Design and optimization of Lorentz motors in a precision active isolator

Abstract

In a precision active isolator, Lorentz motors (LM), a permanent magnet linear synchronous motor (PMLSM), often are used as an actuator. As a key equipment, the performance of the precision active isolator is directly affected by the LM. In the design of LM, many factors must be considered, a comprehensive analysis including active isolation system, structural dynamics, magnetic field and cooling, should be integrated to optimize a LM. First, a dynamical model about an active isolator is built. A measurement method of eccentricity in the precision active isolator is presented for the LMs. A simple decoupling magnetic equivalent circuit (MEC) is proposed to predict MFD distribution of the air-gap in the design of LM, and an water cooling system is presented and analysed. Finally, an optimization combining effective cost function is used to find the optimal motor dimensions. The performance of the optimized LM is described by simulation and experimental results, which also confirm effectiveness of the optimized design.

Keywords

Active isolator lorentz motor optimal design water cooling decoupling magnetic equivalent circuit

1. Introduction

In the fields of precision, especially ultra-precision, such as microelectronics processing and Micro-Electro-Mechanical Systems (MEMS), machines and corresponding measuring devices are very sensitive to vibrational noise. With the rapid advance of manufacturing technology, it has become a serious issue how to isolate external vibration [1]. In the mechanical system, a small amount of vibration (which come from ground) may introduce undesirable noise. Some ground vibrations, such as earthquake, movement of vehicles and other movers, may be the source of the vibrational noise that will affect directly effectiveness and accuracy of operations [2, 3], and may shorten the lifetime of the machine [3]. To isolate the undesirable noise from ground to precision machines, precision vibration isolators are widely applied [4, 5] and are placed between vibrated mass and vibration source, to reduce the transmission of excitation forces. In general, vibration isolators are divided into two types: passive and active [6], and the main difference between them is that external energy is needed for the latter technique. A passive vibrator, in which its damping and stiffness are not adjustable, can only be effective on single-frequency vibrations and there is bad effect of vibration isolation under the changing of external excitation frequency [7]. In isolators, lower stiffness is suitable for isolating ground vibrations [8], but higher stiffness can be applied for supporting a heavy load. It is impossible for a conventional passive vibration isolation system to realize high stiffness and low suspension simultaneously [9], A passive system inevitably involves a tradeoff between lower stiffness and heavy load. To provide a more favourable compromise between static and dynamic stiffness, active isolation solutions such as skyhook damping must be used [10]. So an active system is composed of a passive sub-system and an active sub-system. In the active sub-system, actuators and sensors are essential components and enable damping. In their operation, the electrical signal of the sensors are integrated and amplified by a controller, and the active isolator provides a control effort to operate the actuator. So the heart of an active system is the actuator, as it provides the driving mechanism for translational force of the isolator [11].

Actuators of an active isolator can be separated into several types: mechanical mechanisms, piezoelectric (PZT) actuators [12, 13], pneumatic springs, cylinders and electromagnetic motors [7]. Electromagnetic Lorentz motor (LM) is used widely as an actuator in modern industry, due to its excellent advantages, such as direct drive, high acceleration, fast response time, very high accuracy, low noise, etc. [14, 15, 16]. Over a limited range of travel with fine precision, there has been more research into the correlative technology [17]. In [18], LM is used as an actuator in cell phone cameras. In the servo control of DVD, LMs are applied to drive an objective lens during autofocusing process [19]. In the hard disc driver, a LM actuator plays a very important role [20]. In an active dynamic vibration isolator, LM is also applied successfully [21].

For different applications of LMs, an universal design may be inapplicable due to difficulties in mechanical construction, design constraints, etc. In an precision active isolator system, working environment has strict requirement for the temperature of the LMs, for the coils of LMs are a main heat source. Additionally, the isolator system is thus a necessary requirement for external and installation dimensions. The aim of this paper is to design a LM with small volume, given force coefficient, limited heat dissipation etc. It is clearly evident that the optimal design can satisfy better above requirements.

Figure 1.

Simplified schematic of hybrid vibration isolator system: (a) Simplified schematic; (b) Dimensions and diagrams of the active isolator.

As shown in Fig. 1a, an active isolator system has three isolators. A pneumatic suspension is a passive sub-system and the active sub-system is mainly composed of the LMs, controller and sensors as illustrated in Fig. 2a. In a active isolator, a LM acts on a payload without intermediate as illustrated in Fig. 2a, and it greatly enhances the performance and efficiency. Simultaneously, this also means that the performance of LMs will affect directly the active isolator, and the design and optimization are critical for a precision active isolator.

Figure 2.

Schematic of an active isolation system: (a) Schematic of an active sub-system; (b) Simplified schematic of the active sub-system.

2. Dynamic analysis

A good isolator works as a low pass filter for disturbance transmitting from ground. In this low pass filter, the cut-off frequency is less than 5 Hz. The active isolator system is illustrated in Fig. 2. $M_{2}$ $(M_{2}=M+m)$ and $M_{1}$ represent the mass of an isolated object and the base of the isolator, respectively. $M$ and $m$ are the mass of the platform and the payload, respectively. The spring $k_{1}$ and damper $c_{1}$ represent an passive vibration isolator.

A passive sub-system only can reduce high frequency disturbance. If we want $M_{2}$ to remain static in low frequency, it is simple but effectual to exert an extra force $F_{2}$ (the thrust force of an actuator) to keep $M_{2}$ equilibrium. For a LM actuator only exerts force $F_{2}$ to restrain the disturbance according to the velocity acquired from the sensor (velocity sensor, Geophone GS-11D ), it act as a damper (damping coefficient $C_{2}$ ). Due to the sensors (Geophone GS-11D) are mounted onto the platform and measure directly the absolute velocity, the LM can be expressed as $F_{2}=c_{2}\dot{x}_{2}$ .

To keep $M_{2}$ in equilibrium, the force $F_{2}$ , and the $M_{2}$ and its absolute displacement $x_{2}$ are related by the following equations

$k_{1}(x_{1}-x_{2})+c_{1}(\dot{x}_{1}-\dot{x}_{2})+c_{2}\dot{x}_{2}=M_{2}\ddot{% x}_{2}$ (1)

Here, $x_{1}$ is the absolute displacement of $M_{1}$ .

In general, the damping coefficient $c_{1}$ can be ignored. In a hybrid vibration isolator system, the platform is an equilateral triangle as shown in Fig. 1b. The thrusts of LMs are applied at point A, B and C, and these points form equilateral triangle $\Delta$ ABC. One reference frame, xoy, is attached on the center of the platform, namely the center of $\Delta$ ABC. Let the point where AB cuts CD be called D as shown in Fig. 1b.

While the vibration isolator system is in operation, the center-of-mass of the payload should coincide with the platform to avoid the eccentric load among three isolator. But the quick motion of kinematic mechanisms in a payload causes the center-of-mass of payload to deviate from its original position. In order to compensate for the deviation, two linear permanent-magnet synchronous motors are arranged on the top of platform along the x- and y-axes, separately. Once the center-of-mass of the payload deviates from the original location in xoy frame, the PMLSMs immediately compensate the deviation. So how do the two PMLSMs operate is critical.

As the moving parts of payload are always in continual rapid motion, only the LMs can make a quick response to the movement, an corresponding dynamic torque for the LMs is analyzed as illustrated in Fig. 1b. The analytical expression can be obtained as

$\left\{{\begin{array}[]{l}\frac{1}{3}F_{I}a+\frac{1}{3}F_{\textit{II}}a+mgy_{0% }-\frac{2}{3}F_{\textit{III}}a=0\\ F_{I}b+mgx_{0}-F_{\textit{II}}b=0\\ \end{array}}\right.$ (2)

where CD $=$ a, AD $=$ b. $F_{I}$ , $F_{\textit{II}}$ and $F_{\textit{III}}$ are the thrusts of LMs in three vibration isolators, respectively. $g$ describes acceleration of gravity. $x_{0}$ and $y_{0}$ are the deviation of payload.

Lorentz’s law is used to calculate thrust $F_{i}=k_{\textit{LM}}i$ . Where $F_{i}$ ( $i=\textit{I},\textit{II}$ and III) is thrust of a LM. $k_{\textit{LM}}$ describes force coefficient. $i$ is applied current of coil. If the current flowing in the coil is measured accurately, the corresponding thrust can be obtained indirectly. In the active isolator, the current is sampled by Hall sensors and the current feedback control is designed in driving circuits.

Under above conditions, the deviation of payload $x_{0}$ and $y_{0}$ can obtained from Eq. (2). Then using ( $M+m)x_{01}=mx_{0}$ and ( $M+m)y_{01}=my_{0}$ , the deviation ( $x_{01}$ , $y_{01}$ ) of the mass $M+m$ can be obtained. They are very useful for PMLSMs to compensate the deviation of payload.

3. Optimal design of LM

In the optimal design of LM, the thrust $F$ of LM is an important parameter and depends on many factors and can be expressed as

$\displaystyle F=f(B,g_{air},d,n,T)$ (3)

where $B$ denotes MFD of air gap, $g_{air}$ represents length of air gap in LM, $T$ is working temperature of LM, and $d$ and $n$ denote diameter and number of turns of coil, respectively.

Since there are three isolators in a isolator system as illustrated in Fig. 1, the required force for the active isolator can be obtained from Eq. (1) as

$F^{*}=3F_{t}=M_{2}\dot{x}_{2}$ (4)

Here, $F*$ is the force required to remain the payload at rest, and $F_{t}$ represents the force of one actuator. To gain the ideal thrust of LM, the cost function of LM can be described by the method of least square as $(F_{t}-F)^{2}$ .

3.1 Design and analysis of the water cooling system

As the LM often operates continuously, its temperature would increase because of the heating of coil. If the heat produced from the coil can not be remove timely, it would be transmitted to working environment to cause the demagnetization of LM and affect the performance of optical measuring unit. Furthermore, it would cause the LM overheating. So the peak and normal thrust of LM is limited by the demagnetization or overheating of the winding insulation. To overcome this affection of temperature, an appropriate LM cooling method should be provided in the design stage [22]. Among many passive and active cooling solutions, liquid cooling, which can remove a large amount of heat, has attracted considerable attention in industry.

Figure 3.

Proposed structure of LM: (a) Mover; (b) Photo of the mover; (c) Stator; (d) Simplified layout of LM.

The structure of LM is shown in Fig. 3. The coil is installed in a frame, and the frame have an inlet and outlet of water cooling system as illustrated in Fig. 3a. Four magnets constitute closed loop magnetic lines of force and magnetic lines of force traverse through two air gaps altogether as shown in Fig. 3c and d. A good thermal insulated material such as insulating ceramic covers are used to seal symmetrically on both sides of the frame, and to prevent heat to diffuse outwards. Additionally, in the course of winding the coils, it will guarante the insulation to daub continuously the coils with eposy resin. In the cooling system, the neutral water is used.

As shown in Fig. 3a, the mover has three pairs parallel supports, a core, a frame, and a water cooling system. The inlet, outlet, and the current direction of the water cooling system are shown in Fig. 3a. The supports can not only fix the coil on the frame, but transmit the Lorentz force of the coil to the frame. The coil and core are connected by the supports, when the Lorentz force emerges, it can be transmitted by the supports. As shown in Fig. 3b, the mounting hole and the fixed block are assembled together by interference fit, and a pair of supports are bolted together with three un-conducted magnetic bolts. So the supports, coil and core are formed as one body. When the water cooling system working, they can keep original position, and Lorentz force can be transmitted smoothly. Simultaneously, they are thinner than the frame, and can be completely immersed by cooling water. To show the supports clearly, half of the supports has been removed from mounting position as shown in Fig. 3a.

Additionally, the physical sizes of the designed LM are required by the isolator system. If the structure of LM becomes bigger than the limit, the LM can’t be installed successfully.

In the vibration isolator system, the coil of the LM is responsible for heat dissipation. Water cooling is used to limit heat dissipation. The cooling capacity for LM’s heat dissipation can be evaluated using thermodynamic principles.

$\displaystyle K=\rho_{0}CQ\Delta T$ (5)

Where $\rho_{0}=$ 103 kg/m ${}^{3}$ is the density of cooling water. $C=$ 4.187 kJ/(kg. ${}^{\circ}$ C) is the specific heat capacity of cooling water. $Q$ presents the flow of cooling water. $T$ describes the temperature rise of cooling water.

As the inductance of coil is little and can be ignored, the heat produced by LM can be evaluated according to Joule law

$P_{LM}=\frac{U_{m}^{2}}{R}=\frac{U_{m}^{2}}{\frac{\rho_{c}l}{S}}$ (6)

Where $U_{m}$ is voltage of coil. $R$ is resistance of coil and $\rho_{c}=$ 1.772 $\times$ 10 ${}^{-8}$ $\Omega$ m presents the resistivity of copper, $l$ presents length of coil, $S$ is cross-sectional area of enameled wire.

In order to ensure the system of vibration isolator work normally within temperature, the heat source and cooling should satisfy the condition

$P_{LM}=\frac{U_{m}^{2}}{R}\leqslant K=\rho_{0}CQ\cdot\Delta T$ (7)

3.2 Proposed magnetic analysis of the LM

Figure 3d shows the structure of stator. Here, $g_{air}$ presents the air gap between two magnet poles; $t$ is the gap distance between adjacent poles; $l_{m}$ and $l_{s}$ describe the thickness of the PM and steel-yoke, respectively. $w$ and $w_{m}$ denote the weight of the steel-yoke and PM, respectively (the length of PM in LM is 100 mm).

To avoid magnetic saturation, the geometric size and material properties of steel-yoke have been chosen elaborately ( $l_{m}<l_{s}$ ). With an analysis to Kirchoff’s voltage law in electricity, the MEC [23, 24, 25, 26, 27] is applied to analyze the MFD distribution.

In a magnetic field, every magnetic line of force makes a closed route from the north to the south and don’t cross each other. If the magnetic lines of force are separated into groups, every group doesn’t cross each other. In analysis of the motor, it is difficult and complex to build an accurate MEC modeling including leakage flux [28, 29, 30, 31]. In LM, leakage flux only happens in two edges of PMs. So, magnetic lines of force of LM can be divided into three groups and every group has independent flux sources and loop. The leakage flux only happen in two groups and middle MEC can be nearly ideal if the middle flux and loop are divided narrow (the width of the middle magnetic line of force is set as 2 mm) as shown in Fig. 4a. Because flux leakage occurs mainly near the ends of the magnet pole, the width of the flux source of flux leakage must be smaller than half of the thickness of permanent magnet. Additionally, for weak flux leakage, the width of the middle magnetic line of force must is less than half the permanent magnet, and assumed to satisfy one quadrant of the thickness.

Figure 4.

MEC of LM: (a) The decoupling MEC of LM; (b) The middle MEC of LM; (c) The typical MEC of LM.

Figure 4b depicts the middle sub-MEC and its simplified sub-MEC. Here, $\phi_{sm}$ is the flux source of the magnet pole in the middle sub-MEC; $R_{m}$ is the reluctance corresponding to the flux $\phi_{sm}$ . $R_{g}$ represents the air-gap reluctance in the middle sub-MEC. $R_{s}$ denotes the reluctance in the steel-yoke. According to the width of the corresponding flux tubes of $R_{s}$ , $R_{g}$ and $R_{m}$ , as illustrated in Fig. 4b, applying the formula of resistance yields [32]

$R_{s}=\frac{\frac{w}{5}}{\mu_{s}\mu_{0}\times(w_{m}+t)\times L},\quad R_{g1}=% \frac{g_{air}}{\mu_{0}A_{g}},\quad R_{m}=\frac{l_{m}}{\mu_{rm}\mu_{0}A_{m}}$

Here, $L$ is the length of the PM in the LM (100 mm). $A_{g}$ and $A_{m}$ represent the flux surface areas of the air-gap and PM in the middle sub-MEC, respectively.

By flux division,

$\displaystyle\phi_{g}=\frac{2R_{m}}{2R_{m}+(R_{s}+R_{g})}\phi_{sm}=\frac{2% \times\frac{l_{m}}{\mu_{0}\mu_{rm}A_{m}}}{2\times\frac{l_{m}}{\mu_{0}\mu_{rm}A% _{m}}+R_{s}+\frac{g_{air}}{\mu_{0}A_{g}}}\phi_{sm}$ (8) $\displaystyle B_{g}=\frac{2\times\frac{l_{m}}{\mu_{rm}A_{m}}}{2\times\frac{l_{% m}}{\mu_{rm}A_{m}}+\frac{\frac{w}{5}}{\mu_{s}\times(w_{m}+t)\times L}+\frac{g_% {air}}{A_{g}}}\times B_{sm}$ (9)

Here, $\phi_{g}$ is the air-gap flux that passes through $R_{g}$ . $B_{sm}$ and $B_{g}$ are the MFDs corresponding to fluxes $\phi_{sm}$ and $\phi_{g}$ , respectively. $A_{sm}$ is the flux surface area of magnet pole. $\mu_{0}$ and $\mu_{rm}$ represent air permeability and the relative permeability of the PM, respectively. $H_{c}$ denotes magnetic coercive force.

Using Eq. (9), the effect of $g_{air}$ and t on $B_{g}$ is plotted in Fig. 5, This figure explains that $g_{air}$ plays a prominent role and the relationship is magnetic density in the air gap increases as reducing $g_{air}$ . Although t has similar effect to MFD on air gap, by contrast, it is too slight to sense the change.

3.3 Typical magnetic analysis of the LM

The typical MEC [4, 21, 31] of LM is shown in Fig. 4c. In Fig. 4c, all branches of the MEC are coupled together to become a MEC network, and the analytical accuracy of the air-gap MFD depends on every reluctance in the MEC network. It can have a adverse effect on the accuracy of the air-gap MFD.

$\displaystyle\phi_{g}=\frac{2\frac{R_{1}R_{s2}}{R_{1}+R_{s2}}R_{3}}{2\frac{R_{% 1}R_{s2}}{R_{1}+R_{s2}}R_{3}+(2\frac{R_{1}R_{s2}}{R_{1}+R_{s2}}+R_{3})R_{g-a}}% \times\frac{R_{m}R_{l}}{R_{m}R_{s}+R_{l}R_{s}+R_{m}R_{l}}\times\phi_{sm}$ (10) $\displaystyle R_{s2}=R_{s}+\frac{R_{m}R_{l}}{R_{m}+R_{l}},\quad R_{1}=2R_{mg}+% R_{PM},\quad R_{3}=\frac{2R_{mg}R_{PM}+R_{PM}R_{PM}}{R_{mg}}$ (11)

Figure 5.

The effect of $g_{air}$ and t on Bg.

Here, $R_{mg}$ is the reluctance due to magnet pole-to-pole leakage, $R_{mm}$ denotes the reluctance of the leakage flux between adjacent poles, and $R_{gl}$ and $R_{l}$ are the different leakage reluctances, respectively.

3.4 Optimization of the LM

Genetic algorithm (GA) is a method for solving both constrained and unconstrained optimization problems based on a natural selection process that mimics biological evolution. It is widely used in optimal design problems. The genetic algorithm randomly selects individuals from the current population and uses them as parents to produce the children for the next generation. Over successive generations, the optimization can find out the global optimized solution.

In order to obtain required electromagnetic thrust and satisfy the geometric constraints and temperature condition, the cost function of LM can be described as follow

$f(l_{m},g)=\min(85-F)^{2}=\min\left(85-\frac{2nlI(1-e^{-\frac{t_{0}}{T}})}{2+% \frac{t}{l_{m}}\times\frac{\mu_{R}A_{m}}{\mu_{Rt}A_{t}}+\frac{g_{air}}{l_{m}}% \times\frac{\mu_{R}A_{m}}{A_{g}}}\times\frac{\phi_{s}}{A_{g}}\right)^{2}$ (12)

Here, 85 (N/A) is ideal force constant of a LM. The Eq. (12) represents that the position difference between the object and designed LM is minimal.

Genetic algorithms constitute a class of search algorithms especially suited to solving complex optimization problems according to the ‘survival of the fittest’ principle [33]. The MATLAB optimization toolbox were used to optimize the design parameters. The cost function and constraint of LM are listed in Table 1.

The optimal results are shown as follow: the number of turns of coil $n=$ 623, the air gap $g_{air}=$ 12 mm, the thickness of the permanent magnet $l_{m}=$ 7.5 mm, the thickness of coil is 7 mm, the length of coil is 118 mm.

Table 1

Cost function and constraints.

Parameter	Value	Unit
Cost function	( $85-F$ ) ${}^{2}$
The flow of cooling water	0.4	l/min
Hydraulic maximum pressure	2.5	bar
Cooling water (in)	20	${{}^{\circ}}$ C
Cooling water (out)	25	${{}^{\circ}}$ C
Current of coil	$\leqslant 1$	$A$
Air gap ( $g_{air})$	4 $\leqslant g_{air}\leqslant$ 20	mm
The length of coil	$\leqslant$ 120	mm
The thickness of coil	$\leqslant$ 15	mm
MFD of magnet	0.67	$T$
length $\times$ width (magnet)	100 $\times$ 20	mm
thickness (magnet)	3.5 $\leqslant l_{m}\leqslant$ 11	mm

Figure 6.

Generation iteration process.

Figure 6 shows the convergence of the objective function and the design parameters and condition used during the optimization of the proposed LM.

4. Simulations and experiments

4.1 Effect of the water cooling system

Since the thrust of LM often vary during operation process of the precision active isolator, in order to assess the capacities of the designed cooling water system, a computational fluid dynamics (CFD) software package, Fluent, is used. The variation of heat dissipation at different input current intensity is calculated. the temperature distribution of the LM in the cooling flows, at different current input, is shown in Fig. 7. The results show that the temperature in the LM rises with the increase of driving current under the water cooling system working. For the supports, coil and core of the LM are completely immersed by cooling water, its temperature has a slow rising tendency when the LM works.

Figure 7.

Heat distribution of coil with water cooling: (a) 0.9 A; (b) 1.0 A; (c) 1.1 A; (d) 1.2 A.

At rated current (1 A), the temperature has risen by 5 ${}^{\circ}$ C, and it show that the water cooling system can meet the design requirement. The temperature in supports areas is above the other area, and it is the reason that the flow-path of cooling water in supports areas are thinner than other. In addition, during working process of LM, the LM must supply above rated current for short periods of time such as 1.2 times rating for several seconds. As shown in Fig. 7d, the LM has the capability to the requirement. It can be seen that the temperature of local area can increase by 6 ${}^{\circ}$ C. For insulating ceramic covers are used to seal symmetrically on both sides of the frame, it can prevent heat to diffuse outwards. It validates that the designed water cooling system can satisfy the design requirement.

4.2 Trial design and test

Figure 8 illustrates the schematic diagram and photo of experimental setup for the LMs. As shown in Fig. 8a, the experimental setup is mainly composed of a basic platform, an oscilloscope, a signal generator, a force sensor and a measuring instrument. In experiments, the LM is fixed on the frame in working mode (the LM only produce force and can’t move). The force sensor is used to measure the Lorentz force of the LM. The simplified layout of the experimental installation is shown in Fig. 8c. As shown Fig. 8b and c, the mover of LM is installed on sliding rails to reduce the friction and guide.

Figure 8.

Experimental setup: (a) Schematic of experimental setup; (b) Photo of experimental setup; (c) Simplified layout of experimental installation.

When the LMs are operating in the precision active isolator, the bandwidth of the LMs will affect directly the performance of the precision active isolator. The highest operating frequency of the LMs must be more faster than the precision active isolator ( $\leqslant$ 200 Hz). It is necessary to obtain the Bode diagram of the LMs in the experiment.

Figure 9a shows the variation in the thrust with respect to the current magnitude applied to the LM. The curves indicate that a linear correlation of the thrust force with the current intensity in the LM. It can be seen that the proposed model is more accurate than the typical method, because the proposed model is decoupling and the air gap reluctance can be represented accurately by $R_{g}$ as shown in Fig. 4b. But the situation is reversed in Fig. 4c.

Figure 9.

Plot of thrust force versus current of LM: (a) Plot of average thrust versus current; (b) The Bode diagram of F versus I.

The Bode diagram of the LM is obtained and the results are shown in Fig. 9b. The amplitude-frequency curve and phase-frequency curves show that the optimized LM can easily be implemented to control the low frequency vibration typically within the range 0.1–180 Hz and the delay response in this frequency range is small. Beyond this frequency range, the effect of the stray capacitances and inductances will be increasingly distinct in the Bode diagram and the oscillation of curve becomes serious.

Figure 10.

Experimental setup and results: (a) Experimental setup; (b) Plot of acceleration versus time; (c) Plot of power spectral density (PSD).

To assess the dynamic performance of the design LM in an active isolator, a test setup is built as shown in Fig. 10a. The mass of the platform and the load are 1710 kg and 800 kg separately. The main experiment device is LMS Test.lab system that consists of a hammer, acceleration sensors, the matched software, and some auxiliary equipments as illustrated in Fig. 10a. An accelerometer is placed at point-A of the payload as shown in Fig. 10a. Under different experiment conditions (without using vibration isolation, only passive sub-system and active passive system), experimental results are shown in Fig. 10b. As illustrated in Fig. 10b, the curves indicate clearly that the active sub-system (sensors and LMs) can effectively isolate noise from the ground in time domain. In frequency domain, it can be seen that the resonance frequencies of the passive vibration isolator is about 2.1 and 3.1 Hz, and this will deteriorate the performance of the isolator as shown in Fig. 10c. So the passive vibration isolator amplifies the noise in frequency arrange (about from 0.5 to 3.2 Hz). the active vibration isolator can reduce effectively vibrational noise from ground in frequency domain and restrains the resonance of passive vibration isolator.

5. Results and discussion

In this paper, the optimized design of the LM is performed, which is used as the actuator of an active vibration isolator system. In the process of design, different analysis including dynamics, magnetic field, cooling and the practical condition is integrated together to optimize the LM. The simulation and the experiment are performed to verify the design and the performance of the LM. The results demonstrate that the performance of the optimized LM is good within the frequency range of 0.1–180 Hz. The tests in the active vibration isolator system show that the LM can meet the requirements.

Footnotes

Acknowledgments

The work was supported by the introduction of Talent Fund of Kunming University of Science and Technology under Grant Number of KKSY201401097, and the Analysis and Testing Foundation of Kunming University of Science and Technology (2016M20152203001).

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