300 Building acoustics
3
11high (^13216)
Eh
BBBB≈+=⋅≈. (8.31)
Between these extremes the bending stiffness of the element will be determined by the
properties of the core material. To find the effective bending stiffness we shall start with
a differential equation for a sandwich element developed by Mead (1972). We shall
assume that the element lies in the xz-plane having displacement ξ in the y-direction. The
equation describing the free vibrations may be written
DgDBm mgys^64 ξξ− + − +( ys ) ω^22 ξωX^2 (1− =^2 )ξ0, (8.32)
if we assume harmonic movements at an angular frequency ω. The quantity m is the
total mass per unit area; Dys and B are the total bending stiffness of the face sheets and
the maximum bending stiffness of the complete element, respectively. The quantity g is a
term that contains the bending stiffness of the core. These quantities are given by
()
33 2
11 3 3 1 3 11 3 3
ys 2 2
11 3 311 2 2 3 3
21133,,
12 1^22
11
,.xzEh Eh h h EhEh
DBh
Eh EhGG
mh h h g
hEhEhX
UU U
+ ⎛⎞⎧ ⎫
==⎜⎟++⎨ ⎬
− ⎝⎠⎩⎭+
⎧⎫
=+ + = ⎨⎬+
⎩⎭
(8.33)
As seen from these equations, we may have different shear stiffness G in the x- and z-
direction but for simplicity, we shall assume that these are equal. Expressing the shear
stiffness of the core in the usual way by the modulus of elasticity and Poisson’s ratio, we
get
2 2
2
.
2(1 )
E
G
X
=
+
(8.34)
Assuming a solution of Equation (8.32) of the form
ξξ= 0 e−jkxB, (8.35)
we arrive at the following sixth order equation for the bending wave number:
(^) ()
22
642
BBB
ys ys ys
11
Bm mg
kg k k
DD Dωω
X⎛⎞⎛⎞⎛⎞ 2
+ ⎜⎟⎜⎟⎜⎟+⋅− ⋅− ⋅−=0.
⎜⎟⎜⎟⎜⎟
⎝⎠⎝⎠⎝⎠
(8.36)
It should be mentioned that Ferguson (1986) uses this equation to arrive at an explicit
expression for the critical frequency of the element. As mentioned above, we shall
primarily use it to demonstrate the frequency dependence of the bending stiffness and,
second, show how this affects the bending phase speed.
The polynomial Equation (8.36) may easily be solved numerically and we may then
find the effective bending stiffness Beff and the corresponding phase speed cB from the
following equations: