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Thủy lực học là ngành kĩ thuật nghiên cứu về các vấn đề mang tính thực dụng bao gồm: lưu trữ, vận chuyển, kiểm soát, đo đạc nước và các chất lỏng khác.

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NONLINEAR PHENOMENA IN HYDRAULIC SYSTEMS

Satoru Hayashi Professor Emeritus, Tohoku University, 981-3202, Sendai, Japan

HZK00631@nifty.ne.jp

ABSTRACT

Hydraulic systems include various non-linearities in

static and dynamic characteristics of their components

Consequently, a variety of nonlinear phenomena occur

in the systems This paper deals with intrinsic

nonlinear dynamic behaviors of hydraulic systems

KEYWORDS

Hydraulics, Nonlinear phenomena, Hard

self-excitation, Micro-stick-slip, Chaos

INTRODUCTION

Hydraulic systems consist of various elements: pumps,

actuators, control valves, accumulators, restrictors,

pipelines and the like, which include many types of

nonlinearity, such as pressure-flow characteristics in

control valves, dry friction acting on actuators and

moving parts of valves, collision of valves against

valve seats As a result, various types of nonlinear

phenomena arise caused by these non-linearities It is a

marked feature of nonlinear systems that global

behaviors are sometimes quite different from local

behaviors In such cases, results of linear analysis are

unavailable to estimate global nature of the system

This paper focuses on the nonlinear phenomena

occurring in hydraulic systems, especially, “hard

self-excitation” [8] whose global stability drastically

changes from local one on the basis of the author’s

studies in the past [1]-[7]

HARD SELF-EXCITATION IN

ASYMMET-RICALLY UNDER-LAPPED SPOOL VALVE

[1],[2]

Spool valves are classified into three types, over-lap

valves, zero-lap valves and under-lap valves on the

basis of the relation of the land-width to the port-width

They are used properly according to their applications

Usually in spool valves, the supply side lap is equated

to the exhaust side lap, but the lap of the exhaust side is

often taken smaller than that of the supply side by error

in measurement in working or for stability purpose

This type of spool valve is called ”asymmetrically

under-lapped spool valve” Abnormal oscillations

so-called “hard self-excitation” are excited in hydraulic

servo-systems using this type of spool valve shown in

Fig 1 [1].

“Hard self-excitation” is a kind of a self-excited oscillation that occurs around a stable equilibrium point

by disturbances beyond a critical value and it is distinguished from an ordinary self-excited oscillation which occur around a unstable equilibrium point and is called “soft self-excitation” This situation is

demonstrated in Fig 2, which shows the relation

between soft self-excitation and hard self-excitation by bifurcation maps of amplitude and phase plane trajectories of oscillations, where λ is a related system parameter

Fig 1 Servo-system using asym-metrically lapped spool valve

Fig 2 Types of self-excitation, bifurcation maps and phase trajectories

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Fig 3 shows responses of the cylinder of a hydraulic

system shown in Fig 1 for different magnitudes of step

inputs given to a spool shaft of the system resting at the

neutral position, whose asymmetry lap ratio is

λ(= εe/εs)= 0.047 (εs=0.75mm) and the supply pressure

is Ps = 9.5MPa As shown here, the transient

oscillatory responses (a), (b) and (c) settle down to an

initial equilibrium position for relatively small inputs

This shows the neutral position is locally stable

However, the response (d) for larger inputs beyond a

critical valve develops into a finite amplitude

oscillation This fact shows that the phenomenon is a

typical “hard self-excitation”[1]

Fig 4 indicates a local stability map of the neutral

position of the system, which is calculated from the

following stability criterion Eq (1) [2]



>

+

+





κ κ

0 2

0

2A aMA V B

b M b B

and A is the cross-sectional area of the actuator, B the

damping coefficient, Cx the leakage coefficient, M the

load mass, Q the flow rate of the valve and κthe bulk

modulus of oil

The curve in Fig 4 shows the critical supply

pressure against asymmetry ratio Λ(=1−λ) According

to the map, the system using a symmetrical lapped

valve λ=1 (εs= εe) is locally stable for the supply

pressure Ps =5.9MPa But for the system using a spool valve with asymmetry lap ratio λ=0.047, the neutral position is stable

Equivalent asymmetry ratio (Λ=1−λ) is gradually increases according to the increase of the spool amplitude after the valve begins to move by input disturbances, even though the system is stable at the

neutral position The pressure-flow coefficient b in Eq.

(1) drastically increases as shown in Fig 5 On the

other hand, the flow-gain a changes little As a result,

the system becomes unstable and the oscillation is excited This is the mechanism of the “hard excitation” Taking into consideration this hard self-excitation, the self-excited region is enlarged more than locally unstable region that is between a solid line and a dashed line as shown in Fig 6

Fig 3 Responses of hydraulic

servo-system with asymmetrical spool valve for

different magnitude of step inputs

, 2 1 0 , 2

0 2 0 1

, 0 where

V V V d C e C b b

P

Q P

Q b x

Q a

=

= +

+

=

=

=

Fig 4 Local stability map by asymmetri-cal lap ratio

2 1 0

P

Q

L

=





=

Fig 5 Pressure-flow coefficient

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MICRO-STICK-SLIP IN SERVO-SYSTEM

USING UNDER-LAPPED SPOOL VALVE [3]

It is familiar to most hydraulic engineers that stick-slip

phenomenon occurs in a hydraulic system using an

actuator subjected to dry friction It is shown in this

paper that there exists a quite different type of stick-slip

phenomenon [3] from the above mentioned in a

hydraulic servo-system using a spool valve with the

symmetrical under-lap εe=εs in Fig 1, whose actuator is

affected by dry friction

A global stability map of the system is plotted in Fig.

7, which indicates amplitudes of stable and unstable

self-excited oscillations for each supply pressure Ps

The neutral position is always locally stable for the

system with an actuator affected by dry friction But

self-maintained oscillations are possible to occur for

large disturbances beyond critical values as seen in Fig.

7 This phenomenon is also “hard self-excitation” The

solid lines AB and CD correspond to stable oscillations

and the dashed lines AE and BC to unstable ones This

system has multiple stability construction, that is,

multiple oscillatory solutions for each supply pressure

in a certain region The oscillation on CD is almost

sinusoidal as it is well known However, a stick-slip

oscillation occurs on AB for input disturbances

between the curves AE and CB, which correspond to

amplitudes of unstable oscillatory solutions The

amplitudes are small and do not exceed several

micrometers

Fig 8 shows simulated responses for different

magnitude inputs Fig 8 (c) corresponds to the stable

stick-slip oscillation Unlike the ordinary stick-slip, the

difference between static and dynamic friction is not

essential for occurrence of this stick-slip This is

novelty of the phenomenon

HARD SELF-EXCITATION IN POPPET VALVE CIRCUITS [4],[5],[6]

Poppet valve circuits are classified into two categories: direct-acting type circuits and pilot type circuits [6] It

is familiar to hydraulic engineers that poppet valves are liable to become unstable and excite various

self-oscillations in both circuits Fig 9 shows a pilot type poppet valve circuit Fig 10 indicates a stability map

of the circuit with a 0.6m supply line The abscissa is

the supply pressure Ps and the ordinate the valve lift X.

A hatched line is a boundary of stability and the system

is stable in the lower part of the curve A thin solid line represents a static characteristic for the cracking

pressure Psi = 2 MPa, along which the valve lift moves according to the supply pressure change Ps

Figure 11 indicate responses of the valve for

different magnitude inputs under a same operating

condition that is plotted by a mark A on Fig 10 The

point A is in the locally stable region Thus the

Fig 7 Global stability map of hydraulic servo-system with under-lapped spool valve

Fig 8 Responses of hydraulic system Fig.6 Local and global stability map

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transient response (a) for a small input die downs to a

steady point, while the response (b) for a little larger

input than the case (a) develops into a maintained

oscillation This is a typical hard self-excitation around

a locally stable equilibrium point

Figures 12 and 13 show another case of a hard

self-excitation occurring in a poppet valve circuit with a

0.3m length supply-line A logarithmic scale is taken

to the ordinate of the map to enlarge the small valve lift

region The dashed line represents a static

characteristic corresponding to a cracking pressure Psi =

4MPa As seen in this figure, the small valve lift

region including X=0 is locally stable for any supply

pressure This means that the valve sitting on its seat at

any under-cracking supply pressure is locally stable

But the result of Fig 13 shows that the global behavior

is quite different from this local nature

Figure 13 shows valve responses for different

magnitude disturbances, which are given to the valve

sitting on the seat at an under-cracking supply pressure

plotted by in Fig 12 As seen in Fig 13 (b), a

maintained oscillation is excited for a large disturbance

This is also a typical “hard self-excitation” and this

phenomenon is interesting in that even the valve sitting

steadily on its seat has a possibility of occurrence of

self-excited oscillations

Taking into consideration this “hard self-excitation”,

the self-oscillation region of the system significantly

expands as shown by a heavy line in Fig 10.

CHAOS IN POPPET VALVE CIRCUIT [6],[7]

Another outstanding nonlinear phenomenon occurring

in hydraulic systems is chaotic oscillation [6]

All peak values of each oscillatory wave of

self-excited oscillations occurring in the same system as

Fig 9 are plotted in Fig 14 A dashed line AB

represents a static valve lift against the supply pressure

Ps and the interval AB is a soft self-excitation region

In the much lower supply pressure region under

cracking pressure Psi, there is only one peak value corresponding to the oscillation at each supply pressure This means that the oscillation has only a period (period-one oscillation) As an increase in the supply pressure, the period-one oscillation branch bifurcates into two values (period-two oscillation) and after that the branch bifurcates four values, eight values, and 2N values In the limit, peak values compactly distribute in a limited range of the valve displacement for a supplypressure This implies that the oscillation

is a Faigenbaum type chaos [6]

Furthermore, it is shown that other types of chaotic oscillations, or Lorenz type chaos and intermittent type chaos also appear in direct-acting poppet valve circuits [7]

CONCLUSIONS

Nonlinear dynamic phenomena in hydraulic systems are unique and diverse It is difficult to estimate their global nature from local nature by linear analysis

Fig 11 Responses for different magnitude disturbances

Fig 9 Schematic diagram of pilot type

poppet valve circuit

Fig 10 Local and global stability map of poppet valve circuit for L=0.6m

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Analytical method is virtually impossible to solve

global dynamic problems of these nonlinear systems

Numerical simulation is only one method available to

solve them Almost all results described here were

obtained by use of numerical simulation It is

inevitable to understand global nature about nonlinear

dynamic systems for the purpose of sophisticated

design of hydraulic systems Thus it is expected that a

number of nonlinear problems in hydraulic systems

will be solved by numerical simulation

REFERENCES

[1] Hayashi, S et al., Hard Self-Excitation in Hydraulic Servomechanism with Asymmetrically Under-lapped Spool Valve, Transactions of the Society of Instrument and Control Engineers, 1975, 16(3), 85-90 (in Japanese)

[2] Hayashi, S and Ohashi, T., Stability of Hydraulic Servo-mechanism with Spool Valve, Transactions of the Japan Society of Mechanical Engineers, 1986, 22(4), 459-464 (in Japanese)

[3] Hayashi, S and Iimura, I., Effect of Coulomb Friction on Stability of Hydraulic Servomechanism, Proc FLUCOME’88, 1988, 310-314

[4] Hayashi, S and Ohi, K., Global Stability of a Poppet Valve Circuit, Journal of Fluid Control, 1993, 21(4), 48-63

[5] Hayashi, S et al., Chaos in a Hydraulic Control Valve, Journal of Fluids and Structures, 1997, 11, 693-716

[6] Hayashi, S., Stability and Nonlinear Behavior of Poppet Valve Circuit, Proc FLUCOME’97, 1997, 1, 13-20

[7] Hayashi, S and Mochizuki, T., Chaotic Oscillations Occurring in a Hydraulic Circuit, Proc.1st JHPS Int Symp On Fluid Power, 1989, 475-480

[8] Minorsky, N., Nonlinear Mechanics, Edwards Brothers Inc., 1947, 71

Fig 12 Local stability map of pilot type

poppet valve circuit L=0.3m

Fig 14 Bifurcation map of self-excited oscillations

Fig.13 Responses for different magnitude

disturbances

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