Page 133 - Mechanical Engineers' Handbook (Volume 2)

122 Measurements

q
q im iu (8)
1 S /S gi
go
where q im measured value of effort variable
q undisturbed value of effort variable
iu
S go static output stiffness of measured system
S static stiffness of measuring system
gi
For an elastic force-measuring device such as a load cell S is the spring constant K m .
gi
As an example, consider the problem of measuring the reactive force at the end of a propped
cantilever beam, as in Fig. 3.
According to Eq. (8), the force indicated by the load cell will be
F
F u
m
1 S /S gi
go
3EI
S K m and S go
gi
L 3
The latter is obtained by noting that the deflection at the tip of a tip-loaded cantilever is
given by
PL 3

3EI
The stiffness is the quantity by which the deflection must be multiplied to obtain the force
producing the deflection.
For the cantilever beam
F
F u (9)
m 3
1 3EI/KL
m
or
F F 1
3EI
u
m
KL 3 (10)
m
Clearly, if K 3EI/L , the effect of the load cell on the measurement will be negligible.
3
m
To measure displacement rather than force, the concept of compliance is introduced and
defined as
flow variable
C
g
effort variable dt
Then

Figure 3 Measuring the reactive force at the
tip.