Page 134 - Mechanical Engineers' Handbook (Volume 2)

3 Error Analysis 123

q
q u (11)
m
1 C /C gi
go
If displacements in an elastic structure are considered, the compliance becomes the
reciprocal of stiffness, or the quantity by which the force must be multiplied to obtain the
displacement caused by the force. The cantilever beam in Fig. 4 again provides a simple
illustrative example.
If the deflection at the tip of this cantilever is to be measured using a dial gage with a
spring constant K ,
m
1 L 3
C and C
gi go
K m 3EI
Thus
1 KL 3
m
u
m
3EI (12)
Not all interactions between a system and a measuring device lend themselves to this
type of analysis. A pitot tube, for example, inserted into a flow field distorts the flow field
but does not extract energy from the field. Impedance concepts cannot be used to determine
how the flow field will be affected.
There are also applications in which it is not desirable for a force-measuring system to
have the highest possible stiffness. A subsoil pressure gage is an example. Such a gage, if
it is much stiffer than the surrounding soil, will take a disproportionate share of the total
load and will consequently indicate a higher pressure than would have existed in the soil if
the gage had not been there.

3 ERROR ANALYSIS
It may be accepted as axiomatic that there will always be errors in measured values. Thus
if a quantity X is measured, the correct value q, and X will differ by some amount e. Hence
(q X) e

or
q X e (13)
It is essential, therefore, in all measurement work that a realistic estimate of e be made.
Without such an estimate the measurement of X is of no value. There are two ways of

Figure 4 Measuring the tip deflection.