Building Acoustics

Multilayer elements 317

(^0)

1

,

2

s

f

π m

= (8.49)

where s is the total stiffness per unit area of the elastic layer. For porous layers the
stiffness may be expressed as

(^) skeleton air dyn^0 ,

E P

ss s

ddV

=+=+

⋅

(8.50)

where sskeleton and sair denotes the stiffness of the solid frame and the stiffness of the
enclosed air in the pores, respectively. As shown by the second expression the former
stiffness is expressed by the dynamic modulus of elasticity and the stiffness due to the
enclosed air is calculated by assuming isothermal motion. The quantity P 0 is the
barometric pressure and d is the thickness of the porous layer of porosity V. If the
thickness of the layer is less than approximately 20 mm, we normally observe that the
last term will be the dominating one.
The standard ISO 9052–1 specifies a method for determining the dynamic stiffness
of materials intended for floating floors. A square specimen of dimension 200 mm is
used, loaded by a given mass to make up a simple mass-spring system. Measuring the
resonance frequency of this system determines the dynamic stiffness. Three principal
arrangements are specified, of which two are sketched in Figure 8.34. In arrangement a)
the loading mass is driven dynamically assuming the base is non-moving and in b) the
base is driven and the differential motion between load and base is measured. A third
possibility resembles the latter arrangement but now the load mass is driven. In all cases,
only movements in the vertical direction are assumed.

F

1

F

a) b)

2

(^3)
Figure 8.34 Test arrangement for determining dynamic stiffness according to ISO 9053–1 (some details are
omitted). 1) Mass loading plate. 2) Test specimen. 3) Base (foundation). a) The mass load being driven, the base
is fixed; b) Differential measurements of loading mass and base.
A complication concerning these arrangements is the necessary limited specimen
area. For open-pore porous materials such as mineral wool, the flow resistivity will be a
determining factor as it will determine whether or not the enclosed air will “escape”
during the measurement. For high flow resistivity, r ≥ 100 kPa⋅s/m^2 , one will directly
measure the total stiffness s. At intermediate values, 10 kPa·s/m^2 < r < 100 kPa·s/m^2 ,
sskeleton is determined by the measurement, and the total stiffness is calculated by Equation