Control Systems  Dynamic Synthesis  Method of multiple periods of natural vibrations  Example
Example
The algorithm of control offered in the previous section we'll
consider on the specific rated diagram of a hydraulic drive of progress with the closed circulation of a stream (Fig. 5), modeling transients in
hydraulic system by means of the program of dynamic calculation of any diagrams of volumetric hydraulic drive HYDRA. (see the section
"Hydraulic Systems. Dynamic Analysis").
Fig. 5.
As all necessary parameters of control (Т_{I}, Т_{II}, Т_{III}, q_{p},
... ) are calculated in advance and involved in the program of formation of external indignations PERTR, check of offered algorithm of control
by numerical experiment (which objective to establish a degree of adequacy of the accepted key points) as in system HYDRA varied nonlinear
communications of elements and parameters of a hydraulic drive are considered in most general view, such as is in essence realized: variable
elasticity of cavities of the hydraulic cylinder, loss of pressure on length of pipelines and in local resistances, outflow of a working liquid
from cavities of the pump, work of system of additional charging on dynamics of return valves, etc.
Before to estimate a degree of influence of this or that factor on parameters of control (mainly, on the lowest natural frequency of hydraulic
system), we'll consider main parameters of hydraulic system presented in tab. 1.
The reduced modules of elasticity of pipelines 5 – 6, 10 – 11, 2 – 4, 12 – 1, 28 – 30, 24 – 26 and cavities of the hydraulic cylinder (all –
with a working liquid) are defined according to the formula (7)
previous sections "Algorithm"
where d_{i}, δ_{i} – diameter and thickness of a wall of ith cavity; Е_{lq} –
the module of volumetric elasticity of working liquid; Е_{st} – the module of elasticity of a cavity wall material.
The following values of the resulted modules of elasticity are as a result received:
whence according to the formula (6) previous sections "Algorithm"
we'll receive following values of factors of elasticity of pressure head and drain highways in the beginning of a course of the piston of the
hydraulic cylinder:
Factors of proportionality of making forces of friction in cavities of the hydraulic cylinder k_{1} and k_{2}
are equal [10, 11]:
then according to the equations (9) – (10) previous sections "Algorithm"
or
natural frequency of hydraulic system is equal:
Modeling hydraulic system (Fig. 5) by the program HYDRA gives value of frequency of natural fluctuations 6 rad/s. In
the Appendix it is shown, that taking into account variable elasticity of hydraulic system, losses of pressure on length of pipelines, in local
resistance and leakages practically do not change eigenvalues of hydraulic system. Thus, a difference between values of natural frequency (at
calculation "manually" a = 6.75 rad/s, at modeling by the program HYDRA a = 5.5...6 rad/s) can be explained by oscillation
property of boost system which at an assessment "manually" was not considered. Therefore we’ll accept for the further a = 6 rad/s. Then
according to conditions of (32) previous sections "Algorithm"
To check the condition (36) previous sections "Algorithm" for a
drain pipeline is not necessary here, as at any choice n fulfillment of the condition (35) is provided with boost system that proves to be
true results of modeling (see further). As during movement of the hydraulic cylinder piston k_{elast}_{1} increases, it
is meaningful to admit to the given case n = 1, instead of 2 and by that twice to reduce time of dispersal Т_{I} :
Check of an inequality (50) previous sections "Algorithm": gives:
Thus, it is final Т_{I} = 1.05 s.
As the considered hydraulic system (Fig. 5) is symmetric, we'll accept Т_{III} = Т_{I} = 1.05 s, then according
to expression (49) previous sections "Algorithm":
consequently, total time of regulation _{}
On Fig. 6 results of numerical experiment – modeling of transients in considered hydraulic system by means of the program HYDRA in using control
with the parameters found above are resulted.
Fig. 6.
Here diagrams of variation in time of movement х_{9} and speed v_{9} of the hydraulic cylinder piston (node 9
on Fig. 6), pressure р_{1} and р_{2} (accordingly in nodes 1 and 2), as well as set trapezoidal dependence q (t),
reduced to type:
are presented.
Apparently from Fig. 6, it is not possible to extinguish completely natural fluctuations of system at an appearance on a mode of the "steadystate"
movement. Besides after the end of control fluctuations of the hydraulic cylinder piston relatively some position of the balance
_{},
which differs from the target point х_{9}^{*} = 64 cm at ~ 3.7 mm. That is, accuracy of positioning at
completely damped vibrations of the hydraulic cylinder piston makes about 4 mm. The beginning of control process (0 ≤ t ≤ Т_{I})
is characterized overpressure р_{2} up to 28 MPa, exceeding the set level of the maximal pressure (25 MPa) on ~ 12%.
The specified deviations from character and parameters of transient at the chosen strategy of control take place on two principal causes: availability
of essentially nonlinear effects (including influence of boost system, inconstancy of elastic properties in process of moving the piston) as a
result of which frequency of natural fluctuations "floats", as well as some computing discrepancies in definition of parameters of control (for
example, it was accepted n = 1 though actually the right part of an inequality
(32) the previous section "Algorithm" in this case was equaled ~ 1.052), that has led to a little bit overestimated peak of
pressure р_{2}, etc.
For specification of parameters of control by hydraulic system for the 2nd version of calculation angular speed of the basic pump shaft has been
accepted ω_{н} = 100 rad/s, whence _{}
that is n = 1, consequently, still
Then Т = 1.05 + 0.9 + 1.05 = 3 s.
Results of calculation (computer modeling) are presented on Fig. 7.
Fig. 7.
Here, as well as earlier, v_{9}, х_{9} – speed and movement of the hydraulic cylinder piston (node 9),
р_{1}, р_{2} – pressure in input and output of the basic pump (nodes 1 and 2). In comparison with the 1st rated
case the significantly better agreement of the given and actual parameters of control process is observed. So, overpressure р_{2}
in the given rated case was equal to ~ 23 MPa, error of positioning is less about three (a deviation of the established position
of the piston from the given one х_{9}^{*} = 64 cm have made ~ 1.3 mm), an oscillation property of
transients has decreased. Improvement of parameters of control process is reached due to an insignificant increase of time Т_{II}
and general time of regulation Т (in this case Т = 3 s in comparison with 2.605 s in the previous example). However here again
natural fluctuations are traced neatly, to extinguish which completely it is not possible. Last circumstance as already it was marked above, is
connected with availability nonlinearities in considered hydraulic system (variable rigidity, oscillation property of boost system, etc.), not
considered at synthesis of parameters of the control, lead on linear model. Nevertheless "hit" in a vicinity of a target point of positioning
х_{9}^{*} with a margin error ε ≈ 1.3 mm for the given type of control (without introduction of correcting
links with feedback by position, speed, etc.) can be considered as comprehensible. Increase of accuracy of positioning, including improvement
damping properties of hydraulic system can be achieved really due to introduction of "precise" control by means of in appropriate way chosen
correcting links.
As already it was marked above, in the Appendix it is shown, that
the account of variable factors of elasticity, losses of pressure on length of pipelines and in local resistances, leakages of working liquid
almost does not influence on values of natural frequencies, however existing differences in character and numerical values of parameters of
considered transients (planned values at synthesis of control and "actual" values at numerical experiment) testify to an indispensability of
the account of boost system at construction of algorithm of control, at least a level of the task of the static characteristic of boost return
valves. Therefore let's consider one more rated case, and natural frequency of hydraulic system we'll define in view of the static characteristic
of the return valve in boost system. For this purpose the equation of flows [see the system of equations
(5) in the previous section "Algorithm"] in a drain pipeline of
the hydraulic system we'll write in the form of:
where Q_{b} – boost flow through the return valve equal according to its static characteristic:
where A – angular factor of the static characteristic, р_{0} – pressure in the boost system.
In view of (2) and (3) the mathematical model of dynamics of hydraulic system at dispersal at operation of the return valve in the boost system
will become:
then we'll receive the following characteristic equation:
solving which, we'll receive eigenvalues of the considered hydraulic system. It is easy to see, that at A = 0 (that is without taking into account
the return valve in the boost system) we'll receive the equation which decision is (1). Generally the characteristic equation (5) has roots:
λ_{1} = 0, appropriating a constant component of transient, also three roots of the cubic equation solved usually numerically.
Using values of parameters of the hydraulic system (tab. 1), we'll receive the following characteristic equation:
whence λ_{1} = 0.
It is known, that among roots of the cubic equation, at least, one – real. We'll find it by
iterative method:
where _{}
– value λ_{2} on nth iteration.
With accuracy ε = 0.001 required root is equal λ_{2} = – 375.357 and conforms a rapidly damped component of proper decision
of the equations (4) (time constant ~ 0.003). Eigenvalues defining oscillation property of system, are found from the quadratic
equation received by division of the polynomial (6) on the binomial λ ( λ + 375.357 ):
λ^{2} + 0.043 λ + 29.348 = 0, (8)
λ_{3,4} ≈ – 0.0215 ± 5.42 i, (9)
whence a natural frequency of the hydraulic system is equal а = 5.42 rad/s = 0.862 Hz. The found value practically coincides with the least
observed one (5.46 rad/s) as a result of modeling dynamics of the hydraulic system by the program HYDRA. The factor of damping ν = 0.0215
testifies underdamping of natural fluctuations (logarithmic decrement in this case makes_{
}).
Consequently, on the one hand there is fair an assessment of transient on simplified model
(16) – (18) previous sections "Algorithm", and with another – a consideration of one more rated version for which parameters of control are
equal: а = 5.46 rad/s, ω_{p} = 100 rad/s is of interest; then on conditions
(32) and (50) previous sections "Algorithm":_{
}
and _{}
whence n = 1, and therefore,
Results of modeling of the last rated version are resulted on Fig. 8.
Fig. 8.
Here required accuracy of positioning of 0.5 mm is received(a deviation of an equilibrium position of the hydraulic cylinder piston
_{}
from the given х_{9}^{*} = 64 cm have made 0.25 mm), however because of small damping of the hydraulic system oscillatory
processes after the end of regulation process (t > T) take place. Overpressures do not exceed the given level.
The considered examples show, that at a stage of "rough" control realization of the offered algorithm of multiple periods of natural fluctuations,
is possible in the main, however an availability of some essentially nonlinear effects does not allow determining a natural frequency of hydraulic
system enough precisely. So, changeability of elastic properties of drive in process of movement of the hydraulic cylinder piston leads to
variation of frequency of natural fluctuations, and means, that the actual system is not completely adequate to linear model with constant factors.
Besides a hydraulic system with volumetric regulation is slightly damped. Therefore for achievement of high accuracy of positioning it is
necessary:
1) to increase a damping degree of a hydraulic drive;
2) to enter correcting links at a stage of "precise" control with feedback by position and speeds of the piston, and it may be  on acceleration
(pressures in the hydraulic cylinder cavities).
It is not excluded, that the decision of both problems can be reached by introduction of appropriating feedback.
