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Tasks of mathematical description and experimental modelling of the process of vehicle-soil interaction
Journal of Terramechanics, Vol. 29, No. 2, pp. 187-194, 1992. Printed in Great Britain.
0022-4898/9255.00+0.00 Pergamon Press Ltd © 1992 ISTVS
TASKS OF MATHEMATICAL DESCRIPTION AND EXPERIMENTAL MODELLING OF THE PROCESS OF VEHICLE-SOIL INTERACTION A. V.
MIROSHNICHENKO,*S.
L. SITNIKOV,*V. N. MEHANUK* and V. I. JASKIV*
Summary--This paper deals with the process of interaction of a tracked vehicle with soil. On the example of this interaction, a scheme of modelling of the vehicle's action on the soil has been given. A written and graphical description of the experimental testing unit, permitting modelling of the vehicle's action on the soil, has been given. INTEREST in the process of vehicle-soil interaction is not only explained by an increase in off-road locomotion and a decrease in power costs for locomotion, but also by an actual ecological problem of overcompaction and destruction of the top soil layer caused by the vehicle's action. The character of the mechanical properties of deformable ground and of soils, and the bases of these properties on granular composition, moisture content, density, temperature and other parameters, raise a wide range of problems in mathematical description of the process of interaction with transport systems before research can begin. Accuracy of such description is of vital importance in designing new types of vehicles and in predicting characteristics existing in actual conditions of operation. Additional complications arise due to the influence of the vehicle's action on the behaviour of deformable ground and due to the dependence of its response on kinematic and force factors of the action. In this case the process of the tracked vehicle-deformable ground interaction deserves consideration. Presented in Fig. 1 is a tracked vehicle in the x - z plane of the orthogonal system of x, y, z coordinates. The contact surface A - - f ( x , y) of the vehicle and deformable ground can be divided into the active parts A k ( k - - 1, . . . , m, m is the number of track rollers) and the passive parts Pj (j = 1, m = - 1 ) , in the absence of deformability of the belt along the y-axis. The active parts stand for the areas of contact produced by the belt and the rollers on the interaction surface. The kinematics of these parts is determined by the velocity of motion ore, by the value of the slip coefficient i, and by geometrical parameters of the belt and the rollers. The passive parts are the segments of the track belt between the rollers contacting only deformable ground, and their shape is determined by the force equilibrium of the b e l t - g r o u n d system.
*Soviet Society of Mechanical Engineers, Central Board, Chkalova Street 64, Block 1, 109004 Moscow, Russia. 187
188
A.V.
MIROSHNICHENKO
et al.
vm q
i//~/f(x ~)
f 0
p(x, y,z) ~
Z
Ulmin 0 U3max Un
0 qlmin q4min
~_
ql
~" •
A1 J
IA=I
I A3 I
I A41
I AS Xg
I
] I
I
I
I
I
I
I
I P1, IP21
I
I
I
I
I
I
]
I
IP31
',Pal
X
0.2
1.0 ~
[ ~'~Vm2
2.0 ~ t, S
Vm1 FIG. 1.
The representation allows modelling the vehicle's impact by pressure plate testing of deformable ground through the succession of action with prescribed kinematics (Ak parts). Thus a variation in velocities un, normal to the surface of the interaction is given by Un(X ) = Omfx(X , y)/[1 + f'xZ(X, y)]0.5
(1)
q(x, y*, z) = p'(x, y*, z) - T(x, y*, z)
(2)
and normal stress q by where p(x,y*, z) is stress on the interaction surface, with y = y* being fixed; r(x, y*, z) is the tangential component of p(x, y*, z). The active and passive parts are specified and presented in Fig. 1.
PROCESS OF VEHICLE-SOIL INTERACTION
189
The increase in force T being formed in the contact zone and transmitted by the belt is given by
T(x)
=
s;/s
)
f ' 2(, x , y)]0.5 dy dx, r(x, y, z)[1 + Jx
(3)
where b is the belt width. This results in a decrease in the track belt curvature with the growth of the x-coordinate. This, in turn, results in a decrease in un(x) and in an increase in stresses p, transmitted by the belt, in the inter-roller space (Fig. 1). That is, one can observe the decrease in the velocities' amplitudes Un,k(Un,kmax and Un,kmin) a s well as an increase in qjmin' The time of the vehicle's action on deformable ground depends on the velocity of motion vm and on the track-ground contact length L: L =
[1 + f'~2(x, y)]05 dx.
(4)
In Fig. 1 are given dependencies of the impact time t for the velocities oral = 3.6 km/h (1 m/s), ore2 = 18 km/h (5 m/s), Ore3= 36 km/h (10 m/s) for the quantity xg = 2 m. Frequency of alteration of the active and passive parts in the range of Oral to Ore3 for the five rollers vehicle varies from 2.5 to 25 Hz and is a function of the magnitude of the contact length, the number of track rollers, and the velocity of the vehicle's motion. In reality, the velocity un in the contact zone may be slightly different from the case presented in Fig. 1, which corresponds to a low value of initial tightening of the track and to the vehicle's operation in the absence of a drawbar load. For the case under consideration, the maximum value Un is achieved on the A part, while the value un computed from equation (1) for actual caterpillar machines and for their travel velocities, may reach 10 m/s or even 15 m/s; the time of variation in the velocity of u, to zero value may reach t = 0.02 s. The mode of variation in p ( x , y) has been studied in greater detail, but it varies considerably depending on the properties of the ground being deformed, on design, and on the conditions under which the tracked vehicle operates. Although cyclic recurrence of the variation in q is always inherent in multi-roller vehicles, q does not generally exceed 0.5 MPa. In fact, for the example considered, the basic requirements for the experimental installation intended for simulation of track-ground action have already been defined. Assigning the current testing set-ups employed in terramechanics is different and is aimed at obtaining deformational characteristics either in giving the law of the plate qw = q(t), or in giving the law of the plate displacement z = z(t) (or its velocity), whereas one generally strives for obtaining a2zl3t 2 = 0, for any z over the range of deformation. Now let us consider modelling of the action per se. Two variants are possible here: (1) modelling of kinematic-force action on the plate-ground contact surface in the frequency ranges under study, including kinematic, geometrical and force parameters of the vehicle with a view to studying the response of the deformable ground followed by its mathematical description; (2) modelling of the action that would provide for the required stressed state in the deformable ground.
190
A . V . MIROSHNICHENKO et al.
Actuality and timeliness of the second variant in relation to the problem of constructing an ecologically compatible vehicle is sufficiently evident. Let us consider this variant in detail. A classical approach to defining tension in an elastic half-space resulting from a local action on the surface was suggested by Boussinesq and Herruty, who made use of the theory of potential. It allows determination of all six components of tension at an arbitrary point in a half-space [1]. Applicability of this approach to ground has repeatedly been discussed. Investigations [2] have, in particular, proved adequacy of its use for both dynamic and static loading. If the classical approach is used, allowing for the concentration coefficient v, introduced by Froehlich for deformable ground, then the component Crz(t) of tensions caused by the impact of the vehicle moving under steady-state conditions, with certain assumptions at the point of half-space (Xl, Yl, Zl) being taken into account, will be
v (fpz(X, y)[Zl - Z(X)] vdx dy az(Xl, Yl, Zl, t) : ~-~ JsJ p~+2(t )
(5)
where pz(x, y) is the vertical component of p(x, y); S is the value of the contact surface area, and pg = [(Xl -- V m t -- X) 2 + (Yl -- y ) 2 + (ZI -- z)2]05. (6) In modelling the vehicle's impact by means of plates, the corresponding tension cannot be achieved throughout the entire half-space due to the size difference in the areas of the vehicle and plate contact surfaces. In this case it is necessary to select an area of half-space where tension could be most precisely generated, proceeding from the purpose of the investigation made. If an area degenerates into a fixed point of half-space beneath the plate, then ozw caused by the plate action equals
azw(Zl, xa, Yl, ') - vqw(') 2~
fswf [Zl
Zrn] v dxwdyw
p7 2
,
(7)
where qw(t) is the pressure which is taken to be evenly distributed over the plate area
Sw; P w = [(Xl -- X w ) 2 + (Yl -- Yw) 2 + (Zl -- Zw)2] 0'5.
(8)
Equating z and Zw we obtain the dependence q~(t) for the plate which will reproduce tension at the point with the coordinates (x, y, z), brought about by the vehicle's impact;
qw(t)
=
f ( Pz( x' Y)[Z____2_I-_z(x)!~dxdy JsJ p~+2(x, y, Z, t)
(9)
f%f [Zl - Z;];dxwdy~ Stress tension at a point of half-space may be modelled most precisely. If an area considered cannot be represented by a point of half-space, then it is advisable to determine q~(t) by using mean integral stresses
1 f v f f pz(X, y ) [ z w - z ( x ) f f d x d y ~z(t) = - W JW J s J ~ - ~ v+2 dW PgW(t)
(10)
PROCESS OF VEHICLE-SOIL INTERACTION
191
where W is the region of half-space; Zw, Pgw is the coordinate of the point in the region W and the distance from this point to the surface of the vehicle-ground interaction. In connection with complexity of the process occurring in the vehicle-deformable ground interaction, experimental investigations constitute a basis both for the study and mathematical description and for the development of vehicles in mobile systems as a whole. Also, modelling of the vehicle impact under laboratory conditions will not only enable regularities of a response formation on the interaction surface to be determined, but it will also allow the behaviour of the deformable ground at local impact to be formalized. Testing units used in terramechanics do not allow modelling vehicle's actions in real ranges of velocities, frequencies and forces. Neither do they allow modelling regularities of their variation in the zone of interaction. To effect actions needed and to conduct investigations into deformable ground in relation to the interaction with vehicles, the testing device should meet the following principal requirements; --the velocity of deformation of the ground should attain 15 m/s; --frequency of the harmonic law of loading should amount to 25 Hz (with the amplitude of deformation remaining 30 mm); --the depth of deformation of the ground by the plate should come to 400 mm; --capacity for modelling laws of action, complicated in form (Fig. 1 may serve as an example) F = f(t); F = f(z); F = f(z, t); v = f(t); v = f(z); v = f(z, t); v -- f ( F ) ; --high quality of laws of action being modelled; --the testing device unaffected by unstable parameters of the ground to be deformed. Analysis of different types of actuators showed that an electric-hydraulic tracking actuator meets the prescribed requirements most advantageously. The principal circuit of such an actuator is given in Fig. 2 which depicts the plate (pointer 1) that deforms the ground (pointer 2); pointer 3 stands for the working hydraulic cylinder (HC); 4 is a servo-valve controlling the displacement of the HC piston; 5 is the hydro-accumulative plant which supplies large amounts of hydraulic fluid through the servovalve 4; 6 is the hydrostatic bearing; 7 is the hydraulic brake; 8 is the adaptive system for controlling the hydraulic actuator of the device; 9 is the force meter. In developing and constructing the testing unit (Fig. 2) for the modelling of dynamic actions on the ground, a series of scientific and designing tasks have been solved, making it possible to produce the required technical characteristics set out earlier. One of the main problems was limited functional capabilities of conventional hydraulic cylinders regarding the velocity of the piston displacement. At the velocity of the piston displacement exceeding 3 m/s and at the pressure of liquid exceeding 15 MPa in the contact zone of the compactions of the HC (made of rubber or plastic), the temperature rises over 300 °C. Intensive wear of the compacting element and irreversible relaxation processes of the compaction materials take place [3]. Also, in contact compactions of the HC there occur forces of dry friction which are substantial in quantity. These exert considerable influence on the form of the law of loading followed by the piston. That is why hydrostatic bearings (pointer 6) for the piston and the rod, ensuring guaranteed radial clearance with the walls of the cylinder and excluding dry friction, were used for the working hydraulic cylinder 3 (Fig. 2). Parameters of the hydrostatic bearings were determined by calculating the liquid
192
A.V. MIROSHNICHENKOet al.
/
3
I
17
i
i[i
I
/7
F
m
4/
,,6
8\
)
I/
Plate Ground Hydraulic cylinder Servosystem Hydraulic accumulator Hydrostatic bearing Hydraulic brake Adaptive system Forcemeter
I
"1
2
Fro. 2.
flow in the slots of the hydrostatic bearings from the Reynolds equation Rr 8q)
Sq) ]
+
~-z
h3
8z ]
= 12 #Vr + 6 l~Vz
dz
(11)
where z and cp are the cylindrical coordinates; Pr is the liquid pressure over the surface of the hydrostatic bearings; h is the radial clearance of the hydrostatic bearings; /~ is the coefficient; Vr and Oz are velocities of the piston displacements in the radial and axial directions; R is the radius of the HC rod. Hydrostatic bearings allowed removing limitations on the displacement velocity of the piston and considerably improved the dynamic characteristics of the hydraulic actuator. Another task was to ensure safe and reliable operation of the hydraulic cylinder at piston displacement velocities reaching 15 m/s. Hydrodynamic processes of the liquid flow through the throttling slots of the suitable hydraulic braking unit--7 (Fig. 2), at the end of the HC piston's strokes have been considered. In the mathematical model of the process of the piston's braking, account was taken of the influence of the geometry of the throttling slot on the amount of the liquid consumed, of the liquid
PROCESS OF VEHICLE-SOIL INTERACTION
193
compressibility, and of the variations in viscosity and modulus of volume elasticity of liquid depending on pressure and temperature. The profile of the throttling ring slot of the brake, that allows smooth braking and shock-free stopping of the hydraulic piston of the HC moving at a velocity of 15 m/s, has been determined from the results of the mathematical modelling of the braking process. Combining the hydraulic cylinder constructed on hydraulic supports with the home-made servovalve of "C" series (1250 1/min consumption at a pressure differential of p - - 7 MPa) allowed the principal technical characteristics expected of the testing unit to be obtained. The characteristics are: force on the pressing plate, deformation velocity, frequency of loading and an involved form of the law of action prescribed with the IBM PC. But parameters of the hydraulic actuator will be unstable due to hydraulic, electric, technological, temperature and design factors as well as due to other factors, which in turn results in the variation in the output characteristics of the testing set-up. Considering properties of grounds or soils, a response of the ground being tested to the actuator of the test unit also is of random nature and depends on many interrelated factors [4]. Therefore in modelling the vehicle action the form of the output law of load will be distorted owing to unstable parameters of the hydraulic actuator and to a response of the ground, which may result in a substantial difference from the given one. To account for the error in following the law of action is difficult because of the fact that the ground deformed changes its properties in the process of deformation while deformations cannot be taken into account in advance. Recently there have been systems used that permit enhancing the quality of controlling and exclude the influence of unstable parameters of the system and of external perturbing actions on its output characteristics at the expense of optimizing the controlling laws. To improve dynamic characteristics of the hydraulic actuator of the test-unit, a standard model of the search-free, gradient, self-adjustable system (SAS) of controlling has been chosen from a variety of adaptive control systems, proceeding from the laws of loading being modelled. The structural circuit and its adjustable parameters were obtained on the basis of the theory of automatic control, optimization and sensitivity theory [5]. The algorithm of the action of the self-adjusting control system is based on the variation in parameters being adjusted in the electronic circuits, the criterion being the quality of work of the system as a whole. Proceeding from the notion of gradient, the velocity of variation in the adjustable parameter may be written down in the form ~i = - - / ~ i
9Qc ~E 3E
~0gi '
(12)
where cr is the parameter of the system being adjustable; ~ is the coefficient of the velocity of self-adjustment; Qc is the criterion of the quality of work of the system's operation; e is the error between the standard signal and the output of the hydraulic actuator; i = 1, . . . , n. Constructing the SAS structure and selecting the parameters adjusted is beyond the scope of the present paper [5]; therefore we confined ourselves to considering the results of the theoretical and experimental studies of the test unit. Comparison of the results of calculations and of experimental investigations is given in Fig. 3 in the form of plots for the transition processes. Curve 1 corresponds to the transition process in the hydraulic actuator with the SAS of control; curve 3 shows the
194
A . V . MIROSHNICHENKO et al. m
2 /"
1
i
/
/t~////
Z
I
I
I
t
t
J
0.02
0.04
0.06
0.08
0.10
0.12
l, S F~G. 3.
transition process in the standard model. Experimental transition processes are represented by solid lines (Fig. 3) while dashed lines show computed curves of the transition processes obtained from a nonlinear mathematical model of the hydraulic actuator.
CONCLUSIONS
The principal regularities and parameters of the vehicle's action on the ground have been determined for an example of tracked vehicle--deformable ground interaction. Two ways of modelling the vehicle's action have been proposed, with the investigations into deformable ground being made by means of pressure plates. The principal technical requirements for the testing device for modelling of the vehicle action in investigating deformable grounds have been formulated. The testing device, including the adaptive hydraulic actuator that enables modelling real laws of a vehicle's action on deformable ground, has been developed and constructed.
REFERENCES [1] K. L. JOHNSON, Contact Mechanics (Mechanica contactnogo vzaimodeistviya). "World", Moscow (1989). [2] A. AKAI, S. SHIMOI and T. KxucHi, Model studies on the stress distribution in layered soil systems. Proc. J S C E , 185, 83-94 (1971). [3] G. A. GOt,UBEV, G. M. KUKIN, G. E. LAZAREV and A. V. CHICHINADZE, Contact sealing of rotating shafts (Kontactnye uplotnenija vrashaushihsja valor). State Publishing House of Machinery (1976). [4] N. F. KOSHARNY1, Technical and operational properties of cross-country automobiles (Technicoekspluatacionnye svoistva automobileyi vysokoyi prohodimosti), p. 208. Vysshaya Shkola, Kiev (1981). [5] D. N. PoFov and S. L. SX~IKOV, Selecting of design and parameters of self-controlling hydraulic actuator of the testing unit. Vestnik Mashinostroenija (1) (1990).
0022-4898/9255.00+0.00 Pergamon Press Ltd © 1992 ISTVS
TASKS OF MATHEMATICAL DESCRIPTION AND EXPERIMENTAL MODELLING OF THE PROCESS OF VEHICLE-SOIL INTERACTION A. V.
MIROSHNICHENKO,*S.
L. SITNIKOV,*V. N. MEHANUK* and V. I. JASKIV*
Summary--This paper deals with the process of interaction of a tracked vehicle with soil. On the example of this interaction, a scheme of modelling of the vehicle's action on the soil has been given. A written and graphical description of the experimental testing unit, permitting modelling of the vehicle's action on the soil, has been given. INTEREST in the process of vehicle-soil interaction is not only explained by an increase in off-road locomotion and a decrease in power costs for locomotion, but also by an actual ecological problem of overcompaction and destruction of the top soil layer caused by the vehicle's action. The character of the mechanical properties of deformable ground and of soils, and the bases of these properties on granular composition, moisture content, density, temperature and other parameters, raise a wide range of problems in mathematical description of the process of interaction with transport systems before research can begin. Accuracy of such description is of vital importance in designing new types of vehicles and in predicting characteristics existing in actual conditions of operation. Additional complications arise due to the influence of the vehicle's action on the behaviour of deformable ground and due to the dependence of its response on kinematic and force factors of the action. In this case the process of the tracked vehicle-deformable ground interaction deserves consideration. Presented in Fig. 1 is a tracked vehicle in the x - z plane of the orthogonal system of x, y, z coordinates. The contact surface A - - f ( x , y) of the vehicle and deformable ground can be divided into the active parts A k ( k - - 1, . . . , m, m is the number of track rollers) and the passive parts Pj (j = 1, m = - 1 ) , in the absence of deformability of the belt along the y-axis. The active parts stand for the areas of contact produced by the belt and the rollers on the interaction surface. The kinematics of these parts is determined by the velocity of motion ore, by the value of the slip coefficient i, and by geometrical parameters of the belt and the rollers. The passive parts are the segments of the track belt between the rollers contacting only deformable ground, and their shape is determined by the force equilibrium of the b e l t - g r o u n d system.
*Soviet Society of Mechanical Engineers, Central Board, Chkalova Street 64, Block 1, 109004 Moscow, Russia. 187
188
A.V.
MIROSHNICHENKO
et al.
vm q
i//~/f(x ~)
f 0
p(x, y,z) ~
Z
Ulmin 0 U3max Un
0 qlmin q4min
~_
ql
~" •
A1 J
IA=I
I A3 I
I A41
I AS Xg
I
] I
I
I
I
I
I
I
I P1, IP21
I
I
I
I
I
I
]
I
IP31
',Pal
X
0.2
1.0 ~
[ ~'~Vm2
2.0 ~ t, S
Vm1 FIG. 1.
The representation allows modelling the vehicle's impact by pressure plate testing of deformable ground through the succession of action with prescribed kinematics (Ak parts). Thus a variation in velocities un, normal to the surface of the interaction is given by Un(X ) = Omfx(X , y)/[1 + f'xZ(X, y)]0.5
(1)
q(x, y*, z) = p'(x, y*, z) - T(x, y*, z)
(2)
and normal stress q by where p(x,y*, z) is stress on the interaction surface, with y = y* being fixed; r(x, y*, z) is the tangential component of p(x, y*, z). The active and passive parts are specified and presented in Fig. 1.
PROCESS OF VEHICLE-SOIL INTERACTION
189
The increase in force T being formed in the contact zone and transmitted by the belt is given by
T(x)
=
s;/s
)
f ' 2(, x , y)]0.5 dy dx, r(x, y, z)[1 + Jx
(3)
where b is the belt width. This results in a decrease in the track belt curvature with the growth of the x-coordinate. This, in turn, results in a decrease in un(x) and in an increase in stresses p, transmitted by the belt, in the inter-roller space (Fig. 1). That is, one can observe the decrease in the velocities' amplitudes Un,k(Un,kmax and Un,kmin) a s well as an increase in qjmin' The time of the vehicle's action on deformable ground depends on the velocity of motion vm and on the track-ground contact length L: L =
[1 + f'~2(x, y)]05 dx.
(4)
In Fig. 1 are given dependencies of the impact time t for the velocities oral = 3.6 km/h (1 m/s), ore2 = 18 km/h (5 m/s), Ore3= 36 km/h (10 m/s) for the quantity xg = 2 m. Frequency of alteration of the active and passive parts in the range of Oral to Ore3 for the five rollers vehicle varies from 2.5 to 25 Hz and is a function of the magnitude of the contact length, the number of track rollers, and the velocity of the vehicle's motion. In reality, the velocity un in the contact zone may be slightly different from the case presented in Fig. 1, which corresponds to a low value of initial tightening of the track and to the vehicle's operation in the absence of a drawbar load. For the case under consideration, the maximum value Un is achieved on the A part, while the value un computed from equation (1) for actual caterpillar machines and for their travel velocities, may reach 10 m/s or even 15 m/s; the time of variation in the velocity of u, to zero value may reach t = 0.02 s. The mode of variation in p ( x , y) has been studied in greater detail, but it varies considerably depending on the properties of the ground being deformed, on design, and on the conditions under which the tracked vehicle operates. Although cyclic recurrence of the variation in q is always inherent in multi-roller vehicles, q does not generally exceed 0.5 MPa. In fact, for the example considered, the basic requirements for the experimental installation intended for simulation of track-ground action have already been defined. Assigning the current testing set-ups employed in terramechanics is different and is aimed at obtaining deformational characteristics either in giving the law of the plate qw = q(t), or in giving the law of the plate displacement z = z(t) (or its velocity), whereas one generally strives for obtaining a2zl3t 2 = 0, for any z over the range of deformation. Now let us consider modelling of the action per se. Two variants are possible here: (1) modelling of kinematic-force action on the plate-ground contact surface in the frequency ranges under study, including kinematic, geometrical and force parameters of the vehicle with a view to studying the response of the deformable ground followed by its mathematical description; (2) modelling of the action that would provide for the required stressed state in the deformable ground.
190
A . V . MIROSHNICHENKO et al.
Actuality and timeliness of the second variant in relation to the problem of constructing an ecologically compatible vehicle is sufficiently evident. Let us consider this variant in detail. A classical approach to defining tension in an elastic half-space resulting from a local action on the surface was suggested by Boussinesq and Herruty, who made use of the theory of potential. It allows determination of all six components of tension at an arbitrary point in a half-space [1]. Applicability of this approach to ground has repeatedly been discussed. Investigations [2] have, in particular, proved adequacy of its use for both dynamic and static loading. If the classical approach is used, allowing for the concentration coefficient v, introduced by Froehlich for deformable ground, then the component Crz(t) of tensions caused by the impact of the vehicle moving under steady-state conditions, with certain assumptions at the point of half-space (Xl, Yl, Zl) being taken into account, will be
v (fpz(X, y)[Zl - Z(X)] vdx dy az(Xl, Yl, Zl, t) : ~-~ JsJ p~+2(t )
(5)
where pz(x, y) is the vertical component of p(x, y); S is the value of the contact surface area, and pg = [(Xl -- V m t -- X) 2 + (Yl -- y ) 2 + (ZI -- z)2]05. (6) In modelling the vehicle's impact by means of plates, the corresponding tension cannot be achieved throughout the entire half-space due to the size difference in the areas of the vehicle and plate contact surfaces. In this case it is necessary to select an area of half-space where tension could be most precisely generated, proceeding from the purpose of the investigation made. If an area degenerates into a fixed point of half-space beneath the plate, then ozw caused by the plate action equals
azw(Zl, xa, Yl, ') - vqw(') 2~
fswf [Zl
Zrn] v dxwdyw
p7 2
,
(7)
where qw(t) is the pressure which is taken to be evenly distributed over the plate area
Sw; P w = [(Xl -- X w ) 2 + (Yl -- Yw) 2 + (Zl -- Zw)2] 0'5.
(8)
Equating z and Zw we obtain the dependence q~(t) for the plate which will reproduce tension at the point with the coordinates (x, y, z), brought about by the vehicle's impact;
qw(t)
=
f ( Pz( x' Y)[Z____2_I-_z(x)!~dxdy JsJ p~+2(x, y, Z, t)
(9)
f%f [Zl - Z;];dxwdy~ Stress tension at a point of half-space may be modelled most precisely. If an area considered cannot be represented by a point of half-space, then it is advisable to determine q~(t) by using mean integral stresses
1 f v f f pz(X, y ) [ z w - z ( x ) f f d x d y ~z(t) = - W JW J s J ~ - ~ v+2 dW PgW(t)
(10)
PROCESS OF VEHICLE-SOIL INTERACTION
191
where W is the region of half-space; Zw, Pgw is the coordinate of the point in the region W and the distance from this point to the surface of the vehicle-ground interaction. In connection with complexity of the process occurring in the vehicle-deformable ground interaction, experimental investigations constitute a basis both for the study and mathematical description and for the development of vehicles in mobile systems as a whole. Also, modelling of the vehicle impact under laboratory conditions will not only enable regularities of a response formation on the interaction surface to be determined, but it will also allow the behaviour of the deformable ground at local impact to be formalized. Testing units used in terramechanics do not allow modelling vehicle's actions in real ranges of velocities, frequencies and forces. Neither do they allow modelling regularities of their variation in the zone of interaction. To effect actions needed and to conduct investigations into deformable ground in relation to the interaction with vehicles, the testing device should meet the following principal requirements; --the velocity of deformation of the ground should attain 15 m/s; --frequency of the harmonic law of loading should amount to 25 Hz (with the amplitude of deformation remaining 30 mm); --the depth of deformation of the ground by the plate should come to 400 mm; --capacity for modelling laws of action, complicated in form (Fig. 1 may serve as an example) F = f(t); F = f(z); F = f(z, t); v = f(t); v = f(z); v = f(z, t); v -- f ( F ) ; --high quality of laws of action being modelled; --the testing device unaffected by unstable parameters of the ground to be deformed. Analysis of different types of actuators showed that an electric-hydraulic tracking actuator meets the prescribed requirements most advantageously. The principal circuit of such an actuator is given in Fig. 2 which depicts the plate (pointer 1) that deforms the ground (pointer 2); pointer 3 stands for the working hydraulic cylinder (HC); 4 is a servo-valve controlling the displacement of the HC piston; 5 is the hydro-accumulative plant which supplies large amounts of hydraulic fluid through the servovalve 4; 6 is the hydrostatic bearing; 7 is the hydraulic brake; 8 is the adaptive system for controlling the hydraulic actuator of the device; 9 is the force meter. In developing and constructing the testing unit (Fig. 2) for the modelling of dynamic actions on the ground, a series of scientific and designing tasks have been solved, making it possible to produce the required technical characteristics set out earlier. One of the main problems was limited functional capabilities of conventional hydraulic cylinders regarding the velocity of the piston displacement. At the velocity of the piston displacement exceeding 3 m/s and at the pressure of liquid exceeding 15 MPa in the contact zone of the compactions of the HC (made of rubber or plastic), the temperature rises over 300 °C. Intensive wear of the compacting element and irreversible relaxation processes of the compaction materials take place [3]. Also, in contact compactions of the HC there occur forces of dry friction which are substantial in quantity. These exert considerable influence on the form of the law of loading followed by the piston. That is why hydrostatic bearings (pointer 6) for the piston and the rod, ensuring guaranteed radial clearance with the walls of the cylinder and excluding dry friction, were used for the working hydraulic cylinder 3 (Fig. 2). Parameters of the hydrostatic bearings were determined by calculating the liquid
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PROCESS OF VEHICLE-SOIL INTERACTION
193
compressibility, and of the variations in viscosity and modulus of volume elasticity of liquid depending on pressure and temperature. The profile of the throttling ring slot of the brake, that allows smooth braking and shock-free stopping of the hydraulic piston of the HC moving at a velocity of 15 m/s, has been determined from the results of the mathematical modelling of the braking process. Combining the hydraulic cylinder constructed on hydraulic supports with the home-made servovalve of "C" series (1250 1/min consumption at a pressure differential of p - - 7 MPa) allowed the principal technical characteristics expected of the testing unit to be obtained. The characteristics are: force on the pressing plate, deformation velocity, frequency of loading and an involved form of the law of action prescribed with the IBM PC. But parameters of the hydraulic actuator will be unstable due to hydraulic, electric, technological, temperature and design factors as well as due to other factors, which in turn results in the variation in the output characteristics of the testing set-up. Considering properties of grounds or soils, a response of the ground being tested to the actuator of the test unit also is of random nature and depends on many interrelated factors [4]. Therefore in modelling the vehicle action the form of the output law of load will be distorted owing to unstable parameters of the hydraulic actuator and to a response of the ground, which may result in a substantial difference from the given one. To account for the error in following the law of action is difficult because of the fact that the ground deformed changes its properties in the process of deformation while deformations cannot be taken into account in advance. Recently there have been systems used that permit enhancing the quality of controlling and exclude the influence of unstable parameters of the system and of external perturbing actions on its output characteristics at the expense of optimizing the controlling laws. To improve dynamic characteristics of the hydraulic actuator of the test-unit, a standard model of the search-free, gradient, self-adjustable system (SAS) of controlling has been chosen from a variety of adaptive control systems, proceeding from the laws of loading being modelled. The structural circuit and its adjustable parameters were obtained on the basis of the theory of automatic control, optimization and sensitivity theory [5]. The algorithm of the action of the self-adjusting control system is based on the variation in parameters being adjusted in the electronic circuits, the criterion being the quality of work of the system as a whole. Proceeding from the notion of gradient, the velocity of variation in the adjustable parameter may be written down in the form ~i = - - / ~ i
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where cr is the parameter of the system being adjustable; ~ is the coefficient of the velocity of self-adjustment; Qc is the criterion of the quality of work of the system's operation; e is the error between the standard signal and the output of the hydraulic actuator; i = 1, . . . , n. Constructing the SAS structure and selecting the parameters adjusted is beyond the scope of the present paper [5]; therefore we confined ourselves to considering the results of the theoretical and experimental studies of the test unit. Comparison of the results of calculations and of experimental investigations is given in Fig. 3 in the form of plots for the transition processes. Curve 1 corresponds to the transition process in the hydraulic actuator with the SAS of control; curve 3 shows the
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transition process in the standard model. Experimental transition processes are represented by solid lines (Fig. 3) while dashed lines show computed curves of the transition processes obtained from a nonlinear mathematical model of the hydraulic actuator.
CONCLUSIONS
The principal regularities and parameters of the vehicle's action on the ground have been determined for an example of tracked vehicle--deformable ground interaction. Two ways of modelling the vehicle's action have been proposed, with the investigations into deformable ground being made by means of pressure plates. The principal technical requirements for the testing device for modelling of the vehicle action in investigating deformable grounds have been formulated. The testing device, including the adaptive hydraulic actuator that enables modelling real laws of a vehicle's action on deformable ground, has been developed and constructed.
REFERENCES [1] K. L. JOHNSON, Contact Mechanics (Mechanica contactnogo vzaimodeistviya). "World", Moscow (1989). [2] A. AKAI, S. SHIMOI and T. KxucHi, Model studies on the stress distribution in layered soil systems. Proc. J S C E , 185, 83-94 (1971). [3] G. A. GOt,UBEV, G. M. KUKIN, G. E. LAZAREV and A. V. CHICHINADZE, Contact sealing of rotating shafts (Kontactnye uplotnenija vrashaushihsja valor). State Publishing House of Machinery (1976). [4] N. F. KOSHARNY1, Technical and operational properties of cross-country automobiles (Technicoekspluatacionnye svoistva automobileyi vysokoyi prohodimosti), p. 208. Vysshaya Shkola, Kiev (1981). [5] D. N. PoFov and S. L. SX~IKOV, Selecting of design and parameters of self-controlling hydraulic actuator of the testing unit. Vestnik Mashinostroenija (1) (1990).