The kind of the equations of base hydraulic elements depends, generally, on the assumptions accepted at decision of specific problems. As in this case we consider methods of
The library of equations of hydraulic elements reduced below basically can suppose their various mathematical descriptions under condition of preservation of the concept of a three-node element.
) and output (node
i
) in view of volumetric losses. Thus non-uniformity of the pump flow owing to kinematic features and compressibility of working fluid in sucking and pressure head cavities is not considered. In view of the accepted assumptions the pump mathematical model looks like [1, 2]:
j
, (3)
где
f (q)
–parameter of regulation; – 1
≤ f (q)
≤
1;
а
,
_{
ω
}
а
,
_{
р
}
а
– coefficients of pump hydro mechanical losses depending on angular speed, pressure, and constant of hydro mechanical losses;
u
– transfer number of gear between an engine and a pump;
_{
e
}
k
– coefficient of pump volumetric losses (leakages); for
_{
lea
}
Q
,
_{
i
}
p
the sign "plus" is accepted, for
_{
i
}
Q
,
_{
j
}
p
– the sign "minus". Values
_{
j
}
а
,
_{
ω
}
а
,
_{
р
}
а
,
k
get out under the catalogue or from passport characteristics of mechanical and volumetric efficiency of the certain standard size pump. The hydro mechanical losses depending on pressure, are calculated on the module for an opportunity of consideration of brake modes and flow reversing (when
_{
lea
}
f (q)<
0).
) and output (node
i
) in view of volumetric losses. Without taking into account non-uniformity of flow (it is similar to pump) the equations of the hydraulic motor look like [1, 2]:
j
(4)
where
J
– moment of inertia of the hydraulic motor in view of rotating masses of working mechanism;
_{
m
}
q
– the hydraulic motor maximal geometric volume;
_{
m
}
f (q)
– parameter of 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 of hydro mechanical losses;
u
– transfer number of the working mechanism gear;
_{
mech
}
k
– coefficient of hydraulic motor volumetric losses (leakages); for
_{
lea
}
Q
,
_{
i
}
p
the sign "plus" is accepted, for
_{
i
}
Q
,
_{
j
}
p
– the sign "minus". As well as for a pump, values
_{
j
}
b
,
_{
ω
}
b
,
_{
р
}
b
,
k
choose under 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 the moments are written down in view of a shaft rotation direction (sign
_{
lea
}
ω
)
and opportunities of consideration of a brake mode |
_{
k
}
p
–
_{
i
}
p
|.
_{
j
}
) and output (node
i
) in view of compressibility of fluid in cavities of the cylinder. On a basis of the standard assumption about absence of leakages in hydraulic cylinder with rubber and other soft seals the equations of the hydraulic cylinder dynamics look like [1, 2]:
j
(5)
where
_{
}
– speed of the piston moving;
т
– mass of the hydraulic cylinder mobile parts reduced to a rod;
– the piston working area in cavity
I
adjoining node
(here
i
_{
}
D
– diameter of cylinder;
_{
c
}
D
– diameter of rod in cavity
_{
i
}
I
);
– the piston working area in cavity
I
adjoining node
(here
j
_{
}
D
– diameter of rod in cavity
_{
j
}
II
);
h
– coefficient of viscous friction;
R
– force of friction in seals at absence of pressure;
R
– force on the rod;
– the full piston stroke.
Coefficients of proportionality between pressures in cavities
II
(node
) and force of friction in cylinder seals:
j
+ D
)
_{
i
}
H
/ 2,
k
=
π
f
(
D
_{
c
}
+ D
)
_{
j
}
H
/ 2,
and coefficients of elasticity of cavities with working fluid:
E
,
k
= [
Δ
V
+ (
L
–
z
)
F
] /
_{
j
}
E
,
where
here – volumetric module of elasticity of working fluid; – thickness of the cylinder wall; – an elasticity module of the cylinder wall material .
Hereinafter a function of dry friction
R
sign
v
. Actually, if to write down the equation of movement in a general view:
_{
k
}
where
Such model of friction describes presence of stagnation zone at zero speed of a mobile part, for example, at starting.
) of pipeline, taking place at the following conditions is used:
j
- wave processes are not considered; - losses of pressure on length depend on average value of flows in input and output; - inertial component of a working fluid is not considered. Then the mathematical model of pipeline with a fluid looks like [1, 2]: (7)
где
= ,
here
где
δ
– thickness of the pipeline wall;
Е
–
elasticity module of the pipeline material
.
here Re = 2 |
Q
| / (
_{
j
}
d
) – Reynolds's number,
– kinematic viscosity of a fluid.
(8)
where
,
here
where
– thickness of deadlock pipeline wall;
Е
–
elasticity module of the pipeline material
.
) by known dependence [1, 2]:
j
(9) where – flow coefficient, = (here – coefficient of hydraulic resistance; – throttle through passage section area. Use of flows equation (9) often is as the reason of instability of computing process because of aspiration to infinity of a derivative of a square root in zero (it takes place at small differences of pressures). The equation (9) defines the flow through a throttle in established mode of current of a fluid and, hence, does not consider inertial properties of fluid. More precisely dependence of flow through a throttle is expressed by the differential equation [3]: (10)
where
However the equation (10) is of little use for practical use. The matter is that the length
(11)
which is deprived the specified lacks and asymptotic solution of which coincides with the solution of equation (9). Here
The equations of flows for other kinds of local resistance (tees, valves, pilot operated check valves, directional control valves) are similar to the equations (11).
,
j
at division of flow look like [1, 2]:
k
(12)
where
– flow coefficient in tee branches
,
j
–
i
;
=
(here
– coefficients of hydraulic resistances of tee branches
k
–
i
,
j
–
i
);
– through passage section areas in tee nodes
k
and
j
. The equations at summation of flows are similar (12), but have other values of flow coefficients.
The assumption, that coefficients of hydraulic resistances at change of direction of flow do not vary, is accepted.
k
and
i
) [1, 2]:
j
(13)
where
m
– mass of valve moving part;
F
and
_{
i
}
F
– working areas of
locking-regulating element
from pressure head and drain lines;
_{
j
}
h
– coefficient of viscous friction;
R
– force of dry friction;
_{
fr
}
с
–rigidity of spring;
– preliminary compression of spring;
– 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;
В
– parameter considering inertia of a fluid column.
These equations concern to pressure relief and check valves. The corresponding equations for reducing valve have insignificant differences. In the equations (13) a hydro dynamical force which essential influences only on static characteristic of valve [2] is not considered.
,
s
and a pilot valve with nodes
t
,
i
,
j
. If node
k
is general for both valves, i.e.
j
then mathematical model of the indirect action valve looks like [1, 2]:
s = j
(14)
where
z
– speed and displacement of
locking-regulating element
of pilot valve;
_{
k
}
v
,
_{
t
}
z
– speed and displacement of
locking-regulating element
of main valve;
_{
t
}
т
and
М
– masses of moving parts of pilot and main valves;
– working areas of
locking-regulating element
of pilot valve from pressure head and drain lines;
– working areas of
locking-regulating element
of main valve from pressure head line and cavity between the valves;
h
and
H
– coefficients of viscous friction of pilot and main valves;
– forces of friction in pilot and main valves;
с
and
С
– rigidity of springs of pilot and main valves;
preliminary compression of springs of pilot and main valves;
stroke of mobile parts of pilot and main valves;
G
– conductivity of orifice aperture of main valve;
average diameters of throttling cracks of pilot and main valves;
angles of cone of pilot and main valves;
) and equation of polytropic process in a gas cavity (node
i
) [1, 2]:
j
(15)
where
here
Δ
V
– «dead» volume of cavity with fluid;
here
p
∙
f (q)
= const. In practice it is carried out by means of selection of springs for the 1-st and the 2-nd branches of power regulator characteristic (accordingly,
AO
and
О
D, Fig. 2).
Fig. 2. Static characteristic of power regulator
Then power regulator is described by the following system of equations [1, 2]: (16)
where
h
– coefficient of viscous friction;
– maximal stroke of power regulator plunger.
,
j
designate nodes accordingly of input, output and moving of valve, and by indexes
k
,
r
,
s
– nodes accordingly of input, output and moving of pusher.
Pilot operated check valves are issued in two modifications: with general drain of pusher and valve (in this case
t
=
r
) and with separate drain (then
i
r
^{
}
i
).
Fig. 3. Pilot operated check valve and its connection in hydraulic system.
Depending on, whether there are pusher and valve in contact or not, dynamics of pilot operated check valve is described by two various mathematical models. In the time moment of contact occurrence when pusher starts to influence the valve, there is their impact that leads to necessity of correction of coordinates according to existing dependences of theory of shock systems. At absence of contact of pusher and valve dynamics of pilot operated check valve is described by the following system of equations [1, 2, 4]:
(17)
where
) and output (
r
,
j
);
– forces of friction of valve and pusher;
s
h
– coefficient of viscous friction;
с
,
z
_{
0
}
– rigidity and preliminary compression of spring;
l
and
L
– maximal stroke of valve and pusher;
– flow coefficient, diameter of crack and angle of valve cone.
At approach of pusher and valve contact, when , (18) and at absence of impact both bodies move in common, therefore their movement is described by the system of equations: (19)
where
The given model is incomplete as, considering great speeds and geometry of mobile parts which contact can be considered as the central impact of elastic cores, would be incorrect to consider, that at performance of conditions (18) the system instantly passes from condition described by the equations (17) in condition, described by the equations (19). It speaks about necessity of introduction of transitive shock mode model.
It is possible to apply formulas of classical hypothesis of impact to considered shock system "pusher-valve". Let before impact body in mass of
(20)
where – coefficient of restoration.
Preliminary computations of the accepted shock model at
. Flow through each such local resistance is expressed by equation similar to the equation (11):
s
(21)
where
– area of through passage section of directional control valve channel connecting nodes
in function of moving of spool
s
z
, which maximal value it is equal to
(here
– conditional pass diameter).
In intermediate position of spool channels can be crossed in one node in which, hence, there will be a summation or division of flows of working fluid. Therefore flows in nodes belonging simultaneously various channels of directional control valve, turn out summation of flows through corresponding channels converging in given node. Hence, (22)
where
Change of through passage sections of channels of directional control valve can be approximated, for example, by trapezoidal characteristic, unequivocally defined by four positions of spool: (Fig. 4):
Fig. 4. Dependence of through passage section area of channel on spool displacement.
(23)
) [1]:
k
(24)
where
reduced to diesel engine shaft moment of inertia of rotating details (here
– moment of inertia of diesel engine;
– moment of inertia of pump;
– transfer number of diesel engine gear);
– diesel engine characteristic approximated by finite set of points
– increment of torque moment at maximal fuel feed;
– constant parameters of diesel engine regulator adjustment;
– loading moment of pump reduced to diesel engine shaft;
– coefficient of viscous friction in diesel engine regulator;
– transfer ratio of regulator drive;
,
j
designate on the scheme accordingly nodes of input (power shaft of wheel), output (point of contact of wheel with road) and moving of machine. The considered here model of wheel mover describes rigid communication of wheel with the hydraulic motor, i.e. possible elastic deformations of gear and shaft between hydraulic motor and wheel are not considered.
k
Fig. 5. To conclusion of equations of dynamics of wheel.
In view of the accepted assumptions mathematical model of dynamics of a wheel (wheel mover), Fig. 5
(25)
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;
tangential rigidity of tire;
slipping function (Fig. 5
b
);
mass, speed, displacement and total force of resistance to machine moving;
N
– number of driven wheels (axes).
(26) where (27)
Here
Value of dynamic radius of wheel
(28) where – free radius of wheel; component of machine weight, falling axis; radial rigidity of tire.
(29) In established mode and then (30) i.e. in established mode slipping function is equal to relative slipping of wheel [compare equations (29) – (30) to equations (26) – (27)]. Thus, the mathematical model of wheel (wheel mover) consists of equations (25) and (30).
1) an ideal intensifying (no inertial) link – an adder; 2) the 1-st order aperiodic (inertial) link; 3) the 2-nd order aperiodic link; 4) an oscillatory link; 4a) a conservative link (a special case of an oscillatory link); 5) an ideal integrating link; 6) an inertial integrating link; 7) an ideal differentiating link; 8) an ideal link with introduction of derivative; 9) an inertial differentiating link; 10) the 2-nd order dynamic link (the general case). Mathematical models of the listed linear dynamic links are written in the form of ordinary differential equations, instead of in operational form (in the form of transfer functions) as transient processes are interested for us in time area, instead of in frequency area.
Generally
(31)
where
signals on input of the link;
their time derivatives;
For all other types of dynamic links their equations are received as special cases (31):
(31.1)
(31.2)
(31.3)
(31.4)
(31.5)
(31.6)
(31.7)
(31.8)
(31.9)
Thus, all typical linear links can be incorporated in one generalized element LINK (the identifier of this element in library of base elements) with nodes
(output),
j
(an additional input for the link – adder).
Considering specificity of hydraulic systems, in the right parts of equations of dynamic links as entrance signals pressures can be added:
k
(32)
As in the equations superfluous pressure is considered, performance of the conditions is necessary: (33) (34) where – atmospheric pressure; – flows in input and output of considered cavity; – coefficient of elasticity of cavity with fluid.
In some real elements movement
(35) (36) (37)
where
Let's notice, that the equations for definition of pressure are included into the description of those elements which contain significant in comparison with other elements volumes of working fluid (for example, hydraulic cylinders, pipelines, including deadlock, hydraulic accumulators). Therefore these elements in simplified diagram should be divided by other hydraulic devices in which compressibility of fluid can be neglected and which equations serve for definition of flows (for example, hydraulic cylinder cavity and pipeline, hydraulic accumulator and pipeline, or two consistently connected pipelines, should be divided by throttle or local resistance that does not contradict physical sense). It allows to receive the closed system of equations for definition of pressure and flows in junctions of hydraulic elements. |

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