Chaotic and Hyperchaotic Dynamics of Smart Valves System Subject to a Sudden Contraction

Naseradinmousavi, Peiman; Segala, David B.; Nataraj, C.

doi:10.1115/1.4033610

In this paper, 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 critical applications and missions including municipal piping networks, oil and gas fields, petrochemical plants, and more importantly, the U.S. Navy ships. A comprehensive dynamic analysis is hence needed to be carried out for capturing dangerous behaviors observed repeatedly in practice. Using some powerful tools of nonlinear dynamic analysis including Lyapunov exponents and Poincaré map, a comprehensive stability map is provided in order to determine the safe operational domain of the network in addition to characterizing the responses obtained. Coupled chaotic and hyperchaotic dynamics of two coupled solenoid-actuated butterfly valves are captured by running the network for some critical values through interconnected flow loads affected by the coupled actuators' variables. The significant effect of an unstable configuration of the valve–actuator on another set is thoroughly investigated to discuss the expected stability issues of a remote set due to others and vice versa.

Introduction

Multidisciplinary electromechanical–fluid systems have been widely used in many megascale networks. Municipal piping systems, oil and gas fields, petrochemical plants, and more critically, the U.S. Navy are the immediate ones which need to utilize a reliable, safe, and efficient coupled flow distribution network.

Future smart cities would inevitably need an autonomous flow control network in order to help improve the safety of such a critical system and also to decrease the incremental costs of operation and maintenance. Malfunctions of the flow network have occurred repeatedly resulting in the flow interruption of small towns/districts. Although significant cost and energy would be needed to be spent in order to restore the whole system in addition to human resources required to be recruited. Economical and even social impact of these malfunctions can be expected to be dramatic and consequently, a fully automated flow network is needed to be designed and operated.

The same issues exist for oil and gas fields and petrochemical plants. The industry of oil and gas is one of the most sensitive elements of the global economy and plays important roles even in global politics. The flow control network is the essential part of these fields and is therefore required to be safely designed to minimize the flow interruptions leading to much higher oil/gas production.

The U.S. National Defense and Homeland Security is undoubtedly a highly important priority which requires to be addressed and investigated carefully. The U.S. Navy broadly employs the network of coupled electromechanical valve sets for cooling purposes, mainly for chilled water systems. The proper performance of the network is remarkably effective for other critical units of radar, sonar, defense systems, etc.

We have carried out broad analytical and experimental studies from nonlinear modeling to design optimization of both an isolated and interconnected symmetric butterfly valves driven by solenoid actuators [1–8]. The multidisciplinary couplings, including electromagnetics and fluid mechanics, had to be thoroughly regarded in the modeling phase in order to yield an accurate nonlinear model of such a complex system. A third-order nondimensional dynamic model of the single set was derived to be used in nonlinear dynamic analysis [3] and optimal design [4].

The dynamic analysis expectedly yielded practically observed crisis and transient chaotic dynamics of a single actuated valve for some critical physical parameters. A comprehensive stability map was also presented as an efficient tool to determine the safe domain of operation which in turn could serve for identifying the lower and upper bounds of the design optimization efforts. The design optimization was then carried out [4] to select the optimal actuation unit's parameters coupled with the mechanical and fluid parts in order to significantly reduce the amount of energy consumption (upward of 40%).

Note that the applications addressed earlier contain thousands of actuated valves in which a high level of dynamic coupling has been repeatedly observed in practice. These dynamic couplings among different sets need to be captured through analytical studies. We have developed [5] a novel nonlinear model for two sets of solenoid-actuated butterfly valves operating in series. The closing/opening valves were modeled as changing resistors and the flow between them as a constant one. A sixth-order nonlinear coupled model revealed the high dynamic sensitivity of each element of a set, the valve and the actuator, to another one and vice versa. The power spectrum was used in confirming the same frequency response of a neighbor set due to the external periodic noise applied on another set of the valve and actuator.

In further studies, we optimized the design of coupled actuation units of two sets operating in series [8] subject to a sudden contraction. The pipe contraction imposed an additional resistance to be modeled, and therefore, the coupled dynamic equations derived in Ref. [5] had to be slightly modified. We represent the modeling process here for completeness. We have surprisingly established an interesting coupling between currents of the actuation units through the interconnected flow loads, including hydrodynamic and bearing torques, which affect both the valves' dynamics.

Important nonlinear phenomena in electromechanical systems have also received considerable attention. Sun et al. [9] studied the hyperchaotic behavior of the newly presented simplified Lorenz system by using a sinusoidal parameter variation and hyperchaos control of the forced system via feedback. Banerjee et al. [10] investigated the synchronization of chaos and hyperchaos in first-order time-delayed systems that are coupled using the nonlinear time-delay excitatory coupling by assigning two characteristic time delays: the system delay that is same for both the systems and the coupling delay associated with the coupling path. Many efforts for analyzing hyperchaotic dynamics have been reported in Refs. [11–26].

We have addressed the modeling process subject to the pipe contraction in Ref. [8] and represent here for completeness. The contribution of this work is the inclusion of interconnected electromechanical–fluid nonlinearities between two actuator–valve configurations to thoroughly analyze the effects of an unstable set of the valve–actuator on another one. Through this comprehensive analysis, chaotic and hyperchaotic dynamics of two coupled configurations are captured by exposing the network to some critical values. The responses are then characterized using some powerful tools including Lyapunov exponents and Poincaré map.

Mathematical Modeling

Figure 1(a) shows a pair of symmetric butterfly valves driven by solenoid actuators through rack and pinion arrangements. The rack and pinion mechanism provides a kinematic constraint which connects the dynamics of the valve and actuator. Applying DC voltages, as being used in the Navy ships for chilled water systems, the motive forces give translational motions to the actuators' moving parts (plungers), and subsequently, the valves rotate to desirable angles. Note that a return spring has been a common practice among industries to open the valves.

Fig. 1

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(a) A schematic configuration of two solenoid-actuated butterfly valves subject to sudden contraction and (b) a coupled model of two butterfly valves in series without actuation

Interconnected modeling of such a multiphysics system undoubtedly needs some simplifying assumptions to neglect useless and tremendously time-consuming numerical calculations. The magnetic force resulted from the magnetic field needs an extremely short period of time to reach its maximum value. This period is the so-called “diffusion time” and has an inverse relationship with the amount of current used. Note that using current of 4 A would yield a negligible diffusion time of $τ_{d} \approx 20 ms$ [1] with respect to the nominal operation time of 40 s. We have to also assume dominant laminar flow for both the coupled valves. Note that developing an analytical model is a necessity to carry out the dynamic analysis and optimization which would lead us to make such a commonly used assumption and also to avoid the numerical difficulties involved with a turbulent regime. However, a crucial question needs to be carefully answered with respect to the validity of such an assumption. Using the values of pipe diameter and flow mean velocity listed in Table 1, one can easily distinguish the existence of the turbulent regime which invalidates the assumption we have made. From another aspect, the analytical formulas derived for the flow loads, including the hydrodynamic and bearing torques, have been developed based on the assumption of laminar flow [27,28]. To address the issues discussed above, we have carried out experimental work to measure the sum of the hydrodynamic and bearing torques as the most affecting loads on the valves' and subsequently actuators' dynamics [8]. The experiment yielded the total torque (Fig. 2) for the inlet velocity of $v \approx 2.7 m / s$ and valve diameter of $D_{v} = 2 in .$ validating the laminar flow assumption [29]. The flow torques have shown highly important roles for the dynamics of an isolated solenoid-actuated butterfly valve and we hence expect to observe such effects for the interconnected sets [5].

Fig. 2

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A comparison between the experimental and analytical total torques

Table 1

The system parameters

Ρ

1000 kg/m³

v

0.1 m/s

J_{1, 2}

0.104 \times 10^{- 1}

kg m²

N₂

3000

N₁

3000

C_{11, 22}

1.56 × 10⁶H⁻¹

g_{m 1, m 2}

0.1 m

V_{1, 2}

24 V

D_{v 1}

0.2032 m

D_{v 2}

0.127 m

D_{s 1, s 2}

0.01 m

P_out

2 kPa

k_{1, 2}

1000 N·m⁻¹

C_{21, 22}

6.32 × 10⁸H⁻¹

L₁

2 m

L₂

1 m

μ_f

0.018 kg m⁻¹ s⁻¹

R_{1, 2}

6 Ω

r_{1, 2}

0.05 m

θ

90 deg

P_in

256 kPa

The coupled system is modeled as a set of five resistors. Two changing resistors represent the closing/opening valves, two constant ones indicate head losses between the valves, and another is due to the pipe contraction as shown in Fig. 1(b). The inlet and outlet pressure values are given in Table 1. Using the assumption of the dominant laminar flow, the pressure drops between two valves can be expressed based on the Hagen–Poiseuille [30] and Borda–Carnot [31] formulas (points 1 and 2)

P_{1} - P_{con 1} = \underset{R_{L 1}}{\underset{︸}{\frac{128 μ_{f} L_{1}}{π D_{v 1}^{4}}}} q_{v}

(1)

P_{con 1} - P_{con 2} = \frac{1}{2} K_{con} ρ v_{out}^{2}

(2)

P_{con 2} - P_{2} = \underset{R_{L 2}}{\underset{︸}{\frac{128 μ_{f} L_{2}}{π D_{v 2}^{4}}}} q_{v}

(3)

where q_v is the volumetric flow rate, μ_f indicates the fluid dynamic viscosity,

D_{v 1}

and

D_{v 2}

are the valves' diameters, L₁ and L₂ stand for the pipe lengths before and after contraction,

R_{L 1}

and

R_{L 2}

indicate the constant resistances, and

P_{con 1}

and

P_{con 2}

are the flow pressures before and after contraction. K_con is calculated as follows:

K_{con} = 0.5 (1 - β^{2}) \sqrt{sin (\frac{θ}{2})}

(4)

where β indicates the ratio of minor and major diameters

(D_{v 2} / D_{v 1})

⁠, and θ is the angle of approach. The values listed in Table 1 easily yield K_con = 0.2562. We then rewrite Eq. as follows:

\begin{matrix} P_{con 1} - P_{con 2} = \frac{1}{2} K_{con} ρ v_{out}^{2} \\ = \underset{R_{con}}{\underset{︸}{\frac{8 K_{con}}{π^{2} D_{v 2}^{4}} ρ}} \underset{q_{v}^{2}}{\underset{︸}{\frac{π^{2} D_{v 2}^{4} v_{out}^{2}}{16}}} = R_{con} q_{v}^{2} \end{matrix}

(5)

where R_con is the resistance due to the pipe contraction. The pressure drop between the valves can be derived by adding Eqs. – and

P_{1} - P_{2} = [R_{L 1} + R_{L 2} + R_{con} q_{v}] q_{v}

(6)

The valve's “resistance (R)” and “coefficient (c_v),” as important parameters of the regulating valves, are nonlinear functions of the valve rotation angle to be stated [32] as follows:

R_{i} (α_{i}) = \frac{891 D_{v i}^{4}}{c_{v i}^{2} (α_{i})}, i = 1, 2

(7)

Based on the assumption of laminar flow, the valve's pressure drop is calculated via the following relationship [27]:

Δ P_{i} (α_{i}) = 0.5 R_{i} (α_{i}) ρ v^{2}

(8)

where α indicates the valve rotation angle, ρ is the density of the media, and v stands for the flow velocity. Rewriting Eq. in the standard form gives

Δ P_{i} (α_{i}) = \underset{q_{v}^{2}}{\underset{︸}{\frac{π^{2} D_{v i}^{4} v^{2}}{16}}} \underset{R_{n i} (α_{i})}{\underset{︸}{\frac{8 \times R_{i} (α_{i}) ρ}{π^{2} D_{v i}^{4}}}} = R_{n i} (α_{i}) q_{v}^{2}

(9)

The hydrodynamic (T_h) and bearing (T_b) torques [27,28] have expectedly shown the high sensitivity to the pressure drop obtained via Eq. leading us to rewrite them as follows:

f_{i} (α_{i}) = \frac{16 T_{c i} (α_{i})}{3 π {(1 - \frac{C_{c c i} (α_{i}) (1 - sin (α_{i}))}{2})}^{2}}

(10)

T_{h i} = \frac{16 T_{c i} (α_{i}) D_{v i}^{3} Δ P_{i}}{3 π {(1 - \frac{C_{c c i} (α_{i}) (1 - sin (α_{i}))}{2})}^{2}} = f_{i} (α_{i}) D_{v i}^{3} Δ P_{i}

(11)

T_{b i} = 0.5 A_{d} Δ P_{i} μ D_{s} = C_{i} Δ P_{i}

(12)

where D_s stands for the stem diameter of the valve, μ indicates the friction coefficient of the bearing area, $C_{i} = (π / 8) μ D_{v i}^{2} D_{s}$ ⁠, and T_ci and C_cci are the hydrodynamic torque and the sum of upper and lower contraction coefficients, respectively, depending on the valve rotation angle [1].

The comprehensive stability map we have presented in Ref. [3] was based on a nonlinear analytical model. The analytical model had to be used in the dynamic analysis to investigate the system stability around equilibria by calculating its eigenvalues through the Jacobian matrix; this has led us to identify the safe operational domain to be utilized in the design optimization. The same practice was employed in Ref. [8] with the aid of fitting suitable curves on c_vi and R_ni in order to model the system analytically. For our case study of

D_{v 1}

= 8 in. and

D_{v 2}

= 5 in., the valves' coefficients and resistances are developed as follows:

c_{v 1} (α_{1}) = p_{1} α_{1}^{3} + q_{1} α_{1}^{2} + o_{1} α_{1} + s_{1}

(13)

c_{v 2} (α_{2}) = p_{2} α_{2}^{3} + q_{2} α_{2}^{2} + o_{2} α_{2} + s_{2}

(14)

R_{n 1} (α_{1}) = \frac{e_{1}}{{(p_{1} α_{1}^{3} + q_{1} α_{1}^{2} + o_{1} α_{1} + s_{1})}^{2}}

(15)

R_{n 2} (α_{2}) = \frac{e_{2}}{{(p_{2} α_{2}^{3} + q_{2} α_{2}^{2} + o_{2} α_{2} + s_{2})}^{2}}

(16)

where

e_{1} = 7.2 \times 10^{5}, p_{1} = 461.9, q_{1} = - 405.4, o_{1} = - 1831, s_{1} = 2207, e_{2} = 4.51 \times 10^{5}, p_{2} = 161.84, q_{2} = - 110.53, o_{2} = - 695.1

⁠, and

s_{2} = 807.57

⁠. These fittings were selected with respect to the decremental and incremental profiles of the valves' coefficients and resistances, respectively [5,29]. Applying the mass continuity principle (⁠

q_{in} = q_{out} = q_{v}

⁠) and then rewriting Eq. yield

\frac{P_{in} - P_{1}}{R_{n 1} (α_{1})} = \frac{P_{2} - P_{out}}{R_{n 2} (α_{2})}

(17)

R_{n 1} P_{2} + R_{n 2} P_{1} = R_{n 2} P_{in} + R_{n 1} P_{out}

(18)

The interconnected P₁ and P₂ terms are derived by combining Eqs. and as follows:

P_{1} = \frac{R_{n 2} P_{in} + R_{n 1} P_{out} + R_{n 1} (R_{L 1} + R_{L 2} + R_{con} q_{v}) q_{v}}{(R_{n 1} + R_{n 2})}

(19)

P_{2} = \frac{R_{n 2} P_{in} + R_{n 1} P_{out} - R_{n 2} (R_{L 1} + R_{L 2} + R_{con} q_{v}) q_{v}}{(R_{n 1} + R_{n 2})}

(20)

The dynamic sensitivities of P₁ and P₂ to

R_{n 1}, R_{n 2}, R_{L 1}, R_{L 2}

⁠, and R_con are distinguishable through Eqs. and , as observed in the practice. Any slight dynamic changes of the upstream set of the valve–actuator would be expected to be observed for the downstream one. The hydrodynamic and bearing torques' dependencies on all the resistances are reformulated as follows:

T_{h i} = f_{i} (α_{i}) D_{v i}^{3} Δ P_{i} (R_{n 1}, R_{n 2}, R_{L 1}, R_{L 2}, R_{con})

(21)

T_{b i} = C_{i} Δ P_{i} (R_{n 1}, R_{n 2}, R_{L 1}, R_{L 2}, R_{con})

(22)

f_i is a nonlinear function of the changing T_ci, C_cci, and the valve rotation angle. To carry out a systematic dynamic analysis, the following functions are fitted to the

D_{v i}^{3} f_{i}

of each valve [5,29]:

\begin{matrix} T_{h 1} = \underset{D_{v 1}^{3} f_{1}}{\underset{︸}{(a_{1} α_{1} e^{b_{1} {α_{1}}^{1.1}} - c_{1} e^{d_{1} α_{1}})}} (P_{in} - P_{1}) \\ = (a_{1} α_{1} e^{b_{1} {α_{1}}^{1.1}} - c_{1} e^{d_{1} α_{1}}) \times \frac{\frac{e_{1}}{{(p_{1} α_{1}^{3} + q_{1} α_{1}^{2} + o_{1} α_{1} + s_{1})}^{2}}}{\sum_{i = 1}^{2} \frac{e_{i}}{{(p_{i} α_{i}^{3} + q_{i} α_{i}^{2} + o_{i} α_{i} + s_{i})}^{2}}} \\ \times (P_{in} - P_{out} - (R_{L 1} + R_{L 2} + R_{con} q_{v}) q_{v}) \end{matrix}

(23)

\begin{matrix} T_{h 2} = \underset{D_{v 2}^{3} f_{2}}{\underset{︸}{(a_{1}^{'} α_{2} e^{b_{1}^{'} α_{2}^{1.1}} - c_{1}^{'} e^{d_{1}^{'} α_{2}})}} (P_{2} - P_{out}) \\ = (a_{1}^{'} α_{2} e^{b_{1}^{'} {α_{2}}^{1.1}} - c_{1}^{'} e^{d_{1}^{'} α_{2}}) \times \frac{\frac{e_{2}}{{(p_{2} α_{2}^{3} + q_{2} α_{2}^{2} + o_{2} α_{2} + s_{2})}^{2}}}{\sum_{i = 1}^{2} \frac{e_{i}}{{(p_{i} α_{i}^{3} + q_{i} α_{i}^{2} + o_{i} α_{i} + s_{i})}^{2}}} \\ \times (P_{in} - P_{out} - (R_{L 1} + R_{L 2} + R_{con} q_{v}) q_{v}) \end{matrix}

(24)

where $a_{1} = 0.4249, a_{1}^{'} = 0.1022, b_{1} = - 18.52, b_{1}^{'} = - 17.0795, c_{1} = - 7.823 \times 10^{- 4}, c_{1}^{'} = - 2 \times 10^{- 4}, d_{1} = - 1.084$ ⁠, and $d_{1}^{'} = - 1.0973$ ⁠.

We have previously derived the rate of current and magnetic force terms [1], which are utilized in developing the sixth-order coupled dynamic model [8] as follows. Note that both the motive force and current are highly sensitive to the plunger displacement and subsequently the valve rotation angle

F_{m i} = \frac{C_{2 i} N_{i}^{2} i_{i}^{2}}{2 {(C_{1 i} + C_{2 i} (g_{m i} - x_{i}))}^{2}}

(25)

\begin{matrix} \frac{d i_{i}}{d t} = \frac{(V_{i} - R_{i} i_{i}) (C_{1 i} + C_{2 i} (g_{m i} - x_{i}))}{N_{i}^{2}} \\ - \frac{C_{2 i} i_{i} {\dot{x}}_{i}}{(C_{1 i} + C_{2 i} (g_{m i} - x_{i}))} \end{matrix}

(26)

{\dot{z}}_{1} = z_{2}

(27)

\begin{matrix} {\dot{z}}_{2} = \frac{1}{J_{1}} [\frac{r_{1} C_{21} N_{1}^{2} z_{3}^{2}}{2 {(C_{11} + C_{21} (g_{m 1} - r_{1} z_{1}))}^{2}} - b_{d 1} z_{2} - k_{1} z_{1} \\ + \frac{\frac{(P_{in} - P_{out} - (R_{L 1} + R_{L 2} + R_{con} q_{v}) q_{v}) e_{1}}{{(p_{1} z_{1}^{3} + q_{1} z_{1}^{2} + o_{1} z_{1} + s_{1})}^{2}}}{\sum_{i = 1, 4} \frac{e_{i}}{{(p_{i} z_{i}^{3} + q_{i} z_{i}^{2} + o_{i} z_{i} + s_{i})}^{2}}} \times \\ [(a_{1} z_{1} e^{b_{1} {z_{1}}^{1.1}} - c_{1} e^{d_{1} z_{1}}) - C_{1} \times tan h (K z_{2})]] \end{matrix}

(28)

\begin{matrix} {\dot{z}}_{3} = \frac{(V_{1} - R_{1} z_{3}) (C_{11} + C_{21} (g_{m 1} - r_{1} z_{1}))}{N_{1}^{2}} - \\ \frac{r_{1} C_{21} z_{3} z_{2}}{(C_{11} + C_{21} (g_{m 1} - r_{1} z_{1}))} \end{matrix}

(29)

{\dot{z}}_{4} = z_{5}

(30)

\begin{matrix} {\dot{z}}_{5} = \frac{1}{J_{2}} [\frac{r_{2} C_{22} N_{2}^{2} z_{6}^{2}}{2 {(C_{12} + C_{22} (g_{m 2} - r_{2} z_{4}))}^{2}} - b_{d 2} z_{5} - k_{2} z_{4} + \frac{\frac{(P_{in} - P_{out} - (R_{L 1} + R_{L 2} + R_{con} q_{v}) q_{v}) e_{2}}{{(p_{2} z_{4}^{3} + q_{2} z_{4}^{2} + o_{2} z_{4} + s_{2})}^{2}}}{\sum_{i = 1, 4} \frac{e_{i}}{{(p_{i} z_{i}^{3} + q_{i} z_{i}^{2} + o_{i} z_{i} + s_{i})}^{2}}} \times [(a_{1}^{'} z_{4} e^{b_{1}^{'} {z_{4}}^{1.1}} - c_{1}^{'} e^{d_{1}^{'} z_{4}}) - C_{2} \times tan h (K z_{5})]] \end{matrix}

(31)

\begin{matrix} {\dot{z}}_{6} = \frac{(V_{2} - R_{2} z_{6}) (C_{12} + C_{22} (g_{m 2} - r_{2} z_{4}))}{N_{2}^{2}} - \\ \frac{r_{2} C_{22} z_{5} z_{6}}{(C_{12} + C_{22} (g_{m 2} - r_{2} z_{4}))} \end{matrix}

(32)

where b_d indicates the equivalent torsional damping, K_t is the equivalent torsional stiffness, V stands for the supply voltage, x is the plunger displacement, r indicates the radius of the pinion, C₁ and C₂ are the reluctances of the magnetic path without air gap and that of the air gap, respectively, F_m is the motive force, N stands for the number of coils, i indicates the applied current, g_m is the nominal airgap, J indicates the polar moment of inertia of the valve's disk, and R is the electrical resistance of coil.

Linear Dynamic Analysis

The linearization method is one of the immediate tools to be used in determining the stability of the network around multiple equilibria. The analytical studies of such a six-state system would be tedious and time-consuming, in particular, in the presence of enormous parameters and variables. The numerical method is an optimal approach to calculate the Jacobian matrix shown in Eq. and subsequently the system’s eigenvalues to judge the stability around the equilibria. We select two important critical parameters of the equivalent viscous damping (b_di) and the friction coefficient of bearing area (μ_i) of both the sets to evaluate their effects on the stability/instability of the coupled sets

J = [\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 1732334 & j_{22} & 26 & - 4772 & 0 & 0 \\ 0 & - 1.95 & - 43.17 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ - 1175.93 & 0 & 0 & 1957044 & j_{55} & 0 \\ 0 & 0 & 0 & - 84.26 & 0 & - 43.17 \end{matrix}]

(33)

where $j_{22} = - 96 b_{d 1} - 177.13 μ_{1}$ and $j_{55} = - 96.15 b_{d 2} - 323.71 μ_{2}$ ⁠.

We then obtain the following characteristic equation based on b_di and μ_i:

s^{6} + C o_{1} s^{5} + C o_{2} s^{4} + C o_{3} s^{3} + C o_{4} s^{2} + C o_{5} s + C o_{6} = 0

(34)

where

{Co}_{1} = 96 b_{d 1} + 96 b_{d 2} + 177 μ_{1} + 323 μ_{2} + 86

(35)

\begin{matrix} {Co}_{2} = 8302 b_{d 1} + 8302 b_{d 2} + 15295 μ_{1} + 27951 μ_{2} \\ + 9245 b_{d 1} b_{d 2} + 31125 b_{d 1} μ_{2} + 17032 b_{d 2} μ_{1} \\ + 57340 μ_{1} μ_{2} - 3687463 \end{matrix}

(36)

\begin{matrix} {Co}_{3} = 187998083 b_{d 1} + 166386463 b_{d 2} + 346333714 μ_{1} \\ + 560154376 μ_{2} - 798323 b_{d 1} b_{d 2} - 2687624 b_{d 1} μ_{2} \\ - 1470686 b_{d 2} μ_{1} - 4951194 μ_{1} μ_{2} + 318563300 \end{matrix}

(37)

\begin{matrix} {Co}_{4} = - 16248483272 b_{d 1} - 14382603305 b_{d 2} \\ - 29933270986 μ_{1} - 48420274134 μ_{2} + 17233142 b_{d 1} b_{d 2} \\ + 58016860 b_{d} μ_{2} + 31747229 b_{d 2} μ_{1} + 106879785 μ_{1} μ_{2} \\ + 3383271986600 \end{matrix}

(38)

\begin{matrix} {Co}_{5} = 350750592245 b_{d 1} + 310477025305 b_{d 2} \\ + 646159543021 μ_{1} + 1045247675853 μ_{2} \\ - 292732345689222 \end{matrix}

(39)

{Co}_{6} = 6319208419510455

(40)

Using the numerical approach, the coupled sets' normalized eigenvalues are presented in Fig. 3 revealing an interesting stability map by assuming that b_di's and μ_i's change equally for a critical range of $10^{- 8} \leq b_{d i} = μ_{i} \leq 3 \times 10^{- 1}$ ⁠.

Fig. 3

The interconnected sets' stability map; 10−8≤bdi=μi≤9×10−2 and 10−1≤bdi=μi≤3×10−1 stand for unstable and stable domains, respectively

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The interconnected sets' stability map; $10^{- 8} \leq b_{di} = μ_{i} \leq 9 \times 10^{- 2}$ and $10^{- 1} \leq b_{di} = μ_{i} \leq 3 \times 10^{- 1}$ stand for unstable and stable domains, respectively

Figure 3 shows the instability and stability of the coupled sets for the ranges of $10^{- 8} \leq b_{di} = μ_{i} \leq 9 \times 10^{- 2}$ and $10^{- 1} \leq b_{di} = μ_{i} \leq 3 \times 10^{- 1}$ by presenting positive and negative real parts of eigenvalues, respectively. Such a stability map would help us select some critical values in order to capture practically observed chaotic and hyperchaotic dynamics.

Results

We select two sets of initial conditions for the critical values of

b_{d i} = μ_{i} = 10^{- 7}

⁠, based on the stability map shown in Fig. 3, as follows:

{Initial}_{1} = [20 (deg) 0 0 20 (deg) 0 0]

{Initial}_{2} = [2 (deg) 0 0 2 (deg) 0 0]

Chaotic motions are known to be sensitive to even slight changes of initial conditions and hence examining different initial conditions 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 applications addressed earlier. The second set, Initial₂, is chosen to be close enough to the system's physically feasible equilibrium point and also to avoid numerical singularities.

Figures 4(a) and 4(b) show the phase portraits of the coupled valves for two sets of the initial conditions. Figures 4(a) and 4(b) reveal coupled chaotic and hyperchaotic dynamics. The hyperchaotic attractors by having two or more positive lyapunov exponents [33] are also known to be sensitive to initial conditions, and subsequently, orbits initiated from two close points move expectedly away from each other until the separation reaches the size of attractor. Some power tools of the nonlinear dynamic analysis would potentially help us characterize the responses obtained, in particular, Lyapunov exponents and Poincaré maps shown in Figs. 5(a)–7(b). A chaotic attractor presents one positive Lyapunov exponent [33], as shown in Fig. 5(a) (⁠ $L_{6} = + 0.1535$ ⁠), indicating the chaotic motions of the interconnected valves–actuators configuration. Figure 5(b) presents two positive Lyapunov exponents not only for the sudden pipe contraction (⁠ $θ = 90 \deg$ ⁠) but also for a broad spectrum of the approach angles (⁠ $35 \deg \leq θ \leq 85 \deg$ ⁠), which can be served as a proof of the network's hyperchaotic dynamics. Poincaré map is another power tool to distinguish among periodic, quasi-periodic, and chaotic responses. Note that for an n-dimensional system (⁠ $n \geq 3$ ⁠), this tool may not yield a clear nature of the response to determine whether the motion is chaotic or two-period quasi-periodic [33]. Although the Lyapunov exponents, as discussed earlier, would firmly confirm the chaotic and hyperchaotic motions of the interconnected actuated valves along with irregular Poincaré maps (almost different sets of 635 points) as shown in Figs. 6(a)–7(b) for each set of the upstream and downstream valves and for two sets of the initial conditions.

Fig. 4

(a) The coupled sets' phase portraits for Initial1 and (b) the coupled sets' phase portraits for Initial2

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(a) The coupled sets' phase portraits for Initial₁ and (b) the coupled sets' phase portraits for Initial₂

Fig. 5

(a) The Lyapunov exponents for Initial1 and (b) the positive Lyapunov exponents for Initial2 versus different approach angles (θ)

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(a) The Lyapunov exponents for Initial₁ and (b) the positive Lyapunov exponents for Initial₂ versus different approach angles (θ)

Fig. 6

(a) The poincaré map for Initial1 of the upstream set and (b) the poincaré map for Initial1 of the downstream set

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(a) The poincaré map for Initial₁ of the upstream set and (b) the poincaré map for Initial₁ of the downstream set

Fig. 7

(a) The Poincaré map for Initial2 of the upstream set and (b) the Poincaré map for Initial2 of the downstream set

View large Download slide

(a) The Poincaré map for Initial₂ of the upstream set and (b) the Poincaré map for Initial₂ of the downstream set

Figures 8(a) and 8(b) show the total flow loads, the sum of both the hydrodynamic and bearing torques versus the motive forces for two initial conditions. The squared areas remarkably magnify the differences between the chaotic and hyperchaotic responses of the coupled sets by presenting relatively larger attractor sizes for the hyperchaotic ones.

Fig. 8

(a) The sum of flow loads versus magnetic force of both the upstream and downstream sets for Initial1 and (b) the sum of flow loads versus magnetic force of both the upstream and downstream sets for Initial2

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(a) The sum of flow loads versus magnetic force of both the upstream and downstream sets for Initial₁ and (b) the sum of flow loads versus magnetic force of both the upstream and downstream sets for Initial₂

It is also of a great interest to present the transition from chaos to hyperchaos, in particular for an interconnected system. Figures 9, 10(a), and 10(b) reveal such a transition. Figure 9 shows a broad spectrum of Lyapunov exponents versus the equivalent b_eqi's and μ_i's (as discussed in Sec. 3), which explicitly separates the chaotic and hyperchaotic domains. The chaotic domain, for the range of $2 \times 10^{- 4} \leq b_{d i} = μ_{i} \leq 10^{- 3}$ by presenting one positive Lyapunov exponent, is transmitted to the hyperchaotic one for the range of $10^{- 7} \leq b_{d i} = μ_{i} < 2 \times 10^{- 4}$ ⁠, which presents two positive Lyapunov exponents. As discussed earlier, the attraction domain of a hyperchaotic response is relatively larger than that of the chaotic one, as can be observed in Figs. 10(a) and 10(b). Figures 10(a) and 10(b) present bifurcation diagrams versus the equivalent b_eqi's and μ_i's for the upstream and downstream valves, respectively. The larger domains of attractions of the hyperchaotic responses, for both the valves, are distinguishable with respect to the chaotic ones. The hyperchaotic and chaotic domains of the downstream set are expectedly larger than those of the upstream set.

Fig. 9

View large Download slide

A broad spectrum of Lyapunov exponents versus the equivalent b_eqi's and μ_i's revealing transition from chaos to hyperchaos

Fig. 10

View large Download slide

(a) The bifurcation diagram versus the equivalent b_eqi's and μ_i's for the upstream valve revealing transition from chaos to hyperchaos and (b) the bifurcation diagram versus the equivalent b_eqi's and μ_i's for the downstream valve revealing transition from chaos to hyperchaos

One of the crucial issues which needs to be investigated is the effects of dangerous behavior of a valve–actuator set on another one. Figures 11(a) and 11(b) present this interesting situation in which the upstream set is assumed to be chaotic by exposing to the critical values of $b_{d 1} = μ_{1} = 10^{- 7}$ ⁠, and the downstream set is operated safely using $b_{d 2} = μ_{2} = 10^{- 1}$ ⁠, for the second set of initial condition.

Fig. 11

(a) The phase portrait of the upstream set for Initial2 and (b) the phase portrait of the downstream set for Initial2

View large Download slide

(a) The phase portrait of the upstream set for Initial₂ and (b) the phase portrait of the downstream set for Initial₂

The hyperchaotic motion of the upstream valve is again expected to be observed, but another smaller chaotic attractor, with a positive Lyapunov exponent of $L_{1} = + 0.013$ ⁠, surprisingly emerges (Fig. 11(a)) which is significantly different from the previous case of Fig. 4(b). Its Poincaré map shown in Fig. 12(a) confirms the chaotic dynamics of the upstream set containing irregular points but expectedly different from the map presented in Fig. 7(a). It is of great interest to observe a very weak chaotic motion of the downstream valve as shown in Fig. 11(b), whereas a stable response was logically expected to be seen. Figure 12(b) is the Poincaré map of the downstream set revealing irregular points but too close to its equilibrium point. It is fairly straightforward to conclude that the chaotic motion of the upstream set is transmitted to the downstream one through the media trapped between them and subsequently affects its dynamics. Increasing the chaotic attractor domain of a set would accordingly magnify the domains of neighbor ones which would gradually cause failure of the whole network of thousands of valves–actuators. Such failures have to be avoided to reduce the considerable cost needed to restore the flow line.

Fig. 12

(a) The poincaré map of the upstream set for Initial2 and (b) the poincaré map of the downstream set for Initial2

View large Download slide

(a) The poincaré map of the upstream set for Initial₂ and (b) the poincaré map of the downstream set for Initial₂

Conclusions

This paper represented an interconnected nonlinear model of two actuators and valves subject to the sudden contraction. These dependencies among different components were formalized to yield a sixth-order dynamic model of the whole system. We established a stability map to yield a clear picture of stability/instability of the coupled network for some critical values of the equivalent viscous damping and the friction coefficient of the bearing area.

The coupled chaotic and hyperchaotic dynamics were captured and discussed. Some powerful tools of the nonlinear dynamic analysis were then employed, including Lyapunov exponents and Poincaré map, to characterize the responses obtained. We presented the expected larger hyperchaotic attractor domains in comparison with the chaotic ones. One and two positive Lyapunov exponents were shown to confirm the chaotic and hyperchaotic dynamics of the coupled actuated valves, respectively. The irregular Poincaré maps were also presented to support both the chaotic and hyperchaotic dynamics along with the positive Lyapunov exponents.

The upstream valve–actuator set was intentionally operated with the same initial condition and critical values of the hyperchaotic dynamics to evaluate its effects on a stable downstream set. The dynamics of the upstream set was surprisingly different by revealing a chaotic attractor demonstrated by its Poincaré map and a positive Lyapunov exponent. The downstream set was also affected by the chaotic dynamics of the upstream one by showing the irregular Poincaré map but too close to its equilibrium point.

We are currently focusing our efforts on developing a comprehensive model for n valves and actuators to be operated optimally and safely in series.

Acknowledgment

The experimental work of this research was supported by the Office of Naval Research Grant No. N00014/2008/1/0435. We appreciate this grant and the advice and direction provided by Mr. Anthony Seman III, the program manager.

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Chaotic and Hyperchaotic Dynamics of Smart Valves System Subject to a Sudden Contraction

Introduction

Mathematical Modeling

Linear Dynamic Analysis

Results

Conclusions

Acknowledgment

References

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Ρ	1000 kg/m³	v	0.1 m/s
$J_{1, 2}$	$0.104 \times 10^{- 1}$ kg m²	N₂	3000
N₁	3000	$C_{11, 22}$	1.56 × 10⁶H⁻¹
$g_{m 1, m 2}$	0.1 m	$V_{1, 2}$	24 V
$D_{v 1}$	0.2032 m	$D_{v 2}$	0.127 m
$D_{s 1, s 2}$	0.01 m	P_out	2 kPa
$k_{1, 2}$	1000 N·m⁻¹	$C_{21, 22}$	6.32 × 10⁸H⁻¹
L₁	2 m	L₂	1 m
μ_f	0.018 kg m⁻¹ s⁻¹	$R_{1, 2}$	6 Ω
$r_{1, 2}$	0.05 m	θ	90 deg
P_in	256 kPa

Chaotic and Hyperchaotic Dynamics of Smart Valves System Subject to a Sudden Contraction

Introduction

Mathematical Modeling

Linear Dynamic Analysis

Results

Conclusions

Acknowledgment

References

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