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NONLINEAR PHENOMENA IN HYDRAULIC SYSTEMSSatoru HayashiProfessor Emeritus, Tohoku University, 981-3202, Sendai, JapanHZK00631@nifty.ne.jpABSTRACTHydraulic systems include various non-linearities instatic and dynamic characteristics of their components.Consequently, a variety of nonlinear phenomena occurin the systems. This paper deals with intrinsicnonlinear dynamic behaviors of hydraulic systems.KEYWORDSHydraulics, Nonlinear phenomena, Hard self-excitation, Micro-stick-slip, ChaosINTRODUCTIONHydraulic systems consist of various elements: pumps,actuators, control valves, accumulators, restrictors,pipelines and the like, which include many types ofnonlinearity, such as pressure-flow characteristics incontrol valves, dry friction acting on actuators andmoving parts of valves, collision of valves againstvalve seats. As a result, various types of nonlinearphenomena arise caused by these non-linearities. It is amarked feature of nonlinear systems that globalbehaviors are sometimes quite different from localbehaviors. In such cases, results of linear analysis areunavailable to estimate global nature of the system.This paper focuses on the nonlinear phenomenaoccurring in hydraulic systems, especially, “hard self-excitation” [8] whose global stability drasticallychanges from local one on the basis of the author’sstudies 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-lapvalves, zero-lap valves and under-lap valves on thebasis 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 equatedto the exhaust side lap, but the lap of the exhaust side isoften taken smaller than that of the supply side by errorin measurement in working or for stability purpose.This type of spool valve is called ”asymmetricallyunder-lapped spool valve”. Abnormal oscillations so-called “hard self-excitation” are excited in hydraulicservo-systems using this type of spool valve shown inFig. 1 [1]. “Hard self-excitation” is a kind of a self-excitedoscillation that occurs around a stable equilibrium pointby disturbances beyond a critical value and it isdistinguished from an ordinary self-excited oscillationwhich occur around a unstable equilibrium point and iscalled “soft self-excitation”. This situation isdemonstrated in Fig. 2, which shows the relationbetween soft self-excitation and hard self-excitation bybifurcation maps of amplitude and phase planetrajectories of oscillations, where λ is a related systemparameter.Fig. 1 Servo-system using asym-metrically lapped spool valveFig. 2 Types of self-excitation, bifurcation maps and phase trajectories Fig. 3 shows responses of the cylinder of a hydraulicsystem shown in Fig. 1 for different magnitudes of stepinputs given to a spool shaft of the system resting at theneutral position, whose asymmetry lap ratio isλ(= εe/εs) = 0.047 (εs=0.75mm) and the supply pressureis Ps = 9.5MPa. As shown here, the transientoscillatory responses (a), (b) and (c) settle down to aninitial equilibrium position for relatively small inputs.This shows the neutral position is locally stable.However, the response (d) for larger inputs beyond acritical valve develops into a finite amplitudeoscillation. This fact shows that the phenomenon is atypical “hard self-excitation”[1]. Fig. 4 indicates a local stability map of the neutralposition of the system, which is calculated from thefollowing stability criterion Eq. (1) [2].[ ]>+′′+κκ0202VaMAABbMbBV (1)and A is the cross-sectional area of the actuator, B thedamping coefficient, Cx the leakage coefficient, M theload mass, Q the flow rate of the valve and κ the bulkmodulus of oil. The curve in Fig. 4 shows the critical supplypressure against asymmetry ratio Λ(=1−λ). Accordingto the map, the system using a symmetrical lappedvalve λ=1 (εs= εe) is locally stable for the supplypressure Ps =5.9MPa. But for the system using a spoolvalve with asymmetry lap ratio λ=0.047, the neutralposition is stable. Equivalent asymmetry ratio (Λ=1−λ) is graduallyincreases according to the increase of the spoolamplitude after the valve begins to move by inputdisturbances, even though the system is stable at theneutral position. The pressure-flow coefficient b in Eq.(1) drastically increases as shown in Fig. 5. On theother hand, the flow-gain a changes little. As a result,the system becomes unstable and the oscillation isexcited. This is the mechanism of the “hard self-excitation”. Taking into consideration this hard self-excitation, the self-excited region is enlarged more thanlocally unstable region that is between a solid line and adashed line as shown in Fig. 6.Fig. 3 Responses of hydraulic servo-system with asymmetrical spool valve fordifferent magnitude of step inputs ,210 ,20201,0 whereVVVdCeCbbPQPQbxQa==++=′∂∂=∂∂−=∂∂=Fig. 4 Local stability map by asymmetri-cal lap ratio 210, PPPPQbLL−=∂∂=Fig. 5 Pressure-flow coefficient MICRO-STICK-SLIP IN SERVO-SYSTEMUSING UNDER-LAPPED SPOOL VALVE [3]It is familiar to most hydraulic engineers that stick-slipphenomenon occurs in a hydraulic system using anactuator subjected to dry friction. It is shown in thispaper that there exists a quite different type of stick-slipphenomenon [3] from the above mentioned in ahydraulic servo-system using a spool valve with thesymmetrical under-lap εe=εs in Fig. 1, whose actuator isaffected by dry friction. A global stability map of the system is plotted in Fig.7, which indicates amplitudes of stable and unstableself-excited oscillations for each supply pressure Ps.The neutral position is always locally stable for thesystem with an actuator affected by dry friction. Butself-maintained oscillations are possible to occur forlarge disturbances beyond critical values as seen in Fig.7. This phenomenon is also “hard self-excitation”. Thesolid lines AB and CD correspond to stable oscillationsand the dashed lines AE and BC to unstable ones. Thissystem has multiple stability construction, that is,multiple oscillatory solutions for each supply pressurein a certain region. The oscillation on CD is almostsinusoidal as it is well known. However, a stick-sliposcillation occurs on AB for input disturbancesbetween the curves AE and CB, which correspond toamplitudes of unstable oscillatory solutions. Theamplitudes are small and do not exceed severalmicrometers. Fig. 8 shows simulated responses for differentmagnitude inputs. Fig. 8 (c) corresponds to the stablestick-slip oscillation. Unlike the ordinary stick-slip, thedifference between static and dynamic friction is notessential for occurrence of this stick-slip. This isnovelty of the phenomenon.HARD SELF-EXCITATION IN POPPETVALVE CIRCUITS [4],[5],[6]Poppet valve circuits are classified into two categories:direct-acting type circuits and pilot type circuits [6]. Itis familiar to hydraulic engineers that poppet valves areliable to become unstable and excite various self-oscillations in both circuits. Fig. 9 shows a pilot typepoppet valve circuit. Fig. 10 indicates a stability mapof the circuit with a 0.6m supply line. The abscissa isthe supply pressure Ps and the ordinate the valve lift X.A hatched line is a boundary of stability and the systemis stable in the lower part of the curve. A thin solid linerepresents a static characteristic for the crackingpressure Psi = 2 MPa, along which the valve lift movesaccording to the supply pressure change Ps. Figure 11 indicate responses of the valve fordifferent magnitude inputs under a same operatingcondition that is plotted by a mark A on Fig. 10. Thepoint A is in the locally stable region. Thus theFig. 7 Global stability map of hydraulicservo-system with under-lapped spoolvalveFig. 8 Responses of hydraulic systemFig.6 Local and global stability map transient response (a) for a small input die downs to asteady point, while the response (b) for a little largerinput than the case (a) develops into a maintainedoscillation. This is a typical hard self-excitation arounda locally stable equilibrium point. Figures 12 and 13 show another case of a hard self-excitation occurring in a poppet valve circuit with a0.3m length supply-line. A logarithmic scale is takento the ordinate of the map to enlarge the small valve liftregion. The dashed line represents a staticcharacteristic corresponding to a cracking pressure Psi =4MPa. As seen in this figure, the small valve liftregion including X=0 is locally stable for any supplypressure. This means that the valve sitting on its seat atany under-cracking supply pressure is locally stable.But the result of Fig. 13 shows that the global behavioris quite different from this local nature. Figure 13 shows valve responses for differentmagnitude disturbances, which are given to the valvesitting on the seat at an under-cracking supply pressureplotted by in Fig. 12. As seen in Fig. 13 (b), amaintained oscillation is excited for a large disturbance.This is also a typical “hard self-excitation” and thisphenomenon is interesting in that even the valve sittingsteadily on its seat has a possibility of occurrence ofself-excited oscillations.Taking into consideration this “hard self-excitation”,the self-oscillation region of the system significantlyexpands as shown by a heavy line in Fig. 10.CHAOS IN POPPET VALVE CIRCUIT [6],[7]Another outstanding nonlinear phenomenon occurringin hydraulic systems is chaotic oscillation [6]. All peak values of each oscillatory wave of self-excited oscillations occurring in the same system asFig. 9 are plotted in Fig. 14. A dashed line ABrepresents a static valve lift against the supply pressurePs and the interval AB is a soft self-excitation region.In the much lower supply pressure region undercracking pressure Psi, there is only one peak valuecorresponding to the oscillation at each supplypressure. This means that the oscillation has only aperiod (period-one oscillation). As an increase in thesupply pressure, the period-one oscillation branchbifurcates into two values (period-two oscillation) andafter that the branch bifurcates four values, eightvalues,and 2N values. In the limit, peakvalues compactly distribute in a limited range of thevalve displacement for a supply pressure. This impliesthat the oscillationis a Faigenbaum type chaos [6]. Furthermore, it is shown that other types of chaoticoscillations, or Lorenz type chaos and intermittent typechaos also appear in direct-acting poppet valve circuits[7].CONCLUSIONSNonlinear dynamic phenomena in hydraulic systemsare unique and diverse. It is difficult to estimate theirglobal nature from local nature by linear analysis.Fig. 11 Responses for different magnitudedisturbancesFig. 9 Schematic diagram of pilot typepoppet valve circuitFig. 10 Local and global stability map ofpoppet valve circuit for L=0.6m Analytical method is virtually impossible to solveglobal dynamic problems of these nonlinear systems.Numerical simulation is only one method available tosolve them. Almost all results described here wereobtained by use of numerical simulation. It isinevitable to understand global nature about nonlineardynamic systems for the purpose of sophisticateddesign of hydraulic systems. Thus it is expected that anumber of nonlinear problems in hydraulic systemswill be solved by numerical simulation.REFERENCES[1] Hayashi, S. et al., Hard Self-Excitation in HydraulicServomechanism with Asymmetrically Under-lappedSpool Valve, Transactions of the Society of Instrumentand Control Engineers, 1975, 16(3), 85-90 (inJapanese).[2] Hayashi, S. and Ohashi, T., Stability of HydraulicServo-mechanism with Spool Valve, Transactions ofthe Japan Society of Mechanical Engineers, 1986,22(4), 459-464 (in Japanese).[3] Hayashi, S. and Iimura, I., Effect of CoulombFriction on Stability of Hydraulic Servomechanism,Proc. FLUCOME’88, 1988, 310-314.[4] Hayashi, S. and Ohi, K., Global Stability of aPoppet Valve Circuit, Journal of Fluid Control, 1993,21(4), 48-63.[5] Hayashi, S. et al., Chaos in a Hydraulic ControlValve, Journal of Fluids and Structures, 1997, 11, 693-716.[6] Hayashi, S., Stability and Nonlinear Behavior ofPoppet Valve Circuit, Proc. FLUCOME’97, 1997, 1,13-20.[7] Hayashi, S. and Mochizuki, T., Chaotic OscillationsOccurring in a Hydraulic Circuit, Proc.1st JHPS Int.Symp. On Fluid Power, 1989, 475-480.[8] Minorsky, N., Nonlinear Mechanics, EdwardsBrothers Inc., 1947, 71.Fig. 12 Local stability map of pilot typepoppet valve circuit L=0.3mFig. 14 Bifurcation map of self-excitedoscillationsFig.13 Responses for different magnitudedisturbances . map by asymmetri-cal lap ratio 210, PPPPQbLL−=∂∂=Fig. 5 Pressure-flow coefficient MICRO-STICK-SLIP IN SERVO-SYSTEMUSING UNDER-LAPPED SPOOL. SELF-EXCITATION IN ASYMMET-RICALLY UNDER-LAPPED SPOOL VALVE[1],[2]Spool valves are classified into three types, over-lapvalves, zero-lap valves and under-lap valves