In resulted below equations of statics of hydraulic elements the same designations are accepted, and for the description of the input data of elements the same identifiers and tables are used, as at the dynamic analysis.
) and output (node
i
) in view of volumetric losses. Thus non-uniformity of the pump flow owing to kinematic features and compressibility of liquid in sucking and pressure cavities is not considered. In view of the accepted assumptions the mathematical model of the pump looks like [1, 2]:
j
, (2)
where
f (q)
–
parameter of pump geometric
volume regulation;
– 1
≤
f (q)
≤
1;
ω
– angular speed of engine (diesel engine) shaft;
_{
s
}
а
,
_{
ω
}
а
,
_{
р
}
а
– coefficients of pump hydro mechanical losses depending on angular speed, pressure, and constant value of hydro mechanical losses;
u
– transfer number of gear between engine and pump;
_{
e
}
k
– coefficient of pump volumetric losses; for
_{
lea
}
Q
,
_{
i
}
p
the sign "plus" is accepted, for
_{
i
}
Q
,
_{
j
}
p
– "minus".
_{
j
}
Values of
а
,
_{
р
}
а
,
k
are chosen from the catalogue or from passport characteristics of mechanical and volumetric efficiency of the certain standard size pump.
_{
lea
}
The hydro mechanical losses depending on pressure are calculated modulo for opportunity of consideration of brake modes and reversing flow (when
) and output (node
i
) in view of volumetric losses. Without taking into account non-uniformity of flow (it is similar to pump) the system of equations of hydraulic motor looks like [1, 2]:
j
(3)
where
q
_{
м
}
– maximal geometric volume of hydraulic motor ;
_{
}
f (q)
– parameter of hydraulic motor geometric
volume regulation; – 1
≤ f (q)
≤
1;
М
– loading moment;
_{
l
}
b
,
_{
ω
}
b
,
_{
р
}
b
– coefficients of hydraulic motor hydro mechanical losses depending on angular speed, pressure, and constant value of hydro mechanical losses;
u
– transfer number of the working mechanism gear;
_{
mech
}
k
– coefficient of volumetric losses of hydraulic motor; for
_{
lea
}
Q
,
_{
i
}
p
the sign "plus" is accepted, for
_{
i
}
Q
,
_{
j
}
p
– "minus". As well as for pump, values
_{
j
}
b
,
_{
ω
}
b
,
_{
р
}
b
,
k
are chosen from the catalogue or from passport characteristics of mechanical and volumetric efficiency of the certain standard size hydraulic motor. Hydro mechanical losses in the equation of moments are written down in view of direction of shaft rotation (sign
_{
lea
}
ω
) and opportunities of consideration of a brake mode |
_{
k
}
p
–
_{
i
}
p
|.
_{
j
}
) and output (node
i
). On a basis of standard assumption about absence of leakages in hydraulic cylinder with rubber and other soft seals equations of statics of hydraulic cylinder look like [1, 2]:
j
(4)
where
_{
}
– speed of piston motion;
F
=
π
(
_{
i
}
D
–
D
)/4 – piston working area in cavity
I
adjoining node
(here
i
D
– cylinder diameter of;
_{
c
}
D
– rod diameter in cavity
_{
i
}
I
);
F
=
π
(
_{
j
}
D
–
D
)/4 – piston working area in cavity
II
adjoining node
(here
j
D
_{
j
}
_{
}
– rod diameter in cavity
II
);
h
– coefficient of viscous friction;
R
– force of friction in seals at absence of pressure;
R
– force to rod. Coefficients of proportionality between pressures in cavities
_{
c
}
I
(node
) and
i
II
(node
) and force of friction in seals
j
+ D
)
_{
i
}
H
/ 2,
k
=
π
f
(
D
_{
c
}
+ D
)
_{
j
}
H
/ 2. (5)
Here
) and equations of losses of pressure on length and looks like [1, 2]:
j
(6) where – coefficient of pressure losses on length of pipeline, (7)
here Re = 4 |
d
) – Reynolds’ number,
– kinematic viscosity of liquid;
– density of working liquid;
d
and
_{
p
}
L
– diameter and length of pipeline.
_{
p
}
(8)
) by the known dependence [1, 2]:
j
(9) where – flow coefficient, = (here – coefficient of hydraulic resistance); – area of through passage section of throttle.
,
j
at division of
k
flow look like [1, 2]:
(10)
where
– flow coefficients in tee branches
,
j
–
i
;
=
(here
– coefficients of hydraulic resistances of tee branches
k
–
i
,
j
–
i
);
– areas of through passage sections of tee in nodes
k
and
j
.
k
Equations of tee flows at summation of flows are similar (10), but have other values of flow coefficients. The assumption is accepted, that coefficients of hydraulic resistances at change of flow direction do not vary.
and
i
[1, 2]:
j
(11)
where
F
– working areas of the valve
locking-regulating element
from pressure head and drain line;
_{
j
}
с
– rigidity of spring;
– value of spring preliminary compression;
– stroke of
locking-regulating element
;
– area of through passage section of throttle connected in parallel to the valve;
– average diameter of throttling crack of the valve;
– angle of the valve cone.
The resulted equations concern to pressure relief and check valves. The corresponding equations for reducing valve have insignificant differences. In equations (11) hydro dynamical force is not considered.
,
s
and the pilot valve with nodes
t
,
i
,
j
. If a discharge node
k
is general for both valves, then
j
. Mathematical model of statics of pilot controlled valve looks like [2]:
s = j
(12)
where
– working areas of auxiliary valve
locking-regulating element
from pressure head and drain line;
– working areas of basic valve
locking-regulating element
from pressure head line and cavity between valves;
(13)
) and equation of polytropic process in gas cavity (node
i
) [1, 2]:
j
(14)
where
q
)
where
q
is practically constant, so it is possible to consider that power regulator provides constant taking of power away engine. The static characteristic of power regulator (Fig. 2) looks like piecewise linear function approximating hyperbolic dependence of the pump geometric volume versus pressure:
_{
p
}
p f
(
q
) = const. In practice it is carried out by means of choosing of springs for the 1-st and the 2-nd branches of power regulator characteristic (accordingly,
AO
and
ОD
, Fig. 2).
Fig. 2. Static characteristic of power regulator
Then in statics power regulator is described by the following equation [1, 2]: (15)
where
) [1, 2]:
k
(16)
where
– diesel engine characteristic at minimal fuel consumption, approximated by finite set of points
– increment of torque moment at maximal fuel consumption;
– constant parameters of diesel engine regulator adjustment;
– loading moment of pump reduced to diesel engine shaft;
– transfer ratio of regulator drive;
,
j
designate accordingly nodes in input (power shaft of wheel), output (point of contact of wheel with road) and machine movement. The model of wheel mover considered here describes rigid communication of wheel with the hydraulic motor. In view of the accepted assumptions the static mathematical model of wheel (wheel mover) looks like:
k
(17)
where
– wheel moment in view of losses in gear;
М
_{
n
}
– moment reduced to shaft of hydraulic motor;
– wheel traction reaction (circular force);
r
– dynamic radius of wheel;
– efficiency and transfer number of wheel gear;
angular speeds of hydraulic motor shaft and wheel;
W
– total force of resistance to machine moving;
N
– number of driven wheels (axes).
(18) where (19)
Fig. 3. Function of wheel slipping.
Here
r
depends on static deflection of wheel
under loading:
(20) where – free radius of wheel; component of machine weight, falling axis; radial rigidity of tire. |

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