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The static contact characteristics of heavy-duty tracked vehicle roller and track plate contact structure are analyzed, and the influence mechanism of the roller’s shape parameters on the contact stress is studied. According to the Hertz contact theory, a mathematical model of the roller and track plate contact is established. The contact structure model is established in ANSYS software, and the simulation results are compared with the Hertz theory results to verify each other. In the parameter optimization section for the roller and track plate, based on the Hertz stress calculation formula, a new method is proposed to establish a roller and track plate Kriging model and to globally optimize the model by the genetic algorithm (GA). After that, the relationship among the track roller radius

The crawler tracking mechanism is widely used in heavy construction vehicles. Compared with the wheeled walking device, it can effectively ensure the stability of the whole machine and has the advantages of large traction and low specific pressure to the ground. The track roller is an important component of the crawler walking device, which supports the weight of the whole machine. However, its working environment is harsh, and the load is complex. If it fails, it will affect the overall work safety and work efficiency of the vehicle. Therefore, it is important to research the contact characteristics such as the roller load distribution pattern and relevant influencing factors.

In recent years, many scholars have studied the contact problems of the track roller. Arsic et al. [

In the project, the Hertz contact theory is often used to calculate the roller’s contact stress. The Hertz contact theory is also widely used in other similar structures, such as the wheel-rail contact of the train [

Although many researchers have studied the calculation method of the relationship between the track roller and the load, they have neglected the influence of the structural parameters of the crawler support wheel and the curved surface size of the crawler plate on the maximum contact stress. Therefore, based on studying the contact stress and equivalent stress in the contact area, this paper analyses the relationship between the structural parameters of the crawler support wheel and the crawler plate, establishes the Kriging model, and uses the genetic algorithm to optimize it, to reduce the contact stress between the heavy track roller and the track plate.

In the first part of this paper, the contact mathematical model between the roller and track plate is established by using the Hertz contact theory of arbitrary curved surface contact. Figure

Point contact between the roller and track plate model.

Based on the Hertz contact theory of arbitrary surface contact in modern contact mechanics, the surface near the contact point can be approximated as a paraboloid [

When any surface I and surface II are in elastic contact, since the surface near the contact point can be approximated as a paraboloid, any coordinate with O as the origin can be expressed as follows:

The coordinate system is established as shown in Figure

The roller equation can be expressed as

The track plate equation can be expressed as

In the model,

Considering that the contact between the roller and the track plate is only in a small area near the origin, the equation can be expanded according to the second-order Taylor formula at

Since the plane XOY passes through the tangent point and is a cotangent plane of two surfaces, the first partial derivative is 0; then,

Similarly,

Then, the abovementioned two equations can be rewritten as

Adding equations (

We make

Formula (

In Hertz theory, it is assumed that the contact area between arbitrary continuous surfaces is elliptical. Using the point displacement formula of the elastic half-space surface [

Among them, K(k) and E(k) are the first class and the second class is elliptic integral. Elliptical eccentricity

From the geometric relationship of the contact area deformation,

Bringing (

According to the method of undetermined coefficients,

We define

From (

We are going to explore the maximum stress on the contact point of the roller and find the influence of geometric parameters on stress. To simplify the calculation, only the area near the contact point is modeled for static analysis, and the ANSYS software is used to build the structural 3D solid model. The dimensional parameters related to the roller and the track plate are shown in Table

Size of the track roller and track plate.

Parameter | Notation | Value |
---|---|---|

Track roller radius | R1 | 0.6 m |

Track roller rim radius | r1 | 0.5 m |

Track roller width | B1 | 0.2 m |

Track plate radius | R2 | ∞ |

Track plate rim radius | r2 | 0.8 m |

Track plate width | B2 | 0.2 m |

Elastic modulus | E | 206 GPa |

Figure

Finite-element model.

Under the condition of the abovementioned model setting, the roller’s maximum stress is studied by loading the pressure from 400 kN to 2000 kN. When the normal load is 1200 kN, the pressure distribution in the contact area is shown in Figure

Contact pressure distribution.

Equivalent stress distribution.

To ensure the bearing capacity of the track plate structure is sufficient, the contact stress and equivalent stress value of the load are calculated to ensure that the strength meets the strength theory requirements. Table

Comparison of Hertz contact theory and finite-element simulation results.

Parameter | 400 (kN) | 800 (kN) | 1200 (kN) | 1600 (kN) | 2000 (kN) |
---|---|---|---|---|---|

Hertz theoretical value (MPa) | 1107 | 1395 | 1596 | 1757 | 1893 |

Contact stress simulation value (MPa) | 1020 | 1300 | 1500 | 1670 | 1800 |

Equivalent stress simulation value (MPa) | 634 | 807 | 932 | 1040 | 1110 |

Error between equivalent stress and theoretical value | 7.85% | 6.81% | 6.01% | 4.95% | 4.91% |

Distance between equivalent stress point and contact point (mm) | 6.555 | 8.009 | 8.000 | 9.804 | 9.796 |

Relationship between maximum contact pressure and load.

According to the design manual [

Genetic algorithm parameters.

Parameter | Value |
---|---|

Group number | 50 |

Crossover probability | 0.8 |

Mutation rate | 0.02 |

Repeat times | 11300 |

Related functions | Gaussian function |

Hertz stress calculation formula is relatively complex, including the first kind, the second kind of elliptic integral, and the transcendental equation, and its parameter values are conditionally limited and the calculation time will be relatively long, which cannot guarantee the correctness of the calculation result. Therefore, based on the Hertz stress calculation formula, a roller and track plate Kriging model is established, and the model is optimized by genetic algorithm.

The Kriging method [

Typically, computer-tested data consist of an input

Also, Z(X) is considered a random process.

In the expression, W and

In the abovementioned formula,

To simplify the symbols in the formula, we define

Therefore, the best linear unbiased estimate for the unknown point B is

The mean square deviation of

The parameter

Thus, the construction problem of the optimal Kriging model is transformed into a nonlinear unconstrained optimization problem.

Genetic algorithm is a computational model that simulates the natural selection and genetic mechanism of Darwin’s biological evolution theory. Its main feature is the use of a certain form of coding of decision variables as the object of operation. This way of encoding decision variables allows us to imitate the genetic and evolutionary incentives of organisms in nature and to conveniently apply genetic operators; the fitness function value transformed by the objective function value can be used to determine further the search range of the target function does not require other auxiliary information such as the derivative value of the objective function; the search process of the optimal solution starts from the initial population composed of many individuals, instead of starting from a single individual; it is an adaptive search technology, and its selection and operations such as crossover and mutation are all carried out in a probabilistic manner. Therefore, the genetic algorithm provides a general framework for solving complex system problems. It does not depend on the specific field of the problem and has strong robustness to the type of problem, so it is widely used in various fields [

As shown in Figure

Step 1: the Latin hypercube selection method is used to select the parameters of the Kriging model and the initial sample points.

Step 2: a Kriging model is established and optimized with genetic algorithms to find the best point.

Step 3: the obtained points are checked, and it is determined if they meet the constraints and convergence conditions. If not, the point is deleted. If the point satisfies the constraint condition but does not satisfy the convergence condition, the first step is repeated. Then, the key points will be added to the initial sample site for the next optimized Kriging model. Also, if the point satisfies these two conditions, it becomes the best point.

Optimization flow chart.

This paper aims to optimize the structural parameters of the roller and track plate under the condition that the width B of the roller and the track plate are constant. To meet the strength requirements, minimize the maximum contact stress in the contact area under heavy load conditions. The optimal design of the contact stress between the roller and the track plate is mainly determined by the following: the track roller radius

Range of values for each variable:

When the roller and the track plate are in point contact, the abovementioned verification shows that the maximum stress of the contact area is approximately equal to the calculation result of the Hertz formula. Therefore, with the minimum contact stress as the optimization target, there is

The most basic constraint is the contact strength constraint; that is, the maximum contact stress between the roller and the track plate is less than the allowable contact stress.

According to the abovementioned Kriging model, the parameters of the GA are set as Table

When the load is set to 800 kN, the maximum contact stress for the optimal solution after 11300 iterations is

Generational variation of design variables.

The final structural parameters of the roller and track plate are shown in Table

Optimized structural parameters.

Parameter | Value (m) |
---|---|

Track roller radius _{1} | 2.0011 |

Track roller rim radius _{1} | 2.0000 |

Track plate rim radius _{2} | 0.6158 |

When the roller and the track plate are in point contact, making the

The difference in the radius of the rim gradually decreases, and the relationship between the load and the maximum contact stress is shown in Figure

Influence of rim radius difference on maximum stress.

When the radius of the roller rim is different from the radius of the cylindrical surface of the track plate by 0.02 m, the maximum contact stress with the radius under the load of 1000 kN is as shown in Figure

Relationship between maximum contact stress and rim radius.

When the values of

The roller and the track plate are in plane line contact.

The maximum contact stress of the line contact is taken as the objective function:

When the rim radius

Comparison of maximum stress between plane contact and arc surface contact.

As shown in the abovementioned figure, when the load increases, the maximum contact stress of plane linear contact and arc surface contact increases. When the load is 180 kN, the maximum contact stress of the arc surface contact is equal to the maximum contact stress of the linear contact. When the load is less than 180 kN, the maximum contact stress of the plane line contact is slightly smaller than the arc surface contact. However, as the load increases continuously, after 180 kN, the maximum contact stress of the arc surface contact is significantly smaller than the maximum contact stress of the plane linear contact, and the difference between them is also increasing.

Table

Comparison of stress values between plane contact and arc surface contact.

Parameter | 400 (kN) | 800 (kN) | 1200(kN) | 1600 (kN) | 2000 (kN) |
---|---|---|---|---|---|

Plane contact stress value (MPa) | 350.0 | 494.8 | 606.0 | 699.8 | 785.4 |

Arc surface contact stress value (MPa) | 305.3 | 384.7 | 440.4 | 484.7 | 522.1 |

Difference (%) | 12.78 | 22.25 | 27.31 | 30.73 | 33.52 |

It can be seen from the abovementioned results that when the load on the roller is large, the maximum contact stress of the arc surface contact is smaller than the maximum contact stress of the plane line contact. This is because the contact surface of the roller rim has a certain curvature, and when the difference between the rim radius

Based on Hertz contact theory, the mathematical model of point contact between the roller and track plate is established. Then, the Kriging model and the genetic algorithm are used to obtain the minimum surface contact pressure of the roller rim. On this basis, the relationship between the maximum contact stress and the structural parameters of the roller is discussed, and the structure shape of the roller is optimized.

When the load is 800 kN, the genetic algorithm based on the Kriging model converges after 11300 iterations, and the optimal solution is obtained. At this time, the maximum contact stress is 121.40 MPa, which is 91.30% lower than the maximum contact stress of 1395 MPa under the same load in the second part of the verification model.

The difference of the radius of the arc surface between the roller and the track plate affects the maximum contact stress of the roller. With the decrease of the radius difference of the arc surface, the maximum contact stress decreases gradually. When the radius difference is reduced to a certain extent, the maximum contact stress drops rapidly.

When the contact surface between the roller and the track plate changes from a plane contact to a curved surface contact, the maximum contact stress can be reduced by up to 33.52%.

When the rim radius of the roller and the track plate are similar, the maximum contact stress decreases sharply when the rim radius increases. After the rim radius continues to increase, the maximum contact stress changes to a slow drop.

_{1,2}:

Radius of the track roller and track plate (m)

_{1,2},:

Rim radius of the track roller and track plate (m)

_{1},

_{2}:

Width of the track roller and track plate (m)

_{i},

_{i}:

Parameters used in the derivation of the Hertzian stress formula

Load on the track roller (kN)

Ellipse long semiaxis of the contact area (m)

Ellipse short semiaxis of the contact area (m)

First-class elliptic integral

Second-class elliptic integral

Integrated elliptic integral

Elliptical eccentricity

Material Poisson’s ratio of the track roller and track plate

_{1,2}:

Material elastic modulus of track roller and track plate (GPa)

Displacement of the contact point between the track roller and the track plate (mm)

Input parameters of the Kriging model

Input response of the Kriging model

System deviation in the Kriging model

The

Correlation coefficient in the Kriging model

_{2}:

Process variance in the Kriging model

Spatial correlation coefficient, where W and

An index which indicates how fast the value of the associated space function changes with w-x

Smoothness of the model

Optimal linear unbiased estimation

Least-square estimation of

Mean square deviation of

Maximum unbiased estimation of parameter

Maximum contact stress calculated by the Hertz stress formula (Pa)

Allowable contact static stress (MPa)

Allowable contact dynamic stress (MPa)

Maximum contact stress in line contact (MPa).

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that there are no conflicts of interest.

This work was funded by the Science and Technology Development Fund, Macau SAR (SKL-IOTSC-2018-2020).