- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 986812, 12 pages
Design and Evaluation of a Direct Drive Valve Actuated by Piezostack Actuator
1Department of Mechanical Engineering, Inha University, Incheon 402-751, Republic of Korea
2Department of Mechanical Engineering, Hochiminh University of Industry, 12 Nguyen Van Bao Street, Ward 4, Go Vap District, Ho Chi Minh City 70550, Vietnam
3Division of Automotive Engineering, Ajou Motor College, Chungnam 355-769, Republic of Korea
Received 29 May 2013; Revised 21 July 2013; Accepted 22 July 2013
Academic Editor: Marco Ceccarelli
Copyright © 2013 Juncheol Jeon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper presents performance characteristics of a new type of direct drive valve (DDV) system driven by a piezostack actuator. The flexible beam mechanism is employed to amplify the output displacement from the piezostack actuator. After describing the operational principle of the proposed piezo DDV system, the governing equation of the whole piezo DDV system is then obtained by integrating the equations of the valve components. Based on the proposed model, significant structural components of the piezo DDV system are designed in order to achieve operational requirements (operating frequency: over 100 Hz; flow rate: 20 liter/Min.). An optimal design method is proposed for obtaining the geometry of the flexible beam mechanism by considering spool displacement, required operating frequency, and available space of the valve. After deciding the specific geometric dimensions of the piezo DDV system, a PID control algorithm is designed to enforce the spool position to the desired position trajectories by activating the piezostack actuator. Characteristics and control performances of the proposed piezo DDV system are evaluated using the MATLAB Simulink.
In the hydraulic or pneumatic systems, control valves provide the interface between the power element (such as pump or compressor) and the output device (such as linear or rotary actuator) . Because performances of the valve have decidable influence for the whole system, many kinds of control valves have been actively researched [2–4]. In particular, the electrohydraulic servo valve is widely researched for improving the operational abilities of the valve such as the response time and accurate controllability. The servo valve is a kind of the flow-control valve and is controlled by the feedback signal which is generated automatically through an internal sensing and control mechanism. The servo valves are categorized into two-stage valve and single-stage valve. The two-stage valve consists of two stages as shown in Figure 1: the first stage is the pilot valve; the second stage is fluid valve [5–7]. The pilot valve is generally controlled by the electromagnetic actuator and makes a pressure difference which becomes the actuating force for the fluid valve. Such a two-stage valve can supply very high flow rates because a low power is hydraulically amplified. Therefore, the two-stage valves have been actively employed since the 1960s. However, because the conventional two-stage valve haves bandwidth of about 50 Hz (because of the inertia of the pilot and the fluid system), these valves are not suitable for high-frequency applications . In order to overcome the bandwidth limitation of the two-stage valve, one possible approach is to use the single-stage valve which has been widely researched since the 1990s. Recently, the direct drive valve (DDV) which is a kind of single-stage valve has been actively researched [9, 10]. The main feature of the DDV is that the actuator directly controls the position of the spool. Therefore, the performances of the actuator such as energy density and actuating bandwidth exert influence on the performance of the DDV.
The piezostack actuator featured by high power density, large generated force, and fast response is widely used as an actuator for DDV system [11–13]. The DDV system actuated by piezostack actuator can be categorized into DDV without displacement amplifier and with displacement amplifier. The piezo DDV system without displacement amplifier has wider operating bandwidth than the piezo DDV system with displacement amplifier because there is no additional inertia part (displacement amplifier). However, piezostack actuator has a big weakness that the stroke of the piezostack actuator is very small compared to its length. When the piezostack actuator directly controls the spool without displacement amplifier, the displacement of the spool is generally less than 0.1 mm [7, 12]. With this spool displacement, the piezo DDV system can control the flow rate to be less than 5 liter/min. In order to overcome this weakness, many kinds of displacement amplifiers have been actively researched and adopted for the DDV system [11, 14, 15]. The displacement amplifier can be roughly classified into a couple of types: lever-hinge type and flexible beam type. The lever-hinge type displacement amplifier can be easily adopted in valve system because its mechanism is very simple and its mathematical model can be easily obtained . However, when the tolerance between lever and hinge is rough, the valve system is hard to control the position of the spool due to the dead zone. Therefore, the tolerance between lever and hinge is strict; it means that this type of displacement amplifier is hard to manufacture (expensive). In order to solve this problem, one possible method is to use the flexible beam type displacement amplifier.
Consequently, in this work, a new type of direct drive valve system actuated by piezostack actuator and flexible beam mechanism is proposed. The proposed piezo DDV system consists of the piezostack actuator, flexible beam mechanism, and spool part. The piezostack actuator is used as an actuator and located in one side of the flexible beam mechanism. The flexible beam mechanism plays two kinds of rolls: transferring the displacement and force generated by piezostack actuator, and amplifying the displacement of the piezostack actuator for solving its stroke limitation. After explaining the working principle of the piezo DDV system, the piezo DDV system is mathematically modeled by using the lumped parameter modeling method. In order to evaluate the proposed piezo DDV system, computer simulation is proposed and some typical operational requirements are decided. After the appropriate piezostack actuator is selected, the dimensions of the flexible beam mechanism are optimally obtained for satisfying the operational requirements. The performances of the proposed piezo DDV system such as maximum operating range, step response, operating bandwidth, and position tracking controllability are then evaluated via MATLAB Simulink.
2. Configuration and Working Principle
Figure 2 shows a schematic configuration of the proposed piezo DDV system featuring a piezostack and a flexible beam. As shown in the figure, the piezostack roles as an actuator to make a deflection of the beam which magnifies the motion of the spool part connected to the end of the beam. The spool part consists of spool, spool rod, sleeve, and return spring. The spool rod is kept in contact with the flexible beam by a compressive force generated by the return spring.
When voltage is applied on the piezostack actuator, the piezostack actuator is elongated and pushes the flexible beam. In this work, the commercial piezostack actuator, Pst 150/20/100 VS25 manufactured by PIEZOMECHANIK co. Ltd., is considered as an actuator and its operating voltage is from 0 to 150 V. Then, because the flexible beam rotated counterclockwise, the spool moves towards the right. As a result, the spool valve is in an open state which is shown in Figure 3(a). At the open state, the return spring is deformed due to the actuating force of the piezostack actuator. On the contrary, when the applied voltage is removed, the returning force generated from the return spring makes the spool return to its original position, and the valve is in closed state as shown in Figure 3(b). By using the previous working principle, the proposed piezo DDV system controls the flow rates continuously.
3. Modeling of the Piezo DDV System
The proposed piezo DDV system is a multiphysics system of interacting fluids and structures. In this study, a lumped parameter modeling method is proposed to describe the dynamic behavior of the structural parts (the piezostack, the flexible beam, and the spool part). Additionally, the fluid part is considered by using the flow force which is decided by the pressure drop, effective open area, and properties of the fluid. Figure 4 shows the schematic diagram of the piezostack-driven DDV system. First, the dynamic model of the structural parts is performed by considering dynamic behavior of each structural part such as piezostack, flexible beam, and spool part.
The dynamic model of the piezostack actuator can be mathematically expressed as follows: where is the dynamic effective mass of the piezostack which can be obtained by using the weight of piezostack actuator divided into 3. , , and are the damping coefficient, the stiffness, and the free displacement of the piezostack due to a unit of the applied voltage, respectively. and are the force acting on the flexible beam mechanism from the piezostack and the displacement of the tip point of the piezostack, respectively. The actuating force is transferred to the spool part by the flexible beam mechanism. Due to the flexible beam mechanism, the displacement of the spool is larger than the stroke of the piezostack actuator. In other words, the flexible beam mechanism amplifies the stroke of the piezostack actuator.
In order to obtain the dynamic model of the forced response of the flexible beam, a lumped parameter method is employed . A simplified structure of the flexible beam is shown in Figure 5(a). As shown in the figure, the flexible beam is fixed at the left side of the beam and consists of slender part, middle part, and stiff part. The force from the piezostack actuator and the force from the spool part act on the flexible beam. It is assumed that the first part which is located in the left side of the slender part is fixed and is not included in the simplified flexible beam model for reducing the computation load. Because the first part is very stiff and is clamped to the body, this assumption is reasonable. By using the lumped parameter method proposed by Irvin, the free body diagram of the beam can be obtained as shown in Figure 5(b). In the lumped parameter model, the continuous mass is converted to the lumped masses and the lumped masses are connected to each other by virtual springs. Then, from the free body diagram, the moment equation of the flexible beam can be obtained as follows:
In the aforementioned, and are the force from the piezostack actuator and the force from the spool part. is the base lengths of each full triangle, . , , , and are the lengths, the number of lumps, the stiffness constants of each lump, and lump masses, respectively. In the previous symbols, the subscripts of 1, 2, and 3 mean the stiff, the middle, and the slender parts, respectively. is the net deflection of the th lump, and is the deflection of the beam at the neutral axis of the th lump. It is noted that, because the geometric parameters and material properties of each part of the flexible beam are assumed constants, the lump mass () and stiffness constant () can be obtained as follows: where and are the density and Young’s modulus of the flexible beam material, respectively. and are inertia moments and cross-sectional areas of the th part of the beam, respectively. is the spring deflection which relates to the neutral axis deflection, , as follows:
Because the left side of the beam is fixed as shown in Figure 5(b), is zero from the free body diagram. When the actuating force is applied on the flexible beam with a harmonic signal whose frequency is , the term in (2a), (2b), and (2c) can be rewritten by , which results in equivalent simultaneous linear equations of the flexible beam. Then, the governing equation of the piezo DDV system can be obtained by combining the equivalent simultaneous equations of the flexible beam and dynamic equations of the piezostack and the spool part. However, first of all, it should be satisfied that the tip points of piezostack and spool part are always in contact with the flexible beam. Therefore, the tip point of the piezostack actuator, , can be replaced by and the spool displacement, , which can be obtained by using the dynamics of the spool can be replaced by .
The dynamic model of the spool part can be mathematically expressed as follows: where is the mass of the spool and spool rod. and are the damping coefficient which comes from the frictional force and the spring coefficient of the return spring, respectively. is a flow force of the spool valve which can be calculated by using the pressure drop, effective open area, and properties of the fluid.
In this work, in order to estimate the flow force of the spool valve, three assumptions are made: the spool valve is an overlapping type, the valve throttle area is linearly proportional to the valve opening, and the radial clearance leakage is neglected. The overlapping type means that the spool land length is greater than the valve opening width . Although the positive spool has nonlinearity in the overlap range, this type of valve can be easy to manufacture and has small internal leakage.
Figure 6 shows a schematic of the control volume of the considered spool valve. As shown in this figure, fluid enters the control volume from the bottom through the inlet area, , and exits the control volume on the bottom right through the outlet area, . The outlet area () of the overlapping spool can be calculated as follows: where , , , and are the diameter of the spool, the displacement of the spool, the length of the overlap, and the width of the outlet port, respectively. Then, the flow rate () can be obtained as follows:
From the Reynolds transport theorem, the conservation of the fluid momentum for the control volume can be obtained as follows : where is the fluid velocity vector, is the fluid density, and is a unit vector which indicates normally outward from the control volume surface. is vector of a flow force acting on the spool, which is a combination of the axial directional force, and vertical direction force. In this work, only axial direction force is considered as a flow force and the vertical direction force is ignored (it is reasonable because the vertical direction force influences the frictional force which is already considered in (5)). Then, the flow force can be calculated as follows: where is the length between inlet port and outlet port, is the flow rate through the control volume, and is the jet angle. Equation (8) can be rewritten as follows by using the steady flow force, , and transient flow force, .
Consider the following: where By considering the geometry of the spool and assuming that the pressure drop is constant, the steady and transient flow force can be rewritten more explicitly as follows: where is the discharge coefficient, and is the pressure difference between the inlet port and outlet port. Then, by combining (6), (9), and (12), the dynamic equation of the spool part can be given by
The dynamic model of the whole piezo DDV system can be obtained by combining (1), (2a), (2b), (2c), and (13) by using the block diagram shown in Figure 7. As shown in this figure, the block diagram consists of the dynamic model of the piezostack, the flexible beam, and the spool part. The input voltage () is applied on the piezostack dynamics block, and the piezostack displacement () is obtained from the beam dynamics. By using the input voltage and piezostack displacement, the force acting on the beam generated by the piezostack () is derived (by using (1)). Because the spool rod is always in contact with the flexible beam, as mentioned earlier, the displacement of the end of the beam () is equal to the displacement of the spool (). From the calculated force and the displacement of the beam end, the force from the spool acting on the beam and the piezostack displacement are obtained by solving the dynamic equation of the beam (2a), (2b), and (2c). The force is then considered as an input of the spool part dynamics block (13). By using the displacement of the spool (), the flow rate of the piezo DDV system can be obtained (7).
4. Design of the Piezo DDV System
In order to evaluate the dynamic behavior of the piezo DDV system, computer simulation is proposed based on the lumped parameter model. First, it is necessary to determine the dimensions of the piezo DDV system; to do so, a couple of assumptions are made: the diameter and overlap range of the spool are 5 mm and 0.05 mm, respectively. The pressure drop between input port and output port is 30 bar. Then, the operational requirements can be changeable from the flow rate to the spool displacement which should be larger than 0.5 mm by using (7).
The design procedure is performed as follows in order to achieve operational requirements (operating frequency: over than 100 Hz; flow rate: 20 liter/Min (spool displacement: over than 0.5 mm)).(1)Select a commercial piezostack actuator manufactured by PIEZOMEHANIK co. Ltd. and obtain performance characteristics such as stroke, blocking force, and size of the piezostack actuator .(2)Find the geometric dimensions of the flexible beam mechanism by using the optimal design method.(3)Repeat steps 1 and 2 with other commercial piezostack actuators; hence, several design sets are obtained. (4)Select the best design from the results obtained from step 3 considering the operational requirements and size.
In this paper, the best design set and how to obtain the best design set are explained. Firstly, Pst 150/20/100 VS25 model of the commercial piezostack actuator is considered as an actuator of the piezo DDV system. Then, the optimal geometric dimensions of the flexible beam mechanism are obtained by using finite elements analysis (FEA) integrated with an optimization tool. The optimal design problem of the flexible beam can be stated as follows: find optimal dimensions that are constrained in an available space in order to satisfy the operational requirements mentioned previously. The significant geometric dimensions of the flexible beam are shown in Figure 8. The length of the stiff part (), the length of the middle part (), length of slender part (), fillet radius of the slender part (), plat length of the slender part (), height of the stiff part (), depth of the slender part (), radius of the bottom furrow (), width of the bottom furrow (), and thickness of the beam () are considered as principal design variables.
In the design, it is noted that the overall beam length () is minimized in order to reduce the size of the piezo DDV system. Moreover, for satisfying the operational requirements, a couple of conditions are constrained: the 1st natural frequency of the flexible beam is larger than 300 Hz for operating the piezo DDV system in driving condition of 100 Hz; the maximum displacement of the tip point of the beam is larger than 0.5 mm. In the simulation, point mass and spring effect are added at the tip point of the flexible beam to consider the effects of the spool and return spring.
In this work, the optimal solution of the piezo DDV system is obtained by using the first order method with the golden section algorithm of the ANSYS optimization tool which has been mentioned in detail in several researches [21, 22]. Figure 9 shows the optimal solution of the flexible beam structure. From the figure, it can be found that the total length of the flexible beam () is converged after 24 iterations with a convergence tolerance of 0.1 mm. Therefore, in this work, the solution at the 24th iteration is considered as the optimal geometry of the flexible beam. By using the previous design procedure, the design parameters of the piezo DDV system can be obtained as listed in Table 1.
5. Results and Discussions
In this work, MATLAB Simulink is used for performance evaluation of the proposed piezo DDV system. In every simulation, the Runge-Kutta solver is used and the step size is fixed as 0.0001 seconds. The computer simulations are categorized into two steps. The performances of the piezo DDV system such as maximum operating range, operating bandwidth, and step response are evaluated as a first step. Then, the control performances of the piezo DDV system via a couple of desired spool positions are also simulated as a second step.
The considered piezostack actuator is a kind of unipolar piezostack actuator and its available input voltage is from 0 to 150 V. In order to confirm the controllable position range of the spool and controllable flow rate range at 100 Hz, the input voltage of the piezostack actuator is set as . With this input voltage condition, the maximum displacement of the spool and the flow rate are, respectively, obtained by 0.56 mm and 22.8 liter/min which satisfied the operational requirement as shown in Figure 10. In this figure, when the spool is located in the range between 0 and 0.05 mm, the flow rate is zero. This is because the spool is the overlapping type and the overlap range of the spool is 0.05 mm as early mentioned.
Figure 11 presents the bode plot of the proposed piezo DDV system. Bode plot is obtained by using the dynamic model of the whole piezo DDV system with different input frequencies. The operating bandwidth of the system can be obtained by using the cut-off frequency which means the frequency at which the magnitude of the closed-loop frequency response is 3 dB below its zero-frequency value. As shown in the figure, the cut-off frequency of the proposed piezo DDV system is 181 Hz. Therefore, because the operating bandwidth of the proposed system is from 0 to 181 Hz, the operating requirement of operating bandwidth is satisfied.
In order to obtain the time constant of the piezo DDV system, unit step response is obtained as shown in Figure 12. The time constant represents the time it takes the system’s step response to reach 63.2% of its saturation value. As shown in the figure, the time constant is sec.
In this work, two kinds of signals (sinusoidal signal and chirp signal) are considered as desired trajectories of the spool for confirming the position tracking controllability of the piezo DDV system. In order to control the desired trajectories of the spool, a discrete PID controller is designed as follows: where , , and are the proportional, integral, and derivative gains, respectively. is the sampling period. is the th error which can be obtained as follows: where and are the th desired position of the spool and the th position of the spool, respectively.
Figure 13 shows the sinusoidal position tracking control results of the proposed piezo DDV system. The desired spool position is set at . In this case, the imposed proportional, integral, and derivative gains are determined as , 47, and 0.2, respectively. These gains are determined by using the trial and error method. It is clearly observed that the desired position of the spool is well tracked by the proposed system and PID controller as shown in Figure 13(a). Figure 13(b) shows that the tracking error is less than 3 μm, which is also evident that the PID controller for position tracking control operates the piezo DDV system nicely. The control input of the piezostack actuator shown in Figure 13(c) is less than 150 V which is suitable for the considered piezostack actuator. Additionally, when the spool is operated as , the flow rate of the piezo DDV system is over 20 liter/Min. as shown in Figure 13(d). Therefore, the operational requirement of the flow rate is satisfied.
Figure 14 shows the position tracking control results, when the desired spool position is set as chirp signal from 1 Hz to 100 Hz. In this case, by using the trial error method, the proportional, integral, and derivative gains are determined as , 2400, and 15, respectively. As shown in Figures 14(a) and 14(b), the proposed piezo DDV system is well controlled within negligible error. With this desired spool position, the control input voltage shown in Figure 14(c) is also less than 150 V. The flow rate of the system can be obtained as shown in Figure 14(d). From Figures 13 and 14, it is obviously observed that the proposed piezo DDV system with PID controller has good position tracking controllability.
In this work, a new type of direct drive valve featuring a piezostack actuator and flexible beam mechanism was developed. The proposed piezo DDV system consists of the piezostack actuator, spool part, and flexible beam mechanism which is used for overcoming the stroke limitation of the piezostack actuator. The components of the proposed piezo DDV system were formulated by using the lumped parameter model, and the fluid effect was considered by using the flow force. After deciding the certain operational requirements such as operating bandwidth and maximum flow rate, the proper commercial piezostack actuator was chosen, and the specific dimensions of the flexible beam mechanism were obtained by using the optimal design method. In order to evaluate the proposed system, MATLAB Simulink is used. With maximum input voltage of the chosen piezostack actuator, the spool displacement reached up to 0.56 mm, and the maximum flow rate of the system was 22.8 liter/Min. Moreover, the operating bandwidth and the time constant were obtained by 181 Hz and 1.6 millisecond, respectively. From these results, it was clearly observed that the proposed piezo DDV system was satisfied with the operational requirements in conditions of operating bandwidth and maximum flow rate. A PID controller was then designed and realized to control the position of the spool by controlling the piezostack actuator. In order to evaluate the controllability of the proposed DDV system, a couple of trajectories were considered as the desired signal. By using the proposed piezo DDV system associated with PID controller, it was confirmed through computer simulation that the desired trajectories were well tracked with suitable input voltage. It is finally remarked that the proposed piezo DDV system will be manufactured and experimentally evaluated in the near future as a second phase of this study.
Conflict of Interests
The authors declare that there is no conflict of interests.
- C. Kaddissi, J.-P. Kenné, and M. Saad, “Identification and real-time control of an electrohydraulic servo system based on nonlinear backstepping,” IEEE/ASME Transactions on Mechatronics, vol. 12, no. 1, pp. 12–22, 2007.
- P. A. Sente, F. M. Labrique, and P. J. Alexandre, “Efficient control of a piezoelectric linear actuator embedded into a servo-valve for aeronautic applications,” IEEE Transactions on Industrial Electronics, vol. 59, no. 4, pp. 1971–1979, 2012.
- C. Garvin and A. Mathew, “The application of the method of simultaneous stabilization to the control of a nonlinear servo valve,” IEEE Transactions on Control Systems Technology, vol. 4, no. 6, pp. 654–664, 1996.
- J.-S. Lin and C.-L. Chen, “Buck/boost servo amplifier for direct-drive-valve actuation,” IEEE Transactions on Aerospace and Electronic Systems, vol. 31, no. 3, pp. 960–967, 1995.
- K. Dasgupta and H. Murrenhoff, “Modelling and dynamics of a servo-valve controlled hydraulic motor by bondgraph,” Mechanism and Machine Theory, vol. 46, no. 7, pp. 1016–1035, 2011.
- S. Li and W. Bao, “Influence of magnetic fluids on the dynamic characteristics of a hydraulic servo-valve torque motor,” Mechanical Systems and Signal Processing, vol. 22, no. 4, pp. 1008–1015, 2008.
- S. Karunanidhi and M. Singaperumal, “Mathematical modelling and experimental characterization of a high dynamic servo valve integrated with piezoelectric actuator,” Proceedings of the Institution of Mechanical Engineers I, vol. 224, no. 4, pp. 419–435, 2010.
- G. Changbin and J. A. Zongxia, “A piezoelectric direct-drive servo valve with a novel multi-body contacting spool-driving mechanism: design, modelling and experiment,” Proceedings of the Institution of Mechanical Engineers C, 2013.
- T. Miyajima, T. Fujita, K. Sakaki, K. Kawashima, and T. Kagawa, “Development of a digital control system for high-performance pneumatic servo valve,” Precision Engineering, vol. 31, no. 2, pp. 156–161, 2007.
- M. Taghizadeh, A. Ghaffari, and F. Najafi, “Modeling and identification of a solenoid valve for PWM control applications,” Comptes Rendus—Mecanique, vol. 337, no. 3, pp. 131–140, 2009.
- J. E. Lindler and E. H. Anderson, “Piezoelectric direct drive servovalve,” in Smart Structures and Materials: Industrial and Commercial Applications of Smart Structures Technologies, vol. 4698 of Proceedings of SPIE, pp. 488–496, March 2002.
- M.-L. Zhou, Z.-G. Yang, W. Gao, Y.-T. Tian, C.-L. Shen, and P. Li, “Fuzzy control of a new type of piezoelectric direct drive electro-hydraulic servo valve,” in Proceedings of the International Conference on Machine Learning and Cybernetics (ICMLC '05), pp. 819–824, August 2005.
- P. Sente, C. Vloebergh, F. Labrique, and P. Alexandra, “Control of a direct-drive servo-valve actuated by a linear amplified piezoelectric actuator for aeronautic applications,” in Proceedings of the International Conference on Electrical Machines (ICEM '08), September 2008.
- W. Xu and T. King, “Flexure hinges for piezoactuator displacement amplifiers: flexibility, accuracy, and stress considerations,” Precision Engineering, vol. 19, no. 1, pp. 4–10, 1996.
- Q.-H. Nguyen, S.-B. Choi, and J.-D. Kim, “The design and control of a jetting dispenser for semiconductor electronic packaging driven by a piezostack and a flexible beam,” Smart Materials and Structures, vol. 17, no. 6, Article ID 065028, 2008.
- H. Zeng, R.-B. Yuan, C.-G. Sun, and Z.-C. Zhang, “Study on performance of laminated piezoelectric pneumatic servo valve,” in Proceedings of the 1st International Conference on Advances in Computational Modeling and Simulation (ACMS '11), pp. 1140–1148, December 2011.
- P. V. Irvin, “A general mathematical model for beam and plate vibration in bending modes using lumped parameters,” Tech. Rep., NASA, 1966.
- M. G. Rabie, Fluid Power Engineering, McGraw-Hill Companies, New York, NY, USA, 1st edition, 2009.
- N. D. Manring, Hydraulic Control Sytem, John Wiley & Sons, Hoboken, NJ, USA, 1st ed edition, 2005.
- “Piezo-mechanical Low Voltage Stack and Ring actuators,” Product range and technical data, http://www.piezomechanik.com/en/products/.
- Q.-H. Nguyen, Y.-M. Han, S.-B. Choi, and N. M. Wereley, “Geometry optimization of MR valves constrained in a specific volume using the finite element method,” Smart Materials and Structures, vol. 16, no. 6, pp. 2242–2252, 2007.
- Q.-H. Nguyen and S.-B. Choi, “Optimal design of a vehicle magnetorheological damper considering the damping force and dynamic range,” Smart Materials and Structures, vol. 18, no. 1, Article ID 015013, 2009.