Proceedings of the ASME 2016 Dynamic Systems and Control Con ferenceDSCC 2016

October 12-14, 2016, Minneapolis, USA

DSCC2016-9601

COUPLED CHAOTIC AND HYPERCHAOTIC DYNAMICS OF ACTUATED BUTT ERFLYVALVES OPERATING IN SERIES

Peiman Naseradinmousavi and Mostafa BagheriDepartment of Mechanical Engineering

San Diego State UniversitySan Diego, California

Miroslav Krsti cDepartment of Mechanical and Aerospace Engineering

University of California, San DiegoSan Diego, California

C. NatarajDepartment of Mechanical Engineering

Villanova UniversityVillanova, Pennsylvania

ABSTRACTIn this effort, we focus on determining the safe operational

domain of a coupled actuator-valve configuration. The so-called“Smart Valves” system has increasingly been used in criticalapplications and missions including municipal piping networks,oil and gas fields, petrochemical plants, and more importantly,the US Navy ships. A comprehensive dynamic analysis is henceneeded to be carried out for capturing dangerous behaviors ob-served repeatedly in practice. Using some powerful tools ofnonlinear dynamic analysis including Lyapunov exponents andPoincare map, a comprehensive stability map is provided in or-der to determine the safe operational domain of the networkin addition to characterizing the responses obtained. Coupledchaotic and hyperchaotic dynamics of two coupled solenoid ac-tuated butterfly valves are captured by running the network forsome critical values through interconnected flow loads affectedby the coupled actuators’ variables. The significant effectof anunstable configuration of the valve-actuator on another setisthoroughly investigated to discuss the expected stabilityissuesof a remote set due to others and vice versa.

1 IntroductionMultidisciplinary electromechanical-fluid systems have

been widely used in many megascale networks. Municipal pip-

ing systems, oil and gas fields, petrochemical plants, and morecritically, the US Navy are the immediate ones which need toutilize a reliable, safe, and efficient coupled flow distribution net-work.

Future smart cities would inevitably need an autonomousflow control network in order to help improve the safety of sucha critical system and also to decrease the incremental costsof op-eration and maintenance. Malfunctions of the flow network haveoccurred repeatedly resulting in the flow interruption of smalltowns/districts. Although significant cost and energy would beneeded to be spent in order to restore the whole system in ad-dition to human resources required to be recruited. Economicaland even social impact of these malfunctions can be expectedtobe dramatic and consequently, a fully automated flow networkisneeded to be designed and operated.

The same issues exist for oil and gas fields and petrochemi-cal plants. The industry of oil and gas is one of the most sensitiveelements of the global economy and plays important roles evenin global politics. The flow control network is the essentialpartof these fields and is therefore required to be safely designed tominimize the flow interruptions leading to much higher oil/gasproduction.

The US national defense and homeland security is undoubt-edly a highly important priority which requires to be addressed

1 Copyright c© 2016 by ASME

and investigated carefully. The US Navy broadly employs thenetwork of coupled electromechanical valve sets for cooling pur-poses, mainly for chilled water systems. The proper performanceof the network is remarkably effective for other critical units ofradar, sonar, defense systems, etc.

We have carried out broad analytical and experimental stud-ies from nonlinear modeling to design optimization of both anisolated and interconnected symmetric butterfly valves driven bysolenoid actuators [1–9]. The multidisciplinary couplings, in-cluding electromagnetics and fluid mechanics, had to be thor-oughly regarded in the modeling phase in order to yield an ac-curate nonlinear model of such a complex system. A third-ordernondimensional dynamic model of the single set was derived tobe used in nonlinear dynamic analysis [3] and optimal design[4].

The dynamic analysis expectedly yielded practically ob-served crisis and transient chaotic dynamics of a single actuatedvalve for some critical physical parameters. A comprehensivestability map was also presented as an efficient tool to determinethe safe domain of operation which in turn could serve for iden-tifying the lower and upper bounds of the design optimization ef-forts. The design optimization was then carried out [4] to selectthe optimal actuation unit’s parameters coupled with the mechan-ical and fluid parts in order to significantly reduce the amount ofenergy consumption (upward of %40).

Note that the applications addressed earlier contain thou-sands of actuated valves in which a high level of dynamic cou-pling has been repeatedly observed in practice. These dynamiccouplings among different sets need to be captured through ana-lytical studies. We have developed [5] a novel nonlinear modelfor two sets of solenoid actuated butterfly valves operatinginseries. The closing/opening valves were modeled as changingresistors and the flow between them as a constant one. A sixth-order nonlinear coupled model revealed the high dynamic sen-sitivity of each element of a set, the valve and the actuator,toanother one and vice versa. The power spectrum was used inconfirming the same frequency response of a neighbor set dueto the external periodic noise applied on another set of the valveand actuator.

By taking another step, we optimized the design of coupledactuation units of two sets operating in series [8] subject to a sud-den contraction. The pipe contraction imposed an additional re-sistance to be modeled and therefore, the coupled dynamic equa-tions derived in [5] had to be slightly modified. We representthe modeling process here for completeness. We have surpris-ingly established an interesting coupling between currents of theactuation units through the interconnected flow loads, includinghydrodynamic and bearing torques, which affect both the valves’dynamics.

Important nonlinear phenomena in electromechanical sys-tems have also received considerable attention. Sunet al. [10]studied the hyperchaotic behavior of the newly presented simpli-fied Lorenz system by using a sinusoidal parameter variationand

hyperchaos control of the forced system via feedback. Banerjeeet al. [11] investigated the synchronization of chaos and hyper-chaos in first-order time-delayed systems that are coupled usingthe nonlinear time-delay excitatory coupling by assigningtwocharacteristic time delays: the system delay that is same for boththe systems, and the coupling delay associated with the couplingpath. Many efforts for analyzing hyperchaotic dynamics havebeen reported in [12–27].

We have addressed the modeling process subject to thepipe contraction in [8] and represent here for completeness.The contribution of this work is the inclusion of interconnectedelectromechanical-fluid nonlinearities between two actuator-valve configurations to thoroughly analyze the effects of anun-stable set of the valve-actuator on another one. Through thiscomprehensive analysis, chaotic and hyperchaotic dynamics oftwo coupled configurations are captured by exposing the networkto some critical values. The responses are then characterizedusing some powerful tools including Lyapunov exponents andPoincare map.

k1 bd1

Plunger

Coil

Coil

Yoke

k2 bd2

Plunger

Coil

Coil

Yoke

α1 α2Pin Poutqin qout

qv

(a)

R1

Pin PoutP1 P2

R2

α1 α2

L1 L2

RL1 Rcon RL2

Dv1 Dv2

Pcon1 Pcon2

1 2

(b)

FIGURE 1. (a) A schematic configuration of two solenoid actuatedbutterfly valves subject to the sudden contraction; (b) A coupled modelof two butterfly valves in series without actuation

2 Copyright c© 2016 by ASME

TABLE 1 . The system parameters

ρ 1000kgm3 v 0.1m

s

J1,2 0.104×10-1(kg.m2) N2 3000

N1 3000 C11,22 1.56×106(H-1)

gm1,m2 0.1(m) V1,2 24(Volt)

Dv1 0.2032(m) Dv2 0.127(m)

Ds1,s2 0.01(m) Pout 2(kPa)

k1,2 1000(N.m-1) C21,22 6.32×108(H-1)

L1 2(m) L2 1(m)

µf 0.018 (Kg.m-1.s-1) R1,2 6(Ω)

r1,2 0.05(m) θ 90

Pin 256(kPa)

2 Mathematical ModelingShown in Fig. 1(a) is a pair of symmetric butterfly valves

driven by solenoid actuators through rack and pinion arrange-ments. The rack and pinion mechanism provides a kinematicconstraint which connects the dynamics of the valve and actua-tor. Applying DC voltages, as being used in the Navy ships forchilled water systems, the motive forces give translational mo-tions to the actuators’ moving parts (plungers) and subsequentlythe valves rotate to desirable angles. Note that a return spring hasbeen a common practice among industries to open the valves.

Interconnected modeling of such a multiphysics system un-doubtedly needs some simplifying assumptions to neglect use-less and tremendously time consuming numerical calculations.The magnetic force resulted from the magnetic field needs an ex-tremely short period of time to reach its maximum value. Thisperiod is the so-called “Diffusion Time” and has an inverse re-lationship with the amount of current used. Note that usingthe current of 4 (A) would yield a negligible diffusion time ofτd ≈ 20(ms) [1] with respect to the nominal operation time of40(s). We have to also assume dominant laminar flow for boththe coupled valves. Note that developing an analytical model isa necessity to carry out the dynamic analysis and optimizationwhich would lead us to make such a commonly used assumptionand also to avoid the numerical difficulties involved with a turbu-lent regime. However a crucial question needs to be carefully an-swered with respect to the validity of such an assumption. Usingthe values of pipe diameter and flow mean velocity listed in Table1, one can easily distinguish the existence of the turbulentregimewhich invalidates the assumption we have made. From anotheraspect, the analytical formulas derived for the flow loads, includ-ing the hydrodynamic and bearing torques, have been developedbased on the assumption of laminar flow [28,29]. To address the

0 20 40 60 80−8

−6

−4

−2

0

2

4

α (degree)

Ttot(N

.m)

Expr1Expr2Expr3Theory

FIGURE 2. A comparison between the experimental and analyticaltotal torques.

issues discussed above, we have carried out experimental workto measure the sum of the hydrodynamic and bearing torques asthe most affecting loads on the valves’ and subsequently actua-tors’ dynamics [8]. The experiment yielded the total torque(Fig.2) for the inlet velocity ofv ≈ 2.7

(ms

)and valve diameter of

Dv= 2 (inches) validating the laminar flow assumption [30]. Theflow torques have shown highly important roles for the dynam-ics of an isolated solenoid actuated butterfly valve and we henceexpect to observe such effects for the interconnected sets [5].

The coupled system is modeled as a set of five resistors. Twochanging resistors represent the closing/opening valves,two con-stant ones indicate head losses between the valves, and anotheris due to the pipe contraction as shown in Fig. 1(b). The inletandoutlet pressures are given values in Table 1. Using the assump-tion of the dominant laminar flow, the pressure drops betweentwo valves can be expressed based on the Hagen-Poiseuille [31]and Borda-Carnot [32] formulas (points 1 and 2):

P1−Pcon1 =128µ f L1

πD4v1

︸ ︷︷ ︸

RL1

qv (1)

Pcon1−Pcon2 =12

Kconρv2out (2)

Pcon2−P2 =128µ f L2

πD4v2

︸ ︷︷ ︸

RL2

qv (3)

where,qv is the volumetric flow rate,µ f indicates the fluid dy-namic viscosity,Dv1 andDv2 are the valves’ diameters,L1 andL2

3 Copyright c© 2016 by ASME

stand for the pipe lengths before and after contraction,RL1 andRL2 indicate the constant resistances, andPcon1 andPcon2 are theflow pressures before and after contraction.Kcon is calculated asthe following:

Kcon= 0.5(1−β 2)

√

sin

(θ2

)

(4)

where,β indicates the ratio of minor and major diameters(

Dv2Dv1

)

andθ is the angle of approach. The values listed in Table 1 easilyyield Kcon= 0.2562. We then rewrite Eq. 2 as follows:

Pcon1−Pcon2 =12

Kconρv2out

=8Kcon

π2D4v2

ρ︸ ︷︷ ︸

Rcon

π2D4v2v2

out

16︸ ︷︷ ︸

q2v

= Rconq2v (5)

where,Rcon is the resistance due to the pipe contraction. Thepressure drop between the valves can be derived by adding Eqs.1,2, 3, and 5:

P1−P2 = [RL1+RL2+Rconqv]qv (6)

The valve’s “Resistance (R)” and “Coefficient (cv)”, as importantparameters of the regulating valves, are nonlinear functions of thevalve rotation angle to be stated [33] as follows:

Ri(αi) =891D4

vi

c2vi(αi)

, i = 1,2 (7)

Based on the assumption of laminar flow, the valve’s pressuredrop is calculated via the following relationship [28]:

∆Pi(αi) = 0.5Ri(αi)ρv2 (8)

where,α indicates the valve rotation angle,ρ is the density ofthe media, andv stands for the flow velocity. Rewriting Eq. 8 ina standard form gives,

∆Pi(αi) =π2D4

viv2

16︸ ︷︷ ︸

q2v

8×Ri(αi)ρπ2D4

vi︸ ︷︷ ︸

Rni(αi)

= Rni(αi)q2v (9)

The hydrodynamic (Th) and bearing (Tb) torques [28,29] have ex-pectedly shown the high sensitivity to the pressure drop obtainedvia Eq. 9 leading us to rewrite them as follows.

fi(αi) =16Tci(αi)

3π(

1− Ccci(αi)(1−sin(αi))2

)2 (10)

Thi =16Tci(αi)D3

vi∆Pi

3π(

1− Ccci(αi)(1−sin(αi))2

)2 = fi(αi)D3vi∆Pi (11)

Tbi = 0.5Ad∆PiµDs =Ci∆Pi (12)

where,Ds stands for the stem diameter of the valve,µ indicatesthe friction coefficient of the bearing area,Ci =

π8 µD2

viDs, andTci andCcci are the hydrodynamic torque and the sum of up-per and lower contraction coefficients, respectively, dependingon the valve rotation angle [1].

The comprehensive stability map we have presented in [3]was based on a nonlinear analytical model. The analytical modelhad to be used in the dynamic analysis to investigate the systemstability around equilibria by calculating its eigenvalues throughthe Jacobian matrix; this has led us to identify the safe oper-ational domain to be utilized in the design optimization. Thesame practice was employed in [8] with the aid of fitting suitablecurves oncvi andRni in order to model the system analytically.For our case study ofDv1=8 (in) andDv2=5 (in), the valves’ co-efficients and resistances are developed as follows.

cv1(α1) = p1α31 +q1α2

1 +o1α1+ s1 (13)

cv2(α2) = p2α32 +q2α2

2 +o2α2+ s2 (14)

Rn1(α1) =e1

(p1α31 +q1α2

1 +o1α1+ s1)2(15)

Rn2(α2) =e2

(p2α32 +q2α2

2 +o2α2+ s2)2(16)

where,e1 = 7.2×105, p1 = 461.9, q1 = −405.4, o1 = −1831,s1 = 2207,e2 = 4.51× 105, p2 = 161.84, q2 = −110.53, o2 =−695.1, ands2 = 807.57. These fittings were selected with re-spect to the decremental and incremental profiles of the valves’coefficients and resistances, respectively [5, 30]. Applying themass continuity principle (qin = qout = qv) and then rewritingEq. 9 yields,

Pin −P1

Rn1(α1)=

P2−Pout

Rn2(α2)(17)

Rn1P2+Rn2P1 = Rn2Pin +Rn1Pout (18)

4 Copyright c© 2016 by ASME

The interconnectedP1 and P2 terms are derived by combiningEqs. 6 and 18 as follows:

P1 =Rn2Pin +Rn1Pout+Rn1(RL1+RL2+Rconqv)qv

(Rn1+Rn2)(19)

P2 =Rn2Pin +Rn1Pout−Rn2(RL1+RL2+Rconqv)qv

(Rn1+Rn2)(20)

The dynamic sensitivities ofP1 andP2 to Rn1, Rn2, RL1, RL2, andRcon are distinguishable through Eqs. 19 and 20, as observed inthe practice. Any slight dynamic changes of the upstream setof the valve-actuator would be expected to be observed for thedownstream one. The hydrodynamic and bearing torques’ de-pendencies on all the resistances are reformulated as follows.

Thi = fi(αi)D3vi∆Pi(Rn1,Rn2,RL1,RL2,Rcon) (21)

Tbi = Ci∆Pi(Rn1,Rn2,RL1,RL2,Rcon) (22)

fi is a nonlinear function of the changingTci, Ccci, and the valverotation angles. To carry out a systematic dynamic analysis, thefollowing functions are fitted to theD3

vi fi of each valve [5,30]:

Th1 = (a1α1eb1α11.1− c1ed1α1)

︸ ︷︷ ︸

D3v1 f1

(Pin −P1)

= (a1α1eb1α11.1− c1ed1α1)×

e1(p1α3

1+q1α21+o1α1+s1)2

∑2i=1

ei(piα3

i +qiα2i +oiαi+si)2

× (Pin −Pout− (RL1+RL2+Rconqv)qv) (23)

Th2 = (a′1α2eb′1α1.12 − c′1ed′1α2)

︸ ︷︷ ︸

D3v2 f2

(P2−Pout)

= (a′1α2eb′1α21.1− c′1ed′1α2)×

e2(p2α3

2+q2α22+o2α2+s2)2

∑2i=1

ei(piα3

i +qiα2i +oiαi+si)2

× (Pin −Pout− (RL1+RL2+Rconqv)qv) (24)

where,a1 = 0.4249,a′1 = 0.1022,b1 =−18.52,b′1 =−17.0795,c1 = −7.823× 10−4, c′1 = −2×10−4, d1 = −1.084, andd′

1 =−1.0973.

We have previously derived the rate of current and magneticforce terms [1] which are utilized in developing the sixth-ordercoupled dynamic model [8] as follows. Note that both the motiveforce and current are highly sensitive to the plunger displacement

and subsequently the valve rotation angle.

Fmi =C2iN2

i i2i2(C1i +C2i(gmi− xi))2 (25)

diidt

=(Vi −Ri i i)(C1i +C2i(gmi− xi))

N2i

−C2i i i xi

(C1i +C2i(gmi− xi))(26)

z1 = z2 (27)

z2 =1J1

[r1C21N2

1z23

2(C11+C21(gm1− r1z1))2 −bd1z2− k1z1

+

(Pin−Pout−(RL1+RL2+Rconqv)qv)e1(p1z3

1+q1z21+o1z1+s1)2

∑i=1,4ei

(piz3i +qiz2

i +oizi+si)2

×

[

(a1z1eb1z11.1− c1ed1z1)−C1× tanh(Kz2)

]]

(28)

z3 =(V1−R1z3)(C11+C21(gm1− r1z1))

N21

−

r1C21z3z2

(C11+C21(gm1− r1z1))(29)

z4 = z5 (30)

z5 =1J2

[r2C22N2

2z26

2(C12+C22(gm2− r2z4))2 −bd2z5− k2z4

+

(Pin−Pout−(RL1+RL2+Rconqv)qv)e2(p2z3

4+q2z24+o2z4+s2)2

∑i=1,4ei

(piz3i +qiz2

i +oizi+si)2

×

[

(a′1z4eb′1z41.1− c′1ed′1z4)−C2× tanh(Kz5)

]]

(31)

z6 =(V2−R2z6)(C12+C22(gm2− r2z4))

N22

−

r2C22z5z6

(C12+C22(gm2− r2z4))(32)

where,bd indicates the equivalent torsional damping,Kt is theequivalent torsional stiffness,V stands for the supply voltage,xis the plunger displacement,r indicates the radius of the pinion,C1 andC2 are the reluctances of the magnetic path without airgap and that of the air gap, respectively,Fm is the motive force,N stands for the number of coils,i indicates the applied current,gm is the nominal airgap,J indicates the polar moment of inertiaof the valve’s disk, andR is the electrical resistance of coil.

5 Copyright c© 2016 by ASME

3 Nonlinear AnalysisThe linearization method is one of the immediate tools to

be used in determining the stability of the network around mul-tiple equilibria. The analytical studies of such a six-state sys-tem would be tedious and time consuming, in particular, in thepresence of enormous parameters and variables. The numericalmethod is an optimal approach to calculate the Jacobian matrixshown in Eq. 33 and subsequently the system’s eigenvalues tojudge the stability around the equilibria. We select two impor-tant critical parameters of the equivalent viscous damping(bdi)and the friction coefficient of bearing area (µi) of both the setsto evaluate their effects on the stability/instability of the coupledsets:

J =

0 1 0 0 0 01732334 j22 26 −4772 0 0

0 −1.95 −43.17 0 0 00 0 0 0 1 0

−1175.93 0 0 1957044j55 00 0 0 −84.26 0 −43.17

(33)

where, j22 = −96bd1 − 177.13µ1 and j55 = −96.15bd2 −

323.71µ2. We then obtain the following characteristic equation

00.1

0.20.3

00.1

0.20.3

−30

−20

−10

0

10

λi

beq2Nms

radbeq1

Nms

rad

FIGURE 3. The interconnected sets’ stability map; red and bluecrosses stand for unstable and stable domains, respectively.

based onbdi andµi :

s6+Co1s5+Co2s4+Co3s3+Co4s2+Co5s+Co6 = 0 (34)

0 5 10 15 20 25−80

−60

−40

−20

0

20

40

60

80

αi(degree)

αi

(

rad s

)

α1

α2

(a)

0 5 10 15 20 25 30−150

−100

−50

0

50

100

150

αi(degree)

αi

(

rad s

)

α1

α2

(b)

FIGURE 4. (a) The coupled sets’ phase portraits for Initial1; (b) Thecoupled sets’ phase portraits for Initial2

where,

Co1 = 96bd1+96bd2+177µ1+323µ2+86 (35)

Co2 = 8302bd1+8302bd2+15295µ1+27951µ2

+ 9245bd1bd2+31125bd1µ2+17032bd2µ1

+ 57340µ1µ2−3687463 (36)

Co3 = 187998083bd1+166386463bd2+346333714µ1

+ 560154376µ2−798323bd1bd2−2687624bd1µ2

− 1470686bd2µ1−4951194µ1µ2+318563300 (37)

Co4 = −16248483272bd1−14382603305bd2

− 29933270986µ1−48420274134µ2+17233142bd1bd2

+ 58016860bdµ2+31747229bd2µ1+106879785µ1µ2+

6 Copyright c© 2016 by ASME

0 1000 2000 3000 4000 5000 6000−0.1

−0.05

0

0.05

0.1

0.15

0.2

Time (s)

Lj

L1=−0.0072

L2=−0.0102

L3=−0.0101

L4=−0.0113

L5=−21.569

L6=+0.1535

(a)

12

30 35 40 45 50 55 60 65 70 75 80 85 90

0

0.1

0.2

0.3

0.4

θ

Lj

(b)

FIGURE 5. (a) The Lyapunov exponents for Initial1; (b) The positiveLyapunov exponents for Initial2 vs. different approach angles (θ )

3383271986600 (38)

Co5 = 350750592245bd1+310477025305bd2

+ 646159543021µ1+1045247675853µ2

− 292732345689222 (39)

Co6 = 6319208419510455 (40)

Using the numerical approach, the coupled sets’ normalizedeigenvalues are presented in Fig. 3 revealing an interesting sta-bility map by assuming thatbdis’ andµis’ change equally for acritical range of 10−8 ≤ bdi = µi ≤ 3×10−1.

Shown in Fig. 3 reveals instability and stability of the cou-pled sets for the ranges of 10−8 ≤ bdi = µi ≤ 9× 10−2 and10−1 ≤ bdi = µi ≤ 3× 10−1 by presenting positive and nega-

47

48

49

50

51

52

53

54

α1

α1

(a)

61.5

62

62.5

63

63.5

64

64.5

α2

α2

(b)

FIGURE 6. (a) The poincare map for Initial1 of the upstream set; (b)The poincare map for Initial1 of the downstream set

tive real parts of eigenvalues, respectively. Such a stability mapwould help us select some critical values in order to captureprac-tically observed chaotic and hyperchaotic dynamics.

4 ResultsWe select two sets of initial conditions for the critical values

of bdi = µi = 10−5, based on the stability map shown in Fig. 3,as follows.

Initial1 = [20(deg) 0 0 20(deg) 0 0]

Initial2 = [2(deg) 0 0 2(deg) 0 0]

7 Copyright c© 2016 by ASME

Chaotic motions are known to be sensitive to even slight changesof initial conditions and hence examining different initial con-ditions potentially serve to characterize the responses obtained.The first set, Initial1, would be a realistic option for the so-called“modulating” valves to regulate/reroute flow for many applica-tions addressed earlier. The second set, Initial2, is chosen tobe close enough to the system’s physically feasible equilibriumpoint and also to avoid numerical singularities.

Shown in Figs. 4(a) and 4(b) are the phase portraits of thecoupled valves for two sets of the initial conditions. Figures4(a) and 4(b) reveal coupled chaotic and hyperchaotic dynam-ics. The hyperchaotic attractors by having two or more positivelyapunov exponents [34] are also known to be sensitive to ini-tial conditions and subsequently orbits initiated from twoclosepoints move expectedly away from each other until the separa-tion reaches the size of attractor. Some power tools of thenonlinear dynamic analysis would potentially help us character-ize the responses obtained, in particular, Lyapunov exponentsand Poincare maps shown in Figs. 5(a)-7(b). A chaotic attractorpresents one positive Lyapunov exponent [34], as shown in Fig.5(a) (L6 =+0.1535), indicating the chaotic motions of the inter-connected valves-actuators configuration. Figure 5(b) presentstwo positive Lyapunov exponents not only for the sudden pipecontraction (θ = 90) but also for a broad spectrum of the ap-proach angles (35 ≤ θ ≤ 85), which can be served as a proofof the network’s hyperchaotic dynamics.

Poincare map is another power tool to distinguish amongperiodic, quasiperiodic, and chaotic responses. Note thatfor ann-dimensional system (n≥3), this tool may not yield a clear na-ture of the response to determine whether the motion is chaoticor two-period quasiperiodic [34]. Although the Lyapunov ex-ponents, as discussed earlier, would firmly confirm the chaoticand hyperchaotic motions of the interconnected actuated valvesalong with irregular Poincare maps (almost different setsof 635points) shown in Figs. 6(a)-7(b) for each set of the upstreamanddownstream valves, and for two sets of the initial conditions.

Shown in Figs. 8(a) and 8(b) are the total flow loads, the sumof both the hydrodynamic and bearing torques, vs. the motiveforces for two initial conditions. The squared areas remarkablymagnify the differences between the chaotic and hyperchaoticresponses of the coupled sets by presenting relatively larger at-tractor sizes for the hyperchaotic ones.

One of the crucial issues which needs to be investigated isthe effects of dangerous behavior of a valve-actuator set onan-other one. Figures 9(a) and 9(b) present this interesting situationin which the upstream set is assumed to be chaotic by exposingto the critical values ofbd1 = µ1 = 10−5 and the downstream setis operated safely usingbd2 = µ2 = 10−1, for the second set ofinitial condition.

The hyperchaotic motion of the upstream valve is againexpected to be observed, but another smaller chaotic attractor,with a positive Lyapunov exponent of L1 =+0.013, surprisingly

51

51.5

52

52.5

α1

α1

(a)

64.4

64.5

64.6

64.7

64.8

64.9

65

α2

α2

(b)

FIGURE 7. (a) The Poincare map for Initial2 of the upstream set; (b)The poincare map for Initial2 of the downstream set

emerges (Fig. 9(a)) which is significantly different from the pre-vious case of Fig. 4(b). Its Poincare map shown in Fig. 10(a)confirms the chaotic dynamics of the upstream set containingir-regular points but expectedly different from the map presented inFig. 7(a). It is of great interest to observe a very weak chaoticmotion of the downstream valve as shown in Fig. 9(b) whereas astable response was logically expected to be seen. Figure 10(b)is the Poincare map of the downstream set revealing irregularpoints but too close to its equilibrium point. It is fairly straight-forward to conclude that the chaotic motion of the upstream setis transmitted to the downstream one through the media trappedbetween them and subsequently affects its dynamics. Increas-ing the chaotic attractor domain of a set would accordingly mag-nify the domains of neighbor ones which would gradually cause

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(a)

(b)

FIGURE 8. (a) The sum of flow loads vs. magnetic force of both theupstream and downstream sets for Initial1; (b) The sum of flow loads vs.magnetic force of both the upstream and downstream sets for Initial2.The red and blue lines stand for the upstream and downstream sets, re-spectively.

failure of the whole network of thousands of valves-actuators.Such failures have to be avoided to reduce the considerable costneeded to restore the flow line.

5 ConclusionsThis paper represented an interconnected nonlinear model

of two actuators and valves subject to the sudden contraction.These dependencies among different components were formal-ized to yield a sixth-order dynamic model of the whole system.We established a stability map to yield a clear picture of stabil-ity/instability of the coupled network for some critical values of

0 10 20 30 40−100

−50

0

50

100

α1(degree)

α1

(

rad s

)

(a)

0 10 20 30 40

−100

−50

0

50

100

150

α2(degree)

α2

(

rad s

)

−0.02

0

0.02

(b)

FIGURE 9. (a) The phase portrait of the upstream set for Initial2; (b)The phase portrait of the downstream set for Initial2

the equivalent viscous damping and the friction coefficientof thebearing area.

The coupled chaotic and hyperchaotic dynamics were cap-tured and discussed. Some powerful tools of the nonlinear dy-namic analysis were then employed, including Lyapunov expo-nents and Poincare map, to characterize the responses obtained.We presented the expected larger hyperchaotic attractor domainsin comparison with the chaotic ones. One and two positive Lya-punov exponents were shown to confirm the chaotic and hyper-chaotic dynamics of the coupled actuated valves, respectively.The irregular Poincare maps were also presented to supportboththe chaotic and hyperchaotic dynamics along with the positiveLyapunov exponents.

The upstream valve-actuator set was intentionally operated

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1.1

1.105

1.11

1.115

1.12

1.125

1.13

α1

α1

(a)

−8

−6

−4

−2

0

2

4

6

8x 10

−5

α2

α2

(b)

FIGURE 10. (a) The poincare map of the upstream set for Initial2; (b)The poincare map of the downstream set for Initial2

with the same initial condition and critical values of the hyper-chaotic dynamics to evaluate its effects on a stable downstreamset. The dynamics of the upstream set was surprisingly differentby revealing a chaotic attractor demonstrated by its Poincare mapand a positive Lyapunov exponent. The downstream set was alsoaffected by the chaotic dynamics of the upstream one by showingthe irregular Poincare map but too close to its equilibriumpoint.

We are currently focusing our efforts on developing a com-prehensive model forn valves and actuators to be operated opti-mally and safely in series.

ACKNOWLEDGMENTThe experimental work of this research was supported by

Office of Naval Research Grant (N00014/2008/1/0435). We ap-preciate this grant and the advice and direction provided byMr.

Anthony Seman III, the program manager.

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