**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 sub‐ jected 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 effec‐ tive for shape control of flexible structures such as plate [5]. The second approach is to de‐ vise 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 vari‐ ous vibrating systems. However, it cannot have a favorable performance due to small damp‐ ing effect at the resonant frequency excitation. On the other hand, the passive hydraulic mount

© 2012 Choi et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Choi et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 do‐ main. Thus, the damping and stiffness of the passive mounts are not simultaneously control‐ lable 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 con‐ trol 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 con‐ figuration is complex and its stability may not be guaranteed in a certain operation condi‐ tion. 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 struc‐ tures. 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].

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

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mount has compact structure and operates without frictional components.

The transmitted force (*Ft*) through the proposed MR mount can be represented by

In the above, *kr* and *cr* are stiffness and damping of the rubber element, respectively. *x* is the deflection of the rubber element. The damping force (*Fmr*) due to the flow resistance of the MR fluid in the gap can be represented by considering the plug flow of the MR fluid under

*Ft* =*kr x* + *cr x*˙ + *Fmr* (1)

**Figure 1.** The proposed MR mount.

the mixed mode operation as follows [7].

In this article, a new type of semi-active MR mount shown in the figure 1 is proposed and ap‐ plied 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 for‐ mulated on the basis of non-dimensional Bingham number. After manufacturing an appro‐ priate 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-ac‐ tive actuator, while the beam structure is supported by two passive rubber mounts. The dy‐ namic 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 vibra‐ tion responses of the structural system subjected to external excitations. The controller is ex‐ perimentally 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 dashpot. 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.

**Figure 1.** The proposed MR 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 do‐ main. Thus, the damping and stiffness of the passive mounts are not simultaneously control‐ lable 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 con‐ trol 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 con‐ figuration is complex and its stability may not be guaranteed in a certain operation condi‐ tion. 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 struc‐ tures. The flow operation of the ER or MR mount can be classified into three different modes:

In this article, a new type of semi-active MR mount shown in the figure 1 is proposed and ap‐ plied 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 for‐ mulated on the basis of non-dimensional Bingham number. After manufacturing an appro‐ priate 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-ac‐ tive actuator, while the beam structure is supported by two passive rubber mounts. The dy‐ namic 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 vibra‐ tion responses of the structural system subjected to external excitations. The controller is ex‐ perimentally implemented and control performances such as acceleration of the structural

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 dashpot. 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

shear mode [6], flow mode [10] and squeeze mode [11].

88 Advances in Vibration Engineering and Structural Dynamics

systems are evaluated in frequency domain.

**2. MR Mount**

The transmitted force (*Ft*) through the proposed MR mount can be represented by

$$F\_t = k\_r \mathbf{x} + c\_r \dot{\mathbf{x}} + F\_{mr} \tag{1}$$

In the above, *kr* and *cr* are stiffness and damping of the rubber element, respectively. *x* is the deflection of the rubber element. The damping force (*Fmr*) due to the flow resistance of the MR fluid in the gap can be represented by considering the plug flow of the MR fluid under the mixed mode operation as follows [7].

$$
\varphi\_F = a \varphi\_r^3 + \left(\frac{b}{6} \varphi\_c + c\right) \varphi\_r^2 \tag{2}
$$

input current increases, the damping force also increases. It is also observed that the predict‐ ed field-dependent damping force shows small difference from the measured force in the post-yield velocity region. On the other hand, the damping forces show much large differ‐ ence in the pre-yield region. This is because the model (2,3) is discontinuous and cannot rep‐ resent the hysteretic behavior of the MR mount in the transition from pre-yield to post-yield.

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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

In the above, *cf* and *Fy* are the post-yield damping constant and the yield force, respectively. *I* is the current applied to the MR mount. In order to identify the parameters of the models, a constrained least-mean-squared (LMS) error minimization procedure is used. The cost

> *f* (*tk* )− *f* ^ (*tk* ) 2

ured force shown in figure 2, and *tk* is the time at which the kth sample has been taken. Pa‐ rameters *cf* and *Fy* of the Bingham plastic model are estimated so as to minimize the cost function *J* . The parameter optimization is performed on a test case of the identification data set. It is noted that the optimization procedure is applied to the data of the post-yield re‐ gions for Bingham plastic model. Based on the measured response in Figure 2, the field-de‐ pendent damping constant *cf* and yield force *Fy* are identified by (23 + 25*I*) Nsec/m and

(*tk* ) is the force calculated using the model given by equation (4), *f* (*tk* ) is the meas‐

+ *Fy*(*I*)sgn(*vp*) (4)

(5)

*Fmr* =*cf* (*I*)*vp*

*<sup>J</sup>*(*cf* , *Fy*)=∑

*k*=1 *N*

**Figure 2.** The field-dependent damping force of the MR mount.

function *J* for Bingham plastic model is defined by

implementations.

where *f* ^

where,

$$\varphi\_{\rm F} = \frac{F\_{mr}}{6\pi\eta v\_p L}, \; \varphi\_c = \frac{\pi\_y h}{\eta v\_p}, \; \varphi\_r = \frac{r}{h} \tag{3}$$

In the above, *φF* is the dimensionless damping force, *φc* is the Bingham number, and *φ<sup>r</sup>* is the dimensionless geometric parameter. *η* is the post-yield plastic viscosity, *vp* is the relative velocity between piston and housing, *L* is the annular gap length, *τ<sup>y</sup>* is the field-dependent yield shear stress of the MR fluid, *h* is the annular gap size, and *r* is the piston radius. The dimensionless parameter *φc* represents the ratio of the dynamic yield shear stress to the vis‐ cous shear stress. Moreover, *φc* shows the influence of the magnetic field on the damping force of the MR mount. The dimensionless geometric parameter *φ<sup>r</sup>* is characterized by the piston radius *r* and gap size *h* . The dimensionless damping force *φ<sup>F</sup>* increases as the Bing‐ ham number *φc* and dimensionless geometric parameter *φr* increase. This implies that high yield stress and large piston area are required to generate high damping force of the MR mount. On the other hand, the gap length *L* depends on the dimensionless damping force *φF* only. Therefore, the damping force *Fmr* can be directly scaled by the gap length *L* . It is noted that the damping force *Fmr* can be expressed by the velocity of rubber element and the field-dependent yield shear stress of the MR fluid. The parameters *a* and *c* of equation (2) are chosen to be 1 by considering Newtonian flow behavior of the MR fluid in the absence of magnetic field. And then, the parameter *b* , which reflects the effect of Bingham flow of the MR fluid, is chosen to be 2.47 using a least square error criterion.

The field-dependent yield stress *τy* of the MR fluid (MRF-132LD, Lord Corporation) em‐ ployed in this study has been experimentally obtained by 0.13*H* 1.3 kPa. Here the unit of mag‐ netic field *H* is kA/m. The post-yield plastic viscosity *η* was also experimentally evaluated by 0.59 Pasec. The rheometer (MCR300, Physica, Germany) was used for obtaining the val‐ ue of the yield stress and the post-yield plastic viscosity. By considering both the nondimen‐ sional damping force equations (2,3) and the size of the structural system, an appropriate size of MR mount was designed and manufactured as shown in Figure 1 (b). The piston radius *r* , gap size *h* , and gap length *L* are designed to be 8.5 mm, 1.5 mm, and 10 mm, respective‐ ly. The field-dependent damping force is predicted and experimentally measured by excit‐ ing the MR mount with the sinusoidal signal which has frequency of 15 Hz and amplitude of 0.07m/sec. Figure 2 presents controllable damping force characteristic of the proposed MR mount. In order to measure the field-dependent damping force of the MR mount, the MR mount is placed between the load cell and electromagnetic exciter. When the shaker table moves up and down by a command signal generated from the exciter controller, the MR mount produces the damping force and it is measured by the load cell. As expected, as the input current increases, the damping force also increases. It is also observed that the predict‐ ed field-dependent damping force shows small difference from the measured force in the post-yield velocity region. On the other hand, the damping forces show much large differ‐ ence in the pre-yield region. This is because the model (2,3) is discontinuous and cannot rep‐ resent the hysteretic behavior of the MR mount in the transition from pre-yield to post-yield.

**Figure 2.** The field-dependent damping force of the MR mount.

*φ<sup>F</sup>* =*aφ<sup>r</sup>*

*<sup>φ</sup><sup>F</sup>* <sup>=</sup> *Fmr*

90 Advances in Vibration Engineering and Structural Dynamics

MR fluid, is chosen to be 2.47 using a least square error criterion.

where,

<sup>3</sup> <sup>+</sup> ( *<sup>b</sup>*

<sup>6</sup>*πηvpL* , *<sup>φ</sup><sup>c</sup>* <sup>=</sup>

<sup>6</sup> *<sup>φ</sup><sup>c</sup>* <sup>+</sup> *<sup>c</sup>*)*φ<sup>r</sup>*

*τyh ηvp*

In the above, *φF* is the dimensionless damping force, *φc* is the Bingham number, and *φ<sup>r</sup>* is the dimensionless geometric parameter. *η* is the post-yield plastic viscosity, *vp* is the relative velocity between piston and housing, *L* is the annular gap length, *τ<sup>y</sup>* is the field-dependent yield shear stress of the MR fluid, *h* is the annular gap size, and *r* is the piston radius. The dimensionless parameter *φc* represents the ratio of the dynamic yield shear stress to the vis‐ cous shear stress. Moreover, *φc* shows the influence of the magnetic field on the damping force of the MR mount. The dimensionless geometric parameter *φ<sup>r</sup>* is characterized by the piston radius *r* and gap size *h* . The dimensionless damping force *φ<sup>F</sup>* increases as the Bing‐ ham number *φc* and dimensionless geometric parameter *φr* increase. This implies that high yield stress and large piston area are required to generate high damping force of the MR mount. On the other hand, the gap length *L* depends on the dimensionless damping force *φF* only. Therefore, the damping force *Fmr* can be directly scaled by the gap length *L* . It is noted that the damping force *Fmr* can be expressed by the velocity of rubber element and the field-dependent yield shear stress of the MR fluid. The parameters *a* and *c* of equation (2) are chosen to be 1 by considering Newtonian flow behavior of the MR fluid in the absence of magnetic field. And then, the parameter *b* , which reflects the effect of Bingham flow of the

The field-dependent yield stress *τy* of the MR fluid (MRF-132LD, Lord Corporation) em‐ ployed in this study has been experimentally obtained by 0.13*H* 1.3 kPa. Here the unit of mag‐ netic field *H* is kA/m. The post-yield plastic viscosity *η* was also experimentally evaluated by 0.59 Pasec. The rheometer (MCR300, Physica, Germany) was used for obtaining the val‐ ue of the yield stress and the post-yield plastic viscosity. By considering both the nondimen‐ sional damping force equations (2,3) and the size of the structural system, an appropriate size of MR mount was designed and manufactured as shown in Figure 1 (b). The piston radius *r* , gap size *h* , and gap length *L* are designed to be 8.5 mm, 1.5 mm, and 10 mm, respective‐ ly. The field-dependent damping force is predicted and experimentally measured by excit‐ ing the MR mount with the sinusoidal signal which has frequency of 15 Hz and amplitude of 0.07m/sec. Figure 2 presents controllable damping force characteristic of the proposed MR mount. In order to measure the field-dependent damping force of the MR mount, the MR mount is placed between the load cell and electromagnetic exciter. When the shaker table moves up and down by a command signal generated from the exciter controller, the MR mount produces the damping force and it is measured by the load cell. As expected, as the

, *<sup>φ</sup><sup>r</sup>* <sup>=</sup> *<sup>r</sup>*

<sup>2</sup> (2)

*<sup>h</sup>* (3)

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.

$$F\_{mr} = c\_f(I)v\_p + F\_y(I)\text{sgn}(v\_p) \tag{4}$$

In the above, *cf* and *Fy* are the post-yield damping constant and the yield force, respectively. *I* is the current applied to the MR mount. In order to identify the parameters of the models, a constrained least-mean-squared (LMS) error minimization procedure is used. The cost function *J* for Bingham plastic model is defined by

$$f\left(\mathbf{c}\_{f},\;F\_{y}\right) = \sum\_{k=1}^{N} \left[f\left(t\_{k}\right) - \hat{f}\left(t\_{k}\right)\right]^{2} \tag{5}$$

where *f* ^ (*tk* ) is the force calculated using the model given by equation (4), *f* (*tk* ) is the meas‐ ured force shown in figure 2, and *tk* is the time at which the kth sample has been taken. Pa‐ rameters *cf* and *Fy* of the Bingham plastic model are estimated so as to minimize the cost function *J* . The parameter optimization is performed on a test case of the identification data set. It is noted that the optimization procedure is applied to the data of the post-yield re‐ gions for Bingham plastic model. Based on the measured response in Figure 2, the field-de‐ pendent damping constant *cf* and yield force *Fy* are identified by (23 + 25*I*) Nsec/m and 8.28*I* 1.85 N, respectively. Here, the unit of current *I* is A. These values will be used as system parameters in the controller implementation for the structural system.

## **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 be‐ tween the exciting mass and steel beam structure. When the mass is excited by external dis‐ turbance, 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. *y*(*pj* ) is the displacement at position *pj* of the structural system. The MR mount is placed at position *p*<sup>2</sup> of the beam structure, while two passive rubber mounts are placed at *p*1 and *p*3 to support the beam structure. The governing equation of the motion of the proposed structural system can be obtained using the mode summation method as follows [13]:

$$
\ddot{q}\_i(t) + 2\zeta\_i\omega\_i\dot{q}\_i(t) + \omega\_i^2 q\_i(t) = \frac{Q\_i(t)}{I\_i} + \frac{Q\_{xi}(t)}{I\_i}, \text{ i = 1, 2, \dots, \infty} \tag{6}
$$

where,

$$Q\_i(t) = (\wp\_i(p\_4) - \wp\_i(p\_2))(-F\_{mr}(t))\tag{7}$$

$$\mathcal{Q}\_{ex\_i}(t) = \mathcal{q}\_i(p\_4) F\_{ex}(t) \tag{8}$$

the rotational modes. The 2-node spring elements are used to model the rubber mounts and rubber element of the MR mount. The material property for the spring elements is set by 60kN/m which has been experimentally evaluated. To model the mass of lower part of the MR mount and vibrating mass, the 1-node lumped mass element is used. The material prop‐ erties for the mass elements are 1.5kg and 3kg for the lower part of the MR mount and the

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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 com‐ pared 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 param‐ eter 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 support‐ ed by the rubber mount. The excitation force and frequency are regulated by the exciter con‐ troller. Accelerometers are attached to the mass and beam, and their positions are denoted by *x* = *p*1 , *x* = *p*2 , *x* = *p*3 and *x* = *p*4 . From this, the following system parameters are ob‐

tained : *ω*1 56.9 rad/s, *ω*2 144.8 rad/s, *ω*3 168.4 rad/s, *ζ*1 0.0260, *ζ*2 0.0497, and *ζ*3 0.0630.

vibrating mass, respectively.

**Figure 3.** Configuration of the structural system with the MR mount.

In the above, *qi* (*t*) is the generalized modal coordinate, *φ<sup>i</sup>* (*x*) is the mode shape value at po‐ sition *x* , *ω<sup>i</sup>* is the modal frequency, *ζ<sup>i</sup>* is the damping ratio, and *Ii* is the generalized mass of the *i* th mode. *Qi* (*t*) is the generalized force including the damping force of the MR mount, and *Qexi* (*t*) is the generalized force including the exciting force *Fex*(*t*) . It is noted that the spring and damping effects of the rubber elements are resolved in the modal parameters of the structural system.

In order to determine system parameters such as modal frequencies and mode shapes, mo‐ dal analysis is undertaken by adopting a commercial software (MSC/NASTRAN for Win‐ dows 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 ge‐ ometry of the steel beam is 1500mm (length) 60mm (length) 15mm (thickness). The rubber mounts are placed on the 50mm ( *p*1 ) and 1450mm ( *p*3 ) from left end of the beam. The MR mount is placed on the 600mm ( *p*2 ) from the left end of the beam instead of the center of the beam. This is because that the MR mount located at center of the beam cannot control the rotational modes. The 2-node spring elements are used to model the rubber mounts and rubber element of the MR mount. The material property for the spring elements is set by 60kN/m which has been experimentally evaluated. To model the mass of lower part of the MR mount and vibrating mass, the 1-node lumped mass element is used. The material prop‐ erties for the mass elements are 1.5kg and 3kg for the lower part of the MR mount and the vibrating mass, respectively.

**Figure 3.** Configuration of the structural system with the MR mount.

8.28*I* 1.85 N, respectively. Here, the unit of current *I* is A. These values will be used as 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 be‐ tween the exciting mass and steel beam structure. When the mass is excited by external dis‐ turbance, the force transmitting through the MR mount excites the beam structure. Thus, the

of the beam structure, while two passive rubber mounts are placed at *p*1 and *p*3 to support the beam structure. The governing equation of the motion of the proposed structural system

of the structural system. The MR mount is placed at position *p*<sup>2</sup>

, *i* =1, 2, ⋯, *∞* (6)

(*x*) is the mode shape value at po‐

is the generalized mass of

(*p*2))(− *Fmr*(*t*)) (7)

(*p*4)*Fex*(*t*) (8)

) is the

vibration of the beam structure can be controlled by activating the MR mount. *y*(*pj*

(*p*4)−*φ<sup>i</sup>*

(*t*)=*φ<sup>i</sup>*

is the damping ratio, and *Ii*

(*t*) is the generalized force including the exciting force *Fex*(*t*) . It is noted that the

spring and damping effects of the rubber elements are resolved in the modal parameters of

In order to determine system parameters such as modal frequencies and mode shapes, mo‐ dal analysis is undertaken by adopting a commercial software (MSC/NASTRAN for Win‐ dows 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 ge‐ ometry of the steel beam is 1500mm (length) 60mm (length) 15mm (thickness). The rubber mounts are placed on the 50mm ( *p*1 ) and 1450mm ( *p*3 ) from left end of the beam. The MR mount is placed on the 600mm ( *p*2 ) from the left end of the beam instead of the center of the beam. This is because that the MR mount located at center of the beam cannot control

(*t*) is the generalized force including the damping force of the MR mount,

*Qe xi*

(*t*) is the generalized modal coordinate, *φ<sup>i</sup>*

parameters in the controller implementation for the structural system.

can be obtained using the mode summation method as follows [13]:

**3. Structural System**

92 Advances in Vibration Engineering and Structural Dynamics

displacement at position *pj*

*q* ¨ *i* (*t*) + 2*ζ<sup>i</sup>*

where,

In the above, *qi*

the *i* th mode. *Qi*

the structural system.

sition *x* , *ω<sup>i</sup>*

and *Qexi*

*ωi q*˙*i* (*t*) + *ω<sup>i</sup>* 2 *qi* (*t*)= *Qi* (*t*) *Ii* + *Qexi* (*t*) *Ii*

is the modal frequency, *ζ<sup>i</sup>*

*Qi* (*t*)=(*φ<sup>i</sup>*

> 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 com‐ pared 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 param‐ eter 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 support‐ ed by the rubber mount. The excitation force and frequency are regulated by the exciter con‐ troller. Accelerometers are attached to the mass and beam, and their positions are denoted by *x* = *p*1 , *x* = *p*2 , *x* = *p*3 and *x* = *p*4 . From this, the following system parameters are ob‐ tained : *ω*1 56.9 rad/s, *ω*2 144.8 rad/s, *ω*3 168.4 rad/s, *ζ*1 0.0260, *ζ*2 0.0497, and *ζ*3 0.0630.

*y*(*t*)= *y*˙(*p*4, *t*)*y*˙(*p*2, *t*) *<sup>T</sup>* (12)

Vibration Control of Flexible Structures Using Semi-Active Mount : Experimental Investigation

<sup>2</sup> −2*ζ*3*ω*<sup>3</sup>

<sup>0</sup> *<sup>φ</sup>*1(*p*2) <sup>0</sup> *<sup>φ</sup>*2(*p*2) <sup>0</sup> *<sup>φ</sup>*3(*p*2) (16)

0 1 0 0 0 0

0 0 0 0 0 1

0 (*φ*1(*p*2)−*φ*1(*p*4)) / *I*<sup>1</sup> 0 (*φ*2(*p*2)−*φ*2(*p*4)) / *I*<sup>2</sup> 0 (*φ*3(*p*2)−*φ*3(*p*4)) / *I*<sup>3</sup>

*<sup>C</sup>* <sup>=</sup> <sup>0</sup> *<sup>φ</sup>*1(*p*4) <sup>0</sup> *<sup>φ</sup>*2(*p*4) <sup>0</sup> *<sup>φ</sup>*3(*p*4)

*Cov*(*w*1, *w*<sup>1</sup>

*Cov*(*w*2, *w*<sup>2</sup>

*Cov*(*w*1, *w*<sup>2</sup>

In the above, *x*(*t*) is the state vector, *y*(*t*) is the measured output vector, *A* is the system matrix, *B* is the control input matrix, *C* is the output matrix, and *u*(*t*) is the control input. *w*1 and *w*2 are uncorrelated white noise characterized by covariance matrices *V*1 and *V*2 as

*<sup>T</sup>* )=*V*<sup>1</sup>

*<sup>T</sup>* )=*V*<sup>2</sup>

*<sup>T</sup>* )=0

In order to attenuate unwanted vibration of the flexible structural system, the linear quad‐ ratic 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

0 0 0 0 −*ω*<sup>3</sup>

<sup>2</sup> −2*ζ*1*ω*<sup>1</sup> 0 0 0 0 0 0 0 1 0 0

<sup>2</sup> −2*ζ*2*ω*<sup>2</sup> 0 0

*A*=

follows [14].

**4. Vibration Control**

index (*J*) to be minimized is given by

−*ω*<sup>1</sup>

0 0 −*ω*<sup>2</sup>

*B*=

*u*(*t*)= *Fmr*(*t*) (13)

(14)

95

http://dx.doi.org/10.5772/48823

(15)

(17)

**Figure 4.** Mode shapes of the structural system.

**Figure 5.** Experimental setup for the modal test of MR fluid mount system.

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.

$$
\dot{\hat{x}}(t) = A\mathbf{x}(t) + Bu(t) + \mathbf{w}\_1 \tag{9}
$$

$$y(t) = \mathbf{C}x(t) + \mathbf{w}\_2\tag{10}$$

where

$$\mathbf{x}(t) = \mathsf{L}q\_1(t)\dot{q}\_1(t)q\_2(t)\dot{q}\_2(t)q\_3(t)\dot{q}\_3(t)\mathbf{j}^T \tag{11}$$

Vibration Control of Flexible Structures Using Semi-Active Mount : Experimental Investigation http://dx.doi.org/10.5772/48823 95

$$\mathbf{y}(t) = \mathbf{I} \dot{\mathbf{y}}(p\_{4'}, t) \dot{\mathbf{y}}(p\_{2'}, t) \mathbf{I}^T \tag{12}$$

$$
\mu(t) = \begin{bmatrix} F\_{mr}(t) \end{bmatrix} \tag{13}
$$

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ -\omega\_1^2 & -2\zeta\_1\omega\_1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -\omega\_2^2 & -2\zeta\_2\omega\_2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -\omega\_3^2 & -2\zeta\_3\omega\_3 \end{bmatrix} \tag{14}$$

$$\mathbf{B} = \begin{bmatrix} 0 \\ \left(\wp\_1(p\_2) - \wp\_1(p\_4)\right) / I\_1 \\ 0 \\ \left(\wp\_2(p\_2) - \wp\_2(p\_4)\right) / I\_2 \\ 0 \\ \left(\wp\_3(p\_2) - \wp\_3(p\_4)\right) / I\_3 \end{bmatrix} \tag{15}$$

$$\mathbf{C} = \begin{bmatrix} 0 & \wp\_1(p\_4) & 0 & \wp\_2(p\_4) & 0 & \wp\_3(p\_4) \\ 0 & \wp\_1(p\_2) & 0 & \wp\_2(p\_2) & 0 & \wp\_3(p\_2) \end{bmatrix} \tag{16}$$

In the above, *x*(*t*) is the state vector, *y*(*t*) is the measured output vector, *A* is the system matrix, *B* is the control input matrix, *C* is the output matrix, and *u*(*t*) is the control input. *w*1 and *w*2 are uncorrelated white noise characterized by covariance matrices *V*1 and *V*2 as follows [14].

$$\begin{aligned} \text{Cov}(\mathbf{w}\_{1\prime}, \mathbf{w}\_{1}^{T}) &= \mathbf{V}\_{1} \\ \text{Cov}(\mathbf{w}\_{2\prime}, \mathbf{w}\_{2}^{T}) &= \mathbf{V}\_{2} \\ \text{Cov}(\mathbf{w}\_{1\prime}, \mathbf{w}\_{2}^{T}) &= \mathbf{0} \end{aligned} \tag{17}$$

## **4. Vibration Control**

**Figure 4.** Mode shapes of the structural system.

94 Advances in Vibration Engineering and Structural Dynamics

where

**Figure 5.** Experimental setup for the modal test of MR fluid mount system.

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.

*x*˙(*t*)= *Ax*(*t*) + *Bu*(*t*) + *w*<sup>1</sup> (9)

*x*(*t*)= *q*1(*t*)*q*˙ 1(*t*)*q*2(*t*)*q*˙ <sup>2</sup>(*t*)*q*3(*t*)*q*˙ <sup>3</sup>(*t*) *<sup>T</sup>* (11)

*y*(*t*)=*Cx*(*t*) + *w*<sup>2</sup> (10)

In order to attenuate unwanted vibration of the flexible structural system, the linear quad‐ ratic 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 (*J*) to be minimized is given by

$$J = \int\_0^\infty \left\{ \mathbf{x}^T(t) \mathbf{Q} \mathbf{x}(t) + \boldsymbol{u}^T(t) \mathbf{R} \boldsymbol{u}(t) \right\} dt \tag{18}$$

In the above, *L* is the observer gain matrix, and *O* is the solution of the following observer

Vibration Control of Flexible Structures Using Semi-Active Mount : Experimental Investigation

−1

2(*t*) + *k*4*q* ^˙

<sup>2</sup>(*t*) + *k*5*q* ^

<sup>3</sup>(*t*) + *k*6*q* ^˙

*C P* + *V*<sup>1</sup> =0 (25)

<sup>3</sup>(*t*)) (26)

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97

*AO* <sup>+</sup> *<sup>O</sup>A<sup>T</sup>* <sup>−</sup>*O<sup>C</sup> <sup>T</sup>V*<sup>2</sup>

<sup>1</sup>(*t*) + *k*3*q* ^

Using the estimated states, the control force of the MR mount is obtained as follows.

Riccati equation :

*u*(*t*)= −(*k*1*q*

**Figure 6.** Experimental setup for vibration control.

^

<sup>1</sup>(*t*) + *k*2*q* ^˙

In the above, *Q* is the state weighting semi-positive matirx and *R* is the input weighting pos‐ itive constant. Since the system (*A*, *B*) in equation (9) is controllable, the LQR controller for the MR mount can be obtained as follows.

$$\mathbf{x}(t) = -\mathbf{P}^{-1}\mathbf{B}^{T}\mathbf{x}(t) = -\mathbf{K}\mathbf{x}(t) \tag{19}$$

In the above, *K* is the state feedback gain matrix, and *P* is the solution of the following alge‐ braic Riccati equation :

$$A^T P + PA - PBR^{-1}B^T P + Q = 0\tag{20}$$

Thus, the control force of the MR mount can be represented as

$$\begin{aligned} \mathbf{x}(t) &= -\mathbf{K}\mathbf{x}(t) \\ \mathbf{y} &= -(k\_1q\_1(t) + k\_2\dot{q}\_1(t) + k\_3q\_2(t) + k\_4\dot{q}\_2(t) + k\_5q\_3(t) + k\_6\dot{q}\_3(t)) \end{aligned} \tag{21}$$

In this work, by the tuning method the control gains of the *k*1 , *k*2 , *k*3 , *k*4 , *k*5 and *k*6 are select‐ ed by -536068, 8315.9, -327595, 3706.3, -15.99 and 2359.4, respectively. On the other hand, the proposed MR mount is semi-active. Therefore, the control signal needs to be applied accord‐ ing to the following actuating condition [15]:

$$u(t) = \begin{bmatrix} u(t) & \operatorname{for} u(t)(\dot{y}(p\_\Psi, t) - \dot{y}(p\_{2^\*}, t))0 \\ 0 & \operatorname{for} u(t)(\dot{y}(p\_\Psi, t) - \dot{y}(p\_{2^\*}, t)) \le 0 \end{bmatrix} \tag{22}$$

This condition physically indicates that the actuating of the controller *u*(*t*) assures the incre‐ ment of energy dissipation of the stable system.

Since the states of *qi* (*t*) and *q*˙*<sup>i</sup>* (*t*) of the LQR controller (21) are not available from direct measurement, the Kalman-Bucy filter (KBF) is formulated. The KBF is a state estimator which is optimally considered in the statistical sense. The estimated state, *x* **^**(*t*) , can be ob‐ tained from the following state observer [14].

$$\stackrel{\wedge}{\mathbf{x}}(t) = A\stackrel{\wedge}{\mathbf{x}}(t) + Bu(t) + L \stackrel{\wedge}{\mathbf{u}}(y(t) - C\stackrel{\wedge}{\mathbf{x}}(t))\tag{23}$$

where,

$$\mathbf{L} = \mathbf{O} \mathbf{C}^{T} \mathbf{V}\_{2}^{-1} = \begin{bmatrix} l\_{11} l\_{12} \dots l\_{16} \\ l\_{21} l\_{22} \dots l\_{26} \end{bmatrix}^{T} \tag{24}$$

In the above, *L* is the observer gain matrix, and *O* is the solution of the following observer

Riccati equation :

*J* =*∫* 0 *∞*

*u*(*t*)= −*P* <sup>−</sup><sup>1</sup>

*A<sup>T</sup> P* + *PA*−*PBR* <sup>−</sup><sup>1</sup>

Thus, the control force of the MR mount can be represented as

the MR mount can be obtained as follows.

96 Advances in Vibration Engineering and Structural Dynamics

*u*(*t*)= − *Kx*(*t*)

ing to the following actuating condition [15]:

ment of energy dissipation of the stable system.

tained from the following state observer [14].

(*t*) and *q*˙*<sup>i</sup>*

*x* ^˙(*t*)= *Ax*

*<sup>u</sup>*(*t*)= *<sup>u</sup>*(*t*) 0

braic Riccati equation :

Since the states of *qi*

where,

{*x <sup>T</sup>* (*t*)*Qx*(*t*) + *u <sup>T</sup>* (*t*)*Ru*(*t*)}*dt* (18)

*B<sup>T</sup> x*(*t*)= − *Kx*(*t*) (19)

*B <sup>T</sup> P* + *Q* =0 (20)

In the above, *Q* is the state weighting semi-positive matirx and *R* is the input weighting pos‐ itive constant. Since the system (*A*, *B*) in equation (9) is controllable, the LQR controller for

In the above, *K* is the state feedback gain matrix, and *P* is the solution of the following alge‐

In this work, by the tuning method the control gains of the *k*1 , *k*2 , *k*3 , *k*4 , *k*5 and *k*6 are select‐ ed by -536068, 8315.9, -327595, 3706.3, -15.99 and 2359.4, respectively. On the other hand, the proposed MR mount is semi-active. Therefore, the control signal needs to be applied accord‐

, *foru*(*t*)(*y*˙(*p*4, *t*)− *y*˙(*p*2, *t*))0

This condition physically indicates that the actuating of the controller *u*(*t*) assures the incre‐

measurement, the Kalman-Bucy filter (KBF) is formulated. The KBF is a state estimator

^(*<sup>t</sup>*) <sup>+</sup> *Bu*(*t*) <sup>+</sup> *<sup>L</sup>* (*y*(*t*)−*C<sup>x</sup>*

*l* 12 *l* 22 ⋯ ⋯ *l* 16 *l* 26 *T*

<sup>−</sup><sup>1</sup> = *l* 11 *l* 21

which is optimally considered in the statistical sense. The estimated state, *x*

*<sup>L</sup>* <sup>=</sup>*OC <sup>T</sup>V*<sup>2</sup>

<sup>=</sup> <sup>−</sup>(*k*1*q*1(*t*) <sup>+</sup> *<sup>k</sup>*2*q*˙ 1(*t*) <sup>+</sup> *<sup>k</sup>*3*q*2(*t*) <sup>+</sup> *<sup>k</sup>*4*q*˙ <sup>2</sup>(*t*) <sup>+</sup> *<sup>k</sup>*5*q*3(*t*) <sup>+</sup> *<sup>k</sup>*6*q*˙ <sup>3</sup>(*t*)) (21)

, *foru*(*t*)(*y*˙(*p*4, *<sup>t</sup>*)<sup>−</sup> *<sup>y</sup>*˙(*p*2, *<sup>t</sup>*))≤<sup>0</sup> (22)

(*t*) of the LQR controller (21) are not available from direct

**^**(*t*) , can be ob‐

(24)

^(*<sup>t</sup>*)) (23)

$$\mathbf{A}\mathbf{O} + \mathbf{O}\mathbf{A}^T - \mathbf{O}\mathbf{C}^T\mathbf{V}\_2^{-1}\mathbf{C}\mathbf{P} + V\_1 = \mathbf{0} \tag{25}$$

Using the estimated states, the control force of the MR mount is obtained as follows.

$$u(t) = -\left(k\_1\stackrel{\wedge}{q}\_1(t) + k\_2\stackrel{\wedge}{q}\_1(t) + k\_3\stackrel{\wedge}{q}\_2(t) + k\_4\stackrel{\wedge}{q}\_2(t) + k\_5\stackrel{\wedge}{q}\_3(t) + k\_6\stackrel{\wedge}{q}\_3(t)\right) \tag{26}$$

**Figure 6.** Experimental setup for vibration control.

attached to the beam and mass, and their positions are denoted by (*x* = *p*1) , (*x* = *p*2) , (*x* = *p*3) , and (*x* = *p*4) . Two accelerometers, attached on the positions and , are used for the feedback signals. Velocity signals at these positions are obtained by using integrator cir‐ cuit of charge amplifier. And the other accelerometers are used to measure vibration of the structural system. The velocity signal which denoted by dashed line in Figure 6 is fed back to the DSP (digital signal processor) via an A/D(analog to digital) converter. The state variables required for the LQR (26) are then estimated by the KBF (23), and control voltage is deter‐ mined by means of the LQR controller in the DSP. The control current is applied to the MR mount via a D/A (digital to analog) converter and a current supplier. The sampling rate in the

Vibration Control of Flexible Structures Using Semi-Active Mount : Experimental Investigation

http://dx.doi.org/10.5772/48823

99

Figures 7 and 8 present the measured displacement and acceleration of the structural sys‐ tem. 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 () and beam structure (, ). It is observed from Fig‐ ures 7 and 8 that the resonant modes of the structural system have been reduced by control‐ ling the damping force of the MR mount. Especially, the 2nd and 3rd modes vibration at the position has been substantially reduced by activating the MR mount. It is noted that the

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 appro‐ priate size of the MR mount was designed and manufactured. After experimentally evaluat‐ ing 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 experimen‐ tal 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 justi‐ fying that the proposed semi-active MR mount can be effectively utilized to the vibration con‐

trol of various structural systems such as flexible robot arm and satellite appendages.

grant funded by the Korea government (MEST)(No. 2010-0015090).

This work was partially supported by the National Research Foundation of Korea (NRF)

controller implementation is set by 2kHz.

**5. Conclusion**

**Acknowledgements**

uncontrolled response is obtained without input current.

**Figure 7.** Displacement response of the structural system with the MR mount.

**Figure 8.** Acceleration response of the structural system with the MR mount.

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 (*x* = *p*1) , (*x* = *p*2) , (*x* = *p*3) , and (*x* = *p*4) . Two accelerometers, attached on the positions and , are used for the feedback signals. Velocity signals at these positions are obtained by using integrator cir‐ cuit of charge amplifier. And the other accelerometers are used to measure vibration of the structural system. The velocity signal which denoted by dashed line in Figure 6 is fed back to the DSP (digital signal processor) via an A/D(analog to digital) converter. The state variables required for the LQR (26) are then estimated by the KBF (23), and control voltage is deter‐ mined by means of the LQR controller in the DSP. The control current is applied to the MR mount via a D/A (digital to analog) converter and a current supplier. The sampling rate in the controller implementation is set by 2kHz.

Figures 7 and 8 present the measured displacement and acceleration of the structural sys‐ tem. 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 () and beam structure (, ). It is observed from Fig‐ ures 7 and 8 that the resonant modes of the structural system have been reduced by control‐ ling the damping force of the MR mount. Especially, the 2nd and 3rd modes vibration at the position has been substantially reduced by activating the MR mount. It is noted that the uncontrolled response is obtained without input current.
