WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

The above expressions provide instantaneous values
of the power flow. The energy transported during a
single period of excitation T is defined as:
W = ˆPdt (25)
T
This measure of the energy transported by a travelling
wave is considered as the optimization objective in our
second application example. In the case of a standing
wave considered in the first example, W vanishes since
no energy is transported in this case.
The time-dependence of the solution from (6)–(8) is
inserted into (23)–(24) and the integration is performed
to yield the expressions
Wx = π Dℜ i ˜ψx,x + ν ˜ψy,y ˜ψ ∗
x
+ 1 − ν
π Dℜ i ˜ψx,y + ˜ψy,x ˜ψ
2 ∗
y
− πkGhℜ i ˜ψx −˜w,x ˜w ∗
(26)
Wy =
1 − ν
π Dℜ i ˜ψx,y + ˜ψy,x ˜ψ
2 ∗
x
+π Dℜ i ν ˜ψx,x + ˜ψy,y ˜ψ ∗
y
−πkGhℜ i ˜ψy −˜w,y ˜w ∗
(27)
where the asterisk is used to denote the complex
conjugate.
4 Discretized problem
A standard Galerkin FE approach is used to set up a set
of discretized equations based on (9)–(11):
2
−ω M + iωC + K d = f (28)
in which M, C and K are the mass-, damping- and
stiffness matrices, respectively, d is a vector containing
the complex nodal variables and f is a load vector. The
matrices M, C and K are collected in the system matrix
S =−ω 2 M + iωC + K.
M and K are assembled from the local element matrices
in the usual way. These are based on the local
shape functions defined as:
⎧
⎨
⎩
⎫
˜w ⎬
˜ψx
⎭
˜ψy
= Nde ⎡
= ⎣
NT 1
NT 2
NT 3
⎤
⎦ d e
(29)
and from the B matrix defined as:
⎧
⎫
˜ψx,x
⎪⎨ ˜ψy,y ⎪⎬
˜ψx,y + ˜ψy,x = ∂Nd
˜ψx ⎪⎩
−˜w,x ⎪⎭
˜ψy −˜w,y
e = Bd e ⎡
⎢
= ⎢
⎣
A.A. Larsen et al.
N T 2,x
N T 3,y
N T 2,y + NT 3,x
N T 2 − NT 1,x
N T 3 − NT 1,y
⎤
⎥ d
⎥
⎦
e
(30)
In all examples presented in this paper we use rectangular
4-noded bilinear elements with dimensions
2a by 2b.
Damping is not defined in the continuous problem
but included in the discretized version in the form of
proportional (Rayleigh) damping:
C = αM + βK (31)
where α and β is the mass- and stiffness-proportional
damping coefficients.
The load is specified in the examples and the load
vector is assembled in the usual way.
Inserting the discretized displacements into the
formulas for the energy transport we obtain
Wx = πℜ i d eT e e
Qx d ∗
(32)
Wy = πℜ i d eT e e
Qy d ∗
(33)
in which Qe x and Qey are local element matrices defined
in terms of the shape functions and their derivatives as:
Q e x = D(N2,x + νN3,y)N T 1 − ν
2 +
2 D(N2,y + N3,x)N T 3
−kGh(N2 − N1,x)N T 1
(34)
Q e y = D(νN2,x + N3,y)N T 1 − ν
3 +
2 D(N2,y + N3,x)N T 2
−kGh(N3 − N1,y)N T 1
(35)
The energy transport in the x− and y−direction
through a single element is given as:
W e x =
Wxdy, W e y =
Wydx (36)
e
e
By assuming uniform energy transport through an element
we approximate this as
W e x =
πℜ i
e
d eT e e
Qx d ∗
dy
≈ 2bπℜ i d eT
˜Q e e
x d ∗
(37)
W e y =
πℜ i
e
d eT e e
Qy d ∗
dx
≈ 2aπℜ i d eT
˜Q e e
y d ∗
(38)