文献翻译--履带式越野车.doc

A detailed single-link track model for multi-body dynamic simulation of crawlersDror Rubinstein a,*, James L. Coppock b a Department of Mechanical, Engineering and Mechatronics, Faculty of Engineering, Ariel University Center, Ariel 44837, Israel b John Deere, Construction and Forestry Division, Dubuque, IA 52004, USAAccepted 10 September 2007Available online 26 November 2007AbstractCurrently available models for dynamic simulation of tracked vehicles do not include the necessary detail required of a high-fidelity model of crawlers. The rapid increase in computing speed enables the utilization of more complex models, which may include many bodies and force elements. A three-dimensional multi-body simulation model for simulating the dynamic behavior of a crawler was developed using the LMS-DADS simulation program. The model incorporates detailed description of the track, the suspension system, and the dynamic interaction among its components. Three-dimensional contact force elements are used to describe the interaction of thetrack links with the vehicles rollers, sprocket, and idler. User-defined force elements are used to describe the interaction between each track link and the soil. The normal and tangential forces are calculated using classic soil mechanics equations, such as Bekker and Janosi correlations. The grousers, which are a significant part of any crawler track link, were modeled using McKeys approach. The model includes new elements, such as the plasticity and viscosity properties of the soil. Sinkage and slip are calculated separately for each track link. Simulation results were compared with the experimental results. In some ride conditions, the simulation results indicate forward motion of a track link while in contact with the soil. The existing theories consider backward motion of a track link when the vehicle moves forward and the link is in contact with the soil. This phenomenon was verified in the experimental work. It was concluded that the influence of the track dynamics and the soil-link interaction on the vehicle dynamics can be better predicted with the newly developed model._ 2007 ISTVS. Published by Elsevier Ltd. All rights reserved.Keywords: Multi-body simulation; Tracked vehicle; Track link; Track spud; Crawler1. IntroductionThe suspension unit of a tracked vehicle consists of many components, which need to be optimized for proper design. Optimization based on field tests may significantly increase development cost and time. However, the simulation models for tracked vehicles still need improvement 1,2. Vehicle simulation programs are divided into two categories: the first consists of special-purpose codes and the second is composed of multi-body programs 3. The track simulation is usually based on the behavior of a single track link. In a special-purpose code, such as Rubinstein andGalili 1 and Wong 4, some effects regarding the dynamics of the entire track and the interaction between the track and other suspension units are ignored. A more accurate model of the entire vehicle can be obtained using multibody programs 5. The analyticalempirical theories are widely used for shear stressdisplacement relationships, as in Bekkers model 6 for brittle soils, Janosi and Hanamoto 7 forplastic soils, Wong 8 for brittle and plastic soils, and others 8. Some models are based on constitutive relations; the finite element and discrete element approaches developed by Karafiath 9 and Asaf et al. 10,11, respectively, are examples of such models. In practice, the single track-link models, which are part of all vehicle simulation models, are still based on analyticalempirical theories. The existing simulation models of track vehicles address the grouser effects inadequately. Therefore, these programs are mostly suitable for military-type track vehicles, and not for crawlers. Asaf et al. 10,11 developed a discrete element model of a track link including a grouser. The model was compared with the grouser model of Bekker 6 and the bladesoil interaction model of Mckyes 12. Note that in these papers, the Mckyes model was used to estimate the grouser affects. One of the conclusions of the papers is that it is possible to obtain a reasonable estimation of grouser effects using Mckyes approach. The purpose of this study is to present an interaction model of a crawler track link with the soil and to use the interaction model in a detailed multi-body model of the entire crawler.2. Basic soiltrack interaction model2.1. Soil-link interaction modelThe soil-link interaction model is based on the previous work of Rubinstein and Hitron 5. The accumulated effect of the entire track is achieved using the single-element model with a multi-body program. The previous work deals with a military track link that has a single contact surface. The contact surface of a crawler track link consists of three areas: (a) flat area, (b) grouser 1 area, and (c) grouser2 area. Three points are defined point 1, point 2, and point 3. The locations of the points are in the center of the flat area, grouser 1 area, and grouser 2 area, respectively. A schematic description of a track link of the crawler is shown in Fig. 1. Where:x0 y0, z0 the link mass center coordinate system;R1; R2; R3 location vector to points 1, 2 and 3, respectively;L1 the length of the flat area;L2 the lower length of grouser 1;L3 the upper length of grouser 1;L4 the length of grouser 2; andh1, h2 the height of grousers 1 and 2, respectively. The ground level is described using the XYZ worldcoordinate system. The ground level may be depicted in any profile as the following function:Z=f(X,Y) (1)The track link is spatially oriented, with coordinate system x0y0z0 attached to the link mass center. Where x0 is in the longitudinal direction of the link, y0 is the lateral direction, and z0 is perpendicular to x0 and y0. Another coordinate system, X0Y0Z0, is parallel to x0y0z0, and its origin is intercepted by XYZ origin. The link may penetrate into the soil when the sinkage is measured parallel to the z0-axis. A twodimensional plot of a ground profile with a sinking link is presented in Fig. 2. An upper view on the link is provided in Fig. 3. Note that point i (i = 1, 2,3) may be any of thepoints described in Fig. 1. In these figures: the origin of the x0y0z0 coordinate system (the mass center of the link);gi a point on the ground surface intercepted by a line parallel to the z0-axis;x, y, z the location of point in the XYZ coordinate system;x0; y0; z0 the location of point in the X0Y0Z0 coordinate system;xi, yi, zi the location of point i in the XYZ coordinate system;x0 i; y0 i; z0 i the location of point i in the X0Y0Z0 coordinate system; xgi ; ygi ; zgi the location of point gi in the XYZ coordinate system; x0 gi; y0 gi; z0 gi the location of point gi in X0Y0Z0 coordinate system; b track width; and Di the sinkage of point i in z0 direction. The sinkage distance in the direction of the z0 coordinate is (2)Detailed information regarding the calculation procedures of the values Di and D_ i is provided by Rubinstein and Hitron 5.2.2. Normal forcesThe track-soil model proposed by Bekker 6 is based on the stressdisplacement relationship for a single application of load to the soil. The original formula has been modified by adding a viscous friction element. The normal direction of the pressuresinkage and sinkagevelocity relationship under point i is (3)where pi is the pressure, b the track width, si the smallest size of the track, C the damping per unit area coefficient, and kc, k/, n are the empirically determined constants.According to Bekker, the value of the constant kc of Eq.(3) must be divided by the smallest size of the track. In thiscase: = (4) Note that Bekker originally proposed a vertical sinkage. In our case, the sinkage is perpendicular to the track link plane surface. During track loading, when the link is penetrating into the soil, the two component forces of Eq. (3) point in the same direction. During track unloading or rebound, the force components point in opposite directions. Therefore, a value of zero pressure may be obtained during unloading, while geometrically, the link is still below grade. This is the point at which the link and the soil actually part from one another. The force applied on point i due to the link sinkage is: , =1,2,3 (5)where Ai is the area of the surface of point i. Thus:A1 =L1b; A2= L2b; A3 =L4b.2.3. Shear forcesThe attainable locomotion of the crawler over a terrain is partially based on the shear forces that develop between track links and soil in the longitudinal directions of thelinks. The shear stressdisplacement relation is originally suggested by Janosi as follows: where si is the shear stress under point i, ji the sheardisplacement of point i, c the cohesion, / the angle of internal friction, and k is an empirically determined constant.According to Eq. (6), the shear stress increases as the sheardisplacement increases. The maximum shear stress isHowever, in reality, above a certain value of sheardisplacement, the shear stress decreases. This is due to the soil failure, which changes the soil parameters (c and u). We were unable to find a report in the literature about the sheardisplacement effect on soil properties. A simple approach is proposed, where the sheardisplacement affectsthe maximum shear stress and not the soil parameter. The proposed approach is a modified Janosi approachwhere jmax is the maximum sheardisplacement which affect increasing shear stress; ju the point of ultimate sheardisplacement, above which increasing the sheardisplacement does not affect the shear stress, ju jmax; k1 the constant; and r is the maximum shear ratio, 1 rP 0.In order for smaxi to be a continuous function, the following relation should be maintained: A plot of non-dimensional shear stress s* versus nondimensional sheardisplacement j* is provided in Fig. 4, where: Procedures for defining the sheardisplacement and force are provided by Rubinstein and Hitron 5. 2.4. Grouser effectThe traction force as well as the lateral force due to the grouser sinking into the soil may be a very significant part of the total force. However, the model described abovedoes not take this effect into account. In fact, when initiating the work without considering the grouser affect, the calculated track force was found to be only about 30% of its expected value.The soilgrouser interaction is a very similar case to the soilblade interaction. Therefore, the model proposed by McKyes 12 is suitable for this case. The assumptions of the two-dimensional models introduced by McKyes are better fit to the soilgrouser interaction in a longitudinal direction than to those in a lateral direction. Therefore,in the case of the lateral direction, the model is based on the three-dimensional model proposed by McKyes. The two-dimensional model is (11)where Pi is the grouser force of point i. The sinkage h is obtained as follows:where D2 and D3 are the sinkage of points 2 and 3, respectively; h1 and h2 are shown in Fig. 1. b is the width; for longitudinal direction, b = track width; for lateral directionand the grouser of point 2, b = (L2 + L3)/2, and for the grouser of point 3, b = L4. L2, L3, and L4 are shown in Fig. 1. q is the uniform distributed force; for longitudinaldirection, q = p1b and p1 defined in Eq. (3); for lateral direction, q = 0. v is the velocity; kp, kc, kca, kq, and ka are defined in McKyes 12.In order to achieve a reaction force in the horizontal and vertical direction, the McKyes approach was modified as follows:Detailed information about ji, jmax, ju, k, k1, and r is provided in Section 2.3. The force is applied on a point at 2/3 of the height of the grouser.3. Plastic effect of the soilThe deformation due to the track-link interaction with the soil may be characterized by small elastic and large plastic deformations, as well as failure zones where the soilparticles separate. In fact, we noticed that the plastic sinkage deformation caused by the grouser is about 95% of the total deformation. The distinction between elastic and plastic deformation is not included in the equations used to model the interaction of a track link and soil. For proper modeling, the plastic effect should be added.3.1. Normal forcesWhen the deformations under the track links are relatively small, the plasticity of the soil may be neglected. However, in the small area under the grouser, there can be relatively large soil deformations. We propose to modify Bekkers formulation using different curves for loading and unloading the soil, as shown in Fig. 5. The soil stress under the grouser is developed along the loading curve until certain values of stress and sinkage are reached. The maximum value of this sinkage point is defined as Da and the stress as pa. When the load is decreased, the stresssinkage relation is presented along the unloading curve. The unloading curve also serves as the reloading curve up to the sinkage value of Da. Beyond this point, the loading curve is used, and a new Da value is determined. The unloading curve intercepts the zero stress axis, where the residual value of the sinkage is ap Da, which defines the plastic deformation. The parameter ap describes the plasticity of the soil (0 ap 1). Forap = 0, there is no plastic deformation and for ap = 1, the deformation is purely plastic.We assume that the unloading curve is linear and can be determined in the equation below:The value of K depends on the soil properties ap and Da. It can be shown that the expression of K is obtained as follows: Based on Eqs. (3) and (14), the final formulation of the normal stress during the grouser sinkage is3.2. Shear forcesThe shear stress under the pads was suggested by Janosi(Eq. (5) and modified in this work. Note that it is easy to replace this relation with another, such as the one suggestedby Wong 8. The loading and unloading curves of the shear stress of the modified model are shown in Fig. 6. The soil shear stress under the pads is developed along the loading curve until certain values of shearstress and sheardisplacement are reached. The maximum value of this sheardisplacement point is defined as ja and the stress as sa, where: When the load is decreased, the stressdisplacement relation is presented along the unloading curve. The unloading curve is also used for the reloading curve up to the displacement value of ja. Beyond this point, the loading curve is used, and a new ja value is determined. The unloading curve intercepts the zero stress axis, where the residual value of the sheardisplacement is as ja, which defines the plastic deformation. The parameter as describes the plasticity of the soil (0 6 as 6 1). For as = 0, there is no plastic deformation and for as = 1, the deformation is purely plastic.At the point where the unloading curve intercepts the zero stress line, the value of the sheardisplacement is: j = as ja. At this point, the value of the sheardisplacement is set to zero (j = 0) and all further calculations refer to the origin of the sheardisplacement axis. The shear stress along the unloading curve is A combination of Eqs. (6) and (18) yields the final formulation of the shear stress: As mentioned earlier, when j = asja, the values of j and ja are set to zero. Note that the value of the sheardisplacements j and ja may be positive or negative. In Eqs. (6), (17)(19), positive values are assumed; otherwise absolute values should be used in these equations. The direction of the shear force depends on the direction of the sheardisplacement. Therefore, the force is where A is the contact area, and the parameter as is an additional parameter of the soil properties. One option for determining this parameter is to assume that the slope of the unloading curve S is equal to the initial slope of the loading curve. Thus: The slope S is defined as follows: Therefore the value of as based on this assumption is:The bounds of as in this case depend upon the bounds of ja. Assuming that 0 6 ja 0 and jc for j 0. The value of X is Xb at jb and Xc at jc. Note that Xb 0. When the first contact between the grouser and soil occurs, the values jb and jc are set to zero. From this point, the absolute value of the sheardisplacement is increased until a certain value, which is defined as jb or jc. For j P jb or j 6 jc, the value of X is determined along the loading curve and for jb j jc, the value of X is determined along the unloading curve. When the load is decreased, the stresssinkage relation is presented along the unloading curve. The values of Xb and Xc are: The parameter am describes the plasticity of the soil(0