Excitation and response of dynamic systems - Building Acoustics - page 52

We shall use the model shown in Figure 2.10 to illustrate the behaviour of the
efficiency E given by Equation (2.43) . We shall let the machine be represented by the
mass m 1 and the foundation by the mass m 2 , spring stiffness k 2 and damping coefficient
c 2 . The isolating element has the corresponding spring stiffness k 1 and damping
coefficient c 1 . The efficiency, given on a logarithmic scale, is shown in Figure 2.13 using
two different values for the mass m 2 . For the solid line curve all component data are the
same as used for calculating the input mobility shown in Figure 2.11 . As seen from
Figure 2.13 we do not obtain any isolation for frequencies less than
approximately 2 times the highest natural frequency, which is around 150 Hz. In fact,
below this frequency the conditions are worse than without the isolator. On the other
hand, the influence of the foundation is negligible.
However, by reducing the mass of the foundation by a factor of 10, as shown by the
dashed curve, the situation is quite different. In fact, we do get E > 1.0 (20⋅lg E > 0) in the
interval between the two resonance frequencies but at the same time the frequency must
be well over 200 Hz to obtain good isolation. ( E will be > 1.0 for frequencies above 220
Hz).
2.5.3 Continuous systems
A model using lumped elements will, however, be useless when wave phenomena start
appearing, which applies when the wavelength of an actual wave type becomes
comparable with the physical dimensions of the elements. We then have to deal with
systems having distributed mass and stiffness, the number of freedoms will in principle
be infinite.
The above does not imply that discrete and continuous systems in principle
represent different types of dynamical system exhibiting dissimilar dynamical
characteristics. It should, as indicated above, merely be regarded as two different
mathematical models for the same physical system. The behaviour is analogous even
though the one is described using ordinary differential equations, the other by partial
differential equations.
Comparing with the general description of discrete systems we now mathematically
express the response of a continuous system to a given excitation by its eigenfunctions
and its associated eigenfrequencies (or natural frequencies). The eigenfunctions are
functions of the space coordinates. Analogous to the eigenvectors of discrete systems,
which describe the natural modes of vibration, the eigenfunctions describe the natural
modal shapes of the continuous system. Furthermore, expressing a complex vibration
pattern with these functions is wholly analogous to the use of Fourier series or transforms
on oscillations in the time domain.
We shall treat these subjects in more detail when dealing with wave and wave
phenomena in Chapter 3 and also further on when dealing with sound transmission in
Chapter 6. At this point we shall give a short overview on calculation and measurement
methods relevant to continuous systems.
2.5.3.1 Measurement and calculation methods
For a number of simple structures having idealized boundary conditions we may find
explicit analytical expressions for the eigenfunctions. As an example, we may for a panel
(wall) assume that it is simply supported (or, alternatively, clamped) along the edges. In
approximation, this could be true but such boundary conditions are always an
idealization. In practice, performing calculations of either transfer functions or vibration