Mathematical model of crossing of discrete snow obstacles

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Introduction
In theory the flotation of vehicle is an ability of automobile to cross different road obstacles and to move over the roads without hard surface and off-road. Flotation is a component of mobility -the integrated performance property of transport-technological vehicles (TTV) determining the vehicle ability to perform the assigned task with optimal adaptability to operation conditions and to technical condition of the vehicle, in other words it is the vehicle ability to resist external and internal factors baring from performing assigned task. [1,2] It is customary to separate flotation into profile and supporting ones. Also, the water obstacle crossing is distinguished in particular group.
The analysis of works [3][4][5][6] and authors research made earlier allow to draw conclusion that it is necessary to point the discrete flotation, or the ability to cross the obstacles of discrete type by maneuvering or with considering the dynamics of vehicle. For example, the movement in the forest is essentially the maneuvering among trees and crossing them as the discrete obstacles. Another example of discrete obstacles are the snow drifts. They don't very long but it is impossible to overcome them when moving uniformly so the movement of vehicle in such conditions is possible only by dynamical crossing or in full speed. The figure 1 shows the provided structure of flotation mobility.

Revisiting dynamic crossing of terrain compartments with varying height of snow blanket
Seeing into the subject of moving over the snow road surface it is necessary to consider many factors characterizing both the vehicle and the features of snow blanket. In a majority of works it is customary to estimate vehicle mobility by reference to solving the quasi-static tasks. In this case the parameters of vehicle and supporting base are permanent. Also, the task can be solved by searching the parameters researched. With regard to vehicle moving over the snow generally the specific mass-dimension characteristics of vehicles (total weight, propulsor dimensions) and physical-mechanical and geometrical parameters of snow blanket (height and density) are assigned [6][7][8][9][10]. Actually, such approach is appropriate and unquestionable when defining qualitative characteristics and comparing different models of machines. An example would be moving across the snow field. In the context of providing the flotation mobility the situations when there are both local temporal and space changes of movement conditions or discrete obstacles are of special interest. Moreover, the estimation of length of such movement is to be brought into correlation with movement pattern and dimensions of vehicle. Thus, snow-covered ditch or hillock having in one direction the length of multi-hundred meters and in other one the width comparable with the length of vehicle in fact is discrete obstacle if vehicle crosses it not along but transversally. Meanwhile the share of such terrain parts accounts for a small percentage of all path the loss of flotation mobility is essential in this case. It should be pointed out that the ability to move in straight line is estimated for every vehicle. At the same time the resistance to moving can be different depending upon the type of propulsor and the coursekeeping system. According to authors research the growth of resistance when turning amounts from 30 to 80 percent for different designs of wheeled vehicles. It should be noted that it is necessary to perform calculations exactly for straight-line movement to estimate crossing of the discrete obstacles. Such requirement is based on the practical experience of moving over the snow-covered terrain.
Therefore, however in the case of choosing the effective vehicle there are calculation procedures determining as a result the best vehicle with regard to moving over the snow with permanent or conventionally permanent operational conditions, in further calculations it is necessary to estimate the ability to cross discrete obstacles like as snow drifts, banks etc. for every chosen vehicle. The formation of non-uniformity of height of snow blanket is based on the mechanics of movement of snow particles in snow-drifting and their stop near different barriers [11]. For clarity it is good to consider the classification of discrete snow obstacles as road surface for transport and technological vehicles.

Classification of discrete snow obstacles
In general, the discrete snow obstacles can be separated into natural and artificial, in other words, by the occurrence of directed anthropogenic factor. The natural ones can include snow drifts over the natural objects, and also over the terrain objects appeared as a result of anthropogenic effect not targeted at the formation of snow blanket. The artificial ones are formed by directed effect of a man, for example, over the snow-retaining devices or over the road borders appearing during road cleaning.
Also, the pattern of formation of snow drifts depends largely on the type of formation source of snow masses and on source deformability. It is possible to highlight nondeformable sources such as extensional objects of macroprofile, moats, ditches, forest edges, river banks, precipices, slopes etc. or single objects such as stubs, stones etc. and deformable sources, for example, coppice or grass clumps. In the last case there are no difficulties for movement cause the plants basing the formation of snow blanket bend and break when vehicle moving.
Depending on the type of terrain compartment the different structures of snow blanket can be pointed out. Firstly, the natural structure formed by natural snowfalls and thaw periods and also by natural processes inside the snow blanket. The next types of structure depend largely on the specificity of road cleaning. The borders of roads cleaned by snowplows throwing snow sideways have layer structure consisting of snow fallen in natural way and snow after milling. If roads are cleaned by banking the compacted snow dumps are formed near the borders having high density.
The next classification feature is the composition of snow blanket. Here it can be pointed out the natural composition including snow and ice typical for the majority of terrains, snow fields and cleaned non-trunk roads; and mixture of snow, ice and other components. There can be highlighted the mixtures with particulate matter such as sand or small chipping and mixtures with chemical-mechanical additions such as deicing agents. In both cases the physical-mechanical characteristics of supporting base are different.
The snow varies in physical-mechanical properties depending on its composition and structure. In this case the parameters depend on the structure, texture and chemicalmechanical composition.
Depending on the uniformity of formation source there can be distinguished permanent sources that are changeless during all over the snow period (for example, natural barriers and snow retention objects) and non-permanent ones, for example, the dimensions of snow dump on the border vary while the snow falls or the roads are cleaned, hence the formation sources change. In this case the size of snow drifts will be within reasonable bounds or the road will be cleaned.
For all situations described the form of snow drifts depends on the form of supporting base. In this case there can be pointed out the forms of snow drifts appearing on the plain surfaces usually in some distance away from the formation source and based on the specificity of movement of air masses or appearing on the barriers with irregular shape. At the same time, it is advisable to use the method considered by works [12][13][14][15] describing the situations when the snow drifts and their forms appear depending on the wind and the shape of bottoming surface. The forms and the dimensions of snow drifts have a direct impact on the resistance to vehicle movement.

Formal representation of snow discrete obstacle in relation with ability to cross it by using vehicle
With regard to the geometry of formation of snow drifts there can be distinguished several conditional forms. They vary depending on the form of supporting surface. Also, the estimation of flotation in case of different forms of obstacles should be adequate. In this case it is necessary to establish the criteria that the volume of snow drift in the local area is permanent for different forms. It allows to estimate the mobility during crossing discrete snow obstacles in conventionally similar conditions.
It is provided to describe the form of snow drifts by the next relation: is the beta-function. Depending on the parameters entering in the formula the rate of curve describing the geometry of snow drift is varying. We should consider different situations and compare the diagrams with the real snow drifts for visual clarity.
1) < 1, < 1 -the curve is convex and goes into infinity at the borders. If = the diagram is symmetrical. If = = 1 the lay of snow is uniform. 2) If < 1, ≥ 1 -the curve is convex and goes into infinity at the borders. If ≥ 1, < 1 -the curve is convex and goes into infinity at the borders. Therefore, every snow surface forms the resistance to vehicle moving. As stated before the TTV mobility when snow moving is usually estimated by quasi-statical task, in other words the draft and resistance forces are calculated for uniform motion. Meanwhile the dynamics of crossing the discrete (local) obstacles due to inertial forces isn't considered. Below there is the procedure considering dynamical crossing of discrete snow obstacles.

Resistance to moving during crossing snow drifts
In works [1,[7][8][9][15][16][17][18] there are relations for calculation of resistance to movement. The resistance force is calculated by the next formula: where is propulsor width, is initial stiffness of snow characterizing the specific resistance to compression of snow (it represents the stiffness coefficient in the primary stage of deformation), ℎ max is coefficient characterizing the rate of deformation in case of pressures corresponding to maximum consolidation, max is the maximum pressure under the propulsor. Meanwhile where is the height of snow blanket; is the propulsor width; is the coefficient of snow compressibility; is empirical coefficient.
= 0,0287 (100 ) 1,5 Therefore, this relation is suit for terrain compartment with uniform height of snow blanket. Meanwhile the adhesion is determined by the relation considering many factors, so, in general terms the ultimate draft force by adhesion for wheeled vehicle can be written in the next way where is a component of elementary reaction of friction between tire material and supporting surface, is a component of elementary tangential reaction of shearing resistance [5].
where is the protector saturation coefficient, is the width of tire protector, 0 is the width of reactionless part of protector, is the angle between grouser axis and meridian plane of wheel, ( ) is the relation between the friction coefficient of tire protector and the slipping, ( ) is the relation between the shearing coefficient of snow among the projecting parts of tire protector and the slipping.
The condition of ability of moving or flotation mobility is: where is the draft force by adhesion, is the draft force of propulsor. During calculations the least one is selected from and . But in terms of dynamical crossing of obstacles it is necessary to consider the equality of forces on the motion area: where is the mass of vehicle, is the acceleration (retardation). Because when moving over the snow drift the vehicle is retarded the resistance forces reduce.
Therefore, presetting the movement speed and calculating the values of resistance and adhesion in every following step while crossing the snow drift the place where the vehicle stops can be determined. The condition of the stop is: In the case when such condition hasn't occurred for all over the length of snow drift it can be accepted that the vehicle has sufficient discrete flotation.
It is necessary to point out that not only the parameters of bottoming surface but also the direction of wind, its speed and the presence of thaw periods and snowstorms impact on the forms of discrete snow obstacles. The structure including the ice crusts, the density, the dimensions and the form of snow crystals also depends on those and many other factors. This part of research will be considered in the next works.
In the article the new classification feature of flotation mobility has been provided which is the discrete flotation. The subject of dynamical crossing of terrain compartments with variable height of snow blanket is considered. The classification of discrete snow obstacles is given. The form of discrete snow obstacles depends on geometry of bottoming surface, the direction and the speed of wind, the number and the duration of snowstorms and thaw periods. Depending on these parameters the structure, the texture and the density are different. The form of snow drifts can be described by the relation provided. It considers the form, the scale and the extension of snow drifts. The relation considering the moving dynamics is provided to estimate the discrete flotation mobility during crossing the snow drifts. Further the calculations for real vehicles will be made and the experiments to confirm the theoretical assumptions will be conducted.