Chapter 10

Application of Chaotic Motion

There are many possible applications of chaotic motion, including both industrial equipment and domestic appliances. Some industrial equipment (such as compactors, mixers, and grinders) and some domestic appliances (such as clothes-washers, dishwashers, ventilating fans, heaters, and air-conditioners) have successfully employed chaotic motion to significantly improve their performance.

In this chapter, various promising applications of chaotic motion, including compaction, mixing, washing, HVAC, and grinding, are unveiled and elaborated.

10.1 Chaotic Compaction

Compaction is one of the most important operations in civil and geotechnical engineering. In recent years, being fueled by the advancement of mechatronic design and control strategies, there have been tremendous developments in compaction technology. Modern compaction equipment should be not only powerful, economical, and versatile, but also environmental friendly.

A compactor functions to provide a vibratory compressive stress onto soil, granulates, or powders so that the required densities can be attained. This vibratory compressive stress is due to the centrifugal force of a rotating eccentric mass which can be driven by an electric motor or an internal combustion engine. It should be noted that the electric compactor takes the definite advantages of no smelly gas, quiet operation, and free maintenance over its engine-powered counterpart.

The compaction process is to induce the particles to mechanically vibrate in such a way that the internal frictions among particles, and hence the void content, can be reduced. Since the particles are usually of different sizes and shapes, a fixed frequency vibration is only a compromise to reduce the internal frictions. With the use of a variable frequency vibration, various internal frictions can be individually reduced, hence improving the compaction effectiveness. Because of the wideband spectral power density of chaos, it is anticipated that a chaotic vibration can offer a simultaneous reduction of various internal frictions, hence further improving the compaction effectiveness.

The application of chaos to compaction was initiated in a mechanical vibrator for compacting soft soil (Long, 2001). Figure 10.1 shows the configuration of this mechanically chaotic vibrator which consists of three rotating eccentric masses with mass centers , , and . The eccentric mass 1 is driven by an electric motor or an internal combustion engine at a constant rotational speed, while the eccentric masses 2 and 3 are pivoted on and , respectively. By using this three-mass mechanism for realistic soil compaction, the measured degree of compaction is found to be 12.2 per cent higher than that by the traditional method.

Instead of using a complicated three-mass mechanism to generate chaotic vibration for compaction, a chaotic drive system can perform the same task more easily. Figure 10.2 shows the structure of an electrically chaotic vibrator, which only needs a single eccentric mass driven by an electric motor. The electric motor can be a permanent magnet DC (PMDC) motor or a single-phase AC motor for application to a low-power vibratory plate compactor (Chau and Wang, 2005), whereas it may be a 3-phase AC motor for application to a high-power vibratory roller compactor (Chau and Wang, 2006).

Compared with the mechanically chaotic vibratory compactor, the electrically chaotic vibratory compactor offers the following advantages:

• The structure of this electric drive system is very simple and involves only one eccentric rotating mass.
• By using the electric drive, the generated chaotic vibration can be predefined and analytically formulated. Notice that the three-mass mechanism involves a very complicated interaction between three eccentric rotating masses, and the corresponding complex motion can only be determined using numerical simulation.
• The electrically chaotic vibrator can offer various chaotic vibrations that are suitable for compacting different kinds of particles such as soil, granulates, and powders. Notice that the mechanically chaotic vibrator cannot offer such flexibility or selectivity of chaotic vibrations for different kinds of particles.

The purpose of this section is to present an electrically chaotic compactor system. The key is to chaoize a PMDC drive system by using control means, namely the proportional time-delay (PTD) control (Wang and Chau, 2008) and the chaotic speed reference (CSR) control (Wang and Chau, 2009), which not only need no mechanical auxiliary but also provide a flexible control of chaotic motion. Simulation and experimental results are both used to illustrate the validity of this chaotic compactor.

10.1.1 Compactor System

Figure 10.3 shows the configuration of the vibratory compactor system where the motor shaft is mounted by a solid eccentric mass (Yoo and Selig, 1979). Compared with the mechanical design that uses three interlinked eccentric masses, this compactor has the advantage of simple structure. The dynamical model of this compactor can be expressed as:

(10.1)

(10.2)

(10.3)

(10.4)

(10.5)

(10.6)

where and are respectively the rotational angle and rotational speed; , , , , , and are respectively the feed-in voltage, armature current, torque constant, back EMF constant, armature inductance, and armature resistance of the PMDC motor; and are respectively the viscous damping coefficient and rotor inertia; and are respectively the weight and equivalent radius of the eccentric mass; is the gravitational constant; is the vertical displacement of the plate; is the total weight of the motor, the frame, and the plate; and and are respectively the stiffness and damping of the solid particles (Gethin et al., 2001).

10.1.2 Chaotic Compaction Control

Figure 10.4 shows the diagram of the PMDC drive system. The current control is realized by PI control, and the commanded feed-in voltage is realized by the DC–DC converter. In order to generate chaotic motion in the PMDC drive, there are two possible control methods, namely the PTD control and the CSR control.

As detailed in Chapter 7, the control law of the PTD control is designed as:

(10.7)

where is the current reference, is the time-delay gain, is the time-delay constant, and are proportional gains, and is the nominal rotational speed.

As detailed in Chapter 7, the CSR control is to force to follow a chaotic speed reference . The corresponding control law is represented as:

(10.8)

(10.9)

(10.10)

where is the base speed, is the speed boundary, and is the chaotic sequence generated by the well-known Logistic map with a control parameter .

The chaotic compactor is first compared with a conventional compactor. It adopts the constant feed-in voltage (CFV):

(10.11)

where is a constant voltage. Additionally, the chaotic compactor is also compared with an advanced compactor which uses the sinusoidal speed reference (SSR):

(10.12)

where is the varying frequency.

10.1.3 Compaction Simulation

In order to compare the performance of chaotic and nonchaotic compactors, they adopt the same PMDC motor with the key parameters listed in Table 10.1. Also, they should provide the same average rotor speed in order to enable a fair comparison. Given M_e = 0.296 kg, M_p = 3.404 kg, r = 31.5 mm, , f = 0.5 Hz, , , , , , , , and the eccentric mass inertia , the simulated rotor speed waveforms of the vibratory compactor with different control schemes are shown in Figure 10.5. It can be found that they have the same average value.

Table 10.1 Key parameters of PMDC motor.

Torque constant KT	0.2286 Nm/A
Back EMF constant KE	0.2286 V/rad/s
Armature resistance Ra	3.42 Ω
Armature inductance La	3.4 mH
Viscous damping coefficient B	8 × 10−5 Nm/rad/s
Rotor inertia J	1.588 × 10−4 kgm2

The performance of the compactors using CFV, SSR, PTD and CSR control schemes is assessed by simulation. The average compaction energy density (ACED) is a well-accepted indicator to assess the compaction effectiveness of vibratory compactors (Tran and Muro, 2004). This ACED is defined as:

(10.13)

where is the vertical amplitude of displacement measured on the terrain surface, is the total gravity of the compactor, is the maximum vertical exciting force, is the frequency of compactor motion, is the number of compaction passes, is the horizontal speed of vehicle movement, is the horizontal compaction width and is orthogonal to the vehicle movement, and is the thickness of soil. As (10.13) only describes the average compaction energy density for periodic motion, it cannot be directly used for chaotic motion. Thus, (10.13) is modified by setting equal to and equal to , defining L as the fixed horizontal movement of compaction vehicle and W as the horizontal compaction width, and then integrating the compaction energy density for a period T. Hence, it yields:

(10.14)

(10.15)

where L, W, and are respectively the length, width, and thickness of the solid particles under compaction. During the simulation, is 150 mm, W is 150 mm, and is 63 mm.

Since practical soil and concrete materials have solid particles with different sizes and hence different natural frequencies , five sets of and are chosen to represent five different particles for simulation. Namely, are chosen as 62.55, 69.97, 77.80, 86.05, and 94.72 kN/m, while the corresponding are chosen as 234.06, 247.55, 261.04, 274.53, and 288.02 N/m/s. Thus, according to , the natural frequencies are distributed between 20.7 and 25.5 Hz.

The ACED of the compactors using different control schemes under the same power consumption are simulated, as shown in Figure 10.6. The average ACED values of the compaction with five different solid particles are 75.03 kW/m³ with CFV control, 75.96 kW/m³ with SSR control, 80.60 kW/m³ with PTD control, and 79.24 kW/m³ with CSR control. This indicates that the PTD and CSR compactors have better compaction performance than the conventional CFV compactor and the advanced SSR compactor. Quantitatively, the chaotic CSR compactor offers better ACED than the conventional CFV compactor by 5.6%.

10.1.4 Compaction Experimentation

In order to prove the effectiveness of the electrically chaotic compactor experimentally, the whole system is prototyped, as shown in Figure 10.7. The parameters of the compactor are the same as that for simulation. The mixed solid particles are composed of soybeans, small green beans, small red beans, big red beans, and rice, which will provide a wide range of natural frequencies for compaction. The control schemes are implemented by the dSPACE CP1104 control board. It generates proper PWM signals to drive the MOSFET-based DC–DC converter, which in turn controls the PMDC motor. The voltage and current feedbacks of the motor are respectively measured by a voltage sensor LEM LV25-P and a current sensor LEM LA25-NP, which are also used to compute the power consumption.

Firstly, based on an encoder coupled with the motor shaft, the motor speed waveforms of the compactor using different control schemes are measured as shown in Figure 10.8. The measured nonchaotic results agree well with the simulation results in Figure 10.5. It should be pointed out that the measured chaotic speed waveforms using PTD and CSR controls cannot fully match the simulated results, which is actually the random-like nature of chaos. Nevertheless, the measured chaotic motion boundaries closely match the simulated motion boundaries.

Secondly, the compactors using different control schemes are used to press the mixed solid particles. The compaction of each control is carried out five times, and the average fall height of the solid particles under compaction is physically measured as shown in Table 10.2. During the experiments, the energy consumption is calculated by online integrating the measured motor voltage and current based on the dSPACE control board, while the fall height is directly measured by using a vernier caliper. The outcomes are 23.4 mm with CFV control, 24.5 mm with SSR control, 25.1 mm with PTD control, and 25.2 mm with CSR control. For all compaction processes, the energy consumption is kept at 1 kJ, and the initial height of the solid particles is set at 95 mm. Thus, it experimentally verifies that the chaotic PTD and CSR compactors can offer better compaction performance than the conventional CFV compactor and advanced SSR compactor. Quantitatively, the CSR compactor can create larger fall height than the CFV compactor by 7.3 per cent.

Table 10.2 Comparison of compaction performance.

In order to scale up this chaotic compactor system to the industrial level, the rated torque of the motor should be scaled up while keeping the rated speed near the natural frequencies of solid particles. Then, the weights and sizes of the eccentric mass, frame, and plate can be magnified accordingly. Also, there are two basic criteria to be followed: firstly, the resistive torque generated by the eccentric mass should be sufficiently smaller than the rated motor torque so that high-performance control of the eccentric mass can be guaranteed; secondly, the corresponding exciting force should be sufficiently larger than the weight of the compactor system so that the vibratory compaction can be implemented.

10.2 Chaotic Mixing

Industrial mixers are one of the most important mixing devices in the food, drug, chemical, and semiconductor industries. One of the major problems of conventional mixers is the formation of segregated regions when mixing fluids with low Reynolds numbers (Dong, Johansen, and Engh, 1994). The occurrence and persistence of these segregated regions require extensive energy consumption for mixing. As a result, industrial mixers are among the most ineffective equipment. The industrialists and academics in the USA have estimated that the cost of ineffective industrial mixing is in the order of US$ 1 billion to 10 billion per annum (Harnby, Edwards, and Nienow, 1992). The effects of this ineffectiveness are not only energy wastage (Raynal and Gence, 1997), but can also be disastrous; for example, a nuclear-chemical waste explosion in Russia has been attributed to improper mixing of volatile compounds (Alvarez-Hernández et al., 2002). Although mixing can be improved by increasing the rotational speed, such high-speed operation generally consumes additional energy and is sometimes impractical. Particularly, some shear-sensitive materials for biotechnological applications, such as proteins and other macromolecules, are readily damaged by high shear rates when adopting high rotational speed. Thus, the improvement of mixing is highly desirable and justifiable.

In order to improve the mixing process, time-varying rotation has been proposed (Lamberto et al., 1996), aiming to destroy the segregated regions formed in the mixing process. Among various time-varying schemes, bidirectional rotations with different frequencies are adopted for industrial mixing. In addition, the bidirectional rotation can be further modulated by a sinusoidal wave. The use of time-varying rotation to improve mixing is due to the principle that the flow is continuously perturbed, hence preventing the formation of coherent segregated regions. Conceptually, it is similar to kneading the bread dough where it is stretched and folded repeatedly to create a well mixed result. In recent years, time-varying rotational mixing has been further extended to chaotic mixing because chaos inherently offers the properties of stretching and folding which match with the aforementioned requirement of a good mix (Ottino, 1989). Although there are many studies on chaotic mixing, only a few approaches are considered to be practical. By properly designing the radius of the chamber wall, the radius of twin rotors and the gap size, and then separately controlling the rotational speeds, a twin-screw mixer was developed for chaotic mixing (Jana and Sau, 2004). By moving one of the vanes in the central impeller upward by half the vane height and one adjacent vane downward by the same distance, a perturbed three-impeller design was proposed to create chaotic motion for mixing (Alvarez-Hernández et al., 2002). By continuously varying the angle of the impeller shaft with respect to the vertical axis, a chaotic mixer was also created (Fountain et al., 2000). However, all these chaotic mixers generate the desired chaotic motion by mechanical means, thus suffering from two fundamental problems – complexity and inflexibility. Firstly, the complex configurations significantly increase the cost of the system, enlarge the hardware size, and reduce the operation reliability. Secondly, the inflexible designs definitely limit the applicability and generality since the mixing materials and the mixing tanks are prone to changing. In order to fundamentally solve these problems, the desired chaotic motion for mixing should be produced by electrical means (Chau et al., 2004).

Therefore, the purpose of this section is to newly chaoize a drive system, hence resulting in a controllable chaotic motion, for application in industrial mixing processes. Compared with the aforementioned mechanical means, the chaotic drive system not only produces the desired chaotic mixing but also offers the advantages of mechanical simplicity, high flexibility, and high controllability. As discussed before, the chaotic drive system can be a DC drive system (Ye and Chau, 2007) or an AC drive system (Ye and Chau, 2005a a), while the corresponding chaotic motion can be produced by various control-oriented means (Ye and Chau, 2005b b) or design-oriented means (Ye and Chau, 2005a a). Therefore, in this section, the discussion is focused on the use of a PMDC drive system and the associated means of control.

10.2.1 Mixer System

Figure 10.9 shows the basic configuration of an industrial mixer system, which is composed of a tank stirred by a 6-bladed Rushton turbine with six equally spaced vertical vanes. Notice that the Rushton turbine is only a typical example, and other impellers such as the propeller, paddle, and helical ribbon can be used.

A PMDC motor is used as the agitator, which can be modeled as:

(10.16)

(10.17)

where B is the viscous damping coefficient, J is the rotor inertia, K is the torque constant, L is the armature inductance, R is the armature resistance, T is the motor torque, is the load torque, U is the DC supply voltage, is the motor speed, and i is the armature current.

10.2.2 Chaotic Mixing Control

As detailed in Chapter 7, the PMDC drive system can be chaoized by a time-delay feedback control, as governed by:

(10.18)

(10.19)

where is the torque command, is the current command, is the torque parameter, is the speed parameter, and is the time-delay parameter. It should be noted that all three parameters are adjustable to achieve the desired chaotic motion.

Based on the above derivation, both the armature current and rotor speed of the PMDC motor are used as feedback signals, while the torque command calculated by (10.18) is used to generate the current command for pulse width modulation (PWM) control. Figure 10.10 shows the corresponding control system. Firstly, the measured speed feedback is delayed by a preset value of . Then, the delayed speed is fed into the torque control block in which proper values of and are preset. Hence, it generates and then . Subsequently, the difference between and the measured current feedback is fed into the current control block in which a simple PI control is adopted. Hence, it generates the desired duty ratio for the full-bridge PWM converter that provides bidirectional current control of the PMDC motor.

10.2.3 Chaotic Mixing Simulation

In order to conduct computer simulation, realistic system parameters are adopted. Table 10.3 summarizes some practical data of the PMDC motor. A detailed analysis of the effects of , , and has been discussed in Chapter 7. When selecting , , and , the motor speed and armature current waveforms are as shown in Figure 10.11. It can be seen that the motor operates at a fixed point (which is equivalent to the normal or so-called period-1 operation). When selecting , , and , the motor exhibits chaotic motion. The corresponding speed and current waveforms are shown in Figure 10.12. It can be found that both the amplitude and direction of the motor speed and armature current change with time, thus presenting ergodicity. It is this character of ergodicity that differentiates chaotic mixing and normal constant speed mixing. When selecting , , and , the corresponding chaotic speed and current waveforms are as shown in Figure 10.13. It can be seen that can be used to adjust the speed range of the chaotic motion, which is why it is called the speed parameter. Furthermore, when selecting , , and , the corresponding chaotic speed and current waveforms are shown in Figure 10.14. It can be seen that a change in the time-delay constant has little effect on the speed range. The significance of this parameter lies in the realization of the system. If the parameter is too large, the refresh rate of the control system is too low; and if the parameter is too small, it requires too much computer resource.

Table 10.3 PMDC motor parameters.

Supply voltage U	24 V
Torque constant K	0.05 Nm/A
Armature resistance R	1.1 Ω
Armature inductance L	0.4 mH
Viscous coefficient B	2.2 × 10−3 Nm/rad/s
Rotor inertia J	1.0388 × 10−5 Nm/rad/s2

It should be noted that the above simulation results are under a no-load condition, because the mathematical modeling of fluid dynamics during mixing is too complicated to be handled. The corresponding fluid dynamics involve a set of partial differential equations – known as the Navier–Stokes equations – which can only be solved by using such sophisticated methods as the 3-D finite element method (Fountain et al., 2000). Therefore, the simulation results are used to illustrate the chaoization of the drive system, whereas the experimental results will be used to illustrate the chaotic mixing process in which the mixing load is taken into account.

10.2.4 Chaotic Mixing Experimentation

The mixing system consists of a tank and an impeller spun by a digitally controlled electric drive. The motor is mounted vertically on a stand with its shaft positioned at the center of the tank. The shaft is mounted through a holding plate, which ensures consistent positioning between the experiments and minimizes oscillations of the shaft tip. As shown in Figure 10.15, the adopted motor is a PMDC motor with the parameters listed in Table 10.3; the stand is a drill holder, and the tank is a 1-litre glass beaker.

It should be noted that, when using a mixture with a low or moderate Reynolds number, it is particularly difficult to achieve an effective mix since the corresponding flow is laminar, whereas a mixture with a high Reynolds number can easily achieve an effective or so-called turbulent mix (Chaté, Villermaux, and Chomaz, 1999). Thus, a viscous solvent (light corn syrup) is purposely adopted so that the mixing effectiveness can be evaluated.

There are many methods to evaluate the mixing processes, which can be divided into two categories: intrusive and nonintrusive. The intrusive methods include a probe or tracer that is put in the stirred tank to measure flow velocities, but they disturb the flow patterns that the investigators intend to measure. The nonintrusive methods, such as the laser Doppler anemometer (Mavros, 2001) and the acid-base neutralization reaction (Ascanio et al., 2002), are more attractive since they do not disturb the flow patterns. Increasingly, the acid-base neutralization reaction offers the advantages of simple arrangement and low cost. This method is therefore adopted to evaluate the chaotic mixing.

Firstly, the tank is filled with 200 ml of light corn syrup, 5 ml of pH indicator solution (universal indicator), and a 5-ml solution of 1 N HCl. The solution is mixed until a uniform red color is observed since it is acidic. Next, another well-mixed solution (dark green color) – 100 ml of light corn syrup, 2.5 ml of pH indicator solution, and 2.5 ml of 1 N NaOH – is added into the tank. Although the whole solution is acidic as there is twice as much acid as base, there are dark green regions since diffusion is limited by the viscous solvent (light corn syrup).

The beginning of the mixing process of the acid-base solution is set as time zero. The drive is controlled by a dSPACE digital controller to realize various mixing methods, including constant-speed, periodic, and chaotic motions. The whole experiment is recorded by using a camera focused at the impeller.

To assess whether chaotic mixing is more effective than other mixing methods, it is equivalent to evaluate whether chaotic mixing can offer a more homogeneous mixture under the same amount of energy consumption. The experiment is designed to compare the mixing time needed to achieve homogeneity based on the same input power. Firstly, the chaotic mixing experiment is conducted. The input voltage and current are recorded online until homogeneity is achieved. Hence, the average input power is calculated. Then, the other mixing methods – namely, the conventional constant-speed mixing, the rectangularly bidirectional mixing, and the sinusoidally bidirectional mixing – are also conducted under the same input power for comparison.

As tabulated in Table 10.4, the chaotic mixing (, , and ) takes 4.2 W to fully mix up the aforementioned acid-base solution within 30 s; whereas the constant-speed mixing at 600 rpm requires 360 s, the rectangularly bidirectional mixing with a magnitude of 600 rpm and a frequency of 0.5 Hz requires 33 s, and the sinusoidally bidirectional mixing with an amplitude of 850 rpm and a frequency of 0.5 Hz also requires 33 s, all under the same power of 4.2 W. Thus, it quantitatively verifies that chaotic mixing has the definite advantages of shorter mixing time and, hence, lower energy consumption than the others, particularly the conventional constant-speed mixing.

Table 10.4 Comparison of different mixing methods.

Mixing Method	Time (s)
Constant-speed	360
Rectangularly bidirectional	33
Sinusoidally bidirectional	33
Chaotic	30

The improvement is expected since the constant-speed mixing process involves the formation of a segregated region which is the major obstacle for effective mixing, whereas the chaotic mixing process essentially prevents the formation of segregated regions. Figure 10.16 shows the grayscale level of the constant-speed mixing process, which illustrates that the segregated region (like a donut) persists for a long time. On the other hand, Figure 10.17 shows the grayscale level of the chaotic mixing process, which confirms that no segregated region occurs. For the two bidirectional mixing processes, there are segregated regions formed during the positive or negative sessions, but the change of direction leads to the destruction of larger donuts and, then, the formation of smaller ones. Therefore, these two bidirectional mixing processes still take a longer time than the chaotic mixing, although showing significant improvement over the conventional constant-speed mixing.

As mentioned, the segregated region is the key obstacle for the conventional constant-speed mixing process to achieve effective mixing. It has been identified that the size of this region depends on the Reynolds number of the mixture – in general, the lower the Reynolds number, the larger is the segregated region (Alvarez-Hernández et al., 2002). Thus, chaotic mixing is particularly attractive for mixing those highly viscous fluids that have a low or moderate Reynolds number.

10.3 Chaotic Washing

Commercial and domestic washing machines are among the most energy-consuming devices. Actually, the Welsh Consumer Council has revealed that about one-third of the energy used in the UK is consumed in the home, and the washing machine is one of the major energy-consuming home appliances. Thus, an improvement in washing equipment is highly desirable and justifiable. There are two major kinds of washing machine, namely the clothes-washer and the dishwasher. The former is almost indispensable for all households, while the latter is highly desirable for all restaurants.

Similar to chaotic mixing, the use of chaotic water current can improve the washing power. In recent years, chaotic washing has been proposed for both the clothes-washer (Wang et al., 1996) and the dishwasher (Nomura, Wakami, and Kazuyuki, 1996). However, they generate the desired chaotic motion by mechanical means, thus suffering from complexity and inflexibility. In order to solve these problems, the desired chaotic motion should be produced by electrical means.

10.3.1 Chaotic Clothes-Washer

For washing clothes, Goldstar produced a chaotic washer in 1993, which was claimed to be the first application of chaos theory to the clothes-washer. Its key is to utilize a small pulsator (which stirs the water) that rises and falls randomly as the main pulsator rotates, thus producing chaotic motion. Hence, it enables the adjustment of water whirls chaotically so that the washing effectiveness can be improved – namely, increasing the washing power and preventing the twisting of clothes. Consequently, an improved version was proposed in 1994, as shown in Figure 10.18. The key is to use two fan motors to force the water from the second washing tank through the induce holes into the first washing tank while the first washing tank is rotated by another motor, hence producing chaotic flow water for washing clothes (Wang et al., 1996). Although the above chaotic clothes-washers can provide better performance than a traditional washer, the corresponding mechanisms to produce the desired chaotic water current are complicated and thus costly, leading to offset the merits of the use of chaotic clothes-washing for commercial washers.

As discussed in Chapter 7, the desired chaotic motion can be easily generated by electrical means. Namely, various electric drive systems can be chaoized to provide various chaotic motions by using control-oriented or design-oriented approaches. An electrical chaotic clothes-washer has recently been proposed (Ye, Chau and Niu, 2006). As shown in Figure 10.19, the pulsator of the washing tank is simply spun by a chaotic drive system, hence directly producing the desired chaotic water current for clothes-washing. Rather than using a PMDC drive, the shaded-pole induction drive is adopted by this clothes-washer, since it offers higher efficiency, smaller size, more durability, and is maintenance free. Since the motor parameters are essentially fixed after production, the operating parameters – namely, the supply voltage and frequency – are selected as the bifurcation parameters. This shaded-pole induction drive system and its bifurcation diagrams have been thoroughly discussed in Chapter 7.

When selecting the supply voltage as 220 V and the frequency as 10 Hz, the shaded-pole induction drive can generate the desired chaotic speed for clothes-washing as shown in Figure 10.20. It should be noted that the pattern of chaotic motion can be altered by selecting different sets of supply voltage and frequency parameters.

10.3.2 Chaotic Dishwasher

The first commercially available dishwasher was introduced in 1960. With ever-increasing demands on our living standards, the use of dishwashers is becoming more and more attractive. A conventional dishwasher is shown in Figure 10.21, which consists of a dish basket, a rotatable link with multiple nozzles, and an electric water pump. As depicted in Figure 10.22, the water is pressurized by the pump, then goes through the inside of the link to eject water streams via multiple nozzles. Those nozzles ejecting upward water streams function to wash the dishes, whereas one particular nozzle is purposely designed to eject a horizontal water stream toward the wall. Due to the reaction force of the horizontal water stream, the link rotates on the pivot so that the upward water streams can cover a wide area to clean the dishes. However, the simple rotation of the link confines the coverage of the water streams, hence limiting the effectiveness of dishwashing.

Based on the anticipation that chaotic water streams should offer better dishwashing performance, a chaotic rotatable link mechanism was proposed, aiming to generate chaotic water streams for dishwashing (Nomura, Wakami, and Kondo, 1995). It has already been identified that a double pendulum can exhibit chaotic motion. Thus, the mechanism consists of two links, namely link-1 and link-2 as shown in Figure 10.23. Link-1 rotates on the central pivot, whereas link-2 rotates on a pivot mounted at an end point of link-1. Both the shape and weight distribution of the two links are not symmetrical with respect to the central pivot. A nozzle at an end of link-1 is designed to eject a horizontal water stream, while another nozzle at an end point of link-2 is designed to eject an upward-slant water stream. These two nozzles produce the reaction forces to rotate the links. Since the two links and the water streams affect each other in their motions, the mechanism exhibits a complex motion. When the shape of the links and the direction of the nozzles are properly designed, the complex motion is found to be chaotic (Nomura, Wakami, and Kazuyuki, 1996). Compared to a conventional dishwasher, this chaotic dishwasher can improve the effectiveness of dishwashing by 11 per cent.

Instead of using a complicated two-link mechanism to generate chaotic motion for dishwashing, an electric drive system can perform the same task more easily. Figure 10.24 shows the structure of an electrically chaotic rotatable link, which consists of a conventional link coupled with an electric drive and an electric water pump. The water is first pressurized by the pump, then goes through the inside of the link to eject water streams via multiple nozzles. All nozzles eject upward water streams to wash the dishes, while the motor chaotically rotates the link so that the upward water streams can effectively clean the dishes. Compared with the aforementioned chaotic motion mechanically generated by a two-link mechanism, the electrically generated chaotic motion has the following advantages:

• The rotatable link is simple in structure and can be independently controlled by the electric drive system. Notice that there is no need to spend horizontal or upward-slant water streams by this system.
• By using the electric drive system, the generated chaos motion is well defined and can be analytically formulated. Notice that the aforementioned two-link mechanism involves a complicated interaction between the links and water streams, and the corresponding complex motion is loosely found to be chaotic based on numerical time-series analysis of the measured data.
• The electric drive system can offer different types of chaotic motion, including bidirectional rotation, which are selectable for dishwashing different kinds of tableware. Notice that the two-link mechanism does not offer the necessary flexibility for different chaotic motions, or selectivity for different tableware.

10.4 Chaotic HVAC

HVAC is a well-accepted acronym that stands for heating, ventilating, and air-conditioning, which is particularly important in the design of industrial and office buildings – the so-called building services. In recent years, chaotic motion has been employed by various HVAC systems.

There is a general perception that the room temperature should be kept constant at the optimal value to ensure comfortableness. Thus, the building services engineers have made a great effort, at large expense, attempting to achieve a constant room temperature. However, thermal comfort research has revealed that people prefer imperceptible temperature swings when resting, whereas they desire artificial temperature swings when working (Wyon, 1973). Analysis has also shown that people choose to work harder when there are temperature swings, since they feel that the air is ‘fresher’.

In order to provide appropriate temperature swings about the optimal set point for heating, a chaotic kerosene fan heater was developed in 1992, which was claimed to be the first consumer electronic product in the world using chaos theory (Katayama et al., 1993). In this kerosene fan heater, the operating temperature is controlled in such a way that three types of chaotic temperature swing patterns about the optimal set point are selectable. These chaotic patterns are based on an intermittent chaos which can be represented by a simple one-dimensional Poincaré map (Procaccia and Schuster, 1983):

(10.20)

where u, a, and b are constants, and z is the parameter to control the intermittent period. By selecting u = 1.4, a = 2, and b = − 1, the trajectories with z = 1.5 is plotted in Figure 10.25. The corresponding time series are shown in Figure 10.26. It can be found that the designer can simply tune z to generate the desired time series.

To assess the effectiveness of this chaotic kerosene fan heater, psychological experiments are conducted with about 30 persons (Kuwata et al., 1996). Each person in the experimental room is requested to estimate the comfortableness of the room temperature according to a thermal sensation every five minutes. Table 10.5 lists the commonly used seven-point psychological scale for the measurement of thermal sensation (Fanger, 1970). As a result, the sensation is much more comfortable with chaotic temperature control than that with fixed temperature control.

Table 10.5 Psychological scale for thermal sensation.

Scale	Sensation
−3	Cold
−2	Cool
−1	Slightly cool
0	Neutral
+ 1	Slightly warm
+ 2	Warm
+ 3	Hot

Following the spirit of the aforementioned kerosene fan heater, chaotic temperature control can readily be extended to electric fan heaters and air-conditioners. The key is to electrically generate chaotic rotation of the electric drive in such a way that the rates of warming/cooling air flow, and hence temperature swings, are chaotic to enhance comfortableness. Figure 10.27 shows the schematic diagram of a chaotic fan heater. It consists of a direct resistance heating element, a single-phase AC fan motor and a power controller. This fan motor can be chaoized to provide chaotic rotation by control-oriented approaches such as using time-delay feedback control or design-oriented approaches such as selecting specific applied voltage and frequency (Gao, Chau, and Ye, 2005). On the other hand, Figure 10.28 shows the schematic diagram of a chaotic air-conditioner. It consists of a compressor, an internal heat exchanger (evaporator) coupled with an internal fan motor, an expansion valve, an external heat exchanger (condenser) coupled with an external fan motor, and a control unit. The internal fan motor is chaoized to provide the desired chaotic rotation so that a chaotic temperature swing can be obtained.

For chaotic mixing, a chaotic mixer can enhance the mixing effectiveness in terms of mixture homogeneity and energy consumption. Extending this concept from liquid to air, a chaotic fan can readily be used to enhance the ventilation effectiveness. In addition to chaoizing the AC fan motor, the chaotic nature can be further enhanced by chaoizing the direction of air flow – that is, the DC servomotor of swinging vanes can be further chaoized.

10.5 Chaotic Grinding

Grinding is one of the most important operations in production and manufacturing engineering. It is performed by abrasive particles removing unwanted material to attain the desired geometric and surface properties. The abrasive particles are bonded together to form a revolving body, known as a grinding wheel or grindstone. One of those important grinding operations is the vertical spindle surface grinding. It is an abrasive machining operation for stock removal grinding, and also for surface finish grinding within certain desired tolerances. The goal of stock removal grinding is to remove the unnecessary material as quickly or as cheaply as possible, whereas the goal of surface finish grinding is to provide the quality of surfaces (Shaw, 1996).

Because of the random-like nature of chaotic motion, it is anticipated that the grindstone of a chaotic grinder can provide a more uniform contact surface with the workpiece and hence a better grinding performance. The first application of chaotic motion to grinding was proposed in 1998 (Ito and Narikiyo, 1998). It applies chaos to a vertical spindle surface grinder, as shown in Figure 10.29. The abrasive motor is a DC motor, offering a rated output of 21 W, and a maximum speed of 12 000 rpm under no-load. The motor is controlled in such a way that its rotational speed is governed by:

(10.21)

where x and y are the horizontal displacements on the X–Y table, z is the rotational speed, and the relevant parameters are selected as σ = 10, r = 28, and b = 8/3. As shown in Figure 10.30, the resulting speed chaotically fluctuates between 4580 and 12 000 rpm with a period of around 1.06 s. The grindstone is made of aluminum oxide (Al₂O₃). The relevant grit size is designated by the screen number S (number of openings per linear inch), namely S = 120. The grinding process is under a wet condition and a constant pressure of 650 g.

There are two important parameters by which to evaluate the grinding performance – namely, the abrasive efficiency and the surface roughness. The abrasive efficiency is the ratio of the volume removed to the electric power consumption. The surface roughness is usually based on the peak-to-valley roughness on the European continent and in Japan, or the centerline average roughness in the UK and the USA (Shaw, 1996).

To assess the advantages of chaotic abrasive machining (chaotic speed variation at 4580–12 000 rpm with a period of around 1.06 s), it is compared with different grinding schemes:

• constant speed at 12 000 rpm (the maximum value);
• constant speed at 4580 rpm (the minimum value);
• sinusoidal speed variation (4580–12 000 rpm with a period of 1 s);
• random speed variation (4580–12 000 rpm with a division of 1 s).

As revealed by experiments (Ito and Narikiyo, 1998), chaotic abrasive machining is better than the others in both the amount of volume removed and the abrasive efficiency. In particular, its abrasive efficiency is twice that of constant-speed grinding at 12 000 rpm. Selecting a plane with R_a = 2.3–2.7 µm as the workpiece, the chaotic abrasive machining offers the best surface roughness, namely R_t = 50–100 nm.

Furthermore, in order to observe the better uniformity benefiting from chaotic grinding, a system arrangement is adopted as shown in Figure 10.31, which functions to compare the grinding trajectories. In the system, a permanent magnet synchronous motor (PMSM) is used to drive the workpiece while the grinding wheel is driven by another constant-speed DC motor. By operating the PMSM at constant speed and chaotic speed, the abrasive trajectories on the workpiece under constant-speed grinding and chaotic grinding are shown in Figure 10.32. It can be observed that the constant-speed grinding trajectory can only cover a limited surface area of the workpiece, whereas chaotic grinding can densely spread over the whole surface of the workpiece. This result illustrates that chaotic grinding not only grinds the workpiece surface more evenly, but also removes the workpiece material more effectively, thus achieving better surface uniformity and higher abrasive efficiency.

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