Page 132 - Mechanical Engineers' Handbook (Volume 2)

2 Impedance Concepts 121

replaced with the concepts of static stiffness and static compliance. Consider the idealized
structure in Fig. 2.
To measure the force in member K , an elastic link with a spring constant K is inserted
2
m
in series with K . This link would undergo a deformation proportional to the force in K .If
2
2
the link is very soft in comparison with K , no force can be transmitted to K . On the other
2
1
hand, if the link is very stiff, it does not affect the force in K but it will not provide a very
2
good measure of the force. The measured variable is an effort variable, and in general, when
it is measured, it is altered somewhat. To apply the impedance concept, a flow variable
whose product with the effort variable gives power is selected. Thus,
power
Flow variable
effort variable
Mechanical impedance is then defined as force divided by velocity, or
force
Z
velocity
where force and velocity are dynamic quantities represented by their Fourier transform and
Z is a complex number. This is the equivalent of electrical impedance. However, if the static
mechanical impedance is calculated for the application of a constant force, the impossible
result
force
Z
0
is obtained.
This difficulty is overcome if energy rather than power is used in defining the variable
associated with the measured variable. In that case the static mechanical impedance becomes
the stiffness:
effort
Stiffness S
g
flow dt
In structures,
effort variable
S
g
displacement
When these changes are made, the same formulas used for calculating the error caused by
the loading of an instrument in terms of impedances can be used for structures by inserting
S for Z. Thus

Figure 2 Idealized elastic structure.