1. Introduction
In many dynamic systems such as robot and aerospace areas, flexible structures have been extremely employed to satisfy various requirements for large scale, light weight and high speed in dynamic motion. However, these flexible structures are readily susceptible to the internal/external disturbances (or excitations). Therefore, vibration control schemes should be exerted to achieve high performance and stability of flexible structure systems. Recently, in order to successfully achieve vibration control for flexible structures smart materials such as piezoelectric materials [1-2], shape memory alloys [3-4], electrorheological (ER) fluids [5-6] and magnetorheological (MR) fluids [7] are being widely utilized. Among these smart materials, ER or MR fluid exhibits reversible changes in material characteristics when subjected to electric or magnetic field. The vibration control of flexible structures using the smart ER or MR fluid can be achieved from two different methods. The first approach is to replace conventional viscoelastic materials by the ER or MR fluid. This method is very effective for shape control of flexible structures such as plate [5]. The second approach is to devise dampers or mounts and apply to vibration control of the flexible structures. This method is very useful to isolate vibration of large structural systems subjected to external excitations [6-7]. In this work, a new type of MR mount is proposed and applied to vibration control of the flexible structures.
In order to reduce unwanted vibration of the flexible structure system, three different types of mounts are normally employed: passive, semi-active and active. The passive rubber mount, which has low damping, shows efficient vibration performance at the non-resonant and high frequency excitation. Thus, the rubber mount is the most popular method applied for various vibrating systems. However, it cannot have a favorable performance due to small damping effect at the resonant frequency excitation. On the other hand, the passive hydraulic mount has been developed to utilize dynamic absorber effect or meet large damping requirement in the resonance of low frequency domain [8]. However, the high dynamic stiffness property of the hydraulic mount may deteriorate isolation performance in the non-resonant excitation domain. Thus, the damping and stiffness of the passive mounts are not simultaneously controllable to meet imposed performance criteria in a wide frequency range. The active mounts are normally operated by using external energy supplied by actuators in order to generate control forces on the system subjected to excitations [9]. The control performance of the active mount is fairly good in a wide frequency range, but its cost is expensive. Moreover, its configuration is complex and its stability may not be guaranteed in a certain operation condition. On the other hand, the semi-active mounts cannot inject mechanical energy into the structural systems. But, it can adjust damping to reduce unwanted vibration of the flexible structure systems. It is known that using the controllable yield stress of ER or MR fluid, a very effective semi-active mount can be devised for vibration control of the flexible structures. The flow operation of the ER or MR mount can be classified into three different modes: shear mode [6], flow mode [10] and squeeze mode [11].
In this article, a new type of semi-active MR mount shown in the figure 1 is proposed and applied to vibration control of flexible structures. As a first step to achieve the research goal, the configuration of a mixed-mode MR mount is devised and the mathematical model is formulated on the basis of non-dimensional Bingham number. After manufacturing an appropriate size of MR mount, the field-dependent damping force is experimentally evaluated with respect to the field intensity. The MR mount is installed on the beam structure as a semi-active actuator, while the beam structure is supported by two passive rubber mounts. The dynamic model of the structural system incorporated with the MR mount is then derived in the modal coordinate, and an optimal controller is designed in order to control unwanted vibration responses of the structural system subjected to external excitations. The controller is experimentally implemented and control performances such as acceleration of the structural systems are evaluated in frequency domain.
2. MR Mount
In this work, a new type of the mixed-mode MR mount which is operated under the flow and shear motion is proposed. The schematic configuration of the MR mount proposed in this work is shown in Figure 1 (a). The MR mount consists of rubber element and MR dash-pot. The MR dash-pot is assembled by MR fluid, piston (or plunger), electromagnet coil, flux guide, and housing. The MR fluid is filled in the gap between piston and outer cylindrical housing. The electromagnetic coil is wired inside of the cylindrical housing. The housing can be fixed to the supporting structure, and the plunger is attached to the top end of the rubber element. The rubber element has a role to support the static load and isolate the vibration transmission at the non-resonant and high frequency regions. During the relative motion of the plunger and housing, MR fluid flows through annular gap. Thus, the pressure drop due to flow resistance of MR fluid in the annular gap can be obtainable. At the same time, the MR dash-pot has additional shear resistance due to relative motion of annular gap walls. Therefore, the proposed MR dash-pot operates under both the flow and shear modes. If no magnetic field is applied, the MR dash-pot only produces a damping force caused by the fluid resistance associated with the viscosity of the MR fluid. However, if the magnetic field is applied through the annular gap, the MR mount produces a controllable damping force due to the yield stress of the MR fluid. As it can be seen from Figure 1 (a), the proposed MR mount has compact structure and operates without frictional components.
The transmitted force
In the above,
where,
In the above,
The field-dependent yield stress
The nondimensional form (2,3) of MR mount under mixed mode can be transformed to the Bingham plastic model as follows [12]. This simple model is widely used for the controller implementations.
In the above,
where
3. Structural System
In order to investigate the applicability of the proposed MR mount to vibration control, a flexible structure system is established as shown in Figure 3. The MR mount is placed between the exciting mass and steel beam structure. When the mass is excited by external disturbance, the force transmitting through the MR mount excites the beam structure. Thus, the vibration of the beam structure can be controlled by activating the MR mount.
where,
In the above,
In order to determine system parameters such as modal frequencies and mode shapes, modal analysis is undertaken by adopting a commercial software (MSC/NASTRAN for Windows V4.0). The finite element model of the flexible structure consists of 30 beam elements, 3 spring elements, and 2 mass elements. The nodes of the structural system are constrained in the x, z directions, and the 2-node elastic beam element is used to model the beam. The geometry of the steel beam is 1500mm (length) 60mm (length) 15mm (thickness). The rubber mounts are placed on the 50mm (
Figure 4 presents mode shapes of the first three modes. The first mode is bending mode, while the second and the third modes are combination of rotational and bending modes. In this study, modal parameters of the structural system are identified by experimental modal test. It is evaluated by computer simulation that the effect of the residual modes is quite small compared with dominant mode. Therefore, in this paper, only dominant modes are considered.
Figure 5 shows the schematic configuration of the experimental setup for the modal parameter identification. The mass on the MR mount is excited by the electromagnetic exciter. The MR mount is removed for the modal parameter identification of structure system only. Thus, this test shows the modal test of the passive structural system in which the mass is supported by the rubber mount. The excitation force and frequency are regulated by the exciter controller. Accelerometers are attached to the mass and beam, and their positions are denoted by
By considering first three modes as controllable modes, the dynamic model of the structural system, given by equations (6-8), can be expressed in a state space control model as follows.
where
In the above,
4. Vibration Control
In order to attenuate unwanted vibration of the flexible structural system, the linear quadratic Gaussian (LQG) controller, consisting of linear quadratic regulator (LQR) and Kalman-Bucy filter (KBF), is adopted. As a first step to formulate the LQG controller the performance index
In the above,
In the above,
Thus, the control force of the MR mount can be represented as
In this work, by the tuning method the control gains of the
This condition physically indicates that the actuating of the controller
Since the states of
where,
In the above,
Using the estimated states, the control force of the MR mount is obtained as follows.
In order to implement the LQG controller, an experimental setup is established as shown in Figure 6. The mass supported by the MR mount is excited by the electromagnetic exciter, and the excitation force and frequency are regulated by the exciter controller. Accelerometers are attached to the beam and mass, and their positions are denoted by
Figures 7 and 8 present the measured displacement and acceleration of the structural system. The amplitude of excitations force is set by 1N. The excitation frequency range for the structural system is chosen from 5 to 80Hz. The uncontrolled and controlled responses are measured at the positions of the mass (
5. Conclusion
A new type of the mixed-mode MR mount was proposed and applied to vibration control of a flexible beam structure system. On the basis of non-dimensional Bingham number, an appropriate size of the MR mount was designed and manufactured. After experimentally evaluating the field-dependent damping force of the MR mount, a structural system consisting of a flexible beam and vibrating rigid mass was established. The governing equation of motion of the system was formulated and a linear quadratic Gaussian (LQG) controller was designed to attenuate the vibration of the structural system. It has been demonstrated through experimental realization that the imposed vibrations of the structural system such as acceleration and displacement are favorably reduced by activating the proposed MR mount associated with the optimal controller. The control results presented in this study are quite self-explanatory justifying that the proposed semi-active MR mount can be effectively utilized to the vibration control of various structural systems such as flexible robot arm and satellite appendages.
Acknowledgments
This work was partially supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST)(No. 2010-0015090).
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