Building Acoustics

Excitation and response 47

This transmissibility is shown in Figure 2.9 for four values of the loss factor η.
These values are, according to Equation (2.22), chosen equal to twice the damping ratios
ζ used in Figure 2.8. It should be noted that the transmissibility is independent of the loss
factor at the higher frequencies under the condition where η is not unrealistically high,
thereby violating the assumption η << 1 as indicated in Equation (2.20).
In addition to the transmissibility T other quantities are in use to characterize the
usefulness of elastic supporting systems. An appropriate question one may ask is: What
do I gain by using an elastic system compared to mounting my machine directly on to the
foundation? In that case one must of course model the foundation in a more realistic way
than one of infinite stiffness. We shall define the efficiency E of the isolating system by
the ratio^1

foundationwithout isolator

foundationwith isolator

v

E

v

= (2.37)

Later on we shall give examples using this quantity but this requires us to extend our
knowledge to systems having several degrees of freedom.

2.4.4 Response to a complex excitation

Up until now we have assumed that the exciting force could be described by a simple
harmonic time function, i.e. there is only one frequency operating at a time. We have,
however, pointed out that knowing the transfer function we may calculate the response
for any type of excitation. For this to be possible the condition of linearity must be
fulfilled, the principle of superposition must apply. This means that, when expressing the
excitation in a Fourier series or transform, the response will be the sum of the responses
for each component in the excitation. A periodic excitation will give a periodic response
and the system will resonate when a component in the excitation coincides with one of
the natural frequencies (eigenfrequencies) of the system.
We shall give an example again using the simple mass-spring system in Figure 2.6
but we will exchange the viscous damping with the structural or hysteric one. We
furthermore exchange the harmonic time function of the force with a stochastic one. This
implies describing the force excitation using a spectrum, a spectral density function
GF(f), and our task is to calculate the resulting velocity of the mass. We certainly know
that the velocity v(t) will be a stochastic function but what is the resulting RMS-value?
To answer the question we shall use the general Equation (2.4) linking together the
spectral densities (power spectrum) of the input and output quantities. In our system, the
input quantity is the force F with the power spectrum GF(f), the output being the velocity
v with power spectrum Gv(f). The transfer function H(f) is therefore 1/Zm, i.e. the
mobility M of the system. The task is therefore to calculate the integral

2 2

00

vGff MfGfv()d () ()d.

∞∞

 ==⋅∫∫Ff

(2.38)

In this case the mobility will be

(^1) The term insertion loss is also commonly used, indicating the difference in some quantity when inserting a
new member or device into an existing system.