Magnetorheological damper–based positioning system with power generation

Abstract

This study investigates a newly developed positioning system based on a rotary magnetorheological damper with power generation capability, comprising an electromagnetic power generator (energy extractor) generating electrical power, an magnetorheological damper which alters the damping characteristics of the system and an electrical interface, connected in between the coil of the generator and the damper control coil, which conditions the voltage output from the generator. Structural configurations of the damper and the generator are outlined, and the results of their testing and the testing done on the entire system are summarised, covering the tests conducted with and without the interface, under the idle run and under load in the assumed velocity range. The objective of the work is to examine the performance of the proposed positioning system through experiments. Results of the system testing in the uncontrolled case (passive system) and in the controlled one (semi-active system) in a purpose-built test rig are compared and discussed. Three design options of the electrical system controlling the damper are explored. In the first option, the damper is assumed to be controlled starting from the initial moment of the motor shaft’s motion, and in the second option, the control action begins at the instant when the motor shaft assumes the present position. The third variant is similar to the first one except that an additional condenser is connected behind the Graetz bridge. The results confirm the adequacy of the developed positioning system and demonstrate an improvement in control of the system’s dynamic behaviour.

Keywords

Magnetorheological damper generator positioning system energy harvesting control

Introduction

Positioning systems have received a great deal of researchers’ attention. These systems are implemented to control tool movements (during drilling, milling, cutting, boring operations) or point-to-point motion of the entire machines (such as robot welding tool). They are also incorporated in various pick-place machines (Yihuan et al., 2011). In position control systems, the main objective is to minimise the time required for reaching the predetermined position and the same eliminating the vibration of an object whose position is to be controlled. That the object oscillates around the preset position can be attributed to insufficiently rigid connections between the driving components and the object itself and to clearances or friction in mechanical components. Variable load is a major determinant of the positioning system’s performance, which is of particular importance in drilling machines.

One of the strategies adopted to eliminate these undesirable phenomena involves the modification of the controller’s structural design and of algorithms for control of the component inducing the object’s motion (an electric motor in most cases). For example, Wahyudi and Albagul (2005) and Mohd and Aminudin (2011) demonstrated that when a nominal characteristic trajectory following controller is incorporated in the system for controlling the position of a 2 degree-of-freedom (DOF) object, and tuned accordingly in the open-loop system, the overshoot and control time are significantly reduced. To reduce the object’s vibrations around the target position, Verscheure et al. (2006) suggested that a regulator H_∞ be incorporated in the position control system, thus reducing the time and amplitude of the object’s oscillations. Such solutions can be effective as long as the algorithm of motor control can be varied at any moment. It is often so, however, that modification of the control algorithm is not feasible. In such case, adaptive control systems are good solutions because the controller settings are adjusted by on-line identification when the system is operated. Alongside conventional controllers and feedback loops, adaptive control systems are equipped with filters allowing the unit’s resonance frequencies to be detected. These filters are usually installed ahead of the torque controller which uses the signal of current taken up by the motor (variations of the current level correspond to variations of the motor loading moment). Adequacy of such solutions was confirmed in works by Bong and Chung (2002) and Miyazaki and Ozaki (2005).

Vibrations of the object to be position-controlled can be reduced by incorporating actuators (dampers) based on smart materials, such as piezoelectric materials or magnetorheological (MR)/electrorheological (ER) fluids. Applications of MR/ER actuators in vibration control systems were explored in works of Sun and Thomas (2011), Weber et al. (2013), Gołdasz and Dzierżek (2016), Rosół and Martynowicz (2016), Abdeljaber et al. (2016), Sun et al. (2016) and Naito et al. (2009).

As regards MR dampers in such applications, Jolly (1999) revealed that a liner damper supporting a pneumatic cylinder led to reduction of the object’s oscillation around the preset position. Yadmellat et al. (2014) and Fernández and Chang (2016) suggested that the robot arm control system should incorporate a cylindrical MR clutch with permanent magnets. Thus, the robot grip can be safely manoeuvred into its target position using just one direct current (DC) motor. Yang et al. (2012) showed that providing MR dampers in manipulator joints led to reduction of the jitters effect during the positioning of the robot arm.

Bai et al. (2011) demonstrated the application of an MR damper in the motor speed control system. Another most undesirable phenomena frequently occurring in rotating machinery and impairing the position control are torsional vibrations and vibrations occurring at critical speeds. Test results showed that damping torque control of a rotary damper leads to reduction of torsional vibration amplitudes (Pręgowska et al., 2013) and of vibrations at critical speed (Wang and Meng, 2005). Rotary MR dampers are widely used in manipulators to improve the accuracy of position control and to reduce vibration. Altering the variable viscosity coefficient and variable impedance control through MR damper control leads to improvement of dynamic properties of an artificial muscle manipulator (Nagai et al., 2011; Tomori et al., 2013). Shin et al. (2013) demonstrated that incorporating a rotary MR damper in a pneumatic manipulator interacting with a DC motor results in improvement of the manipulator’s operation precision and efficiency.

Other noteworthy actuators that can be used in such applications are MR dampers which do not require an external power source and can be controlled using energy harvested from vibrations (Cho et al., 2007; Li et al., 2013; Sapinski, 2011, 2014; Snamina and Sapinski, 2011; Wang and Wang, 2010). Moreover, some newly developed MR dampers have also self-sensing capability as for example that is proposed in the work of Hu et al. (2017).

The position control system proposed in this study incorporates a rotary MR damper power supplied with energy harvested from vibrations through the use of an electromagnetic generator. Key components of the position control system include an alternating current (AC) motor, a rotary MR damper and a power generator. The motor, connected to the controlled object via an elastic element (elastic shaft), acts upon the object to be positioned to move it to the target position. As soon as the shaft covers a specified distance, it stops. Once the movements of the motor shaft cease, the object will for some time oscillate around the preset position. To improve the dynamic features of the system, new elements are incorporated: an MR damper, a generator and a conditioning system connected between the damper coil and that in the generator. Thus, the object’s position can be controlled through altering the anti-torque in the MR damper using the energy harvested from the object’s vibration.

This work is organised as follows. Sections ‘MR damper’ and ‘Generator’ summarise the design structure of the MR damper and the power generator and provide the experimental characteristics. Section ‘MR damper–generator system’ investigates the MR damper–generator system by way of experimental testing. Section ‘Test rig’ focuses on the test rig and the measurement and control equipment used in the experiments. Tests results of a proposed MR damper–based positioning system are discussed in section ‘Experimental tests’, demonstrating its adequacy and improvement of control of the system’s dynamics. Final conclusions are given in section ‘Summary’.

MR damper

The damper assembly is depicted in Figure 1. The damper comprises a double-ended shaft (1) made of non-magnetic steel, mounted in a sectional housing (2) on rolling bearings. The housing (2) is made of magnetic steel. On the shaft there are four discs (4) moving between three immobile discs (5) mounted in the damper housing. Both moving and immobile discs are made of magnetic steel, and the space between the discs is filled with a MR fluid. Rubber seals (6) protect the system from fluid leakage. Discs and MR fluid are inside the carcass (7) of the control coil (8) having 306 turns of windings of wire 0.04 mm in diameter. Inductance of the control coil L_d = 0.810 H, resistance R_d = 15.32 Ω. Figure 2 shows geometry of the damper, whereas dimensions of its major components are summarised in Table 1.

Figure 1.

Damper assembly cut-out view.

Figure 2.

Geometry of the damper.

Table 1.

Dimensions of the damper components.

Housing	Height, h_h	31 mm
	Diameter, d_h	111 mm
Coil	Height, h_c	15 mm
	Thickness, t_c	12 mm
Shaft	Diameter, d_s	10 mm
Moving discs	Inside diameter, d_mi	12 mm
	Outside diameter, d_mo	56 mm
Immobile discs	Inside diameter d_ii	15.2 mm
	Outside diameter, d_io	61 mm

In Figures 3 to 6, we present selected characteristics of the damper which were obtained under the sinusoidally varying input φ = Asin(2πft) and for constant current levels I in the control coil: 0, 0.1, 0.2, 0.3, 0.4 A. The amplitude of the applied input was A = 0.36 rad, and its frequency fell in the range (1, 12) Hz with step of 1 Hz. Figure 3 illustrates the dependence of damping torque on displacement T_d(φ), and Figure 4 shows plots of the damping torque versus velocity T_d(ω) relationship for frequencies 1 and 10 Hz. Figures 5 and 6 plot the relationships rms damping torque T_d and between rms energy dissipated by the damper E_dh and rms velocity and current.

Figure 3.

Torque T_d versus position φ: (a) 1 Hz and (b) 10 Hz.

Figure 4.

Torque T_d versus velocity ω: (a) 1 Hz and (b) 10 Hz.

Figure 5.

rms values of torque T_d versus rms values of velocity ω and current i.

Figure 6.

rms values of dissipated energy E_dh versus rms values of velocity ω and current i.

Apparently, T_d(φ) and T_d(ω) plots reveal a hysteresis. The damping torque value tends to increase with increasing current level (for I = 0 A, T_d = 0.5 N m, for I = 0.4 A, T_d = 5 N m). The actual pattern of T_d(φ) and T_d(ω) plots changes with increasing frequency, which is manifested by reduction of energy dissipated by the damper and the maximal rms E_dh = 4.75 J.

Generator

The structure of the generator is shown in Figure 7. A cylindrical rotor (1) is made of non-magnetic material and embedded on its inside wall is a thin-walled cylinder (2) made of ferromagnetic metal plate with 28 permanent magnets (3) uniformly distributed over its circumference. The magnets are configured into 14 pairs of magnetic poles. The rotor is mounted on ball bearings (4) and set on the shaft (5) to which the stator (6) is attached. The stator comprises 14 pairs of rectangular frames (7) (to match the number of pairs of magnetic poles), made of ferromagnetic steel plates. Frames are arranged radially on the shaft and attached to the carcass made of an electro-insulating material. Each frame consists of two half-sections connected inside the coil (8) and shaped such that each half-section corresponds to one magnetic pole (the frames being shifted with respect to one another by about 13°). Inside the frames there is a coil (306 turns of windings of wire 0.2 mm in diameter), wound upon the carcass in the direction normal to the frames. Inductance of the generator coil is L_g = 0.031 H and its resistance R_g = 2.7 Ω. The coil core is made from frame sections inside the coil being in contact with the shaft.

Figure 7.

Generator assembly cut-out view.

Selected characteristics of the generator obtained under the idle run are shown in Figures 8 and 9; those obtained under load (the load being the control coil in the MR damper) are given in Figures 10 and 11. Figure 8 plots the emf e versus relative time for various values of rotational velocity, and the relationship between the rms value of electromotive force and velocity ω is shown in Figure 9. For rotational velocity ω = 6.28 rad/s, the emf frequency is about 14 Hz (which is the consequence of 14 pairs of magnetic poles), and the value of rms e increases linearly with increasing ω (przy ω = 50 rad/s; e = 48.54 V). Figure 10 plots the voltage u and current i versus time, for ω = 31.42 rad/s, and the dependence between rms u and rms i on velocity ω is illustrated in Figure 11, revealing a phase shift between the voltage and current plots approaching 50.35°. The values of rms u and i tend to increase with increasing velocity (for ω = 50 rad/s rms u = 37.95 V, rms i = 0.27 A).

Figure 8.

Time history of emf e.

Figure 9.

rms values of emf e versus velocity ω.

Figure 10.

Time histories of voltage u and current i.

Figure 11.

rms values of voltage u and current i versus velocity ω.

MR damper–generator system

The next stage testing was done on the MR damper and generator system mounted on a joint shaft having high stiffness. The following system configurations were considered: when the damper coil and the generator coil were not connected (NS), when the damper coil was directly connected to the generator coil (DS) and when the damper coil was connected to a generator coil via a Graetz bridge (GS). Tests were conducted in two stages. In the first stage, the input was applied with the time constant velocity (ω = const) which varied in the range (0.1, 50) rad/s. The damping torque T_dg dependence on velocity ω in the steady state conditions is illustrated in Figure 12, revealing that at the same rotational velocity, the damping torque value in the case DS would slightly increase in relation to NS. For ω = 50 rad/s, this increase is the most significant, approaching 0.22 N m. In the configuration GS, the rms value of T_dg increased (for ω = 50 rad/s rms T_dg = 2.9 N m) when compared to NS and DS. In the second stage, harmonic input was applied φ = Asin(2πft) with the amplitude A = 0.36 rad and frequency f varying in the range (1, 15) Hz, and again the cases NS, DS and GS were considered. Figure 13 shows the damping torque versus displacement plot T_dg(φ) while the relationship between the damping torque and velocity T_dg(ω) at 1 and 10 Hz is shown in Figure 14. Figures 15 and 16 plot the energy supplied to the control coil in the damper E_gd and energy dissipated in the damper E_dh. It appears that at frequency 1 Hz, the values of damping torque T_dg are similar in each of the three cases: NS, DS, GS. Apparently, the presence of the Graetz bridge in the GS configuration is the major determinant of the actual T_dg level. Both E_gd and E_hd tend to increase with increasing velocity. At ω = 23.4 rad/s, energy dissipated in the damper reaches the value E_gd = 0.24 J, which accounts for 5.9% of E_dh.

Figure 12.

rms value of torque T_dg versus velocity ω.

Figure 13.

Torque T_dg versus position φ: (a) 1 Hz and (b) 10 Hz.

Figure 14.

Torque T_dg versus velocity ω: (a) 1 Hz and (b) 10 Hz.

Figure 15.

Energy provided to the control coil E_gd versus velocity ω.

Figure 16.

Energy dissipated E_dh versus velocity ω.

The scheme of the considered positioning system incorporating an MR damper and a generator is shown in Figure 17. The equation of motion can be written as

I_{P} {\overset{\cdot\cdot}{φ}}_{p} = - k_{s} (φ_{p} - φ_{m}) - c_{s} ({\overset{\cdot}{φ}}_{p} - {\overset{\cdot}{φ}}_{m}) - c {\overset{\cdot}{φ}}_{p} - T_{d} - T_{g}

(1)

where T_d is damping torque in the damper, T_g is damping torque in the generator, I_p is mass moment of inertia of an object to be positioned, c_s is material damping and c is equivalent damping ratio of the remaining elements of the system.

Figure 17.

Scheme of the positioning system.

It is apparent that damping torques T_d and T_g counteract the object’s motions (energy dissipation). The input to the system is the angular displacement φ_m, and the output parameter is angular displacement φ_p of the object whose position is controlled. The system considered in this study has the following elements: a source of mechanical energy (an electric motor), a potential energy accumulator (elastic shaft), an accumulator of kinetic energy (the object to be positioned), an energy-dissipating element (MR damper) and a device that converts kinetic energy of the object being positioned into electric power (generator).

Test rig

The main components of the experimental setup include a synchronous motor interacting with a controller, the object to be positioned, an MR damper and a generator. A general view of the test rig is shown in Figure 18. Inside the motor housing (on the shaft) there is an encoder 1. The object whose position is controlled is connected to the motor via an elastic shaft. The object’s displacement is measured with the encoder having the resolution of 1024 impulses per rotation. An MR damper and a generator are connected to the object via a clutch. The mass moment of inertia of the positioned object equals I_p = 25 × 10⁻⁴ kg m². Elasticity modulus of the shaft is k_s = 15.66 N m/rad. The mass moment of inertia I_p of the elastic shaft is neglected as it is rather small in relation to that of the controlled object. The resultant mass moment of inertia of other components in the setup (clutches, discs in the MR damper, discs in the encoder 2) is equal to I_d = 6 × 10⁻⁴ kg m². Sub-assemblies giving rise to I_d are fitted between the motor and the MR damper and connected with one another by rigid elements.

Figure 18.

Positioning system – general view.

The measurement and control module (see Figure 19) being a part of the positioning system comprises a motor control block, a purpose-built energy management block used for data acquisition and control of the MR damper. The synchronous motor operates in the step/direction mode with position and torque controllers implemented inside the drive system being in the ‘on’ position. The input to the motor control block is position φ_set. The motor is operated in the open-loop control system wherein the number of impulses corresponds to the shaft’s angular path, and their frequency corresponds to angular velocity. The number of impulses per one rotation is 1024. The motor control program, developed in LabVIEW, generates two impulse sequences using the I/O board. The first sequence determines the modulus absolute value for the subsequent step (0 or 1), the other determines the direction. The impulses are fed to the controller/drive input. Using the cascade of position and torque regulators, it controls the motor such that the preset position φ_set should be implemented.

Figure 19.

Diagram of the measurement and control module.

The energy management block incorporates two integrated parts: a conditioning unit and a measurement and control unit complete with an electronic analogue signal conditioning system and a STM32F4 processor board. The measurement block uses high-precision instrumentation amplifiers. The STM32F4 (STM Microelectronics, 2013) processor boards is based on a 32-bit processor ARM Cortex-M4 (a float point operation unit). The control unit incorporates an analogue-to-digital (A/D) converter to measure analogue quantities from the energy-conditioning system. Instantaneous positions (the distances covered by the motor shaft φ_m and by the object φ_p) are measured with the time–counter systems converting the sequences of impulses from encoders into instantaneous values of the angular positions, with the movement direction detection functionality. Detection of the movement direction is necessary when the positioned object tends to oscillate. Relays K are controlled using pulse width modulation (PWM) signals. Communication between PC and the STM32F4 processor board is effected via a serial port universal asynchronous receiver/transmitter (UART). Because of engineering constraints and the value of induced voltage in the generator operated under the idle run, the value of velocity considered in the tests was 21 rad/s. The sampling frequency in signal measurements was 0.5 Hz.

Experimental tests

The performance of the positioning system was tested in two operation modes: the passive mode – absence of control (NE – when the damper is not powered, DG – when a Graetz bridge is provided in between the damper coil and the generator coil) and in the semi-active mode (the damper control in accordance with the selected control algorithm). In both cases, the set value of angular position was φ_set = 6.28 rad.

Figure 20 shows the diagram of the system operated in the semi-active mode. The value of current i_c derived by the control algorithm is compared to the instantaneous current level i in the circuit, measured via a voltage drop on resistor R_S(Texas Instruments, 2015). The resultant control error ε_i determines whether the relay K is on or off. The relay switches on when voltage u₁ = 0 V is applied to its control terminal, and it is off when voltage becomes u_h = 3.3 V. Since the system is capable of handling voltages up to 100 V (it is reasonable to anticipate high instantaneous voltage levels associated with inductance at the instant when the current suddenly ceases in the circuit), the system AQV255 (Panasonic Industrial Devices, 2017) will act as the relay. A diode D₀ is provided to implement quick discharge of energy accumulated in the coil.

Figure 20.

Diagram of the energy management block.

The conceptual design of the positioning system is shown in Figure 21. Energy recovered from the object’s oscillations while it is positioned is converted in the conditioning system and then utilised to power (control) the MR damper.

Figure 21.

Diagram of the positioning system.

Figure 22 illustrates three configurations of the conditioning system (ES1, ES2, ES3) during the positioning system testing. In the case of ES1 configuration, it is assumed that the damper control begins from initial moment of the motor shaft’s movement. In the case of ES2, the control is assumed to be switched on at the instant when the shaft assumed its preset position (φ_m = φ_set). In the ES3 configuration, a condenser is provided behind the Graetz bridge, and damper control begins at the initial moment of the movement (unlike the ES1 structure).

Figure 22.

Structures of the conditioning unit: (a) ES1, (b) ES2 and (c) ES3.

To determine how the presence of a damper and a generator should affect the accuracy of position control, the plots of position φ_p versus time were obtained for the following cases (see Figure 23):

The damper and generator are absent, not connected to the object (Case M);

The damper is connected to the object, yet it is not powered (Case TO);

The damper and generator are present though their coils are not inter-connected (Case NE).

Figure 23.

Time history of the object position φ_p.

Evaluation and comparison of the positioning system’s performance rely on two indicators: time t_u, after which the object’s vibration should not exceed ±1% of the preset position φ_set (starting from the initial moment of the movement) and the absolute overshoot φ_pmax. Plots in Figure 23 indicate that the time t_u and overshoot φ_pmax become 1.33 s, 0.434 rad (Case M); 0.72 s, 0.331 rad (Case TO); and 0.52 s, 0.335 rad (Case NE). Damping provided by connecting a damper and a generator (Case NE) led to the reduction of the time t_u by 60.9% and overshoot φ_pmax by 22.8% in relation to the case M. The position φ_p(t) obtained in the case NE will be henceforth considered as the reference level.

Passive mode

Two cases of passive mode of operation were considered: NE and DG. As regards the case DG, three structural configurations, ES1, ES2 and ES3, of the conditioning unit were considered (see Figure 23). In the configurations ES1 and ES3, the relay K remains off from the initial moment of the movement; in the case ES2, the relay is off when φ_p = φ_set. Figure 24 plots the object’s displacement in the case NE and DG and alsothe object’s ω_p and current i.

Figure 24.

Time histories of the object position φ_p: (a) ES1, (b) ES2 and (c) ES3.

The shortest time t_u and the smallest overshoot φ_pmax is obtained for ES1 (Table 2). It appears (see Figure 25) that for ES1, the current level in the circuit remains constant (about 0.5 A) for nearly the whole time when the object reaches its preset position. In the case ES1, the MR damper adds the torque, and its operation is desirable only at instants when the object’s energy ought to be dissipated. In the remaining time, it provides additional damping torque to the motor, which is undesirable.

Table 2.

Values of t_u and φ_pmax; case DG.

		ES1	ES2	ES3
t_u	s	0.35	0.41	0.38
φ_pmax	rad	0.006	0.315	0.124

Figure 25.

Time histories of the velocity ω_p and current i: (a) ES1, (b) ES2 and (c) ES3.

Semi-active mode

Following a thorough analysis of φ_m(t) and φ_p(t) and their derivatives, the analogue and discrete versions of the damper control algorithm were suggested. Continuous control algorithms (CA1, CA2) rely on the relay K implementing the present patterns of current i, assuming that the value i should fall in the predetermined range (0, 0.6) A. The maximal current level i_max was derived from the experimental test data of the generator, the load being the damper coil. Discrete algorithms (DA1, DA2) determine the time instants in which the value of current i becomes either 0 or 0.6 A. The algorithm CA1 operates such that the damper is switched on at instants when the absolute value of the object’s velocity $| ω_{p} |$ is greater than the absolute value of the motor shaft velocity $| ω_{m} |$ . In the remaining time, the damper is off. Such mode of the damper’s operation results in dissipation of the object’s energy, thus allowing the difference between the two positions φ_m(t) and φ_p(t). The algorithm CA1 is governed by the equation

T_{dg} = {\begin{matrix} c_{T} | ω_{p} - ω_{m} |; | ω_{m} | < | ω_{p} | \\ 0; | ω_{m} | \geq | ω_{p} | \end{matrix}

(2)

To obtain the required level of damping torque T_dg, the value of current i (for CA1) should be governed by the equation

i = {\begin{matrix} G_{3} | ω_{p} - ω_{m} |; | ω_{m} | < | ω_{p} | \\ 0; | ω_{m} | \geq | ω_{p} | \end{matrix}

(3)

where G₃ is a coefficient having a fixed value 0.006 A s/rad.

The algorithm DA1 is governed by the equation

i = {\begin{matrix} i_{\max}; | ω_{m} | < | ω_{p} | \\ 0; | ω_{m} | \geq | ω_{p} | \end{matrix}

(4)

The underlying assumption in algorithm CA2 relying on positions φ_m, φ_p and velocities ω_m, ω_p obtained accordingly is that the damper ought to be switched on exclusively when the object’s position increases its value in relation to that of the motor shaft position φ_m. At time instants when $| ε_{φ} | = | φ_{p} - φ_{m} |$ , the damper ought to be on, while at the instants when the absolute value of position $| ε_{φ} |$ decreases, the damper ought to be off, so that the positioned object can reach the motor shaft position in as short time as possible. At such moments, therefore, the damper ought to generate the minimal possible damping torque, (in the ideal case $T_{dg} = 0$ ). The algorithm CA2 is governed by the equation

T_{dg} = {\begin{matrix} c_{T} | ω_{p} |; \begin{matrix} φ_{p} > φ_{m} and ω_{m} > 0 \\ φ_{p} > φ_{m} and ω_{m} > 0 \end{matrix} \\ 0; in other cases \end{matrix}

(5)

In order to implement the torque determined by CA2, the current in the control coil should be governed by the equation

i = {\begin{matrix} G_{4} | ω_{p} |; \begin{matrix} φ_{p} > φ_{m} and ω_{m} > 0 \\ φ_{p} > φ_{m} and ω_{m} > 0 \end{matrix} \\ 0; in other cases \end{matrix}

(6)

where G₄ is a coefficient having a fixed value 0.006 A s/rad.

The algorithm DA2 is expressed by the equation

i = {\begin{matrix} i_{\max}; \begin{matrix} φ_{p} > φ_{m} and ω_{m} > 0 \\ φ_{p} > φ_{m} and ω_{m} > 0 \end{matrix} \\ 0; in other cases \end{matrix}

(7)

The values of coefficients G₃ and G₄ were chosen experimentally. In the first step, the research examines the semi-active mode of operation of the position control system for ES1. Figure 26 shows time histories of the angular position φ_p for all control algorithms.

Figure 26.

Time histories of the object position φ_p: (a) continuous control and (b) discrete control; ES1.

Table 3 compiles the values of t_u and φ_pmax, read out from plots in Figure 26. It appears that the lowest values of t_u and φ_pmax were obtained in the case DA1.

Table 3.

Values of t_u and φ_pmax; structure ES1.

		NE	DA1	DA2	CA1	CA2
t_u	s	0.51	0.37	0.4	0.38	0.43
φ_pmax	rad	0.335	0.073	0.273	0.147	0.330

Figure 27 shows the time histories of angular velocity and current level for CA1 and CA2 and Figure 28 for DA1 and DA2. It is readily apparent that the peak value of i in the circuit will not exceed 0.55 A. At the instant the motor shaft reaches the preset position (φ_p = φ_set), the object’s velocity is the largest in the case CA1 (19.2 rad/s).

Figure 27.

Time histories of velocity $ω_{m}, ω_{p}$ and current i: (a) CA1 and (b) CA2; ES1.

Figure 28.

Time histories of velocity $ω_{m}, ω_{p}$ and current i: (a) DA1 and (b) DA2; ES1.

In the next stage, the positioning system’s performance was investigated in the ES2 configuration. Time histories of position φ_p for CA1 and CA2 are shown in Figure 29(a), for DA1 and DA2 in Figure 29(b).

Figure 29.

Time histories of the object position φ_p: (a) continuous control and (b) discrete control; ES2.

Plots in Figure 29 indicate that from the moment t = 0 s, until the moment the control is switched on (φ_p = φ_set), in each case the object reaches exactly the same position within subsequent time instants. Throughout this time interval, the generator operates under the idle run. Starting from the moment when φ_p = φ_set, the generator begins to operate under load, the control of the damper begins and the relay K (Figure 18). Table 4 summarises the values of t_u and φ_pmax, read out from the plots in Figure 29.

Table 4.

Values of t_u and φ_pmax; structure ES2.

		NE	DA1	DA2	CA1	CA2
t_u	s	0.51	0.42	0.43	0.42	0.47
φ_pmax	rad	0.337	0.302	0.329	0.311	0.331

The shortest time t_u was obtained in the case DA1 and CA1 (0.42 s). In the case DA1, the overshoot φ_pmax proved to be the smallest (0.302 rad). Figure 30 illustrating the rate of positioning error decreases clearly indicating that all trajectories tend towards the identical point and that in the case DA1, this point is reached within the shortest time. Figure 31 shows time histories of velocity and current levels for CA1 and CA2; the results obtained in the case DA1 and DA2 are compiled in Figure 32.

Figure 30.

Position error derivative ${\overset{\cdot}{ε}}_{φ}$ versus position error $ε_{φ}$ ; ES2.

Figure 31.

Time histories of velocity $ω_{m}, ω_{p}$ and current i: (a) CA1 and (b) CA2; ES2.

Figure 32.

Time histories of velocity $ω_{m}, ω_{p}$ and current i: (a) DA1 and (b) DA2; ES2.

It is clearly seen (Figures 31 and 32) that in the case ES2, the current flow in the circuit begins at the instant when the motor shaft reaches the preset position. The maximal value of i equals 0.45 A (DA1). It is slightly higher than in the case DG, which is attributable to the generator’s temporarily switching to the idle run mode (DA1) before the current i reaches its peak value, and the relay is disconnected, in accordance with the control algorithm. Time histories of position φ_p obtained in the case ES3 featuring a 10000 F condenser connected behind the Graetz bridge are shown in Figure 33(a) (CA1, CA2) and Figure 33(b) (DA1, DA2).

Figure 33.

Time histories of object position φ_p: (a) continuous control and (b) discrete control; ES3.

Table 5 compiles the values of t_u and φ_pmax read out from time histories of displacement. It appears that the lowest values of t_u and φ_pmax were obtained in the case DA1 and CA1.

Table 5.

Values of t_u and φ_pmax; structure ES3.

		NE	DA1	DA2	CA1	CA2
t_u	s	0.51	0.38	0.43	0.38	0.43
φ_pmax	rad	0.335	0.2	0.32	0.201	0.335

Figures 34 and 35 plot the current i in the control coil, showing that while the object approaches the target position, the variability pattern of i would be similar in each case (CA1, CA2, DA1, DA2). The choice of the control algorithm affects the variability pattern of i is revealed after the motor shaft reaches its target position (φ_set). The registered current level i is the greatest (0.47 A) in the case DA1.

Figure 34.

Time histories of the object velocity $ω_{m}, ω_{p}$ and current i: (a) CA1 and (b) CA2; ES3.

Figure 35.

Time histories of the object velocity $ω_{m}, ω_{p}$ and current i: (a) DA1 and (b) DA2; ES3.

Comparison of the positioning system’s performance is based on the following indicators

I_{AE} = \int_{t_{I}}^{t_{I} + t_{E}} | ε_{φ} (t) | dt

(8)

I_{SE} = \int_{t_{I}}^{t_{I} + t_{E}} {(ε_{φ} (t))}^{2} dt

(9)

I_{λ} = \int_{t_{I}}^{t_{I} + t_{E}} ({(ε_{φ} (t))}^{2} + λ {({\overset{\cdot}{ε}}_{φ} (t))}^{2}) dt

(10)

They are expressed such as to allow the evaluation of the control system in terms of selected criteria. Integrands can be interpreted as an instantaneous cost of loss. Introducing a derivative of the control error ${\overset{\cdot}{ε}}_{φ}$ into integrand expresses an additional ‘fee’ charged for significant changes of the control error. As the square root of the control error is involved, $ε_{φ}$ will rapidly tend to be zero. Control strategy relying on $λ ({\overset{\cdot}{ε}}_{φ} (t))^{2}$ prevents the occurrence of strong oscillations (large instantaneous values of the control error derivative). The coefficient λ expresses the weight determining the fluidity of control error variations. Quality indicators normalised with respect to the case NE are shown in Figure 36.

Figure 36.

Quality coefficients.

Time t_I, for which φ_m = φ_set, is taken to be the lower limit of integration while the upper limit is defined as time t_E = 0.8 s, after which the object should not oscillate around the preset position. In order that the factors $(ε_{φ} (t))^{2}$ and $({\overset{\cdot}{ε}}_{φ} (t))^{2}$ should be assigned a similar weight when calculating I_λ, the value of λ is assumed to be 0.01 s². Apart from factors I_AE, I_SE, $I_{λ}$ another factor I_TU is introduced which determines the value of t_u (see Tables 3 to 5). Its value is the greatest in the NE case; the lowest value is registered in the case DG (Figure 36(a)). The lowest value of I_AE is registered in the case DA-ES1 and DA1-ES2, while I_SE in the configuration ES1 reaches the lowest value in the case DG and in the ES3 configuration in the case DA1 and CA1. The lowest value of overshoot φ_max alongside the lowest value of the control error derivative ${\overset{\cdot}{ε}}_{φ} (t)$ is obtained in the case DG for ES1 and in the case CA1 for ES2 and ES3 (see Figure 36(d)).

Summary

The primary goal of this study was to examine the newly developed MR damper–based positioning system through experiments. In particular, the work includes test results of individual system components as well as the damper-generator system under the idle run and under load. The performance of the system was tested in uncontrolled case (passive system) and in the controlled case (semi-active system). The main advantage of the system is its self-power capability, that is, it is adaptable to external excitations without requiring an external power supply.

The results of tests showed that the efficiency of the generator is sufficient for powering (controlling) the damper and indicated that the applied conditioning units (in particular the Graetz bridge) limit the impact of inductive load on the generator’s operation, resulting in higher current levels. Moreover, both the conditioning and control systems enable fast switching between the passive and semi-active mode. The authors found that the system in passive and semi-active mode exhibits improved dynamic behaviour in relation to case with no damper. It was observed that the damper, the generator and the energy management block incorporated in the positioning system enable to control effectively position of the object without interfering in mechanical structure and motor control. The conditioning system allows to switch in between passive mode and semi-active mode very fast. The greatest efficiency of the system in passive mode was achieved when the Graetz bridge was placed between the power generator coil and the damper coil. In semi-active mode, the system was most effective when the continuous (CA1) and discrete (DA1) algorithm were employed.

Further research efforts will be focused on application of the investigated positioning system in a special clutch or machine safety systems.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by AGH University of Science and Technology under research program no. 11.11.130.958.

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