|
Planning of the hydraulic cylinder piston braking law Mathematical model of the considered system looks like:
where
m
– mass of mobile parts of hydraulic cylinder reduced to a rod;
Е – the reduced volumetric module of elasticity of a working liquid in an elastic environment:
The executed from model (1) dynamic calculations of transient processes arising at braking in hydraulic cylinders, in view of their geometry, simplified scheme of loading (impeding or passing), various forms of throttle cracks, reduced masses, etc., had testing character, have shown, that the choice of this or that dependence
For the solution of the problem we’ll accept some assumptions: 1) prior to beginning of the piston braking its speed, pressure in cavities of the hydraulic cylinder and a charge of working liquid are constant; 2) considering, that the piston braking occurs on rather small movement, it is possible to consider the reduced mass and force on a rod of the hydraulic cylinder as constants;
3) it is expedient to give such law of the piston movement at braking, at which speed
Proceeding from it, we’ll plan the law of the piston movement
Here
Let's impose restriction on the maximal pressure
whence in view of (1) follows:
where
Здесь
From (4) follows:
Let's note, that giving as much as possible admissible pressure
Having excluded
At the given law of movement of the piston values
Having six boundary conditions (2), we’ll plan the piston movement law by means of a polynomial of the fifth degree:
Whence, differentiating, we’ll receive
Using boundary conditions (2), we come to system of the linear algebraic equations relatively unknown coefficients
As an initial position of the piston
which in view of (9) and (10) gives:
Then
whence follows, that at
Then from (6) follows:
whence it is final
Hence, having given the braking time
Т
from a condition (16) and using (12), we’ll receive
|
Contents
>> Engineering Mathematics
>> Hydraulic Systems
>> Dynamic Synthesis of Hydraulic Devices
>> Planning of the piston braking law
– areas of pressure head and drain cavities of the hydraulic cylinder;
– force of dry friction in seals, equal to
– force of dry friction at absence of pressure;
– coefficients of proportionality;
R
– external force on a rod;
– pressures in pressure head and drain cavities of hydraulic cylinder;
– a piston movement and speed;
– a piston initial position and a full stroke;
– coefficients of elasticity of a liquid in cavities of the hydraulic cylinder, equal:
– diameter and thickness of a wall of the cylinder;
– an elasticity module of the cylinder wall material;
– "dead" volumes of cavities of the cylinder;
a working liquid flow in a pressure head cavity of the cylinder;
a throttle flow coefficient;
density of a working liquid;
pressure behind a throttle;
through passage section area of throttle as function of piston movement.
strictly decreasing function that allows to accept
during braking
, having accepted following boundary conditions:
,
– a final piston speed;
:
– pressure of adjustment of a pressure relief valve of hydraulic system. Let’s consider, that the charge on an input in the hydraulic cylinder changes in conformity with the static characteristic of the valve:
, it is necessary to provide performance of a condition
as differently the piston braking is impossible.
from the equations (1), we’ll receive:
and
, entering into the formula (7), can be defined either by numerical integration of the third equation of system (1), or analytically as the equation relatively
is practically
is contained, and the charge
is described by
of the polynomial (9), solving which we’ll receive:
is in advance not known, then for its definition, not raising a degree of a polynomial (9), we’ll impose an additional boundary condition:
we have
i.e. the demanded monotony of
is provided and
is reached at
and is equal:
.