Research Article Vehicle Unsteady Dynamics

Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2013, Article ID 153257, 13 pages http://dx.doi.org/10.1155/2013/153257

Research Article Vehicle Unsteady Dynamics Characteristics Based on Tire and Road Features Bin Ma and Hong-guo Xu College of Traffic, Jilin University, Changchun 130022, China Correspondence should be addressed to Hong-guo Xu; [email protected] Received 8 September 2013; Accepted 28 September 2013 Academic Editor: Fenyuan Wang Copyright © 2013 B. Ma and H.-g. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. During automotive related accidents, tire and road play an important role in vehicle unsteady dynamics as they have a significant impact on the sliding friction. The calculation of the rubber viscoelastic energy loss modulus and the true contact area model is improved based on the true contact area and the rubber viscoelastic theory. A 10 DOF full vehicle dynamic model in consideration of the kinetic sliding friction coefficient which has good accuracy and reality is developed. The stability test is carried out to evaluate the effectiveness of the model, and the simulation test is done in MATLAB to analyze the impact of tire feature and road self-affine characteristics on the sport utility vehicle (SUV) unsteady dynamics under different weights. The findings show that it is a great significance to analyze the SUV dynamics equipped with different tire on different roads, which may provide useful insights into solving the explicit-implicit features of tire prints in systematically and designing active safety systems.

1. Introduction Traffic accident is increasing because various causes including road environment, vehicle dynamics, and social loss are also increasing. Many themes have been studied about active safety systems (e.g., antilock brake systems, traction control systems, and vehicle dynamic systems) in modern cars in order to prevent traffic accidents, and many of them have strongly depended on the tire forces and accuracy of the vehicle dynamic model [1, 2]. For the tire is the only component contacting with the road surface, the vehicle stability highly depends on the factors of tire features and road surface friction coefficient. Furthermore, sport utility vehicle (SUV) has the difference dynamic characters for the height of the mass center is higher and changes in a wide range. So, it is significant and could explore the tire feature and road character influence on the vehicle dynamics to analyze the SUV stability characteristics at high speed under different heights of mass center when it is equipped with different tires on different road pavements. In the field of contact mechanics between rubber and rough fractal surfaces, the multiasperity contact theories were developed and further refined based on considering the top of bigger asperities consisting of a multiscale distribution. The

relation between the true contact area and the normal force approached on a linear relation in the large separation situations is widely approved [3–6]. Subsequently, the method has been greatly modified by means of taking the asperity spheres radius of curvature as a function of the asperity height [7]. Then the rough contact model can be used to analyze the tireroad interaction mechanism. Rubber, compared with the rigid body friction, exhibits unusual sliding friction characteristics. Theoretical and numerical studies on rubber friction law [8–10] give a deeper insight into the physical mechanisms involved in rubber friction properties and find that the macrotexture mostly contributes to hysteresis friction, and the microtexture is determined by the amount of contact patches. Later, a simple rubber friction coefficient which can be used in models of tire (and vehicle) dynamics [11] and a kinetic rubber friction coefficient which is the function of sliding velocity and can be used to develop a sliding mode controller for ABS braking system [12] are proposed, respectively. Nevertheless, the effect of tire feature and road self-affine characteristics is not considered. In the past studies, the 8 DOF vehicle model was proposed through the various modeling assumptions, and its validity and effect limitations for prediction of roll behavior

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Advances in Mechanical Engineering

are discussed [13]. Some vehicle dynamic models such as 2 DOF and 3 DOF [14] were established to analyze the vehicle rollover, but no model is found to be fully satisfactory in accuracy which is used to predict the vehicle yaw rate and vehicle lateral acceleration simultaneously. About tire forces estimate [15–18], estimation method of tire forces and the dynamic states is employed to discuss the vehicle stability control and the influence on run-off-road risk. Although the breakthrough progress has been made in previous researches, the effects of tire, road surface on and kinetic fiction coefficient to the dynamic tire force are not taken into account; moreover, the fewer freedom degree of unsteady vehicle model also has not adequacy of accuracy, not to mention to analysis the effect to the SUV unsteady dynamics under different weight. Thus, a 10 DOF full SUV dynamics model combined with the kinetic sliding friction coefficient is employed to study the impact of tire feature and road self-affine characteristics on the tire force and vehicle dynamics when they are in nonsteady state. This study will render some insights into providing theoretical support to test and evaluate of the SUV dynamics when it is driving in different roads and useful insights into solving the explicit-implicit features of tires print in systematically as well as designing the active safety systems of SUV based on the computer simulation.

2. The Kinetic Rubber Sliding Friction Coefficient

𝑉 . 𝜆

En

𝜂1

𝜂2

𝜂n

Figure 1: The schematic of Maxwell-Wiechert model.

Rubber Contact rubber d

𝜆⊥

𝜆‖

Figure 2: The schematic of truncate length on rough pavement.

𝛿̇ (𝑡) = [𝐸 (𝜔) + 𝑖𝐸󸀠󸀠 (𝜔)] 𝜀 (𝑡) .

𝑁

𝜔𝜏𝑅 /𝑝2 3𝜌𝑅𝑇 𝑒 , ∑ 𝑀e 𝑝=1 (1 + 𝜔2 𝜏𝑅2 /𝑝4 )

(3)

Suppose that each parallel Maxwell’s unit mode is action at Rolls mode, solely, and 𝑝𝑖 = 1, 𝑖 = 1, 2, 3, . . .. Then, the constitutive equations of dynamic stress response in viscoelastic materials can be expressed as 𝑛

𝐸󸀠󸀠 (𝜔) = ∑ 𝑖=0

𝐸𝜏𝑖 𝜔 𝑛𝐸𝜏𝜔 = . 1 + 𝜏𝑖2 𝜔2 1 + 𝜏2 𝜔2

(4)

Considering 𝑏 = 𝜏, 𝑎 = 𝑛𝐸𝜏 and is introduced into 𝑑, expression (4) can be rewritten as

(1)

According to the extended rouse theory [10], the loss modulus 𝐸󸀠󸀠 (𝜔) for a monodisperse polymer can be calculated as 𝐸󸀠󸀠 (𝜔) =

E2

dynamic stress response in viscoelastic materials for a single Maxwell model can be expressed as

2.1. Simplify the Rubber Viscoelastic Characteristics. The molecular nature of rubber friction is a topic of considerable practical importance. Rubber has a low elastic modulus but high extensibility and has the kinetic sliding friction feature, that is due to the viscoelastic energy dissipation via the internal molecular friction caused by time-dependent forces on the rubber surface exerted by asperities of the substrate when rubber block slides on a hard rough surface [8–11]; thus, the kinetic sliding friction coefficient is changing with the sliding velocity, and the corresponding excitation frequency is given by 𝜔=

E1

𝐸󸀠󸀠 (𝜔) =

𝜔𝑎 . (𝑑 + 𝜔2 𝑏2 )

(5)

After simplifications, 𝜏 and 𝐸 are the only unknown parameters and can be obtained easily. Then, the simplified viscoelastic energy loss modulus 𝐸󸀠󸀠 (𝜔) can be calculated as a function of the spatial frequency 𝜔.

(2)

where 𝑝 represent the normal modes of motion. The extended model of Maxwell-Weichert is the Maxwell elements which are composed as parallel mode, as shown in Figure 1. According to the expansion of Rolls theory [19], the incentive model is Rolls mode when 𝑝 = 1. Based on the rubber viscoelastic theory, the constitutive equations of

2.2. Characterization of Self-Affine Surfaces. As it was mentioned earlier [3–7], the road surfaces characterize the selfaffine behaviors between the parallel crossover length 𝜉|| and the perpendicular crossover length 𝜉⊥ ; otherwise, characterize the spine behaviors [8]. The schematic of crossover length of the self-affine road surfaces is shown in Figure 2. For self-affine surfaces, a power-law dependence on the height difference correlation function 𝐶𝑖 (𝜆) below a certain



3

length scale is used to character scaling properties. The function of 𝐶𝑖 (𝜆) can be expressed as 𝐶 (𝜆) = 𝜆2⊥ (

2𝐻

𝜆 ) , 𝜆 ||

for 𝜆 min < 𝜆 < 𝜆 || .

(6)

When a tire is sliding on a rough road, due to mutual incentives between road surface asperities and rubber surface and the hysteretic energy losses arising from the rubber deformation by surface asperities, the sliding fiction coefficient is variation when a viscoelastic slab sliding at constant velocity 𝑉 on a rigid rough surface. The surface roughness power spectrum 𝑆(𝜔) can be available to characterize the incentives between road pavement and contacting rubber at a certain slip velocity. The 𝑆(𝜔) can be calculated as 𝑆 (𝜔) =

𝐻𝜆2⊥ 𝜆 || (

𝜔 −𝛽 ) , 𝜔min

𝜔min < 𝜔 < 𝜔max .

(7)

Here, 𝛽 = 7 − 2𝐷.

(8)

2.3. The Simplified Real Contact Area Model. Generally speaking, tire and rough surfaces is not contact completely in the apparent contact area when they are squeezed together, and the real contact area can be decomposed into a finite number of microterms; see Figure 2. Suppose that the curvature of spheres is depending on the asperity height and summits of very large heights behave as perfectly spherical [5–7]; the model for the relation between the real contact area and load that will be used for the calculations is given by 𝐴𝑐 = 𝐴0

+∞ 1 𝑥2 ∫ 𝑥 (𝑥 − 𝑡) exp (− ) d𝑥, 2 2√2𝜋 𝑡

𝐹 = 𝐴0

√2 𝐸 √𝑚2 3 1 − ]2 𝜋√𝜋

×∫

+∞

𝑡

(9)

[𝑥 (𝑥 − 𝑡)]3/2 exp (−

𝜔max

𝜔min

𝑆 (𝜔) 𝜔2 d𝜔,

𝑥2 ) d𝑥, 2

for 𝜔 =

2𝜋V . 𝜆

(10)

Based on Carbone and Bottiglione’s [6] discussion about the large separation and considers that is the Hertz theory states, the real contact area can be simplified as 1 𝑡 𝐴 𝑐 = 𝐴 0 erfc ( ) , √2 4 𝑚 1/2 1 𝐸 𝑡 𝐹= ( 2 ) erfc ( ) 𝐴 0 . 2 √2 41−] 𝜋

Road parameter 𝐷 𝜆 ⊥ [𝜇m] 𝜆 || [𝜇m]

(11)

Then the true contact area 𝐴 𝑐 is the function of the normalized distance 𝑡 and the macroscopic contact area 𝐴 0 .

Granite 2.37 310 2490

Asphalt 2.39 430 1440

Considering the relation of the normal force and the true contact area is the same in the large separation condition, as Bush et al.’s [3] hypothesis, the expression can be written as 𝐹=

𝑚 1/2 𝜋 𝐸 ( 2 ) 𝐴 𝑐. 2 2 1−] 𝜋

(12)

From above, we can draw the conclusion that when the applied normal force increases, the true contact area increases with the decrease of the separation between the surfaces at the interface. 2.4. The Simplified Kinetic Rubber Sliding Friction Coefficient. The modeling of kinetic rubber sliding friction coefficient on self-affine surfaces has been treated by several authors based on the hysteretic energy losses arising from the rubber deformation by surface asperities [20–22]; after appropriate simplification of real contact area and viscoelastic energy loss, the calculation of the kinetic sliding friction coefficient also has two components, the hysteretic friction coefficient and the adhesion friction coefficient. Even in a typical case, the increasing temperature can result in a decrease in rubber friction with increasing sliding velocity for V > 0.01 m/s [9]; this can be ignored in vehicle road applications. Hence, we emphasize that the friction coefficient depends on the sliding velocity, normal force, tire feature, and road self-affine characteristics; the calculation can be expressed as 𝜇𝑆 = 𝜇Hys + 𝜇Adh =

where 𝑚2 is the second momenta of roughness spectra [10]. The function of 𝑚2 can be expressed as 𝑚2 = ∫

Table 1: Road self-affine parameter and tire parameter.

𝜏𝐴 ⟨𝛿⟩ 𝜔max 𝜔𝐸󸀠󸀠 (𝜔) 𝑆 (𝜔) d𝜔 + 𝑠 𝑐 . ∫ 2𝜎0 𝑉 𝜔min 𝜎0 𝐴 0

(13)

This kinetic sliding friction coefficient model considers the effect of the tire rubber material and pavement fractal feature and can be able to characterize the variation of the different tires sliding friction coefficient on different roads as a function of load and the sliding velocity. 2.5. Simulation 2.5.1. Analysis of the True Contact Area. The typical granite and asphalt surface affine parameters are determined in Table 1. The simulation test is progressed to indicate the relation between normalized distances and the rough road true contact area. The typical macroscopic contact areas are set to 0.05 m2 , 0.04 m2 , and 0.035 m2 . The tire feature parameters are listed in Table 2, and the simulation results are presented in Figures 3 and 4. Figures 3 and 4 show that the relation between the rough road true contact area and the normalized distances is


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Advances in Mechanical Engineering ×10−3 10

×105 10

9 8 A c (m2 )

F (Pa)

8 6 4

7 6 5

2 0

4 3 0

2

4

6

8

10

0.2

0

0.4

t

Tire 1, asphalt Tire 1, granite

1

Figure 4: Relations between the normalized distances and true contact area.

Table 2: Tire properties parameters. Type 2 63∼68 10∼13 21

1.2 1 us

Type 1 61∼66 9∼12 20

0.8

Asphalt A 0 = 0.035 Granite Asphalt

A 0 = 0.04 Granite Asphalt A 0 = 0.05 Granite

Tire 2, asphalt Tire 2, granite

Figure 3: Relations between the normalized distances and the contact pressure.

Tire Shore hardness/HA Tensile stress/MPa Tensile strength/MPa

0.6 t

approximately linear which is quite similar to that of Heinrich and Klüppel [23]; meanwhile, the road surface self-affine features, the normal force, and tread rubber feature have a great impact on the normalized distance [24]. At the same normal force, since the wave length of granite pavement is shorter and its density of profile is bigger than asphalt and the shore hardness of tire 2 is smaller, the normalized distance of granite pavement is smaller and tire 1 is bigger, respectively. Meanwhile, due to the fact that the vertical cut-off length of granite pavement is shorter and the level cutoff length is bigger, the contact spots and the true contact area of granite pavement are bigger while the normalized distance is smaller. Owing to those, the true contact area decreases with the normalized distances increasing for different roads, and the contact pressure tends to zero when the normalized distance is large enough. 2.5.2. Analysis of the Kinetic Sliding Friction Coefficient. On general conditions, the macroscopic contact pressure is changed, associated with the sliding velocity and the contact pressure. The typical contact pressure between the tire and road is about 0.3 Mpa. Progress of the simulation test within the range of 0.2∼0.4 Mpa despite the 0.4 MPa is difficult to be achieved. The results of simulations are presented in Figures 5 and 6. Figures 5 and 6 show that the tire feature, the normal force, and road surface self-affine characteristics have a significant influence on kinetic rubber sliding friction coefficient. For the two road conditions, the two tire sliding frictions are less than 1.2 and decrease nonlinearly as the load increases. Meanwhile, the kinetic rubber sliding friction coefficient of asphalt is higher, and it is decreasing with the increasing

0.8 0.6 0.4

2

4

6

8

10 12 V (m/s)

Tire 1, granite, 0.2 MP Tire 1, asphalt, 0.2 MP Tire 1, granite, 0.3 MP

14

16

18

20

Tire 1, asphalt, 0.3 MP Tire 1, granite, 0.4 MP Tire 1, asphalt, 0.4 MP

Figure 5: Friction coefficients of different tires on asphalt pavement.

normal force within the range of 0.2∼0.4 Mpa; the asphalt has a 0.3, and the granite has a 0.25 decrease, respectively. Later, the kinetic rubber sliding friction coefficient tends to a constant value when the velocity is higher than 10 m/s. Since the parallel and perpendicular crossover length of asphalt pavement profile is smaller and the spatial frequency is higher than granite pavement profile at the same sliding velocity, the asphalt pavement has higher kinetic rubber sliding friction coefficient for the same tire at certain normal force, namely, 0.2∼0.3. For different tires at certain incentives, the tire 2 has smaller kinetic sliding friction coefficient; that is may be due to the fact that tire 2 has smaller elastic modulus and spatial frequency. Therefore, on asphalt pavement tire 1 has higher performance which the elastic modulus is bigger.

3. The Improved 10 DOF Full Vehicle Model An improved 10 DOF unsteady full vehicle model that includes nonlinear effects is developed to analyze the unsteady



5 Z

1.2 z󳰀󳰀 1 us

z󳰀

z Φ

𝜃

0.8 0.6 0.4

y 2

4

6

8

10 12 V (m/s)

Tire 2, granite, 0.2 MP Tire 2, asphalt, 0.2 MP Tire 2, granite, 0.3 MP

14

16

18

Φ

20 𝜓 𝜓

Tire 2, asphalt, 0.3 MP Tire 2, granite, 0.4 MP Tire 2, asphalt, 0.4 MP

Y

X 𝜃

x󳰀

Figure 6: Friction coefficients of different tires on asphalt pavement.

dynamics characteristics accurately when the SUV applied a combined steering and braking maneuver. The improved model includes 6 DOF at the vehicle lumped mass center of gravity including the longitudinal, lateral, vertical, pitch, roll, and yaw dynamics and 4 DOF at each of the four wheels, including the wheel spin. The effect of load transfer is taken into account when the vehicle roll and pitch; meanwhile, a stationary roll center is also assumed. The model has three coordinate frames, body-fixed coordinate frame 𝑥𝑦𝑧 attached to sprung mass C.G, coordinate frame 𝑥󸀠 𝑦󸀠 𝑧󸀠 attached to tire-ground contact point, and inertia fixed coordinate frame 𝑋𝑌𝑍. The schematic of three coordinate frames is shown in Figure 7. For a complete vehicle system, take the research in reality and make the following assumptions:

x

x󳰀󳰀

Figure 7: Schematic of three coordinates frame and the converter law.

𝑚 (𝑤̇ + 𝜔𝑥 V − 𝜔𝑦 𝑢) = ∑ 𝐹𝑧 − 𝑔,

(16)

𝐽𝑥 𝜔̇ 𝑥 = 𝑚𝑠 ℎ𝑐𝑔 𝑎𝑦 + 𝑚𝑠 𝑔𝜑ℎ𝑐𝑔 − (𝐾𝜑𝑓 + 𝐾𝜑𝑟 ) 𝜑 − (𝐶𝜑𝑓 + 𝐶𝜑𝑟 ) 𝜑,̇

(17)

𝐽𝑦 𝜔̇ 𝑦 = −𝑚𝑠 ℎ𝑐𝑔 𝑎𝑥 + 𝑚𝑠 𝑔𝜃ℎ𝑐𝑔 − (𝐾𝜃𝑓 + 𝐾𝜃𝑟 ) 𝜃 − (𝐶𝜃𝑓 + 𝐶𝜃𝑟 ) 𝜃,̇

(18)

𝐽𝑧 𝜔̇ 𝑧 = 𝑎 (𝐹𝑥𝑟𝑓 + 𝐹𝑥𝑙𝑓 ) sin 𝛿 − 𝑏 (𝐹𝑦𝑟𝑟 + 𝐹𝑦𝑙𝑟 ) + 𝑎 (𝐹𝑦𝑟𝑓 + 𝐹𝑦𝑙𝑓 ) cos 𝛿 −

𝑐 (𝐹 − 𝐹𝑥𝑟𝑟 ) 2 𝑥𝑙𝑟

𝑐 (𝐹 − 𝐹𝑥𝑟𝑓 ) cos 𝛿 2 𝑥𝑙𝑓 𝑐 + (𝐹𝑦𝑙𝑓 − 𝐹𝑦𝑟𝑓 ) sin 𝛿. 2

(1) the two front-wheel steering angles are identical;

−

(2) the front and rear suspensions are represented simply by their respective equivalent stiffness and damping coefficients; the vehicle body is modeled as being rigid;

y󳰀

(19)

Here, ∑ 𝐹𝑥 = (𝐹𝑥𝑟𝑓 + 𝐹𝑥𝑙𝑓 ) cos 𝛿

(3) ignore the changes of the wheel radius; (4) consider the vertical velocity only;

− (𝐹𝑦𝑟𝑓 + 𝐹𝑦𝑙𝑓 ) sin 𝛿 + 𝐹𝑥𝑟𝑟 + 𝐹𝑥𝑙𝑟 ,

(5) consider the linearization of the trigonometric terms with the small Cardan angle assumption. Figure 8 shows the schematic of the 10 DOF full vehicle model. 3.1. Sprung Mass Model. According to the rigid body kinematics and dynamics principle, the equations of motion of 10 DOF model of vehicle body with the above assumptions can be obtained as given subsequently. Consider 𝑚 (𝑢̇ + V𝜔𝑧 − 𝑤𝜔𝑦 ) = ∑ 𝐹𝑥 + 𝑔 sin 𝜃,

(14)

𝑚 (V̇ + 𝑢𝜔𝑧 − 𝑤𝜔𝑥 ) = ∑ 𝐹𝑦 − 𝑔 sin 𝜃,

(15)

∑ 𝐹𝑦 = (𝐹𝑥𝑟𝑓 + 𝐹𝑥𝑙𝑓 ) sin 𝛿

(20)

+ (𝐹𝑦𝑟𝑓 + 𝐹𝑦𝑙𝑓 ) cos 𝛿 + 𝐹𝑦𝑟𝑟 + 𝐹𝑦𝑙𝑟 , ∑ 𝐹𝑧 = 𝐹𝑧𝑟𝑓 + 𝐹𝑧𝑙𝑓 + 𝐹𝑧𝑟𝑟 + 𝐹𝑧𝑙𝑟 . The wheels are each modeled by their rotary inertia, angular acceleration, and radius; the schematic of wheel dynamics model is shown in Figure 9. The motion for the four wheels can be calculated as follows: 1 (−𝐹𝑥𝑖𝑗 𝑟0 − 𝑇𝑏𝑖𝑗 ) , 𝑖𝑗 = 𝑟𝑓, 𝑙𝑓, 𝑟𝑟, 𝑙𝑟, 𝜔̇ 𝑖𝑗 = (21) 𝐽𝑖𝑗 where 𝑖𝑗 subscript denotes 𝑟𝑓, 𝑙𝑓, 𝑟𝑟, and 𝑙𝑟.


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Advances in Mechanical Engineering 𝛿

𝛿 Fxlf

Fxrf

𝜑

Fyrf u

Fylf

a v

ms

𝜔z

Fxlr Fylr

Fxrr

ms ay

b

Fyrr

x

ms g

y Fzrf

C

Fzlf

(a) Pitch motion

(b) Roll motion

z x

𝜃 Fzrf

ms ax

Fzlf

hcg mur

muf

(c) Pitch motion

Figure 8: Vehicle model with 10 DOF.

The longitudinal tire forces can be expressed as

−Tb

𝐹𝑥 = 𝜔

r0

Vx

𝜎𝑥 𝜇𝑥 𝐹𝑧 𝐹 (𝜎𝑥 ) , 𝜎∗ 𝜇𝑥𝑜 𝐹𝑧𝑜 𝑥𝑜 eq

𝑥 𝑥 𝐹𝑥𝑜 (𝜎eq ) = 𝐷𝑥𝑜 sin(𝐶𝑥 arctan (𝐵𝑥𝑜 𝜎eq 𝑥 − 𝐸𝑥 (𝐵𝑥𝑜 𝜎eq 𝑥 − arctan (𝐵𝑥𝑜 𝜎eq )))) . (22)

Fxij

Figure 9: Simplified wheel dynamics model.

The lateral tire forces can be expressed as Indeed, although simplification and linearization of certain key equations on the vehicle responses are carried out, the improved 10 DOF full vehicle model is also fairly complex and valid for applications which do not involve wheel lift-off. 3.2. Nonlinear Tire Model. The longitudinal and lateral forces of the tire are calculated using the MAGIC model [25], which is a nonlinear tire model could combining with the change of the kinetic sliding friction coefficient and normal force. This model has been widely studied and used for nonlinear simulations all nonlinear mechanical properties of tire.

𝐹𝑦 =

𝜎𝑦 𝜇𝑦 𝐹𝑧 𝜎∗ 𝜇𝑦𝑜 𝐹𝑧𝑜

𝑦 ), 𝐹𝑦𝑜 (𝜎eq

(23)

𝑦 𝐹𝑦𝑜 (𝜎eq ) 𝑦 = 𝐷𝑦𝑜 sin(𝐶𝑦 arctan (𝐵𝑦𝑜 𝜎eq


𝑦 − 𝐸𝑦 (𝐵𝑦𝑜 𝜎eq 𝑦 − arctan (𝐵𝑦𝑜 𝜎eq )))) . (24)


7

Here,

𝐹𝑧𝑟𝑟 = 𝑥 𝜎eq =

𝐶𝐹𝜅 (𝐹𝑧 ) 𝜇𝑥𝑜 𝐹𝑧𝑜 𝜎, 𝐶𝐹𝜅𝜅 𝜇𝑥 𝐹𝑧

+

(25)

𝐶 (𝐹 ) 𝜇𝑦𝑜 𝐹𝑧𝑜 ∗ 𝑦 𝜎eq = 𝐹𝛼 𝑧 𝜎 . 𝐶𝐹𝛼𝛼 𝜇𝑦 𝐹𝑧

𝐹𝑧𝑙𝑟 =

The tire forces are determined by the tire properties and slip models, and the tire models are based on the slip angles. The slip angles of the front and rear wheels are calculated as follows: 𝛼𝑟𝑓 = 𝛿 − arctan (

𝛼𝑙𝑓 = 𝛿 − arctan (

𝛼𝑟𝑟 = − arctan (

𝛼𝑙𝑟 = − arctan (

(V + 𝑎𝜔𝑦 ) (𝑢 + (𝑐/2) 𝜔𝑦 ) (V + 𝑎𝜔𝑦 ) (𝑢 − (𝑐/2) 𝜔𝑦 ) (V − 𝑏𝜔𝑦 )

(𝑢 + (𝑐/2) 𝜔𝑦 ) (V − 𝑏𝜔𝑦 ) (𝑢 − (𝑐/2) 𝜔𝑦 )

), (26)

),

).

𝑉𝑥 − 𝑟0 𝜔 . 𝑉𝑥

(27)

𝑚𝑔𝑏 𝐵 𝑏𝐴 + 𝑚𝑢1 𝑔 + − 2 (𝑎 + 𝑏) 𝑐 (𝑎 + 𝑏) 2 (𝑎 + 𝑏)

𝐹𝑧𝑙𝑓 =

(𝐾𝜑𝑓 𝜑 + 𝐶𝜑𝑓 𝜑)̇ 𝑐

+

(𝐶𝜃𝑓 𝜃 + 𝐶𝜃𝑓 𝜃)̇ 𝑎

,

𝑚𝑔𝑏 𝑏𝐴 𝐵 + 𝑚𝑢2 𝑔 − − 2 (𝑎 + 𝑏) 𝑐 (𝑎 + 𝑏) 2 (𝑎 + 𝑏) −


+

(𝐶𝜃𝑓 𝜃 + 𝐶𝜃𝑓 𝜃)̇ 𝑎

(𝐶𝜃𝑓 𝜃 + 𝐶𝜃𝑓 𝜃)̇ 𝑏

,

𝑚𝑔𝑏 𝐵 𝑎𝐴 + 𝑚𝑢4 𝑔 − + 2 (𝑎 + 𝑏) 𝑐 (𝑎 + 𝑏) 2 (𝑎 + 𝑏) −


−

(𝐶𝜃𝑓 𝜃 + 𝐶𝜃𝑓 𝜃)̇ 𝑏

. (28)

𝐵 = 𝑚 (V̇ − 𝑢𝜔𝑧 ) ℎ𝑐𝑔 ,

3.3. Weight Transfer Model. In order to calculate the tire force more accurately, the weight transfer effects cannot be neglected under the combined steering and braking maneuver, and it includes lateral and longitudinal weight transfers. The lateral weight transfer is based on the vehicle’s roll dynamics and the unsprung weight; the longitudinal weight transfer is based on the vehicle’s pitch dynamics and the normal force acting on each wheel caused by the static distribution of the masses. The sum of the normal forces at the four tires including the load transfer is determined as

+

𝑐

−

𝐴 = 𝑚 (𝑢̇ + V𝜔𝑧 ) ℎ𝑐𝑔 ,

Due to the normal force having a good consideration of the lateral and longitudinal weight transfers and the kinetic sliding friction coefficient, the model can accurately describe the tire force changing when the vehicle is during dynamic maneuvers.

𝐹𝑧𝑟𝑓 =

(𝐾𝜑𝑓 𝜑 + 𝐶𝜑𝑓 𝜑)̇

Here,

),

The longitudinal slip is calculated as 𝜎𝑥 =

𝑚𝑔𝑏 𝐵 𝑎𝐴 + 𝑚𝑢3 𝑔 + + 2 (𝑎 + 𝑏) 𝑐 (𝑎 + 𝑏) 2 (𝑎 + 𝑏)

,

(29)

where 𝑚𝑢𝑖 , 𝑖 = 1, 2, 3, 4 denotes each un-sprung mass. 3.4. Validation. Before conclusions can be drawn on the accuracy of the model, the 10 DOF full vehicle model is validated with measurement system of stability for comparison of yaw rate changes under combined steering and braking maneuver with the 50 km/h initial velocity. The steering angle is taking a value of −1.5 deg; the braking torque is set to 900 N⋅m for front tire while 650 N⋅m for rear wheel; the SUV structural parameters are determined in Table 3; the measurement system of stability is presented in Figure 10; the yaw rates are computed and presented in Figure 11. From Figure 11, it is observed clearly that the simulations of the suggested model approach a fair agreement with the experimental results at response time, change trends, and maximum value. Meanwhile, the maximum yaw rate reached about 0.1 rad/s, and the vehicle is in the stable state as Figure 11 shows. So, even after the small angle assumption and ignoring the inertia forces of the un-sprung masses, the 10 DOF unsteady full vehicle model which considers the kinetic sliding friction coefficient can effectively analyze the vehicle unsteady dynamics characteristics reliability.

4. Simulation Results and Discussion Most of standard highways in our country have three lanes in one direction and are 13 meters wide (each lane is 3.75 m wide; see Figure 12). The comparative on-road vehicle trajectories are evaluated through the PC-Crash when applying a different steering and identical braking maneuver; the results are shown in Figure 12. Generally speaking, the vehicle has a high speed once the accident happened, so the initial velocity is kept at 90 km/h and the step wheel steer angle is set to 1.5 deg, 2 deg, and 3 deg. One can see that the difference of lateral offset distance is relatively small when the road wheel steer angle is kept at 2 deg and 3 deg, namely, 2.6 m. Meanwhile, the additional sideslip moment is generated on rear wheel when the road wheel steer angle is kept at 3 deg and the vehicle is unsteady. Therefore, in order to ensure that the vehicle is always on road and steady, the steering angle should be smaller than 3 deg under this specific operation.


8


Gyroscope

Data processing

VBOX unit

Display device

Steering angle device

Whole test system

Braking force device

Data acquisition

Figure 10: Measurement system of stability. Table 3: SUV structural parameters. 𝑎 1.15 𝑚 1650

Parameter Value Parameter Value

𝑏 1.62 𝑚𝑢 50

ℎ𝑐𝑔 0.56 𝐽𝑥 978

0.02 Yaw rate (rad/s)

0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.12

0

0.5

1

1.5

2 2.5 3 Braking time (s)

3.5

4

4.5

Measurement results Simulation results

Figure 11: Validation with yaw rate.

Taking into account the nonlinear characteristic of the vehicle model and ensuring the accuracy of the simulation at high velocity and different load, the constant steering angle is taking a value of −2 deg and and the constant braking torque is kept at 1200 N⋅m for front wheel and kept at 800 N⋅m for rear wheel, not changed with load, respectively; the initial braking velocity is set to 90 km/h. The impact of tire feature and road self-affine characteristics on the CG acceleration, slip angles, and vehicle dynamic characteristics has been compared in MATLAB, and the results are analyzed as follows. 4.1. The CG Acceleration Difference Characteristics. In simulation experiments, we find that the C.G longitudinal acceleration and the C.G lateral acceleration have significantly

𝑐 1.54 𝐽𝑦 3262

𝑚𝑠 1450 𝐽𝑧 3262

𝑟0 0.35 𝐽𝑤 4.5

difference and identical at about 1 s (shown in Figure 13), although the extent of difference is slightly at another period. Therefore, we conclude that the tire feature and road selfaffine characteristics have a significant effect on both the C.G acceleration at limitation station with different loads under the combined steering and braking maneuver situations. Meanwhile, it is noticeable that the dynamic variation trend of C.G acceleration is consistent with the corresponding tire force, which are the leading factors that predominate the C.G acceleration (according to (14) and (15)), respectively. Here, we divide acceleration difference under load, tire feature, and road self-affine characteristics effect into two different periods: period Ι and period Π, and it is obvious that the main difference is in period Ι. In period Ι, we are able to observe that the acceleration decreases as the load increases, which gives us a proof that the vehicle has a better stability equipped with the same tire as the load increases while the braking performance decreases inversely; the reason could be that the CG height decreases and the momentum increases due to the load’s increase. It can also be seen that, on the asphalt road surface, the stability of the vehicle with tire 1 is superior to that of the vehichle with tire 2 than on granite, with or without load, respectively. Moreover, the peak lateral acceleration is reached at about 1 s and persists for more than 1 s, namely, 5 m/s2 . In other words, the vehicle has the worst stability at this period when the vehicle is equipped with tire 2 on granite pavement while it has a better stability with tire 1 on asphalt pavement. 4.2. The Slip Angle Difference Characteristics. As can be seen from Figure 14, the influences of tire feature and road



9

3.5 m

3.5 m 4.6 m

8.6 m

45.8 m

45.5 m

(a) Steering angle is fixed at 1.5 deg

(b) Steering angle is fixed at 2 deg

3.5 m 11.2 m 45.1 m

(c) Steering angle is fixed at 3 deg

Figure 12: Vehicle responses trajectories during steering and braking maneuvers.

1 Longitudinal acceleration (m/s2 )

Longitudinal acceleration (m/s2 )

1 0 −1 −2 −3 −4 −5 −6

0

1

2

3 4 Braking time (s)

5

0 −1 −2 −3 −4 −5 −6

6

0

Asphalt, tire 1, load Asphalt, tire 2, load

Asphalt, tire 1, no load tire Asphalt, tire 2, no load tire


5

6

Asphalt, tire 1, load Granite, tire 1, load

(b) Longitudinal acceleration on different road

1

1 Lateral acceleration (m/s2 )

Lateral acceleration (m/s2 )

2

Asphalt, tire 1, no load road Granite, tire 1, no load road

(a) Longitudinal acceleration with different tire

0 −1 −2 −3 −4

1

0

1

2



5

6


(c) Lateral acceleration with different tire

0 −1 −2 −3 −4

0

1

2




(d) Lateral acceleration on different road

Figure 13: The impact of tire feature and road self-affine characteristics on C.G acceleration.


5

6

Advances in Mechanical Engineering 0.01

0.01

0

0

−0.01

−0.01

Slip angle (rad)

Slip angle (rad)

10

−0.02 −0.03 −0.04 −0.05

−0.02 −0.03 −0.04

0

1

2


5

−0.05

6



2


5

6


(b) The slip angles of front wheel on different roads

0.01

0.01

0

0 Slip angle (rad)

Slip angle (rad)

1


(a) The slip angles of front wheel with different tires

−0.01 −0.02

−0.01 −0.02 −0.03

−0.03 −0.04

0

0

1

2



5


(c) The slip angles of rear wheel with different tires

6

−0.04

0

1

2



5

6


(d) The slip angles of rear wheel on different roads

Figure 14: The impact of tire feature and road self-affine characteristics on the slip angle.

self-affine characteristics on the slip angle of front and rear wheel in the first period are also significantly and identical, basically. The effect of the slip angle difference can also been divided into two periods. Note that, at the beginning of the first period, the front wheel slip angle is not zero since there is a steering angle at the road wheels between the driving direction of front wheel and the C.G motion tangential direction when the vehicle is steering. Meanwhile, the slip angle have significantly difference and identical at about 1 s (shown in Figure 14) although the extent of difference is slightly different at other period. Overall, during vehicle motion process, the slip angle is the function of the lateral force and is the main factor (see (23)); therefore the trend of slip angle dynamic change is consistent with the CG lateral acceleration. The CG height has a decisive role in the lateral acceleration due to the slip angle increase while the lateral acceleration decreased due to the load increase. Totally, the vehicle has better stability when the slip angle is smaller.

4.3. Vehicle Dynamics Difference Characteristics. Figure 15 shows that the effect of tire feature and road self-affine characteristics on the vehicle braking stability is significant in the first period while unnotable at the second period with loaded or not. Meanwhile, the vehicle stability is enhanced with the load increase. At high sliding velocity (at initial braking phase), since the two kinetic sliding friction coefficients of two kinds of road pavement are higher and the lateral force has weak difference, the vehicle dynamic parameters have weak discrepancy. When applying braking maneuver, the vehicle velocity decreases and the difference of the tire force is enhanced relatively, so the roll rate and yaw rate of the vehicle C.G have an obvious fluctuation. That is because the relative velocity is reduced and the kinetic corresponding sliding friction coefficient decreases to a certain value and the lateral force is bigger enough and changes drastically. It is also illustrated that, compared with tire 2, the roll rate and yaw rate of the vehicle C.G when equipped with tire 1 are



11

0

0

−0.02 −0.04 Yaw rate (rad/s)

Yaw rate (rad/s)

−0.05

−0.1

−0.15

−0.06 −0.08 −0.1 −0.12 −0.14

−0.2

0

1

2



5

−0.16

6

Asphalt, tire 1, full load tire Asphalt, tire 2, full load tire

0

(a) Yaw rate difference with different tires


5

6

Granite, tire 1, full load road Asphalt, tire 1, full load road

(b) Yaw rate difference on different roads

0.04

0.02

0.02

0

0

Roll rate (rad/s)

Roll rate (rad/s)

2

Granite, tire 1, no load road Asphalt, tire 1, no load road

0.04

−0.02 −0.04

−0.02 −0.04 −0.06

−0.06 −0.08

1

0

1

2



5

6

Asphalt, tire 1, full load tire Asphalt, tire 2, full load tire

(c) Roll rate difference with different tires

−0.08

0

1

2


Granite, tire 1, no load road Asphalt, tire 1, no load road

5

6

Granite, tire 1, full load road Asphalt, tire 1, full load road

(d) Roll rate difference on different roads

Figure 15: The impact of tire feature and road self-affine characteristics on the vehicle dynamics.

arguably smaller since the kinetic sliding friction coefficient of tire 1 is lower on the same road pavement and at high sliding velocity (see Section 2.5.2). Since the kinetic sliding friction coefficient on asphalt pavement is always bigger under the same tire-road interface friction situations, the vehicle on asphalt pavement has better stability but it is also in stable limit state. Moreover, the vehicle’s transition to unsteady state due to the tire force provided by granite pavement is not big enough to maintain the vehicle stability when it is equipped with tire 2. To sum up, the vehicle has better stability with high performance of tire and running on high sliding friction road.

5. Conclusion A 10 DOF full vehicle dynamics model combined with the kinetic sliding friction coefficient which has good reality is developed and validated by the measurement system of

stability. The effects of tire feature and road self-affine characteristics on the kinetic sliding friction coefficient, vehicle dynamic characteristics are analyzed by the simulation test which is carried out with the utility of MATLAB. The simulation results indicate that the 10 DOF full vehicle dynamics model has a good accuracy and reliability under the stablelimited conditions. Meanwhile, the following conclusions are obtained. (1) The kinetic rubber sliding friction coefficient is decreasing with the increase of normal force and sliding velocity in the case of a passenger car. (2) Tire feature and road self-affine features have a significant but contrary influence on the C.G acceleration. Moreover, the CG height has a decisive role in the lateral acceleration.


12

Advances in Mechanical Engineering (3) The vehicle has better stability but worse braking performance with high performance of tire and running on high sliding friction road when the load is increased. (4) The lateral tire force is the main reason for causing the vehicle stability difference characteristics after a combined steering and braking maneuver is applied.

It is also worthy to point out that this finding also provides an analytical tool to analyze the dynamic tire force accurately and solve the explicit-implicit features of tire prints systematically.

Nomenclature 𝑉: 𝜆: 𝜆 min : 𝜌: 𝑅: 𝑇: 𝑁𝑒 :

Sliding velocity Length scale of asperity Minimum wave length Density of the bulk polymer Universal gas constant temperature Number of subunits between two successive entanglements Molecular mass of a subunit 𝑀𝑒 : Relaxation time of the first mode (the 𝜏𝑅 : Rouse mode) of motion (𝑝 = 1) 𝐸: elastic modulus 𝜔: Spatial frequency Minimum incentive spatial frequency 𝜔min : Maximum incentive spatial frequency 𝜔max : 𝐹: The normal force ]: Rubber Poisson’s ratio Macroscopic contact area 𝐴 0: 𝑡: Normalized distance between rubber and rough asperities ⟨𝛿⟩: Thickness of the motivate rubber layer Interfacial shear stress 𝜏𝑠 : 𝐸󸀠󸀠 (𝜔): Rubber viscoelastic energy loss depending on frequency 𝑚: Gross mass of the vehicle Sprung mass 𝑚𝑠 : 𝑎: Distance of C.G from front axle 𝑏: Distance of C.G from rear axle 𝑐: Track width Vertical distances of the roll centers ℎ𝑐𝑔 : below the sprung mass C.G 𝛿: Road wheel steer angle Velocity of the wheel center. 𝑉𝑥 : Angular velocity of wheel rotation 𝜔𝑖𝑗 : (rad/s) Rotational inertia of each wheel (kg⋅m2 ) 𝐽𝑖𝑗 : Nominal wheel radius (m) 𝑟0 : External torque applied at wheel (N⋅m) 𝑇𝑏𝑖𝑗 : The equivalent normal force (3000 N) 𝐹𝑧𝑜 : Normal force 𝐹𝑧 : Longitudinal and lateral acceleration 𝑎𝑥 , 𝑎𝑦 : 𝜃, 𝜑: Pitch angle and roll angle 𝜔𝑥 , 𝜔𝑦 , 𝜔𝑧 : Roll rate/pitch rate/yaw rate of C.G in body-fixed coordinate

𝜎𝑥 , 𝜎𝑦 , 𝜎∗ : 𝐽𝑥 , 𝐽𝑦 , 𝐽𝑧 :

Theoretical slip ratio Roll inertia, pitch inertia, and yaw inertia 𝑢, V, 𝑤: Longitudinal/lateral/vertical velocities of C.G 𝐹𝑥𝑟𝑓 , 𝐹𝑥𝑙𝑓 , 𝐹𝑥𝑟𝑟 , 𝐹𝑥𝑙𝑟 : Tire longitudinal forces 𝐹𝑦𝑟𝑓 , 𝐹𝑦𝑙𝑓 , 𝐹𝑦𝑟𝑟 , 𝐹𝑦𝑙𝑟 : Tire lateral forces 𝐹𝑧𝑟𝑓 , 𝐹𝑧𝑙𝑓 , 𝐹𝑧𝑟𝑟 , 𝐹𝑧𝑙𝑟 : Tire normal forces 𝐶𝐹𝜅 (𝐹𝑧 ), 𝐶𝐹𝛼 (𝐹𝑧 ): Shape factor which changes with the normal force Shape factor on nominal load 𝐶𝐹𝜅𝜅 , 𝐶𝐹𝛼𝛼 : Stiffness factor 𝐵𝑥𝑜 , 𝐵𝑦𝑜 : Front and rear suspension damping 𝐶𝜃𝑓 , 𝐶𝜃𝑟 : of the pitch angle Front and rear suspension damping 𝐶𝜑𝑓 , 𝐶𝜑𝑟 : of the roll angle Front and rear suspension stiffness of 𝐾𝜃𝑓 , 𝐾𝜃𝑟 : the pitch angle Front and rear suspension stiffness of 𝐾𝜑𝑓 , 𝐾𝜑𝑟 : the roll angle Peak value 𝐷𝑥𝑜 , 𝐷𝑦𝑜 : Curvature factor 𝐸𝑥 , 𝐸𝑦 : 𝑥 𝑦 The equivalent slip angle 𝜎eq , 𝜎eq : Improved kinetic sliding friction 𝜇𝑥 , 𝜇𝑦 : coefficient.

Conflict of Interests The authors do not have any conflict of interests with the content of the paper.

Acknowledgments This research was supported partly by Chinese national natural science foundation (51078167) and transport vehicle safety operation technology transportation sector key laboratory open foundation (vehicle operating risk based on the analysis of the typical traffic accident).

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