C.2.1 Outline of determination of peak sensitivity of a one-axis accelerometer
The peak sensitivity of a one-axis accelerometer is defined in the three-dimensional space by the following 1 x 3 matrix. A definition of each element is shown in Table C.1.
(PSzx PSzy PSzz) (C.1)
Table C.1 – Definition of elements in one-axis accelerometer peak sensitivity
Term Definition
PSzx Peak sensitivity defined by Z-axis output induced by X-axis input PSzy Peak sensitivity defined by Z-axis output induced by Y-axis input PSzz Peak sensitivity defined by Z-axis output induced by Z-axis input
2
3
f1 f2 f3 f4 f5 f6 1
IEC 074/11
Key
1 frequency 2 peak sensitivity
3 maximum allowable limit from a straight line
Figure C.1 – Peak sensitivity as a function of each frequency bandwidth from DC to fn
Figure C.1 shows the fundamental concept of the peak sensitivity. The seismic system in an accelerometer has a resonant frequency. Normally, an accelerometer is allowed to be used in the frequency bandwidth below these resonant frequencies for the reason that the sensitivity becomes larger if the frequency component of the impact acceleration includes the frequency that triggers the resonance. The wider the bandwidth of the impact acceleration waveform, the larger the sensitivity becomes. As the horizontal line in Figure C.1 shows, as long as the bandwidth is sufficiently below the minimum resonant frequency, sensitivity remains constant.
The peak sensitivity shall be measured using the input acceleration that does not generate the resonance. Therefore, the peak sensitivity shall always be accompanied by the frequency bandwidth in which the peak sensitivity value is defined and the allowable deviation from the constant is shown, as Figure C.1 illustrates.
PSzx, PSzy and PSzz can be obtained using the following procedures, as shown in Table C.2.
Table C.2 – Peak sensitivity of one-axis accelerometer
Procedure no. Description of each step
1 Target accelerometer is subjected to the acceleration with peak Ap and frequency bandwidth from DC to f1
2 Target accelerometer is subjected to the acceleration with peak Ap and frequency bandwidth from DC to f2
ã
ã
ã
ã
ã
ã
N Target accelerometer is subjected to the acceleration with peak Ap and frequency bandwidth from DC to fn
The procedure for deriving peak sensitivity with the defined frequency bandwidth is as follows:
a) Activate the target accelerometer by the impact acceleration with the constant peak value either both in plus and minus side or one side at each shot to the target accelerometer.
b) Frequency bandwidth should be controlled at each shot. The first shot should cover the bandwidth from DC to f1, the second shot should cover the bandwidth from DC to f2 and the n-th shot should cover the bandwidth from DC to fN.
c) At each n-th shot (n = 1ãããN), repeat the same calculation.
C.2.2 Calculation of peak sensitivity components of a one-axis accelerometer
Three linearly independent sinusoidal acceleration vectors are used for the matrix sensitivity calculation given in Annex A. The same is true for the peak sensitivity calculation. Let three shock accelerations ai,1(t), ai,2(t) and ai,3(t), be defined. The following table shows the definition of the direction cosine between the three shock acceleration vectors and the rectangular co-ordinate system fixed to the target accelerometer.
Table C.3 – Relationship of direction cosine and the co-ordinate system of the target accelerometer
Excitation X-axis Y-axis Z-axis
)
i,1(t
a cosα1 cosβ1 cosγ1
)
i,2(t
a cosα2 cosβ2 cosγ2
)
i,3(t
a cosα3 cosβ3 cosγ3
Since three shock accelerations ai,1(t) , ai,2(t) and ai,3(t) are linearly independent, the following must hold:
1 1 1
2 2 2
3 3 3
cos cos cos
cos cos cos 0
cos cos cos
α β γ
α β γ
α β γ
≠ (C.2)
Reasonable techniques for generating shock accelerations ai,1(t), ai,2(t) and ai,3(t) are to use a three-dimensional vibration generator under a good control for the low acceleration and the elastic pulse reflection for the high acceleration. Assuming the output axis of the target accelerometer is the Z-axis, this leads to the Equations (C.3):
[ ] [ ] [ ] [ ] [ ] [ ]
=
i3 oz3
i2 oz2
i1 oz1
zz zy zx
3 3
3
2 2
2
1 1
1
cos cos
cos
cos cos
cos
cos cos
cos
a L a L
a L a L
a L a L S
S S γ β
α
γ β
α
γ β
α
(C.3)
Equations (C.3) define the components of sensitivity matrix of 1 × 3 as a function of angular frequency, since determinant of the coefficient matrix is not zero. Because the peak sensitivity is defined using the signals in the time-domain, following is derived:
{ }
{ }
1 1
zx 1 i1 zx 1 i1
i+
zx
i1 1 i1 1
L S L a t L S L a t
PS a t a t
ω α ω α
α α
− − ≥
= ≥
max [ ( )cos [ ( )]] : [ ( )cos [ ( )]] 0
max ( )cos : ( )cos 0 (C.4)
{ }
{ }
1 1
zx 1 i1 zx 1 i1
i zx
i1 1 i1 1
L S L a t L S L a t
PS a t a t
ω α ω α
α α
− −
− ≤
= ≤
max [ ( )cos [ ( )]] : [ ( )cos [ ( )]] 0
max ( )cos : ( )cos 0 (C.5)
{ }
{ }
1 1
zy 1 i1 zy 1 i1
i+
zy
i1 1 i1 1
L S L a t L S L a t
PS a t a t
ω β ω β
β β
− − ≥
= ≥
max [ ( )cos [ ( )]] : [ ( )cos [ ( )]] 0
max ( )cos : ( )cos 0 (C.6)
{ }
{ }
1 1
zy 1 i1 zy 1 i1
i zy
i1 1 i1 1
L S L a t L S L a t
PS a t a t
ω β ω β
β β
− −
− ≤
= ≤
max [ ( )cos [ ( )]] : [ ( )cos [ ( )]] 0
max ( )cos : ( )cos 0 (C.7)
{ }
{ }
1 1
zz 1 i1 zz 1 i1
i zz
i1 1 i1 1
max [ ( )cos [ ( )]] : [ ( )cos [ ( )]] 0 max ( )cos : ( )cos 0
L S L a t L S L a t
PS a t a t
ω γ ω γ
γ γ
− −
+ ≥
= ≥ (C.8)
{ }
{ }
L S L a t L S L a t
PS a t a t
ω γ ω γ
γ γ
− −
− ≤
= ≤
1 1
zz 1 i1 zz 1 i1
zzi
i1 1 i1 1
max [ ( )cos [ ( )]] : [ ( )cos [ ( )]] 0
max ( )cos : ( )cos 0 (C.9)
where PSzxi+, PSzxi−, PSzyi+, PSzyi−, PSzzi+ and PSzzi− are the peak sensitivities on the + side by Z-axis output versus X-axis input, peak sensitivity on the − side by Z-axis output versus X-axis input, peak sensitivity on the + side by Z-axis output versus Y-axis input, peak sensitivity on the − side by Z-axis output versus Y-axis input, peak sensitivity on the + side by Z-axis output versus Z-axis input and peak sensitivity on the − side by Z-axis output versus Z-axis input, respectively.
a) In the final stage, plot the peak sensitivity PSzxi+ or PSzxi− as a function of frequency bandwidth. The deviation of peak sensitivity from the straight line exceeds the prescribed allowable limit at the frequency f6 in Figure C.1. The frequency f6 determines the maximum bandwidth available for the peak acceleration measurement using the target accelerometer.
b) The same procedures should be applied to PSzyi+, PSizy−, PSizz+ and PSzzi−.
c) Then change the peak value Ap and make a similar measurement. This step clarifies the maximum peak acceleration within the requested frequency bandwidth. In other words, this measurement will lead to the dynamic linearity of peak sensitivity within the specified frequency bandwidth.
NOTE 1 Accurate evaluation of the peak sensitivity matrix requires the control of the bandwidth of the impact acceleration in terms of the resonant frequency of a target accelerometer. The more important this is, the higher the peak acceleration is. It should be noted that bandwidth of impact acceleration with a high peak is very much dependent on the impulse acceleration duration time.
NOTE 2 It is very important to set a co-ordinate system fixed to every target accelerometer in order to calibrate under this scheme that acceleration is a vector quantity.