2 �
EI(x) d2φ j(x)
dx2 �
− λ (p) �
ρ(x)S(x)φ j(x) .
(22.166)
The weak formulation of the problem is given by
�L
p�
�
d2 φi(x) a j
dx
0
j=1
2
�
EI(x) d2φ j(x)
dx2 �
− λ (p) �
ρ(x)S(x)φ j(x) dx = 0 . (22.167)
The goal of the method is to find the coefficients a j for
which the residue R � Φ (p) (x) � is equal to zero. Thus
(22.167) becomes
p�
kija j − λ (p)
p�
mija j = 0
j=1
kij =
0
j=1
for i = 1, 2,...p , (22.168)
where the mass and stiffness coefficients are
�L
φi(x) d2
dx2 �
EI(x) d2φ j(x)
dx2 �
dx (22.169)
and
�
L
mij =
0
φi(x)ρ(x)S(x)φ j(x) dx . (22.170)
Equation (22.168) can be written equivalently in matrix
form:
[K − λM] a = 0 , (22.171)
where a is a vector with dimension p and K and M
are matrices with dimension p × p. The problem to be
solved is therefore equivalent in form to the eigenvalue
problem for a p-component discrete system (22.49).
Prestressed Beam or Stiff String
The flexural motion for a beam subjected to tension
(or prestressed beam) is governed by the same equation
as a string with stiffness. For a homogeneous and
isotropic beam of constant section subjected to a uniform
tension T,wehave
ρS ∂2v ∂t2 = T ∂2v ∂x2 − EI∂4 v
. (22.172)
∂x4 For a propagating wave of the form y(x, t) = ei(ωt−kx) ,
we obtain the dispersion relationship
ω 2 = k 2 c 2
�
1 + EI
T k2
�
. (22.173)
For a stiff string of length L, the usual order of magnitude
generally implies ε = EI
TL2 ≪ 1sothat(22.173)
can be written:
ω 2 = k 2 c 2� 1 + εk 2 L 2� . (22.174)
For a stiff string with hinged ends the boundary
conditions impose
sin kL = 0 sothat kn L = nπ (22.175)