The position of speed bump in front of truck scale based on vehicle vibration performance

Abstract

Speed bump set before truck scale is to prevent the error of weighing measurement caused by truck driver non-standard operation through the weighing platform. However, the vehicle vibration produced by the speed bump will affect the weighing result if the speed bump is closer to truck scale, and can’t play a role in restricting driver’s operation if the speed bump is farther to truck scale. This paper researches the reasonable position of speed bump based on triaxial truck vibration characteristics. Simulation model of vibration in the process of triaxial truck passing the speed bump is built based on the speed hump model and dynamics model of 5DOF 1/2 triaxial truck. Then vehicle vibration characteristics are researched at the speed of 1.8 km/h, 3.6 km/h, 5.4 km/h, 7.0 km/h, and obtains the vibration decay time with different speed, which determine the minimum distance between speed bump and truck scale. The accuracy of the model is tested on the commercial vehicles safety test platform. The results of simulation and experiment indicate that model of vibration in the process of triaxial truck passing the speed bump can accurately calculate the minimum distance between speed bump and truck scale with different speed, which maximum error is 5.81%.

Keywords

Vehicle dynamics speed bump position truck scale three-axle truck

1 Introduction

The road cargo turnover grows rapidly for the rapid development of China transportation [1]. To prevent the overloading transport, the Chinese expressway has basically adopted weight fees mode for trucks. The weight fees are a mode of collecting tolls that measuring the trucks’ axle load calculates the total weight of vehicles, then collects tolls according to the total weight of vehicles and the actual mileage [2]. In order to avoid tolls the driver takes the jumping scale or the rushing scale and other non-standard weighing operation behavior, for example the driver takes hard acceleration while the front axle is passing the weighting platform or the driver takes hard deceleration while the rear axle is passing the weighting platform. This driving behavior makes the center of the vehicle gravity shift, and the weighing mass is far less than the actual weight and avoid tolls. It also reduces weighing accuracy greatly, and seriously affects the popularity of expressway weighting charging mode [3, 4].

According to current literature, there are two main methods for improving weighing accuracy. The first method is to improve the data acquisition and processing algorithms of the weighing system, to eliminate the weighing errors which are caused by vehicle motion. In literature [5] EMD-RBF network was used to process the signal of vehicle dynamic weighing system to eliminate interference of the vehicle movement for the weighing platform. In literature [6] Levenberg-Marquardt nonlinear fitting method was used to reduce low frequency dynamic load interference of the vehicle dynamic weighing accuracy. In literature [7] adaptive inverse filter was used to suppress the noise signal of the vehicle dynamic weighing system. These methods eliminate the impact of the speed and vibration on weighing accuracy by using the signal processing, but they do not research the errors caused by the non-standard driving behavior. Therefore, these methods can only solve the weighing errors caused by non-human factors. The second method is to install the speed bump before the weighing device, to interfere with hard acceleration and hard deceleration while vehicle is passing the weighing platform.

Although the speed bump could restrict the non-standard behavior, at the same time the speed bump will produce vibrations of vehicle body, wheels and suspension system [8, 9]. If the mounting position of the speed bump is too close to the weighing platform, when the vehicle is traveling to the weighing platform, the vehicle vibration caused by the speed bump is not completely eliminated. It is bound to affect the weighing result. On the other hand if the mounting position of the speed bump is too far from the weighing platform, the interferential effect will not work. So the distance between speed bump and the weighing platform will directly affect the weighing accuracy of the weighing platform. This article researches on the triaxial truck vibration characteristics when the triaxial truck passes through the speed bump, and obtains a reasonable location of the speed bump. According to the existing literature, there are few research papers about a reasonable location of the speed bump. Literature [10 –12] studied the vibration characteristics of the vehicle while the vehicle passed the speed bump. According to the vibration characteristics of the vehicle while the vehicle passed the speed bump, Literature [10] obtained the vehicle impact load. Literature [11] studied the impact of the speed bump height and width on the vehicle ride comfort. According to the vehicle model, Literature [12] analyzed the relationship between the vehicle speed, the cross-sectional dimension of the speed bump and the maximum vibration acceleration.

2 Methodology

Effect factors of weighing error are studied through analyzing the dynamic characteristic and force in weighing process of triaxial truck. Then vibration characteristic in the process of triaxial truck passing the speed bump and the reasonable position of speed bump in front of truck scale are researched based on the speed hump model and dynamics model of 5dof 1/2 triaxial truck. Finally based on the safe operation platform of commercial vehicle, this paper does the experimental research of the vehicle vibrating characteristics to verify the accuracy of the model.

2.1 Effect of triaxial truck dynamic characteristic on weighing error

2.1.1 The analysis of triaxial truck motion

In order to describe the vehicle motion state, fix the x, y, z three dimensional moving coordinate to the vehicle coordinate system, and the origin O in the vehicle center of gravity. As shown in Fig. 1, it is the triaxial truck motion state model, this model includes the longitudinal motion along the x-axis and the roll motion around the x-axis; the lateral motion along the y-axis and pitching motion around the y-axis; the vertical motion along the z-axis and the yawing motion around the z-axis [13]. The motion is the manifestation of force, the force and moment generated by the triaxial truck motion will affect the weighing accuracy.

Fig.1

Model of vehicle motion state and dynamic characteristics.

In the graph, the roll motion around the x-axis generates the wheel load transfer between the left wheel and right wheel; the pitching motion around the y-axis generates the axle load transfer. The yawing motion around the z-axis and the longitudinal, lateral motion along the x, y-axis are the movement in the horizontal plane. The dynamic weighing process is carried out by weighing each axis, and the weighing sensor is not sensitive to the force and moment that are caused by the level movement. So the yawing motion around the z-axis, longitudinal motion along the x-axis, the lateral motion along y-axis can be ignored, just considering the pitching motion around the y-axis and vertical motion along the z-axis.

2.1.2 Mechanical analysis of triaxial truck weighing process

The vehicle dynamic weighing system calculates the total weight of vehicle by detecting the number of axle and axle load [8]. In order to study the influence of vehicle motion on the weighing accuracy, this paper established the mechanical model of triaxial truck weighing process, as shown in Fig. 2.

Fig.2

Mechanical model of triaxial truck weighing process.

Take the first axle moment:

$\begin{matrix} {Gah}_{1} / g + G_{mr} (L_{V} + L_{H}) \\ = Gh sin θ + {GL}_{V} cos θ \end{matrix}$ (1)

In the Equation (1), G is the vehicle mass; a is the acceleration; h₁ is the gravity center height when the first axle passes through the weighing platform; L_V and L_H are the distance between gravity center and the front axle, rear axle; G_mr is the ground support force of second, third axle balance beam; θ is the road gradient angle; h is the gravity center height. Take the balance beam moment for the second, third axle: $\begin{matrix} G_{f} (L_{V} + L_{H}) + {Gh}_{2} sin θ \\ = {Gah}_{2} / g + {GL}_{H} cos θ \end{matrix}$ (2) $G_{mr} = G_{m} + G_{r}$ (3)

In the Equations (1 and 2), G_f, G_m, G_r is vertical acting force of front wheel, middle wheel and rear wheel; h₂ is the gravity center height when the middle and rear axle passes through the weighing platform. Suppose that gravity center heights are the same when the each axle passes through the weighing platform, and the road is smooth, so h₁ = h₂ = h, θ = 0, according to the Equations (1 and 2): $Gah / g + G_{mr} (L_{V} + L_{H}) = {GL}_{V}$ (4) $G_{f} (L_{V} + L_{H}) = Gah / g + {GL}_{H}$ (5)

Suppose that the vehicle passes through the weighing platform with a smooth, low speed, so a = 0, according to the Equations (4 and 5): $G_{mr} (L_{V} + L_{H}) = {GL}_{V}$ (6) $G_{f} (L_{V} + L_{H}) = {GL}_{H}$ (7)

Then according to the Equations (3, 6, 7): $G_{f} + G_{m} + G_{r} = G$ (8)

In order to ensure the accuracy of dynamic weighing system, the following conditions must be met: (1) the road is flat, the road and the weighing platform are at the same horizontal plane, and the gravity center height is same when the triaxial truck passes through the weighing platform. (2) the triaxial truck passes through the weighing platform with a smooth, low speed, and longitudinal acceleration is zero at weighing process. When these two conditions are met, vehicle weight can be calculated according to the Equation (8).

2.2 Vibration analysis when the triaxial truck passes through the speed bump

When the triaxial truck passes through the speed bump, it will cause vibrations of wheels, axles, body and suspension systems [14]. The vibration will attenuate with the vehicle traveling distance and time going. If the speed bump is close to the weighing platform, when the vehicle travels to the weighing platform after passing through the speed bump, and the vehicle vibration caused by speed bump does not completely eliminate, which will applied dynamic load on the weighing platform. So it will affect weighing results. This paper established the speed bump model and the triaxial truck dynamics model, based on the mathematical model to study vibration characteristics of the triaxial truck while the triaxial truck passes through a speed bump, and to analyze the reasonable distance between the speed bump and the weighing platform.

2.2.1 The speed bump model

The weighing platform of expressway generally uses the rubber hump speed bump. The surface profile of the rubber hump speed bump can be approximated thought as the arc, therefore the arc can be used to simulate the shape of the rubber hump speed bump [15]. As shown in Fig. 3, it is the static model of the rubber hump speed bump. In the Fig. 3, A is the height of the speed bump, b is the width of the speed bump, o is the center of circle, r is the radius.

Fig.3

Static model of speed bump.

When the vehicle passes through the speed bump with speed v, the dynamic incentive of the speed bump is x_v (t), and the equation of motion is the Equation (9). ${(vt - \frac{b}{2})}^{2} + (x_{v} (t) + r - A)^{2} = r^{2}$ (9) $r = \frac{(b^{2} + 4 A^{2})}{8 A}$ (10)

According to the Equations (9 and 10), the dynamic incentive of the speed bump x_v (t) can be expressed as:

$\begin{matrix} x_{v} (t) & = & \sqrt{{(\frac{{(\frac{b}{2})}^{2} + A^{2}}{2 A})}^{2} - {(vt - \frac{b}{2})}^{2}} \\ - \frac{{(\frac{b}{2})}^{2} - A^{2}}{2 A} (0 \leq t \leq b / v) \end{matrix}$ (11)

2.2.2 Dynamics model of triaxial truck

The vehicle dynamics model is a complex spring damper mass system [16], when the vehicle passes through the speed bump, the vehicle vibration system receives the input of dynamic excitation, the vehicle system will generate vibrations under this dynamic excitation, and the degree of vibration is influenced by the vibration frequency, amplitude, strength, direction and duration of the force and other factors. This paper studies the effects of vibration on the weighing accuracy after the vehicle passes through the speed bump. Therefore, for the establishment of the vehicle dynamics model, it need assumption that [17 –24]:

Assuming that the vehicle is traveling straight along the x-axis on a level surface, and the vehicle is about symmetry of the longitudinal axis (x-axis);

Only considering vertical vibration and pitching motions in the vehicle dynamics model;

Ignoring the elasticity and damping of the body, and the damping effect of the tire;

Supposing the same traveling trace for the front and rear wheels of the vehicle, that means that the phase difference of the front and rear incentives is only related to the vehicle wheelbase;

Ignoring effects of engine vibration, drive train vibration, tread, road roughness on the vehicle vibration;

Supposing that the vehicle passes through the speed bump with the slow speed, and the wheel is always kept in contact with the speed bump, which means the vehicle speed $v \leq \sqrt{gr}$ .

Based on the above assumptions, the complex vehicle model can be reduced to a dynamics model of five degrees of freedom 1/2 triaxial truck. The vehicle structure can be considered that it is composed by the sprung mass and unsprung mass, so in this paper the sprung mass vibration mainly considers the vertical motion and pitching motion, the unsprung mass vibration mainly considers vertical movements of the front wheel, the middle wheel and the rear wheel.

Fig. 4 shows the dynamics model of five degrees of freedom 1/2 triaxial truck, and the five degrees of freedom respectively are: the vertical displacement of the center of sprung mass Z₂, the pitch angle of the sprung mass θ₂, the vertical displacement of the front wheel Z_1V, the vertical displacement of the middle wheel Z_1m, the vertical displacement of the rear wheel Z_1H.

Fig.4

Dynamics model of five degrees of freedom 1/2 triaxial truck.

The dynamics equation of the front suspension system is:

$\begin{matrix} m_{1 V} {\ddot{Z}}_{1 V} \\ = - C_{1 V} (Z_{1 V} - Z_{0 V}) + C_{2 V} (Z_{2 V} - Z_{1 V}) \\ + d_{2 V} ({\dot{Z}}_{2 V} - {\dot{Z}}_{1 V}) \end{matrix}$ (12)

The dynamics equation of intermediate axle system is:

$\begin{matrix} m_{1 m} {\ddot{Z}}_{1 m} \\ = - C_{1 m} (Z_{1 m} - Z_{0 m}) + \frac{C_{2 H}}{2} (Z_{2 H} - Z_{1 mH}) \\ + \frac{d_{2 H}}{2} ({\dot{Z}}_{2 H} - {\dot{Z}}_{1 mH}) \end{matrix}$ (13)

The dynamics equation of the rear axle system is:

$\begin{matrix} m_{1 H} {\ddot{Z}}_{1 H} \\ = - C_{1 H} (Z_{1 H} - Z_{0 H}) + \frac{C_{2 H}}{2} (Z_{2 H} - Z_{1 mH}) \\ + \frac{d_{2 H}}{2} ({\dot{Z}}_{2 H} - {\dot{Z}}_{1 mH}) \end{matrix}$ (14)

The vertical vibration dynamics equation of sprung mass is:

$\begin{matrix} m_{2} {\ddot{Z}}_{2} = - C_{2 V} (Z_{2 V} - Z_{1 V}) - d_{2 V} ({\dot{Z}}_{2 V} - {\dot{Z}}_{1 V}) \\ - C_{2 H} (Z_{2 H} - Z_{1 mH}) - d_{2 H} ({\dot{Z}}_{2 H} - {\dot{Z}}_{1 mH}) \end{matrix}$ (15)

The pitch motion dynamics equation of sprung mass is:

$\begin{matrix} J_{2 X} {\ddot{θ}}_{2} & = & - C_{2 V} (Z_{2 V} - Z_{1 V}) L_{V} - d_{2 V} ({\dot{Z}}_{2 V} \\ - {\dot{Z}}_{1 V}) L_{V} + C_{2 H} (Z_{2 H} - Z_{1 mH}) L_{H} \\ + d_{2 H} ({\dot{Z}}_{2 H} - {\dot{Z}}_{1 mH}) L_{H} \end{matrix}$ (16)

The geometric relationship of the vehicle dynamics model is: $Z_{2 V} = Z_{2} + L_{V} θ_{2}$ (17) $Z_{2 H} = Z_{2} - L_{H} θ_{2}$ (18) $Z_{1 mH} = \frac{1}{2} (Z_{1 m} + Z_{1 H})$ (19)

Assuming that X₁ = Z_1V; $X_{2} = {\dot{Z}}_{1 V}$ ; X₃ = Z_1m; $X_{4} = {\dot{Z}}_{1 m}$ ; X₅ = Z_1H; $X_{6} = {\dot{Z}}_{1 H}$ ; X₇ = Z₂; $X_{8} = {\dot{Z}}_{2}$ ; X₉ = θ₂; $X_{10} = {\dot{θ}}_{2}$ ; X₀₁ = Z_0V; X₀₂ = Z_0mX₀₃ = Z_0H.

According to the Equations (12–19), they can be formed the first-order differential matrix equation: $A \dot{X} = BX + {CX}_{0}$ (20) $A = {\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & m_{1 V} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & m_{1 m} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & m_{1 H} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & m_{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{2} \end{matrix}}$

$\begin{matrix} B = {\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ - (C_{1 V} + C_{2 V}) & - d_{2 V} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & - (\frac{C_{2 H}}{4} + C_{1 m}) & - \frac{d_{2 H}}{4} & - \frac{C_{2 H}}{4} \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - \frac{C_{2 H}}{4} & - \frac{d_{2 H}}{4} & - (\frac{C_{2 H}}{4} + C_{1 H}) \\ 0 & 0 & 0 & 0 & 0 \\ C_{2 V} & d_{2 V} & \frac{C_{2 H}}{2} & \frac{d_{2 H}}{2} & \frac{C_{2 H}}{2} \\ 0 & 0 & 0 & 0 & 0 \\ C_{2 V} L_{V} & d_{2 V} L_{V} & - \frac{1}{2} C_{2 H} L_{H} & - \frac{1}{2} d_{2 H} L_{H} & - \frac{1}{2} C_{2 H} L_{H} \end{matrix} \end{matrix}$

$\begin{matrix} \begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & C_{2 V} & d_{2 V} & C_{2 V} L_{V} & d_{2 V} L_{V} \\ 0 & 0 & 0 & 0 & 0 \\ - \frac{d_{2 H}}{4} & \frac{C_{2 H}}{2} & \frac{d_{2 H}}{2} & - \frac{C_{2 H}}{2} L_{H} & - \frac{d_{2 H}}{2} L_{H} \\ 1 & 0 & 0 & 0 & 0 \\ - \frac{d_{2 H}}{4} & \frac{C_{2 H}}{2} & \frac{d_{2 H}}{2} & - \frac{C_{2 H}}{2} L_{H} & - \frac{d_{2 H}}{2} L_{H} \\ 0 & 0 & 1 & 0 & 0 \\ \frac{d_{2 H}}{2} & - (C_{2 V} + C_{2 H}) & - (d_{2 V} + d_{2 H}) & - (C_{2 V} L_{V} - C_{2 H} L_{H}) & - (d_{2 V} L_{V} - d_{2 H} L_{H}) \\ 0 & 0 & 0 & 0 & 1 \\ - \frac{1}{2} d_{2 H} L_{H} & - (C_{2 V} L_{V} - C_{2 H} L_{H}) & - (d_{2 V} L_{V} - d_{2 H} L_{H}) & - (C_{2 H} L_{H}^{2} + C_{2 V} L_{V}^{2}) & - (d_{2 V} L_{V}^{2} + d_{2 H} L_{H}^{2}) \end{matrix}} \end{matrix}$

$\begin{matrix} C = {\begin{matrix} C_{1 V} & 0 & 0 \\ 0 & C_{1 m} & 0 \\ 0 & 0 & C_{1 H} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}} \\ X = [X_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, \\ {X_{7}, X_{8}, X_{9}, X_{10}]}^{T} \\ X_{0} = {[X_{01}, X_{02}, X_{03}]}^{T} \end{matrix}$

In the Equation (20): m_1V is the unsprung mass of the front axle; m_1m is the unsprung mass of the middle axle; m_1H is the unsprung mass of the rear axle; m₂ is the sprung mass; θ₂ is the pitch angle of the center of sprung mass; J_2x is the moment of inertia around the center of sprung mass pitching motion; Z₂ is the vertical displacement of the body; Z_1V is the vertical displacement of the front wheel; Z_1m is the vertical displacement of the middle wheel; Z_1H is the vertical displacement of the rear wheel; Z_2V is the vertical displacement of the front suspension; Z_2H is the vertical displacement of the rear suspension; Z_0V is the ground roughness of the front wheel; Z_0m is the ground roughness of the middle wheel; Z_0H is the ground roughness of the rear wheel; C_2V is the front suspension stiffness; d_2V is front suspension damping; C_1V is the front tire stiffness; C_2H is the after suspension stiffness; d_2H is rear suspension damping; C_1H is the rear tire stiffness; C_1m is the middle tire stiffness.

2.3 Simulations and experiments

2.3.1 Simulation model of vibration characteristics

As shown in Fig. 5, according to the rubber hump speed bump model and the triaxial truck dynamics model, based on Matlab/Simulink, this paper establishes the vibration characteristics of simulation model while the triaxial truck passes through the speed bump.

Fig.5

Vibration characteristics simulation model when triaxial truck passing speed bump.

This paper selects the triaxial autodumper of Shaanxi Automobile Delong 3000 SX3255DR434C as the simulation prototype, Tables 1 and 2 show the vehicle parameters, suspension and wheel parameters.

Table 1

Parameters of vehicle model

Number	Parameter	Value
1	m (kg)	12100
2	m_1V (kg)	290
3	m_1m (kg)	800
4	m_1H (kg)	800
5	m₂ (kg)	10130
6	J_2x (kg*m²)	2544.2
7	L_V (mm)	2994.28
8	L_H (mm)	1433.22
9	L_S (mm)	1345

Table 2

Parameters of suspension system and wheels

Number	Parameter	Value
1	C_2V (N/m)	426087
2	d_2V (N.s/m)	7000
3	C_2H (N/m)	3362745
4	d_2H (N.s/m)	84000
5	C_1V (N/m)	1209000
6	C_1H (N/m)	1209000
7	C_1m (N/m)	1209000

According to China JT/T713-2008 “road speed bump” standards and market research, this paper chooses the representative and versatile B-type speed bump as the speed bump model. The size of B-type speed bump is: Height A = 0.05 m, width B = 0.38 m. According to the speed bump size, its radius is $r = \frac{(b^{2} + 4 A^{2})}{8 A} = 0.386 m$ (21)

According to the speed bump size and radius, the excitation function for each wheel of axle is below.

The excitation function for the wheel of front axle is

$\begin{matrix} X_{01} & = & x_{v} (t) = \sqrt{0.148996 - {(vt - 0.19)}^{2}} \\ - 0.336 (0 \geq t \geq 0.38 / v) \end{matrix}$ (22)

The excitation function for the wheel of middle axle is

$\begin{matrix} X_{02} (t) = X_{01} (t - \frac{L_{V} + L_{H} - \frac{L_{S}}{2}}{V}) \\ = \sqrt{0.148996 - {(v \times (t - 1.930591) - 0.19)}^{2}} \\ - 0.336 (1.930591 \geq t \geq 0.38 / v + 1.930591) \end{matrix}$ (23)

The excitation function for the wheel of rear axle is

$\begin{matrix} X_{03} (t) = X_{01} (t - \frac{L_{V} + L_{H} + \frac{L_{S}}{2}}{V}) \\ = \sqrt{0.148996 - {(v \times (t - 2.622108) - 0.19)}^{2}} \\ - 0.336 (2.622108 \geq t \geq 0.38 / v + 2.622108) \end{matrix}$ (24)

2.3.2 Experimental research

Based on the safe operation platform of commercial vehicle, this paper does the experimental research of the vehicle vibrating characteristics. In order to simulate the excitation signal generated by the triaxial truck through the speed bump, each wheel is fixed to the excitation platform, through inputting signals of vertical displacements to the excitation platform. The excitation platform is mounted accelerometer, displacement sensor and force sensor, they can be used to collect the vertical displacement of the unsprung mass, acceleration and force, therefore, during the experiment only needing to collect the acceleration of the center of the sprung mass. This paper uses KISTLER K-BEAM 8310A10 acceleration sensor to measure the acceleration of the center of the sprung mass, and uses DEWE- 3010 data logger to collect signals.

3 Results and discussion

3.1 Simulation results and analysis

When the vehicle passes through the speed bump with high speed, the jumping phenomenon may occur, therefore, this paper researches that the vehicle passes through speed bump below the critical speed, and the critical speed is $v_{c} = \sqrt{gr} = 1.945 m / s = 7.0 km / h$ . This paper mainly researches the vibration characteristics of the vehicle when the vehicle passes through the speed bump at 1.8 km/h, 3.6 km/h, 5.4 km/h, 7.0 km/h. It can be obtained by the simulation that the vertical displacement, velocity and acceleration of the front wheel; the vertical displacement, velocity and acceleration of the middle wheel; the vertical displacement, velocity and acceleration of the rear wheel; the vertical displacement, velocity and acceleration of sprung mass; the pitch angle, pitch angular velocity and pitch angular acceleration of sprung mass around the center of mass. The acceleration is the manifestation of force, and force is a major factor generated weighing errors, so this article researches the acceleration decay process after the vehicle passes through the speed bump. For example, when the vehicle speed is 5.4 km/h, the simulation results are shown in Figs. 6–11, and simulation results of acceleration, acceleration decay time are shown in Tables 3 and 4.

Fig.6

Excitation signal as passing speed bump at 5.4 km/h.

Fig.7

Front wheel vertical vibration acceleration as passing speed bump at 5.4 km/h.

Fig.8

Middle wheel vertical vibration acceleration as passing speed bump at 5.4 km/h.

Fig.9

Rear wheel vertical vibration acceleration as passing speed bump at 5.4 km/h.

Fig.10

Sprung mass vertical vibration acceleration as passing speed bump at 5.4 km/h.

Fig.11

Angle acceleration of pitch as passing speed bump at 5.4 km/h.

Table 3

Simulation result of acceleration of five degrees of freedom for triaxial truck passing speed bump at different speed

Velocity (km/h)	1.8	3.6	5.4	7.0
Maximum vertical acceleration of the front wheel (m/s²)	11.32	15.91	17.08	20.04
Maximum vertical acceleration of the middle wheel (m/s²)	17.01	20.85	23.34	29.91
Maximum vertical acceleration of the rear wheel (m/s²)	14.90	17.49	21.78	28.28
Maximum vertical acceleration of the center of sprung mass (m/s²)	12.41	15.50	17.73	21.67
Maximum pitch angle acceleration of sprung mass around the center of mass (°/s²)	10.19	12.84	15.66	19.31

Table 4

Simulation result of acceleration decay time of five degrees of freedom for triaxial truck passing speed bump at different speed

Velocity (km/h)	1.8	3.6	5.4	7.0
Vertical acceleration decay time of the front wheel (s)	12.41	8.09	6.45	5.64
Vertical acceleration decay time of the middle wheel (s)	12.49	7.94	6.22	5.51
Vertical acceleration decay time of the rear wheel (s)	12.64	8.17	6.53	5.67
Vertical acceleration decay time of the center of sprung mass (s)	12.80	8.22	6.67	5.81
Pitch angle acceleration decay time of sprung mass around the center of mass (s)	12.40	7.97	6.35	5.66
Minimum distance between the speed bump and the weighing platform (m)	6.40	8.22	10.01	11.30

Simulation results show that time intervals of decay to the steady state value (5% error) are basically the same for the vertical acceleration of the front wheel, the vertical acceleration of the middle wheel, the vertical acceleration of the rear wheel, the vertical acceleration of the sprung mass, pitch angle acceleration of sprung mass around the center of mass and other parameters. The maximum time interval difference is 6.75%, and the vertical acceleration time interval of the center of sprung mass is biggest. Therefore, this article chooses the sprung mass vertical acceleration decay process as the criterion, to calculate the reasonable location of the speed bump. It means that the criterion is according to the time interval of the vertical acceleration of the sprung mass that delays to the steady state value, and the traveling distance in the time interval. This traveling distance is the reasonable distance between the speed bump and the weighing platform.

3.2 Experiments results and analysis

The Fig. 12 shows the physical map of the experiment; the Figs. 13–16 show the comparison of simulation & experimental results for vertical acceleration of the front wheel, the middle wheel, the rear wheel and the center of the sprung mass while the vehicle passes through the speed bump with 5.4 km/h. The dotted line is the simulation results, and the solid line is the experimental results. The Table 5 shows the comparison of simulation & experimental results as passing speed bump at 5.4 km/h, and the Table 6 shows the comparison of simulation & experimental results triaxial truck passing speed bump at different speed.

Fig.12

Physical picture of test vehicle and test bench.

Fig.13

Comparison of simulation & experimental results for front wheel vertical acceleration.

Fig.14

Comparison of simulation & experimental results for middle wheel vertical acceleration.

Fig.15

Comparison of simulation & experimental results for rear wheel vertical acceleration.

Fig.16

Comparison of simulation & experimental results for vertical acceleration of barycenter position of sprung mass.

Table 5

Comparison of simulation & experimental results as passing speed bump at 5.4 km/h

Item	Simulation results	Experimental results	Error
Velocity (km/h)	5.4	5.4	0
Maximum vertical acceleration of the front wheel (m/s²)	17.08	16.86	–1.30%
Maximum vertical acceleration of the middle wheel (m/s²)	23.34	24.48	4.66%
Maximum vertical acceleration of the rear wheel (m/s²)	21.78	22.65	3.84%
Maximum vertical acceleration of the center of sprung mass (m/s²)	17.73	18.47	4.01%
Vertical acceleration decay time of the front wheel (s)	6.45	6.17	–4.54%
Vertical acceleration decay time of the middle wheel (s)	6.22	6.18	–0.65%
Vertical acceleration decay time of the rear wheel (s)	6.53	6.21	–5.15%
Vertical acceleration decay time of the center of sprung mass (s)	6.67	6.30	–5.87%
Minimum distance between the speed bump and the weighing platform (m)	10.00	9.45	–5.82%

Table 6

Comparison of simulation & experimental results triaxial truck passing speed bump at different speed

	Velocity (km/h)	Vertical acceleration of the center of sprung mass (m/s²)	Vertical acceleration decay time of the center of sprung mass (s)	Minimum distance (m)	Error
Simulation results	1.8	12.41	12.80	6.40	1.99%
Experimental results	1.8	14.71	13.06	6.53
Simulation results	3.6	15.50	8.22	8.22	–4.05%
Experimental results	3.6	16.15	7.90	7.90
Simulation results	5.4	17.73	6.67	10.00	–5.82%
Experimental results	5.4	18.47	6.30	9.45
Simulation results	7.0	21.67	5.81	11.30	–5.81%
Experimental results	7.0	24.71	5.49	10.68

Through simulations and experiments, this paper researches the vibration characteristics of the triaxial truck when it passes through the speed bump with 1.8 km/h, 3.6 km/h, 5.4 km/h, 7.0 km/h. Here the following conclusions can be got.

According to the comparison of simulation & experimental results as passing speed bump with 5.4 km/h as shown in the Table 5, the maximum relative error is 4.66%. It shows that the triaxial truck dynamics simulation model can truly reflect the actual vibrating characteristics of the vehicle.

According to the comparison of simulation & experimental results for vertical acceleration decay time of the front wheel, the middle wheel, the rear wheel and the center of the sprung mass while the vehicle passes through speed bump with 5.4 km/h, the maximum relative error is 5.87%. It shows that the parameters of the triaxial truck dynamics simulation model and the parameters of the actual vibration characteristics are the same.

The vertical acceleration decay times of the front wheel, the middle wheel, the rear wheel and the center of the sprung mass basically are the same, the biggest difference is 6.75%, and the vertical acceleration time interval of the center of sprung mass is biggest. Therefore, choosing the sprung mass vertical acceleration decay process as the criterion, to calculate the reasonable location of the speed bump.

When the vehicle passes through the speed bump with different speed, and with the increase of vehicle speed, the vertical acceleration and the vertical acceleration decay times of the front wheel, the middle wheel, the rear wheel and the center of the sprung mass will also increase, and the distance between the speed bump and the weighing platform will also increase. When the vehicle speed is 7.0 km/h, the distance between the speed bump and the weighing platform is biggest, the simulation and experimental results are 11.30 m/10.68 m.

4 Conclusions

The reasonable position between the speed bump and the weighting platform is researched in the paper based on vibration characteristics of triaxial truck. First of all, factors which influence weighing error are analyzed and restricted according to the vehicle driving state and the mechanical model of weighing process. And then simulation model of vibration process for triaxial truck passing speed bump is built based on the hump speed bump model and dynamics model of five degrees of freedom for triaxial truck. Vibration characteristics of vehicle after passing speed bump at the speed of 1.8 km/h, 3.6 km/h, 5.4 km/h and 7.0 km/h is researched. Vibration characteristics experiment of vehicle when passing speed bump is also researched based on the commercial vehicle safety operation platform. Finally, comparison is made on the simulation results and experiment results. It shows that with the rising of vehicle speed, the distance between speed bump and weighting platform is also increases. The reasonable position between speed bump and weighting platform is obtained when vehicle passes speed bump at the speed of 7.0 km/h. Simulation result and experiment result of the distance between speed bump and weighting platform is respectively 11.30 m and 10.68 m. The results of simulation and experiment indicate that model of vibration in the process of triaxial truck passing the speed bump can accurately calculate the minimum distance between speed bump and truck scale with different speed, which maximum error is 5.81%.

Footnotes

Acknowledgments

This work was supported by Natural Science Foundation of China (Grant: 51507013), Specialized Research Fund for the Doctoral Program of Higher Education (Grant: 20110205110008), Natural Science Foundation of Shaanxi Province (Grant: 2016JQ5012), the Science and Technology research project of Shaanxi Province (Grant: 2016GY-043), Chang’an university (Grant: 310822151025 & Grant: 310822162019).

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