Continuous measurement of tire deformation using long-gauge strain sensors

Mechanical Systems and Signal Processing 142 (2020) 106782

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Continuous measurement of tire deformation using long-gauge strain sensors Wenju Zhao b, Cheng Zhang b, Jian Zhang a,b,⇑ a b

Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, 210096 Nanjing, China School of Civil Engineering, Southeast University, 210096 Nanjing, China

a r t i c l e

i n f o

Article history: Received 14 December 2019 Accepted 2 March 2020

Keywords: Intelligent tire Tire deformation measurement The long-gauge Fiber Bragg Grating sensor Modified conjugate beam method

a b s t r a c t Measurement of tire deformation is a hot spot in the research of intelligent tire, which not only provides deep insights into the mechanism of generating tire forces and moments, but also has a significant influence on assessing the health condition of bridges. This paper proposed a tire deformation measurement method by monitoring tire hoop strain with long-gauge Fiber Bragg Grating (FBG) sensors, which can achieve real-time continuous measurement of the tire deformation by the modified conjugate beam method. Taking advantage of the FBG sensor’s high sensitivity, high durability and macro–micro, the modified conjugate beam method transforms the problem of solving the tire deformation into a more convenient solution to the bending moment of the conjugate structure. Intelligent bridge impact vehicle experiments and high-speed experiment were carried out to validate the proposed method, from which the estimated tire deformations under different wheel loads and tire pressures show good agreements with reference measurements. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction Measurement of tire deformation not only helps to gain insight into the physics of tire-road interactions, such as friction, water skiing and rolling resistance, but also is important to vehicle safety, handling and fuel consumption [1–3]. Furthermore, when it comes to bridge structure health monitoring, real-time rapid structural dynamic analysis and damage identification based on vehicle-bridge coupling force need to be solved urgently. Currently, some researchers had carried out plenty of methods of the rapid test and bridge diagnosis based on the vehicle-bridge coupling theory [4,5]. However, they mainly focused on the accuracy of the vehicle-bridge coupling model and the analysis of the bridge response. Zhang et al. proposed an effective means for damage detection of bridges using the contact-point response of a moving test vehicle [6]. Taking a moving vehicle as a continuous exciter, Zhou and Zhang proposed a structural flexibility identification theory using the continuous wheel forces and the acceleration response of the bridge [7]. Currently, researches on the measurement of vehicle-bridge-interaction force can be broadly categorized into measurements that use vehicle body response (the rotations and displacements) and measurements that only use tire responses, such as the hub’s deformations, tire pressure and tire strains [8–10]. Therefore, as an important information of the only link between vehicle and bridge, tire deformation can not only be used to identify the vehicle-bridge-interaction force, but also can be used in frequency domain to identify the damage of bridge.

⇑ Corresponding author. E-mail address: [email protected] (J. Zhang). https://doi.org/10.1016/j.ymssp.2020.106782 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

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Currently, the research on identification of tire vertical deformation mainly from the two aspects of tire models and tire deformation measurement technologies. Various tire models which were used to measure the tire deformation had been studied in recent years. The early physical tire models were mainly string models, beam models, brush models, and combinations or variants of the above three models. Although these physical models can help to qualitatively understand the mechanism of the six-component force of the tire, it was difficult to be practically applied in the field of vehicle dynamics simulation due to the lack of quantitative description accuracy. As the most commonly used empirical model, the Magic Formula (MF) tire model was a set of trigonometric function formulas which were used to fit the steady-state tire sixcomponent force values [11]. It had a high sensitivity and had been integrated with the vehicle dynamics simulation tools. However, the disadvantage of it was that too many fitting parameters need to be estimated and the highly cost of parameter fitting. Therefore, a structured model that can not only reflect the tire structural characteristics, but also better balance the complexity and calculation accuracy was a popular trend. The representative examples were the Ftire model [12], the SWIFT tire model [13], and the Rmod-K model [14]. The theoretical basis of these models can be considered as the ring model, that is, the tire was simplified to the ring on the elastic basis, the elastic foundation represents the sidewall stiffness, and the ring represents the belt of the tire, which was suitable for the radial tire. Various tire sensors had been developed to measure tire deformation on specific parts of the tire. Tuononen et al. developed an optical test system to measure tire deformation, which based on the relative position of the laser sensor located on the tire lining and the two-dimensional position detector mounted on the rim [15]. Matsuzaki et al. monitored tire deformation by mounting the camera on the tire rim [16]. With the superiority of high sensitivity and none power supply, Eun et al. used a passive surface acoustic wave strain sensor (SAW) to measure the tire tread deformation, but it is sensitive to temperature, humidity and pressure [17]. Rectifying the toroidal shape of the tire, and assuming the tire tread has the characteristics of an infinite of the sidewalls, Roveri et al. measured the tire deformation by using the Fiber Bragg Gratings (FBG) sensors and light spectrum analyzer [18]. Yi et al. made a small model vehicle for experimental research, in which the polyvinylidene fluoride piezoelectric film material was glued to the inner surface of the tire to measure the deformation of the tire’s tread and sidewall [19]. It is noteworthy to mention that the aforementioned methods in the literatures mainly were point-type measurements, that is, only a sensor was mounted on the inner or outer side of the tire, and only specific parts of the tire were measured under rotating. Specially, the experimental setups were very complicated and expensive for continuous real-time measurement of the tire deformation based on the above methods. The long-gauge FBG sensor is a sensor with long gauge length which refers to the effective length that the sensor can measure. Compared with a point-type sensor with a short-length gauging, the gauge length of long-gauge sensors is generally designed between 10 and 300 cm, which can not only reflect the change in the selected parameter’s physical quantity in a certain region of the measured structures, but also in characteristic scaled range [20]. Moreover, the long-gauge sensors can be connected in series to form an FBG sensor array for area sensing. The above features enable the developed sensor to have the advantage of measuring both local and global information of the structure. Therefore, it provided an excellent opportunity for developing the strain modal identification theory, such as identifying strain flexibility [21]. Currently, it had been widely applied in large bridges and tunnel structures, especially concrete structures [22,23]. Deflection can be directly obtained from strain based on integration methods. However, the integration method was sensitive to the environmental and other sources of noise. The classical conjugate beam method was first presented by Mohr in 1860. Substantially, in order to determine a beam’s deformation, the same amount of computation as the moment-area theorems are needed. However, when it comes to cases with support settlement action or temperature variation, this method is no longer applicable. With the development of long-gauge fiber optic sensing technology, an improved conjugated beam method for straight beam had been developed to calculate the deflection deformation [24]. Moreover, this method had been used successfully not only in simple structures, including simply supported beams and continuous beams under a pure bending moment, but also in Timoshenko beams with shear deformation [25]. To overcome these above limitations, this paper proposed a method of continuously measuring the tire deformation using long-gauge strain sensors. The structure of this paper is organized as follows: In Section 2, the framework of the proposed method is introduced. Moreover, the long-gauge FBG sensing and the theories of the modified conjugate beam method for curved beam are reviewed. Section 3 describes tire static experiments to verify this proposed method; it displays the experimental results and discusses the potential of the proposed method in real conditions. In Section 4, a tire high-speed experiment was preformed to validate the effectiveness of the proposed method at high velocities. Section 5 discusses the optimal sensor placement of the proposed method to pursue the maximum cost performance for the practical engineering applications. Finally, the conclusions drawn from the results of the study are summarized in the conclusion section.

2. Theory 2.1. Framework The framework of continuous measurement of tire deformation using long-gauge FBG strain sensors is shown in Fig. 1. Firstly, monitoring the tire strains in the circumferential direction using the long-gauge FBG sensors. Then, based on the modified conjugate beam method for curved beam, the problem of solving the tire vertical deformation can be transformed into a bending moment problem of solving the conjugate structure. Its theoretical foundation includes two parts: the first

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W. Zhao et al. / Mechanical Systems and Signal Processing 142 (2020) 106782

Fig. 1. Framework of the proposed method.

part is the simplification of the tire model. Simplify the tire into a SWIFT tire model for analysis, which can translate tire into a rigid ring on a flexible basis. The carcass belt of the tire is connected to the rigid body rim by longitudinal, vertical and circumferential spring damping systems that characterize the sidewall and compressed air characteristics. The tire deformation is mainly manifested by the deformation of the rigid ring under external force. Hence, the deformation of the tip of the rigid ring is very small. Therefore, the spring damping system can be simplified to an external force acting on the rigid ring, and the tire model can be simplified as a rigid ring structure with a fixed end at the top under random loading. The second part, which is the core of the proposed method, is that the conventional conjugate beam method was modified to be suitable for curved beam based on the SWIFT tire model. As shown in Fig. 1, assuming that a virtual beam is equal to the original beam, the absolute values of curvature distribution at any point along the original beam are equal to the absolute values of uniform load distribution in the conjugated beam. Thus, the deformation of the original beam is equal to the moment of the conjugated beam in the same section. Therefore, the problem of solving the tire vertical deformation can be transformed into a bending moment problem of solving the conjugate structure.

(a) Fixing end Fiber sheath

FBG

Connector

Gage length

Plastic tube

Fixing end

Connector

(b) Rotation

Deformation

Rotation Deformation

(c)

Fig. 2. Packaged long-gauge FBG sensor: (a) schematic picture; (b) the concept; (c) actual sensor.

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2.2. Long-gauge FBG sensing technology This section focuses on the principle and experimental apparatus of long-gauge FBG sensing. To overcome the limitation of the traditional point-type strain gauges, a kind of long-gauge FBG sensor as shown in Fig. 2 had been developed [20,21]. The essential component of this sensor is the handling of an embedded tube, inside which bare optic fiber with fiber-optic sensor (FOS) is sleeved and fixed at two ends. The recoated FOS must be pretension enough to measure the strain caused by pressure. Using composite materials to connect the recoated bare fiber makes the FBG sensor more durable to extreme environments and convenient to install. Moreover, since the FBG sensor is well protected, it can extend the sensor’s service life. Therefore, compared to traditional strain gauges and bare fibers, the unique advantages of long-gauge FBG sensors make it more suitable for monitoring tires. In particular, it is more convenient and easier to install on the outer surface of the tire. As shown in Fig. 2(b), the strain output from the long-gauge sensor has a direct relation with rotational angle, which is



expressed as follows: em ¼ h hi
hj =L, where em refers to long-gauge strain, hi and hj refer to the rotational angles at two ends of the long-gauge sensor respectively, h refers to the distance from the sensor location to the neutral axis of the beam section, L refers to the gauge length. There is a direct relationship among output values of strain, displacement, angular, load, and other parameters for the area where the regionally distributed sensing is selected for monitoring. The actual long-gauge FBG sensor is shown in Fig. 2(c). 2.3. Theoretical derivation of the modified conjugate beam method In this section, the SWIFT tire model was analyzed, which is that the carcass belt of the tire is connected to the rigid body rim by longitudinal, vertical and circumferential spring damping systems that characterize the sidewall and compressed air characteristics. The tire deformation is mainly manifested by the deformation of the rigid ring under external force. It is noteworthy stressing the fact that the deformation of the top end is very small compared to that of the contact patch, which means the deformation of the top end can be ignored. Therefore, the tire can be furtherly simplified as a rigid ring structure with a fixed end at the top. 2.3.1. Extraction of bending strain As illustrated in Fig. 3(a), choosing a micro-segment of the tire, the arc length is ds, the central angle is dh, the section height is h and the radius of curvature of the center of the section is r. Thus, establish force balance relationship of the tire micro-segment and omit high-order traces,


dN
Qdh þ qs ds ¼ 0

ð1Þ


Ndh þ dQ þ qr ds ¼ 0

ð2Þ

dM
Qds ¼ 0

ð3Þ

where qr and qs are the average wiring load and the tangential average wiring load acting on the outside of the microsegment. N, Q and M are the axial forces, shear forces and bending moments acting on the section, respectively. Substituting Eq. (3) into Eqs. (1) and (2) and assuming qs ¼ 0, the above equations can be written as:

dN þ

dM ¼0 r

ð4Þ

s r

B

s M+dM

M N

N+dN Q r

dθ

(a)

Q+dQ

B’

A’

r

B’’

A B’’’

A’’

dθ

(b)

Fig. 3. The micro-segment equilibrium state in cylindrical coordinates: (a) internal force; (b) geometric.

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2




N d M þ 2 þ qr ¼ 0 r ds

ð5Þ

From Eq. (4), the axial force N and the bending moment M can be calculated as:

N¼


M þC r

ð6Þ

where C is constant in the tangential direction, and its value is determined by the boundary condition. Substituting Eq. (6) into Eq. (5), we can obtain:

  2 d M M C ¼0 þ 2 þ qr
2 ds r r

ð7Þ

The axial force N can be divided into two part: N 1 ¼
M=r result from the bending moment M and N 2 ¼ C is generated by a circular uniform load pointing to the center of the section. Therefore, the strain of the micro-segment can be divided into three parts:

e ¼ eM þ eN1 þ eN2

ð8Þ

where eM is the bending strain component which is caused by M, eN1 is the axial strain component which is caused by N 1 and eN2 is axial strain component which is caused by N2 . In accordance with the principles of elastic mechanics of materials, N 1 and N 2 can be rewritten as:

R rN 1 RE N1 ¼
dA ¼
AeN1 
EAeN1 rþh=2

ð9Þ

M ¼ EIeM rþh=2 =ð
h=2Þ

where E is elastic modulus, A is the area of the section, I is moment of second moment of area, eN1 rþh=2 and eM rþh=2 are the axial strain and the bending strain of the upper end of the section, respectively. From Eq. (9), we can obtain:

eN1 rþh=2 

h M e rþh=2 6r

ð10Þ

From Eq. (10), it can be seen that the axial strain eN1 is very small when compared to the bending strain eM , so it can be ignored. Then only the strains caused by N 2 and M are considered in the calculation, Eq. (8) can be simplified as:

e ¼ eN2 þ eM

ð11Þ

Because of the fact that N 2 is equal along the hoop, so e does not change along the hoop. And consider of the tire structure is a closed structure, so the total area of the bending moment envelope equal to 0. Therefore, we can obtain the following equation. N2

n X

el ¼ nleN2

ð12Þ

i¼1

where n is the number of units. Analyze the left and right sides of Eq. (12), the axial strain component eN2 can be defined as follows.

eN2 ¼

n 1X e n i¼1

ð13Þ

In order to eliminate the axial strain, substituting Eq. (13) into Eq. (11), the bending strain is defined as follows.

eM ¼ e


n 1X e n i¼1

ð14Þ

2.3.2. Calculation of the deformation For the micro-segment of the tire, as shown in Fig. 3(b), AB and A00 B000 represent the central axis before and after deformation of the segment shown in Fig. 3(a), respectively. The overall deformation can be divided into three parts: uM is the tangential deformation which represents AB to A0 B0 , v M is radial deformation which represents A0 B0 to A00 B00 and wM is bending deformation which represents A00 B00 to A00 B000 . The tangential deformation is very small, so it can be ignored. When only the radial deformation is taking into consideration, the deformation geometrical relationship can be defined as,


 v M ds Dds ¼ rdh
r
v M dh ¼ v M dh ¼ r

ð15Þ

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where ds is the length of A0 B0 , Dds is the deformation from A0 B0 to A00 B00 . Assuming that the micro segment is deformed, the curvature k around the center can be calculated from the angular increment of the unit arc length.

1 dh þ k¼ 0¼ r Considering



@v M @s

 2 M vM þ @@sv2 ds
@@s

ds
Dds

¼

1 r

þ vr2 þ ddsv2 þ vr
M 2 1
vr M

2 M

M

d2 v M ds2

ð16Þ

v M =r is very small, Eq. (15) can be further simplified as:

1 vM d vM þ 2 þ r r ds2 2

k

ð17Þ

Further simplification of Eq. (17), the curvature increment can be rewritten as:

1 1 vM d vM
¼ 2 þ r0 r r ds2 2

Dk ¼

ð18Þ

In accordance with the principles of elastic mechanics of materials, the bending moment is defined as:

M ¼
EI  Dk ¼
EI

vM

d vM þ r2 ds2 2

! ð19Þ

Rewriting Eq. (18), the following equation can be obtained.

d vM vM M þ 2 þ ¼0 EI ds2 r 2

ð20Þ

Compared Eqs. (20) with (7), the mathematical forms of them are identical on the same section as long as the following equation holds.

qr


C M ¼ r EI

ð21Þ

Considering of the value C must satisfy all load forms, assuming that the external load of the structure is zero or a uniform confining pressure, which means that the bending moment distribution of the actual structure is zero, and the value of M is also zero for tire’s each point. According to Eq. (21), only if C ¼ 0, the bending moments of the points all to be zero. Therefore, the conjugate beam structure is a ring with free end on the top and Eq. (21) can be further simplified as qr ¼ M=EI ¼ e=ðh=2Þ. It should be noted that if the two equations have the same identical solutions, the special solutions must be the same, which means that the two equations should have the same boundary conditions. The boundary condition of the original structure is the solid end, that is, v M ¼ 0; dv M =ds ¼ 0 at the support, and M ¼ 0; dM=ds ¼ 0 to the conjugate structure correspondingly. Thereby, the problem of solving the tire deflection in the radical direction can be transformed into the problem of solving the bending moment of the conjugate structure. Assuming p point is within the gauge length of the jth long-gauge FBG sensor, the deflection in the radical direction of tire’s any point p caused by the bending moment can be calculated as:

v p ¼
hr2

j
1 X i¼1

    1 ah qri sin j
i þ þ a h
hr 2 qrj sin 2 2

ð22Þ

where h is the corresponding angle of one long-gauge FBG sensor, qri is the radial uniform force of the ith sensor gauge length, a is the proportion of the left part of the two parts in which the jth segment is divided with p point and a 2 ð0; 1Þ. Therefore, the tire deflection in the vertical direction caused by the bending moment is defined as follows,


 M d ¼ max v p

ð23Þ

The tire vertical deformation is consisting of two components: the vertical deformation caused by the bending moment and the vertical deformation caused by the axial compression. In which the former can be calculated by the modified conjugate beam method, and the latter can be expressed as follow.

d ¼ r eN N

ð24Þ

Finally, the total vertical displacement of the tire is obtained as: M

N

d¼d þd

ð25Þ

In addition, in order to measure the length of the tire contact patch, which is the deformed area that is in contact with the road, the first-order differential values of the tire circumferential strain, which represent the tire circumferential direction, is used. And the first and second peaks of the first-order differential values of the strain coincide with the starting and ending edges of the contact patch, respectively.

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3. Intelligent bridge impact vehicle experiment To validate the effectiveness of the proposed method, experiment of the tire installed in the bridge impact vehicle was performed in the lab. As shown in Fig. 4(a), the experimental vehicle is an intelligent test vehicle which is independently developed by our research group for rapid bridge health monitoring. The experimental tire was mounted on the left rear wheel of the vehicle for testing. And the size of the experimental tire is 195/70R15C. Considering that the grounding angle of the tire is generally 15–20 degrees, 20 long-gauge FBG sensors with a gauge length of 10 cm were mounted on the experimental tire. The sensors can be attached to the inner or outer surface of the tire when pasted along the surface of the tire. It should be noted that the advantage of sticking it to the inner surface of the tire is that the protection of the sensor in the dynamic test. However, the disadvantage is that it requires drilling on the rim and being equipped with a special sealing device, also a danger of damaging the sensor when the tire is disassembled may happen. Pasting it on the outer surface of the tire is an easier way to implement, but care should be taken to avoid direct contact between the sensor and the ground. Therefore, in order to simplify the experiment, the long-gauge FBG sensors were attached to the center groove of the outer surface of the tire, and packaged them with hard glue. The package surface was lower than the outer surface of the tire, which could prevent direct contact with the ground, as shown in Fig. 4(b). A steel plate with a thickness of 2 cm was placed on the bottom of the tire as a gasket to easily measure the tire contact patch. Synchronously, an industrial camera, which model is UI-3370CP-M-GL, was placed on the opposite side of the experiment tire for comparison (Fig. 4(a)). With digital image correlation technique, the industrial camera can easily measure the deformation of the rim relative to the ground when the tire is stationary [26,27]. As shown in Fig. 5, the intelligent bridge impact vehicle experiment was divided into two parts. Part 1: tire static test, that is, a hydraulic jack was used to apply a stepped load to the experimental tire (Fig. 5(a)). Part 2: tire static impacting test. Considering the functional requirements of the bridge intelligent impact vehicle, the impact device was controlled by the controller to impact the panel of the vehicle, and the impacting force was transmitted from the frame to the experimental tire, which can make the tire vibration.

3.1. Tire static test results Fig. 6(a) shows the representative tire strain signal in the circumferential direction with tire pressure 400 kPa and load 4 kN, which was denoted by a blue line and had an ‘M’ shape. The red circles A and E represent the tire maximum strain values in the circumferential direction, and the blue circle C represents the minimum strain value, which is normally located in the middle of the tire contact area. Fig. 6(b) shows the first-order differential of the tire strain signal, which had an inverted ‘N’ shape. The green circles B and D represent the minimum and maximum values of the first-order differentials of the strains, respectively. As shown in Fig. 6(c), the distance between points B and D represents the contact patch length, and the distance between the midpoints of AB and DE represents the length of tire deformation. Based on the improved

Fig. 4. Tire experiment configuration: (a) overview; (b) the experimental tire; (c) the impact device.

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Fig. 5. Experimental condition: (a) tire static test; (b) tire static impacting test.

Fig. 6. Tire static experimental results (load: 4kN, velocity: 0 km/h, tire pressure: 400 kPa): (a) Tire circumferential strain; (b) Tire first-order strain differential signal; (c) Concept map; (d) Tire circumferential deformation.

conjugate beam method, Fig. 6(d) shows tire deformation in the radical direction using the measured tire strain signal. It can be seen that the partial strains away from the contact area of the tire are close to zero. Substantially, the calculated tire deformations are close to zero, which further illustrates that the effectiveness of the theoretical assumption that the tire as a rigid ring with the fixed end. In order to validate the effectiveness of the proposed method, the tire static tests under different vehicle loads and tire pressures were performed. With the modified conjugate beam method, Fig. 7(a) shows the calculated tire vertical deformation time histories using the measured tire circumference strains under unloading conditions. It is found that the tire deformations calculated by the proposed method have a good agreement with which attained by the DIC method. In addition, Fig. 7(b) shows the maximum and mean errors between the above two methods. Compared with the values calculated by the DIC method, the proposed method has a good performance and the maximum error between the above two methods was 3.8%.

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Fig. 7. Tire static experimental results under tire unloading: (a) Tire vertical deformations with different tire pressures; (b) The maximum and mean errors.

Making a first-order difference to the measured tire circumferential strain, the contact patch length between the tire and the ground can be obtained. Fig. 8(a) shows the measurements of the tire static footprint. The bottom part of the tire was painted with ink and then pressed against a sheet of paper, which lied on the ground. As shown in Fig. 8(b), the measured and calculated values under different vehicle loads were denoted by the blue and red line, respectively. It is found that the maximum error between them is 4.2 mm, which is less than 5%. 3.2. Impact test results Fig. 9(a and b) show the tire strain time histories measured by the long-gauge FBG sensors which were mounted on the top and bottom of the tire under the impact force of 3.5kN, respectively. It can be found that the maximum strain change at the top and bottom of the tire were 25 le and 1750 le, respectively, which means that a large gap between them. In addition, Fig. 9(c) further shows the strain difference ratio of the above two sensor positions. It is found that under the impact force, the maximum ratio of the strain change between them was 1.001%, which means that the strain change at the top of the tire can be negligible compared to the strain change at the bottom of the tire. With the modified conjugate beam method, Fig. 10(a) shows the dynamic tire vertical deformations, which were accurately estimated under different impact forces using the measured tire circumference strains. Obviously, the values in the time domain were basically consistent with the DIC method. Fig. 10(b) shows the frequency domain diagram of the tire deformation. The identified first order frequency was 4.626 Hz and 4.578 Hz respectively, the identified second order frequency was 7.263 Hz and 7.324 Hz respectively, and the errors between them were very small, which further validated the high precision of the proposed method. 4. High-speed experiment To validate the proposed method at high velocities, a tire high-speed experiment was performed in the lab. As shown in Fig. 11, the experimental tire was tested on a rigid ring of 300 mm diameter, which were rotated relatively to each other with

Fig. 8. Static measurements of the tire footprint: (a) overview; (b) the measurement results under different tire loads and tire pressures.

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Fig. 9. Tire strains under impacting load; (a) strain time history measured by the sensor on the top; (b) strain time history measured by the sensor on the bottom; (c) strain difference ratio of the above two position.

Fig. 10. Tire deformation of the contact area under impacting load; (a) time domain; (b) frequency domain.

a certain vertical preload. An AC motor was used as the driving device, and the tire velocity was adjustable through a frequency modulator. A laser displacement sensor (model: Panasonic HL-G1) was used to measure the tire deformation in the vertical direction for comparison. Also, to ensure uninterrupted data transmission during the tire rotation, a fiber optic rotary coupler, a means to pass signals across rotating interfaces, was interposed between the tire and the data acquisition system, the experimental results were shown as follows. Fig. 12(a) shows the time history of the tire circumferential strain under tire rotating. The blue contour indicated the tire footprint line, which represents the long-gauge FBG sensors located in the grounding area. Fig. 12(b) shows the tire strain data in the tire circumferential direction in the initial moment (t = 0), that is the strain data in y-axis direction. It is found that the tire circumferential strain at any moment in tire moving state is the same as that in tire stationary state, and it has a M-shaped. In term of tire velocity, based on the tire strain data measured by any one sensor, that is the strain data in x-axis direction, the time history between the two minimum strain values was 0.786 s. Given the spatial distance between two consecutive peaks, which was roughly2pr, the tire velocity can be obtained, that is 9.4 km/h, as shown in Fig. 12(c).

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Fig. 11. Overview of the tire high-speed experiment.

Fig. 12. Tire high-speed experimental results: (a) Tire circumferential strain signal under rotating; (b) Tire circumferential strain the initial moment; (c) Tire circumferential deformation signal under rotating.

In addition, Fig. 13 shows the concluded tire vertical deformations under tire rotating with different tire pressures using the measured tire circumferential strains. It is found that the proposed method has a good agreement with the laser-based method.

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Fig. 13. Tire vertical deformations under tire rotating with different tire pressures.

5. Discussion of the sensor optimization layout In order to ensure the accuracy of the experiment, 20 long-gauge FBG sensors with a gauge length of 10 cm were used in the above tire static and high-speed experiments. It should be noted that in the actual vehicle engineering applications, how to use the minimum number of sensors to meet the test accuracy requirements to pursue the maximum cost performance is an urgent problem. Therefore, the sensor optimization layout was discussed in this section, and three sensor programs were considered, respectively. Case 1: using 20 long-gauge FBG sensors with a gauge length of 10 cm; Case 2: using 10 long-gauge FBG sensors with a gauge length of 20 cm; Case 3: using 5 long-gauge FBG sensors with a gauge length of 40 cm.

Fig. 14. The results of the tire static experiment with different gauge length: (a) the tire circumferential strains; (b) the first-order difference derivative of the tire strains; (c) the tire circumferential deformation; (d) the errors between gauge length of 20 cm, 40 cm and gauge length of 10 cm.

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Tire circumferential strains of the above three sensor programs under different wheel loads and tire pressures were compared in Fig. 14(a). The smaller number of sensors, the smaller minimum strain peak (C00 < C0 < C). The main reason is that the long-gauge FBG sensor measuring the average strain within the gauge length. When the number of sensors is very small, the gauge length of the sensor maybe larger than the contact patch length of the tire. Moreover, the strain of the tire is mainly concentrated in the contact area, so that the sensor with a larger gauge length will attain a smaller strain value. The firstorder tire strain differentials of the above three sensor programs were compared in Fig. 14(b). The contact patch length calculated by Case1 and Case2 was relatively close, that is BD  B0 D0 . However, the contact patch length calculated by Case3 is significantly larger (B00 D00 > BD). With the modified conjugate beam method, the calculated tire deformation under the above three sensor programs using the measured tire circumferential strains were shown in Fig. 14(c). Compared with Case 1 using 20 long-gauge FBG sensors, the calculated values of Case 2 and Case 3 gradually decrease, which has the same regularity as the measured minimum strain peak of the tire. Also, Fig. 14(d) shows the errors between the proposed method with different gauge lengths. It is found that longer the gauge length, higher the error. In summary, in order to improve the cost performance and meet the measurement accuracy requirements, a long-gauge FBG sensor with a gauge length less than the grounding length of the tire can be selected for layout. However, for sensors with a gauge length larger than the contact patch length (BD), it is not advisable despite the reduced cost. 6. Conclusions In this paper, an innovative method based on monitoring tire circumferential strain with long-gauge FBG sensors to continuously measure tire deformation using the modified conjugate beam method is proposed. The major conclusions are as follows. Considering the unique advantages of high precision, high durability and macro–micro, the long-gauge FBG sensor can be conveniently used to monitor tire strain information. Furthermore, based on the SWIFT tire model, the tire deformation can be calculated using the measured tire circumferential strain with the modified conjugate beam method. An intelligent bridge impact vehicle experiment and a high-speed tire test were used to identify the proposed method. It is found that the proposed method can be robust to different tire pressure and load tested. The static and high-speed experiment results indicate that the proposed method can have better performance in real conditions by comparing with that of the traditional method. Outdoor vehicle tests will be performed in the future work. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The research presented was financially supported by the National Key R&D Program of China (No.: 2018YFC0705601) and National Natural Science Foundation of China (Grant No.: 51778134).Contributions by the anonymous reviewers are also highly appreciated. References [1] M. Matilainen, A. Tuononen, Tyre contact length on dry and wet road surfaces measured by three-axial accelerometer, Mech. Syst. Sig. Process. 52–53 (2015) 548–558. [2] A.J.C. Schmeitz, A.P. Teerhuis, Robustness and applicability of a model-based tire state estimator for an intelligent tire, Tire Sci. Technol. 46 (2) (2018) 105–126. [3] Y. Jorge, G.P. Daniel, D. 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