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1 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 3, MARCH Application of Voice Coil Motors in Active Dynamic Vibration Absorbers YiDe Chen, ChyunChau Fuh, and PiCheng Tung Abstract A dynamic vibration absorber reduces the influence of a force whose excitation frequency nearly coincides with the natural frequency of a rotating machine. However, the performance of this type of passive absorber can be affected by changes in the environment. In this paper, we describe a voice coil motor (VCM) that can serve as the actuator in an active dynamic vibration absorber which can be regulated for different conditions. With a VCM, suitable controllers can be designed for periodic excitation force rejection by using the characteristics of the notch filter in combination with the rootlocus theorem. We have evaluated the performance of the active vibration absorber by both simulations and experiments. Index Terms Active vibration absorber, notch filter, rootlocus theorem, voice coil motors. I. INTRODUCTION THE integrated circuit (IC) industry, both domestic and foreign, has had a higher growth rate than others. Since the IC industry requires highly developed technology and highprecision manufacturing, its corresponding production machinery and measuring devices are very sensitive to vibrational noise. A small amount of vibration may introduce undesirable noise and act as a source of vibration in the mechanical system, which affects the accuracy and may shorten the lifetime of the machine. Thus vibration control, that is attenuation of undesirable vibration from various sources, has become an important research topic. The most common method for the control of vibration has been to use vibration isolators or vibration absorbers to reduce or eliminate vibrations. The former method uses an isolator between the vibrated mass and the source of the vibration, to reduce the transmission of the excitation forces. In general, vibration isolators [1] can be divided into two types: passive and active. The main difference between them is that the active type provides power for the performance of vibration control. The dynamic vibration absorber [2] is designed to reduce the influence of a force whose excitation frequency is nearly coincidental with the natural frequency of the system. An ideal undamped dynamic vibration absorber [3] consists of an auxiliary mass and a stiff spring, tuned to the excitation frequency necessary to cause the steady amplitude of the system to be near zero at that frequency. However, if the system operates at other frequencies, Manuscript received June 22, 2004; revised November 30, Y.D. Chen and P.C. Tung are with the Department of Mechanical Engineering, National Central University, Taoyuan 32054, Taiwan, R.O.C. ( C.C. Fuh is with the Department of Mechanical and Mechatronic Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan, R.O.C. ( Digital Object Identifier /TMAG or if the excitation force has several varying frequencies, the amplitude of the vibration of a dynamically vibration absorber system may become large. On the other hand, an active dynamic vibration absorber [4], [5] can be tuned according to the system characteristics, to meet the desired requirements. Hence, active vibration control can be used to solve vibration problems, in spite of its higher cost and complexity. The actuators of active dynamic absorbers can be separated into several types: mechanical mechanisms, piezoelectric actuators [6], pneumatic springs [7], cylinders [8], electromagnetic motors [9], and electrical linear motors (linear actuators [10], [11]). The last two have some advantages, such as a faster response time and greater precision, so there has been more research into the correlative technology. It is more complex to control because magnetic forces belong to an unstable nonlinear system, and electromagnetic motors need the addition of a gear transmission to convert the rotational motion into translational motion. In contrast, the employment of a linear motor can avoid these drawbacks, as well as having the desirable features of low noise and low vibration. Their simple structure and direct drive make them easy to maintain, and they provide the ability of acceleration and deceleration. The voice coil motor (VCM) [12] is one type of linear motor, originally used in amplifiers. VCM actuators are usually used in occasions that rapid and controlled motions of devices are required. In general, VCM has many applications, such as in the servo control of the DVD [13], the hard disc [14], and the camera lens. Thus, in our experiments, the VCM is chosen to be the absorber actuator as it performs well in terms of vibration rejection. Periodic disturbances occur in many engineering control applications, commonly in rotating machinery. Several methods are available for the rejection of sinusoidal disturbances. One common method, based on the internal model principle (IMP) [15], states that a model of the system generating the disturbance must be included in the feedback system. Such an approach is based on notch filtering [16], [17], which adds a notch filter at the synchronous frequency into the loop. Another method is adaptive feedforward cancellation (AFC) [18], which can be used to estimate the magnitudes of unknown sinusoidal disturbances. In this paper, the method proposed for the design of a suitable controller by which a periodic excitation force can be rejected is based on the characteristics of the notch filter. The method uses a simple rootlocus theorem to design controllers which correspond to the different frequencies of the excitation forces. The advantage of this method is that the addition of a notch controller only influences the designed frequency response /$ IEEE
2 1150 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 3, MARCH 2005 Fig. 1. Schematic of the VCM structure. Fig. 3. Experimental platform. TABLE I SPECIFICATIONS OF THE PLATFORM WITH AN ACTIVE ABSORBER Fig. 2. Chart of the distribution of the magnetic lines of force. II. SYSTEM DESCRIPTION AND THE MATHEMATICAL MODEL A. Voice Coil Motor The VCM is a direct drive motor that utilizes a permanent magnetic field and coil winding to produce a force proportional to the current applied to the coil. Its simple structure makes it easy to maintain. It can provide superior acceleration and deceleration via the magnetic force. The VCM structure used in the experiments is shown in Fig. 1 and a chart of the distribution of the magnetic lines of force is shown in Fig. 2. The magnetic field is produced by permanent magnets placed on the both sides of a permeable material such as silicon steel or lowcarbon steel. The Lorentz force equation can be used to compute the thrust on the coil when it is electrified in the magnetic field. The thrust decides the direction of the coil s movement where represents the thrust on the coil, is the ratio of the effective length to the whole length of the coil in the magnetic field, is the current of the coil, and is the magnetic flux density. When and are vertical to each other, the direction of can be decided by Fleming s lefthand rule. Under this condition, (1) can be rewritten as (1) (2) where is called the force constant. Thus, the thrust on the coil can be controlled by regulating the input current. The current can be regulated by the control signal, expressed as a voltage command, so the relation between control force and control signal can be shown as where is a constant between voltage and current, and is a constant. B. System Description To reject vibration induced by an excitation force, an active dynamic absorber is added to the platform. The VCM is used as an actuator for the dynamic absorber. The experimental platform, with the active dynamic absorber, is illustrated in Fig. 3 and the specifications of the platform are given in Table I. The coil remains vertical by using a linear guideway. To analyze the system dynamics more easily, the model, including the active dynamic absorber, is simplified shown in Fig. 4. The assumption here is that the force generated by the actuator will be substituted for the excitation force. The mass, the stiffness, and the damping coefficient composing the upper components, together form the active dynamic vibration absorber of the model. The active dynamic vibration absorber can provide a controlling force, which controls the vibration. Before adding this active vibration absorber into a system, the original lower components must be modeled, that is the mass, the stiffness (3)
3 CHEN et al.: APPLICATION OF VOICE COIL MOTORS IN ACTIVE DYNAMIC VIBRATION ABSORBERS 1151 Fig. 4. Schematic of the platform with an active dynamic absorber. Fig. 5. (a) Block diagram of the LTI plant perturbed by a periodic excitation force. (b) Block diagram of the feedback control using the notch controller to reject the periodic excitation force., and the damping coefficient of the original plant. The plant is affected by an excitation force. C. Mathematical Model By employing a freebody diagram and Newton s second law of motion, the governing equations can be deduced. The original 2degree system, composed of the lower components, can be simplified into where the states represent the position and the velocity responses of the original platform. When an active vibration absorber is added to the system, the dynamic equation can be rewritten as where and the two states represent the position and the velocity responses of the active dynamic vibration absorber, respectively. The signals and represent the excitation force and the control force, respectively; see Fig. 4. The relation between the position output, the control force, and the excitation force can be expressed via the transfer function as Thus, represents the characteristic equation of the plant (4) (5) (6) (7) Fig. 5(a) shows the organization of a lineartimeinvariant system perturbed by a periodic excitation force, represented by where and and are the Fourier coefficients of the synchronous excitation force at the frequency. III. PERIODIC EXCITATION FORCE REJECTION METHOD In order to counter the effects of an excitation force with a suitable control force, the notch controller is adopted. The notch controller discussed in this paper applies a notch filter combined with the rootlocus theorem. In general, a notch filter [19] is used to reduce any system response to some assigned frequency. For instance, it is often applied for the suppression at highfrequency disturbances in communications technologies. From the frequencydomain standpoint, the notch filter does not affect properties of the other frequencies of the system, only the assigned frequency. The notch filter transfer function is a function of both the Laplace variable and the excitation force frequency. The standard form is where the two coefficients are defined as and. The parameter is the damping ratio of the notch filter. For the situation, is a common notch filter which reduces the response of the system and can be aimed at a specific frequency. The attenuation provided by the magnitude of is (8) (9) (10) On the other hand, the notch filter will amplify the system s response for the situation. Thus, it can be seen that the variable decides the gain ratio and the direction of the notch in the magnitude plot. With different directions, the notch
4 1152 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 3, MARCH 2005 filter can amplify or reduce the system response. The other variable can determine the notch width for the designed frequency. Under experimental conditions, a notch controller cannot be added between the excitation force and the plant, that is to say the influence from outside on the platform cannot be directly reduced by the common notch filter. Thus, a novel method, different from the common notch filter, is presented. The opposite notch filter application, namely, the situation, is chosen. The controlling force is designed as the amplified system response, as shown in Fig. 5(b). We let the control force cancel the excitation force to decrease the effect from outside to the system. Unfortunately, the step whereby the notch filter is added into the system alters the closedloop system s stability. The transfer function for this closedloop system, including the notch filter, can be described as follows: Fig. 6. Frequency response diagram of the experimental system. signal, and represents the amplitude of the output signal. As seen from (3) and (6), the transfer function can be written as (11) Apparently, becomes small when is large. This means that can be employed to reduce the influence from the excitation force on the system. However, the system s stability, as affected by, should be taken care of simultaneously. To avoid instability, the rootlocus theorem is employed to design the most appropriate notch controller. The characteristic equation for determining variables and can be separately expressed as (13) where and are the natural frequencies of the system. The frequencies and are obviously defined in Fig. 6. By adjusting parameter, we can change the gain. Similarly, by adjusting parameters and, we can change the damping ratio. A curve, fitting, approximating the real system responses, can be calculated by regulating the parameters, and. Via the parameter estimation method, the transfer function can be expressed as (12) By using the rootlocus theorem, suitable values for the notch filter parameters and can be obtained to allow good performance and to maintain the stability of the system. Since the parameters and are coupled, one of them must be given before the root locus can be applied. The parameter that can be regulated to avoid some problems such as modeling errors, uncertainties, and disturbances is given in advance. Then the criterion for choosing the parameter is based on finding the optimal dominant root under different frequencies. The optimal dominant root let the system response be fastest. IV. SIMULATION AND EXPERIMENTAL RESULTS Although the system model type is known from (5), the parameters of the dynamic equation are unknown. Before the design at the controller can be completed, the characteristics of the system must be known. Since the system is a minimum phase system, only the magnitude response is considered. Thus, with suitable sine waves having different frequencies as the input, the frequency response diagram can be obtained by measuring the output, namely, the displacement of the platform, as shown in Fig. 6. The symbol represents the amplitude of the input (14) This model, based on the system identification results, is adopted for the simulation and the experiments. In the proposed method, parameters and should be decided on first. The parameter,,influences the magnitude of the system response and the direction of the notch filter. Parameter determines the notch width aimed the design frequency, that is to say, the regulation at parameter allows one to avoid some problems such as modeling errors, uncertainties, and disturbances. According to the parameter, the error range of the corresponding design frequency value can be decided. In the simulation and the experiments, is set as The addition of a notch filter to the plant changes the system s stability; therefore, the rootlocus method should be employed to design an appropriate notch filter. The criterion for choosing the parameter is based on finding the optimal dominant root under different frequencies. When the excitation frequency is 25 rad/s, the rootlocus plot is shown as Fig. 7. For faster system responses, the optimal dominate root is obtained when the parameter is set to be Then the addition of a suitable notch filter to this plant leads to the system responses shown as Fig. 8. To maintain
5 CHEN et al.: APPLICATION OF VOICE COIL MOTORS IN ACTIVE DYNAMIC VIBRATION ABSORBERS 1153 TABLE II EXPERIMENTAL PARAMETERS FOR THE DESIGN NOTCH CONTROLLER WITH DIFFERENT FREQUENCIES OF EXTERNAL FORCE Fig. 7. Rootlocus plot in function of K at! =25(rad/s). Fig. 9. Resolution chart of the eddycurrent sensor. Fig. 8. Frequency response diagram of the system and the system with a notch filter aimed at! =25(rad/s). system stability under different frequencies, different values for parameters are calculated, and are shown in Table II. In the simulation and the experiments, for generating a periodic excitation force, the force is set as and the sampling time is s. A control force is added to eliminate the periodic excitation force 3 s after starting the system. In the experiments, the displacement of the middle platform is detected by eddycurrent gap sensors (IAS10A24IL). The curve showing the relation between the position and the voltage can be seen in the resolution chart in Fig. 9. By fitting to the curve, we obtain the following expression for the position as a function of the voltage : (15) The initial reference mark set as 2.7 V, that is to say, the distance between the initial location of the middle platform and the eddycurrent gap sensor is 7.62 mm. When an excitation force with a frequency of 15 rad/s affects the system, the resultant simulation and experimental results are illustrated in Figs. 10 and 11. Fig. 10 presents the simulation results for a system with a notch controller influenced by an excitation force. Fig. 11 shows the experimental results Fig. 10. Simulation results for a notch controller system for the rejection of an excitation response when! =15(rad/s). contrasted with Fig. 10. On the premise that the uncertainty is ignored in the experimental process, the simulation result and experimental result are similar and reasonable. The parameters of the notch controller according to the different frequencies are calculated in advance and shown in Table II. By the relation between the position and the voltage in (15), the experimental frequency responses for the uncontrolled system and a notch controller system at different frequencies are shown in Fig. 12. The results show that a notch controller can help to alleviate periodic
6 1154 IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 3, MARCH 2005 at the parameter. Simultaneously, the tolerant error range of the corresponding frequency can be adjusted by the other parameter. From the design viewpoint, the notch controller has the advantage of being directly perceivable through the senses; however, the employment of the VCM as an actuator can accomplish an effective dynamic vibration absorber. Fig. 11. Experimental results for a notch controller system for the rejection of an excitation response when! =15(rad/s). Fig. 12. Experimental results of the frequency responses for an uncontrolled system and a notch controller system, at different frequencies. excitation force problems and to maintain the system s stability by using the rootlocus theorem. V. CONCLUSION In this paper, a VCM is chosen to be the actuator in an active dynamic absorber for the achievement at vibration control. A notch controller provides the control force, which cancels the excitation force, thus decreasing the outside effects on the system. According to the simulation and the experimental results, vibrations can be effectively reduced by using this active dynamic absorber. The use of the notch controller can lead to a quick reduction of the influence of an excitation force. The notch controller achieves a better performance by the regulation REFERENCES [1] S. S. Rao, Mechanical Vibrations, 4th ed. Upper Saddle River, NJ: Pearson, 2004, pp [2] B. G. Korenev and L. M. Reznikov, Dynamic Vibration Absorbers Theory and Technical Applications. New York: Wiley, 1993, pp [3] B. A. Francis and W. M. Wonham, The role of transmission zeros in linear multivariable regulators, Int. J. Control, vol. 22, no. 5, pp , [4] S.J. Huang and R.J. Lian, A dynamic absorber with active vibration control, J. Sound Vib., vol. 178, no. 3, pp , Dec [5] T. Mizuno, M. Moriya, and K. Araki, Robust disturbance cancellation in an active dynamic vibration absorber system, Control Eng. Practice, vol. 3, no. 6, pp , Jun [6] J. X. Gao and L. Cheng, Modeling of a high performance piezoelectric actuator assembly for active and passive vibration control, Smart Mater. Struct., vol. 13, no. 2, pp , Apr [7] G. L. Giliomee and P. S. Els, Semiactive hydropneumatic spring and damper system, J. Terramechanics, vol. 35, no. 2, pp , Apr [8] X. Pan and C. H. Hansen, Active control of vibration transmission in a cylindrical shell, J. Sound Vib., vol. 203, no. 3, pp , Jun [9] D.H. Cho and H.J. Kim, Modeling of electromagnetic excitation forces of small induction motor for vibration and noise analysis, in IEE Proc.: Elect. Power Appl., vol. 145, May 1998, pp [10] H.P. Kelly, Linear drives, Industrial Robot, vol. 20, no. 6, pp. 8 11, [11] A. M. Madni, J. B. Vuong, M. Lopez, and R. F. Wells, A smart linear actuator for fuel management system, BEI Technol., vol. 16, pp , [12] A. Babinski and T. C. Tsao, Acceleration feedback design for voice coil actuated direct drive, in Amer. Control Conf., vol. 5, 1999, pp [13] C.L. Chu, K.C. Fan, and Y.J. Chen, A compensation method for the hysteresis error of DVD VCM, Meas. Sci. Technol., vol. 15, no. 4, pp , Apr [14] R. Oboe, F. Marcassa, P. Capretta, and F. C. Soldavini, Realization of a hard disk drive head servopositioning system with a voltagedriven voicecoil motor, Microsyst. Technol., vol. 9, no. 4, pp , Mar [15] G. Feng and M. Palaniswamy, A stable adaptive implementation of the internal model principle, IEEE Trans. Autom. Control, vol. 37, no. 8, pp , Aug [16] C. R. Knospe, Stability and performance of notch filter controllers for unbalance response, in Proc. Int. Symp. Magn. Suspension Technol.. Hampton, VA, 1991, NASA Conf. Pub [17] R. Herzog, P. Buhler, C. Gahler, and R. Larsonneur, Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings, IEEE Trans. Contr. Syst. Technol., vol. 4, no. 5, pp , Sep [18] H. S. Na and Y. Park, An adaptive feedforward controller for rejection of periodic disturbances, J. Sound Vib., vol. 201, no. 4, pp , [19] G. M. L. Gladwell, Vibration control of active structures, Sol. Mech. Appl., vol. 50, pp , 1997.
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