INTERPRETING THE MAIN POWER CHARACTERISTICS CHOICE OF THE WHEEL VEHICLES GUIDED CUSHIONING SYSTEM

Department of Engineering Mechanics, Hetman Petro Sahaidachnyi National Army Academy, Lviv, Ukraine Department of Automobiles, Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine Department of Transport Technologies, Lviv Polytechnic National University, Lviv, Ukraine Department of Transport Technologies and Mechanics, Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine Department of Maintenance and Repair of Machines, Central Ukrainian National Technical University, Kropyvnytskyi, Ukraine

of the non-linear elastic forces of absorbers on vertical rolling vibrations and stability of the vehicle's movement along the curvilinear parts of the road has been developed only for the simplest physical and corresponding non-linear mathematical models of the WV dynamics [7,[10][11][12]. The static deformation and nonlinearity of the system are the main parameters for progressive (regressive) suspensions. That suspension is more comfortable for transportation of dangerous goods and passengers.
The obtained results, which can be the basis for creation of the software of the guided suspension, show that the elastic absorbers, with non-linear law of the recovering force change, affect not only the numerical characteristics of the cushioning mass (CM) vibrations, but provide them with the qualitative new property -their own frequency depends on the amplitude, as well. Thus, the WV with such a cushioning system along the road with the ordered bumps obtains new characteristics. If one is to look at the process from a mathematical point of view, it is non-periodic, since the oscillations amplitude of the damping device due to the impact of the obstacle attenuates and the natural frequency of oscillations depends on its amplitude. These cyclic periodic signals have been outlined by [13]. Furthermore, the critical

Introduction
Guided or semi-guided suspension system [1][2][3][4] has been widely used for improving of the wheel vehicle (WV) smooth movement for the last decades. It makes possible to adjust the main characteristics of the suspension system so that the dynamic loading on people or cargo would be minimum. The basis of "adjustment" of these characteristics is the physical and corresponding to it mathematical model of the WV dynamics along the bump road, as well as the response of the latter to this or that type of motion perturbation. However, the mathematical instrument for the analytical investigation of the WV dynamics under their different physical models [5][6] is developed generally for the small vibrations under linear dependence or recovering forces on the deformation of the elastic elements (tires in particular) or the damper devices deformation rate. The adaptive control process is carried out in addition to the speed of the spring part, which is an important special case of the regulated suspension [6][7][8][9]. In addition, it is necessary to investigate the relationship between the parameters v 1 and v 2 for which the nonlinear suspension has similar characteristics to the linear one. In some papers, the method of the analytical investigations provide the CM oscillatory motion (they will be analyzed below), T and / d dt i T -deformation and the rate of deformation of the elastic absorbers or damping devices; • maximum values of the damping devices resistance forces are sufficiently smaller than those maximum values of the absorbers elastic forces; • maximum deformations of the elastic forces are much smaller than those of maximum elastic deformations, because of that they are ignored; • The CM perturbation is taken into account by the initial conditions in certain mathematical models; • horizontal displacements of the WV CM and WV mass centre are much smaller than the vertical ones and can be ignored. Presented above makes possible to state that the relative WV CM motion is plane-parallel, so its position is specified by the position of the mass centre (point 0) and the angle of rotation t {^h of the mentioned part around the centre O. Position of the mass centre at any time (taking into account the mentioned above restrictions) can be easily found, relatively the static position by the coordinate -z t h. Thus, to build the mathematical model of the CM relative motion the 2-nd order Lagrange equation can be used. According to the chosen generalized coordinates the CM kinetic energy in its relative motion T looks like where M-mass of the cushioned area, IC -its inertia moment with respect to the lateral axis, which crosses the masses centre and is parallel to the transient motion vector speed.
As to the generalized forces, which correspond to the generalized chosen coordinates, they can be found, if they are primarily expressed by the WV geometric parameters and mentioned coordinates: • vectors of relative displacements , 1 2 T T v v h of the elastic absorbers connection points to WV from its initial position to the arbitrary one speed of the stable motion along the curvilinear areas of the road sufficiently decreases, which is caused by the CM vibrations.
The objective of the work was to obtain the analytical dependencies, which could be the basis for creation of the software product for the more general law than that, for example in [7], of the non-linear recovering force of elastic absorbers, non-conservative in particular.

Materials and methods
For the WV, physical model of which is presented in Figure 1 and consists of the cushionless -1, cushioned -2 parts interrelating between each other by the elastic absorbers -3 and damping devices -4, it is necessary to obtain such analytical dependencies, which would be the basis for creation of the software product of the guided cushioning system with the purpose: • to minimize the dynamic loadings on transported people and cargo; • to provide maximum speed of stable motion during the WV manoeuvring, going round the bump or along the curvilinear areas of the rod. The main assumptions, as to the physical model and the motion of the investigated object, are: • all the points of the cushioned mass move in the vertical surfaces perpendicular to the speed vector of the WV movement, the latter being constant; • the mass of the cushionless area is sufficiently smaller than that cushioned one and is being ignored while solving a task; • force characteristics of the right-hand and left-hand sides of the cushioning system are similar and are described by dependencies 2 -the projections on the axis OY and OZ of the active forces acting on the right and left areas of the CA WV; y 1 , z 1 and y 2 , z 2 -coordinates of these forces application points, that is, , sin c os y l z l If corresponding values are substituted in the last dependence, one will obtain: Following the similar way as for the active components of the recovering force one will obtain the passive components of the generalized forces , , In the presented above ratio , , functions specified according to the assumptions dealing with the damping device resistance forces. Thus, the vibration process of the cushioned mass is described by the system of differential equations in which the right-hand parts are found according to the Equations (6)- (8).
-static deformation of the elastic elements, , , , l l 1 2 1 2 b b -geometric parameters, which specify the location of elastic elements relatively the masses center, , k j v v -unit vectors directed along the axis stable reading system OZ and OY correspondingly ( Figure 1). In the case of symmetric location of the cushioned area weight centre relatively the elastic elements l 1 = l 2 and 1 Equations (2)-(4), as well as the assumptions as to the power characteristics of the cushioning system (See 3), make possible to present the active force components acting on the CM ( F1 v and F2 v ) as follows: The projection sum of the mentioned above forces and the cushioned area weight force on the vertical axis (CZ) will be nothing else but the active component of the generalized force, which corresponds to the generalized coordinate z that is, The active component of the generalized force, which corresponds to the generalized coordinates { that is, Q a { is found based on the following [14]: -from one side, the work of active forces acting on the CM, at the possible displacement of the mentioned generalized coordinate d{ , is found by the ratio A Q a a where MC a -the sum of active forces moments acting on the cushioned area B142 S O K I L e t a l .
phase, ã^h -natural frequency of vibration. The latter, being dependent on amplitude for the most of non-linear vibration systems, is found by relation If in the expression for the natural frequencies of the vertical vibrations more convenient notion "rigidity" is used, that is the static deformation, one will obtain Reliability of the presented above results can be described as follows: at v 0 1 = the known values can be found, which deal with the non-linear conservative [12], and at v v 0 1 2 = = -linear characteristics of the elastic absorbers. Besides that, at v v 1 2 = the CM natural frequency does not depend on the amplitude. It means that in the case of the non-linear non-conservative cushioning system (CS) force, which is changed according to the law The dependencies obtained above are the basis for creation of the software product for control of the nonlinear non-conservative recovering force of the absorbers to provide the most comfortable conditions of people transporting. It is known that the most comfortable conditions of transporting are those, when the vibration frequency of the WV CM varies within . .
Hz, [17][18]. That is why, if the main factor of the frequency control is the static deformation of the cushioning system, it is found as the function of the vertical vibrations amplitude according to the relation where semi-period according to the phase of the used special Ateb-functions.
In Figure 2 it is shown, according to Equation (14), the laws of the CM static deformation change (the main parameter of the guided CS, at which the frequency of the natural CM vibrations is of the most comfortable for transportation of dangerous goods and passengers.
Presented graphic dependencies show that in order to meet the ergonomic requirements as to the frequency of the vertical vibrations, in the case of the great CM vibration amplitudes .
To obtain the analytical solution of the obtained system of non-linear differential equations is not possible.
That is why let us analyze only the CM vertical and rolling vibrations separately.

The cushioning mass (CM) vertical vibrations
These vibrations occur when their perturbation is caused by the bump road, similar under the right and the left wheels. In the mathematical model of the CM dynamics it equals 0 / { . The mentioned factor, as well as the conditions combined with the power factors acting on the WV CM, make possible to present the first part of the differential system in Equation (9) It should be noted that the maximum value of the right-hand part of the differential equation is a small value in comparison to the maximum value of the function h . All the mentioned above makes possible to apply the general rules of the non-linear differential equations perturbation theory with "small" right-hand part [15]. Effectiveness of application of the mentioned methods depends on the possibility to build a real solution of the non-perturbated option of the equation in question. For the non-linear differential CM vertical vibrations this equation looks like Both the perturbated and non-perturbated differential equations have to describe the CM vibration processes and the necessary condition for this type of solution in Equation (11) is non-conjugated with respect to the generalized coordinate non-conservative recovering force that is, where a -amplitude, According to dependencies in Equation (15), dependence of the amplitude of the initial perturbation a a x 0 1 =^hh on the motion speed V is presented in Figure  4 and on the static deformation in Figure 5, at different values of the WV CS power.
The presented dependencies show that the amplitude of the WV CM initial vibration perturbation during getting into the bump: • is smaller for the greater motion speed; • greater values of the parameter v2 , correspond to the greater values of the initial perturbation amplitude in the case of the progressive or regressive power characteristics; • for the case v 0 1 2 and v 0 2 2 ; v 0 1 1 and v 0 2 2 , the amplitude of the initial perturbation is greater than the amplitude of the CS initial perturbation with the CS linear characteristics (at all other parameters being constant), and for the cases v is of the great practical importance, which is seen in Figure 3. The ergonomic requirements for such a CS as to the vibration frequencies are provided, when the static deformation is stable and does not depend on the CM vibration amplitudes (Figure 3).
The dependencies obtained above can be the basis for finding the amplitude of the initial perturbation of the vertical vibrations under the condition when the bump road under the right and left wheels are of similar profiles. If one assumes that the latter are described by relation The graph dependencies show that the qualitative nature of the CM vibration damping remains, but the amplitude damping rate greatly depends on the power characteristics of the absorbers.

The CM rolling vibrations
These vibrations occur under the condition when the CM motion perturbation was caused by getting on single bump by the right (left) WV tire. The mathematical model of the mentioned WV area vibration can be easily obtained from Equations (7) where sin sin cos The differential equation (18) is similar to Equation (10), according to its structure, that is why the method of its solution is not mentioned here, but the main relations, which describe the main characteristics of the rolling vibration, are presented. Thus, the natural frequency of these vibrations X , as a function of the rolling vibrations amplitude a{ is found by relation perturbation with the linear power characteristics and for the regressive one and the cases v 0 1 1 and v 0 2 1 ; v 0 1 2 and v 0 2 1 -it is smaller. As to the effect of the "small" non-linear resistance forces of the WV CM dynamics, it is demonstrated in the change in time of both the amplitude and the vibration frequencies. To state the law of the mentioned parameters change, advantage of the Vander Paul method was employed to Equation (10) and the following was obtained [19][20][21][22]: .
where t -CM inertia radius with respect to the longitudinal axis, which crosses the masses centre. In Figure 7 the dependence of the natural frequency of the rolling vibrations in H3 It is considered that the road curvature radius is much greater than the WV base, that is why it is assumed that the curvature radius of the WV points in the transient motion is constant.
If in expression for the moment of the inertia forces relatively the CM motion one uses only the maximum values of the used Ateb-functions, Equation (20) would look like: .
The last dependence specifies the critical value of the stable motion speed taking into account the rolling vibrations as follows In Figure 8, for some WV geometric and power characteristics, the dependence of the critical speed of the stable motion against the capsizing on the amplitude of the rolling vibrations is presented.
As it was expected, the CM rolling vibrations cause the decrease of the stable motion critical speed. Besides that, at critical speed of the stable motion due to the capsizing: From the other side, the most dangerous, because of the WV capsizing during the movement along the curvilinear areas of the road, are the rolling vibrations [23][24][25][26][27][28][29]. That is why to find the critical speed of the stable motion of the WV with non-conservative power CS characteristics, along the curvilinear area of the road, only the rolling CM vibrations will be taken into account. In this case, from the kineticstatic equations for the system cushionless-cushioned mass for finding the critical speed of the stable motion against capsizing V cr , one obtains: the progressive power characteristics of the cushioning system at ; . v v 0 0 1 1 < < 1 2 = and non-concentrative at .
; v v 0 4 0 0 1 < < < < 1 2 -. As to the motion along the road with insufficient bumps -the progressive CS and non-concentrative with values , v 0 4 1 < < 1 are recommended to be chosen, v 0 1 < < 2 at the static deformation - m. The developed model allows to obtain the parameters of the resonant mode of oscillations, which have their own limiting velocity of the inequality path. The suspension has been designed to minimize the effects of overload caused by the movement of the vehicle on a pavement with irregularities with a frequency that meets the ergonomic conditions for the greatest comfort of transporting people. The following studies for this class of suspensions have been conducted by the authors in subsequent publications.

Conclusions
The presented method for investigation of the WV dynamics with the non-linear CS power characteristics made possible to obtain the analytical dependencies, which can be the basis for creation of the software product of the guided WV in order to: meet the ergonomic requirements related to the people transportation; minimize the dynamic loadings on the transported loads; improve the motion stability along the curvilinear areas of the bump road. As to the quantitative recommendations, resulted in the obtained data in order to meet the ergonomic requirements related to the frequency of the vertical vibrations, it is proposed, in the case of the WV movement along the sufficiently bump road, to choose the CS with the static deformation .