Analysis on influence of static calibration on the axle-group weigh-in-motion system accuracy


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 69 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 69 - 74 

 
 

ANALYSIS ON INFLUENCE OF STATIC CALIBRATION ON THE  

AXLE-GROUP WEIGH-IN-MOTION SYSTEM ACCURACY 
 

Lai Zhengchuang1,2, Yang Xiaoxiang1,3, Yao Jinhui1,2 

 
1 College of Mechanical Engineering and Automation, Fuzhou University, Fuzhou, China, lzcfjjl@126.com 

2 Fujian Key Laboratory of Force Measurement (Fujian Institute of Metrology), Fuzhou, China, lzcfjjl@126.com 
3 Quanzhou Normal University, Quanzhou, China, yangxx@fzu.edu.cn 

 

Abstract: 

The axle-group weigh-in-motion system has two 

functions: static weighing and dynamic weighing. 

According to the weighing model, the accuracy of 

dynamic weighing is affected by the static 

performance. This paper analyses the size of various 

factors affecting the static performance, such as 

sensor tilt installation, platform deformation, 

platform tilt installation, and these errors will lead 

to sensor swing, bearing head tilt, gravity line of 

action and sensor axis direction is not consistent, 

thus affecting the static weighing accuracy. 

However static calibration is the best way to reduce 

or even eliminate the above errors. The dynamic 

truck scale of different manufacturers with or 

without static calibration is used in the test process. 

The results show that the dynamic performance 

index can meet the requirements only after the static 

calibration is used. 

Keywords: Static Calibration; Axle-Group 

weigh-in-motion; Dynamic Weighing; Weighing 

Accuracy 

1. INTRODUCTION 

The Axle-Group weigh-in-motion system consists 

of static truck scale platform and dynamic weighing 

technology, mainly including load carrier (scale 

platform), strain type weighing sensor, junction box 

and weighing instrument [1]. Its quality model can 

be expressed as follows: 

𝑦(𝑡) = 𝑊 + ∑ 𝐴𝑖 sin(𝑊𝑖𝑡 + 𝜗𝑖)𝑖   (1) 

where 𝑊 is the weighing value and 𝐴𝑖, 𝑊𝑖 and 𝜗𝑖 
are the amplitude, frequency and phase of 

interference signal. 𝑊  is the basis of dynamic 
weighing data and determines the basic weighing 

accuracy. Its model is expressed as: 

𝑊 = 𝑘𝑛𝑣,   𝑊𝑠 = 𝑘𝑛𝑣 ∙ (∑𝜔𝑖𝑚𝑖

𝑁

𝑖=1

) (2) 

where 𝑁 is the number of support points, 𝑚𝑖 is the 
load value (kg) allocated to each load cell in the 

static mode and 𝜔𝑖 the weight coefficient allocated 
to each load cell. 𝑛𝑣 is the running speed and 𝑘𝑛𝑣 is 

the quality conversion coefficient between the static 

part and the dynamic part based on the data 

processing algorithm in the 𝑛th running speed. 𝑊𝑠 
is the static weighing data and also the only 

parameter that can control the error in the 

calibration process of the above model, which plays 

an important role in the accuracy of the dynamic 

weighing instrument. Therefore, the static weighing 

performance needs to be guaranteed in the 

calibration. 

According to the full-bridge method of tension 

and compression elastic elements, theoretically the 

output strain value of the sensor is only related to 

the magnitude of the axial force 𝐹𝑛 along the strain 
direction of the elastic body axis, and is not related 

to the tangential force and the additional bending 

moment generated by the tangential force 𝐹𝜏  [2]. 
The relative error of the measurement is: 

𝑒 =
𝐹𝑛 − 𝐹𝑁
𝐹𝑁

= cos(𝛼1 + 𝛼2 + 𝛼3) − 1 

(𝑖 = 1,2,3,4) 

(3) 

where 𝛼1  is the inclination angle of the upper 
indenter, 𝛼2 is the inclination angle of the sensor 
strain elastomer axis and 𝛼3 is the angle between 
the external force direction and the axis of the upper 

indenter. 

2. ANALYSIS OF THE FACTORS 
INFLUENCING THE ACCURACY OF 

STATIC WEIGHING  

2.1. The Impact of the Tilt Installation of the 
Column Load Cell on the Weighing 

Accuracy 

The axle group dynamic truck scale supported by 

four points is selected as the research object. When 

the supported sensors have different angles of 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 70 

rotation, the weighing platform has a horizontal 

frictional force on the sensors, as shown in Figure 1. 

The sensor strains the elastomer to roll purely 

between the upper and lower indenters, taking the 

contact point when the elastomer is not deflected as 

the reference point, and when the elastomer deflects 

by angle 𝛼21 , the horizontal displacement of the 
upper indenter can be expressed as follows. 

𝑆21 = 2𝑅𝛼21 − (2𝑅 − ℎ)sin𝛼21 (4) 

where 𝛼21  is the inclination of the axis of the 
strained elastomer and 𝑆21  is the horizontal 
displacement of the contact point caused by the 

deflection of the elastomer. Meanwhile, there is the 

following geometric relationship: 

𝑂𝐻 = (2𝑅 − ℎ)sin𝛼21 (5) 

𝐻𝐾 = 2𝑅 − (2𝑅 − ℎ)cos𝛼21 (6) 

Assume that the two sensors on the left have 

equal installation angles and the two sensors on the 

right are equal. When the deflection angle of the 

elastomer is small, the rolling friction resistance is 

less than the maximum rolling friction resistance 

couple, and the weighing platform does not produce 

translation, so the friction force of the weighing 

platform against the elastomer is zero. When the 

inclination of the elastomer is greater than the 

critical angle, the weighing platform has a tendency 

to move in translation under the action of the 

restoring force, and the weighing platform generates 

friction against the elastomer. Suppose the initial 

angular displacements during installation are 𝛼01 
and 𝛼02, respectively, and the state after loading is 
shown in Figure 1. 

 

Figure 1: Supporting structure diagram of tilt 

installation of column load cell 

Taking the force direction shown in Figure 1 as 

the positive direction. According to the moment 

balance relationship between the elastomer and the 

support point O, we can obtain: 

𝐹𝑁1(2𝑅 − ℎ)sin𝛼21 − 2𝑀𝑓1 − 𝐹𝑆1[2𝑅 −

(2𝑅 − ℎ)cos𝛼21] = 0  
(7) 

𝐹𝑁2(2𝑅 − ℎ)sin𝛼22 − 2𝑀𝑓2 − 𝐹𝑆2[2𝑅 −

(2𝑅 − ℎ)cos𝛼22] = 0  
(8) 

𝐹𝑆1 = 𝐹𝑆2 (9) 

with 𝐹𝑁1 = 0.25𝑀𝑔 + 0.5𝐹(𝑙 − 𝑎) 𝑙⁄  and 𝐹𝑁2 =
0.25𝑀𝑔 + 0.5𝐹𝑎 𝑙⁄ . When the angle is small, 
cos𝛼21 = cos𝛼22 ≈ 1. 

The deflection of the elastomer causes the 

horizontal displacement of the elastomer and the left 

and right reference points of the weighing platform 

to be equal. 

𝑆21 − 𝑆01 = −(𝑆22 − 𝑆02) (10) 

𝑆01 and 𝑆02 are the horizontal displacement of 
the reference point caused by the sensor deflection 

during initial installation. 

When the initial installation tilt angle is satisfied: 
(𝑅 − 0.5ℎ)sin𝛼01 ≤ 𝛿  and (𝑅 − 0.5ℎ)sin𝛼02 ≤
𝛿 , the torque on the elastomer is not enough to 
overcome the rolling friction couple, and the 

weighing platform does not produce friction. 

When the inclination angle of the sensor on one 

side is greater than the critical angle, the elastomer 

has a rolling tendency, and the weighing platform 

generates horizontal frictional force. The value and 

direction of the rolling friction couple on both sides 

of the elastomer depend on the rolling tendency. 

Suppose that the inclination angle of the right 

elastomer satisfies: (𝑅 − 0.5ℎ)sin𝛼02 > 𝛿 , and 
the load is continuously increased, so that the rolling 

friction of the elastic bodies on both sides reaches 

the maximum rolling frictional couple, that is, 

𝑀𝑓2 = 𝛿𝐹𝑁2. When the torque is greater than the 

maximum rolling friction couple, the elastomer will 

turn to the left until a new equilibrium state is 

formed. At this time, the rolling friction of the 

elastic bodies on both sides is equal to the maximum 

rolling friction couple. From equations (7)-(9), we 

can obtain: 

𝐹𝑆2 =
2𝐹𝑁2[(𝑅 − 0.5ℎ)sin𝛼22 − 𝛿]

ℎ
 (11) 

(𝑅 − 0.5ℎ)(𝐹𝑁1 sin𝛼21 − 𝐹𝑁2 sin𝛼22)+
𝛿(𝐹𝑁2 + 𝐹𝑁1) = 0  

(12) 

(2𝑅 − ℎ)(sin𝛼21 + sin𝛼22 − sin𝛼01 −
sin𝛼02) = 2𝑅(𝛼21 + 𝛼22 − 𝛼01 − 𝛼02)  

(13) 

We can obtain the value of rotation angle of 

elastomer 𝛼21 and 𝛼22 by simultaneous equations 
(12) and (13). 

According to the above inference, we found that 

the rotation angle of the elastomer has nothing to do 

with their respective inclination angles before being 

loaded but is related to the sum value of the 

inclination angles. During the loading process, the 

sum of the inclination angles remains unchanged. 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 71 

When the sensor elastomer generates the above-

mentioned rotation angle, the relative error 

magnitude of the weighing result of each sensor can 

be obtained by equation (3). After loading, the 

inclination of the elastomer will not restore the state 

before loading and will affect the subsequent 

weighing. In addition, the presence of friction also 

reduces the weighing value. 

2.2. Deformation of the Weighing Platform 
Affect on the Weighing Accuracy 

If the rigidity of the weighing platform is small, 

and the external load is large, the weighing platform 

will produce excessive deformation. This 

deformation will cause the angular displacement of 

the elastomer and affect the weighing accuracy, as 

shown in Figure 2. 

 

Figure 2: Schematic diagram of load-bearing 

deformation of weighing platform 

When the weighing platform is loading, the 

distance between the contact points of the elastic 

bodies on both sides changes due to the deformation 

of the weighing platform, and the elastic bodies on 

both sides will deflect in the opposite direction. 

When the rotation reaches equilibrium, the 

contact point of the elastomer reaches the maximum 

rolling friction couple, as shown in Figure 3. 

According to the loading and deformation process 
 

 

Figure 3: Loading model of column load cell when the 

weighing platform is deformed 

of the weighing platform, the direction of the rolling 

friction of the elastomer is the opposite direction 

when the figure is 0 < 𝑎 < 𝑙 2⁄ , and the direction is 
the same as the figure when 𝑙 2⁄ < 𝑎 < 𝑙. 

According to the moment balance relationship of 

the elastomer to the fulcrum O, we can obtain: 

𝐹𝑁1[(2𝑅 − ℎ)sin𝛼21 + 𝑅sin𝛼11]−
2𝑀𝑓1 − 𝐹𝑆1ℎ = 0  

(14) 

𝐹𝑁2[(2𝑅 − ℎ)sin𝛼22 + 𝑅sin𝛼12]−
2𝑀𝑓2 − 𝐹𝑆2ℎ = 0  

(15) 

𝐹𝑆1 = 𝐹𝑆2 (16) 

The bending deformation of the neutral layer of 

the weighing platform reduces the horizontal 

distance of the neutral axis position of the original 

support point cross section by 𝑠1 + 𝑠2. According to 
the displacement relationship of the initial contact 

point, a supplementary equation is obtained. 

2𝑅(𝛼21 + 𝛼22)− (2𝑅 − ℎ)(sin𝛼21 +
sin𝛼22) = (ℎ2 + ℎ3)(sin𝛼11 + sin𝛼12)−
(𝑠1 + 𝑠2)  

(17) 

From equations (14)-(17), the relationship 

between the angle of the elastomer and the 

magnitude and position of the load can be obtained. 

Treat the weighing platform as a simply 

supported beam supported on both sides. Ignore the 

initial deformation of the weighing platform and the 

rolling friction couple of the elastomer.  

{
 
 

 
 𝛿𝑚 =

𝐹𝑎(2𝑎𝑙 − 𝑎2)3 2⁄

9√3𝑙𝐸𝐼
          (0 ≤ 𝑎 ≤

𝑙

2
)

𝛿𝑚 =
𝐹(𝑙 − 𝑎)(𝑙2 − 𝑎2)3 2⁄

9√3𝑙𝐸𝐼
   (
𝑙

2
≤ 𝑎 ≤ 𝑙)

 (18) 

{
 

 𝛼11 =
𝐹(𝑙 − 𝑎)(2𝑎𝑙 − 𝑎2)

6𝑙𝐸𝐼
  (0 ≤ 𝑎 ≤

𝑙

2
)

𝛼11 =
𝐹(𝑙 − 𝑎)(𝑙2 − 𝑎2)

6𝑙𝐸𝐼
      (

𝑙

2
≤ 𝑎 ≤ 𝑙)

 (19) 

{
 

 𝛼12 =
𝐹𝑎(2𝑎𝑙 − 𝑎2)

6𝑙𝐸𝐼
  (0 ≤ 𝑎 ≤

𝑙

2
)

𝛼12 =
𝐹𝑎(𝑙2 − 𝑎2)

6𝑙𝐸𝐼
      (

𝑙

2
≤ 𝑎 ≤ 𝑙)

 (20) 

{
 
 

 
 
𝑥 = 𝑙 − √

𝑙2 − 𝑎2

3
      (0 ≤ 𝑎 ≤

𝑙

2
)

𝑥 = √
𝑙2 − (𝑙 − 𝑎)2

3
   (
𝑙

2
≤ 𝑎 ≤ 𝑙)

 (21) 

𝜃𝑚 = arcsin
𝛿𝑚
𝑥

 (22) 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 72 

𝑠1 = 𝑥(1 − cos𝜃𝑚) (23) 

𝑠2 = (𝑙 − 𝑥)(1 − cos𝜃𝑚) (24) 

where 𝛿𝑚  is the maximum deflection of the 
weighing platform, 𝐸𝐼 is the bending stiffness of the 
weighing platform beam, 𝑙 is the supporting length 
of the weighing platform, 𝑎  is the horizontal 
distance from the load to the left end support point, 

𝑥 is the horizontal distance from the position of 𝛿𝑚 
to the left end support point, 𝑠1  is the left side 
horizontal displacement of the neutral layer at the 

cross section of the supporting point, 𝑠2  is the 
horizontal displacement of the neutral layer at the 

cross section of the right supporting point. 

As can be seen from Figure 4, when the load 

begins to move away from the support point, the 

inclination angle of the upper head increases, and 

decreases slowly when approaching the centre, and 

quickly decreases to zero after passing the centre. 

 

Figure 4: Curve of inclination of upper indenter on left 

elastomer 

 

Figure 5: Curve of left elastomer inclination 

As can be seen from Figure 5, when the load 

begins to move away from the support end, the 

elastomer of the support end deflects to the right by 

a small angle first, and then deflects to the left. The 

extreme point is on the other side of the midpoint, 

and then decrease to zero. At the same time, the 

greater the load, the greater the tilt angle of the 

upper head and the outward rotation angle of the 

sensor elastomer. When the sensor elastomer 

generates the above-mentioned rotation angle, the 

relative error magnitude of the weighing result of 

each sensor can be obtained by equation (3). 

2.3. The Influence of Tilting Installation of 
Weighing Platform on the Weighing 

Accuracy 

When the weighing platform is installed 

obliquely, the gravity of the weighing platform to 

the column load sensor will deviate from the axis 

direction of the strained elastomer. Due to the 

rolling friction between the strained elastomer and 

the upper and lower indenters, the strained 

elastomer will change from a static state to a rolling 

state as the tilt angle of the weighing platform 

increases, as shown in Figure 6. 

 

Figure 6: Weighing platform tilt installation 

When the tilt angle of the weighing platform 

exceeds the critical angle, the moment of the 

weighing platform against the gravity of the sensor 

exceeds the maximum rolling friction of the 

elastomer, and the elastomer produces a tilt angle. 

The tilt angle of the elastomer is approximately 

linearly related to the tilt angle of the weighing 

platform. Under the same tilt angle of the weighing 

platform, the greater the height of the elastomer, the 

smaller the radius of the ball head, and the greater 

the tilt angle of the elastomer. When the sensor 

elastomer produces the above-mentioned rotation 

angle, the relative error magnitude of the sensor 

weighing result can be obtained from equation (3). 

In addition, the elevation difference of the sensor 

installation and the non-linear characteristics of the 

sensor itself have an influence on the result of the 

weighing. 

3. THE CALIBRATION SITUATION OF 
AXLE GROUP WIM SYSTEM 

In section 2, we analysed the main factors 

affecting the weighing accuracy, such as tilt 

installation of the column load cell, stiffness and 

tilting installation of the weighing platform, the 

elevation difference of the sensor installation and 

the non-linear characteristics of the sensor itself. 

These influencing factors will cause the output of 

the WIM system to show obvious nonlinearity, as 

shown in Figure 7. Line 1 is the ideal output of 

WIM, line 2 is the output after static calibration, 

line 3 is the output before static calibration. 

According to equations (1) and (2), 𝑊𝑠 is the static 
weighing value, which affect the dynamic output 

directly. Therefore, it is necessary to carry out static 

calibration to ensure static accuracy, in order to 

ensure the results of dynamic weighing. 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 73 

Real weight

O
ut

pu
t o

f 
W

IM

0

1

2

3

 

Figure 7: Principle of static calibration influence on 

dynamic accuracy 

4. EXPERIMENT 

In order to verify the effect of static calibration 

on the dynamic weighing results. Three different 

manufacturers’ axle group weigh-in-motion 

instruments (factory A, factory B, factory C) were 

selected to be calibrated by means of three methods 

respectively. 

First method: without any static calibration 

method (no calibration). Second method: Only 

using the reference vehicle calibrate the total weight 

of the tested weighing instrument (single-point 

calibration). Third method: The static calibration of 

full scale and full performance are carried out by 

Load Measurement Apparatus according to 

regulations (static full-scale calibration), as shown 

in Figure 8. 

 

Figure 8: Static calibration test 

 

Figure 9: Dynamic weighing test 

After static calibration, we chose 2-axle rigid, 

3-axle rigid, 4-axle articulated trailer and 6-axle 

articulated trailer at a speed of 5 km/h to test the 

dynamic weighing result, as shown in Figure 9. 

none Single-point Static full-scale
-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

 

 

T
o
ta

l 
v
e
h
ic

le
 w

e
ig

h
in

g
 e

rr
o
r

Calibration method

 2-axle truck

 3-axle truck

 4-axle truck

 6-axle truck

 
(a) 

none Single-point Static full-scale

-5.0%

-4.0%

-3.0%

-2.0%

-1.0%

0.0%

1.0%

 

 

T
o
ta

l 
v
e
h
ic

le
 w

e
ig

h
in

g
 e

rr
o
r

Calibration method

 2-axle truck

 3-axle truck

 4-axle truck

 6-axle truck

 
(b) 

none Single-point Static full-scale
-3.0%

-2.0%

-1.0%

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

 

 

T
o

ta
l 
v
e

h
ic

le
 w

e
ig

h
in

g
 e

rr
o

r

Calibration method

 2-axle truck

 3-axle truck

 4-axle truck

 6-axle truck

 
(c) 

Figure 10: Test results of dynamic weighing; (a) test 

result of factory A, (b) test result of factory B, (c) test 

result of factory C 

The testing result of the three manufacturers’ 

axle group weigh-in-motion are shown in Figure 10 

(a), (b) and (c). It can be seen from Figure 10, only 

when the static measurement performance fully 

meets the requirements of medium accuracy truck 

scales, the error of the dynamic part is 

(+0.1 ~ +0.4) %, it can meet the legal error of 

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ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 74 

±0.5 %. If any index of partial load or weighing fails 

in the static measurement performance, the error of 

the dynamic part is (-5.0 ~ +4.2) %, which far 

exceeds the legal maximum allowable error of 

±0.5 %, and cannot meet the level 1 dynamic road 

vehicle requirements for automatic weighing 

instruments. 

5. SUMMARY 

There are many factors that affect the static 

weighing accuracy of the axle group weigh-in-

motion system, including sensor tilt, scale platform 

deformation, scale platform tilt sensor height 

difference and other factors, and the static weighing 

data is the decisive data of the dynamic weighing 

accuracy. When dynamic verification is carried out, 

the static performance calibration must be carried 

out first. The test data show that the accuracy of 

dynamic weighing can meet the error requirements 

of regulations only when all indexes of static 

performance meet the requirements. 

6. REFERENCES 

[1] R. Bajwa, E. Coleri, R. Rajagopal, P. Varaiya, C. 
Flores, “Development of a cost-effective wireless 

vibration weigh-in-motion system to estimate axle 

weights of trucks”, Computer-Aided Civil and 

Infrastructure Engineering, vol. 32, no. 6, 

pp. 443-457, 2017. 

[2] J. Richardson, S. Jones, A. Brown, E. O’Brien, D. 
Hajializadeh, “On the use of bridge weigh-in-

motion for overweight truck enforcement”, 

International Journal of Heavy Vehicle Systems, 

vol. 21, no. 2, pp. 83-104, 2014. 

[3] A. T. Papagiannakis, “Calibration of weigh-in-
motion systems through dynamic vehicle 

simulation”, Journal of Testing and Evaluation, 

vol. 25, no. 2, pp. 197-204, 1997. 

[4] U. Sonmez, N. Sverdlova, R. Tallon, D. J. 
Klinikowski, D. Streit, “Static calibration 

methodology for weigh-in-motion systems”, 

International Journal of Heavy Vehicle Systems, 

vol. 7, no. 2-3, pp. 191-204, 2000. 

[5] P. Lou, H. Nassif, D. Su, P. Truban, “Effect of 
overweight trucks on bridge deck deterioration 

based on weigh-in-motion data”, Transportation 

Research Record, vol. 2592, pp. 86-97, 2016. 

 

 

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