Modeling of logging machine stability operating with a vertically held tree

. The article presents the results of mathematical modeling of the stability of a logging machine with a vertically held tree when it is removed from the canopy of the forest. The calculation schemes for two positions of a vertically held tree are described. The substantiation of the idealization for the calculation schemes of a vertically held tree is given. The dynamic stability equations of a logging machine with a vertically held tree are constructed on the basis of second kind Lagrange equations system. Due to analyze the interrelationships between dynamic parameters and processes of a logging machine with a vertically held tree, motion differential equations were integrated taking into account external influences and initial conditions. Using the MATLAB program, the integration operations of second kind Lagrange equations system were carried out for two cases: 1) The vertical tree is rigidly fixed and repeats the deviations of the logging machine; 2) The tree is perfectly held vertically with any deviations of the logging machine. The input parameters for the calculation were the rigidity of the connections, the parameters of the logging machine, the gripping cutting device (GCD) and the tree. The parameters of the slope of the logging machine were set using the implementation of a Gaussian random process with a given correlation and variance. The dynamic stability study of a logging machine with a vertical tree showed that the ideal tree alignment in a vertical position leads to a decrease in the inertia moment of the system and to a significant decrease in the frequency of angular and linear oscillations compared with a rigidly fixed tree.


Introduction
To reduce the negative impact of logging it is advisable to remove the tree from the canopy of the forest in a vertical position.Now manipulator logging machines are used for this which have high productivity and a large mass [1][2][3].But with low-volume logging it is not economically profitable to use powerful logging equipment.For example, for thinning logging it is advisable to use wheeled tractors of medium and small traction class with specialized logging equipment [4][5][6][7][8][9].
When a low-power equipment is used for carrying out a tree in an vertical position from the forest canopy, the tree and the tractor are exposed to disturbing factors affecting stability.As a result of this impact a logging machine with a vertically held tree becomes unstable.
Ensuring the stability of a logging machine with a vertically held tree by compensating for disturbing factors is possible in various ways but all of them will be reduced to the principle of keeping the tree in an vertical position under any external influence.
To assess the stability of a logging machine with a vertically held tree we will conduct a mathematical modeling in which the tree retains its vertical position regardless of the deviations of the machine.

Mathematical modeling
The effectiveness of the logging machine stability with a vertically held tree is revealed through the consideration of two calculation schemes (Fig. 1-2).Assume that the system has only two degrees of freedom -vertical and angular oscillations.
To simplify and speed up the calculation we idealize the system and set the tree in the form of an absolutely rigid rod.Such a simplification is possible, with a certain linearisation of the system (we consider the angular oscillations to be small) taking into account the additional degree of freedom (associated with the elasticity of the tree being moved) does not affect the first two generalized coordinates due to the decomposition of the system of equations into independent groups.
To study the dynamic stability of the system under consideration we use the main tool for solving problems of dynamics, namely the differential equations of a mechanical system motion obtained by applying Lagrangian formalism.
Based on the fact that the Lagrangian is the difference between the kinetic and potential energies of the system, the kinetic and potential energy of the system (shown in Figure 5) have the following form: where J_Mis the inertia moment of the logging machine relative to the transverse horizontal axis passing through the center of gravity of the logging machine; l is the specific length of the elastic suspension elements in the unloaded state; k_k1, k_k2is the specific rigidity of the elastic suspension elements, m d is the mass of the tree concentrated in the center of gravity; h cgT is the height of the tree gravity center; g is the acceleration of gravity L is the center distance of the logging machine; a is the distance from the far wheel to the gravity center of the logging machine; h cgM is the gravity center height of the logging machine; b is the distance from the near wheel to the axis of the tree; α is the inclination angle of the logging machine and the tree.
The system of second kind Lagrange equations for the scheme under consideration is given: Fig. 2 shows a scheme where the tree is always in a vertical position with working perfectly alignment system, i.e. the angular inclinations of the logging machine do not affect the tree.Kinetic and potential energy of the system is: Partial derivatives of kinetic and potential energies are presented: The system of second kind Lagrange equations for the scheme under consideration is given: When considering the dynamic stability of a logging machine with a vertical tree, the ideal operation of the tree alignment system in a vertical position leads to a decrease in the inertia moment of the system by an amount (( −  + ) 2 + ℎ  2 )  relative to the transverse horizontal axis passing through the gravity center of the logging machine.Under these conditions it is possible to consider the mass of a tree located at a height hcgT of the tree gravity center relative to the height of the machine gravity center applied at the same z level.This allows us to conclude that the dynamic stability of the system is qualitatively increased.

Computational experiment
Due to analyze the interrelationships between dynamic parameters and processes of a logging machine with a vertically held tree, motion differential equations were integrated taking into account external influences and initial conditions.To do this using the visual construction of a model block diagram from the standard blocks library in the MATLAB program, the integration operations of the second kind Lagrange equations system were carried out.
On the basis of equations ( 7) and ( 14) the corresponding block diagrams were compiled (Fig. 3-4) for two cases: 1.The vertical tree is rigidly fixed and repeats the deviations of the logging machine.
2. The tree is perfectly held vertically with any deviations of the logging machine.
The input parameters for the calculation were the rigidity of the connections, the parameters of the logging machine, the gripping cutting device (GCD) and the tree.The parameters of the logging machine inclination were set using the implementation of a Gaussian random process with a given correlation and variance of the process which was found in the framework of experimental studies [10].As a result of the numerical solution the following oscillation simulation graphs were obtained (Fig. 5 -6).

Conclusions
When considering the dynamic stability of a logging machine with a vertical tree, the ideal alignment of the tree in a vertical position leads to a decrease in the inertia moment of the system in absolute values.
Analysis of the angular oscillations graphs of a logging machine with a vertically held tree shows that the use of tree alignment reduces the frequency of vibrations compared to a rigidly fixed tree.Linear oscillations graphs of a logging machine with a vertically held tree show that the ideal vertical alignment has a slight advantage with the specified parameters.
Graphs of changes in linear and angular oscillations of a logging machine with a vertically held tree show their dependence on the design and elastic-deforming characteristics of the supporting elements of the running system of the machine, the distribution of its weight along the supports.

Fig. 1 .Fig. 2 .
Fig. 1.Dynamic stability calculation scheme of a logging machine with a tree deviated from the vertical.

Fig. 3 .Fig. 4 .
Fig. 3. Model block diagram of a vertically held tree that is fixed and repeats the deviations of the logging machine