Keywords

1 Introduction

The dynamic load acting on the vehicle is the cause of vibration, loss of dynamics safety, vehicle damage and road destruction. The specifications are used to evaluate the dynamic load acting on vehicle as following formulas:

  1. (i)

    The maximum dynamic load factor acting on the chassis (ηmax): This specification is used to evaluate the smooth movement and durability of the vehicle [1,2,3].

    $$ \eta_{\max } = \frac{{F_{C\max } + F_{K\max } }}{{F_{\text{G}} }};\;\eta_{\max } \le \,\,1.5 $$
    (1)
  2. (ii)

    The maximum dynamic load factor acting on the tire (Kdmax): This specification is used to evaluate the dynamics safety, smooth movement and durability of the vehicle [1,2,3].

    $$ K_{d\max } = \frac{{F_{CL\max } }}{F_G } + 1;\;K_{d\max } \le \,\,2.5 $$
    (2)
  3. (iii)

    The minimum dynamic load factor acting on the tire (Kdmin): This specification is used to evaluate the vehicle dynamics safety and tire transmission capacity [1,2,3].

    $$ K_{d\min } = \frac{{F_{CL\min } }}{F_G } + 1;\;0 \le K_{d\min } \le 1 $$
    (3)

Kdmin = 0.5 within the warning limit; Kdmin = 0 is the intervention limit.

2 The Three-Dimensional Dynamics Model

The method of separating the structure of the multibody system is used to build a three-dimensional (3D) dynamics model of DVM 2.5 truck is shown in Fig. 1 [4,5,6].

Fig. 1.
figure 1

Dynamics model of the DVM 2.5 truck

Full size image

The Newton–Euler equations are used to build the system of dynamics equations for DVM 2.5 truck as follows [4,5,6] (Table 1):

$$ m\ddot{x} = F_{x1j} c{\text{os}}\delta_{1j} - F_{y1j} \sin \delta_{1j} + \,F_{x2j} ;\,j = 1\,is\,left\,wheel;\;i = 2\,is\,right\,wheel $$
(4)
$$ m\ddot{y} = F_{x1j} \sin \delta_{1j} + F_{y1j} c{\text{os}}\delta_{1j} \, + \,F_{y2j} ;\;j = 1\,is\,left\,wheel;\;i = 2\,is\,right\,wheel\, $$
(5)
$$ J_z \ddot{\psi } = (F_{x1j} \sin \delta_{1j} + F_{y1j} c{\text{os}}\delta_{1j} )l_1 + (F_{xi2} - F_{xi1} )b_i - F_{y2j} l_2 $$
(6)
$$ m\ddot{z} = F_{Cij} + F_{Kij} ;\,\,\,\,i = 1 \div 2;\,\,j = 1\,\,\,is\,left\,wheel;\,\,i = 2\,\,\,is\,right\,wheel\, $$
(7)
$$ J_y \ddot{\varphi } = (F_{C1j} + F_{K1j} )l_1 - (F_{C2j} + F_{K2j} )l_2 + M_{1j} + M_{2j} ;j = 1\;is\;lef\;wheel;\;i = 2\;is\;right\;wheel $$
(8)
$$ J_x \ddot{\beta } = (F_{Ci2} + F_{Ki2} - F_{Ci1} - F_{Ki1} )w_i ;\;i = 1 \div 2 $$
(9)
$$ m_{A1} \ddot{z}_{A1} = F_{CLij} + F_{KLij} - F_{Cij} - F_{Kij} ;\,\;i = 1 \div 2;\;j = 1\;is\,left\,wheel;\,\,i = 2\,is\,right\,wheel $$
(10)
$$ m_{A1} \ddot{y}_{A1} = F_{y1j} \,;\,\;j = 1\,is\,left\,wheel;\,\,i = 2\,is\,right\,wheel $$
(11)
$$ J_{Ax1} \ddot{\beta }_{A1} = (F_{C11} + F_{K11} - F_{C12} - F_{K12} )w_1 + (F_{CL12} + F_{KL12} - F_{CL11} - F_{KL11} )b_1 \, - F_{y11} (r_{11} + \xi_{A11} ) - F_{y12} (r_{12} + \xi_{A12} ) $$
(12)
$$ m_{A2} \ddot{z}_{A2} = F_{CL2j} + F_{KL2j} - F_{C2j} - F_{K2j} ;\,\;j = 1\,is\,left\,wheel;\,\,i = 2\,is\,right\,wheel $$
(13)
$$ m_{A2} \ddot{y}_{A2} = F_{y2j} ;\,\;j = 1\,is\,left\,wheel;\,\;i = 2\,is\,right\,wheel $$
(14)
$$ J_{Ax2} \ddot{\beta }_{A2} = (F_{C21} + F_{K21} - F_{C22} - F_{K22} )w_2 + (F_{CL22} + F_{KL22} - F_{CL21} - F_{KL21} )b_2 - F_{y21} (r_{21} + \xi_{A21} ) - F_{y22} (r_{22} + \xi_{A22} ) $$
(15)
$$ J_{Ayij} \ddot{\varphi }_{ij} = M_{Aij} - M_{Bij} - F_{xij} r_{dij} ;\,i = 1 \div 2;\,j = 1\,is\,left\,wheel;\;\,i = 2\,is\,right\,wheel $$
(16)
Table 1. A list of symbols and abbreviations
Full size table

3 Survey Results and Discussions

Matlab-Simulink software is used to consider the tire vertical stiffness on dynamic load acting on DVM 2.5 truck. The DVM 2.5 truck is fully loaded and is run at speeds V0 = [30, 40, 50, 60] kph on a class E road according to ISO 8608:2016 is shown in Fig. 2 [7].

Fig. 2.
figure 2

Road surface according to ISO 8608:2016 standards

Full size image

The tire vertical stiffness of the DVM 2.5 truck is CL = 652 kN/m. When surveying, the tire vertical stiffness value is changed as CL0 = [0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5] CL, the survey results are as follows:

Fig. 3.
figure 3

Maximum dynamic load factor acting on the chassis

Full size image

When the DVM 2.5 truck is run on a class E road at speeds V0 = [30 ÷ 60] kph, if the tire vertical stiffness is increased from CL0 = [0.5 ÷ 1.5] CL, the maximum dynamic load factor acting on the chassis at the front axle is about η1max = [0.68 ÷ 0.88] shown in Fig. 3a. According to Fig. 3a, the value of the maximum dynamic load factor acting on the chassis at the front axle is \(\eta_{1\max } \,\, \le \,\,\eta_{\max } = \,\,1.5\), the truck has smooth movement and durability.

When the DVM 2.5 truck is run on a class E road at speeds V0 = [30 ÷ 60] km/h, if the tire vertical stiffness is increased from CL0 = [0.5 ÷ 1.5] CL, the maximum dynamic load factor acting on the chassis at the rear axle is about η2max = [0.46 ÷ 0.91] shown in Fig. 3b, \(\eta_{2\max } \,\, \le \,\,\eta_{\max } = \,\,1.5\), the truck has smooth movement and durability.

Figure 4a is a graph showing the maximum dynamic load factor acting on the front tires when the DVM 2.5 truck is run on a class E road at speeds V0 = [30 ÷ 60] kph. According to Fig. 4a, the tire vertical stiffness is increased from CL0 = [0.5 ÷ 1.5] CL, the maximum dynamic load factor acting on the front tire is about Kd1max = [1.62 ÷ 1.96], \(K_{d1\max } \,\, \le \,\,K_{d\max } = \,\,2.5\). The front tire of the DVM 2.5 truck has smooth movement and durability, and dynamics safety.

Figure 4b is a graph showing the maximum dynamic load factor acting on the rear tires when the truck is run on a class E road at speeds V0 = [30 ÷ 60] kph. When the tire vertical stiffness is increased from CL0 = [0.5 ÷ 1.5] CL, the maximum dynamic load factor acting on the rear tires is increased, Kd2max = [1.56 ÷ 2.1], \(K_{d2\max } \,\, \le \,\,K_{d\max } = \,\,2.5\). The rear tires of the DVM 2.5 truck has smooth movement and durability, and dynamics safety.

Fig. 4.
figure 4

Maximum dynamic load factor acting on the tires

Full size image

The minimum dynamic load factor acting on the front tires when the DVM 2.5 truck is run on a class E road at speeds V0 = [30 ÷ 60] kph shown in Fig. 5a. When the tire vertical stiffness is increased from CL0 = [0.5 ÷ 1.5] CL, the minimum dynamic load factor acting on the front tires drops, Kd1min = [0.36 ÷ 0]. When the DVM 2.5 truck is run at V0 = [30 ÷ 50] kph, Kd1min is within the warning limit, 0 < Kd1min ≤ 0.5. The front tires of the DVM 2.5 truck have dynamics safety and tire transmission capacity. When the DVM 2.5 truck is run at a speed of V0 = 60 kph, the value of the minimum dynamic load factor acting on the front tires Kd1min = 0, Kd1min within the intervention limit. The front tires of the DVM 2.5 truck don’t have dynamics safety and tire transmission capacity.

The minimum dynamic load factor acting on the rear tires is shown in Fig. 5b. When the tire vertical stiffness is increased from CL0 = [0.5 ÷ 1.5] CL, the minimum dynamic load factor acting on the rear tires drops, Kd2min = [0.54 ÷ 0]. When CL0 = [0.5 ÷ 1.4] CL, Kd2min is within the warning limit, 0 < Kd2min ≤ 0.5. The rear tire of the DVM 2.5 truck has dynamics safety and tire transmission capacity. When CL0 = 1.5C, Kd2min is within the intervention limit, Kd2min = 0. The rear tires of the DVM 2.5 truck don’t have dynamics safety and tire transmission capacity.

Fig. 5.
figure 5

Minimum dynamic load factor acting on the tires

Full size image

4 Conclusions

When the tire vertical stiffness is increased, the maximum dynamic load value on the tires, chassis and road rises, and the transmission capacity drops. Therefore, the smooth movement and durability, and dynamics safety of the DVM 2.5 truck drop. A fully loaded DVM 2.5 truck is run on a class E road according to ISO 8608:2016, the speeds must be less than 50 kph and the tire vertical stiffness must be less than 912 kN/m (or CL0 ≤ 1.4CL), the DVM 2.5 truck is ensured dynamics safety and durability.