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Procedia Engineering 00 (2014) 000–000
www.elsevier.com/locate/procedia
“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology,
APISAT2014
Implication of Dynamic Unbalance to the Inertial Accelerometer
Calibration with Vibrafuge
Xue-Ming Donga, Wei Guanb,*, Xiao-Feng Mengb
a
Changcheng Institute of Metrology and Measurement (CIMM), Huanshan villiage, Wenquan Town, Beijing 100095, China
Science and Technology on Inertial Laboratory of Beihang University (BUAA), No. 37, Xueyuan Road, Beijing 100191, China
b
Abstract
In practical navigation and guidance applications, inertial accelerometers are always subjected to multiple accelerations at the
same time. But inertial accelerometers are usually calibrated in the laboratory with either pure vibration or constant acceleration
according to IEEE standards and ISO standards. In this case, there are inconsistent characteristics of sensors working in the
laboratory and operational conditions because the input accelerations are different. In order to decrease the inconsistency,
calibration of sensors with vibrafuge, a machine which can produce multicomponent inputs, is proposed. But there is a critical
problem for the vibrafuge is the dynamic unbalance due to uneven distribution of mass. In order to build a high-performance
vibrafuge for metrological purpose, we analyze the implication of dynamic unbalance of the vibrafuge to the output acceleration
in this paper. The structure of the vibrafuge is introduced to explain how the dynamic unbalance affects the vibrafuge. Kinetics
analysis is presented to show the implication of the dynamic unbalance. By simulation and examples, we estimate the influence
of the dynamic unbalance.
© 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
Keywords: accelerometer; calibration; centrifuge; dynamic unbalance; vibrafuge
1. Introduction
Conventional calibration methods for inertial accelerometers are based on either single constant acceleration or
vibration. But sensors working in the real world are surrounded by multiple accelerations, including constant
* Corresponding author. Tel.: +086 18710150124; fax: +086-010-82338221.
E-mail address :guanfuzi@aspe.buaa.edu.cn
1877-7058 © 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
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Xue-Ming Dong/ Procedia Engineering 00 (2014) 000–000
acceleration and vibration. In order to improve the application accuracy of accelerometers, we have to test them with
multiple inputs since different inputs of the accelerometers will cause different performances. Hence we proposed to
calibrate the accelerometer with constant and sine acceleration. A review of available equipments which can
produce constant and sine accelerations is presented in [1]. They are rate table with tilted axis of rotation, double
centrifuge, and vibrafuge. Among them, the vibrafuge is a special machine. It is applied to environmental testing at
present [2,3]. It consists of a centrifuge and a vibration exciter, and is able to provide large constant accelerations
and high-frequency sine accelerations (above 20 Hz)[4,5].
Fig. 1. Structure of a vibrafuge
As shown in Fig. 1, the vibration table is mounted on the centrifuge. The centrifuge produces the constant
acceleration while the vibration table produces the sine acceleration. In the calibration, we mount the sensor under
calibration on the vibration table. In this case, the dynamic unbalance of the centrifuge is an inevitable problem in
the investigation because the accelerometer will move along with the vibration table periodically and the distribution
of the mass is unbalance [6]. Such unbalance will impose periodic force and vibration on bearings, and cause
damage to the centrifuge [7]. Moreover, the dynamic unbalance will affect the output of the vibrafuge. In this paper,
we investigate the dynamic unbalance of the vibrafuge, and evaluate its implications.
2. Dynamic unbalance
This section analyzes the dynamic unbalance of the aforementioned vibrafuge. The unbalance mass is moving
along with the vibration table, we can describe the vibration as:
(1)
x  A sin  2 ft   
where A is the vibration amplitude of the vibration table; f is the vibration frequency of the vibration table, and  is
the initial phase of the vibration table. As illustrated in Fig. 2, Or is the intersection of the rotation plane and the
spindle. Om is the effective center of mass of the unbalance mass m. In this context, we define an effective unbalance
mass me by
mR  mR0  me R0

 R  R0  x
m  mx R
0
 e
(2)
where R is the distance from the effective center of mass of the accelerometer to O r; R0 is the distance from the
balance point of the vibration table to Or; The effective unbalance mass induces cause radial force Fr and axial force
Fz. The distance between the Om and Or is R. In order to analyze the influence of the unbalance to the spindle, we
will transfer the additional force and moments on the middle point of the spindle (Omz). According to kinetics
analysis, we write the following expressions with respect to [6]:
 Fr  me R0 2

 Fz  me g
M  h F  R  F
r
z

(3)
Xue-Ming Dong/ Procedia Engineering 00 (2014) 000–000
3
Fig. 2. Additional force and moments caused by the unbalance mass
Fig. 2. Additional force and moments caused by the unbalance mass
where m is the effective unbalance mass, and Or Omz  h . The spindle will have a radial displacement r1 relating
to the Z axis due to the additional force:
r1 
Fr me R0 2

,
kr
kr
(4)
where kr is the radial rigidity, and an angular displacement relating to the Z axis due to the additional moments:
 
M me R0 2 h  me gR0

,
k
k
(5)
where k is the angular rigidity. Because of this angular displacement, another radial displacement of the spindle is:
r2   d  h  tan    ,
(6)
where d is a fixed vertical distance from the effective center of mass of the accelerometer under test to the rotation
plane of the centrifuge. Then the dynamic working radius of the centrifuge is
r  r1  r2 
 m R  2 h  me gR0 
me R0 2
  d  h  tan  e 0



kr
k


(7)
3. Implication of the dynamic unbalance
According to [3], output acceleration of a vibrafuge is composed of the constant acceleration produced by the
centrifuge and the sine acceleration produced by the vibration table. The acceleration component of the input axis of
the accelerometer under test is
a0   2 R  ( 2  4 2 f 2 ) A sin t
(8)
In this context, we analyze the implication of the dynamic unbalance to the output acceleration of the centrifuge.
According to the principle of the centrifuge, that part of output acceleration is:
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Xue-Ming Dong/ Procedia Engineering 00 (2014) 000–000
a  2R
(9)
In the form of relative error, we can write the ratio of acceleration error a to a as
a
 R
2

a

R
(10)
where  is the angular velocity error. In this paper, we take it as zero. And, the radius error
R  r
(11)
Therefore, we can analyze the influence of the dynamic unbalance to the output acceleration based on (1), (7), and
(10) as:
 m R  2 h  me gR0
R  me R0 2

  d  h  tan  e 0

k
R  kr


 mx 2
  2 h  g  

  d  h  tan  mx
 

k

 
 kr
 


 
 R0  x 
(12)
 R0  x 
For typical values,   15 rad s, h  0.5 m, m  1 kg, A  12.5 mm, and k  1107 Nm/rad , we can estimate the
maximum angular displacement mx
 2h  g
k
as 1.53125 107 rad . Hence the angular is small, we have
  2h  g 
 2h  g
tan  mx
  mx

k 
k

(13)
and
R  mx 2
 2h  g 

  d  h  mx

k 
R  kr
 R0  x 
(14)
 k
(15)
For convenience, the following definitions are used:

b  m 2 kr , c  m  d  h   2 h  g
and we have

R0 
R bx  cx
x

 b  c 
  b  c  1 

R
R0  x
R0  x
 R0  x 
(16)
x A
(17)
Referring to (1), we know
and

 b  c  1 


R0  R
R0 
  b  c  1 


R0  A  R
 R0  A 
Hence the relative radius error is bounded by
(18)
Xue-Ming Dong/ Procedia Engineering 00 (2014) 000–000
0
 A 
R
 b  c  

R
 R0  A 
5
(19)
To show the numerical level of the relative radius error, we presented more typical values as follows:
kr  5 106 N m, d  0.1 m, R0  2 m, f  20 Hz,   0 . With respect to (1) and (16), we calculated the relative
radius error by LabVIEW. As shown in Fig. 3, the maximum relative radius error is around 3.21541 107 . When
we calibrate inertial accelerometers with precision as high as 1  10 8 , this relative error caused by the dynamic
unbalance cannot be omitted.
Fig. 3. Simulation results of the relative radius error.
4. Conclusion
To reduce the inconsistency of the sensors working in field and in laboratory, we propose calibrating
accelerometers with multiple accelerations. Vibrafuge, which is used to do environmental tests, is able to be applied
to produce constant and sine vibrations. However, dynamic unbalance is an inevitable problem. In this paper, we
analyzed the implication of dynamic unbalance to the output acceleration of the centrifuge. By simulation, the level
of the relative radius error is given. Analysis and methods presented in this paper can be a reference to the future
study of this topic.
Acknowledgements
The wok was financially supported by the National Natural Science Foundation Innovation Group of China
(Grant No. 61121003), the Changjiang Scholars and Innovative Research Team in University(IRT1203).
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