Page 50 - Mechanical Engineers' Handbook (Volume 2)

3 Energy, Power, Impedance 39

tions for these elements; they are the transfer functions of the elements and are expressed
in terms of the Laplace operator s. The familiar F M a, for example, becomes, in power-
variable terms, F M(dv/dt); it transforms as F(s) Msv(s), so
dF
(Z translation mass mass Ms (11)
)
dv mass
Because these involve the Laplace operator s, they can be manipulated algebraically to
derive combined impedances. The reciprocal of the impedance, the admittance, is also useful.
Formally, admittance is defined as
1 d(flow)
Admittance: Y (12)
Z d(effort)

3.3 Combining Impedances and/or Admittances
Elements in series are those for which the flow variable is common to both elements and
the efforts sum. Elements in parallel are those for which the effort variable is common to
both elements and the flows sum. By analogy to electrical resistors, we can deduce that the
impedance sum for series elements and the admittance sum for parallel elements form the
combined impedance or admittance of the elements:
Impedances in series:

Z Z Z total (common flow) (13)
1
2
1 1 1

Y 1 Y 2 Y total
Impedances in parallel:
Y Y Y total (common effort) (14)
2
1
1 1 1

Z Z Z
1 2 total
When applying these relationships to electrical or fluid elements, there is rarely any
confusion about what constitutes series and parallel. In the mobility analogy, however, a pair
of springs connected end to end are in parallel because they experience a common force,
regardless of the topological appearance, whereas springs connected side by side are in series
because they experience a common velocity difference.* For a pair of springs end to end,
the total admittance is
s s s

k total k 1 k 2
so the impedance is
k total k k
1 2
s s(k k )
1
2
For the same springs side by side, the total impedance is

*For many, the appeal of the Firestone analogy is that springs are equivalent to inductors, and there can
be no ambiguity about series and parallel connections. End to end is series.