Effect of the angles of the earth-moving machine moldboard on the cross slope of the graded surface

Investigations of the effect of the moldboard angle, the earth-moving machine of the motor grader type, on the angle of the cross slope of the formed earth surface was carried out. It is established, that the slope in the cross direction of the surface being graded, depends on the angles of installation of the moldboard blade element. The slope of the surface being graded, is not affected by the dimensions of the moldboard and the construction of the earth-moving machine. Analytical and graphical functional dependences of the angle of inclination of the moldboard blade element to the horizontal plane are obtained, which are necessary to ensure a given angle of the cross slope of the surface being graded. The results of the research can be applied in the field of development and tuning of on-board systems for indicating and controlling the operating modes of an earth-moving machine.


Introduction
To earth-moving machines (EM), widely used in road-building and agriculture for the profiling of earth surfaces, movement and leveling of ground, a number of requirements are produced. The work of many researchers is devoted to the solution of particular problems. The operation of the machine should be economical [1,2], and the control system should provide the convenience of control [3]. Also, a given accuracy of ground treatment should be ensured [4]. The construction of the EM should be fairly simple and reliable in operation [5,6]. A significant number of research works solve the problem of increasing productivity [7,8,9]. The kinematic analysis of the working equipment of the EM is one of the most important elements in the solution of most of the above problems [10,11]. The basis for the known methods of studying the kinematics of EM is laid flat calculation schemes, that do not take into account the spatial character of the constructions. Mathematical models of control systems for the moldboard working body of the EM, the use of which allows the formation of a certain profile of the roadbed (surface being graded) in the cross direction, as a rule, does not take into account the effect of the moldboard angle. This causes significant errors in the cross slope of the roadbed [3,10]. At an moldboard angle is meant the angle of rotation of the blade relative to the machine about the vertical axis. The derivation of the analytical dependencies of the blade tilt angle (angle of inclination of the moldboard to the horizontal plane) is actual. The use of dependencies in control systems of the EM will allow to provide the given values of the angle of the cross slope of the roadbed.

Theoretical theses
In figure 1, a, the image of the moldboard in the fixed coordinate system O 0 X 0 Y 0 Z 0 is given. The EM moves along the axis O 0 X 0 . In figure 1, b, shows the calculation scheme for illustrating the subsequent theses. The following designations are accepted on the scheme: Y po -coordinate of the center of the blade in the vertical direction; γ po -blade tilt angle to the horizontal plane; φ -the moldboard angle; γ f -the actual angle of the cross slope of the roadbed, at the moldboard angle φ and at the blade tilt angle γ po . Additional notations are introduced: L 1p =O 0 A; L 2р =O 0 A po ; L 3p =O 0 A f ; Δγ=γ−γ f -deviation of the given cross profile from the actual in angular expression (error of the cross profile); Δγ k =γ−γ po -correction of the blade angular position, required to ensure the slope of the graded surface in the cross direction γ. A number of theses were formulated.

Thesis 1
The angle γ of the cross slope of the surface, graded with the EM, with the value of the moldboard angle φ≠90°, can be provided by setting the blade tilt angle relative to the horizontal plane γ po : Proof From figure 1, b, it can be seen that Substituting (1) and (2) into (3) and carrying out the transformations, we get: For values of the moldboard angle φ, and the blade tilt angle to the horizontal plane γ, the surface that is formed by the EM will have a slope γ f in the cross direction, equal to tg arctg sin

Proof
From figure 1, b, it can be seen that Substituting (1) and (6) in (7), and carrying out the transformations, we get: The functional dependences of Δγ k and Δγ have the form: -arctg sin t ( ) g whence arctg 0.5 sin 2 c ( ) tg cos where arctg 0.5 sin 2 ctg cos ( ) The value of the correction Δγ n does not depend on the size of the moldboard, the type of the EM, its geometric parameters, the magnitude of the cross profile error Δγ, and is determined only by the values of the angles of the cross slope γ and moldboard φ. The validity of effect 2.1 follows directly from (15).

Thesis 3
If there is an error in the cross slope of the surface Δγ (at the moldboard angle φ≠90°), in order to compensate for the error, it is necessary to change the blade tilt angle relative to the horizontal plane by Δγ k : where K g ′ is the coefficient of compensation of the moldboard angular position: ( ) arctg sin tg tg arctg sin Proof From (10) and (11) we can write ( ) arctg sin tg tg arctg sin from which expression (16) is directly obtained.  The validity of effect 3.2 follows from the symmetry of the function (17) with respect to the values of the angles of the cross slope and moldboard γ=0° and φ=90°, respectively.

The experimental results
In figure 2, as an example, the functional dependences of γ po and γ f , obtained in the range -30°≤γ≤30° for fixed values of φ are given.