The kind of base elements equations always depends on the assumptions accepted at the solution of the specific problems. As in the given case methods of
The resulted below library of equations of typical elements basically can suppose their various mathematical description at condition of preservation of the concept of a three-node element.
Characteristics of elements of mechanical and hydro mechanical drives resulted on Fig. 3, are approximated by a final set of points
, where
Fig. 3. Characteristics of base elements of mechanical and hydro mechanical transfers:
) [1, 2]:
k
(9)
where
– the diesel engine characteristic at the minimal fuel feed in view of a brake branch, approximated by a final set of points
– an increment of the torque moment at the maximal fuel feed;
+
– the diesel engine external characteristic at maximal fuel consumption in view of a brake branch (Fig. 3
At simulation of transient processes often it is necessary to pass in area of partial (regulator) characteristics of a diesel engine that in real conditions is provided with change of
(10) where a function of regulation of force of preliminary compression of a spring,
,
j
, takes place, i.e.
k
(11) where transfer ratios of a gear branches .
Let efficiencies of a gear in branches
; then a friction total losses in a gear, reduced to node
(12)
where
absolute values of nominal moments transferred accordingly in branches
(in nodes
).
k
(13)
where
(14)
The moment of an opposite sign operates in node
where
a constructive constant of frictional clutch;
pressure in the mechanism of pressing of frictional pair in function of time;
a coefficient of friction in function of the module of relative angular speed (Fig. 3
The moment of an opposite sign operates in node
(16) and besides
Here
c
);
angular speeds of pump (node
) and turbine (node
i
) wheels of hydraulic torque converter.
j
Coefficient of transformation by definition is equal to: (17) Then (18)
If hydraulic torque converter is executed structurally with overtaking clutch, then in blocking mode of pump (node
) wheels (in detail blocking mode see in the section
«Blocking of frictional clutches and hydraulic torque converters»
) we’ll receive:
j
(19) where the turbine wheel moment, defined from blocking equations (see here ); the reactor blades loss moment :
If pump and turbine wheels of hydraulic torque converter are blocked by a friction clutch, then in blocking mode pump (node
) wheels (in detail blocking mode see in the section
«Blocking of frictional clutches and hydraulic torque converters»
) we’ll receive:
j
where
, (21) and at equality of angular speeds (22)
(23)
where
d
).
As at , then a blocking mode in hydro dynamical clutch is not present, as at equality of angular speeds the moment developed by hydro dynamical clutch, is equal to zero.
,
j
designate accordingly nodes of an input
k
(a power shaft of a wheel), an output
i
(a point of contact of a wheel with road) and machine movement
j
.
k
Fig. 4. To a conclusion of equations of a wheel dynamics.
In view of the accepted assumptions mathematical model of a wheel (wheel mover) dynamics, Fig. 4, looks like [1, 2]: (24)
where
n
-th wheel (on wheels of the
n
-th axis);
W
– a summer force of resistance to machine moving;
speed and moving of the machine;
reduced moment of inertia of rotating masses of the
n
-th axis;
М
_{
n
}
– an active moment of the
n
-th axis;
dynamical radius of the
n
-th wheel (wheels of the
n
-th axis);
braking moment on the
n
-th axis shaft, enclosed in input node;
N
– number of driving wheels (axes).
(25) where (26)
Here
The value of dynamic radius of wheel
(27) where – a free radius of wheel; a part of the machine weight falling an axis; a radial rigidity of tire.
(28)
(29) i.e. in the settled modea function of slipping is equal to relative slipping of a wheel [compare the equations (28) – (29) with the equations (25) – (26)]. Thus, the mathematical model of a wheel (wheel mover) consists of the equations (24) and (28).
of differential axes are either initial (at a branching of a power stream), or final (at summation of a power streams) – Fig. 5.
j
Fig. 5. The kinematic scheme of differential.
Angular speeds of differential axes are connected with an entrance shaft speed by following kinematic dependence: (30) whence (31) where a transfer number of differential, a transfer ratio of a gear between an entrance shaft and driver. For symmetric differential , and then (32)
Torque moments in nodes
,
j
(Fig. 5) are connected by relations:
k
(33)
Then, knowing the moment
, it is easy to define moments in nodes of differential axes
. On the other hand, integrating equations of dynamics of differential axes (2), we’ll receive
, whence, using the equation (31), we’ll define
. Hence, it is necessary to define value
for what it is expediently
an entrance shaft of differential
always
.
as elastic
It means, that the node
and
i
are either initial (if in the scheme there are elements which first nodes coincide with nodes
j
and
i
of differential), or final (if in the scheme there are elements which second nodes coincide with nodes
j
and
i
of differential), and the node
j
is not included into one of sites and
k
. It does not limit a generality of the problem solution, but allows to receive rather easily values interesting us.
necessarily is a node of an elastic shaft
Let's write down the equations of dynamics of differential axes in view of its geometry, kinematics and operating forces and moments (Fig. 5).
Input preconditions and the basic idea of a conclusion of these equations belong to Dr. L.B.Zaretsky. We’ll enter a number of additional designations:
tangential forces of interaction in gearings an entrance gear and differential;
radiuses of gearings of cogwheels of the gear and differential;
own moments of inertia of cogwheels of the gear and differential;
the moments of loading reduced to nodes
of differential axes. Then dynamics of differential can be described by the following system of equations:
j
(34) Considering, that , and also including small enough , after of some simple algebraic transformations, having excluded we’ll finally receive: (35) Thus, the mathematical model of differential consists of the equations (31), (33) and (35). |

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