Control Systems - Dynamic Synthesis - Method of multiple periods of natural vibrations - Algorithm
Let's consider a simplified rated diagram of progressive hydraulic drive with the closed circulation of a stream presented on Fig. 3.
Let’s write system of differential equations of dynamics in view of elasticity of working liquid, accepting the following assumptions:
1). Leakages of working liquid in hydraulic system are absent, therefore work of boosting system is not considered.
2). Losses of pressure on length of pipelines and in local resistances negligible.
3). Reduced mass and load on the hydraulic cylinder rod, as well as frequency of rotation of the pump shaft are constant.
4). Parameters of working liquid (density, viscosity, module of volumetric elasticity) are constant.
5). Dynamics of the pump flow control mechanism is not considered.
The account of dynamics of the pump flow control mechanism does not represent basic difficulties, as this problem is maybe easily solved after
(in parallel) considered, when required dependence of working volume of the pump q(t) (or even its current numerical value) is
already known. For this purpose it is necessary to record mathematical model of the control system which output is q(t).
The mathematical model of considered hydraulic system (Fig. 3) in view of the accepted assumptions looks like:
where m – reduced to a rod mass of mobile parts of the executive mechanism; р1, р2 – pressures in
pressure head and drain cavities of the hydraulic system; x, v – the hydraulic cylinder piston movement and speed;
R0fr – a constant force of dry friction in sealing of the hydraulic cylinder at absence of pressure in
cavities; k1, k2 – factors of proportionality in variable components of force of dry friction, depending
on pressure; R – a constant load on the hydraulic cylinder rod; ωp – an angular speed of the pump shaft;
F – the working area of the hydraulic cylinder piston; q(t) – the pump working volume in function of time (actually a
control signal); kelast1, kelast2 – factors of elasticity of the hydraulic system
where L – the maximal stroke of the hydraulic cylinder piston; dр i, lр i – diameter and length
of i-th pipeline; ΔV – a "dead" volume of the hydraulic cylinder cavity; Еred i – the reduced
module of volumetric elasticity of i-th pipeline with a working liquid:
where Elq – module of volumetric elasticity of working liquid; δр i – thickness of i-th pipeline
wall; Est – the module of elasticity of the pipeline wall material; Еred – reduced module of elasticity
of the hydraulic cylinder cavity, defined according to the formula similar (7).
Let's put for simplicity v > 0, that is sign v = 1.
Differentiating on time the first of equations of the system (5), in view of the accepted assumptions we’ll receive:
Substituting in (8)
from last two equations of the system (5), we’ll receive:
Despite of dependence (6) of factors of elasticity kelast1 and kelast2 on х,
within the limits of the accepted assumptions and for actual values F, k1, k2 and other parameters,
it is easy to show (see the Appendix), that the expression standing
in the equation (9) as multiplier before v, changes slightly (~13 %), therefore as a first approximation it is possible to consider the
equation (9) as linear:
Having accepted the initial conditions:
Let's receive the decision of the equation (10) which in view of initial conditions (11) looks like :
Proceeding from conditions of accuracy of positioning, it is necessary to choose control φ(t) so that by the end of regulation
to extinguish natural fluctuations.
Let's consider a class of linear control φ(t) (generally – piecewise linear if to consider, that for each stage of movement the
time interval (0, t) is "sliding", that is its left border conforms t = 0; differently, initial conditions (11) for
2-nd and 3-rd stage of movement are replaced with values of phase variables in the end of the previous stage).
So, let q(t) = αt + β, then
Substituting (13) in (12), we’ll receive:
or it is final:
Substituting (17) in the equation (5) for р1 and integrating on t, we’ll receive:
where р10 = р1 (0).
Let's consider separately the basic phases of movement and control.
The initial conditions at dispersal:
v0 = 0, w0 = 0, β = 0. (19)
In this case we have:
Let's enter the restrictions:
First of these restrictions (21) is entered, proceeding from requirements of hydraulic system operation (pressure should not be below some minimum
level defined by conditions of suction of working liquid on the basic pump input) and failure of a pressure relief valve, otherwise a desirable
control process by a hydraulic drive will be deformed. The restrictions (22) reflect purely geometrical opportunities of the variable displacement
Let TI – time of control at dispersal, then TI = qp / α. At
t = TI we’ll receive:
Let's demand, that
If to accept
(to ensure that, at the end of dispersal ),
Then at t = TI we have:
Let’s find max p1 (t) at (27) on temporal segment 0 ≤ t ≤ TI :
– the period of function p1 (t), then its maximum on [0, TI] is equal:
From the condition (21) we’ll receive:
On the other hand, at absence of boost system pressure p2 in a drain pipeline changes similarly (18):
where p20 = p2 (0).
At dispersal, in view of the initial conditions (19), we’ll receive:
whence proceeding from the restriction (21) in view of (27), we’ll receive:
Thus, from (32) and (36) follows, that at dispersal:
Once again we emphasize, that validity of formulas (34) – (37) takes place only at absence of boost system. At its availability there is
no the need in these conditions.
For this stage of movement the new initial conditions are specified:
– values of appropriating phase coordinates in the end of the previous stage of movement (dispersal).
Substituting (38) in the equations (16), (17) and (18), accordingly we’ll receive:
whence apparently, that on the temporal segment
an uniform movement of the hydraulic drive takes place. The value TII (duration of the steady-state movement) will be defined
The initial conditions at the stage of braking are:
If to accept similarly (27) at dispersal, that at braking
that substituting (42) in the equations (16), (17) and (18), we’ll receive from the condition (21) similarly previous:
If to take as n and k their general meanings for all modes, then
Duration of braking in this case will make:
then movements of the piston at dispersal Δ х1 and braking Δ х3 are accordingly equal:
Then, if хТ* – the given final position of the piston,
whence in view of (48) we’ll receive:
If to designate the right part of an inequality (46) through ,
then in view of (50) the inequality should take place:
from which follows, that if it is not carried out (for example, at small movements )
it is necessary to enter κqp instead of qp where 0 ≤ κ ≤ 1, that is to reduce qp
until the inequality (51) will be carried out.
General time of movement Т is equal:
and the moment of the braking beginning is:
At t ≥ TIII we’ll receive: α = 0, β = 0, v0 = 0, w0 = 0,
that is the system "stops".
Let's note that from set of all admissible values n, that is satisfying the condition (46), it is necessary to choose the least one. It
will allow under other equal conditions time minimizing of regulation Т, as well as will allow possibility to provide performance of the
On it the stage of "rough" control is ended.
The described control algorithm is illustrated on Fig. 4, where v(t) and x(t) – idealized cycle charts of
speeds and movements of the executive hydraulic cylinder piston.