JRC JLN-550 -, Instruction Manual

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APPENDIX 29
ε
'
δ
= 100 (cos
δ
+ tan
θ
sin
δ
– 1) %
or
ε
'
δ
= 100 (cos
δ
– tan
θ
sin
δ
– 1) % (10.5)
Thus, it can be seen that the error due to balanced array is always negative, independently of the
beam radiation angle.
The following calculate an error when an offset of 5
°
degrees has occurred:
ε
δ
= - 0.38 %
ε
'
δ
= + 2.15 or -1.85 %
The balanced array has a smaller error than the single beam method, and is therefore more
advantageous.
(3) Error due to pitching and rolling
When pitching and rolling have occurred, we
get the same result as when
δ
in Eq. 10.4 and
Eq. 10.5 is assigned with the following values:
δ → δ
(t) =
δ
m
sin
ω
t
δ
m
= Maximum deflection angle
ω
= Angular frequency of motion
The average Doppler shift frequency in
balanced array can be expressed by the
following:
f'
d
= f'
d1
–f'
d2
=
1
T
∫
−
2
T
2
T
cos θ cos δ (t) dt
=
4Vf
0
C cos
θ ⋅
ω
2
π
∫
ω
π
ω
π
−
cos ( δ
m
sin ω t) dt
=
4Vf
0
C J
0
( δ
m
) (10.6)
The average error is obtained as follows:
ε δ
m
= 100 {J
0
(
δ
m
) –1} (10.7)
The above-mentioned relationship is shown in Fig. 10.4.
(4) Error due to deviation of radiation angle from reference value
The following equations show, with respect to axes N
1
and N
2
, the Doppler shift frequency is generated the
acoustic wave is radiated when there is a deviation of
∆δ with respect to θ as shown in Fig. 10.5:
F ′
d1
=
2Vf
0
C cos (
θ ± ∆δ )
F ′
d2
=
2Vf
0
C cos (
θ ± ∆δ )
F ″
d
= F ″
d1
– F ″
d2
=
4Vf
0
C cos (
θ ± ∆δ ) (10.8)
ε
d
= 100 {2cos ( θ ± ∆δ ) –1} % (10.9)
Assume that
∆δ ≤
0.1
°
, then we get:
ε
d
≤
0.3 %