An ultrasonic levitation journal bearing able to control spindle center position

Mechanical Systems and Signal Processing 36 (2013) 168–181

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An ultrasonic levitation journal bearing able to control spindle center position Su Zhao a,n, Sebastian Mojrzisch b, Joerg Wallaschek b a b

School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore Institute for Dynamics and Vibration Research, Leibniz University Hannover, Germany

a r t i c l e in f o

abstract

Article history: Received 1 December 2011 Received in revised form 30 April 2012 Accepted 18 May 2012 Available online 12 June 2012

A novel active non-contact journal bearing based on squeeze film levitation is presented. Two qualities distinguish the proposed design from the previous ones: significantly improved load capacity and the ability of precision spindle position control. Theoretical models to calculate load carrying forces induced by squeeze film ultrasonic levitation are studied and validated by experimental results. Dynamic behavior of the ultrasonic transducer is investigated using electro-mechanical equivalent circuit model. Levitation forces generated by each transducer are individually controlled by a state feedback controller with auto-resonant (self-excited) frequency control. Active control of the spindle center position is achieved with positioning accuracy of the spindle center in the range of 100 nm. The load capacity achieved by the proposed bearing is dramatically improved compared to previously reported approaches. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Non-contact bearing Ultrasonic transducer Squeeze film levitation Near-field acoustic levitation State feedback control Auto-resonant

1. Introduction In application areas such as computer hard disc drive, machine tools for micro and nano-scale machining, high precision and high speed motions are required. Traditional ball bearings cannot meet the requirements any more due to problems such as wear, heat generation and so on. Non-contact bearings such as electromagnetic bearings, hydrodynamic/ hydrostatic bearings, aerodynamic/aerostatic bearings, have been intensively investigated and developed. The non-contact feature allows these bearings to achieve high precision, low friction, low wear and/or lubricant-free operations. However, a continuous supply of a large volume of clean lubricant is required for air bearings and hydrostatic bearings, which leads to high running cost. And, the requirement of an external compressor excludes this type of bearing from many applications. Magnetic bearings cannot be used for magnetically sensitive configurations due to the strong magnetic flux. Therefore, it is of great interest to find other concepts for realizing non-contact bearings which can overcome some of these problems. Squeeze film type ultrasonic levitation has been found to be a promising alternative solution to construct non-contact bearings and has been investigated since several decades [1]. The so called squeeze film effect is observed when a flat surface oscillates closely to a conformal surface in high frequency. A non-symmetrical force per cycle is generated with a greater force value occurring on the compressing part of the cycle. Due to the nature of squeeze film levitation, it can be directly applied to support non-contact linear motions between two flat surfaces. Hence, majority of the previous investigations focused on applications

n

Corresponding author. Tel.: þ 65 67904377; fax: þ65 67935921. E-mail address: [email protected] (S. Zhao).

0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2012.05.006

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requiring non-contact linear motions such as non-contact linear guides [2–6], non-contact transportation systems [7–10], non-contact thrust bearings [11] and ultrasonic clutch [12]. However, a journal bearing is aimed to provide support to the spindle and carry loads in radial directions around the rotation axis. Squeeze film effect must be generated along the circumference of the spindle, which increases the complicity of the design and construction of the system. Several configurations for squeeze film journal bearings have been reported. In 1964, Salbu [1] introduced the concept of constructing a non-contact bearing using squeeze film effect. Salbu used magnetic actuators to generate oscillations and the operating frequency was within the audible range. Several designs of squeeze film bearings using bulk piezoelectric ceramics can be found in early US patents filed in 1960s, invented by Warnock [13], Farron and John [14], and Emmerich [15]. These designs used bulk piezoelectric materials to create uniform vibration amplitude over the entire bearing surfaces. The transducers were rather massive and required high power to generate sufficient vibration amplitude. Scranton and Robert [16] suggested using piezo-tubes with a flexural vibration mode as the bearing sleeve. However, only basic concepts were sketched in Scranton’s patent. Following Scranton’s idea, Wiesendanger [3] presented a rotational bearing using a tubular piezoelectric bending element mounted in a steel sleeve. The size and load capacity of such bearings are limited by the size of the available piezo-tubes. Only low vibration amplitude and load capacity was achieved. Ha et al. [17] presented an aerodynamic journal bearing which used squeeze film levitation to lift the spindle at the starting phase. Piezoelectric stack actuators and flexure hinges were used to deform the bearing sleeve. The bearing had a bore diameter of 30.12 mm and a length of 25 mm, and could support a static load of 2.18 N with a minimum film thickness of 1:5 mm. Recently, the same research group reported a new configuration for squeeze film journal bearing for spindles of 30 mm diameter. Flexural piezoelectric elements were attached to tubular bearing sleeves made of aluminum alloy to excite flexural vibration modes. The maximum load capacity of 5.6 N was obtained for the bearing with bore diameter of 30 mm and length of 50 mm. The reported bearings could not carry the selfweight of the spindles. Similar concept was used earlier by Hu et al. [18] to develop a non-contact ultrasonic motor with an ultrasonically levitated rotor. Two Langevin transducers driven by two AC voltages with a phase difference of 90 degree were bolted on the stator with an interval of a quarter wavelength to generate traveling waves. Although the maximum load capacity could be higher, it was reported to successfully levitate the rotors weighted up to 2.7 N (diameter of 56 mm) and to drive them at rotation speed up to 3000 rpm. Despite different ways to excite vibrations, it can be concluded that a common approach was shared by all the previous reported squeeze film journal bearings, which is to use tubular vibrators as the sleeves of the journal bearings. The sleeves were often excited to vibrate in flexural resonant modes. The advantage of using tubular vibrator is that the system is compact and easy to fabricate. However, the output power of bending piezoelectric elements is limited by their size and lack of prestress. Hence the achievable load capacity is also limited. According to the model and experimental results presented previously, squeeze film ultrasonic levitation can provide load capacity of up to 7 N/cm2 [19,20]. It is comparable to common air bearings which commonly have load capacity in the range of 10–20 N/cm2. In principle, squeeze film bearings have most of the advantages offered by aerostatic bearings. However, all the previously reported tubular vibrator based squeeze film bearings had limited load capacities much lower than the load capacity measured on a piston vibrator. The inadequacy of load capacity excludes squeeze film journal bearings from most of the practical applications. This paper presents a novel design which directly utilizes the radiation surfaces of three high power ultrasonic transducers to levitate the spindle. Each transducer is an individual piston-like vibrator whose vibration amplitude can be controlled. Section 2 starts with a review on the theoretical models for calculating the load capacity of squeeze film levitation, followed by an experimental setup to validate the models. Based on the theoretical studies, the design and control of a novel non-contact journal bearing using squeeze film ultrasonic levitation will be presented in Section 3. Initial experimental results obtained from the prototype bearing will be presented in Section 5. 2. The load-carrying force induced by squeeze film ultrasonic levitation A schematic diagram of a typical squeeze film levitation system is shown in Fig. 1. It consists of two flat plates with the lower plate being the radiator which oscillates normally against the upper plate (levitated object). The distance between

z L

y x

a0

h(t) = h0 + a0 sin t

Squeezed gas film

Fig. 1. Schematic diagram of squeeze film levitation.

h0

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the radiator and the levitated object is usually in micrometer range, much smaller than the sound wavelength in air. Thus, while the radiator is oscillating in a high frequency, a thin gas film is rapidly compressed and expanded in the gap. The time-averaged mean pressure in the gap has a positive value which is higher than the environmental pressure, which is caused by the second-order effects possessed by the gas film between two surfaces. 2.1. Theoretical modeling Mathematical models for calculating the load-carrying force of squeeze film levitation have been studied and well documented by several researchers [1,19,21–25]. In this study, only the most classic models are studies to find out the critical design parameters for high load capacity. A common approach is to calculate the pressure in squeeze film levitation using the acoustic radiation pressure theory. The Rayleigh radiation pressure in an ideal gas on a perfectly reflecting target was derived by Chu and Apfel [26] as   1þ g sinð2khÞ p¼ /ES ð1Þ 1þ 2kh 2 Here /ES is the time averaged energy density which can be expressed as /ES ¼ ða20 =4Þðra o2 =sin2 khÞ

ð2Þ

k represents the wave number, g a specific heat ratio (gamma¼1.4 for air), ra the density of air, o the angular velocity of the wave, a0 the vibration amplitude and h the distance between vibration source and target. In squeeze film levitation, the levitation distance is very small compared to the wavelength of sound in free field. It ranges from several to several tens micrometers, therefore, sin kh can be approximated as kh. Eq. (1) is the linearized equation describing the radiation pressure in squeeze film levitation [19]

P¼

a2 1þg ra c2 02 4 h

ð3Þ

where P represents the time averaged pressure generated by squeeze film action and c the speed of sound in air. The pressure P in squeeze film levitation is reversely proportional to the square of the levitation distance and proportional to the square of the vibration amplitude a0. Other approaches start from the theory of gas film lubrication theory and solves variants of the Reynolds equation numerically or analytically [1,22,6]. In 1-D case, the Reynolds equation valid for squeeze film levitation can be written as [1] ! 3 @ ph @p @ðphÞ ð4Þ ¼ @x 12Z @x @t where r represents the density of the fluid, h the gap distance, p the pressure, Z the absolute viscosity, t the time, and x the coordinate of the length direction of the gap. The following dimensionless parameters are defined: P¼

p , p0

H¼

h , h0

X¼

x , L

T ¼ ot,

s¼

12oZL2 2

p0 h0

where s is named squeeze number, L the characteristic length of the gas film. Substituting above dimensionless parameters in Eq. (4), we obtain [27]   @ @P @ðPHÞ PH3 ¼s ð5Þ @X @X @T Eq. (5) is the second order partial differential equation that governs the time-dependent, laminar, Newtonian, isothermal and compressible thin film flow. It can be solved analytically for the special case of large squeeze number [1,22,3]. The time and space averaged mean pressure in the gap was obtained as [1] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ3=2E2 ð6Þ P¼ 1E2 where E represents excursion ratio, which is the ratio between the vibration amplitude and the gap distance, E ¼ a0 =h0 . This approximate solution gives a mean pressure along the entire length of the gap and is independent of the squeeze number. It is suitable only for conditions with high squeeze number. In order to obtain the pressure distribution along the gap, the Reynolds equation has to be solved numerically [28,5]. In the current paper Eq. (5) is solved numerically using the finite difference method by employing mathematical tool MATLAB. The initial pressure in the gap at T¼0 is assumed to be equal to the ambient pressure p0. The boundary condition at the edge of the plate is set as pressure release which means that the pressure near the edge always approaches to p0. At the center of the plate (X¼0) there should be no pressure gradient along the x-direction, so that the pressure gradient across the complete gap is always differentiable and no local extreme exists. This means P 0 ¼ P 1 at all time points. To sum up, the initial condition and boundary conditions are listed as following: Initial condition : PðX,T ¼ 0Þ ¼ 1

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Boundary condition 1 : PðX ¼ 1=2,TÞ ¼ 1 Boundary condition 2 :

@PðX ¼ 0,TÞ ¼0 @X

The pressure distribution along half of the gap length can be obtained as a function of time. For a typical levitation system with length L¼30 mm, vibration amplitude a0 ¼ 5 mm, driving frequency f¼ 20 kHz and gap distances h0 ¼ 50 mm, the time-dependent pressure distribution along the gap is calculated and shown in Fig. 2. The corresponding squeeze number is calculated as 97. It can be seen that the pressure in the gap oscillates with time according to the movement of the vibrating surface. The peak values are higher than the absolute anti-peak values, which leads to a positive time-averaged pressure. As specified in the boundary condition 1, the pressure at the edge of the plate is always equal to atmosphere pressure. The pressure increases towards the center of the gap. 2.2. Experimental validation An experimental setup as shown in Fig. 3 is constructed to validate squeeze film levitation force. A l=2 aluminum cylinder with a diameter of 50 mm is mounted on a Langevin type ultrasonic transducer to provide a piston-like vibrator. A Phase Locked Loop (PLL) controller is implemented on a dSPACE 1103 real-time system to drive the transducer in resonance. A broad band amplifier (ENI 1040L 400 Watt RF amplifier) is used to supply the power. A plate is mounted on a load cell with an accuracy of 0.01 N through a ball joint to keep it parallel to the vibrating surface during the measurement. The measurement is conducted with a vibration amplitude of 10 mm on the surface at a frequency of 20 kHz. The gap

1.2

P

1.1 1 0.9 0.8 8

6 T

4

2

0 0

0.1

0.2

0.3

0.4

0.5

L

Fig. 2. Numerical simulation results for the time-dependent pressure distribution along the gap.

Vertical stage Force sensor Ball joint a0

Displacement sensor Horn

Ultrasonic transducer

Fig. 3. Experiment setup for measuring squeeze film levitation force.

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distance is slowly reduced from 300 mm by adjusting the vertical stage until the two surfaces get into contact (contact is indicated by loud noise generated by subharmonics). The displacement is measured using a linear encoder (model LE/25/S from Solartron Metrology) with accuracy of 0:4 mm. The measured squeeze film levitation forces for different gap distances are plotted in Fig. 4 together with simulation results from the acoustic radiation model (Eq. (3)), the approximated (Eq. (6)) and the numerical solution of the general Reynolds equation (Eq. (5)). For the analytical solution of Reynolds equation and the acoustic radiation pressure, the calculated mean pressure is multiplied by the area of the radiator to get the mean levitation force. The distributed pressure obtained from the numerical solution of Reynolds equation is integrated over the surface to get the mean levitation force. The measured force is plotted in Fig. 4 together with the theoretical calculation results. It can be seen that all models give the correct tendency for the levitation force with respect to the levitation distance. At small gap distance (squeeze number s 4100), the measured pressure fits well to the numerical and analytical results of Reynolds equation. It can be concluded that for a squeeze film levitation system with high squeeze numbers (s 4 100), the levitation force can be predicted precisely using the Reynolds equation. For very high squeeze number (s 4 1000), the approximated analytical solution of Reynolds equation can be applied instead of solving it numerically, since they agree to each other very well at this level of squeeze number. When the gap distance increases (lower squeeze number), all the presented models fail to predict the levitation force, because edge effect becomes significant and boundary condition 1 does not hold anymore. Li et al. [25] showed that for small squeeze numbers (s o36), more advanced models which took account of gas inertia effect [29] and edge effect could lead to more accurate results. A maximum levitation force of 115 N (5.86 N/cm2) is measured at input power of about 50 W and mean gap distance of 18 mm. The corresponding squeeze number is around 450. Some points/areas start to contact when further reducing the gap distance due to the surface roughness and form errors. The force can be further increased using polished surfaces and higher vibration amplitudes [19,30]. High squeeze number helps to build up the levitation pressure effectively and increase load capacity [28], therefore, for squeeze film journal bearing applications, higher squeeze numbers (s 4100) are essential. Thus, Eq. (6) can be used to guide the design of squeeze film bearings. It can be seen from Eq. (6) that the time averaged pressure increases while E approaches 1. From the definition of excursion ratio E , we can see that E ¼ 1 if a0 ¼ h0 . Theoretically, a0 ¼ h0 means that at the peak the oscillation the gap between the two surfaces is zero. In reality, such a situation is impossible to achieve since some points/areas will get into contact first due to the surface roughness and form errors as well as the misalignment between two surfaces. In other words, there will be always a small mean gap d between two surfaces. The maximum achievable excursion ratio Emax is then a0 =ðd þa0 Þ. The minimum mean gap d is usually a constant finite value for a certain system. For example, in the experiment setup shown in Fig. 3, contact happens at a mean gap distance of 18 mm and vibration amplitude of 10 mm (E ¼ 0:56 and d ¼ 8 mm). In practice, E can only approach one when d approaches zero or a0 gets much greater than d. The minimum mean gap d can be reduced only with higher manufacturing accuracy. For a given input power, higher a0 can be achieved by optimizing the design of ultrasonic vibration system including the ultrasonic transducer, the horn and the radiator. In conclusion, in order to achieve high load capacity in a squeeze film levitation setup, the squeeze number has to be designed to be as high as possible (at least higher than 100); the surface roughness should be as low as possible to minimize d; and the vibration amplitude should as high as possible.

Fig. 4. Comparison of the measured and calculated levitation forces.

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3. Design of the bearing Based on the discussion in the previous section, a novel design is proposed as shown in Fig. 5. Instead of using tubular vibrator, the bearing journal is directly formed by the radiation surfaces of three individually driven Langevin piezoelectric transducers mounted on a housing, in a circle with 1201 between each other. Different from previously presented approaches, piston-like vibrations are used for the generation of squeeze film levitation. Piston-like vibration mode has several advantages compared to flexural modes. Firstly, it is less sensitive to load, i.e., vibration amplitude does not drop much while load is increased. Secondly, the squeezed film is uniform, which helps to develop pressure more efficiently. The center lines of all the transducers point to the rotation center of the spindle. Each transducer has a arced concave radiation surface which covers 1001 of a cylindrical surface. The three radiation surfaces form the bearing inner shell, which has a diameter slightly larger than the spindle. When the transducers are driven in their resonant frequency and vibrate in the first longitudinal mode, piston-like vibrations are generated at the radiation surfaces. Squeeze film levitation effect takes place between the piston vibrators and the spindle surfaces. Repelling force is generated in the air gap which pushes the spindle from all around and held it at the equilibrium position. Each radiation surface is an independent vibrating surface. The load carrying forces generated by each transducer can be adjusted by modulating the vibration amplitude of the corresponding transducer. Unlike electromagnetic bearings, active control is not necessary, since the bearing is an inherently stable system. 3.1. The high power ultrasonic transducers The piezoelectric transducers are the key parts of the squeeze film bearing system. They generate high frequency mechanical vibration for creating strong and stable squeeze film levitation. Although, commercialized standard piezoelectric transducers are available, they cannot fulfill the specific requirements of a bearing system. Langevin type transducers are specially designed and fabricated for the proposed journal bearing [31]. 3.1.1. Design and manufacturing For high power and high intensity vibration, the Langevin type transducer is preferable due to its excellent electroacoustic efficiency and compact size. It is composed of head and tail masses, a central bolt for pre-stressing and piezoelectric ceramic rings in the middle. For this application, a half wavelength Langevin type piezoelectric transducer is selected as the vibration generator. The designed working frequency is 20 kHz at the first longitudinal vibration mode. The expected unloaded vibration amplitude is to be 15 mm at the center of the radiation surface. The piezoelectric ceramic rings used in this research are manufactured by PI (PI Ceramic GmbH), namely PIC-181. PIC-181 is a modified lead zirconate—lead titanate material with a high mechanical quality factor and a high Curie temperature. An analytical model is used for a first design of the transducer, which gives a starting design for subsequent numerical optimization. The materials selected for the front and back cover are Titanium alloy (TiV4) and stainless steel, instead of the most commonly used aluminum–PZT–steel combination [32]. The selected combination provides the best balance of acoustic impedance matching and mechanical strength. The nodal plane of the transducer is designed to be within the front cover (radiator), so that the transducer can be modeled as two individual 1/4 wavelength systems divided by the

Fig. 5. Schematic diagram of the proposed bearing system.

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nodal plane. A schematic diagram of the transducer is shown in Fig. 6. By solving the frequency equations of the 1/4 wave length transducer and the stepped horn on the left and right side of the nodal plane, unknown lengths of the transducer parts can be calculated. It is known that the Finite Element Method (FEM) provides reliable indications of the natural frequency and the vibration mode [32]. By employing ANSYS software, a FEM model is built according to the dimension parameters obtained from the analytical model. By repeating the modal analysis and modifying the dimensional parameters, the transducer is tuned to match the designed natural frequency and vibration mode. A concave radiation surface to match the spindle surface is also modified on the front cover. The transducers are manufactured according to the dimensions from FEM optimization. After assembling, the frequency response of all three transducers is measured using an impedance analyzer. The measured results show that the resonant and anti-resonant frequencies at the desired vibration mode are 20, 20.1 and 20.8 kHz, respectively. Despite slight differences among the three transducers, the measurements match with the simulation results quite well. Under unloaded condition, the vibration amplitude at the middle of the radiation surface is measured as 16 mm using a fiber-optic laser interferometer (OFV-552 from Polytec) when the input power is approximately 60 W. The vibration amplitude is much higher compared to the previous presented squeeze film bearings. And, it can be further increased by increasing the input power. However, this will lead to severe heat generation which will cause thermal extension on the transducers. This is undesirable since it will reduce the designed gap distance and in extreme cases cause jam between bearing and spindle. Therefore, cooling fans have been installed to dissipate the heat generated by the transducers and keep the temperature of the bearing constant. 3.1.2. Dynamic behavior As discussed in Section 2.1, when the gap distance is fixed, levitation force is positively proportional to the vibration amplitude of the transducer. In order to achieve active adjustment of the spindle center position, the levitation force will be controlled through modulation of the vibration amplitude of the transducer. Therefore, the dynamic behavior of the ultrasonic transducers has to be studied. The dynamic behavior of a linear system subjected to harmonic excitation can usually be described with sufficient accuracy by superposing only a few of the eigenmodes. Each eigenmode dominates the vibration behavior of the system in the range of the respective resonant frequency. Since ultrasonic transducers are driven in the range of one of its resonant frequencies, its behavior can most often be described with reasonable accuracy by a model with only one degree of freedom [33]. Based on electro-mechanical analogies the vibration behavior of a piezoelectric actuator operating in the vicinity of one of its resonant frequencies can be described by an equivalent mechanical or electrical model as shown in Fig. 7 [34]. This model represents the mechanical and electrical behavior and illustrates the relations between electrical input and mechanical output.

Rear cover

PZT

L3

Nodal plane

L2 L1 L4

Front cover

L5

Fig. 6. Design of the power ultrasonic transducer with nodal plane inside the front part.

Fig. 7. Equivalent models of a piezoelectric transducer mechanical (left) and electrical (right).

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175

In the models, m, cm, dm and Cp are modal mass, modal stiffness, modal damping and electrical capacitance, respectively. The electromechanical transformation factor a describes the transmission ratio of electrical and mechanical quantities. The input voltage and charge are represented by u and q. The modal displacement and mechanical load are represented by x and F. It can be seen that current is proportional to velocity and voltage proportional to force. With knowledge of a the velocity amplitude at the end of the transducer can be derived from the electrical current. According to the models, the dynamics of the system can be described by mx€ þdx_ þcm x ¼ au þ F

ð7Þ

1 ðqaxÞ ¼ u Cp

ð8Þ

Parameters of the model are identified through a measurement of frequency sweep response on the transducer. Electrical and mechanical admittance are measured at the unloaded transducer (F¼0). For the transducer used in this contribution the model parameters are identified as listed in Table 1. With these parameters the frequency response can be calculated using the models [35]. The calculated frequency response is plotted together with the measured results in Fig. 8. The upper figure shows the magnitude change of the electrical admittance, which reaches to maximum at the electrical resonant frequency f0 and a minimum at the antiresonance frequency. The lower figure shows the phase between current and voltage. The calculated results show good agreement with the measured ones, which proofs that the transducer’s dynamics near resonance can be represented precisely by the proposed model. It is worth mentioning that for a weakly damped system such as the ultrasonic levitation transducer, the electrical and mechanical resonant frequencies are close enough to be considered as identical. In practice, the resonance frequency of every individual transducer varies slightly due to manufacturing tolerances. Moreover, the resonant frequency is also subject to change during operation from change of load, temperature, input power and so on. Therefore, it is necessary to implement a resonance tracking scheme that can adjust the driving frequency during operation. In this study auto-resonant (self-excited) controller is employed due to its good dynamical behavior [36]. In an autoresonant controller the oscillation frequency is generated by the system and is typically a resonance frequency of the

Table 1 Equivalent circuit parameters for the presented transducer. Parameter

Value

Unit

m dm cm Cp

95.15 71.5 1.56  109 15.8  10  9 1.34 20,355

g Ns/m N/m F N/V Hz

a f0

−1

|Yel| [A/V]

10

Measurement Simulation

−2

10

−3

10

−4

10

1.95

2

2.05

2.1

2.15

f [Hz]

2.2 4

x 10

phase(Yel) [°]

100 50 0 −50 −100 1.95

2

2.05

2.1

2.15

f [Hz]

Fig. 8. Frequency response of the ultrasonic transducer.

2.2 4

x 10

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system. The oscillation starts due to a positive feedback, which causes an instability and oscillation, respectively. Auto-resonant controllers have been used and proved to be effective in ultrasonic cutting and drilling processes [37,38]. 3.2. The rotor-bearing system A spindle-bearing system is constructed in the lab to investigate the proposed squeeze film journal bearing (see Fig. 9). For the convenience of the investigation, two eddy current displacement sensors with resolution of 0:1 mm at 10 kHz (driver ECL101, probe U5 from LION Precision) are installed to monitor the actual position of the spindle center. A hollow steel cylinder is used as the spindle. The rotor of an outer rotor brushless DC motor (maximum speed 20,000 rpm) is mounted in the spindle. The stator of the motor is fixed on the back cover of the housing which drives the spindle to rotate and keeps the axial position of the short spindle by the magnetic force without mechanical contact. The diameter of the spindle is 49.94 mm which forms a mean bearing clearance of 30 mm with the bearing journal. With a vibration amplitude of 15 mm the squeeze number can be calculated as 300, which is high enough to build up the pressure efficiently. 4. Control of the bearing The previously reported ultrasonic bearings were mostly passive and with only one degree of freedom [3,17]. The job of the control system was basically to drive the ultrasonic transducers in resonance. No feedback control was used except for the bearing presented by Oiwa and Suzuki [4], who used simple position feedback control. However, Oiwa and Suzuki did not control the transducer’s driving frequency but only its driving amplitude. The bearing proposed in the current paper involves multiple transducers and should perform spindle position control in X –Y plane. Thus, beside resonance control for each transducer, spindle position control algorithm must be implemented. Fig. 10 shows the free body diagram and the equivalent model of the bearing. The Cartesian coordinates are located in the center of the spindle. The inputs of the spindle position control system are the desired position in the Cartesian coordinates; the outputs should be the driving signal for each of the three ultrasonic transducers. Therefore, the Cartesian coordinates are transformed by the Jacobian matrix JAB to the coordinates of the actuators, with a ¼ 1201:    cosðaÞ sinðaÞ  !   x   dB ¼ J AB qA ¼  cosðaÞ sinðaÞ  ð9Þ   y   0 1 With this coordinate transformation it is possible to design a controller for each degree of freedom separately. In this study, the control method proposed by Mojrzisch and Wallaschek [39] is employed. Mojrzisch and Wallaschek used

Fig. 9. Prototype of the proposed squeeze film bearing.

Fig. 10. Free body diagram and equivalent model of the bearing system.

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177

Fig. 11. Control system of the ultrasonic bearing system.

auto-resonant controller to drive the transducers in an ultrasonic levitation system. It was found that the transducer’s response to amplitude changes was of the first-order-lag kind. Due to this fact the amplitude control feedback can be very high, which is of great importance for fast amplitude modulation. It can be seen from Fig. 7 that charge q is related to displacement x as x ¼ aq. Since both q and x are sinusoidal signals of the same frequency, it can be derived that the vibration velocity v is equal to ai, where i represents the driving current. Since we are interested in the amplitude of the high frequency signals, the vibration amplitude of the transducer can be described using electrical current amplitude as following: x^ ¼

a^i o0

ð10Þ

where o0 ¼ 2pf 0 is the angular frequency. The prerequisite is that the transducer is always driven near its resonant frequency, which is full filed by employing the auto-resonant controller. Since current is easier to measure and control, the vibration amplitude modulation is done indirectly by controlling the current. The levitation pressure is modeled by Eq. (3) and linearized around the operating point (the levitation distance)   @P  @P f ðx,iÞ ¼ ð11Þ hþ  , i ¼ kh h þ ki i^ @h0 OP @i^0 OP This gives a linear dependency of levitation force on the levitation gap h and driving current i. The overlaid statefeedback controller by Mojrzisch and Wallaschek [39] is employed and extended to two degrees of freedom for resonant and x–y position control. Fig. 11 shows the overall control of the ultrasonic bearing system. The bearing with resonance and amplitude control is put in one block as a subsystem. The resonance control is done by an auto-resonant circuit, whose output amplitude is set by an amplitude controller. The amplitude controller is designed as PI-current-controller whose gains are determined by experimental tuning. Due to the fact that the amplitude control path is a first order lag type [39], the P-Gain of amplitude controller can be set to high values without risk of instability or oscillation. The dynamic design of the controller is done by pole placement. The measured spindle displacements from the eddy current sensors are fed to the state-feedback controllers, which estimate the velocity using Luenberger observer. The output of each controller is the force needed to move the spindle to the desired position in x and y-direction. These force values are coupled into the actuators coordinate system and converted to three current amplitudes. Using auto-resonant systems and power amplifiers the appropriate vibration amplitudes are set on the transducers and levitation forces in the gaps are generated, respectively. 5. Performance of the bearing 5.1. Load capacity In order to evaluate the load capacity of the proposed bearing, the levitation forces at different gap distances in vertical direction are measured using the eddy current sensor and a load cell (maximum load 500 N, accuracy 0.01 N). External load is applied on the spindle in the gravity direction through the load cell. The load is increased gradually and measured by the load cell until contact happens between the bearing and the spindle (when loud noise from contact is heard). The results are plotted in Fig. 12 together with the calculated results from the presented mathematical models by utilizing the actual parameters. Throughout the measurement the vibration amplitude of the transducer maintained constant by keeping constant current amplitude using the amplitude controller. The vibration amplitude is measured as 15 mm using a fiber-optic laser interferometer (OFV-552 from Polytec). The input power to the transducer is around 50 W. It can be seen from Fig. 12 that the load-carrying force increases while the gap distance is reduced. The slope of the force curve represents the stiffness of

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120 Reynolds - numerical Reynolds - analytical Acoustic radiation Measurement

Levitation force [N]

100 80 60 40 20

0 20

30

40

50 60 70 Distance [µm]

80

90

100

Fig. 12. Load capacity versus levitation distance at constant vibration amplitude of 20 mm.

the bearing which also increases with smaller gap distances. The measurement results agree well with the numerical results of Reynolds equation. A load-carrying force of 51 N (6.37 N/cm2) is obtained when the gap distance is 28 mm at the bottom transducer. The input power is not affected much by decreasing the gap distance. Seeing the trend in Fig. 12, higher bearing force may be achieved by further reducing the gap distance. But this is limited by the manufacturing accuracy of the bearing. Contact starts to happen at certain point when the gap distance is further reduced. Nevertheless, the achieved load capacity is already considerably larger than that of all other squeeze film bearing presented before [2,5,6,17] whose load capacities are usually within a few Newton (lower than 1 N/cm2).

5.2. Energy efficiency It has been found that the power consumption is not affected much by the change of levitated load. The power consumed by the bearing is mainly for exciting and maintaining the vibration amplitude of the ultrasonic transducers. The transducer is radiating in air, and there is no mechanical contact with the levitated object. Thus, little mechanical power is conducted out of the transducer. Most of the mechanical power is transformed into heat. For steady levitation, only the bottom transducer is needed to overcome gravity force. In our experiment, a vibration amplitude of 6 mm on the bottom transducer is sufficient to levitate the spindle. The power needed is less than 10 W. Since the transducers behave as resistive loads when they are driven in there resonance, the power can be calculated as i2 R. As discussed in Section 3.1.2, current is linearly proportional to the vibration amplitude. Thus, the power consumption increases quadratically with higher vibration amplitude.

5.3. Rotating accuracy The run-out error of the levitated spindle is measured when the spindle is stationary at the beginning and then starts to rotate at a low speed (about 50 rpm). The movements of the spindle surface are measured by the displacement sensor placed in the vertical direction. The results are shown in Fig. 13. In the first two seconds, the spindle is not rotating. The levitation is very stable with error motions that can hardly detected by the used eddy current sensors which have a bandwidth of 80 kHz and resolution of 0:2 mm. No oscillation is observed on the spindle in the frequency range which the ultrasonic transducers are driven (20 kHz). When the spindle starts rotating at 2 s, periodic error of a few micrometers is measured at which is due to the form error and surface roughness of the spindle. The run-out error of the center of the spindle remains very small. It is worth to mention that the load-carrying force of the proposed bearing is the average force from every compress– release cycle between the bearing and the spindle. For driving frequency of 20 kHz, the time needed for one compress– release cycle is 50 ms. When the spindle is rotating, a lateral movement is generated between the bearing and the spindle surfaces. The lateral movement has the line speed of the rotating spindle. For a rotation speed of 10,000 rpm, the line speed of spindle (50 mm diameter) is 8.33 m/s. The spindle rotates 31 in one compress–release cycle, and the relative lateral movement between two surfaces is 0.42 mm, which is much smaller compared to the circumference of the bearing. Therefore, the spindle can be considered approximately as stationary within one compress–release cycle. It has been shown that squeeze film levitation is still functional even when the linear velocity between the surfaces is higher than the vibration velocity [18]. The proposed bearing is suitable for both low and high speed applications.

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5 4 3

Position [µm]

2 1 0 -1 -2 -3 -4 -5 0

1

2

3

4 t [s]

5

6

7

8

Fig. 13. Spindle run-out error during levitation.

3

Y-Position [µm]

2 1 0 -1 -2 -3 -3

-2

-1

0 1 2 X-Position [µm]

3

4

5

-3

-2

-1

0 1 2 X-Position [µm]

3

4

5

3

Y-Position [µm]

2 1 0 -1 -2

Fig. 14. Open loop response of the ultrasonic levitation bearing (a) original results and (b) low-pass filtered results.

5.4. Position control In the stat-feedback controller design in the previous section it is assumed that the two degrees of freedom can be actuated separately without influencing each other. This is done by a transformation from Cartesian coordinates to the actuator coordinates. To validate this assumption the open loop system is measured. In Fig. 14 the response to 1 mm steps

S. Zhao et al. / Mechanical Systems and Signal Processing 36 (2013) 168–181

Y-Position [µm]

180

2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -2

-1

0 X-Position [µm]

1

1.6

1.7 1.8 X-Position [µm]

1.9

2

-0.65

Y-Position [µm]

-0.7 -0.75 -0.8 -0.85 -0.9 -0.95 -1 -1.05 1.4

1.5

2

2.1

Fig. 15. Closed loop response of the ultrasonic levitation bearing (low-pass filtered) (a) step size 1 mm and (b) step size 100 nm.

in x and y-direction is plotted. As can be seen from the position response of the open loop system, the coupling between x and y axes is obvious. Besides, the reference position is not set accurately, As state-feedback controllers are not able to eliminate static deviation from the reference position. To overcome this problem a PI position controller is added to the control system. The results obtained with PI overlaid state-feedback controller are shown in Fig. 15. The upper plot shows response to 1 mm steps in x and y-direction. With the use of PI position control the position trajectory follows very close to the reference steps. The steady state precision can be seen in the lower plot from Fig. 15. In this measurement the reference was changed in 100 nm steps. The reference position is set accurately by the controller due to the absence of friction force. It is worth mentioning that the displacement that can be adjusted is much smaller than the vibration amplitude of the transducers, which is in the range of 10 mm. Similarly, using the presented control system, the spindle center can be kept stationary when external load is introduced. This means that the presented bearing is a smart bearing which is able to perform active spindle run-out error compensation similar to the fast tool servo systems [40,41]. However, the bandwidth and maximum force output are still limited. 6. Conclusion The development and performance of a novel squeeze film journal bearing is presented which directly utilizes the radiation surfaces of three high power ultrasonic transducers to support the spindle without mechanical contact. The proposed bearing does not require external pressurized air or liquid supply. The bearing is naturally stable even without feedback control. A steel spindle with a diameter of 50 mm has been successfully levitated and driven at low rotational speed. Each of the three ultrasonic transducers can provide load carrying force up to 51 N at an input power of 50 W. The achieved overall load capacity of the presented bearing is considerably larger than that of the squeeze film bearing presented previously, whose load capacities are usually within a few Newton. Classic models for calculation of levitation force are reviewed. The calculated results agree well with the measured ones at high squeeze numbers, which proofs that these models are suitable for design of squeeze film bearings. Dynamic behavior of the ultrasonic transducers is investigated analytically. Auto-resonant controller is implemented to drive the transducers in resonance. Precision control of spindle center position is achieved by current amplitude modulation using the PI overlaid state-feedback controller. Due to absence of friction, the position of the spindle can be

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