Excitation and response 37
of transfer functions we shall use and many of these functions have special names. In this
chapter, we shall draw on examples from mechanical systems. The analogous acoustic
quantities will be introduced in the following chapter.
For mechanical systems we shall define the following quantities; see also ISO
2041:
Mechanical impedance, the complex ratio of force F, applied to a point in the system,
and the resulting velocity v:
(^) mech
Ns
m
F
Z
v⎛⋅
= ⎜
⎝⎠
⎞
⎟ (2.9)
This definition presupposes simple harmonic motion. If this is not the case, the variables
must be interpreted as functions of frequency, i.e. Fourier transforms. The force and
velocity may be taken at the same or different points. One normally distinguishes
between these two cases by applying different names. In the former case the name point
impedance, or more precisely driving point impedance, is used whereas one uses the
name transfer impedance in the latter case. Do note that the impedance is not the transfer
function when interpreting the force as the excitation. In that case the inverse quantity
represents the transfer function, which is called
Mechanical mobility and thus
(^) mech
m
Ns
v
M
F
⎛
= ⎜
⎞
⋅ ⎟
⎝⎠
(2.10)
The quantity is in some cases referred to as admittance. We shall also define the
Transmissibility, which is the non-dimensional ratio of the response amplitude of a
system in steady-state forced motion to the excitation amplitude. The ratio may be one of
forces, displacements, velocities or accelerations.
Analogous quantities are used for mechanical moments, where one finds moment
impedance and moment mobility. Then the force and velocity are replaced by moment
and angular velocity. It should also be mentioned that one may also find data represented
as apparent mass, i.e. the velocity v in Equation (2.9) is replaced by the acceleration a.
2.3.3.1 Driving point impedance and mobility
The fundamental physical characteristics of mechanical systems are mass, stiffness and
damping. We shall in the first place assume that our masses, springs and dampers are
ideal concentrated (or lumped) elements. When driving these elements using a harmonic
force F⋅exp(jω t) the resulting velocity v will also be harmonic and we may create the
Table 2.1 below.
The fundamental and important difference between the impedances, maybe looking
at the impedance as resistance against movement, increase with frequency for a mass
whereas decrease with a spring. For a viscous damper the damping is proportional to
velocity, thereby making the impedance frequency independent. The damping in elastic
media, e.g. metal, rubber and plastics, is however better characterized as hysteretic. This
kind of damping is proportional to the displacement and described, as shown later on, by
using complex spring stiffness, for materials using a complex modulus of elasticity.