Analysis and optimization of coupled vibration between substructures of a multi-axle vehicle

Abstract

The vibration from transport vehicles may negatively affect the ride comfort in the cab and the product safety in the carriage. A 15 degrees of freedom vehicle model consisted of a cab, a carriage, a chassis, and mounts and suspensions between them are introduced to present the vibration behavior of a three-axle vehicle. Two indices, the coupling factor and the vibration attenuation factor, are employed to quantify the correlation between structures and the vibration isolation capability of subsystems. A sensitivity analysis is carried out with a 43-factor optimal Latin hypercube design, concerning the effect of mass properties, geometries, stiffness, and damping on the vibration coupling effect and the vibration attenuation of subsystems. Results show obvious trade-offs between the vibration coupling effect and the vibration attenuation among different subsystems. Based on the significant factor identified, multi-objective optimizations are conducted to improve the vibration performance of both the cab and the carriage and simultaneously reduce the correlations between structures using different algorithms. Comparison between different optimal results indicates that a compromise can be achieved between the ride comfort and the cargo safety based on a lightweight constraint.

Keywords

Multi-body dynamic modelling coupled vibration ride comfort sensitivity analysis multi-objective optimization

1. Introduction

In road transportation, the vibration of the carriage has great influence on the cargo safety, which is as important as the ride comfort in the cab. Moreover, in some particular cases, such as medical transport or sophisticated equipment delivery, the vibration at the carriage is the paramount concern.

Since the vehicle can be considered as a highly coupled dynamic system, any changes in its substructure size and weight, mount and suspension properties, and the arrangement of the connection may affect the vehicle dynamics as well as the vibration performance. To optimize the ride comfort and handling properties of the vehicle, a proper method of modeling should be considered firstly. A multibody model of city bus suspension was presented by Kowarska et al. (2011) which could be used as an input for optimization of the bus suspension. Equivalent modeling methods are also an available way for vehicle modeling. Zhang et al. (2018) proposed a model-based dynamic equivalence of force/moment at the center of gravity whose performance has been proved by simulation and comparison with other equivalence methods. Various necessary fundamental researches before the optimization of the suspension system have been done, such as the effect of controllable damper on semi-active suspension control strategies (Qin et al., 2017b), and road excitation classification based on system response (Qin et al., 2017a). When it comes to the optimization of suspension, much work has been done concerning the improvement of ride comfort (Van Deusen, 1973; Elmadany and Abduljabbar, 1990; Cole and Cebon, 1996; Valášek et al., 1998a, 1998b; Cole, 2001; Nakano, 2004; Jarimopas et al., 2005; Ieluzzi et al., 2006; Polach and Hajžman, 2006; Rajamani, 2006; Singh et al., 2006). Although the spring and the damper of the suspension are the main filters to isolate the vibration transmitted from the road roughness, the elastic connections between the substructures, such as the powertrain mounts, the cab mounts, and the carriage mounts, also significantly affect the local vibrations. Various studies have been conducted in designing the mount shape (Kim and Kim, 1997; Ward, 2002), optimizing the stiffness and damping of the mount (Baum et al., 1977; Lee et al., 2002; Arzanpour et al., 2006; Hartley and Choi, 2012), searching for the optimal mounting position (Flower, 1978; El Hafidi et al., 2010; Xu et al., 2011), and proposing novel mounting arrangement (Shital et al., 2015).

Nevertheless, either the engine mount or the cab mount is investigated just focusing on a specific individual subsystem. This means that the coupled effect between different subsystems has not been accounted for. However, the chassis, the cab, and the carriage of a cargo vehicle can be considered as a multi-supported system, so the correlations in vibration between the three parts should be considered in the design or optimization of each single part. The reason is that, in a coupled multi-body system, any modifications in structure or subsystem may introduce adverse effect in other structures. Thus, it is important to know the coupling effect between structures in advance of optimization in order to avoid unexpected influences in the whole system.

In this paper, a coupling effect identification method is introduced and applied in a three-axle vehicle. The coupling effects between the cab and the chassis and between the carriage and the chassis are estimated, as well as the vibration attenuation capability of the two subsystems. A sensitivity analysis is carried out to identify the effects of three sorts of design parameters, including mass properties, mount and suspension parameters, and geometries, on the subsystem coupling and the vibration attenuation. Then multi-objective optimizations aiming at reducing the coupling effect and increasing the vibration attenuation of subsystems are carried out using different algorithms. A detailed comparison and discussion on the optimal results are presented.

2. Mathematical model

2.1. Vehicle model

The three-axle vehicle model is developed, as shown in Figure 1, consisting of a cab (A, in red), a carriage (B, in blue), a chassis (C, in green), the front, middle and rear axles (E, F, and G respectively, in grey), and the ground. The cab is supported on the chassis by four mounts (red hollow spots), and the carriage is connected to the chassis through eight mounts (blue hollow spots). The axles are linked to the chassis by the front and rear suspensions (grey hollow spots), and the axles are jointed to the ground by tyres. All the connections between substructures are considered elastic and simplified as spring-damper parallels except tyres (only stiffness is considered). The substructures are regarded as rigid bodies with masses and moments of inertias integrated on their centers of gravity ( shown as solid spots). Vertical (z), roll (α) and pitch (β) motions of the cab, the carriage, and the chassis, and vertical and roll motions of the axles are accounted for such that 15 degrees of freedom are included in the equations of motion.

2.2. Performance indices

Each substructure of the vehicle is subjected to two sorts of load: the external excitations; and the internal forces from adjacent parts. In this section, the performance indices are derived by taking the cab–chassis subsystem for example, and those in the carriage–chassis case follow the same form. According to the state space theory (Preumont, 1997), the state equations of the cab–chassis subsystem can be written as

\begin{matrix} {\begin{matrix} {\overset{\cdot}{Q}}_{A} = D_{A} Q_{A} + \sum_{i = 1}^{n} A_{CA}^{i} Q_{C} + B_{A} U_{A} \\ {\overset{\cdot}{Q}}_{C} = D_{C} Q_{C} + \sum_{j = 1}^{m} A_{AC}^{j} Q_{A} + B_{C} U_{C} \end{matrix} \end{matrix}

(1)

where Q _A and Q _C are state variables of the cab and the chassis, D _A and D _C are the state matrices of the cab and the chassis, A _CA and A _AC are the interaction matrices of the states of the cab and the chassis, B _A and B _C are the external excitations loaded on the cab and the chassis, U _A and U _B are the internal forces applied on the cab and the chassis, i and j are the number of supports of the cab and the carriage, and n = 4 and m = 8 in this case. Q _A, Q _C, D _A, D _C, A _CA, and A _AC take the forms as

Q_{A} = {\begin{matrix} {\overset{\cdot}{q}}_{A} \\ q_{A} \end{matrix}}, and q_{A} = (z_{A}, α_{A}, β_{A})^{T}

(2)

Q_{C} = {\begin{matrix} {\overset{\cdot}{q}}_{C} \\ q_{C} \end{matrix}}, and q_{C} = (z_{C}, α_{C}, β_{C})^{T}

(3)

D_{A} = [\begin{matrix} - M_{A}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{A}^{k, l}^{T} T^{k, l}^{T} C_{M}^{k, l} T^{k, l} E_{A}^{k, l} & - M_{A}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{A}^{k, l}^{T} T^{k, l}^{T} K_{M}^{k, l} T^{k, l} E_{A}^{k, l} \\ I & 0 \end{matrix}]

(4)

D_{C} = [\begin{matrix} - M_{C}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{C}^{k, l}^{T} T^{k, l}^{T} C_{M}^{k, l} T^{k, l} E_{C}^{k, l} & - M_{C}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{C}^{k, l}^{T} T^{k, l}^{T} K_{M}^{k, l} T^{k, l} E_{C}^{k, l} \\ I & 0 \end{matrix}]

(5)

A_{AC} = [\begin{matrix} - M_{C}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{C}^{k, l}^{T} T^{k, l}^{T} C_{M}^{k, l} T^{k, l} E_{A}^{k, l} & M_{C}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{C}^{k, l}^{T} T^{k, l}^{T} K_{M}^{k, l} T^{k, l} E_{A}^{k, l} \\ 0 & 0 \end{matrix}]

(6)

A_{CA} = [\begin{matrix} - M_{A}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{A}^{k, l}^{T} T^{k, l}^{T} C_{M}^{k, l} T^{k, l} E_{C}^{k, l} & M_{A}^{- 1} \sum_{k = 1}^{p} \sum_{l = 1}^{q} E_{A}^{k, l}^{T} T^{k, l}^{T} K_{M}^{k, l} T^{k, l} E_{C}^{k, l} \\ 0 & 0 \end{matrix}]

(7)

where E _A and E _C are the displacement transform matrices of the cab and the chassis, T is the transform matrix from the local coordinates to the global coordinates, M _A and M _C are the mass matrices of the cab and the chassis, and K _M and C _M are the principle stiffness and damping matrices of the mounts connecting the cab and the chassis. The combination of the index of summation, k,l, means the lth support of the kth substructure (p = 1 and q = n for cab, and p = 2 and q = m for carriage).

Thus, the state equation of the cab–chassis subsystem can be rewritten as

\begin{matrix} {\begin{matrix} {\overset{\cdot}{Q}}_{A} \\ {\overset{\cdot}{Q}}_{C} \end{matrix}} = [\begin{matrix} D_{A} & \sum_{i = 1}^{n} A_{CA}^{i} \\ \sum_{j = 1}^{m} A_{AC}^{j} & D_{C} \end{matrix}] {\begin{matrix} Q_{A} \\ Q_{C} \end{matrix}} \\ + [\begin{matrix} B_{A} \\ B_{C} \end{matrix}] {\begin{matrix} U_{A} \\ U_{C} \end{matrix}} \end{matrix}

(8)

To assess the influence of substructure variable on the cab–chassis subsystem, the state matrix can be divided into two components as

\begin{matrix} D = [\begin{matrix} D_{A} \\ D_{C} \end{matrix}] + [\begin{matrix} \sum_{i = 1}^{n} A_{CA}^{i} \\ \sum_{j = 1}^{m} A_{AC}^{j} \end{matrix}] \\ = D_{sub} + \sum_{i = 1, j = 1}^{n, m} A_{i, j} \end{matrix}

(9)

where D indicates the state matrix of the coupled subsystem, D _sub is the independent part, and A _i,j is the interacted part. Accordingly, a coupled effect factor, f_c, is introduced as (Zhang et al., 2016)

f_{c, i} = ‖ A_{i, j} ‖ F / ‖ A_{i, j} ‖ F ‖ D ‖ F ‖ D ‖ F

(10)

where the ‖ ‖_F is the Frobenius norm. The value of f_c quantified the correlation between substructures of a subsystem, so a higher f_c means a stronger correlation between substructures. Furthermore, to observe the subsystem performance in vibration attenuation, an inspection matrix (Anderson et al., 1986) is introduced in the form of

H (D) = \int_{0}^{\infty} exp (D^{T} t) exp (D t) d t

(11)

Therefore, the vibration attenuation of a subsystem can be assessed by taking the ratio of the inspection matrices with and without the contribution of the target subsystem. A vibration attenuation factor, f_v, can be written as

{f_{v,}}_{i} = ‖ H (D + \sum_{j = 1}^{n} A_{i, j}) ‖ 2 / ‖ H (D + \sum_{j = 1}^{n} A_{i, j}) ‖ 2 ‖ H (D + \sum_{j = 1, j \neq i}^{n} A_{i, j}) ‖ 2 ‖ H (D + \sum_{j = 1, j \neq i}^{n} A_{i, j}) ‖ 2

(12)

With a high value of f_v, the vibration transmitted to the target substructure would be low, and vice versa.

2.3. Model validation

A vehicle road test was conducted in proving ground. The vertical vibration at the carriage center were measured when the vehicle was traveling on a flat straight road at 40, 50, 60, 70, and 80 km/h. The data were recorded for 120 seconds in length with 512 Hz sampling rate and each case was repeated 3 times.

The road roughness used in the simulation is generated according to the B-class road in ISO 8608: 2016 Standard (International Organization for Standardization, 2016). In this paper, the road roughness data for the left and right tracks are different so as to introduce roll movement of the vehicle. A 1000 m straight road is obtained as shown in Figure 2. The experimental and the calculated root-mean-square (RMS) accelerations at different speeds are listed in Table 1. The variable values of the calibrated model are taken as the baseline values (as given in Table 2) in the following design of the experiment.

Table 1.

Comparison of root-mean-square (RMS) acceleration between test and calculation.

Speed (m/s)	RMS acceleration (m/s²)
Speed (m/s)	Calculation	Test	Relative error
40	1.23	1.12	0.10
50	1.44	1.27	0.13
60	1.56	1.40	0.11
70	1.51	1.59	−0.05
80	1.81	1.61	0.12

Table 2.

Design variables of the three-axle vehicle.

Variable	Unit	Definition	Baseline	Range
m _A	kg	Mass of the cab	2285	1599–2970
I _xA	kgm²	Moment of inertia of the cab about the longitudinal direction	313	220–408
I _yA	kgm²	Moment of inertia of the cab about the lateral direction	478	335–622
m _B	kg	Mass of the carriage	10000	7000–13000
I _xB	kgm²	Moment of inertia of the carriage about the longitudinal direction	1257	880–1635
I _yB	kgm²	Moment of inertia of the carriage about the lateral direction	1519	1064–1975
m _C	kg	Mass of the chassis	8685	6080–11291
I _xC	kgm²	Moment of inertia of the chassis about the longitudinal direction	721	505–937
I _yC	kgm²	Moment of inertia of the chassis about the lateral direction	14981	10487–19476
K _A1-2	N/m	Stiffness of the mount of the cab	11873	8312–15436
C _A1-2	Ns/m	Damping of the mount of the cab	5292	3705–6880
K _B1-B4	N/m	Stiffness of the mount of the carriage	4830668	3381468–6279869
C _B1-B4	Ns/m	Damping of the mount of the carriage	1298	909–1688
K _D	N/m	Stiffness of the front suspension	595389	416773–774006
C _D	Ns/m	Damping of the front suspension	61289	42902–79676
K _EF	N/m	Stiffness of the rear suspension	1552193	1086535–2017851
C _EF	Ns/m	Damping of the rear suspension	61289	42902–79676
l _Ax	m	Longitudinal distance between centers of gravity (CGs) of the cab and the chassis	2.886	2.020–3.752
l _Bx	m	Longitudinal distance between CGs of the carriage and the chassis	0.117	0.082–0.153
l _Dx	m	Longitudinal distance from the front suspension to the CG of the chassis	2.610	1.828–3.394
l _Dy	m	Lateral distance from the front suspension to the CG of the chassis	0.744	0.521–0.967
l _EFx	m	Longitudinal distance from the rear suspension to the CG of the chassis	1.464	1.025–1.904
l _EFy	m	Lateral distance from the rear suspension to the CG of the chassis	0.744	0.521–0.967
l _A1-2x	m	Longitudinal distance from the cab mount to the CG of the cab	1.090	0.763–1.417
l _A1-2y	m	Lateral distance from the cab mount to the CG of the cab	0.692	0.484–0.899
l _B1,4x	m	Longitudinal distance from the front/rear carriage mounts to the CG of the carriage	2.062	1.444–2.681
l _B2,3x	m	Longitudinal distance from the middle carriage mounts to the CG of the carriage	1.021	0.715–1.328
l _B1-4y	m	Lateral distance from the carriage mount to the CG of the carriage	1.094	0.766–1.423

3. Design of experiment

3.1. Design variables

In this study, three sorts of design variables are considered in the sensitivity analysis: the substructure mass properties; the elastic connection properties; and the vehicle geometries. The mass properties include the mass and the moments of inertia about the longitudinal and lateral directions of the cab, the carriage, and the chassis. The equivalent stiffness and damping of the mount and the suspensions are used as the elastic connection properties. The geometries mean the relative distances between substructures and connecting points. In total, 43 design parameters are used in the sensitivity analysis and the parameter range is set as ± 30% over the baseline value, as listed in Table 2.

3.2. Design method

In the sensitivity analysis, four responses are studied including the coupling factors and the vibration attenuation factors of the cab–chassis subsystem and the carriage–chassis subsystem. The optimal Latin hypercube technique is employed to conduct the design of the experiment in order to allow more points and combinations to be involved so that the nonlinearity of the effect of the design parameter on the response could be detected. Additionally, to make sure that the design of the experiment is reproducible, fixed seeds are used to generate the design chart. Finally, a 43-factor optimal Latin hypercube design is conducted with 5000 runs. Quadratic regression is used to build the relationship between the response and the design parameter.

The relationship between the substructure coupling and the subsystem vibration attenuation is also studied. Based on the sampling results, the response surface method is used to find out the influence of substructure coupling on the subsystem vibration attenuation.

4. Effect of design parameter

4.1. Parameter significance

The contributions of the design parameters were identified according to their standardized effects on the responses using the quadratic regression. Pareto charts of the first 20 terms of the standardized effects on the f_c and f_v were calculated and given in absolute values as shown in Figures 3 and 4, respectively. The horizontal bars in Figures 3 and 4 can be considered as the contributions, and the dashed line is the reference limit calculated based on the t-statistic with the confidence level of 0.95. A bar exceeding the reference limit means that the corresponding design parameter is statistically significant.

For the coupling effect of the cab–chassis subsystem, 11 design parameters have a statistically significant effect in which the contributions from the longitudinal distance between the front cab mount and the cab CG, the stiffness of the front cab mount, and the moment of inertia of the cab about the lateral direction are much larger than those of other design parameters (see Figure 3 (a)). Influences of the other parameters of the coupling effect of the cab–chassis subsystem are given in Figure 3 (b). Similar information concerning the vibration attenuation of the carriage–chassis subsystem is given in Figure 4.

4.2. Effect on substructure coupling

After identifying the contributions of design parameters to the four responses, the main effects of statistically significant parameters were investigated to study how they affect the corresponding response. The main effects were fitted continuously according to the parameter range and are shown in Figures 5 –9.

From Figure 5, it is found that the increases of all the mass properties decrease the coupling effect in the cab–chassis subsystem, and the decreasing effect caused by the moment of inertia of the cab about the lateral direction is the greatest. For the mount properties, to increase the rear mount stiffness can reduce the coupling effect of the cab–chassis subsystem, but to increase the stiffness and damping of the front mount increases the coupling effect of the cab chassis subsystem. In terms of the geometry parameters, the coupling effect of the cab–chassis subsystem increases with increasing distances between the mounts and the CG of the cab, and between the CG of the cab and the CG of the chassis except for the longitudinal distance between the rear mount of the cab and the CG of the cab. Information about the carriage–chassis subsystem is given in Figure 6.

4.3. Effect on subsystem vibration attenuation

The main effects of the significant design parameters on the vibration attenuation of the cab–chassis and the carriage–chassis subsystems are shown in Figure 7 and Figure 8. It can be found from Figure 7 that many mass properties are involved. To increase the mass properties can decrease the vibration attenuation of the cab–chassis subsystem, except for the lateral moment of inertia. In other words, the mass reductions in the cab, the carriage, and the chassis can benefit the ride quality of the vehicle. For the mount properties, only the stiffness of the front middle mount of the carriage and the damping of the front mount of the cab can significantly affect the vibration attenuation of the cab–chassis subsystem. To increase the stiffness of the front middle mount of the carriage can decrease the vibration attenuation effect, while to increase the damping of the front mount of the cab can increase the vibration attenuation effect. Information about the carriage–chassis subsystem is given in Figure 8.

4.4. Relationship between vibration coupling and vibration attenuation

It is found that the significant parameters vary according to the response of interest and different parameters have very different effects on the response as well. So, the relationship between the coupling effect and the vibration attenuation of the subsystem is developed using the response surface method, as shown in Figure 9.

Figure 9 (a) indicates that to obtain a high vibration attenuation of the cab–chassis subsystem, a high coupled cab–chassis subsystem and a low coupled carriage–chassis subsystem should be achieved at the same time. However, Figure 9 (b) shows that a preferable vibration attenuation of the carriage–chassis cannot be obtained with a low coupling effect of the carriage–chassis subsystem, no matter how large the coupling effect of the cab-carriage subsystem is. As it is known, a system with low-coupled structures is easy for the vibration control, and vice versa. Meanwhile, the higher vibration attenuation performance that a vehicle system has, the better the ride quality and the higher cargo safety will be. Consequently, trade-off effects exist not only between the vibration attenuation and the coupling effect of a specific subsystem but also between the vibration attenuations of the two subsystems. Compromises should be made to achieve better ride vibration in the cab and in the carriage at the same time.

5. Optimization

5.1. Objective function

The objective of the optimization in this section is to find out a set of parameters that can achieve low coupling effect of the subsystem and high vibration attenuation of the subsystem. The parameter identification is formulated as the following optimization problem

Find: ξ = [m_A, I_xA, I_yA, m_B, I_xB, I_yB, m_C, I_xC, I_yC, K_A1, C_A1, K_A2, K_B1, K_B2, K_B3, C_B3, K_B4, C_B4, C_D, l_Ax, l_A1x, l_A1y, l_A2x, l_A2y, l_B1x, l_B1y, l_B2x, l_B2y, l_B3x, l_B4x, l_B4y, l_Dx, l_EFx]^T

Minimize: f_{c,cab–chassis} and f_{c,carriage–chassis}

Maximize: f_{v,cab–chassis} and f_{v,carriage–chassis}

Subject to: $ξ_{i}^{LB} < ξ_{i} < ξ_{i}^{UB} \begin{matrix} (i = 1, 2, \dots, 33) \end{matrix}$

where LB and UB represent the lower and upper limits of the design parameter corresponding to 70% and 130% of the baseline value.

5.2. Optimization algorithms

Four multi-objective optimization algorithms are employed: the multi-objective particle swarm (MOPS); the nondominated sorting genetic algorithm-II (NSGA-II); the neighborhood cultivation genetic algorithm (NCGA); and the adaptive simulated annealing (ASA).

The MOPS is conducted with 33 particles of 0.9 inertia, the global increment and the particle increment are 0.9, and the maximum velocity is 0.1. For the NSGA-II, the population size is 60, the number of generations is 100, the cross-probability is 0.9, and the distribution indices of the crossover and the mutation are 10 and 20, respectively. The population size and the generation number of the NCGA are the same as the NSGA-II, and the crossover rate and the mutation rate are 1 and 0.01, respectively. In the ASA, the relative rates of parameter annealing, cost annealing, parameter quenching, and cost quenching are all set as 1, and the initial parameter temperature is also 1.

5.3. Optimization results

The optimized responses and the corresponding parameter sets are listed in Table 3 and Table 4. It is found that to find the optimal parameter set the MOPS runs 3103 iterations in total, the NSGA-II runs 5966 iterations, the NCGA runs 4710 iterations, and the ASA runs 6253 iterations. The MOPS is the fastest algorithm to reach the convergence, and the ASA takes the longest time to find the global optimal.

Table 3.

Optimal responses based on different algorithms.

	Multi-objective particle swarm	Nondominated sorting genetic algorithm-II	Neighbourhood cultivation genetic algorithm	Adaptive simulated annealing
f _{c,cab–chassis}	210	79	145	112
f _{c,carriage–chassis}	36728	29118	32449	27037
f _{v,cab–chassis}	1.001	1.939	1.089	1.003
f _{v,carriage–chassis}	0.916	2.921	1.489	1.187

Table 4.

Optimal parameter sets based on different algorithms.

	Multi-objective particle swarm	Nondominated sorting genetic algorithm-II	Neighbourhood cultivation genetic algorithm	Adaptive simulated annealing
m_A (kg)	2970	2759	1662	2929
I_xA (kgm²)	220	387	404	353
I_yA (kgm²)	335	406	573	605
m_B (kg)	13000	7054	7878	12774
I_xB (kgm²)	880	1577	1516	1634
I_yB (kgm²)	1975	1966	1961	1974
m_C (kg)	11291	6081	6413	11117
I_xC (kgm²)	937	910	934	934
I_yC (kgm²)	17396	19428	17613	18465
K_A1 (N/m)	15436	8370	9838	11037
C_A1 (Ns/m)	3705	5354	6668	4024
K_A2 (N/m)	15436	13839	8471	14724
K_B1 (N/m)	3381468	3390877	3489888	3390221
K_B2 (N/m)	3381468	3386525	3882502	3387411
K_B3 (N/m)	3849947	3436253	3821050	3399402
C_B3 (Ns/m)	1077	1623	1656	972
K_B4 (N/m)	6279869	3434664	3397478	3409460
C_B4 (Ns/m)	1119	1603	1687	1645
C_D (Ns/m)	70423	63200	73696	49182
l_Ax (m)	2.02	2.09	2.45	3.67
l_A1x (m)	0.76	0.78	1.13	0.79
l_A1y (m)	0.48	0.53	0.55	0.88
l_A2x (m)	1.42	1.15	1.06	1.24
l_A2y (m)	0.90	0.54	0.60	0.56
l_B1x (m)	1.44	1.48	1.45	1.45
l_B1y (m)	0.77	0.77	0.89	0.77
l_B2x (m)	0.72	0.82	0.83	0.73
l_B2y (m)	0.77	0.77	0.78	0.77
l_B3x (m)	0.72	0.72	0.72	0.72
l_B4x (m)	1.44	1.49	1.50	1.45
l_B4y (m)	0.77	0.77	0.79	0.77
l_Dx (m)	2.22	1.87	1.96	2.34
l_EFx (m)	1.90	1.03	1.03	1.15

From the target point of view, the NSGA-II provides the best combination of the performance indices which give the smallest f_c for the cab–chassis subsystem, the second smallest f_c for the carriage–chassis subsystem, and the highest f_v for both subsystems.

On the other hand, the optimized results should be compared according to the design parameters which would affect other performances of the vehicle. It can be seen from Table 4 that the NCGA provides the lightest cab which is much lower compared with the results from the other three algorithms. The mass of chassis given by the NSGA-II and the NCGA are similar and much smaller than those from the MOPS and the ASA. In addition, to obtain optimal vibration performances, both the MOPS and the ASA choose to add some weight in the main structures, while the genetic algorithms successfully reduced the mass and the vibration at the same time. Light weight is also important in the assessment of the vehicle design, which can affect the fuel efficiency and the emission of the vehicle strongly. Thus, in this respect, the results from the NSGA-II and the NCGA are more preferable and, in particular, the NCGA gives the lightest vehicle design.

The vertical accelerations of the cab and the carriage are the most important vibration performances for the vehicle. So, the RMS of the vertical vibration at different speeds are compared between the optimizations and the baseline vehicle, as shown in Figure 10. For the vertical vibration of the cab, all optimal designs show lower vibrations compared with the baseline vehicle regardless of vehicle speed, and the RMS acceleration increases apparently with increasing vehicle speed. The RMS accelerations of the MOPS are the lowest at the whole speed range from 40 to 80 km/h, and the increment of RMS acceleration due to the increase of speed is the smallest as well. The RMS vibrations of the ASA and the NSGA-II are very close and they are a bit higher than that of the MOPS. Although the NCGA give the highest RMS accelerations compared to the other three optimizations, the vibration of the NCGA is still much lower than that of the baseline vehicle. For the vibration of the carriage, all the optimizations still show better ride performances compared with the baseline vehicle. The ASA and the MOPS provide the lowest RMS accelerations, followed by the NCGA and the NSGA-II. However, the vibration of the carriage varies a lot according to the vehicle speed, especially in the ASA and the MOPS cases. A drop of the RMS acceleration of the MOPS can be found due to the increase of speed from 50 to 60 km/h, and a similar condition can be seen in the ASA case since the speed increases from 70 to 80 km/h. At 70 km/h, the MOPS, the ASA, and the NCGA show very close RMS accelerations of the carriage.

6. Conclusion

Based on the dynamic model of a three-axle vehicle, two performance indices are proposed for assessing the coupled vibration and the vibration attenuation between substructures. A sensitivity analysis is carried out to identify the effects of mass properties, stiffness, damping, and geometries on the vibration coupling and attenuation. Optimizations aiming at reducing the coupling effects and increasing the vibration attenuation simultaneously are conducted using four algorithms: MOPS; NSGA-II; NCGA; and ASA.

Trade-off effects of the design parameters on the coupling factor and the vibration attenuation are found, which indicates that compromises should be made in the vehicle ride optimization. Using the proposed coupling factor and vibration attenuation factor as objectives, the ride vibrations at different locations of the vehicle are optimized effectively.

The optimal design parameters obtained by the MOPS and the ASA show better performances with respect to the RMS accelerations of the cab and the carriage in the vertical direction. However, in those parameter sets, the masses of the cab and the chassis are even higher than the baseline values, which may not be preferred due to the possible increase of the fuel consumption and emissions. The NCGA and the NSGA-II also provide much higher ride qualities regarding the vibration of the cab and the carriage – the former shows lower vibration at the carriage and the latter gives better vibration at the cab.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the Project 51705357 supported by the National Natural Science Foundation of China.

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