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The thrird chapter outlines the demand on the mathematical model of operation and movement of lift-load trucks based on the problems connected with the longitudinal vertical surface presented in the first chapter. The calculated structural model of the lift-load truck with the load-lifter, as well as carriage with the fork lifter alignment and load have been carried out on the basis of generally accepted presuppositions (Fig.).
Fig. Calculation diagram of lift load truck
Further on the following air coordinates system has been used :
X0О0Z0 – non-moving system, where Z0 axis is parallel to the line of the force of gravity and oppositely directed axis X0 – is perpendicular to the axis Z0 and directed to the movement of lift-load truck.
X1О1,3Z1, X3О1,3Z3 - are movable, the first is firmly connected with the load-lifter ; and the second with the body, and both are lying on the surface X0О0Z0; their common origin – point О1,3 which lies along the turn axis of carriage wheels; the axis Z1,Z3 –are parallel to longitudinal axis of symmetry of load-lifter.
The system of X2O2Z2 – is movable air coordinate system that is firmly connected with the carriage and lies on the surface of arrangement of XoOoZo , axis Z2 – is parallel to the axis Z1 and coincides with the straight line along which the carriage moves.
Both the lift-load truck and carriage of load-lifter move along the surface XoOoZ10. Thus, it was decided to take 1, 3 for integrated coordinates – the angles of their rotation about an axis that passes through their common center O1,3 and coordinates  , of this center.
The carriage moves forward along the lift-load truck and the load moves along the fork. That is why coordinates  and  have been chosen as integrated.
The rotation movement of the driver’s elements together with the carrying wheels are characterized by the integrated coordinates  .
The figure shows:
C1, C2, C3, C4 – are the mass centers of the lift-loader, carriage, lift-load truck and load respectives.
On the basis of Langrage equation of the second type the system of equations of the model has been made which is as following:
 ,
де  ;  ;
 ;  ;
 ;  ;
 ;  ;
C13, C12 – extension clamping up rigidity of hydraulic cylinder between load-lifter and lift-load truck, hydraulic cylinder and chain of carriage lifting correspondingly.
 ,  ,  ,  - tangential and radius rigidity of track and guide wheel , radius rigidity of buffer.
The complete explanation of symbols used in the equations (1)- (7) has been given in the dissertation.
The system of equations (1) - (7) is supplied by generalized coordinates and speeds at the initial moment of time and provides a mathematical model of operation and movement of lift-load truck.
Seven equations of the system correspond to seven generalized coordinates. These equations relate to non-linear class of integral-differential equations, as they include non-linear both derivatives of unknown functions and their integrals (equations (1) and (7) ).
The equations of the system are non-linear because the derivatives from generalized coordinates and generalized coordinates themselves become non-linear unit of the system of equation, i.e.  and  into trigonometric functions their derivatives of the first order – into indicative; generalized coordinates  ,  - into indicative functions,  ,  ,  - into breaking functions of the first order; generalized coordinates Zo13 and Zo2 – are included into the integral with upper altering limit of integrity.
The use of computers for investigation with the help of the model have been motivated; the method of numerical solving of the system of equations (1) - (7) has been chosen; the structure of applied programs’ kit has been described; relations between kit’s modules and modules itself have been shown.
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