# Multiobjective Lightweight Optimization Design Method for a Dump Truck Carriage under Multiple Working Conditions

Authors: Kun Zhang; Jiabao Zeng; Qianxi Zhang; Tonglin Pu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

To reduce the transportation cost and improve the fuel economy of a dump truck in road transport, vehicle lightweight optimization with the objectives such as the mass, stiffness, and sixth-order frequency of the carriage is adopted in this study. First, the carriage is divided into finite element meshes, and the working conditions of the carriage are analyzed under the conditions of full-load and uniform transportation, 0° lift, and 30° lift. Then, a comprehensive model consisting of full-load uniform transport and 0° lifting dual conditions is established, and 18 plate thicknesses are defined as design variables. The samples derived from the design variables of the carriages via the optimal Latin hypercube method are employed to analyze, in which the design variable that critically impacts the optimization objective is screened out from the sample for research. In addition, the dual-case approximation model constructed by a variety of methods is executed for accuracy comparison. And a moving least squares method combining linear and specific numerical values with the highest accuracy is selected to build a dual-case approximation model for the carriage. Ultimately, on the basis of the constructed double-condition approximation model, a multiobjective genetic algorithm is utilized to optimize the mass, bending stiffness, and sixth-order frequency. Compared with other methods, the multiobjective genetic algorithm can maintain the optimal solution, and the stored optimal solution can also be introduced into other groups to obtain another optimal solution. The simulation results, obtained from the scenarios of the carriage under the conditions of full-load uniform transportation, 0° lift and 30° lift, reveal that, on the basis of keeping the shape and structure of the original carriage unchanged, the mass is reduced by 1.215 t, the overall bending stiffness of the carriage is increased by 16300 N·mm−1, and the sixth-order modal frequency is optimized.


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## Body

## 1. Introduction

The issues of fossil fuel emissions and global climate change have attached wide concerns in the world. Therefore, reducing fossil fuel consumption has posed a huge challenge for many industries [1–3].With the continuous improvement of environmental protection policies, vehicle emission requirements are also further enhanced. How to meet the emission requirements under the environmental protection policy has become a challenge for all vehicle companies. Vehicle lightweight, a potentially effective solution to mitigate emissions, is widely employed by vehicle companies. Studies have revealed that when the mass of the vehicle is shrunk by 100 kg, the braking and handling stability of the vehicle will be improved, and the fuel consumption and emissions are reduced by 5–8% and 4.5%, respectively. In addition, the fuel utilization rate increased by 3.8% correspondingly [4–6].Up to now, there are still a lot of studies focused on lightweight. In the past decades, various weight reduction methods have been intensively studied to achieve vehicle weight reduction. Lee combines the optimal design of lightweight materials and rigid materials to achieve a 10% reduction in the weight of the instrument panel [7]. Additionally, the GPA and AHP methods are employed to select the body lightweight materials and determine the weight of each attribute, respectively [8]. In [9], a cost-effective and reliable joining technique for multimaterial designs, such as the joining of aluminum to press-hardened boron steel and welding of resistance elements, is investigated. To solve the problem of unique surface smoothness and low-dimensional accuracy of Ti 49.4 Ni 50.6 (at. %) shape memory alloy (SMA), Takale and Chougule improved the surface smoothness and dimensional accuracy of SMA by studying a series of processing processes [10]. By comparing the accuracy of the approximate model method under multiple working conditions, a lightweight design of the cargo compartment is executed by Li and Ma, and a 7.47% mass reduction is achieved [11]. The genetic algorithm is utilized to study concrete bridges in [12]. The results display that effective and well-calibrated structure identification is successfully realized by the inversion technique of the genetic algorithm. Moreover, Ye et al. have improved vehicle acceleration, braking, and handling to a certain extent by optimizing the design of load-bearing frame components [13]. And Sookchanchai et al. developed a topology optimization method for automotive suspension control arms; it is worth noting that this approach has practical guidance for other automotive stamping lightweight designs [14]. The optimal Latin equation design and response surface model are employed to optimize the thickness of each pipe frame by Wang et al., and the results show that 14.38% weight reduction on the premise of the brake pedal intrusion is reduced by 25.31% and realize the 20.02% reduction of the cockpit of peak acceleration[15]. Meschut et al. proposed a multiobjective particle swarm optimization algorithm to find the optimal layout of parallel P2 hybrid electric vehicles [16]. Long et al. proposed a multiresponse weighted adaptive sampling (MWAS) method based on a hybrid surrogate model, which can significantly improve the efficiency of car seat model optimization [17]. The detailed model and calculation process of GMM-SOMVSA-CO to solve the RBMDO problem is put forward by Meng et al. so that the uncertainties brought by other disciplines do not need to be considered in the process of single discipline analysis and optimization [18].As the important component of the dump truck and main load bearing, the traditional carriages are designed as thicker and heavier as possible to ensure safety and service life. However, it will lead to the waste of materials and a dramatic increase in the cost of transportation and energy consumption. Fortunately, the lightweight design of the carriage will improve the economy, power, braking, and safety of the dump truck [19]. In the initial study, the lightweight of dump trucks is achieved by the replacement of raw materials with higher strength steel with various optimization algorithms. With the continuous maturity of automobile manufacturing materials, how to more effectively improve the utilization rate of materials has become more prominent in the role of theoretical algorithms. Thus, the utilization of different algorithms or the improvement of the original algorithm is the focus of lightweight.Nevertheless, there are still many problems in the lightweight design of dump truck compartments. It can be found that most of the research studies on dump truck carriages only take the carriage quality as a single objective to optimize the design, which leads to many problems in the manufactured carriages in the past ten years. For example, problems such as bumps, cracks, and fractures that may occur in the carriage owing to insufficient stiffness will cause the carriage’s service life to plummet. Furthermore, the shrunk of the carriage service life contributed by the reduction in the vehicle mass will increase the vehicle manufacturing cost to a certain extent. Thus, the carriage stiffness is a significant factor for optimization objective design. Additionally, the variation of vibration frequency is affected by the change in the quality of the carriage. To prevent the resonance of the carriage from affecting the life, safety, and cost of the carriage, the frequency is also studied and analyzed in this study.The methods to achieve lightweight can be classified into four categories: (1) the materials with higher stiffness, better energy absorption, and lower density are utilized to replace traditional materials individually or as a whole [20], (2) a more mature processing technology is adopted to guarantee the material properties do not change while operating the plates connected [21], (3) the objective optimization is carried out in the aspects of topological, structural, and size via leveraging finite element techniques [22, 23], and (4) a novel approximate model construction method is adopted to improve the accuracy of the approximate model. In addition, the lightweight design of the vehicle is realized by integrating it into a new or improved algorithm [24, 25]. Currently, the study on lightweight is implemented via adopting a variety of methods combined and selecting high-strength steel to replace traditional materials, as well as executing the optimization for the size of the objective. Note that the processing techniques, employed to connect a variety of materials, will be reasonably selected based on the nature of the connection between different materials. Additionally, for larger components, topology optimization is generally adopted first, then size optimization is carried out via optimal design. Ultimately, structural optimization is executed according to actual requirements to improve the utilization rate of energy and materials [26].The criteria for the lightweight design of the dump truck compartment is whether the economic efficiency of the vehicle has been improved after taking the lightweight, which includes the terms of bending stiffness, service life, and vehicle cost. By comparing the accuracy of the dual-condition carriage models constructed by different approximation methods, a method combining linear and specific numerical moving least squares (MLSM) is selected to fit the approximate model of the carriage in this study. Additionally, the mass, stiffness, and frequency of the carriage are analyzed utilizing a multiobjective genetic algorithm and lightweight optimization design.


## 2. Finite Element Establishment and Rigid Strength Analysis of Dump Truck Carriage

The finite element establishment and rigid strength analysis of dump truck are explained in the following sections.


### 2.1. Establishment of the Finite Element Model of the Carriage

In this study, UG software is employed to model the carriage, and it is seamlessly imported into HyperMesh for finite element meshing. First, the finite element model preprocessing is applied to simplify the parts with small mass and complex structure. Then, a finite element model with a grid size of 10 mm for the treated carriage is performed, and the welding between the panels and the quality of the grid is checked. Ultimately, the actual thickness, material properties, and loading force of the model are added to the finite element model to prepare for the working condition analysis.


### 2.2. Static and Dynamic Working Conditions and Modal Analysis for Carriage

Dump trucks have the characteristics of long single transportation time, heavy load, and severe working environment during operation. Thus, as the main bearing component of the dump truck, it is essential to analyze the working conditions of the carriage in practical. In this study, three wording conditions of the carriage are studied, i.e., relatively static full-load and uniform transportation, static and dynamic 0° lift, and dynamic 30° lift.


#### 2.2.1. Stiffness and Strength Analysis of the Carriage under the Full-Load Working Condition

Generally, the dump truck carriage works under full-load conditions, and the service life of the carriage is most tested during long-distance transportation. Since the dump truck is mainly leveraged to load small goods such as sand and gravel, the sand is employed to carry out the force analysis on the full-load condition of the carriage in this study. In fully loaded conditions, the dump truck is mainly in a state of uniform motion. Thus, it can be assumed that the carriage remains relatively stationary under specific conditions. By constraining the translational degrees of freedom in theX, Y, and Z directions of the main longitudinal beam, the force diagram under the full-load condition of the carriage is shown in Figure 1:Figure 1
Full-load working condition of carriage.The force analysis of the finite element model of the carriage is done as shown in Figures2 and 3.Figure 2
Strain diagram of carriage full load.Figure 3
Stress diagram of carriage full load.In Figures2 and 3, it can be observed that the maximum strain of the carriage under the loaded condition is 10.84 mm and the maximum stress is 287.5 MPa. The strain is mainly concentrated at the two ends of the carriage floor: one end is at the contact surface between the carriage and the lifting mechanism and the other end is at the contact surface between the carriage and the subframe. The results derived from Table 1 show that the carriage meets the vehicle manufacturer’s design requirements, in which the whole carriage is made of A610L material, the maximum deformation is not more than 15 mm, the safety factor is 1.2, and the maximum yield stress is 425 MPa.Table 1
Strain and stress diagram of carriage full-load working condition.
Overall carriageFront plateBase plateBackboardSide panelsMaximum stress (MPa)287.50200.95287.50100.98159.70Maximum strain (mm)10.848.7610.844.745.44


#### 2.2.2. Analysis of 0° Lifting Condition of Carriage

In this study, the 0° lifting condition is analyzed when the carriage is fully loaded. When 0° is lifted, the carriage is subjected to the vertical upward lifting force of the lifting mechanism, which includes a full load of goods and the dead weight of the carriage. The main longitudinal beam of the carriage constrains theX-direction translation and Y- and Z-direction rotational degrees of freedom; the side plate constrains the Y-direction translation; the front plate is all restrained except for releasing the Y-direction rotational degrees of freedom. The forces applied are shown in Figure 4.Figure 4
The carriage 0° lift working condition diagram.In Figure5, Q is the total length of the bottom plate, which is valued at 3.5 m; Q1 : Q2 = 2 : 3; the rated load of the carriage is 5 t, and the dead weight of the carriage is 3 t. The lifting force is loaded into the finite element model and the working conditions are analyzed, and the strain stresses in Figures 5 and 6 are derived.Figure 5
Force diagram at the moment of lifting 0°.Figure 6
Strain diagram of carriage 0° working condition.It can be seen from Figures6 and 7 that when the carriage is lifted at 0°, the maximum strain occurs on the contact surface between the carriage and the subframe, and the maximum stress occurs on the contact surface between the lifting mechanism and the carriage. Since the contact area between the lifting mechanism and the subframe and the carriage is the stressed part, strain and stress concentration are generated on these two sides, where the maximum strain is 13.59 mm and the maximum stress is 444.31 MPa. Compared under the full-load condition, the strain and stress in the 0° condition have an increased to a certain extent. The 0° working condition data table is shown in Table 2.Figure 7
Carriage 0° working stress.Table 2
Strain and stress table for 0° working condition.
Overall carriageFront plateBase plateBackboardSide panelsMaximum stress (MPa)444.312.47444.314.556.04Maximum strain (mm)13.5990.8813.59148.10246.80


#### 2.2.3. Analysis of Working Conditions When the Carriage Is Lifting at 30°

When the lift mechanism lifts the carriage to 30°, nearly half of the cargo will be unloaded, and the carriage is mainly subjected to the lift force of the lift mechanism and the support force of the subframe. As shown in Figure4, the loading force is added to the carriage. Additionally, the results of the 30° lift working condition analysis of the carriage are presented in Figures 7 and 8.Figure 8
Strain diagram of carriage 30° working condition.Figures8 and 9 describe the stress-strain diagram of the carriage under the working condition of 30°, in which the strain concentration and stress concentration, respectively, exist at the contact surface between the lifting mechanism and the subframe, and at the welding point between the subframe and the carriage. Note that the maximum strain is 10.27 mm and the maximum stress is 362.20 MPa. The 30° working condition data are shown in Table 3.Figure 9
Carriage 30° working condition stress.Table 3
30° working condition strain and stress table.
Overall carriageFront plateBase plateBackboardSide panelsMaximum strain (mm)10.274.0710.273.726.84Maximum stress (MPa)362.20126.60362.2080.74201.20


#### 2.2.4. Free Mode Analysis of the First 8th Order of the Carriage

Dump trucks work on complex and harsh roads with low surface flatness leveling, and the requirements of the dynamic performance of the dump truck are higher. Therefore, the structural vibration obtained by the modal analysis can well reflect the dynamic performance of the dump truck.The frequency and vibration pattern are determined by a modal analysis of the carriage to prevent resonance. Generally, the first eight orders are analyzed in the carriage because when the eighth orders are greater than 40 Hz, there is no need to analyze the following orders. Modal analysis is the analysis of the vibration characteristics of a system. For a linear system, when the linear conditions do not change, the dynamic characteristics of the structure can be expressed as(1)Mx¨+Cx˙+Kx=ft,where M, C, and K represent the mass matrix, damping matrix, and stiffness matrix of the system structure, respectively, X is the displacement vector of the system, x˙ and x¨ denote the velocity and acceleration vectors, and f (t) is the excitation force vector.The inherent frequency of theith order can be denoted as(2)fi=ωi2π;i=1,2,…,n.The results of the first 8th order modal analysis of the carriage in Hyperview are shown in Figure10.Figure 10
Modal diagram of the first eight orders of the carriage. (a) 1st order carriage mode. (b) 2nd order carriage mode. (c) 3rd order carriage mode. (d) 4th order carriage mode. (e) 5th order carriage mode. (f) 6th order carriage mode. (g) 7th order carriage mode. (h) 8th order carriage mode.The first eight orders of modal frequencies are shown in Table4.Table 4
Carriage front eighth-order free mode.
No.First stageSecond stageThird stageFourth stageFifth stageSixth stageSeventh stageEighth stageFrequency (Hz)14.6419.9722.0029.1132.4434.1638.6242.01The 6-cylinder and 4-stroke diesel engine is adopted in this study, and the rotational speed of the engine at idle speed is 800 r/min. The frequency at engine idle speed can be expressed as(3)fd=2nz60τ=2×800×660×4=40Hz.The magnitude of the excitation frequency of the carriage road surface depends on the vehicle speed and road surface type. Generally, the driving speed of the dump truck in the sand mine area is 50 km/h, and the road adhesion coefficient is 0.4. Thus, the frequency can be calculated as(4)fL=v3.6Lmin=503.6×0.4=37.722Hz.It can be seen from Figure10 that the value of the sixth-order frequency analysis of the carriage is 34.126 Hz, which is relatively close to the road excitation frequency. Therefore, as a constraint condition for carriage optimization, the sixth-order frequency needs to be not equal to 34.722 Hz.


## 2.1. Establishment of the Finite Element Model of the Carriage

In this study, UG software is employed to model the carriage, and it is seamlessly imported into HyperMesh for finite element meshing. First, the finite element model preprocessing is applied to simplify the parts with small mass and complex structure. Then, a finite element model with a grid size of 10 mm for the treated carriage is performed, and the welding between the panels and the quality of the grid is checked. Ultimately, the actual thickness, material properties, and loading force of the model are added to the finite element model to prepare for the working condition analysis.


## 2.2. Static and Dynamic Working Conditions and Modal Analysis for Carriage

Dump trucks have the characteristics of long single transportation time, heavy load, and severe working environment during operation. Thus, as the main bearing component of the dump truck, it is essential to analyze the working conditions of the carriage in practical. In this study, three wording conditions of the carriage are studied, i.e., relatively static full-load and uniform transportation, static and dynamic 0° lift, and dynamic 30° lift.


### 2.2.1. Stiffness and Strength Analysis of the Carriage under the Full-Load Working Condition

Generally, the dump truck carriage works under full-load conditions, and the service life of the carriage is most tested during long-distance transportation. Since the dump truck is mainly leveraged to load small goods such as sand and gravel, the sand is employed to carry out the force analysis on the full-load condition of the carriage in this study. In fully loaded conditions, the dump truck is mainly in a state of uniform motion. Thus, it can be assumed that the carriage remains relatively stationary under specific conditions. By constraining the translational degrees of freedom in theX, Y, and Z directions of the main longitudinal beam, the force diagram under the full-load condition of the carriage is shown in Figure 1:Figure 1
Full-load working condition of carriage.The force analysis of the finite element model of the carriage is done as shown in Figures2 and 3.Figure 2
Strain diagram of carriage full load.Figure 3
Stress diagram of carriage full load.In Figures2 and 3, it can be observed that the maximum strain of the carriage under the loaded condition is 10.84 mm and the maximum stress is 287.5 MPa. The strain is mainly concentrated at the two ends of the carriage floor: one end is at the contact surface between the carriage and the lifting mechanism and the other end is at the contact surface between the carriage and the subframe. The results derived from Table 1 show that the carriage meets the vehicle manufacturer’s design requirements, in which the whole carriage is made of A610L material, the maximum deformation is not more than 15 mm, the safety factor is 1.2, and the maximum yield stress is 425 MPa.Table 1
Strain and stress diagram of carriage full-load working condition.
Overall carriageFront plateBase plateBackboardSide panelsMaximum stress (MPa)287.50200.95287.50100.98159.70Maximum strain (mm)10.848.7610.844.745.44


### 2.2.2. Analysis of 0° Lifting Condition of Carriage

In this study, the 0° lifting condition is analyzed when the carriage is fully loaded. When 0° is lifted, the carriage is subjected to the vertical upward lifting force of the lifting mechanism, which includes a full load of goods and the dead weight of the carriage. The main longitudinal beam of the carriage constrains theX-direction translation and Y- and Z-direction rotational degrees of freedom; the side plate constrains the Y-direction translation; the front plate is all restrained except for releasing the Y-direction rotational degrees of freedom. The forces applied are shown in Figure 4.Figure 4
The carriage 0° lift working condition diagram.In Figure5, Q is the total length of the bottom plate, which is valued at 3.5 m; Q1 : Q2 = 2 : 3; the rated load of the carriage is 5 t, and the dead weight of the carriage is 3 t. The lifting force is loaded into the finite element model and the working conditions are analyzed, and the strain stresses in Figures 5 and 6 are derived.Figure 5
Force diagram at the moment of lifting 0°.Figure 6
Strain diagram of carriage 0° working condition.It can be seen from Figures6 and 7 that when the carriage is lifted at 0°, the maximum strain occurs on the contact surface between the carriage and the subframe, and the maximum stress occurs on the contact surface between the lifting mechanism and the carriage. Since the contact area between the lifting mechanism and the subframe and the carriage is the stressed part, strain and stress concentration are generated on these two sides, where the maximum strain is 13.59 mm and the maximum stress is 444.31 MPa. Compared under the full-load condition, the strain and stress in the 0° condition have an increased to a certain extent. The 0° working condition data table is shown in Table 2.Figure 7
Carriage 0° working stress.Table 2
Strain and stress table for 0° working condition.
Overall carriageFront plateBase plateBackboardSide panelsMaximum stress (MPa)444.312.47444.314.556.04Maximum strain (mm)13.5990.8813.59148.10246.80


### 2.2.3. Analysis of Working Conditions When the Carriage Is Lifting at 30°

When the lift mechanism lifts the carriage to 30°, nearly half of the cargo will be unloaded, and the carriage is mainly subjected to the lift force of the lift mechanism and the support force of the subframe. As shown in Figure4, the loading force is added to the carriage. Additionally, the results of the 30° lift working condition analysis of the carriage are presented in Figures 7 and 8.Figure 8
Strain diagram of carriage 30° working condition.Figures8 and 9 describe the stress-strain diagram of the carriage under the working condition of 30°, in which the strain concentration and stress concentration, respectively, exist at the contact surface between the lifting mechanism and the subframe, and at the welding point between the subframe and the carriage. Note that the maximum strain is 10.27 mm and the maximum stress is 362.20 MPa. The 30° working condition data are shown in Table 3.Figure 9
Carriage 30° working condition stress.Table 3
30° working condition strain and stress table.
Overall carriageFront plateBase plateBackboardSide panelsMaximum strain (mm)10.274.0710.273.726.84Maximum stress (MPa)362.20126.60362.2080.74201.20


### 2.2.4. Free Mode Analysis of the First 8th Order of the Carriage

Dump trucks work on complex and harsh roads with low surface flatness leveling, and the requirements of the dynamic performance of the dump truck are higher. Therefore, the structural vibration obtained by the modal analysis can well reflect the dynamic performance of the dump truck.The frequency and vibration pattern are determined by a modal analysis of the carriage to prevent resonance. Generally, the first eight orders are analyzed in the carriage because when the eighth orders are greater than 40 Hz, there is no need to analyze the following orders. Modal analysis is the analysis of the vibration characteristics of a system. For a linear system, when the linear conditions do not change, the dynamic characteristics of the structure can be expressed as(1)Mx¨+Cx˙+Kx=ft,where M, C, and K represent the mass matrix, damping matrix, and stiffness matrix of the system structure, respectively, X is the displacement vector of the system, x˙ and x¨ denote the velocity and acceleration vectors, and f (t) is the excitation force vector.The inherent frequency of theith order can be denoted as(2)fi=ωi2π;i=1,2,…,n.The results of the first 8th order modal analysis of the carriage in Hyperview are shown in Figure10.Figure 10
Modal diagram of the first eight orders of the carriage. (a) 1st order carriage mode. (b) 2nd order carriage mode. (c) 3rd order carriage mode. (d) 4th order carriage mode. (e) 5th order carriage mode. (f) 6th order carriage mode. (g) 7th order carriage mode. (h) 8th order carriage mode.The first eight orders of modal frequencies are shown in Table4.Table 4
Carriage front eighth-order free mode.
No.First stageSecond stageThird stageFourth stageFifth stageSixth stageSeventh stageEighth stageFrequency (Hz)14.6419.9722.0029.1132.4434.1638.6242.01The 6-cylinder and 4-stroke diesel engine is adopted in this study, and the rotational speed of the engine at idle speed is 800 r/min. The frequency at engine idle speed can be expressed as(3)fd=2nz60τ=2×800×660×4=40Hz.The magnitude of the excitation frequency of the carriage road surface depends on the vehicle speed and road surface type. Generally, the driving speed of the dump truck in the sand mine area is 50 km/h, and the road adhesion coefficient is 0.4. Thus, the frequency can be calculated as(4)fL=v3.6Lmin=503.6×0.4=37.722Hz.It can be seen from Figure10 that the value of the sixth-order frequency analysis of the carriage is 34.126 Hz, which is relatively close to the road excitation frequency. Therefore, as a constraint condition for carriage optimization, the sixth-order frequency needs to be not equal to 34.722 Hz.


## 2.2.1. Stiffness and Strength Analysis of the Carriage under the Full-Load Working Condition

Generally, the dump truck carriage works under full-load conditions, and the service life of the carriage is most tested during long-distance transportation. Since the dump truck is mainly leveraged to load small goods such as sand and gravel, the sand is employed to carry out the force analysis on the full-load condition of the carriage in this study. In fully loaded conditions, the dump truck is mainly in a state of uniform motion. Thus, it can be assumed that the carriage remains relatively stationary under specific conditions. By constraining the translational degrees of freedom in theX, Y, and Z directions of the main longitudinal beam, the force diagram under the full-load condition of the carriage is shown in Figure 1:Figure 1
Full-load working condition of carriage.The force analysis of the finite element model of the carriage is done as shown in Figures2 and 3.Figure 2
Strain diagram of carriage full load.Figure 3
Stress diagram of carriage full load.In Figures2 and 3, it can be observed that the maximum strain of the carriage under the loaded condition is 10.84 mm and the maximum stress is 287.5 MPa. The strain is mainly concentrated at the two ends of the carriage floor: one end is at the contact surface between the carriage and the lifting mechanism and the other end is at the contact surface between the carriage and the subframe. The results derived from Table 1 show that the carriage meets the vehicle manufacturer’s design requirements, in which the whole carriage is made of A610L material, the maximum deformation is not more than 15 mm, the safety factor is 1.2, and the maximum yield stress is 425 MPa.Table 1
Strain and stress diagram of carriage full-load working condition.
Overall carriageFront plateBase plateBackboardSide panelsMaximum stress (MPa)287.50200.95287.50100.98159.70Maximum strain (mm)10.848.7610.844.745.44


## 2.2.2. Analysis of 0° Lifting Condition of Carriage

In this study, the 0° lifting condition is analyzed when the carriage is fully loaded. When 0° is lifted, the carriage is subjected to the vertical upward lifting force of the lifting mechanism, which includes a full load of goods and the dead weight of the carriage. The main longitudinal beam of the carriage constrains theX-direction translation and Y- and Z-direction rotational degrees of freedom; the side plate constrains the Y-direction translation; the front plate is all restrained except for releasing the Y-direction rotational degrees of freedom. The forces applied are shown in Figure 4.Figure 4
The carriage 0° lift working condition diagram.In Figure5, Q is the total length of the bottom plate, which is valued at 3.5 m; Q1 : Q2 = 2 : 3; the rated load of the carriage is 5 t, and the dead weight of the carriage is 3 t. The lifting force is loaded into the finite element model and the working conditions are analyzed, and the strain stresses in Figures 5 and 6 are derived.Figure 5
Force diagram at the moment of lifting 0°.Figure 6
Strain diagram of carriage 0° working condition.It can be seen from Figures6 and 7 that when the carriage is lifted at 0°, the maximum strain occurs on the contact surface between the carriage and the subframe, and the maximum stress occurs on the contact surface between the lifting mechanism and the carriage. Since the contact area between the lifting mechanism and the subframe and the carriage is the stressed part, strain and stress concentration are generated on these two sides, where the maximum strain is 13.59 mm and the maximum stress is 444.31 MPa. Compared under the full-load condition, the strain and stress in the 0° condition have an increased to a certain extent. The 0° working condition data table is shown in Table 2.Figure 7
Carriage 0° working stress.Table 2
Strain and stress table for 0° working condition.
Overall carriageFront plateBase plateBackboardSide panelsMaximum stress (MPa)444.312.47444.314.556.04Maximum strain (mm)13.5990.8813.59148.10246.80


## 2.2.3. Analysis of Working Conditions When the Carriage Is Lifting at 30°

When the lift mechanism lifts the carriage to 30°, nearly half of the cargo will be unloaded, and the carriage is mainly subjected to the lift force of the lift mechanism and the support force of the subframe. As shown in Figure4, the loading force is added to the carriage. Additionally, the results of the 30° lift working condition analysis of the carriage are presented in Figures 7 and 8.Figure 8
Strain diagram of carriage 30° working condition.Figures8 and 9 describe the stress-strain diagram of the carriage under the working condition of 30°, in which the strain concentration and stress concentration, respectively, exist at the contact surface between the lifting mechanism and the subframe, and at the welding point between the subframe and the carriage. Note that the maximum strain is 10.27 mm and the maximum stress is 362.20 MPa. The 30° working condition data are shown in Table 3.Figure 9
Carriage 30° working condition stress.Table 3
30° working condition strain and stress table.
Overall carriageFront plateBase plateBackboardSide panelsMaximum strain (mm)10.274.0710.273.726.84Maximum stress (MPa)362.20126.60362.2080.74201.20


## 2.2.4. Free Mode Analysis of the First 8th Order of the Carriage

Dump trucks work on complex and harsh roads with low surface flatness leveling, and the requirements of the dynamic performance of the dump truck are higher. Therefore, the structural vibration obtained by the modal analysis can well reflect the dynamic performance of the dump truck.The frequency and vibration pattern are determined by a modal analysis of the carriage to prevent resonance. Generally, the first eight orders are analyzed in the carriage because when the eighth orders are greater than 40 Hz, there is no need to analyze the following orders. Modal analysis is the analysis of the vibration characteristics of a system. For a linear system, when the linear conditions do not change, the dynamic characteristics of the structure can be expressed as(1)Mx¨+Cx˙+Kx=ft,where M, C, and K represent the mass matrix, damping matrix, and stiffness matrix of the system structure, respectively, X is the displacement vector of the system, x˙ and x¨ denote the velocity and acceleration vectors, and f (t) is the excitation force vector.The inherent frequency of theith order can be denoted as(2)fi=ωi2π;i=1,2,…,n.The results of the first 8th order modal analysis of the carriage in Hyperview are shown in Figure10.Figure 10
Modal diagram of the first eight orders of the carriage. (a) 1st order carriage mode. (b) 2nd order carriage mode. (c) 3rd order carriage mode. (d) 4th order carriage mode. (e) 5th order carriage mode. (f) 6th order carriage mode. (g) 7th order carriage mode. (h) 8th order carriage mode.The first eight orders of modal frequencies are shown in Table4.Table 4
Carriage front eighth-order free mode.
No.First stageSecond stageThird stageFourth stageFifth stageSixth stageSeventh stageEighth stageFrequency (Hz)14.6419.9722.0029.1132.4434.1638.6242.01The 6-cylinder and 4-stroke diesel engine is adopted in this study, and the rotational speed of the engine at idle speed is 800 r/min. The frequency at engine idle speed can be expressed as(3)fd=2nz60τ=2×800×660×4=40Hz.The magnitude of the excitation frequency of the carriage road surface depends on the vehicle speed and road surface type. Generally, the driving speed of the dump truck in the sand mine area is 50 km/h, and the road adhesion coefficient is 0.4. Thus, the frequency can be calculated as(4)fL=v3.6Lmin=503.6×0.4=37.722Hz.It can be seen from Figure10 that the value of the sixth-order frequency analysis of the carriage is 34.126 Hz, which is relatively close to the road excitation frequency. Therefore, as a constraint condition for carriage optimization, the sixth-order frequency needs to be not equal to 34.722 Hz.


## 3. Multiobjective Optimal Design under the Construction of the Dual Working Condition Approximation Model

Through the above analysis, it can be observed that the stiffness and strength of the carriage are severe under the full-load condition and the 0° lift condition, and the sixth-order frequency need to be constrained. Therefore, optimal analysis is carried out for the dual operating conditions established comprehensively under the full-load condition and the 0° operating condition. Then, the dual working condition model and modal model are imported in the Hyperstudy, in which 18 carriage thicknesses are defined as design variables for multiobjective optimization of car mass and stiffness.


### 3.1. Latin Hypercube Sampling

LHS is a space-filling method for arbitrary dimensions and has no number of times of sampling in space. In this study, LHS is leveraged to sample the dual working conditions and modes of the carriages to ensure that each sampling point has no other test points in the same row and in the same row. In addition, 260 optimization analysis on three working condition models are performed by LHS, and the results of working condition diagram, modal diagram, mass diagram, and stiffness diagram are shown in Figures11–14.Figure 11
Iteration diagram of dual working conditions.Figure 12
Modal iteration diagram.Figure 13
Mass iteration diagram.Figure 14
Rigidity iteration diagram.Due to a large number of design variables for the carriage, it is necessary to screen out the design variables that have a great influence under the working conditions of the carriage and the optimization objective of the carriage. According to the sensitivity Figures15–17, the design variables of 63 mm and 8 mm are selected as the predictions for the mode, mass, and stiffness. The relationship between the predicted design variables and the response is shown in Figures 18–20.Figure 15
Modal sensitivity diagram.Figure 16
Mass sensitivity graph.Figure 17
Stiffness sensitivity graph.Figure 18
Modal 3D prediction diagram.Figure 19
Mass 3D prediction diagram.Figure 20
3D prediction of stiffness.Table5 shows the table of design variables that have a high impact on the carriage optimization objective using Latin hypercube extraction screening.Table 5
Latin hypercube sampling.
LHS123…2582592608 mm6.769.038.91…5.285.558.6163 mm64.9963.3051.76…35.0625.4164.676 mm5.073.425.77…6.763.054.3587.288.385.03…5.708.338.2265.347.175.97…7.784.224.456 mm5.447.575.38…3.894.204.95Mass (t)2.81012.96502.6740…2.02051.57512.8872Stiffness (N·mm−1)4932.7225657.7116900.57…11639.3012991.6527754.31Modal (Hz)32.5632.7532.49…27.3125.7431.73Full-load strain (mm)9.4710.529.53…27.5622.9523.30Full-load stress (MPa)241.69312.69385.94…287.63447.72306.250° strain (mm)50.689.7414.92…21.4819.249.010° stress (MPa)1195.09374.80459.59…608.92623.87371.34


### 3.2. Selection of Methods for Approximate Model Construction

The main methods for constructing approximate models at this stage are radial basis functions, HyperKriging, least square regression, and moving least square method. By comparing the accuracy obtained by the above four different approximation model methods, an approximation method more suitable for the carriage model is selected. The results of precision are shown in Table6.Table 6
Accuracy of the four approximation methods for constructing the approximation model.
Approximation methodMassEIMTFull-load strainFull-load stress0° strain0° stressRBF0.9970.9810.9080.9770.9170.9030.927HK10.9990.9100.9120.9660.9830.988LSR10.9800.9070.9100.9140.9810.910MLSM10.9820.9200.9340.9060.9890.916It can be found in Table6 that the fitting accuracy of the MLSM approximation method is higher and is adopted to construct the approximate model of the dual working conditions of the carriage. The construction of the carriage model utilizing the MLSM method with linear parameters will lead to the issue of lower accuracy for modal and full-load strain. Therefore, a customized numerical value is constructed for the response with low precision, and the precision results are shown in Table 7.Table 7
MLSM quadratic construction of approximate models using specific parameters.
ResponseApproximation methodResponse selectionPrecisionMass (t)MLSMLinear-2.001.00EI (N·mm−1)MLSMCustom-10.000.998MT (Hz)MLSMCustom-5.930.932Full-load strain (mm)MLSMCustom-5.660.974Full-load stress (MPa)MLSMCustom-8.320.9790° strain (mm)MLSMCustom-8.840.9940° stress (MPa)MLSMCustom-8.160.973As shown in Table7, the accuracy of the approximation model for the dual working conditions of the carriage is improved by changing the MLSM method. In addition, the actual and simulated results of the finite element model can be reflected by the scatter plot, i.e., the closer the scatter plot to the y = x function, the higher the fitting accuracy of the model. The accuracy graph of the response value is shown in Figure 21.Figure 21
Simulation accuracy graph. (a) Mass scatter plot. (b) Stiffness scatter diagram. (c) Modal scatter plot. (d) Scatter plot of full-load strain. (e) Full-load stress scatter diagram. (f) 0° strain scatter plot. (g) 0° stress scatter plot.
(a)(b)(c)(d)(e)(f)(g)The calculated formula for the coefficient of determinationR2 of the approximate model fitting function is expressed as(5)R2=∑p=1ntesty^p−y¯2∑p=1ntestyp−y¯2,where the random test sample ntest is utilized to construct an approximation of the accuracy of the model, P and yp are the target predicted response value and the actual target value of the pth sample point, respectively, and y denotes the mean value of the actual target response of the sample point.It can be concluded that the closerR2 is to 1, the higher the accuracy of the approximate model is. According to the numerical results of the approximate model accuracy of each target response value displayed in Table 7, the decisive coefficients of the 7 approximate models constructed are all greater than 0.90, which meets the requirements of engineering analysis.


### 3.3. Multiobjective Optimization Design of Dump Truck Compartment

In order to reduce the mass while ensuring the improvement of stiffness, a multiobjective optimization design is carried out on the mass and stiffness of the carriage. Under the condition of full load and 0° lift, the model is not equal to 34.722 Hz, the strain is not more than 15 mm, and the stress is not more than 425 MPa as constraints, the mathematical model of the carriage optimization algorithm can be expressed as(6)minmmass,maxEIsubjectedtodisp≤15mmstress≤425MPaMT≠34.722HZ,where m represents the mass, EI is the bending stiffness, disp is the strain under dual conditions, and stress and MT are the stress under dual conditions and sixth-order mode, respectivelyBased on the constructed MLSM approximation model with dual working conditions, the genetic algorithm is used to carry out the multiobjective optimization design of the carriage. The multiobjective genetic algorithm transforms the multiobjective into a single objective for the optimal design according to a weighted approach with the following expression:(7)fitnessI=∑k=1Kωk×fkI,where ωk is the weighting of the kth objective and fk is the normalized kth objective value. To obtain the solution set of the multiobjective genetic algorithm based on Pareto ranking, firstly, the distance between various individuals needs to be obtained and the formula is shown in (1):(8)dI1,I2=∑k=1KfkI1−fkI2fkmax−fkmin2,where fkmax is the maximum target value and fkmin is the minimum target value. For I, the individual niche count is obtained by calculating the adaptation of the niche count for an individual with the following equation:(9)fitnessI′fitnessIncI.In the obtained 16448 iteration results, the optimal solution set with a large gap is screened as shown in Table8. Furthermore, the Pareto frontier diagrams of the stiffness and the total mass of the carriage obtained by multiobjective optimization are illustrated in Figures 22 and 23.Table 8
Multiobjective optimal solution set for multiple working conditions.
No.12345678Base plate crossbeam (mm)5.135.115.005.105.085.039.929.50Main longitudinal beam (mm)26.1530.6620.0020.2523.1230.2140.0048.20Base plate (mm)3.023.103.003.053.033.105.967.98Side panels (mm)5.485.095.005.125.485.098.295.17Crossbar (mm)4.454.073.004.254.234.074.677.60Longitudinal beam (mm)6.556.133.005.726.466.133.086.47Mass (t)1.59511.68361.28871.41361.50961.66632.42942.8273Stiffness (N·mm−1)34699.0934703.5533940.4934659.7134695.4334701.4533125.5034304.89Modal (Hz)23.9225.1021.2421.7722.8524.8534.5534.72Full-load strain (mm)12.0911.8714.4714.0613.2912.4111.919.39Full-load stress (MPa)330.86313.78410.04394.70381.34343.90317.57308.350° strain (mm)7.838.048.048.167.968.048.128.460° stress (MPa)319.52318.84308.38316.34320.73318.40327.48361.31Figure 22
Mass and stiffness Pareto front plot.Figure 23
Mass and modal Pareto front plot.It can be found from Table9 that if the ultimate goal is to ensure that the carriage is rigid enough to minimize the mass of the carriage, case 1 is the best choice. If the maximum stiffness of the carriage is required and the mass of the carriage is reduced, scheme 2 can be selected; if the mass stiffness of the carriage needs to be well uniform, schemes 3, 4, and 5 can be selected; moreover, to ensure the thickness of the carriage plate, schemes 6, 7, and 8 are a preferable choice. To achieve the goal of greatly reducing the quality of the carriage and improving the utilization rate of materials, scheme 2 with better comprehensive performance is selected and the whole circle inspection is carried out to verify whether the requirements of stiffness and strength of the carriage are satisfied. The weight of the carriage is 2.9 t after removing the complex and relatively negligible mass of the components. And the results are illustrated in Table 9.Table 9
Comparison of results before and after multiobjective optimization for multiple working conditions.
Before optimizationAfter optimizationIntegral circleChange amount (%)Base plate crossbeam (mm)85.115−37.5Main longitudinal (mm)6330.6631−50.79Base plate (mm)63.13−50Side panels (mm)85.095−37.5Crossbar (mm)64.074−33.33Longitudinal beam (mm)66.1360Mass (t)2.91.68361.6850−41.90Stiffness (N·mm−1)1840034703.553470088.59Modal (Hz)34.7225.1025.5−26.56Full-load strain (mm)10.8411.8712.1411.99Full-load stress (MPa)287.49313.78321.9932.480° strain (mm)13.598.049.78−28.040° stress (MPa)444.31318.84319.90−28.00Figures20–23 show the stiffness and strength of the carriage after the carriage is fully rounded.The comparison results of stiffness and strength before and after full-circle optimization are shown in Table9. It can be found in Figure 24 that the maximum deformation of the full-load uniform transport carriage before and after the optimization of the full circle changes from 10.84 mm to 12.14 mm, with a change rate of 11.99%. In Figure 25, the maximum strain of the fully loaded uniform transport carriage changes from 287.49 MPa to 321.99 MPa, with an increase of 34.5 MPa. And it can be seen from Figures 26 and 27 that the strain under 0° lifting conditions changes from 13.59 mm to 9.78 mm; the stress changes from 444.31 MPa to 319.90 MPa, which satisfies the stiffness and strength conditions under the safety factor of 1.2.Figure 24
Full-load strain after rounding.Figure 25
Full-load stress after rounding.Figure 26
Strain of 0° lift after rounding.Figure 27
Stress at 0° lift after rounding.


## 3.1. Latin Hypercube Sampling

LHS is a space-filling method for arbitrary dimensions and has no number of times of sampling in space. In this study, LHS is leveraged to sample the dual working conditions and modes of the carriages to ensure that each sampling point has no other test points in the same row and in the same row. In addition, 260 optimization analysis on three working condition models are performed by LHS, and the results of working condition diagram, modal diagram, mass diagram, and stiffness diagram are shown in Figures11–14.Figure 11
Iteration diagram of dual working conditions.Figure 12
Modal iteration diagram.Figure 13
Mass iteration diagram.Figure 14
Rigidity iteration diagram.Due to a large number of design variables for the carriage, it is necessary to screen out the design variables that have a great influence under the working conditions of the carriage and the optimization objective of the carriage. According to the sensitivity Figures15–17, the design variables of 63 mm and 8 mm are selected as the predictions for the mode, mass, and stiffness. The relationship between the predicted design variables and the response is shown in Figures 18–20.Figure 15
Modal sensitivity diagram.Figure 16
Mass sensitivity graph.Figure 17
Stiffness sensitivity graph.Figure 18
Modal 3D prediction diagram.Figure 19
Mass 3D prediction diagram.Figure 20
3D prediction of stiffness.Table5 shows the table of design variables that have a high impact on the carriage optimization objective using Latin hypercube extraction screening.Table 5
Latin hypercube sampling.
LHS123…2582592608 mm6.769.038.91…5.285.558.6163 mm64.9963.3051.76…35.0625.4164.676 mm5.073.425.77…6.763.054.3587.288.385.03…5.708.338.2265.347.175.97…7.784.224.456 mm5.447.575.38…3.894.204.95Mass (t)2.81012.96502.6740…2.02051.57512.8872Stiffness (N·mm−1)4932.7225657.7116900.57…11639.3012991.6527754.31Modal (Hz)32.5632.7532.49…27.3125.7431.73Full-load strain (mm)9.4710.529.53…27.5622.9523.30Full-load stress (MPa)241.69312.69385.94…287.63447.72306.250° strain (mm)50.689.7414.92…21.4819.249.010° stress (MPa)1195.09374.80459.59…608.92623.87371.34


## 3.2. Selection of Methods for Approximate Model Construction

The main methods for constructing approximate models at this stage are radial basis functions, HyperKriging, least square regression, and moving least square method. By comparing the accuracy obtained by the above four different approximation model methods, an approximation method more suitable for the carriage model is selected. The results of precision are shown in Table6.Table 6
Accuracy of the four approximation methods for constructing the approximation model.
Approximation methodMassEIMTFull-load strainFull-load stress0° strain0° stressRBF0.9970.9810.9080.9770.9170.9030.927HK10.9990.9100.9120.9660.9830.988LSR10.9800.9070.9100.9140.9810.910MLSM10.9820.9200.9340.9060.9890.916It can be found in Table6 that the fitting accuracy of the MLSM approximation method is higher and is adopted to construct the approximate model of the dual working conditions of the carriage. The construction of the carriage model utilizing the MLSM method with linear parameters will lead to the issue of lower accuracy for modal and full-load strain. Therefore, a customized numerical value is constructed for the response with low precision, and the precision results are shown in Table 7.Table 7
MLSM quadratic construction of approximate models using specific parameters.
ResponseApproximation methodResponse selectionPrecisionMass (t)MLSMLinear-2.001.00EI (N·mm−1)MLSMCustom-10.000.998MT (Hz)MLSMCustom-5.930.932Full-load strain (mm)MLSMCustom-5.660.974Full-load stress (MPa)MLSMCustom-8.320.9790° strain (mm)MLSMCustom-8.840.9940° stress (MPa)MLSMCustom-8.160.973As shown in Table7, the accuracy of the approximation model for the dual working conditions of the carriage is improved by changing the MLSM method. In addition, the actual and simulated results of the finite element model can be reflected by the scatter plot, i.e., the closer the scatter plot to the y = x function, the higher the fitting accuracy of the model. The accuracy graph of the response value is shown in Figure 21.Figure 21
Simulation accuracy graph. (a) Mass scatter plot. (b) Stiffness scatter diagram. (c) Modal scatter plot. (d) Scatter plot of full-load strain. (e) Full-load stress scatter diagram. (f) 0° strain scatter plot. (g) 0° stress scatter plot.
(a)(b)(c)(d)(e)(f)(g)The calculated formula for the coefficient of determinationR2 of the approximate model fitting function is expressed as(5)R2=∑p=1ntesty^p−y¯2∑p=1ntestyp−y¯2,where the random test sample ntest is utilized to construct an approximation of the accuracy of the model, P and yp are the target predicted response value and the actual target value of the pth sample point, respectively, and y denotes the mean value of the actual target response of the sample point.It can be concluded that the closerR2 is to 1, the higher the accuracy of the approximate model is. According to the numerical results of the approximate model accuracy of each target response value displayed in Table 7, the decisive coefficients of the 7 approximate models constructed are all greater than 0.90, which meets the requirements of engineering analysis.


## 3.3. Multiobjective Optimization Design of Dump Truck Compartment

In order to reduce the mass while ensuring the improvement of stiffness, a multiobjective optimization design is carried out on the mass and stiffness of the carriage. Under the condition of full load and 0° lift, the model is not equal to 34.722 Hz, the strain is not more than 15 mm, and the stress is not more than 425 MPa as constraints, the mathematical model of the carriage optimization algorithm can be expressed as(6)minmmass,maxEIsubjectedtodisp≤15mmstress≤425MPaMT≠34.722HZ,where m represents the mass, EI is the bending stiffness, disp is the strain under dual conditions, and stress and MT are the stress under dual conditions and sixth-order mode, respectivelyBased on the constructed MLSM approximation model with dual working conditions, the genetic algorithm is used to carry out the multiobjective optimization design of the carriage. The multiobjective genetic algorithm transforms the multiobjective into a single objective for the optimal design according to a weighted approach with the following expression:(7)fitnessI=∑k=1Kωk×fkI,where ωk is the weighting of the kth objective and fk is the normalized kth objective value. To obtain the solution set of the multiobjective genetic algorithm based on Pareto ranking, firstly, the distance between various individuals needs to be obtained and the formula is shown in (1):(8)dI1,I2=∑k=1KfkI1−fkI2fkmax−fkmin2,where fkmax is the maximum target value and fkmin is the minimum target value. For I, the individual niche count is obtained by calculating the adaptation of the niche count for an individual with the following equation:(9)fitnessI′fitnessIncI.In the obtained 16448 iteration results, the optimal solution set with a large gap is screened as shown in Table8. Furthermore, the Pareto frontier diagrams of the stiffness and the total mass of the carriage obtained by multiobjective optimization are illustrated in Figures 22 and 23.Table 8
Multiobjective optimal solution set for multiple working conditions.
No.12345678Base plate crossbeam (mm)5.135.115.005.105.085.039.929.50Main longitudinal beam (mm)26.1530.6620.0020.2523.1230.2140.0048.20Base plate (mm)3.023.103.003.053.033.105.967.98Side panels (mm)5.485.095.005.125.485.098.295.17Crossbar (mm)4.454.073.004.254.234.074.677.60Longitudinal beam (mm)6.556.133.005.726.466.133.086.47Mass (t)1.59511.68361.28871.41361.50961.66632.42942.8273Stiffness (N·mm−1)34699.0934703.5533940.4934659.7134695.4334701.4533125.5034304.89Modal (Hz)23.9225.1021.2421.7722.8524.8534.5534.72Full-load strain (mm)12.0911.8714.4714.0613.2912.4111.919.39Full-load stress (MPa)330.86313.78410.04394.70381.34343.90317.57308.350° strain (mm)7.838.048.048.167.968.048.128.460° stress (MPa)319.52318.84308.38316.34320.73318.40327.48361.31Figure 22
Mass and stiffness Pareto front plot.Figure 23
Mass and modal Pareto front plot.It can be found from Table9 that if the ultimate goal is to ensure that the carriage is rigid enough to minimize the mass of the carriage, case 1 is the best choice. If the maximum stiffness of the carriage is required and the mass of the carriage is reduced, scheme 2 can be selected; if the mass stiffness of the carriage needs to be well uniform, schemes 3, 4, and 5 can be selected; moreover, to ensure the thickness of the carriage plate, schemes 6, 7, and 8 are a preferable choice. To achieve the goal of greatly reducing the quality of the carriage and improving the utilization rate of materials, scheme 2 with better comprehensive performance is selected and the whole circle inspection is carried out to verify whether the requirements of stiffness and strength of the carriage are satisfied. The weight of the carriage is 2.9 t after removing the complex and relatively negligible mass of the components. And the results are illustrated in Table 9.Table 9
Comparison of results before and after multiobjective optimization for multiple working conditions.
Before optimizationAfter optimizationIntegral circleChange amount (%)Base plate crossbeam (mm)85.115−37.5Main longitudinal (mm)6330.6631−50.79Base plate (mm)63.13−50Side panels (mm)85.095−37.5Crossbar (mm)64.074−33.33Longitudinal beam (mm)66.1360Mass (t)2.91.68361.6850−41.90Stiffness (N·mm−1)1840034703.553470088.59Modal (Hz)34.7225.1025.5−26.56Full-load strain (mm)10.8411.8712.1411.99Full-load stress (MPa)287.49313.78321.9932.480° strain (mm)13.598.049.78−28.040° stress (MPa)444.31318.84319.90−28.00Figures20–23 show the stiffness and strength of the carriage after the carriage is fully rounded.The comparison results of stiffness and strength before and after full-circle optimization are shown in Table9. It can be found in Figure 24 that the maximum deformation of the full-load uniform transport carriage before and after the optimization of the full circle changes from 10.84 mm to 12.14 mm, with a change rate of 11.99%. In Figure 25, the maximum strain of the fully loaded uniform transport carriage changes from 287.49 MPa to 321.99 MPa, with an increase of 34.5 MPa. And it can be seen from Figures 26 and 27 that the strain under 0° lifting conditions changes from 13.59 mm to 9.78 mm; the stress changes from 444.31 MPa to 319.90 MPa, which satisfies the stiffness and strength conditions under the safety factor of 1.2.Figure 24
Full-load strain after rounding.Figure 25
Full-load stress after rounding.Figure 26
Strain of 0° lift after rounding.Figure 27
Stress at 0° lift after rounding.


## 4. Conclusion

(1)
By synthesizing the stiffness and strength analysis of various working conditions of the carriage, the optimal Latin hypercube sampling and approximate model are leveraged to construct a comprehensive model of the carriage with full-load uniform transport and 0° lifting dual working conditions.(2)
Compared with different approximate model construction methods, it is found that the MLSM model has higher modeling accuracy. Furthermore, to ensure the accuracy of the subsequent multiobjective optimization design, the MLSM construction method is improved by changing the constructed single value to multivalue construction.(3)
The results of the dual working conditions and modal optimization design of the finite element model of the carriage show that the optimization effect of the carriage is obvious, in which the final mass is reduced by 1.215 t and the stiffness is increased by 16300 N·mm−1. Additionally, the results verify the optimized full-circle carriage model meets the design requirement. In this study, the analysis under a variety of severe working conditions has been done and a preferable result is obtained. Nevertheless, to achieve a more realistic scenario, more severe working conditions will be integrated into the lightweight design analysis in our future work.



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*Source: 2870005-2022-12-12.xml*