Civil Engineering Reference
In-Depth Information
CHAPTER 2
Excitation and response of dynamic systems
2.1
INTRODUCTION
The main purpose of Chapter 1 was to give a general description of different types of
oscillating motion, how it can be described and how it can be measured when
representing the motion of a real physical system. We also presupposed that when
working with such a system, either in acoustics or vibration, we could convert the actual
physical oscillation variable (force, acceleration, sound pressure etc.) into an electrical
signal to be used in the further processing. The concept of signal and oscillation are in
this connection synonymous.
In general, oscillations in a physical elastic system will arise when dynamic forces
or moments excite the system. A pure acoustic system could be an air-filled enclosure or
a collection of such enclosures; a room inside a building, a reactive silencer (exhaust
silencer for a car), a driver's cabin and so on. Mechanical systems are often associated
with solid structures such as beams, plates or shells. Dealing with building acoustics the
actual system is normally a combined or coupled system containing acoustical and
mechanical elements. This may easily be illustrated using the transmission of sound
energy from your neighbour's TV or hi-fi system. The loudspeakers set up a sound field
in your neighbour's room, which excites the separating wall into mechanical oscillation.
This motion will cause motion in the air next to the wall, thereby setting up a sound field
in the room that in turn will excite our eardrums.
Our interest will therefore be concerned with the coupling between an oscillation
variable describing the excitation or input to the system and the corresponding variable
describing the response or output. In the following we shall use these words alternatively
because it is quite common to talk about the input-output relationship of a system, in
particular when dealing with electric circuits.
Assuming that our physical system is linear and that the physical parameters are
constant, we may always define a
transfer function
, a frequency function giving the
relationship between the input and output variables. Several transfer functions have their
own names; impedance and mobility are important examples. Assuming that the
parameters are constant means that the system properties are independent of time, i.e. the
system is
time
invariant.
Linearity means the
principle of superposition
is valid, which
implies that if the excitation contains several frequency components, expressed by a
Fourier series or transform, the response will be the summed response caused by each
component alone.
The assumption that the parameters are constant in time may often be a reasonable
one, at least when the time span of the measurements is relatively short. The assumption
concerning linearity could be more critical as all physical systems will give a non-linear
response when driven too hard. The transition to non-linearity will normally occur
gradually, which does not lessen the problem. Nevertheless, we may in most cases
