Research on an Electromagnetic Actuator for Vibration Suppression and Energy Regeneration

: This paper proposes an electromagnetic actuator that concurrently realizes two working functions of vibration suppression and energy regeneration. The actuator consists of four permanent magnetic rings, three soft iron rings, three coils, and three springs. The design of the electromagnetic actuator is based on ﬁnite element method (FEM) analysis, and the prototype is based on this analysis. Based on the prototype, the characteristics of the electromagnetic actuator, which has an output force–current coe ﬃ cient of 39.49 N / A, are explored. A control algorithm with a position controller and an acceleration controller are applied to the actuator. When an impulse excitation is input to the electromagnetic actuator, the acceleration of the controlled object decreases from 114.26 m / s 2 to 3.14 m / s 2 here. Moreover, when the sinusoidal excitation with a 3 mm amplitude and 5 Hz frequency is input to the electromagnetic actuator, the vibration amplitude of the controlled object is 0.045 mm, suppressed within 1.46% when compared with the input signal. The peak value of the regenerated electromotive force is 1.97 V here, and the actuator e ﬃ ciency for regenerating energy is 11.59%. The experimental results with multiple frequencies and amplitudes also show that the amplitude of the controlled object can be suppressed within 5.5%, and that the ratio of the electromotive force (EMF) to the input amplitude is 0.13. The results indicate that this electromagnetic actuator can suppress vibrations e ﬀ ectively and regenerate energy from vibrations. − 1.9 A and 1.9 A with an interval of 0.1 A to ensure the normal operation of the coil. The output force range of the electromagnetic actuator is approximately − 72.78 N to 72.56 N. The simulation result shows that the output force–current relationship has high linearity and that its output force–current coe ﬃ cient is 38.3 N / A. Actuators by a position controller and an acceleration controller. Impulse excitation response and sinusoidal excitation response analyses were carried out. The experimental results show that the vibration acceleration of the controlled object can be suppressed from 114.26 m / s 2 to 3.14 m / s 2 , which is within 2.75% under the impulse excitation response. For the sinusoidal excitation response, the input signal under a 3 mm amplitude and 5 Hz frequency was set as the input of the excitation object, where the peak value of the displacement of the controlled object was 0.045 mm, accounting for 1.46% of that of the excitation object. The peak value of the regenerated EMF was 1.97 V, where the e ﬃ ciency for regenerating energy


Introduction
Vibration is a common phenomenon of a mechanical system [1] that possesses properties which partly affect the stability and safety of equipment and machine operation. Examples of these mechanical systems include vehicles [2,3], motors [4], elevators, robots [5], construction machinery [6], wind power systems [7], flywheel systems, rail transportation, and many other mechanisms in other fields [8]. Excessive or unwanted vibration may also have a negative effect on the mechanism. For mitigating vibrations of mechanisms, Xu et al. [9] proposed a new vibration isolation and mitigation device which is used to restrain platform vibration under a wide frequency excitation. Le et al. [10] proposed a vibration isolation system which improves the vibration isolation validity of a driver seat. Wu et al. [11] presented a magnetic levitation (maglev) vibration isolation system, and verified the nonlinear dynamic model and the isolation performance, which was proven to be highly effective. Moradi et al. [12] proposed a tunable vibration absorber to suppress regenerative chatter in the milling of cantilever plates, based on the mode summation approach. Rosbi et al. [13] proposed an optimally designed dynamic absorber, using the suppression method of the nonlinear vibration to suppress the occurrence of subharmonic vibration in the natural frequency and the damping ratio. Stelzer et al. [14] developed a semi-active magnetorheological isolator with a low cost that can be used to mount components that generate vibrations. The isolator is applicable for automobiles, engine mounts, pumps, and the isolation of aviation and naval components. Yang et al. [15] studied the vibration power flow and force transmission characteristics of a 2-degree of freedom (DOF) nonlinear system to test the performance of a nonlinear isolator. Wang et al. [16] proposed a two-stage quasi-zero stiffness vibration isolation system, with a low roll-off rate in the effective frequency range of vibration isolation. Therefore, vibration is one of the important factors in machining accuracy, machine operation, and machine life, where reducing vibrations is beneficial with regards to improving machining accuracy and reducing the risk of hidden dangers.
In addition to changing the mechanical structure to alleviate vibrations, vibrations can also be reduced by designing control and theoretical methods that are suitable for the vibration system. Zhang et al. [17] proposed a hybrid error criterion-based frequency-domain least mean square (LMS) control method, for suppressing the vibrations of milling processes using piezoelectric actuators and sensors, which is an effective and reliable way to suppress vibrations and improve the machining quality, based on the simulation and experimental results. Lou et al. [18] proposed a novel optimal trajectory planning approach applied to a flexible piezoelectric manipulator system, which is a combination of this approach and the feedback control of piezoelectric actuators. Zhou et al. [19] proposed an adaptive vibration control method using a pair of mirror finite impulse response filters, to reduce the amplitude of the rear gap vibration and improve the stability and ride comfort of a maglev train. Kotake et al. [20] proposed a vibration manipulation function, applied in a one-dimensional overhead traveling crane to remove the residual vibrations of a hoisted load, and validated the abilities of the proposed function in the crane operation by simulations and experiments. Shao et al. [21] proposed a fault-tolerant fuzzy H∞ control design approach for the presence of sprung mass variations, actuator faults, and control input constraints, to suppress the vibrations of the active suspension of in-wheel motor-driven electric vehicles.
The other fundamental flaw associated with mechanisms is that they expel undesired kinetic energy as heat, meanwhile, energy loss generated by vibrations is also a hot topic of current research. Ze et al. [22] proposed a new capacitive energy recycling converter to improve the power generation performance of a doubly salient permanent magnet generator. Warner et al. [23] proposed three actuator models considered for energy regeneration potential of the knee joint. Manna et al. [24] designed a novel energy-harvesting device, using magnetic levitation to produce an oscillator with a tunable resonance, and investigated the energy-harvesting potential of this prototypical nonlinear system. Elvin et al. [25] presented an experimentally validated electromagnetic energy harvester, the results of which show a normalized power density of 1.7 mW/[(m/s 2 ) 2 cm 3 ] when operating at a resonance frequency of 112. 25 Hz. Glynne-Jones et al. [26] described miniaturized generators for converting ambient vibration energy into electrical energy, as applied in powering intelligent sensors. These generators have been tested with a vehicle engine and have produced a peak power of 3.9 mW with an average power of 157 W. Xu et al. [27] put forward a braking system using only electric motors/generators as the actuators, the generated energy of which can be fed back to the onboard energy storage system as much as possible. Zhang et al. [28] proposed an energy-efficient torque allocation scheme to increase traction efficiency and brake energy-recovery. Yan et al. [29] put forward an energy regeneration implementation scheme to achieve an energy-saving goal of active suspension with self-power supply potential in a practical application.
To obtain energy from vibrations, energy regenerating mechanisms have been generally divided into three main categories, namely, rotation energy regeneration mechanisms, linear energy regeneration mechanisms, and energy regeneration mechanisms based on energy regenerating materials [30]. Rotation energy regeneration mechanisms convert reciprocating linear vibrations into bidirectional rotary motion to produce electricity [31]. Liu et al. [32] proposed an energy-harvesting shock absorber based on a mechanical motion rectifier, which is able to harvest an average power of 13.3 W for a representative period of 8 s on a paved road at 40 mph. Linear energy regeneration mechanisms adopt a coil, or multiple coils, to cut the magnetic flux to generate electric energy and produce a back electromotive force [33,34]. Buren et al. [35] presented a prototype linear electromagnetic generator that supplied power to body-worn sensor nodes with an output power of 2-25 W while walking. Arroyo et al. [36] presented and optimized a new electromagnetic harvester with a nonlinear energy extraction circuit (the synchronized magnetic flux extraction technique), the experimental results of which show that a rectified power of 1.6 mW is harvested at 1g excitation acceleration at 100 Hz frequency over a 10 Hz bandwidth. Tang et al. [37] presented a tubular linear electromagnetic transducer, applied in large-scale vibration energy-harvesting from vehicle suspensions, tall buildings, or long-span bridges. The experiment results show that the prototype (63.5 mm outer diameter, 305 mm compressed length) can harvest 2.8 W of power under a 0.11 m/s relative velocity, with a damping coefficient of 940 N·s/m. Sapiński [38] proposed an energy-harvesting linear magnetorheological (EH-LMR) damper to obtain energy from external excitations, using an electromagnetic energy extractor with self-powered and self-sensing capabilities. The energy regenerating mechanism uses the characteristics of energy regenerating materials to regenerate energy [39]. Liu et al. [40] designed, microfabricated, and characterized a piezoelectric microelectromechanical systems (MEMS) energy-harvester which has a wideband and steadily increased power from 19.4 nW to 51.3 nW within the operating frequency bandwidth of 30-47 Hz at gravity acceleration of 1.0 g. Liu et al. [41] studied a cantilevered energy-harvesting device, with a root mean square (RMS) harvesting power up to 13.3 mW and a RMS power density of up to 3.7 mW/cm 3 /g, based on an iron-gallium alloy magnetostrictive material for low frequencies, verifying the electricity-generating capability of the harvester prototype. The energy generated by vibrations can be collected by the energy regenerating mechanisms mentioned above, and stored in a battery through a series of energy regenerating circuits. Because the linear energy regeneration mechanism has the advantages of high efficiency, non-contact, no lubrication, and being non-pollutive, this scheme was adopted for energy regeneration in this study.
This paper presents a novel electromagnetic actuator, that has been designed to produce energy via vibrations while suppressing said vibrations, which can be installed in many kinds of mechanisms to improve machining accuracy and reduce energy loss. Firstly, the electromagnetic actuator is proposed, and the vibration suppression function and energy regeneration function are explained in detail. Then, the proposed electromagnetic actuator is designed based on the static magnetic field simulation with finite element method (FEM) analysis, and its prototype is fabricated. Based on this prototype, characteristics of the electromagnetic actuator are explored via measuring experiments. Furthermore, a control algorithm with a position controller and an acceleration controller is applied to the proposed actuator. Finally, an experimental control system is built, and experiments of the impulse and sinusoidal excitation responses are carried out to verify the feasibility and effect of the electromagnetic actuator.

Structure
As shown in Figure 1, the electromagnetic actuator consists of a controlled object, three coils, four permanent magnet rings (PM rings), four soft iron rings, a soft iron outer shell, three springs, an acceleration sensor (acc. sensor), and a displacement sensor (disp. sensor). Three springs are used to support the controlled object and three coils are fixed on the controlled object. The soft iron outer shell, three springs, and four PM rings are fixed on a stator. The axial-magnetized PM rings are adopted here to generate a concentrated magnetic flux in the surrounding air gap, and the arrangement of the PM rings is shown in Figure 1.
The coils are facing the soft iron rings to cut as much magnetic flux as possible. Here, coil 2 is used for suppressing vibration, and coil 1 and coil 3 are used for synchronously regenerating energy. The acc. sensor and the disp. sensor are used to measure the acceleration and the displacement of the controlled object, respectively. The coils are facing the soft iron rings to cut as much magnetic flux as possible. Here, coil2 is used for suppressing vibration, and coil1 and coil3 are used for synchronously regenerating energy. The acc. sensor and the disp. sensor are used to measure the acceleration and the displacement of the controlled object, respectively.

Working Functions
As shown in Figure 2, the proposed electromagnetic actuator has a vibration suppression function and energy regeneration function, which are used at the same time during vibration.
The vibration suppression function is used to decrease useless vibrations of the controlled object. The disp. sensor and acc. sensor measure the displacement and acceleration of the controlled object, respectively, which is fed back to a controller, where the amplified control current is applied to coil2 and the electromagnetic actuator can generate an output force for suppressing vibrations of the controlled object.
The energy regeneration function collects the kinetic energy from vibrations of the controlled object. Because of the relative motion between the controlled object and the PM rings, coil1 and coil3 cut magnet flux in the air gap of the electromagnetic actuator. An electromotive force (EMF) can be produced and stored in the battery by a series of circuits, such as a voltage boosting circuit, rectifier circuit, filter circuit, and voltage stabilizing circuit.

Design of the Electromagnetic Actuator
For the electromagnetic actuator, it is necessary to realize a compact size and a large output force. The output force of the electromagnetic actuator is mainly affected by the PM rings, coil2, and soft iron ring1. The PM rings were not made according to demand and use products from a magnetic materials company (NeoMag Co., Ltd.). In the design of the PM rings, their selection requirements

Working Functions
As shown in Figure 2, the proposed electromagnetic actuator has a vibration suppression function and energy regeneration function, which are used at the same time during vibration.
The coils are facing the soft iron rings to cut as much magnetic flux as possible. Here, coil2 is used for suppressing vibration, and coil1 and coil3 are used for synchronously regenerating energy. The acc. sensor and the disp. sensor are used to measure the acceleration and the displacement of the controlled object, respectively.

Working Functions
As shown in Figure 2, the proposed electromagnetic actuator has a vibration suppression function and energy regeneration function, which are used at the same time during vibration.
The vibration suppression function is used to decrease useless vibrations of the controlled object. The disp. sensor and acc. sensor measure the displacement and acceleration of the controlled object, respectively, which is fed back to a controller, where the amplified control current is applied to coil2 and the electromagnetic actuator can generate an output force for suppressing vibrations of the controlled object.
The energy regeneration function collects the kinetic energy from vibrations of the controlled object. Because of the relative motion between the controlled object and the PM rings, coil1 and coil3 cut magnet flux in the air gap of the electromagnetic actuator. An electromotive force (EMF) can be produced and stored in the battery by a series of circuits, such as a voltage boosting circuit, rectifier circuit, filter circuit, and voltage stabilizing circuit.

Design of the Electromagnetic Actuator
For the electromagnetic actuator, it is necessary to realize a compact size and a large output force. The output force of the electromagnetic actuator is mainly affected by the PM rings, coil2, and soft iron ring1. The PM rings were not made according to demand and use products from a magnetic materials company (NeoMag Co., Ltd.). In the design of the PM rings, their selection requirements were that the inner and outer diameters are the same. Four PM rings are magnetized along the axis direction and placed to face the same magnetic poles.  The vibration suppression function is used to decrease useless vibrations of the controlled object. The disp. sensor and acc. sensor measure the displacement and acceleration of the controlled object, respectively, which is fed back to a controller, where the amplified control current is applied to coil 2 and the electromagnetic actuator can generate an output force for suppressing vibrations of the controlled object.
The energy regeneration function collects the kinetic energy from vibrations of the controlled object. Because of the relative motion between the controlled object and the PM rings, coil 1 and coil 3 cut magnet flux in the air gap of the electromagnetic actuator. An electromotive force (EMF) can be produced and stored in the battery by a series of circuits, such as a voltage boosting circuit, rectifier circuit, filter circuit, and voltage stabilizing circuit.

Design of the Electromagnetic Actuator
For the electromagnetic actuator, it is necessary to realize a compact size and a large output force. The output force of the electromagnetic actuator is mainly affected by the PM rings, coil 2 , and soft iron ring 1 . The PM rings were not made according to demand and use products from a magnetic materials company (NeoMag Co., Ltd.). In the design of the PM rings, their selection requirements were that the inner and outer diameters are the same. Four PM rings are magnetized along the axis direction and placed to face the same magnetic poles.
Meanwhile, the height and thickness of coil 2 and the height of soft iron ring 1 affect the output of the proposed actuator. Here, an orthogonal experiment method is used to analyze the influences of the height and thickness of coil 2 and the height of soft iron ring 1 on the output force of the proposed actuator. The height and thickness of coil 2 and the height of soft iron ring 1 are determined as three factors of the orthogonal experiment method, and each factor has three levels, as shown in Table 1. In the simulations of the output force, coil 2 has a 0.5 mm diameter arranged in a single turn. The coil 2 current is calculated according to the maximum current of section 10 A/mm 2 . The coercivity of the PM rings is 868 KA/m and the residual flux density of the PM rings is 1.17 T. The magnetic permeability of the soft iron rings is 1.17 T. Assuming that any two factors cannot interact with each other, size optimization simulations of the actuator based on the FEM analysis by orthogonal experiment method were carried out to find the optimal combination of the maximum output force of the electromagnetic actuator, as listed in Table 2. The analysis by the orthogonal experiment method is shown in Table 3. K jk (k = 1, 2, 3) is the sum of the simulation results with the same level k in the jth column. K jpk is the mean value of the simulation results with the same level k in the jth column. R j is the range of K jpk . As can be observed, the combination (A3B3C2) constitutes the optimal parameters, which are the height of coil 2 (40 mm), the thickness of coil 2 (8 mm), and the height of soft iron ring 1 (10 mm), and the output force is 72.78 N. The heights of PM ring 2 and soft iron ring 2 are smaller than that of PM ring 1 and soft iron ring 1 , due to the overall sizes. The soft iron outer shell is split into two parts due to the simple assembly of the PM rings and soft iron rings. It must be noted that the safe distance between the coils and the soft iron shell must be ensured, and friction between the coils and other parts is avoided. The optimized configuration of the electromagnetic actuator is shown in Figure 3 and its main parameters are listed in Table 4. tuators 2020, 9, x FOR PEER REVIEW 6 of gs and soft iron rings. It must be noted that the safe distance between the coils and the soft iron she st be ensured, and friction between the coils and other parts is avoided. The optimized configuratio the electromagnetic actuator is shown in Figure 3 and its main parameters are listed in Table 4.  Height of soft iron ring1 10 hi2 Height of soft iron ring2 5 hc1 Coil1 and coil3 height 20 hc2 Coil2 height 40 wc Coils thickness 8 The magnetic flux distribution was explored in order to design the electromagnetic actuator, a own in Figure 4. The magnetic flux of the PM rings starts from the N pole and returns to the S pol ssing through the soft iron rings, air gap, coils, and soft iron outer shell.  The magnetic flux distribution was explored in order to design the electromagnetic actuator, as shown in Figure 4. The magnetic flux of the PM rings starts from the N pole and returns to the S pole, passing through the soft iron rings, air gap, coils, and soft iron outer shell.
Magnetic loops are shown in Figure 4a. The magnetic flux distribution was found by magnetic field analysis of FEM, as shown in Figure 4b, where the magnetic flux density in PM ring 1 is the largest, namely, about 1.2 T. The coils are facing the soft iron rings and the average value of the magnetic flux density in the air gap is about 0.3 T. It must be ensured that the magnetic flux of the soft iron materials is in an unsaturated state. The magnetic flux density of coil 2 is proportional to the output force of the electromagnetic actuator, therefore, the magnetic flux density in the air gap is one of the important factors that determines the overall performance of the electromagnetic actuator. Coil2 height 40 wc Coils thickness 8 The magnetic flux distribution was explored in order to design the electromagnetic actuator, as shown in Figure 4. The magnetic flux of the PM rings starts from the N pole and returns to the S pole, passing through the soft iron rings, air gap, coils, and soft iron outer shell.   In Figure 5, the relationship between the output force and the coil 2 current of the electromagnetic actuator has been simulated by Ansys and Maxwell. Here, coil 2 adopts a single-turn enameled wire with a 0.5 mm diameter, and its number of turns is 1280. The coil 2 current is applied between −1.9 A and 1.9 A with an interval of 0.1 A to ensure the normal operation of the coil. The output force range of the electromagnetic actuator is approximately −72.78 N to 72.56 N. The simulation result shows that the output force-current relationship has high linearity and that its output force-current coefficient It must be ensured that the magnetic flux of the soft iron materials is in an unsaturated state. The magnetic flux density of coil2 is proportional to the output force of the electromagnetic actuator, therefore, the magnetic flux density in the air gap is one of the important factors that determines the overall performance of the electromagnetic actuator. In Figure 5, the relationship between the output force and the coil2 current of the electromagnetic actuator has been simulated by Ansys and Maxwell. Here, coil2 adopts a single-turn enameled wire with a 0.5 mm diameter, and its number of turns is 1280. The coil2 current is applied between −1.9 A and 1.9 A with an interval of 0.1 A to ensure the normal operation of the coil. The output force range of the electromagnetic actuator is approximately −72.78 N to 72.56 N. The simulation result shows that the output force-current relationship has high linearity and that its output force-current coefficient is 38.3 N/A.

Fabrication of the Prototype
The prototype of the electromagnetic actuator has been fabricated, as shown in Figure 6. An excitation device consists of a cam and a motor, which has been adopted to generate a sinusoidal excitation signal to the actuator. The amplitude and frequency of the sinusoidal excitation can be changed by replacing the various cams and adjusting the motor (SGMJV-A5A3A21, Yaskawa Electric Corp.) speed.

Fabrication of the Prototype
The prototype of the electromagnetic actuator has been fabricated, as shown in Figure 6. An excitation device consists of a cam and a motor, which has been adopted to generate a sinusoidal excitation signal to the actuator. The amplitude and frequency of the sinusoidal excitation can be changed by replacing the various cams and adjusting the motor (SGMJV-A5A3A21, Yaskawa Electric Corp., Kitakyushu, Japan) speed.
The electromagnetic actuator is mounted on the excitation device. An eddy current displacement sensor (eddy current displacement sensor 1 ) (PU-20, AEC Corp., Carrollton, TX, USA) and acceleration sensor (acceleration sensor 1 ) (TYPE 7820A, Aco co.,LTD, Tokyo, Japan) are fixed on the controlled object for measuring its displacement and acceleration. Eddy current displacement sensor 2 , displacement sensor 3 (PU-20, AEC Corp.) and acceleration sensor 2 (TYPE 7820A, Aco co.,LTD) are installed on the excitation device for measuring its displacements and acceleration. Liner shafts and sliding bearings play a role of guidance here. The PM rings are made of Nd 2 Fe 14 B. The three coils are single-turn enameled wires with a 0.5 mm diameter, which have 612, 1262, and 617 turns, respectively. The controlled object weights 1.37 kg, with overall dimensions of Φ78 × 146 mm, and the mass without the excitation device is 5.01 kg, where the overall dimensions are about 140 mm × 140 mm × 173 mm. Figure 5. Output force-current relationship of the electromagnetic actuator.

Fabrication of the Prototype
The prototype of the electromagnetic actuator has been fabricated, as shown in Figure 6. An excitation device consists of a cam and a motor, which has been adopted to generate a sinusoidal excitation signal to the actuator. The amplitude and frequency of the sinusoidal excitation can be changed by replacing the various cams and adjusting the motor (SGMJV-A5A3A21, Yaskawa Electric Corp.) speed. The electromagnetic actuator is mounted on the excitation device. An eddy current displacement sensor (eddy current displacement sensor1) (PU-20, AEC Corp.) and acceleration sensor (acceleration sensor1) (TYPE 7820A, Aco co.,LTD) are fixed on the controlled object for measuring its displacement and acceleration. Eddy current displacement sensor2, displacement sensor3 (PU-20, AEC Corp.) and acceleration sensor2 (TYPE 7820A, Aco co.,LTD) are installed on the excitation device for measuring its displacements and acceleration. Liner shafts and sliding bearings play a role of guidance here. The PM rings are made of Nd2Fe14B. The three coils are single-turn enameled wires with a 0.5 mm

Characteristics of the Electromagnetic Actuator
The characteristics of the electromagnetic actuator determine its overall performance, and therefore, this section mainly studies and analyzes the electromagnetic actuator from the characteristics of the magnetic flux density and the output force. There are two experiments to measure and analyze these two characteristics of the electromagnetic actuator.
Experimental devices for measuring the magnetic flux density are shown in Figure 7a, composed of a gauss meter (Model 6010, Toyo Corp., Kariya-shi, Japan), height gauge, translation stage, and stator of the electromagnetic actuator. The stator and translation stage are fixed on a workbench, and the height gauge is fixed on the translation stage. The gauss meter's head is moved along the z-axis by the height gauge and the translation stage is used for movement along the y-axis. The axial range for measuring the magnetic flux density is from 0 to 104 mm, with an increment of 1 mm in the z-axis, and the radial positions for measuring the magnetic flux density are 1 mm, 3 mm, and 5 mm from the outer diameter of the PM rings. The length of the gauss meter's head is 60 mm and less than the height of the air gap; therefore, the magnetic flux density of the whole air gap is obtained based on the positional symmetry of the PM rings and soft iron rings.

Characteristics of the Electromagnetic Actuator
The characteristics of the electromagnetic actuator determine its overall performance, and therefore, this section mainly studies and analyzes the electromagnetic actuator from the characteristics of the magnetic flux density and the output force. There are two experiments to measure and analyze these two characteristics of the electromagnetic actuator.
Experimental devices for measuring the magnetic flux density are shown in Figure 7a, composed of a gauss meter (Model 6010, Toyo Corp.), height gauge, translation stage, and stator of the electromagnetic actuator. The stator and translation stage are fixed on a workbench, and the height gauge is fixed on the translation stage. The gauss meter's head is moved along the z-axis by the height gauge and the translation stage is used for movement along the y-axis. The axial range for measuring the magnetic flux density is from 0 to 104 mm, with an increment of 1 mm in the z-axis, and the radial positions for measuring the magnetic flux density are 1 mm, 3 mm, and 5 mm from the outer diameter of the PM rings. The length of the gauss meter's head is 60 mm and less than the height of the air gap; therefore, the magnetic flux density of the whole air gap is obtained based on the positional symmetry of the PM rings and soft iron rings.
The results for measuring the magnetic flux density are shown in Figure 7b. The experimental devices for measuring the output force are shown in Figure 8a. The devices consist of a PC with a digital signal processor (DSP (DS1104, dSPACE Corp.)) board, a power amplifier, a force sensor, an amplifier of the force sensor, and the proposed actuator. The force sensor (LUR-A-100NSA1, Japan KYOWA Corp.) was connected to the controlled object, which was fixed on The results for measuring the magnetic flux density are shown in Figure 7b. The maximum values of the N pole are located at axial displacements of 13 mm and 93 mm, and the minimum value of the S pole is at the axial displacement of 53 mm. The values of the magnetic flux density in radial displacements of 1 mm, 3 mm, and 5 mm at axial displacements of 13 mm were 531 mT, 363 mT, and 271 mT, respectively, which are the same as the ones at an axial displacement of 93 mm. The values of the magnetic flux density at radial displacements of 1 mm, 3 mm, and 5 mm at an axial displacement of 53 mm were −570 mT, −471 mT, and −390 mT, respectively. The magnetic flux densities of all curves did not change when the axial displacement changed from 22 mm to 42 mm, and from 64 mm to 84 mm. The experimental results show that the magnetic flux densities at the soft iron rings are relatively at their maximum.
The experimental devices for measuring the output force are shown in Figure 8a. The devices consist of a PC with a digital signal processor (DSP (DS1104, dSPACE Corp., Wixom, MI, USA)) board, a power amplifier, a force sensor, an amplifier of the force sensor, and the proposed actuator. The force sensor (LUR-A-100NSA1, Japan KYOWA Corp., Osaka, Japan) was connected to the controlled object, which was fixed on a bench supported two pillars. A signal generated by the DSP was amplified and applied to coil 2 . The output force values were recorded on the amplifier of the force sensor. The coil 2 current was changed from −1 A to 1 A, with an increment of 0.1 A.
Actuators 2020, 9, x FOR PEER REVIEW 9 of 17 output force values were recorded on the amplifier of the force sensor. The coil2 current was changed from −1 A to 1 A, with an increment of 0.1 A.
The results for measuring the output force are shown in Figure 8b. It can be seen that the relationship between the output force and the coil2 current is linear, and the output force-current coefficient is 39.49 N/A, which shows that the output force can be easily controlled by inputting the coil2 current. The error between the experiment results and the simulation results is within 3.0%.

Dynamics Model
The control model of the electromagnetic actuator is shown in Figure 9. It is assumed that only the z-axis motion of the controlled object is considered, and that the interference of the z-axis motion to other directions is neglected. The dynamics model in the z-axis is given as follows: where z is the displacement of the controlled object, m is the mass of the controlled object, c is the damping coefficient, kz is the parallel stiffness of three springs, fc is the output force of the electromagnetic actuator, fm is the external disturbance force, u is the displacement of the input excitation, ki is the output force-current coefficient of the electromagnetic actuator, and i is the current of the coil2. These parameters were established using the experimental and simulated results, as shown in Table 5. The results for measuring the output force are shown in Figure 8b. It can be seen that the relationship between the output force and the coil 2 current is linear, and the output force-current coefficient is 39.49 N/A, which shows that the output force can be easily controlled by inputting the coil 2 current. The error between the experiment results and the simulation results is within 3.0%.

Dynamics Model
The control model of the electromagnetic actuator is shown in Figure 9. It is assumed that only the z-axis motion of the controlled object is considered, and that the interference of the z-axis motion to other directions is neglected. The dynamics model in the z-axis is given as follows: where z is the displacement of the controlled object, m is the mass of the controlled object, c is the damping coefficient, k z is the parallel stiffness of three springs, f c is the output force of the electromagnetic actuator, f m is the external disturbance force, u is the displacement of the input excitation, k i is the output force-current coefficient of the electromagnetic actuator, and i is the current of the coil 2 . These parameters were established using the experimental and simulated results, as shown in Table 5.
where z is the displacement of the controlled object, m is the mass of the controlled object, c is the damping coefficient, kz is the parallel stiffness of three springs, fc is the output force of the electromagnetic actuator, fm is the external disturbance force, u is the displacement of the input excitation, ki is the output force-current coefficient of the electromagnetic actuator, and i is the current of the coil2. These parameters were established using the experimental and simulated results, as shown in Table 5.    Figure 10 is a block diagram of the electromagnetic actuator control system in the z-axis, which includes an acceleration controller and a position controller. The acceleration controller is composed of a regulator consisting of a 1st order numerator, 1st order denominator, and an integrating compensator. The position controller is the same as the acceleration controller. To improve the response speed of the output force, a local current feedback loop using a proportional-integral (PI) controller has been added.

Controller Design
Actuators 2020, 9, x FOR PEER REVIEW 10 of 17  Figure 10 is a block diagram of the electromagnetic actuator control system in the z-axis, which includes an acceleration controller and a position controller. The acceleration controller is composed of a regulator consisting of a 1st order numerator, 1st order denominator, and an integrating compensator. The position controller is the same as the acceleration controller. To improve the response speed of the output force, a local current feedback loop using a proportional-integral (PI) controller has been added. Assuming that the bandwidth of the current feedback loop is infinite [i.e., that (s)/I r (s) = 1] and that the integrating compensators can be neglected, the transfer function of the acceleration loop (A/A r ) is shown in Equation (3). Based on this, the controller parameters a 0a , b 1a , and b 0a can be automatically defined, such that the denominator of the transfer function (A/A r ) approaches (m + k i b 1a )×(s + p 1 ) 3 after deciding only on the value of pole p 1 , as shown in Equation (4). The control principle of the position loop is the same as for the acceleration loop. The transfer function of the position loop (Z/Z r ) is shown in Equation (5), and adjusting the value of pole p 2 can change a 0z , b 1z , and b 0z , as shown in Equation (6). The parameters of the integral compensators in the acceleration loop and the position loop were adjusted by experiments here. When p 1 and p 2 were set to −31.4 and −314, and the values of the PI controller parameters k pi and k ii were set to 60 and 1000, respectively, the decided controller parameters were recorded, and these are shown in Tables 6 and 7. Compared with the proportion integral derivative (PID) controller, this controller has the advantages of having a simple structure, fewer adjustment parameters, and the characteristic of saving time. 3.0 k ia 0

Experimental Control System
As shown in Figure 11, the schematic of the electromagnetic actuator shows signal loops of the control system for various physical quantities. The vibration excitation is carried out by the motor driving the cam, where the vibration generated by the cam is then transmitted to the excitation object, electromagnetic actuator, and controlled object. The electromagnetic actuator is controlled by a DSP (DS1104, dSPACE Corp.), and the sampling rate was set to 10 kHz. Two acceleration sensors and three displacement sensors were connected to the A/D converter of the DSP. The control signal acts as an output to the D/A converter of the DSP via a power amplifier applied to coil 2 . Coil 1 and coil 3 are connected in series, and the voltage signal is transmitted to the A/D converter of the DSP for regenerating the EMF. Actuators 2020, 9, x FOR PEER REVIEW 12 of 17 Figure 11. Schematic of the control system.

Impulse Excitation Response
In the impulse excitation experiment, the acceleration signals of the excitation object and controlled object are measured. A rubber hammer hits the excitation object, the impulse excitation response results of the electromagnetic actuator are shown in Figure 12. The peak to peak value of the acceleration of the excitation object is 114.26 m/s 2 , and the peak to peak value of the acceleration of the controlled object is 3.14 m/s 2 . It can be seen that the acceleration of the controlled object is smaller than that of the excitation object, and that the acceleration ratio of the controlled object to the excitation object is 2.75%. The experimental results show that the proposed actuator can evidently decrease the external impulse disturbance.

Sinusoidal Excitation Response
To verify the vibration suppression and energy regeneration effect of the electromagnetic actuator, the sinusoidal excitation signal with a 3 mm amplitude and 5 Hz frequency was used as the input signal, the experimental results of which are shown in Figure 13. In Figure 13b-d, the fundamental frequency components of the displacement of the controlled object, output force, and EMF were extracted by a low-pass filter with a cutoff frequency of two-times that of the excitation frequency.  Figure 11. Schematic of the control system.

Impulse Excitation Response
In the impulse excitation experiment, the acceleration signals of the excitation object and controlled object are measured. A rubber hammer hits the excitation object, the impulse excitation response results of the electromagnetic actuator are shown in Figure 12. The peak to peak value of the acceleration of the excitation object is 114.26 m/s 2 , and the peak to peak value of the acceleration of the controlled object is 3.14 m/s 2 . It can be seen that the acceleration of the controlled object is smaller than that of the excitation object, and that the acceleration ratio of the controlled object to the excitation object is 2.75%. The experimental results show that the proposed actuator can evidently decrease the external impulse disturbance.
Actuators 2020, 9, x FOR PEER REVIEW 12 of 17 Figure 11. Schematic of the control system.

Impulse Excitation Response
In the impulse excitation experiment, the acceleration signals of the excitation object and controlled object are measured. A rubber hammer hits the excitation object, the impulse excitation response results of the electromagnetic actuator are shown in Figure 12. The peak to peak value of the acceleration of the excitation object is 114.26 m/s 2 , and the peak to peak value of the acceleration of the controlled object is 3.14 m/s 2 . It can be seen that the acceleration of the controlled object is smaller than that of the excitation object, and that the acceleration ratio of the controlled object to the excitation object is 2.75%. The experimental results show that the proposed actuator can evidently decrease the external impulse disturbance.

Sinusoidal Excitation Response
To verify the vibration suppression and energy regeneration effect of the electromagnetic actuator, the sinusoidal excitation signal with a 3 mm amplitude and 5 Hz frequency was used as the input signal, the experimental results of which are shown in Figure 13. In Figure 13b-d, the fundamental frequency components of the displacement of the controlled object, output force, and EMF were extracted by a low-pass filter with a cutoff frequency of two-times that of the excitation frequency.

Sinusoidal Excitation Response
To verify the vibration suppression and energy regeneration effect of the electromagnetic actuator, the sinusoidal excitation signal with a 3 mm amplitude and 5 Hz frequency was used as the input signal, the experimental results of which are shown in Figure 13. In Figure 13b-d, the fundamental frequency components of the displacement of the controlled object, output force, and EMF were extracted by a low-pass filter with a cutoff frequency of two-times that of the excitation frequency.
Actuators 2020, 9, x FOR PEER REVIEW 12 of 17 Figure 11. Schematic of the control system.

Impulse Excitation Response
In the impulse excitation experiment, the acceleration signals of the excitation object and controlled object are measured. A rubber hammer hits the excitation object, the impulse excitation response results of the electromagnetic actuator are shown in Figure 12. The peak to peak value of the acceleration of the excitation object is 114.26 m/s 2 , and the peak to peak value of the acceleration of the controlled object is 3.14 m/s 2 . It can be seen that the acceleration of the controlled object is smaller than that of the excitation object, and that the acceleration ratio of the controlled object to the excitation object is 2.75%. The experimental results show that the proposed actuator can evidently decrease the external impulse disturbance.

Sinusoidal Excitation Response
To verify the vibration suppression and energy regeneration effect of the electromagnetic actuator, the sinusoidal excitation signal with a 3 mm amplitude and 5 Hz frequency was used as the input signal, the experimental results of which are shown in Figure 13. In Figure 13b-d, the fundamental frequency components of the displacement of the controlled object, output force, and EMF were extracted by a low-pass filter with a cutoff frequency of two-times that of the excitation frequency. The peak value of the input displacement was 3.045 mm, which was caused by a manufacturing error of the cam. In Figure 13a, the peak value of the base harmonic of the controlled object displacement is 0.045 mm, accounting for 1.46% of the excitation object displacement in Figure 13b, and there is a quarter period of phase difference between the displacement of the excitation object and the controlled object. The output force of the electromagnetic actuator was calculated and deduced by measuring the coil2 current. The peak value of the base harmonic of the output force was 17.15 N, as shown in Figure 13c. Figure 13d shows that the peak value of the base harmonic of the EMF for the energy regeneration function was 1.97 V. Figure 13e-h show the results of fast fourier transform (FFT) analysis for the input displacement of the excitation object, displacement of the controlled object, output force, and EMF, respectively. The displacement of the excitation object contains the fundamental frequency components of 5 Hz, 10 Hz, and its other harmonics, which are caused by the machining error of the cam. The displacement of the controlled object, output force and EMF contain the fundamental frequency components of 5 Hz, 10 Hz and their other harmonics, which are brought about by the machining error of the cam and pulse width modulation (PWM) of the motor. These amplitudes at a 5 Hz fundamental frequency are much higher than the ones at their harmonic frequencies, and therefore, the experimental results were filtered to eliminate the influences of the harmonics.
To estimate the efficiency of energy regeneration of the electromagnetic actuator, energy supplied to the electromagnetic actuator in a period of 0.2 s, and regenerated energy in a period of 0.2 s, were calculated. The efficiency of the proposed actuator is a ratio of regenerated energy in a period of 0.2 s to energy supplied to the electromagnetic actuator in a period of 0.2 s. According to the output force of the electromagnetic actuator and the velocity calculated by the displacement of the controlled object, energy supplied to the electromagnetic actuator in a period of 0.2 s is 0.164 J. Assuming that the external resistance is 0 Ω, the series resistances of coil1 and coil3 are 20.45 Ω, therefore, regenerated energy in a period of 0.2 s is 0.019 J, and the efficiency of the proposed electromagnetic actuator is 11.59%. The generated electromotive force is relatively small and cannot The peak value of the input displacement was 3.045 mm, which was caused by a manufacturing error of the cam. In Figure 13a, the peak value of the base harmonic of the controlled object displacement is 0.045 mm, accounting for 1.46% of the excitation object displacement in Figure 13b, and there is a quarter period of phase difference between the displacement of the excitation object and the controlled object. The output force of the electromagnetic actuator was calculated and deduced by measuring the coil 2 current. The peak value of the base harmonic of the output force was 17.15 N, as shown in Figure 13c. Figure 13d shows that the peak value of the base harmonic of the EMF for the energy regeneration function was 1.97 V. Figure 13e-h show the results of fast fourier transform (FFT) analysis for the input displacement of the excitation object, displacement of the controlled object, output force, and EMF, respectively. The displacement of the excitation object contains the fundamental frequency components of 5 Hz, 10 Hz, and its other harmonics, which are caused by the machining error of the cam. The displacement of the controlled object, output force and EMF contain the fundamental frequency components of 5 Hz, 10 Hz and their other harmonics, which are brought about by the machining error of the cam and pulse width modulation (PWM) of the motor. These amplitudes at a 5 Hz fundamental frequency are much higher than the ones at their harmonic frequencies, and therefore, the experimental results were filtered to eliminate the influences of the harmonics.
To estimate the efficiency of energy regeneration of the electromagnetic actuator, energy supplied to the electromagnetic actuator in a period of 0.2 s, and regenerated energy in a period of 0.2 s, were calculated. The efficiency of the proposed actuator is a ratio of regenerated energy in a period of 0.2 s to energy supplied to the electromagnetic actuator in a period of 0.2 s. According to the output force of the electromagnetic actuator and the velocity calculated by the displacement of the controlled object, energy supplied to the electromagnetic actuator in a period of 0.2 s is 0.164 J. Assuming that the external resistance is 0 Ω, the series resistances of coil 1 and coil 3 are 20.45 Ω, therefore, regenerated energy in a period of 0.2 s is 0.019 J, and the efficiency of the proposed electromagnetic actuator is 11.59%. The generated electromotive force is relatively small and cannot charge the battery, and therefore, the electromotive force can be increased by a voltage boosting circuit, or by considering a new structure to improve the energy regeneration efficiency of the proposed actuator in the future.
The vibration control and energy regeneration experiments were carried out when the sinusoidal excitation signals were respectively set to various amplitudes and frequencies. All the experimental results are expressed by the amplitude ratio and the ratio of the EMF to the input amplitude, as shown in Figure 14. The amplitude ratio is defined as the ratio of the amplitude of the controlled object to the excitation object in the fundamental frequency component, and the ratio of the EMF to the input amplitude is defined as the ratio of the regenerated EMF to the input amplitude of the excitation object in the fundamental frequency component. When the input amplitudes were 1 mm, 2 mm, and 3 mm, the lowest frequency was 1 Hz and the highest frequencies were 7 Hz, 10 Hz, and 15 Hz, respectively. As shown in Figure 14a, the amplitude ratios are in the range of 0% to 5.5%. According to the trends of these three curves, as the amplitude and frequency of the excitation signals increase, the effect of vibration suppression decreases. In the future, we plan to change the controller of the actuator to improve the vibration suppression effect. As shown in Figure 14b, the ratios of the EMF to the amplitude of the three curves from 1 mm to 3 mm input amplitudes are 0.128, 0.125, and 0.127. When the frequency increases, the ratio of the EMF to the input amplitude also increases.
Actuators 2020, 9, x FOR PEER REVIEW 14 of 17 charge the battery, and therefore, the electromotive force can be increased by a voltage boosting circuit, or by considering a new structure to improve the energy regeneration efficiency of the proposed actuator in the future. The vibration control and energy regeneration experiments were carried out when the sinusoidal excitation signals were respectively set to various amplitudes and frequencies. All the experimental results are expressed by the amplitude ratio and the ratio of the EMF to the input amplitude, as shown in Figure 14. The amplitude ratio is defined as the ratio of the amplitude of the controlled object to the excitation object in the fundamental frequency component, and the ratio of the EMF to the input amplitude is defined as the ratio of the regenerated EMF to the input amplitude of the excitation object in the fundamental frequency component. When the input amplitudes were 1 mm, 2 mm, and 3 mm, the lowest frequency was 1 Hz and the highest frequencies were 7 Hz, 10 Hz, and 15 Hz, respectively. As shown in Figure 14a, the amplitude ratios are in the range of 0% to 5.5%. According to the trends of these three curves, as the amplitude and frequency of the excitation signals increase, the effect of vibration suppression decreases. In the future, we plan to change the controller of the actuator to improve the vibration suppression effect. As shown in Figure 14b, the ratios of the EMF to the amplitude of the three curves from 1 mm to 3 mm input amplitudes are 0.128, 0.125, and 0.127. When the frequency increases, the ratio of the EMF to the input amplitude also increases.
(a) (b) Figure 14. Vibration control and energy regeneration experimental results: (a) Amplitude ratio of the controlled object to excitation object; (b) ratio of the EMF to the input amplitude.

Conclusions
This paper has proposed a new type of electromagnetic actuator that may be applied in mechanisms with low vibration and a low frequency, such as in transportation machinery, mining machinery, construction machinery, and other areas. The proposed actuator has two working functions, namely, a vibration suppression function and an energy regeneration function. These functions can relieve vibrations generated by a vibrating component and collect a part of the energy dissipated by the vibrating component. The structure of the electromagnetic actuator was designed by the static magnetic field simulation of FEM analysis, and the prototype of the actuator was fabricated. On this basis, the characteristics of the electromagnetic actuator were explored, finding an output force-current coefficient of 39.49 N/A. Moreover, the control system of the experiments was constructed by a position controller and an acceleration controller. Impulse excitation response and sinusoidal excitation response analyses were carried out. The experimental results show that the vibration acceleration of the controlled object can be suppressed from 114.26 m/s 2 to 3.14 m/s 2 , which is within 2.75% under the impulse excitation response. For the sinusoidal excitation response, the input signal under a 3 mm amplitude and 5 Hz frequency was set as the input of the excitation object, where the peak value of the displacement of the controlled object was 0.045 mm, accounting for 1.46% of that of the excitation object. The peak value of the regenerated EMF was 1.97 V, where the efficiency for regenerating energy is 11.59%. Under multiple frequencies and amplitudes, the amplitude ratio

Conclusions
This paper has proposed a new type of electromagnetic actuator that may be applied in mechanisms with low vibration and a low frequency, such as in transportation machinery, mining machinery, construction machinery, and other areas. The proposed actuator has two working functions, namely, a vibration suppression function and an energy regeneration function. These functions can relieve vibrations generated by a vibrating component and collect a part of the energy dissipated by the vibrating component. The structure of the electromagnetic actuator was designed by the static magnetic field simulation of FEM analysis, and the prototype of the actuator was fabricated. On this basis, the characteristics of the electromagnetic actuator were explored, finding an output force-current coefficient of 39.49 N/A. Moreover, the control system of the experiments was constructed by a position controller and an acceleration controller. Impulse excitation response and sinusoidal excitation response analyses were carried out. The experimental results show that the vibration acceleration of the controlled object can be suppressed from 114.26 m/s 2 to 3.14 m/s 2 , which is within 2.75% under the impulse excitation response. For the sinusoidal excitation response, the input signal under a 3 mm amplitude and 5 Hz frequency was set as the input of the excitation object, where the peak value of the displacement of the controlled object was 0.045 mm, accounting for 1.46% of that of the excitation object. The peak value of the regenerated EMF was 1.97 V, where the efficiency for regenerating energy is 11.59%. Under multiple frequencies and amplitudes, the amplitude ratio of the controlled object and excitation object was within 5.5%, and ratio of the EMF to the input amplitude was 0.13, which can predict the trend of the regenerated energy.
This electromagnetic actuator has great potential to be adopted as a viable development for occasions with millimeter-sized vibrations. The next step is to conduct a thorough study of the amplitude ratio of the controlled object to excitation object, and the ratio of the EMF to the input amplitude, to improve the control effect at high frequencies in the active control function. In the energy regeneration function, the efficiency of the proposed electromagnetic actuator will be verified by experiments, and this efficiency needs to be further improved.