**Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper**

Hao Chen, Zhi Sun and Limin Sun

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/45732

### **1. Introduction**

16 Will-be-set-by-IN-TECH

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Since the erection of the Stromaund Bridge of Sweden in 1956, cable-stayed bridge, as an efficient and economic bridge type to surmount a long-distance obstacle, has attracted more and more interests both from bridge engineering community and from the society and government. Nowadays, the cable stayed bridge is the most competitive type for the bridge with the span of 300-1000 meters. For a cable supported bridge, which is generally quite flexible and of low damping, its vibration under ambient excitation (such as the wind and ground motion excitation) and operational loading (such as the vehicle and train loads) is quite critical for its safety, serviceability and durability. Vibration control countermeasures, such as the installation of the energy dissipating devices, are thus required [1, 2]. Structural active control, which applies a counter-force induced by a control device to mitigate structural vibration, has been widely proposed for the vibration control of cable stayed bridges and proven to be efficient by many researchers [3-6].

Although the vibration response of a fully erected cable-stayed bridge should be controlled, a cable-stayed bridge under construction, which is of low damping and not as stable as the completed structure, is generally more vulnerable to dynamic loadings. During the construction stage, the cable pylons were generally erected firstly and the cable and main girders are then hang on the pylons symmetrically in a double-cantilever way. With the increase of the cantilever length, the bridge is more and more flexible. When the girder is on its longest double-cantilever state, the bridge is the most vulnerable to the external disturbance (such as the ambient wind fluctuation and ground motions). Moreover, if the cables, pylons and main girder of the bridge are all steel components and thus the damping of the bridge is very low, its vibration under ambient excitations will be quite large. The vibration reduction countermeasures are thus in great demand. Frederic

© 2012 Sun et al., licensee InTech. This is an open access chapter 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 Sun 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.

conducted several mock-up tests representing a cable-stayed bridge during the construction stage [7]. Since the control objective was set to reduce the girder vertical vibration response or cable parametric vibration response, the active tendon was installed as the control device. While for a cable stayed bridge under uncontrollable ambient excitations, structure will vibrate not only in the vertical direction but also in the transverse direction. Moreover, since the bridges are generally designed to carry the vertical loads, the unexpected transverse loads, especially the transverse dynamic loads, will induce structural safety and durability problems. It is thus of crucial importance to install some vibration reduction devices to control bridge transverse vibration. For this type of structure vibration control problem, the active mass damper (AMD) or the active tuned mass damper (ATMD) will be a competent candidate [8, 9].

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 197

**Figure 1.** The elevation view (a) and side view (b) of the test model (unit: mm)

**Table 2.** Sensor sensitivity and location distances from the tip end of the side span

**Table 1.** Cable parameters

Sensitivity

Cable No. 1# 2# 3# Young's Modulus (Mpa) 451.7 327.0 216.0 Length with spring (m) 1.62 1.17 0.77 Spring Force (N) 32.0 13.9 7.2

Num. 1 2 3 4 5 6 7 8

(mv/g) 134.6 140.8 127.4 134.6 149.2 138.3 138.5 130.8 Location (m) 0.06 0.32 0.62 1.02 2.06 2.46 2.76 3.02

In this chapter, the general procedure and key issues on adopting an active control device, the active mass damper (AMD), for vibration control of cable stayed bridges under construction are presented. Taking a typical cable stayed bridge as the prototype structure; a lab-scale test structure was designed and fabricated firstly. A baseline FEM model was then setup and updated according to the modal frequencies measured from structural vibration test. A numerical study to simulate the bridge-AMD control system was conducted and an efficient LQG-based controller is designed. Based on that, an experimental implementation of AMD control of the transverse vibration of the bridge model was performed.

### **2. Model structure description and vibration test**

The lab-scale bridge model studied in this chapter is designed according to a prototype cable-stayed bridge, the Third Nanjing Yangtze River Bridge located in Jiangsu Province of China. Since the prototype bridge is of all the characteristics of a modern cable-stayed bridge, the test model is assumed to be a good test bed to study the feasibility of active structural control applied to cable-stayed bridges under construction. The test model was designed and fabricated to simulate the longest double cantilever state during the construction stage of the prototype bridge. Since structural dynamic response control is the main focus of this chapter, the test model is preliminarily designed according to the dynamic scaling laws. However, concerning the restriction of test conditions, some modifications were made during the detailed design of the model bridge [10]. Fig. 1 shows the dimension of the designed model bridge. The bridge is composed of a 1.433 meters high cable pylon, a 3.08 meters long main girder, and six couples of stay cables. The cross section of the main girder is a rectangular of 16 mm wide and 10 mm high. At two ends of the main girder, the 3.6 kg and 3.8 kg weight AMD orbits were installed on the side span and main span respectively. The stay cables are made of steel wire with the diameter of 1 mm. At the upper end of each cable, an original 30 cm long spring was installed and adjusted to simulate the cable force. The cable forces were adjusted to provide supporting force to the main girder and to force it to match the designed layout of the test bridge. Table 1 shows the length, Young's Modulus and computed cable force for the cables. All of the components were made of steel. Since the model consists of only one cable pylon and no other piers, it is symmetric with respect to the cable pylon.

tuned mass damper (ATMD) will be a competent candidate [8, 9].

**2. Model structure description and vibration test** 

symmetric with respect to the cable pylon.

conducted several mock-up tests representing a cable-stayed bridge during the construction stage [7]. Since the control objective was set to reduce the girder vertical vibration response or cable parametric vibration response, the active tendon was installed as the control device. While for a cable stayed bridge under uncontrollable ambient excitations, structure will vibrate not only in the vertical direction but also in the transverse direction. Moreover, since the bridges are generally designed to carry the vertical loads, the unexpected transverse loads, especially the transverse dynamic loads, will induce structural safety and durability problems. It is thus of crucial importance to install some vibration reduction devices to control bridge transverse vibration. For this type of structure vibration control problem, the active mass damper (AMD) or the active

In this chapter, the general procedure and key issues on adopting an active control device, the active mass damper (AMD), for vibration control of cable stayed bridges under construction are presented. Taking a typical cable stayed bridge as the prototype structure; a lab-scale test structure was designed and fabricated firstly. A baseline FEM model was then setup and updated according to the modal frequencies measured from structural vibration test. A numerical study to simulate the bridge-AMD control system was conducted and an efficient LQG-based controller is designed. Based on that, an experimental implementation

The lab-scale bridge model studied in this chapter is designed according to a prototype cable-stayed bridge, the Third Nanjing Yangtze River Bridge located in Jiangsu Province of China. Since the prototype bridge is of all the characteristics of a modern cable-stayed bridge, the test model is assumed to be a good test bed to study the feasibility of active structural control applied to cable-stayed bridges under construction. The test model was designed and fabricated to simulate the longest double cantilever state during the construction stage of the prototype bridge. Since structural dynamic response control is the main focus of this chapter, the test model is preliminarily designed according to the dynamic scaling laws. However, concerning the restriction of test conditions, some modifications were made during the detailed design of the model bridge [10]. Fig. 1 shows the dimension of the designed model bridge. The bridge is composed of a 1.433 meters high cable pylon, a 3.08 meters long main girder, and six couples of stay cables. The cross section of the main girder is a rectangular of 16 mm wide and 10 mm high. At two ends of the main girder, the 3.6 kg and 3.8 kg weight AMD orbits were installed on the side span and main span respectively. The stay cables are made of steel wire with the diameter of 1 mm. At the upper end of each cable, an original 30 cm long spring was installed and adjusted to simulate the cable force. The cable forces were adjusted to provide supporting force to the main girder and to force it to match the designed layout of the test bridge. Table 1 shows the length, Young's Modulus and computed cable force for the cables. All of the components were made of steel. Since the model consists of only one cable pylon and no other piers, it is

of AMD control of the transverse vibration of the bridge model was performed.

**Figure 1.** The elevation view (a) and side view (b) of the test model (unit: mm)


**Table 2.** Sensor sensitivity and location distances from the tip end of the side span

Vibration tests under forced excitations were conducted to identify the dynamic characteristics of the bridge model. Eight accelerometers (as described in Table 2) were distributed along the main girder both on the side span and the main span to collect structural acceleration responses at a sampling frequency of 50 Hz (as shown in Fig. 2a). A transverse impulse force was acted on the cantilever end of the side span to excite the structure. Concerning that the bridge model is symmetrical about the cable pylon and thus of repeated or close frequency modes, a modal identification algorithm of the capacity to identify the close modes, the wavelet based modal identification method developed by the research group, is used [11]. During the analysis, the mother wavelet function adopted is the complex Morlet function with the central frequency of 300 Hz and the scale increment is set to be 0.25 during the analysis. Fig. 2b shows the wavelet scalgram of a set of response measurement on the tip end of the side span. As shown in the figure, structural transverse vibration responses were dominated by two modes at the scale of 168.0 and 176.5, which correspond to the vibration modes of the natural frequencies of 1.701 Hz and 1.786 Hz, respectively. This figure also shows that the adopted modal identification method can separate these two close modes successfully. Structural modal parameters can then be estimated and the results are shown in Table 3.

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 199

The first transverse antisymmetrical bending (TAB)

[1.00 0.84 0.72 0.28 0.00 -0.32 -0.70 -0.86 -0.99]

 [1.00 0.74 0.47 0.17 0.00 -0.14 -0.38 -0.60 -0.81]

[1.00 0.74 0.46 0.17 0.00 -0.17 -0.46 -0.74 -1.00]

mode

The first transverse symmetrical

bending (TSB) mode

0.30 0.70 0.90 0.99]

0.13 0.38 0.60 0.81]

0.17 0.46 0.74 1.00

Updated Model 0.9722 0.9718 \* MAC (modal assurance criteria) is defined to be a correlation coefficient of two mode shape vectors.

**Table 3.** The computed modal parameters before and after model updating compared with the tested

Taken eigenvalue analysis of the numerical model, structural natural frequencies and mode shape vectors can be computed. Table 4 shows the computed modal parameters of the transverse bending modes. As shown in the table, since the sum of the effective mass of the first transverse anti-symmetric bending mode (TAB) and the first transverse symmetric bending mode (TSB) is 80.1% of the sum of the effective mass of all transverse modes, these 2 transversal modes dominant the transverse vibration of the bridge. Fig. 3 shows the mode

No. Frequency (Hz) Mode description Effective mass (kg) 1 1.848 The first TAB mode 6.00 2 1.856 The first TSB mode 6.00 3 19.802 The second TAB mode 0.94 4 19.848 The second TSB mode 0.94

**Table 4.** The computed natural frequencies for the first 4 transverse modes of the main girder

first TSB and TAB modes have a good match with the modal test results.

Comparing the natural frequencies obtained from the eigenvalue analysis on the numerical model and the vibration test on the test bridge, some differences can be observed (as shown in Table 3). A model updating process is thus conducted to get an accurate baseline numerical model. The results of the sensitivity analysis show that the vertical and transverse vibration modes are sensitive to the tip mass magnitude, while the Young's module of the cable is critical for vertical bending modes. So these two parameters were updated: the tip masses of two spans were updated from 3.8kg and 3.6kg to 4.16kg, respectively, and the spring modulus was updated from 220N/m to 203N/m. Table 4 shows the natural frequencies and mode shapes of the updated model. As shown, the modal parameters of the

MAC\* Initial Model 0.9620 0.9620

Test [1.00 0.86 0.70 0.32 0.00

Initial Model [1.00 0.74 0.46 0.17 0.00

Updated Model [1.00 0.74 0.46 0.17 0.00

shape of these two transverse bending modes.

Test 1.701 Hz 1.786 Hz Initial Model 1.848 Hz 1.856 Hz Updated Model 1.787 Hz 1.786 Hz

Frequency

Mode Shape

mode parameters

**Figure 2.** Collected (a) free decay acceleration response and (b) its wavelet scalogram

### **3. Numerical modeling and model updating**

For structural active control, a baseline numerical model is generally required for controller design. A FEM bridge model is thus setup in ANSYS according to the design diagram of the bridge model. The cable is modeled using a 3D uniaxial tension-only truss element. Equivalent modulus for the cables without spring are computed using Ernst formula and then series wound equivalent modulus for the cable with spring can be established. Other structural elements are all modeled using 3D elastic beam element. Cables are connected to the main girder using rigid beam element. Additional masses were modeled using isotropic mass element. The cable pylons and the main girder are linked by coupling the horizontal projective intersection points of the lowest transverse beam of the pylon with the main girder. The six DOFs at the feet of the pylon are fixed.


\* MAC (modal assurance criteria) is defined to be a correlation coefficient of two mode shape vectors.

198 Advances on Analysis and Control of Vibrations – Theory and Applications

estimated and the results are shown in Table 3.

**Figure 2.** Collected (a) free decay acceleration response and (b) its wavelet scalogram

For structural active control, a baseline numerical model is generally required for controller design. A FEM bridge model is thus setup in ANSYS according to the design diagram of the bridge model. The cable is modeled using a 3D uniaxial tension-only truss element. Equivalent modulus for the cables without spring are computed using Ernst formula and then series wound equivalent modulus for the cable with spring can be established. Other structural elements are all modeled using 3D elastic beam element. Cables are connected to the main girder using rigid beam element. Additional masses were modeled using isotropic mass element. The cable pylons and the main girder are linked by coupling the horizontal projective intersection points of the lowest transverse beam of the pylon with the main

**3. Numerical modeling and model updating** 

girder. The six DOFs at the feet of the pylon are fixed.

Vibration tests under forced excitations were conducted to identify the dynamic characteristics of the bridge model. Eight accelerometers (as described in Table 2) were distributed along the main girder both on the side span and the main span to collect structural acceleration responses at a sampling frequency of 50 Hz (as shown in Fig. 2a). A transverse impulse force was acted on the cantilever end of the side span to excite the structure. Concerning that the bridge model is symmetrical about the cable pylon and thus of repeated or close frequency modes, a modal identification algorithm of the capacity to identify the close modes, the wavelet based modal identification method developed by the research group, is used [11]. During the analysis, the mother wavelet function adopted is the complex Morlet function with the central frequency of 300 Hz and the scale increment is set to be 0.25 during the analysis. Fig. 2b shows the wavelet scalgram of a set of response measurement on the tip end of the side span. As shown in the figure, structural transverse vibration responses were dominated by two modes at the scale of 168.0 and 176.5, which correspond to the vibration modes of the natural frequencies of 1.701 Hz and 1.786 Hz, respectively. This figure also shows that the adopted modal identification method can separate these two close modes successfully. Structural modal parameters can then be

**Table 3.** The computed modal parameters before and after model updating compared with the tested mode parameters

Taken eigenvalue analysis of the numerical model, structural natural frequencies and mode shape vectors can be computed. Table 4 shows the computed modal parameters of the transverse bending modes. As shown in the table, since the sum of the effective mass of the first transverse anti-symmetric bending mode (TAB) and the first transverse symmetric bending mode (TSB) is 80.1% of the sum of the effective mass of all transverse modes, these 2 transversal modes dominant the transverse vibration of the bridge. Fig. 3 shows the mode shape of these two transverse bending modes.


**Table 4.** The computed natural frequencies for the first 4 transverse modes of the main girder

Comparing the natural frequencies obtained from the eigenvalue analysis on the numerical model and the vibration test on the test bridge, some differences can be observed (as shown in Table 3). A model updating process is thus conducted to get an accurate baseline numerical model. The results of the sensitivity analysis show that the vertical and transverse vibration modes are sensitive to the tip mass magnitude, while the Young's module of the cable is critical for vertical bending modes. So these two parameters were updated: the tip masses of two spans were updated from 3.8kg and 3.6kg to 4.16kg, respectively, and the spring modulus was updated from 220N/m to 203N/m. Table 4 shows the natural frequencies and mode shapes of the updated model. As shown, the modal parameters of the first TSB and TAB modes have a good match with the modal test results.

**Figure 3.** The baseline FEM model (a) setup in ANSYS and the computed mode shape (b) of the first transverse anti-symmetric and symmetric bending mode

### **4. Control system simulation**

Based on the baseline numerical model updated according to dynamic measured structural modal parameters, a system simulation study is conducted. For a bridge-AMD system, its governing equation of motion is

$$\dot{\mathbf{C}} \cdot \mathbf{M}\_s \ddot{\mathbf{X}}(t) + \mathbf{C}\_s \dot{\mathbf{X}}(t) + \mathbf{K}\_s \mathbf{X}(t) = \mathbf{F}\_\varepsilon(t) + \mathbf{D}^T \mathbf{T}^T f\_d(t) \tag{1}$$

$$m\_d \ddot{\mathbf{x}}\_d(t) + f\_d(t) = 0 \tag{2}$$

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 201

1 () () *r*

is the generalized mode shape matrix composed of the mode shape

, *<sup>T</sup> C C s s*

(5)

*r rr r r x Ax Bu Ew* (6)

*r rr r r z Cx Du Ew* (7)

*r rr r r y Cx Du Ew* (8)

and *<sup>T</sup> K K s s*

*i Xt Y t* 

*i i*

and the full order model can

*e e F F* is the

are

*f t* is the

is the excitation force vector acting on the structure; *D I* 0 0 is the AMD location

actuation force on the structure applied by the AMD;*T* is the transfer matrix from the affiliated nodes of AMD to the principle nodes of the structure; *m diag m m m <sup>d</sup> d di dl* <sup>1</sup> is the mass matrix of the AMDs, in which *mdi* is the mass of the *i*th AMD; ( ) *<sup>d</sup> x t* are absolute displacement of the AMD to the bridge structure, respectively; *dc* is the viscous coefficient of the AMD; *db* is the force-voltage coefficient of the AMD and *u t*( ) is the control input

Since the numerical model of a complex structure is generally of a large amount of DOFs, for example in this study the FEM model obtained from the last section is of 1188 DOFs, this will induce great computation difficulty to design the controller according to this so called full order model. A reduced order model is thus required. In this study, the critical modal reduction method is adopted for this purpose because this method can greatly reduce structural DOFs and the reduced order model is of clear physical meaning [12, 13]. Concerning a structure whose vibration is dominated by the first *r* modes, its dynamic

thus be reduced to an *r-*order reduced order model. The governing equation of motion for

() ( ) () () () () ( ) *T TT T TT T TT Ms sd Y t C c D T TD Y t K Y t D T c x t F t D T b u t <sup>s</sup> dd e <sup>d</sup>*

() () ( ) *m x t c x t c TD Y b u t dd dd d <sup>d</sup>*

where, *Y t*( ) is the modal response vector superposed by the modal responses of the *r*th

generalized excitation vectors. In this study, since the vibration responses of the first two transversal vibration modes are the most accountable for structural transverse vibration, structural modal mass, stiffness and damping matrices are computed using the modal parameters of these two modes. Correspondingly, the governing equation of motion for the

*z zz*

*y yy*

structural generalized mass, damping and stiffness matrices, respectively; *<sup>T</sup>*

 

(4)

matrix, in which 0 and *I* are zero and identity matrix with appropriate sizes; ( ) *<sup>d</sup>*

voltage.

response can be approximately expressed as

vectors of the first *r* principle modes; *<sup>T</sup> M M s s*

reduced order bridge-AMD system can be derived.

For this reduced order bridge-AMD system, its state space equation is

this reduced order model is

principle modes;

$$f\_d(t) = -c\_d[\dot{\chi}\_d(t) - TD\dot{X}] + b\_d \mu(t) \tag{3}$$

where, *Ms* , *Cs* , and *Ks* are the mass, damping and stiffness matrices of the bridge structure; <sup>1</sup> ( ) *Xt x x x i n* is a *n*-dimensional vector, in which *xi* is the displacement response vector of the *i*th DOF of the structure and *n* is the number of the DOF of the structure; ( ) *<sup>e</sup> F t*

is the excitation force vector acting on the structure; *D I* 0 0 is the AMD location matrix, in which 0 and *I* are zero and identity matrix with appropriate sizes; ( ) *<sup>d</sup> f t* is the actuation force on the structure applied by the AMD;*T* is the transfer matrix from the affiliated nodes of AMD to the principle nodes of the structure; *m diag m m m <sup>d</sup> d di dl* <sup>1</sup> is the mass matrix of the AMDs, in which *mdi* is the mass of the *i*th AMD; ( ) *<sup>d</sup> x t* are absolute displacement of the AMD to the bridge structure, respectively; *dc* is the viscous coefficient of the AMD; *db* is the force-voltage coefficient of the AMD and *u t*( ) is the control input voltage.

200 Advances on Analysis and Control of Vibrations – Theory and Applications

**Figure 3.** The baseline FEM model (a) setup in ANSYS and the computed mode shape (b) of the first

Based on the baseline numerical model updated according to dynamic measured structural modal parameters, a system simulation study is conducted. For a bridge-AMD system, its

() () 0 *mx t f t dd d* (2)

where, *Ms* , *Cs* , and *Ks* are the mass, damping and stiffness matrices of the bridge structure;

vector of the *i*th DOF of the structure and *n* is the number of the DOF of the structure; ( ) *<sup>e</sup> F t*

is a *n*-dimensional vector, in which *xi* is the displacement response

() () () () () *T T MXt CXt KXt F t DT f t s s se d* (1)

() [ () ] () *d dd <sup>d</sup> f t c x t TDX b u t* (3)

transverse anti-symmetric and symmetric bending mode

**4. Control system simulation** 

governing equation of motion is

<sup>1</sup> ( ) *Xt x x x i n*

Since the numerical model of a complex structure is generally of a large amount of DOFs, for example in this study the FEM model obtained from the last section is of 1188 DOFs, this will induce great computation difficulty to design the controller according to this so called full order model. A reduced order model is thus required. In this study, the critical modal reduction method is adopted for this purpose because this method can greatly reduce structural DOFs and the reduced order model is of clear physical meaning [12, 13]. Concerning a structure whose vibration is dominated by the first *r* modes, its dynamic

response can be approximately expressed as 1 () () *r i i i Xt Y t* and the full order model can

thus be reduced to an *r-*order reduced order model. The governing equation of motion for this reduced order model is

$$
\hat{M}\_s \ddot{\mathbf{Y}}(t) + (\hat{\mathbf{C}}\_s - \mathbf{c}\_d \boldsymbol{\phi}^T \mathbf{D}^T \mathbf{T}^T \mathbf{T} \mathbf{D} \boldsymbol{\phi}) \dot{\mathbf{Y}}(t) + \hat{\mathbf{K}}\_s \mathbf{Y}(t) + \boldsymbol{\phi}^T \mathbf{D}^T \mathbf{T}^T \mathbf{c}\_d \dot{\mathbf{x}}\_d(t) = \hat{\mathbf{F}}\_c(t) + \boldsymbol{\phi}^T \mathbf{D}^T \mathbf{T}^T \mathbf{b}\_d \mathbf{u}(t) \tag{4}
$$

$$m\_d \ddot{\mathfrak{x}}\_d(t) - c\_d \dot{\mathfrak{x}}\_d(t) + c\_d T D \dot{\mathfrak{y}} \dot{Y} = -b\_d \mathfrak{u}(t) \tag{5}$$

where, *Y t*( ) is the modal response vector superposed by the modal responses of the *r*th principle modes; is the generalized mode shape matrix composed of the mode shape vectors of the first *r* principle modes; *<sup>T</sup> M M s s* , *<sup>T</sup> C C s s* and *<sup>T</sup> K K s s* are structural generalized mass, damping and stiffness matrices, respectively; *<sup>T</sup> e e F F* is the generalized excitation vectors. In this study, since the vibration responses of the first two transversal vibration modes are the most accountable for structural transverse vibration, structural modal mass, stiffness and damping matrices are computed using the modal parameters of these two modes. Correspondingly, the governing equation of motion for the reduced order bridge-AMD system can be derived.

For this reduced order bridge-AMD system, its state space equation is

$$
\dot{\boldsymbol{x}}\_r = \boldsymbol{A}\_r \boldsymbol{\mathfrak{x}}\_r + \boldsymbol{B}\_r \boldsymbol{\mathfrak{u}} + \boldsymbol{E}\_r \boldsymbol{\mathfrak{w}} \tag{6}
$$

$$
\omega\_r = \mathbf{C}\_r^z \mathbf{x}\_r + D\_r^z \boldsymbol{\mu} + E\_r^z \mathbf{w} \tag{7}
$$

$$\mathbf{y}\_r = \mathbf{C}\_r^y \mathbf{x}\_r + D\_r^y \boldsymbol{\mu} + E\_r^y \boldsymbol{w} \tag{8}$$

where, *<sup>r</sup> x* is an *a* dimensional state-space vector, 2 *a rl* , *r* and *l* are the number of DOF for the reduced order structure and the number of the installed AMD, respectively; *Ar* is an *a*×*a* system matrix; *u* is the control input vector for the *l* AMDs; *Br* is an *a*×*l* AMD location matrix; *w* is the generalized modal excitation vector; *Er* is an *a*×*r* excitation matrix; *<sup>r</sup> z* is an *l*  dimensional control output vector; *<sup>r</sup> y* is a *q* dimensional observer output vector. The system matrices can be expressed as:

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 203

(9)

study [10]. During the controller design process, the excitation is assumed to be a stationary

<sup>0</sup> lim ( ) *<sup>t</sup> T T r r <sup>t</sup> J E z Qz u Ru dt*

where *<sup>r</sup> z* , system output variables, are set to be the transverse displacement or acceleration response at the tip ends; *Q* , a square matrix of the same order as *<sup>r</sup> z* , equals to the multiplication of an identity matrix with a parameter *q*; *u* is a control force variable; *R* , the active force weight matrix, is set to be an identity matrix of the same order as the number of AMD applied. The design of the *LQG* controller is to adjust the weight parameter *q* via optimizing the performance of the system with compensator under the limitation of energy supply. The design of the controller relies on the full state feedback vector *Xr* . Since limited sensors are mounted on the structure, this full state vector cannot be directly measured but be estimated from the sensor measurements. When the excitation forces *w* and the measurement noise *v* are uncorrelated white noise process, the Kalman-Bucy filter is

To obtain a good controller for experimental implementation, a series of numerical analysis with different value of weight parameter *q* are conducted. During the numerical analysis, structural modal damping coefficients are set to be the same as the real measured modal damping ratios. A scaled *El Centro* earthquake time history, whose dominant frequency band covers the first 2 transverse modal frequencies of the bridge, is adopted to excite the bridge in the numerical study. Two AMD placement strategies are employed for the comparison of optimal actuator location. These two strategies are the strategy of one AMD placed at the tip end of the main span (named S1) and the strategy of two AMDs placed at the tip ends of both spans (named S2). The AMDs are simulated to be the two electric servotype AMD carts provided by Quanser Inc. with the following expression of the actuation

( ) 8.246 ( ) 1.42 ( )

To simulate a more practical control condition during experimental implementation, the following constraint condition is adopted: 1) The discrete digital computation is employed for the controller computation with the sampling frequency of 500Hz; 2) The precision of the A/D converter is set to be 12-bits and the range of the input voltage is set to be 5 V; 3) The measurement noise of a root mean square (RMS) value of 0.015 V is added into each channel of the acceleration responses, which corresponds to the 0.3% of voltage range of the A/D converter; 4) The maximum actuation voltage is set to be 5 V with the corresponding RMS voltage of 1.67 V and the maximum actuation displacement is set to be 0.08 m with the

*<sup>d</sup> <sup>d</sup> f t x t ut* (10)

( ) 12.576 ( ) 1.73 ( ) *<sup>d</sup> <sup>d</sup> f t x t ut* (11)

white noise, and the following cost function is set as the control objective:

employed to get an estimation of the state vector *Xr* [15].

1

2

corresponding RMS displacement of 0.027 m.

force

$$\mathbf{x} = \begin{Bmatrix} Y \\ \dot{Y} \\ \dot{X}\_d \end{Bmatrix}; \ A\_r = \begin{bmatrix} 0 & I & 0 \\ -\hat{M}\_s^{-1}\hat{K}\_s & -\hat{M}\_s^{-1}\hat{C} & -\hat{M}\_s^{-1}\dot{\phi}^T D^T T^T c\_d \\ 0 & m\_d^{-1}c\_d H & m\_d^{-1}c\_d \end{bmatrix}; \ B\_r = \begin{bmatrix} 0 \\ \hat{M}\_s^{-1}\hat{H}\dot{B}\_d \\ -m\_d^{-1}\dot{b}\_d \end{bmatrix}; \ E\_r = \begin{bmatrix} 0 \\ \hat{M}\_s^{-1} \\ 0 \end{bmatrix};$$

$$\mathbf{C}\_r^z = \alpha \begin{bmatrix} I & I \\ 0 & I \end{bmatrix} A\_r \begin{bmatrix} 0 \\ \end{bmatrix}; \ D\_r^z = \alpha \begin{bmatrix} I & I \\ 0 & I \end{bmatrix} B\_r \begin{bmatrix} 0 \\ \end{bmatrix}; E\_r^z = \alpha \begin{bmatrix} 0 \\ 0 & I \end{bmatrix} E\_r \begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix};$$

$$\mathbf{C}\_r^y = \beta \begin{bmatrix} I & I \\ 0 & I \end{bmatrix} A\_r \begin{bmatrix} 0 \\ \end{bmatrix}; D\_r^y = \beta \begin{bmatrix} I & I \\ 0 & I \end{bmatrix} B\_r \begin{bmatrix} I & I \\ 0 & I \end{bmatrix}; E\_r^y = \beta \begin{bmatrix} 0 \\ \end{bmatrix} \begin{bmatrix} 0 \\ 0 & I \end{bmatrix} E\_r$$

where \* *<sup>T</sup> C C Hc H S d* ; *<sup>T</sup> H TD I* ; *0* and *I* are zero and identity matrix with appropriate sizes; and are appropriately selected weighting matrix to adjust the optimize objective of the controller, respectively. The control output vector *<sup>r</sup> z* and the observer output vector *<sup>r</sup> y* are the displacement, velocity and acceleration responses of the bridge structure or the AMDs.

The control output and observer output matrices *<sup>z</sup> Cr* , *<sup>z</sup> Dr* , *<sup>y</sup> Cr* , and *<sup>y</sup> Dr* are determined according to the sensor and actuator placement. The sensor placement should basically satisfy the following observability criteria *<sup>r</sup> y r A I rank a C* , where is an arbitrary

complex number [14]. For a system with *n* modes of repeated or close frequency, *n* sensors are required for full state response measurement. In this study, to provide redundant channels to collect structural response, eight accelerometers, the same as aforementioned in the modal test, are installed along the main girder during the control process. The actuators are placed on the girder by checking the following controllability criteria , *r r a rank A I B* . In this study, different schemes for AMD placement will be discussed in detail in the following sections.

### **5. Controller design**

The design of a controller is very important for the success of structural active vibration control. In this study, the *LQG* control algorithm is adopted since this control algorithm can offer excellent control performance and is of good robustness as shown in some preliminary study [10]. During the controller design process, the excitation is assumed to be a stationary white noise, and the following cost function is set as the control objective:

202 Advances on Analysis and Control of Vibrations – Theory and Applications

1 1\* 1

; <sup>0</sup>

; <sup>0</sup>

*I*

satisfy the following observability criteria *<sup>r</sup>*

*I*

*r*

*r*

*<sup>T</sup> H TD I* 

*r ss s s d*

0 0

*I A M K M C M DTc*

1 1

*m cH m c*

*z r*

*y r*

*d d d d*

system matrices can be expressed as:

0

*<sup>C</sup> I A* 

*<sup>C</sup> I A* 

*z r*

*y r*

> and

;

where \* *<sup>T</sup> C C Hc H S d* ;

bridge structure or the AMDs.

appropriate sizes;

, *r r a rank A I B* 

**5. Controller design** 

in detail in the following sections.

*d*

*Y x Y x*

 

 

where, *<sup>r</sup> x* is an *a* dimensional state-space vector, 2 *a rl* , *r* and *l* are the number of DOF for the reduced order structure and the number of the installed AMD, respectively; *Ar* is an *a*×*a* system matrix; *u* is the control input vector for the *l* AMDs; *Br* is an *a*×*l* AMD location matrix; *w* is the generalized modal excitation vector; *Er* is an *a*×*r* excitation matrix; *<sup>r</sup> z* is an *l*  dimensional control output vector; *<sup>r</sup> y* is a *q* dimensional observer output vector. The

;

*z r*

*y r*

; *0* and *I* are zero and identity matrix with

are appropriately selected weighting matrix to adjust the

1

*m b*

*d d*

*r*

*r*

<sup>1</sup>

0

 

0

*r s E M*

;

0 ; *r sd*

 

0 ; <sup>0</sup>

0 ; <sup>0</sup>

, where

is an arbitrary

*<sup>E</sup> I E* 

*<sup>E</sup> I E* 

*B M Hb*

*T TT*

; <sup>0</sup>

; <sup>0</sup>

*I*

*I*

optimize objective of the controller, respectively. The control output vector *<sup>r</sup> z* and the observer output vector *<sup>r</sup> y* are the displacement, velocity and acceleration responses of the

The control output and observer output matrices *<sup>z</sup> Cr* , *<sup>z</sup> Dr* , *<sup>y</sup> Cr* , and *<sup>y</sup> Dr* are determined according to the sensor and actuator placement. The sensor placement should basically

complex number [14]. For a system with *n* modes of repeated or close frequency, *n* sensors are required for full state response measurement. In this study, to provide redundant channels to collect structural response, eight accelerometers, the same as aforementioned in the modal test, are installed along the main girder during the control process. The actuators are placed on the girder by checking the following controllability criteria

The design of a controller is very important for the success of structural active vibration control. In this study, the *LQG* control algorithm is adopted since this control algorithm can offer excellent control performance and is of good robustness as shown in some preliminary

. In this study, different schemes for AMD placement will be discussed

*r*

*r*

*y r*

*A I rank a C* 

<sup>1</sup>

*<sup>D</sup> I B* 

*<sup>D</sup> I B* 

$$J = \lim\_{t \to \infty} E\left[\int\_0^t (\mathbf{z}\_r^{\top} \mathbf{Q} \mathbf{z}\_r + \mathbf{u}^T R \mathbf{u}) dt\right] \tag{9}$$

where *<sup>r</sup> z* , system output variables, are set to be the transverse displacement or acceleration response at the tip ends; *Q* , a square matrix of the same order as *<sup>r</sup> z* , equals to the multiplication of an identity matrix with a parameter *q*; *u* is a control force variable; *R* , the active force weight matrix, is set to be an identity matrix of the same order as the number of AMD applied. The design of the *LQG* controller is to adjust the weight parameter *q* via optimizing the performance of the system with compensator under the limitation of energy supply. The design of the controller relies on the full state feedback vector *Xr* . Since limited sensors are mounted on the structure, this full state vector cannot be directly measured but be estimated from the sensor measurements. When the excitation forces *w* and the measurement noise *v* are uncorrelated white noise process, the Kalman-Bucy filter is employed to get an estimation of the state vector *Xr* [15].

To obtain a good controller for experimental implementation, a series of numerical analysis with different value of weight parameter *q* are conducted. During the numerical analysis, structural modal damping coefficients are set to be the same as the real measured modal damping ratios. A scaled *El Centro* earthquake time history, whose dominant frequency band covers the first 2 transverse modal frequencies of the bridge, is adopted to excite the bridge in the numerical study. Two AMD placement strategies are employed for the comparison of optimal actuator location. These two strategies are the strategy of one AMD placed at the tip end of the main span (named S1) and the strategy of two AMDs placed at the tip ends of both spans (named S2). The AMDs are simulated to be the two electric servotype AMD carts provided by Quanser Inc. with the following expression of the actuation force

$$\int\_{\cdot}^{1} f(t) = 8.246 \dot{\alpha}\_d(t) + 1.42 \mu(t) \tag{10}$$

$$\left(f\_d^2(t) = 12.576\dot{x}\_d(t) + 1.73u(t)\right) \tag{11}$$

To simulate a more practical control condition during experimental implementation, the following constraint condition is adopted: 1) The discrete digital computation is employed for the controller computation with the sampling frequency of 500Hz; 2) The precision of the A/D converter is set to be 12-bits and the range of the input voltage is set to be 5 V; 3) The measurement noise of a root mean square (RMS) value of 0.015 V is added into each channel of the acceleration responses, which corresponds to the 0.3% of voltage range of the A/D converter; 4) The maximum actuation voltage is set to be 5 V with the corresponding RMS voltage of 1.67 V and the maximum actuation displacement is set to be 0.08 m with the corresponding RMS displacement of 0.027 m.

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 205

**Figure 5.** Excitation (a), driving voltage (b), main span tip acceleration (c), and side span tip acceleration (d) time histories of the bridge under El Centro seismic excitation for S1 control

**Figure 4.** The ratio of the controlled acceleration RMS value to uncontrolled RMS value at the tip ends of the side span (a) and main span (b) with respect to the weighting parameter q for S1 control

For S1 control, a series of numerical studies simulating the control system with one AMD cart installed on the tip end of the main span of the bridge, whose actuation force is expressed as Eq. (10), are conducted when the weighting parameter *q* varies from 0.01 to 10. Fig. 4 shows the relative RMS ratio of the controlled acceleration response to the uncontrolled acceleration response at the tip end of both main span and side span with the varying of *q*. As shown in the figure, *q* = 0.398 are set for S1 to achieve an optimal control performances. This figure also tells that for S1 control, the tip acceleration response of the main span can be well controlled; however, a good control performance for the tip acceleration at the side span cannot be achieved by adjusting the weight parameter. Fig. 5 shows the controlled and uncontrolled tip acceleration response of the side span and the main span when *q* is set to be 0.398. This figure also tells that the well designed controller can greatly reduce the acceleration response of the tip end of the main span but cannot mitigate the vibration of the side span.

For S2 control, the control system with two AMD carts, whose actuation force expressions are shown as Eq. (10) and (11), installed on the tip ends of both spans of the bridge, is simulated. Numerical studies are conducted when the weighting parameter *q* varies from 0.01 to 100. Fig. 6 shows the relative RMS ratio of the controlled acceleration response to the uncontrolled acceleration response at the tip end of both spans with the varying of *q*. As shown in the figure, when *q* = 9.1, the control system provides the most optimal control performances. This figure also tells that for S2 control, the tip acceleration response of both the main span and the side span can be well reduced. Fig. 7 shows the controlled and uncontrolled tip acceleration time response of the side span and the main span when *q* is set to be 9.1. This figure also tells that the well designed controller can greatly reduce the acceleration response at the tip end of both spans.

mitigate the vibration of the side span.

acceleration response at the tip end of both spans.

**Figure 4.** The ratio of the controlled acceleration RMS value to uncontrolled RMS value at the tip ends of the side span (a) and main span (b) with respect to the weighting parameter q for S1 control

For S1 control, a series of numerical studies simulating the control system with one AMD cart installed on the tip end of the main span of the bridge, whose actuation force is expressed as Eq. (10), are conducted when the weighting parameter *q* varies from 0.01 to 10. Fig. 4 shows the relative RMS ratio of the controlled acceleration response to the uncontrolled acceleration response at the tip end of both main span and side span with the varying of *q*. As shown in the figure, *q* = 0.398 are set for S1 to achieve an optimal control performances. This figure also tells that for S1 control, the tip acceleration response of the main span can be well controlled; however, a good control performance for the tip acceleration at the side span cannot be achieved by adjusting the weight parameter. Fig. 5 shows the controlled and uncontrolled tip acceleration response of the side span and the main span when *q* is set to be 0.398. This figure also tells that the well designed controller can greatly reduce the acceleration response of the tip end of the main span but cannot

For S2 control, the control system with two AMD carts, whose actuation force expressions are shown as Eq. (10) and (11), installed on the tip ends of both spans of the bridge, is simulated. Numerical studies are conducted when the weighting parameter *q* varies from 0.01 to 100. Fig. 6 shows the relative RMS ratio of the controlled acceleration response to the uncontrolled acceleration response at the tip end of both spans with the varying of *q*. As shown in the figure, when *q* = 9.1, the control system provides the most optimal control performances. This figure also tells that for S2 control, the tip acceleration response of both the main span and the side span can be well reduced. Fig. 7 shows the controlled and uncontrolled tip acceleration time response of the side span and the main span when *q* is set to be 9.1. This figure also tells that the well designed controller can greatly reduce the

**Figure 5.** Excitation (a), driving voltage (b), main span tip acceleration (c), and side span tip acceleration (d) time histories of the bridge under El Centro seismic excitation for S1 control

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 207

**Figure 7.** Driving voltage of the main span (a) and side span (b), and tip acceleration time histories of main span (c) and side span (d) of the bridge under *El Centro* seismic excitation for S2 control

**Figure 6.** The ratio of the controlled acceleration RMS value to uncontrolled RMS value at the tip ends of the side span (a) and main span (b) with respect to the weighting parameter q for S2 control

The control performance comparison of these two AMD placing strategies tell that for the cable-stayed bridge studied, which is of two dominant transverse vibration modes with close frequencies, the single AMD control strategy (S1) can only reduce structural vibration response of the AMD-instrumented span, and the double AMD control strategy (S2) can achieve a good control performance for reducing structural response of both spans. These observations are verified by checking the controllability criteria. If the control system is of two close eigenvalues, at least two actuators are required to ensure the system is controllable. Moreover, since structural dominant transverse modes are anti-symmetric and symmetric bending modes, the shift of the AMD position along one span of the bridge will only proportionally vary the coefficients of *<sup>r</sup> B* . If two AMDs are placed at one span of the bridge, the corresponding two columns of *<sup>r</sup> B* are linear dependant. Therefore, the two AMDs must be placed at both the side span and the main span respectively to ensure the controllability of the bridge.

### **6. Experimental implementation**

To verify the feasibility of the AMD control for transverse vibration reduction of cablestayed bridge under construction, an experimental study on the fabricated test model is conducted in the Bridge Testing Laboratory of Tongji University. During the experiment, the S2 control strategy was adopted according to the conclusion obtained from the above numerical simulation study. Fig. 8 shows the layout of the experimental setup. As shown in the figure, the control system includes a data acquisition system, a central control computer, and two AMD carts. The data acquisition system consists of eight accelerometers, whose sensitivity is checked using dynamic calibration method; Dspace signal amplifier and filter; a general purpose data acquisition and control board MultiQ-3, which has 8 single ended analog inputs, 8 analog outputs, 16 bits of digital input, 16 bits of digital output, 3 programmable timers and up to 8 encoder inputs decoded in quadrature (option 2E to 8E). The central control computer is of 512 Mb memory and 1.0 GHz Intel Celeron processor. The

controllability of the bridge.

**6. Experimental implementation** 

**Figure 6.** The ratio of the controlled acceleration RMS value to uncontrolled RMS value at the tip ends of the side span (a) and main span (b) with respect to the weighting parameter q for S2 control

The control performance comparison of these two AMD placing strategies tell that for the cable-stayed bridge studied, which is of two dominant transverse vibration modes with close frequencies, the single AMD control strategy (S1) can only reduce structural vibration response of the AMD-instrumented span, and the double AMD control strategy (S2) can achieve a good control performance for reducing structural response of both spans. These observations are verified by checking the controllability criteria. If the control system is of two close eigenvalues, at least two actuators are required to ensure the system is controllable. Moreover, since structural dominant transverse modes are anti-symmetric and symmetric bending modes, the shift of the AMD position along one span of the bridge will only proportionally vary the coefficients of *<sup>r</sup> B* . If two AMDs are placed at one span of the bridge, the corresponding two columns of *<sup>r</sup> B* are linear dependant. Therefore, the two AMDs must be placed at both the side span and the main span respectively to ensure the

To verify the feasibility of the AMD control for transverse vibration reduction of cablestayed bridge under construction, an experimental study on the fabricated test model is conducted in the Bridge Testing Laboratory of Tongji University. During the experiment, the S2 control strategy was adopted according to the conclusion obtained from the above numerical simulation study. Fig. 8 shows the layout of the experimental setup. As shown in the figure, the control system includes a data acquisition system, a central control computer, and two AMD carts. The data acquisition system consists of eight accelerometers, whose sensitivity is checked using dynamic calibration method; Dspace signal amplifier and filter; a general purpose data acquisition and control board MultiQ-3, which has 8 single ended analog inputs, 8 analog outputs, 16 bits of digital input, 16 bits of digital output, 3 programmable timers and up to 8 encoder inputs decoded in quadrature (option 2E to 8E). The central control computer is of 512 Mb memory and 1.0 GHz Intel Celeron processor. The

**Figure 7.** Driving voltage of the main span (a) and side span (b), and tip acceleration time histories of main span (c) and side span (d) of the bridge under *El Centro* seismic excitation for S2 control

AMD carts (as shown in Fig. 8) are electric servo type [16]. They are 0.645 kg weight. Their track length is 32 cm and the max safe input voltage is 5V. The system is designed under the constraint of avoiding the AMD to knock the baffle. If the AMD position exceeds the safe range of the orbit, the system will be shut down.

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 209

performance of the AMD control system under different excitation schemes. For the first case, case C1, the tip ends of both the main span and the side span were pulled in the same direction. After the steel wires were cut, the transverse symmetric bending mode of the bridge was excited. For case C2, the tip ends of the two spans were pulled in the opposite direction to excite the transverse anti-symmetric bending mode of the bridge. Case C3 simulated bridge free vibration under an initial displacement of the main span. Both antisymmetric and symmetric transverse bending modes of the bridge would thus be excited.

For test case C1, Fig. 9 shows the AMD driving voltage, recorded tip acceleration responses with or without AMDs, and their Fourier spectrums. As shown in these figures, when the power of the AMD carts was still turned off, structural acceleration responses were already reduced. When the power is turned on, the free decay ratios of structural responses were further increased. That verifies the performance of the active control system on structural response reduction. The comparison of the uncontrolled and AMD controlled Fourier spectrum magnitude of structural responses also verifies this statement. Moreover, as shown, when the AMD carts were installed, the peaks of Fourier spectrum were shifted to the left-hand-side, which meant that the natural frequencies of the system were decreased. Table 5 shows the peak and RMS acceleration response of the structure with and without AMD. As shown in the table, after the AMD was mounted, the peak and RMS accelerations recorded at the tip point of the side span were reduced 44.1 % and 82.1 %, respectively. For the tip point of the main span, the recorded peak and RMS accelerations were reduced 31.1 % and 81.4 %, respectively. For test cases C2 and C3, the peak and RMS acceleration responses of the structure with and without AMD were recorded. As shown in Table 5, the transverse vibration responses for these sensormounted points were also efficiently reduced: For case C2, structural peak and RMS acceleration responses were respectively reduced 28.3% and 65.4% for the tip point of the side span and 22.0% and 68.4% for the tip point of the main span; For case C3, structural peak and RMS acceleration responses were reduced 65.8% and 85.6% respectively for the tip point of the side span and 40.5% and 76.7% for the tip point of the main span. Moreover, concerning that the controller is designed via numerical studies on a reduced order model, the good control performances obtained on the bridge model tells that the

control algorithm adopted in this study is of good robustness.

No AMD

RMS acc. of sensor 1

> With AMD

C1 4.431 2.479 1.826 0.326 4.312 2.970 1.810 0.336 C2 4.198 3.012 1.341 0.464 3.563 2.779 1.327 0.419 C3 5.186 1.773 1.800 0.260 5.833 3.473 1.820 0.424

**Table 5.** Peak and RMS acceleration at the tip ends of the bridge with and without AMDs for the

Peak acc. of sensor 8

> With AMD

No AMD RMS acc. of sensor 8

> With AMD

No AMD

Peak acc. of sensor 1

> With AMD

No AMD

Case

experimental cases

**Figure 8.** The experimental setup (a) and the AMD devices (b)

During the experiment, the tip ends of the main girder were pulled transversely using steel wires firstly to generate an initial displacement in this direction. The wires were then cut suddenly and the bridge started to vibrate due to this initial potential energy imported. Several seconds later, the power of the AMD control system was turned on and structural vibration response was recorded. Three case studies were conducted to check the performance of the AMD control system under different excitation schemes. For the first case, case C1, the tip ends of both the main span and the side span were pulled in the same direction. After the steel wires were cut, the transverse symmetric bending mode of the bridge was excited. For case C2, the tip ends of the two spans were pulled in the opposite direction to excite the transverse anti-symmetric bending mode of the bridge. Case C3 simulated bridge free vibration under an initial displacement of the main span. Both antisymmetric and symmetric transverse bending modes of the bridge would thus be excited.

208 Advances on Analysis and Control of Vibrations – Theory and Applications

range of the orbit, the system will be shut down.

**Figure 8.** The experimental setup (a) and the AMD devices (b)

During the experiment, the tip ends of the main girder were pulled transversely using steel wires firstly to generate an initial displacement in this direction. The wires were then cut suddenly and the bridge started to vibrate due to this initial potential energy imported. Several seconds later, the power of the AMD control system was turned on and structural vibration response was recorded. Three case studies were conducted to check the

AMD carts (as shown in Fig. 8) are electric servo type [16]. They are 0.645 kg weight. Their track length is 32 cm and the max safe input voltage is 5V. The system is designed under the constraint of avoiding the AMD to knock the baffle. If the AMD position exceeds the safe

> For test case C1, Fig. 9 shows the AMD driving voltage, recorded tip acceleration responses with or without AMDs, and their Fourier spectrums. As shown in these figures, when the power of the AMD carts was still turned off, structural acceleration responses were already reduced. When the power is turned on, the free decay ratios of structural responses were further increased. That verifies the performance of the active control system on structural response reduction. The comparison of the uncontrolled and AMD controlled Fourier spectrum magnitude of structural responses also verifies this statement. Moreover, as shown, when the AMD carts were installed, the peaks of Fourier spectrum were shifted to the left-hand-side, which meant that the natural frequencies of the system were decreased. Table 5 shows the peak and RMS acceleration response of the structure with and without AMD. As shown in the table, after the AMD was mounted, the peak and RMS accelerations recorded at the tip point of the side span were reduced 44.1 % and 82.1 %, respectively. For the tip point of the main span, the recorded peak and RMS accelerations were reduced 31.1 % and 81.4 %, respectively. For test cases C2 and C3, the peak and RMS acceleration responses of the structure with and without AMD were recorded. As shown in Table 5, the transverse vibration responses for these sensormounted points were also efficiently reduced: For case C2, structural peak and RMS acceleration responses were respectively reduced 28.3% and 65.4% for the tip point of the side span and 22.0% and 68.4% for the tip point of the main span; For case C3, structural peak and RMS acceleration responses were reduced 65.8% and 85.6% respectively for the tip point of the side span and 40.5% and 76.7% for the tip point of the main span. Moreover, concerning that the controller is designed via numerical studies on a reduced order model, the good control performances obtained on the bridge model tells that the control algorithm adopted in this study is of good robustness.


**Table 5.** Peak and RMS acceleration at the tip ends of the bridge with and without AMDs for the experimental cases

Transverse Vibration Control for Cable Stayed Bridge Under Construction Using Active Mass Damper 211

simulation and experimental study verified that the proposed AMD control technique is applicable and efficient for the control of transverse vibration of cable-stayed bridge under

For the cable-stayed bridge studied in this chapter, structural vibration test showed that the bridge was of two dominant transverse bending modes with close frequencies. The numerical study verified that for the control of such a structure with repeated frequencies; at least two AMDs should be installed for a good control performance. Moreover, the

For the control of a linear, time-invariant system, an accurate and complete system models are generally required. However, for the bridge-AMD system studied in this chapter, it is very difficult, if not impossible, to set up such a numerical model due to the complex layout of the bridge structure. Since structural vibration responses are generally governed by some dominant vibration modes and the objective of structural vibration control is to reduce but not to completely restrain structural vibration responses, this study verified that a reduced order model constructed from the critical modes is good enough for the controller design to achieve an excellent vibration reduction performance. Considering the differences between the numerical model and the real structure, the control algorithm adopted in this study must be robust to the vibration property change of the controlled structure. The results of experimental studies show that the vibration of the test model can be well controlled using the controller designed from the numerical studies. That means the *LQG* control algorithm is

This study is an initial work on AMD control of transverse vibration of a cable stayed bridge under construction before it can be used for real applications. Although the experimental study verified the efficiency of the adopted AMD control for structural response reduction under given excitation, some primary issues for real application of the AMD control technique, such as how to deal with time delay, how to reduce the requirement on power supply of the control system, and et al., are not addressed in this study. Further laboratory studies or field applications on some real bridges will be conducted in the coming future to discuss these

issues and make the technique more efficient and practical for real applications.

No. 09QH1402300). These supports are greatly appreciated.

*Institute of Engineering Mechanics, China Earthquake Administration, Sanhe, Hebei, China* 

*State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China* 

This research is partially supported by the National High-tech R&D Program of China (863 Program) (Grant No. 2006AA11Z109), and Shanghai Rising Star Tracking Program (Grant

placement of those two AMDs should be carefully studied.

of good robustness for real application.

**Author details** 

Zhi Sun and Limin Sun

**Acknowledgement** 

Hao Chen

construction.

**Figure 9.** The driving voltage (a) and tip acceleration (b) time histories and Fourier spectrum (c) of responses for control case C1

### **7. Concluding remarks**

In this chapter, the active mass dampers are implemented for vibration control of a lab-scale cable stayed bridge in double cantilever construction state. The results of both numerical simulation and experimental study verified that the proposed AMD control technique is applicable and efficient for the control of transverse vibration of cable-stayed bridge under construction.

For the cable-stayed bridge studied in this chapter, structural vibration test showed that the bridge was of two dominant transverse bending modes with close frequencies. The numerical study verified that for the control of such a structure with repeated frequencies; at least two AMDs should be installed for a good control performance. Moreover, the placement of those two AMDs should be carefully studied.

For the control of a linear, time-invariant system, an accurate and complete system models are generally required. However, for the bridge-AMD system studied in this chapter, it is very difficult, if not impossible, to set up such a numerical model due to the complex layout of the bridge structure. Since structural vibration responses are generally governed by some dominant vibration modes and the objective of structural vibration control is to reduce but not to completely restrain structural vibration responses, this study verified that a reduced order model constructed from the critical modes is good enough for the controller design to achieve an excellent vibration reduction performance. Considering the differences between the numerical model and the real structure, the control algorithm adopted in this study must be robust to the vibration property change of the controlled structure. The results of experimental studies show that the vibration of the test model can be well controlled using the controller designed from the numerical studies. That means the *LQG* control algorithm is of good robustness for real application.

This study is an initial work on AMD control of transverse vibration of a cable stayed bridge under construction before it can be used for real applications. Although the experimental study verified the efficiency of the adopted AMD control for structural response reduction under given excitation, some primary issues for real application of the AMD control technique, such as how to deal with time delay, how to reduce the requirement on power supply of the control system, and et al., are not addressed in this study. Further laboratory studies or field applications on some real bridges will be conducted in the coming future to discuss these issues and make the technique more efficient and practical for real applications.

### **Author details**

210 Advances on Analysis and Control of Vibrations – Theory and Applications

**Figure 9.** The driving voltage (a) and tip acceleration (b) time histories and Fourier spectrum (c) of

In this chapter, the active mass dampers are implemented for vibration control of a lab-scale cable stayed bridge in double cantilever construction state. The results of both numerical

responses for control case C1

**7. Concluding remarks** 

Hao Chen *Institute of Engineering Mechanics, China Earthquake Administration, Sanhe, Hebei, China* 

Zhi Sun and Limin Sun *State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China* 

### **Acknowledgement**

This research is partially supported by the National High-tech R&D Program of China (863 Program) (Grant No. 2006AA11Z109), and Shanghai Rising Star Tracking Program (Grant No. 09QH1402300). These supports are greatly appreciated.

### **8. References**

[1] Boonyapinyo, V., Aksorn, A., and Lukkunaprasit, P. (2007). Suppression of aerodynamic response of suspension bridges during erection and after completion by using tuned mass dampers. *Wind and Structures*, *An Int\'l Journal*, 10(1): 1-22.

**Chapter 10** 

© 2012Blanco-Ortega et al., licensee InTech. This is an open access chapter 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.

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.

© 2012Blanco-Ortega et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons

**Automatic Balancing of Rotor-Bearing Systems** 

Rotating machinery is commonly used in many mechanical systems, including electrical motors, machine tools, compressors, turbo machinery and aircraft gas turbine engines. Typically, these systems are affected by exogenous or endogenous vibrations produced by unbalance, misalignment, resonances, bowed shafts, material imperfections and cracks. Vibration can result from a number of conditions, acting alone or in combination. The vibration problems may be caused by auxiliary equipment, not just the primary equipment. Control of machinery vibration is essential in the industry today to increase running speeds and the requirement for rotating machinery to operate within specified levels of vibration.

Vibration caused by mass imbalance is a common problem in rotating machinery. Rotor imbalance occurs when the principal inertia axis of the rotor does not coincide with its geometrical axis and leads to synchronous vibrations and significant undesirable forces transmitted to the mechanical elements and supports. A heavy spot in a rotating component will cause vibration when the unbalanced weight rotates around the rotor axis, creating a centrifugal force. Imbalance could be caused by manufacturing defects (machining errors, casting flaws, etc.) or maintenance issues (deformed or dirty fan blades, missing balance weights, etc.). As rotor speed changes, the effects of imbalance may become higher. Imbalance can severely reduce bearing life-time as well as cause undue machine vibration. Shaft misalignment is a condition in which the shafts of the driving and driven machines are not on the same centre-line generating reaction forces and moments in the couplings. Flexible couplings are used to reduce the misalignment effects and transmit rotary power

Many methods have been developed to reduce the unbalance-induced vibration by using different devices such as active balancing devices, electromagnetic bearings, active squeeze film dampers, lateral force actuators, pressurized bearings and movable bearings (see, e.g., Blanco et al., 2003, 2007, 2008, 2010a, 2010b; Chong-Won, 2006; Dyer et al., 2002; El-Shafei,

Jorge Colín-Ocampo, Marco Oliver-Salazar, Gerardo Vela-Valdés

Andrés Blanco-Ortega, Gerardo Silva-Navarro,

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48621

**1. Introduction** 

without torsional slip.


## **Automatic Balancing of Rotor-Bearing Systems**

Andrés Blanco-Ortega, Gerardo Silva-Navarro, Jorge Colín-Ocampo, Marco Oliver-Salazar, Gerardo Vela-Valdés

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48621

### **1. Introduction**

212 Advances on Analysis and Control of Vibrations – Theory and Applications

[1] Boonyapinyo, V., Aksorn, A., and Lukkunaprasit, P. (2007). Suppression of aerodynamic response of suspension bridges during erection and after completion by using tuned

[2] Ubertini, F. (2010). Prevention of suspension bridge flutter using multiple tuned mass

[3] Warnitchai, P., Fujino, Y., Pacheco, B.M. and Agret, R. (1993). An experimental study on active tendon control of cable-stayed bridges. *Earthquake Engineering and Structural* 

[4] Achkire, Y. and Preumont, A. (1996). Active tendon control of cable-stayed bridges.

[5] Dyke, S.J., Caicedo, J.M., Turan, G., Bergman, L.A., Hague, S. (2003). Phase I benchmark control problem for seismic response of cable-stayed bridges. *Journal of Structural* 

[6] Caicedo, J.M., Dyke, S.J., Moon, S.J., Bergman, L.A., Turan, G., Hague, S. (2003). Phase II benchmark control problem for seismic response of cable-stayed bridges. *Journal of* 

[7] Frederic, B. and Andre, P. (2001). Active tendon control of cable-stayed bridges: a largescale demonstration. *Earthquake Engineering & Structural Dynamics*, 30: 961–979. [8] Shelley, S.J., Lee, K.L., Aksel, T. and Aktan, A.E. (1995). Active control and forced vibration studies on highway bridges. *Journal of Structural Engineering*, *ASCE*, 121(9):

[9] Korlin, R. and Starossek, U. (2007). Wind tunnel test of an active mass damper for bridge

[10] Chen, H. (2007). Experimental study on active control of cable-stayed bridge under

[11] Sun, Z., Hou, W., Chang, C. C. (2009). Structural system identification under random excitation based on asymptotic wavelet analysis. *Engineering Mechanics*, 26(6), 199-204

[12] Schemmann, A.G. (1997). Modeling and active control of cable-stayed bridges subject to

[13] Xu, Y. and Chen, J. (2008). Modal-based model reduction and vibration control for uncertain piezoelectric flexible structures. *Structural Engineering and Mechanics*, 29(5):

[14] Laub, A.J. and Arnold, W.F. (1984). Controllability and observability criteria for multivariable linear second-order models, *IEEE Transactions on Automatic Control*, AC-

[15] Wu, J.C., Yang, J.N., and Schmitendorf W.E. (1998). Reduced-order H∞ and *LQR* control

[16] Quanser Consulting Inc. (2002). Active Mass Damper: Two-Floor (AMD-2), User

decks. *Journal of Wind Engineering and Industrial Aerodynamics*, 95: 267-277.

multiple-support excitation. PhD thesis. Standford University.

for wind-excited tall buildings. *Engineering Structures*. 20(3): 222-236.

mass dampers. *Wind and Structures*, *An Int\'l Journal*, 10(1): 1-22.

dampers. *Wind and Structures, An Int\'l Journal*, 13(3): 235-256.

*Earthquake Engineering and Structural Dynamics*, 25(6): 585-597.

**8. References** 

*Dynamics*, 22: 93-111.

*Engineering, ASCE*, 129(7): 857–872.

construction. Master Thesis. Tongji University.

*Structural Control,* 10: 137–168.

1306-1312.

(in Chinese).

489-504.

29(2).

Manual.

Rotating machinery is commonly used in many mechanical systems, including electrical motors, machine tools, compressors, turbo machinery and aircraft gas turbine engines. Typically, these systems are affected by exogenous or endogenous vibrations produced by unbalance, misalignment, resonances, bowed shafts, material imperfections and cracks. Vibration can result from a number of conditions, acting alone or in combination. The vibration problems may be caused by auxiliary equipment, not just the primary equipment. Control of machinery vibration is essential in the industry today to increase running speeds and the requirement for rotating machinery to operate within specified levels of vibration.

Vibration caused by mass imbalance is a common problem in rotating machinery. Rotor imbalance occurs when the principal inertia axis of the rotor does not coincide with its geometrical axis and leads to synchronous vibrations and significant undesirable forces transmitted to the mechanical elements and supports. A heavy spot in a rotating component will cause vibration when the unbalanced weight rotates around the rotor axis, creating a centrifugal force. Imbalance could be caused by manufacturing defects (machining errors, casting flaws, etc.) or maintenance issues (deformed or dirty fan blades, missing balance weights, etc.). As rotor speed changes, the effects of imbalance may become higher. Imbalance can severely reduce bearing life-time as well as cause undue machine vibration. Shaft misalignment is a condition in which the shafts of the driving and driven machines are not on the same centre-line generating reaction forces and moments in the couplings. Flexible couplings are used to reduce the misalignment effects and transmit rotary power without torsional slip.

Many methods have been developed to reduce the unbalance-induced vibration by using different devices such as active balancing devices, electromagnetic bearings, active squeeze film dampers, lateral force actuators, pressurized bearings and movable bearings (see, e.g., Blanco et al., 2003, 2007, 2008, 2010a, 2010b; Chong-Won, 2006; Dyer et al., 2002; El-Shafei,

© 2012Blanco-Ortega et al., licensee InTech. This is an open access chapter 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. © 2012Blanco-Ortega 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.

2002; Green et al., 2008; Guozhi et al. 2000; Hredzak et al., 2006; Sheu et al., 1997; Zhou y Shi, 2001, 2002). These active balancing control schemes require information of the eccentricity of the involved rotating machinery. On the other hand, there exists a vast literature on identification and estimation methods, which are essentially asymptotic, recursive or complex, which generally suffer from poor speed performance (see, e.g., Ljung, 1987; Soderstrom, 1989; and Sagara and Zhao, 1989, 1990).

Automatic Balancing of Rotor-Bearing Systems 215

**2. Active balancing and vibration control of rotating machinery** 

mass in opposite direction to compensate the residual unbalance.

**Figure 1.** Inertial disk and eddy current probe displacement sensor.

one spirally sliding arm (Chong-Won, 2006; Zhou y Shi, 2001).

**Figure 2.** Diagram of the automatic balancer using two masses.

Many methods for passive balancing have been proposed, such as single plane, two planes or multi-plane balancing. These off-line balancing methods are very common in industrial applications. In these methods, the rotor is modeled as a rigid shaft that without elastic deformation during operation. Rotors operating under 5000 rpm can be considered rigid rotors. For flexible rotors the modal balancing and influence coefficient methods were developed for off-line balancing. Figure 1 shows an inertial disk to be balanced by adding a

Thearle (Thearle, 1932) developed a machine for dynamically balancing rotating elements or high speed rotors (figure 2), where an out-of-balance mass of a rotating element or body can quickly and easily be located, providing the exact amount and location of the balancing mass that should be placed or removed to reduce the vibration. The balancing machine contains a balancing head with a clutch which is first opened to release a set of balls to naturally take place in the balancing positions. Subsequently, the clutch is closed producing a clamping of the balls in the adjusted positions, while the body is being rotated above its critical speed and then released. Other automatic balancing devices have been proposed; essentially using one of the four balancing methods; two angular arms, two sliding arms, one angular and sliding arm, or,

Passive, semi-active and active control schemes have been proposed in order to cancel or attenuate the vibration amplitudes in rotating machinery. In passive control the rotating machinery is modified off-line, e.g. the rotor is stopped to adjust some of its parameters such as mass, stiffness or damping. Balancing consist of placing correction masses onto the rotating shaft (inertial disk) so that centrifugal forces due to these masses cancel out those caused by the residual imbalance mass.

Active vibration control (AVC) changes the dynamical properties of the system by using actuators or active devices during instantaneous operating conditions measured by the appropriate sensors. The main advantage of active control (compared to passive control) is the versatility in adapting to different load conditions, perturbations and configurations of the rotating machinery and hence, extending the system's life while greatly reducing operating costs.

Semiactive vibration control devices are increasingly being investigated and implemented. These devices change the system properties such as damping and stiffness while the rotor is operating. This control scheme is based on the analysis of the open loop response. Semiactive control devices have received a great deal of attention in recent years because they offer the adaptability of active control without requiring the associated large power sources.

This chapter deals with the active cancellation problem of mechanical vibrations in rotorbearing systems. The use of an active disk is proposed for actively balancing a rotor by placing a balancing mass at a suitable position. Two nonlinear controllers with integral compensation are proposed to place the balancing mass at a specific position. Algebraic identification is used for on-line eccentricity estimation as the implementation of this active disk is based on knowledge of the eccentricity. An important property of this algebraic identification is that the eccentricity identification is not asymptotic but algebraic, in contrast to most of the traditional identification methods, which generally suffer of poor speed performance. In addition, a velocity control is designed to drive the rotor velocity to a desired operating point during the first critical speed.

The proposed results are strongly based on the algebraic parameter identification approach for linear systems reported in (Flies and Sira, 2003), which requires a priori knowledge of the mathematical model of the system. This approach has been used for parameter and signal estimation in nonlinear and linear vibrating mechanical systems, where numerical simulations and experimental results show that the algebraic identification provides high robustness against parameter uncertainty, frequency variations, small measurement errors and noise (Beltran et al., 2005, 2006, 2010).

### **2. Active balancing and vibration control of rotating machinery**

214 Advances on Analysis and Control of Vibrations – Theory and Applications

Soderstrom, 1989; and Sagara and Zhao, 1989, 1990).

desired operating point during the first critical speed.

and noise (Beltran et al., 2005, 2006, 2010).

caused by the residual imbalance mass.

operating costs.

2002; Green et al., 2008; Guozhi et al. 2000; Hredzak et al., 2006; Sheu et al., 1997; Zhou y Shi, 2001, 2002). These active balancing control schemes require information of the eccentricity of the involved rotating machinery. On the other hand, there exists a vast literature on identification and estimation methods, which are essentially asymptotic, recursive or complex, which generally suffer from poor speed performance (see, e.g., Ljung, 1987;

Passive, semi-active and active control schemes have been proposed in order to cancel or attenuate the vibration amplitudes in rotating machinery. In passive control the rotating machinery is modified off-line, e.g. the rotor is stopped to adjust some of its parameters such as mass, stiffness or damping. Balancing consist of placing correction masses onto the rotating shaft (inertial disk) so that centrifugal forces due to these masses cancel out those

Active vibration control (AVC) changes the dynamical properties of the system by using actuators or active devices during instantaneous operating conditions measured by the appropriate sensors. The main advantage of active control (compared to passive control) is the versatility in adapting to different load conditions, perturbations and configurations of the rotating machinery and hence, extending the system's life while greatly reducing

Semiactive vibration control devices are increasingly being investigated and implemented. These devices change the system properties such as damping and stiffness while the rotor is operating. This control scheme is based on the analysis of the open loop response. Semiactive control devices have received a great deal of attention in recent years because they offer the adaptability of active control without requiring the associated large power sources. This chapter deals with the active cancellation problem of mechanical vibrations in rotorbearing systems. The use of an active disk is proposed for actively balancing a rotor by placing a balancing mass at a suitable position. Two nonlinear controllers with integral compensation are proposed to place the balancing mass at a specific position. Algebraic identification is used for on-line eccentricity estimation as the implementation of this active disk is based on knowledge of the eccentricity. An important property of this algebraic identification is that the eccentricity identification is not asymptotic but algebraic, in contrast to most of the traditional identification methods, which generally suffer of poor speed performance. In addition, a velocity control is designed to drive the rotor velocity to a

The proposed results are strongly based on the algebraic parameter identification approach for linear systems reported in (Flies and Sira, 2003), which requires a priori knowledge of the mathematical model of the system. This approach has been used for parameter and signal estimation in nonlinear and linear vibrating mechanical systems, where numerical simulations and experimental results show that the algebraic identification provides high robustness against parameter uncertainty, frequency variations, small measurement errors Many methods for passive balancing have been proposed, such as single plane, two planes or multi-plane balancing. These off-line balancing methods are very common in industrial applications. In these methods, the rotor is modeled as a rigid shaft that without elastic deformation during operation. Rotors operating under 5000 rpm can be considered rigid rotors. For flexible rotors the modal balancing and influence coefficient methods were developed for off-line balancing. Figure 1 shows an inertial disk to be balanced by adding a mass in opposite direction to compensate the residual unbalance.

**Figure 1.** Inertial disk and eddy current probe displacement sensor.

Thearle (Thearle, 1932) developed a machine for dynamically balancing rotating elements or high speed rotors (figure 2), where an out-of-balance mass of a rotating element or body can quickly and easily be located, providing the exact amount and location of the balancing mass that should be placed or removed to reduce the vibration. The balancing machine contains a balancing head with a clutch which is first opened to release a set of balls to naturally take place in the balancing positions. Subsequently, the clutch is closed producing a clamping of the balls in the adjusted positions, while the body is being rotated above its critical speed and then released. Other automatic balancing devices have been proposed; essentially using one of the four balancing methods; two angular arms, two sliding arms, one angular and sliding arm, or, one spirally sliding arm (Chong-Won, 2006; Zhou y Shi, 2001).

**Figure 2.** Diagram of the automatic balancer using two masses.

The use of piezoelectric actuators as active vibration dampers in rotating machines has been considered in the past. Palazzolo, et al. (Palazzolo et al., 1993) first used the piezoelectric pusher for active vibration control in rotating machinery as it is shown in Figure 3.a. The pusher is soft mounted to the machine case to improve the electromechanical stability and connected to the squirrel cage-ball bearing supports of a rotating shaft, to actively control the unbalance, transient and subsynchronous responses of the test rotor, using velocity feedback. The piezoelectric actuators are modeled as dampers and springs. Recently, Carmignani et al. (Carmignani et al., 2001) developed an adaptive hydrodynamic bearing made of a mobile housing mounted on piezoelectric actuators to attenuate the vibration amplitudes in constant speed below the first critical speed. The actuators, arranged at 90° on a perpendicular plane to the shaft axis, exert two sinusoidal forces with a tuned phase angle to produce a balancing or, alternatively, a dampering effect. The authors presented experimental and numerical results.

Automatic Balancing of Rotor-Bearing Systems 217

*chamber Sealed*

*Spring*

(ER) fluids are materials that respond to an applied magnetic or electric field with a dramatic change in rheological behavior. To attenuate the vibration amplitudes around the first critical speed an on/off control is proposed to control the large amplitude around the

Hathout and El-Shafei (Hathout and El-Shafei, 1997) proposed a hybrid squeeze film damper (HSFD), (see Figure 4.b), to attenuate the vibrations in rotating machinery for both sudden unbalance and transient run-up through critical speeds. El-Shafei (El-Shafei, 2000) have implemented different control algorithms (PID-type controllers, LQR, gain scheduling, adaptive and bang-bang controllers) for active control of rotor vibrations for HSFDsupported rotors. Controlling the fluid pressure in the chamber, the bearing properties of

**Figure 4.** Fluid film bearings: a) using rheological fluids and b) using a pressure chamber.

*Squirrel cage*

Sun y Kroedkiewski (Sun and Krodkiewski, 1997, 1998) proposed a new type of active oil bearing, see Figure 5.a. The active bearing is supplied with a flexible sleeve whose deformation can be changed during rotor operation. The flexible sleeve is also a part of a hydraulic damper whose parameters can be controlled during operation as well. The oil film and the pressure chamber are separated by the flexible sealing. The equilibrium position of the flexible sleeve and the bearing journal is determined by load and pressure, which can be controlled during operation. Parameters of this damper can also be varied during operation to eliminate the self exciting vibration and increase the stability of the equilibrium position

*a) b)*

*bearing Ball*

*Fluid inlet*

*rotor*

Recently, Dyer et al., (Dyer et al., 2002) developed an electromagnetically actuated unbalance compensator. The compensator consists of two rings as shown in Figure 5.b. These two rings are not balanced and can be viewed as two heavy spots. These two rings are held in place by permanent magnetic forces. When the balancer is activated, an electric current passes through the coil and the rings can be moved individually with respect to the spindle by the electromagnetic force. The combination of these two heavy spots is equivalent to a single heavy spot whose magnitude and position can change to attenuate the

first critical speed.

*Electrorheological fluid*

stiffness and damping can be changed.

*rotor*

*Ball bearing*

of the rotor-oil bearing system.

vibration amplitudes.

Active Magnetic Bearings (AMBs) are the mostly used devices but their use in the industrial field is still limited due to a low stiffness and the need of additional conventional bearings for fault emergency. An AMB system is a collection of electromagnets used to suspend an object and stabilization of the system is performed by feedback control, see Figure 3.b. In recent decades, AMBs has been widely used as a non-contact, lubrication-free, support in many machines and devices. Many researchers (Lee, 2001; Sheu-Yang, 1997) have proposed a variety of AMBs that are compact and simple-structured. The AMB system, which is openloop unstable and highly coupled due to nonlinearities inherited in the system such as the gyroscopic effect and imbalance, requires a dynamic controller to stabilize the system.

**Figure 3.** a) Piezoelectric actuator and b) active magnetic bearing.

Another device for AVC in rotating machinery is the one based on fluid film bearings. The dynamics of a rotor system supported by fluid film bearings is inherently a nonlinear problem and these fluid film bearings have been used in combination with other devices, such as piezoelectric actuators, magneto or electro-rheological fluids, etc. (see Figure 4).

Guozhi *et al*. (2000) proposed the use of a fluid bearing with rheological fluids to reduce the vibrations around the first critical speed. Magnetorheological (MR) or electrorheological (ER) fluids are materials that respond to an applied magnetic or electric field with a dramatic change in rheological behavior. To attenuate the vibration amplitudes around the first critical speed an on/off control is proposed to control the large amplitude around the first critical speed.

216 Advances on Analysis and Control of Vibrations – Theory and Applications

**Figure 3.** a) Piezoelectric actuator and b) active magnetic bearing.

experimental and numerical results.

The use of piezoelectric actuators as active vibration dampers in rotating machines has been considered in the past. Palazzolo, et al. (Palazzolo et al., 1993) first used the piezoelectric pusher for active vibration control in rotating machinery as it is shown in Figure 3.a. The pusher is soft mounted to the machine case to improve the electromechanical stability and connected to the squirrel cage-ball bearing supports of a rotating shaft, to actively control the unbalance, transient and subsynchronous responses of the test rotor, using velocity feedback. The piezoelectric actuators are modeled as dampers and springs. Recently, Carmignani et al. (Carmignani et al., 2001) developed an adaptive hydrodynamic bearing made of a mobile housing mounted on piezoelectric actuators to attenuate the vibration amplitudes in constant speed below the first critical speed. The actuators, arranged at 90° on a perpendicular plane to the shaft axis, exert two sinusoidal forces with a tuned phase angle to produce a balancing or, alternatively, a dampering effect. The authors presented

Active Magnetic Bearings (AMBs) are the mostly used devices but their use in the industrial field is still limited due to a low stiffness and the need of additional conventional bearings for fault emergency. An AMB system is a collection of electromagnets used to suspend an object and stabilization of the system is performed by feedback control, see Figure 3.b. In recent decades, AMBs has been widely used as a non-contact, lubrication-free, support in many machines and devices. Many researchers (Lee, 2001; Sheu-Yang, 1997) have proposed a variety of AMBs that are compact and simple-structured. The AMB system, which is openloop unstable and highly coupled due to nonlinearities inherited in the system such as the gyroscopic effect and imbalance, requires a dynamic controller to stabilize the system.

Another device for AVC in rotating machinery is the one based on fluid film bearings. The dynamics of a rotor system supported by fluid film bearings is inherently a nonlinear problem and these fluid film bearings have been used in combination with other devices, such as piezoelectric actuators, magneto or electro-rheological fluids, etc. (see Figure 4).

Guozhi *et al*. (2000) proposed the use of a fluid bearing with rheological fluids to reduce the vibrations around the first critical speed. Magnetorheological (MR) or electrorheological Hathout and El-Shafei (Hathout and El-Shafei, 1997) proposed a hybrid squeeze film damper (HSFD), (see Figure 4.b), to attenuate the vibrations in rotating machinery for both sudden unbalance and transient run-up through critical speeds. El-Shafei (El-Shafei, 2000) have implemented different control algorithms (PID-type controllers, LQR, gain scheduling, adaptive and bang-bang controllers) for active control of rotor vibrations for HSFDsupported rotors. Controlling the fluid pressure in the chamber, the bearing properties of stiffness and damping can be changed.

**Figure 4.** Fluid film bearings: a) using rheological fluids and b) using a pressure chamber.

Sun y Kroedkiewski (Sun and Krodkiewski, 1997, 1998) proposed a new type of active oil bearing, see Figure 5.a. The active bearing is supplied with a flexible sleeve whose deformation can be changed during rotor operation. The flexible sleeve is also a part of a hydraulic damper whose parameters can be controlled during operation as well. The oil film and the pressure chamber are separated by the flexible sealing. The equilibrium position of the flexible sleeve and the bearing journal is determined by load and pressure, which can be controlled during operation. Parameters of this damper can also be varied during operation to eliminate the self exciting vibration and increase the stability of the equilibrium position of the rotor-oil bearing system.

Recently, Dyer et al., (Dyer et al., 2002) developed an electromagnetically actuated unbalance compensator. The compensator consists of two rings as shown in Figure 5.b. These two rings are not balanced and can be viewed as two heavy spots. These two rings are held in place by permanent magnetic forces. When the balancer is activated, an electric current passes through the coil and the rings can be moved individually with respect to the spindle by the electromagnetic force. The combination of these two heavy spots is equivalent to a single heavy spot whose magnitude and position can change to attenuate the vibration amplitudes.

$$(M + m\_1)\ddot{x} + c\dot{x} + kx = p\_x(t)$$

$$(M + m\_1)\ddot{y} + c\dot{y} + ky = p\_y(t)$$

$$J\_e\ddot{\varphi} + c\_\theta\dot{\varphi} = \tau\_1 + p\_\theta(t)\tag{1}$$

$$m\_1r\_1^2\ddot{\omega} + 2m\_1r\_1\dot{r}\_1\dot{\alpha} + m\_1gr\_1\cos\alpha = \tau\_2$$

$$m\_1\ddot{r}\_1 - m\_1r\_1\dot{\alpha}^2 + m\_1g\sin\alpha = F$$

$$p\_{\boldsymbol{x}}(t) = Mu[\ddot{\boldsymbol{\varphi}}\sin(\boldsymbol{\varphi} + \boldsymbol{\beta}) + \dot{\varphi}^2 \cos(\boldsymbol{\varphi} + \boldsymbol{\beta})] + m\_1 r\_1 [\ddot{\boldsymbol{\varphi}}\sin(\boldsymbol{\varphi} + \boldsymbol{a}) + \dot{\varphi}^2 \cos(\boldsymbol{\varphi} + \boldsymbol{a})]$$

$$p\_{\boldsymbol{y}}(t) = Mu[\dot{\boldsymbol{\varphi}}^2 \sin(\boldsymbol{\varphi} + \boldsymbol{\beta}) - \ddot{\boldsymbol{\varphi}} \cos(\boldsymbol{\varphi} + \boldsymbol{\beta})] + m\_1 r\_1 [\dot{\boldsymbol{\varphi}}^2 \sin(\boldsymbol{\varphi} + \boldsymbol{a}) - \ddot{\boldsymbol{\varphi}} \cos(\boldsymbol{\varphi} + \boldsymbol{a})]$$

$$p\_{\boldsymbol{\varphi}}(t) = Mu[\dot{\boldsymbol{\varphi}}\sin(\boldsymbol{\varphi} + \boldsymbol{\beta}) - \ddot{\boldsymbol{\varphi}} \cos(\boldsymbol{\varphi} + \boldsymbol{\beta})] + m\_1 r\_1 [\ddot{\boldsymbol{\varphi}}\sin(\boldsymbol{\varphi} + \boldsymbol{a}) - \ddot{\boldsymbol{\varphi}} \cos(\boldsymbol{\varphi} + \boldsymbol{a})]$$

$$f\_{\boldsymbol{\varphi}} = f + Mu^2 + mr^2$$

$$\begin{aligned} \dot{z}\_1 &= z\_2\\ \dot{z}\_2 &= \frac{1}{\lambda} \Big( \frac{1}{\mu\_e} (b^2 + f\_e \mathcal{M}\_e) f\_1 + \frac{ab}{\mathcal{M}\_e} f\_2 + a \Big( \tau\_1 - c\_q \dot{z}\_6 \Big) \Big) \\\\ \dot{z}\_3 &= z\_4 \end{aligned}$$

$$\begin{aligned} \dot{z}\_4 &= \frac{1}{\lambda} \Big( \frac{ab}{\mathcal{M}\_e} f\_1 + \frac{1}{\mathcal{M}\_e} Q\_e \mathcal{M}\_e - a^2 \mathcal{I} \Big) f\_2 + b \Big( \tau\_1 - c\_q \dot{z}\_6 \Big) \Big) \\\\ \dot{z}\_5 &= z\_6 \end{aligned}$$

$$\begin{aligned} \dot{z}\_6 &= \frac{1}{\lambda} \Big( -af\_1 - bf\_2 - M\_e \Big( \tau\_1 - c\_q \dot{z}\_6 \Big) \Big) \\\\ \dot{z}\_7 &= z\_6 \end{aligned} \tag{2}$$

$$\begin{aligned} \dot{z}\_8 &= \frac{1}{m\_1} (F - g m\_1 \sin z\_9 + m\_1 z\_7 z\_{10}^2) \\\\ \dot{z}\_9 &= z\_{10} \end{aligned}$$

$$\begin{aligned} \dot{z}\_{10} &= \frac{1}{m\_1 z\_7^2} (\tau\_2 - g m\_1 z\_7 \cos z\_9 - 2m\_1 z\_7 z\_8 z\_{10}) \\\\ \dot{y} &= z\_1^2 + z\_3^2 \end{aligned}$$

$$\begin{aligned} f\_1 &= c\_{\varphi} z\_2 + kz\_1 - M z\_6^2 u\_y - m\_1 r\_y z\_6^2, \quad f\_2 = c z\_4 + kz\_3 - M z\_6^2 u\_x - m\_1 r\_x z\_6^2, \quad a = -M u\_x - m\_1 r\_{\chi\nu}, \\ b &= M u\_\chi + m\_1 r\_\chi, \\ b &= M u\_\chi + m\_1 r\_\chi, \end{aligned}$$

$$R = \frac{\sqrt{\mathbf{z}\_1^2 + \mathbf{z}\_3^2}}{u} \tag{3}$$


$$F = m\_1 \upsilon\_2 + g m\_1 \,\mathrm{sen}\,\mathrm{z}\_9 - m\_1 \mathrm{z}\_7 \mathrm{z}\_{10}^2 \tag{4}$$

$$
\pi\_2 = m\_1 z\_7^2 v\_3 + g m\_1 z\_7 \cos z\_9 + 2 m\_1 z\_7 z\_8 z\_{10} \tag{5}
$$

$$\begin{aligned} \boldsymbol{\upsilon}\_{2} &= \boldsymbol{\ddot{\upsilon}}\_{2}^{\*}(\boldsymbol{t}) - \boldsymbol{\chi}\_{22} [\boldsymbol{\dot{\psi}}\_{2} - \boldsymbol{\dot{\psi}}\_{2}^{\*}(\boldsymbol{t})] - \boldsymbol{\chi}\_{21} [\boldsymbol{\dot{\chi}}\_{2} - \boldsymbol{\chi}\_{2}^{\*}(\boldsymbol{t})] - \boldsymbol{\chi}\_{20} \int\_{0}^{t} [\boldsymbol{\dot{\chi}}\_{2} - \boldsymbol{\chi}\_{2}^{\*}(\boldsymbol{\sigma})] d\boldsymbol{\sigma} \\\\ \boldsymbol{\upsilon}\_{3} &= \boldsymbol{\ddot{\psi}}\_{3}^{\*}(\boldsymbol{t}) - \boldsymbol{\chi}\_{32} [\boldsymbol{\dot{\psi}}\_{3} - \boldsymbol{\dot{\psi}}\_{3}^{\*}(\boldsymbol{t})] - \boldsymbol{\chi}\_{31} [\boldsymbol{\dot{\chi}}\_{3} - \boldsymbol{\chi}\_{3}^{\*}(\boldsymbol{t})] - \boldsymbol{\chi}\_{30} \int\_{0}^{t} [\boldsymbol{\dot{\psi}}\_{3} - \boldsymbol{\chi}\_{3}^{\*}(\boldsymbol{\sigma})] d\boldsymbol{\sigma} \end{aligned}$$

$$e\_2^{(3)} + \chi\_{22}\ddot{e}\_2 + \chi\_{21}\dot{e}\_2 + \chi\_{20}e\_2 = 0$$

$$e\_3^{(3)} + \chi\_{32}\ddot{e}\_3 + \chi\_{31}\dot{e}\_3 + \chi\_{30}e\_3 = 0$$

$$p\_2(\mathbf{s}) = \mathbf{s}^3 + \chi\_{22}\mathbf{s}^2 + \chi\_{21}\mathbf{s} + \chi\_{20}\mathbf{s}$$

$$p\_3(\mathbf{s}) = \mathbf{s}^3 + \chi\_{32}\mathbf{s}^2 + \chi\_{31}\mathbf{s} + \chi\_{30}\mathbf{s}$$

$$\left[f + (Mu^2 + m\_1r\_1^2)\right] \quad \dot{\mathbf{z}}\_6 + c\_\varphi \mathbf{z}\_6 = \mathbf{r}\_1$$

$$\mathbf{y}\_1 = \mathbf{z}\_6\tag{6}$$

$$
\tau\_1 = [f + (Mu^2 + m\_1 r\_1^2)]\nu\_1 + c\_\varphi z\_6
$$

$$
\nu\_1 = \dot{\nu}\_1^\*(t) - \chi\_{11}[\nu\_1 - \chi\_1^\*(t)] - \chi\_{10} \int\_0^t [\nu\_1 - \chi\_1^\*(\sigma)]d\sigma. \tag{7}
$$

$$
\ddot{e}\_1 + \chi\_{11}\dot{e}\_1 + \chi\_{10}e\_1 = 0\tag{8}
$$

$$(M + m\_1)\dot{z}\_2 + cz\_2 + kz\_1 = Mu[\dot{z}\_6\operatorname{sen}(\dot{z}\_5 + \beta) + \dot{z}\_6^2\cos(\mathbf{z}\_5 + \beta)] + $$

$$+ m\_1r\_1[\dot{z}\_6\operatorname{sen}(\dot{z}\_5 + a) + \dot{z}\_6^2\cos(\mathbf{z}\_5 + a)]$$

$$(M + m\_1)\dot{z}\_4 + cz\_4 + kz\_3 = Mu[\dot{z}\_6^2\operatorname{sen}(\dot{z}\_5 + \beta) - \dot{z}\_6\cos(\mathbf{z}\_5 + \beta)] + \tag{9}$$

$$+ m\_1r\_1[\dot{z}\_6^2\operatorname{sen}(\dot{z}\_5 + a) - \dot{z}\_6\cos(\mathbf{z}\_5 + a)]$$

$$\int^{\langle 2\rangle} \left[ (\mathcal{M} + m\_1) t^2 \frac{d\mathbf{z}\_2}{dt} + ct^2 \mathbf{z}\_2 + kt^2 \mathbf{z}\_1 \right] =$$

$$\int^{\langle 2\rangle} \mathcal{M}ut^2 \frac{d}{dt} [\mathbf{z}\_6 \,\mathrm{sen}(\mathbf{z}\_5 + \beta)] + \int^{\langle 2\rangle} m\_1 r\_1 t^2 \frac{d}{dt} [\mathbf{z}\_6 \,\mathrm{sen}(\mathbf{z}\_5 + a)]$$

$$\int^{\langle 2\rangle} \left[ (\mathcal{M} + m\_1) t^2 \frac{d\mathbf{z}\_4}{dt} + ct^2 \mathbf{z}\_4 + kt^2 \mathbf{z}\_3 \right] \tag{10}$$

$$= \int^{\langle 2\rangle} \mathcal{M}ut^2 \frac{d}{dt} [\mathbf{z}\_6 \cos(\mathbf{z}\_5 + \beta)] + \int^{\langle 2\rangle} m\_1 r\_1 t^2 \frac{d}{dt} [\mathbf{z}\_6 \cos(\mathbf{z}\_5 + a)]$$

$$\begin{aligned} \left(M+m\_1\right)\left[t^2z\_1 - 4\int tz\_1 + 2\int^{\left(^{2}\right)}z\_1\right] + c\_\theta & \left[\int t^2z\_1 - 2\int^{\left(^{2}\right)}tz\_1\right] + k\int^{\left(^{2}\right)}t^2z\_1 = \\\\ &=Mu\left[t^2z\_6\sin\left(z\_5 + \beta\right) - 2\int^{\left(^{2}\right)}tz\_6\sin\left(z\_5 + \beta\right)\right] +\\ &+m\_1r\_1\left[\int t^2z\_6\sin\left(z\_5 + a\right) - 2\int^{\left(^{2}\right)}tz\_6\sin\left(z\_5 + a\right)\right] \end{aligned} \tag{11}$$

$$\begin{aligned} \left(M+m\_1\right)\left[t^2z\_3 - 4\int tz\_3 + 2\int^{\left(^{2}\right)}z\_3\right] + c\_\theta\left[\int t^2z\_3 - 2\int^{\left(^{2}\right)}tz\_3\right] + k\int^{\left(^{2}\right)}t^2z\_3 = \\\\ &=-Mu\left[t^2z\_6\cos(z\_5 + \beta) - 2\int^{\left(^{2}\right)}tz\_6\cos(z\_5 + \beta)\right] +\\ &+m\_1r\_1\left[\int t^2z\_6\cos(z\_5 + a) - 2\int^{\left(^{2}\right)}tz\_6\cos(z\_5 + a)\right] \end{aligned}$$

$$A(\mathfrak{t})\theta = b(\mathfrak{t})\tag{12}$$

$$A(t) = \begin{bmatrix} a\_{11}(t) & a\_{12}(t) \\ -a\_{12}(t) & a\_{11}(t) \end{bmatrix}, b(t) = \begin{bmatrix} b\_1(t) \\ b\_2(t) \end{bmatrix}.$$

$$a\_{11} = M \left[ \int t^2 \mathbf{z}\_6 \, \text{sen} \, \mathbf{z}\_5 - 2 \int^{\text{(2)}} t \mathbf{z}\_6 \, \text{sen} \, \mathbf{z}\_5 \right]$$

$$a\_{12} = M \left[ \int t^2 \mathbf{z}\_6 \cos \mathbf{z}\_5 - 2 \int^{\text{(2)}} t \mathbf{z}\_6 \cos \mathbf{z}\_5 \right]$$

$$b\_1 = (M + m\_1) t^2 \mathbf{z}\_1 + \int (ct^2 \mathbf{z}\_1 - 4(M + m\_1) t \mathbf{z}\_1)$$

$$\begin{aligned} \mathbf{b}\_2 &= (M + m\_1)t^\omega \mathbf{z}\_3 + \int (ct^\omega \mathbf{z}\_3 - 4(M + m\_1)tz\_3) \\\\ &+ \int (m\_1 \mathbf{z}\_6 \mathbf{z}\_7 t^2 \cos(\mathbf{z}\_5 + a)) - \int^{(2)} 2m\_1 \mathbf{z}\_6 \mathbf{z}\_7 t \cos(\mathbf{z}\_5 + a) \end{aligned}$$

$$\begin{aligned} u\_{\eta e} &= \frac{u\_{11} u\_{11} - u\_2 u\_{12}}{a\_{11}^2 + a\_{12}^2} \\ u\_{\xi e} &= \frac{b\_{11} a\_{12} + b\_2 a\_{11}}{a\_{11}^2 + a\_{12}^2} \\ u\_e &= \sqrt{u\_{\eta e}^2 + u\_{\xi e}^2} \\ \beta\_e &= \cos\left(\frac{u\_{\eta e}}{u\_e}\right) \end{aligned} \quad \forall t \in \left(t\_0, t\_0 + \delta\_0\right] \tag{13}$$

**Figure 10.** Rotor speed and control torque.

Fig. 11 shows the dynamic behavior of the active disk controllers when the balancing mass is driven to the equilibrium position ��̅= � �� ��� ��=�� � ��. In this position the active disk cancels the unbalance, as it is shown in the Fig. 12. The controllers are implemented when the eccentricity has been estimated.

Automatic Balancing of Rotor-Bearing Systems 227

0 1 2 3

t [s]

0 1 2 3

t [s]

**Figure 11.** Dynamic response of the balancing mass: radial position (y2=z₇), angular position (y3=z₉),

0



2 [Nm]

0

<sup>1</sup> x 10-3

1

2

y3 [rad]

3

4

**Figure 12.** Unbalance response with automatic balancing and without active disk.

0 50 100 150 200 250 300

Automatic balancing using active disk

Unbalance response without active disk

z 6 [rad/s]

control force (F) and moment force (τ2).

0

5

10

15

R [m/m]

20

25

30

0.02 0.025 0.03 0.035 0.04


F [N]

0

0.01

0.02

y2 [m]

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> 0.015

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> -0.02

t [s]

t [s]

226 Advances on Analysis and Control of Vibrations – Theory and Applications

**Figure 10.** Rotor speed and control torque.

0

0.2

1 [Nm

0.4

100

200

y1 [rad/s]

300

the eccentricity has been estimated.

is driven to the equilibrium position ��̅= �

Fig. 11 shows the dynamic behavior of the active disk controllers when the balancing mass

0 50 100 150

t [s]

<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>0</sup>

t [s]

cancels the unbalance, as it is shown in the Fig. 12. The controllers are implemented when

��� ��=�� � ��. In this position the active disk

��

**Figure 11.** Dynamic response of the balancing mass: radial position (y2=z₇), angular position (y3=z₉), control force (F) and moment force (τ2).

**Figure 12.** Unbalance response with automatic balancing and without active disk.

### **6. Conclusions**

The active vibration control of rotor-bearing systems using active disks for actively balancing a rotor is addressed. This approach consists of locating a balancing mass at a suitable position. Since this active control scheme requires information of the eccentricity, a novel algebraic identification approach is proposed for the on-line estimation of the eccentricity parameters. This approach is quite promising, in the sense that from a theoretical point of view, the algebraic identification is practically instantaneous and robust with respect to parameter uncertainty, frequency variations, small measurement errors and noise. Thus the algebraic identification is combined with two control schemes to place the balancing mass in the correct position to cancel the unbalance of the rotor. A velocity control is designed to take the rotor velocity to a desired operating point over the first critical speed in order to show the vibration cancellation. The controllers were developed in the context of an off-line prespecified reference trajectory tracking problem. Numerical simulations were included to illustrate the proposed high dynamic performance of the active vibration control scheme proposed.

Automatic Balancing of Rotor-Bearing Systems 229

Blanco, A.; Silva, G. and Gómez, J. C. (2003). Dynamic Stiffness Control and Acceleration Scheduling for the Active Balancing Control of a Jeffcott-Like Rotor System, *Proceedings of The tenth International Congress on Sound and Vibration*, pp. 227-234, Stockholm,

Blanco, A.; Beltran, F. and Silva, G. (2007). On Line Algebraic Identification of Eccentricity in Active Vibration Control of Rotor Bearing Systems, *4th International Conference on Electrical and Electronics Engineering*, pp. 253 - 256, ISBN 978-1-4244-1166-5, México, Sept.

Blanco, A.; Beltrán, F. and Silva, G. (2008). Active Disk for Automatic Balancing of Rotor-Bearing Systems. *American Control Conference, ACC 2008*. pp. 3023 - 3028, ISBN 978-1-

Blanco, A.; Beltrán, F.; Silva, G. and Oliver, M. A. (2010). Active vibration control of a rotorbearing system based on dynamic stiffness, *Revista de la Facultad de Ingeniería.* 

Blanco, A.; Beltrán, F.; Silva, G. and Méndez, H. (2010). Control de Vibraciones en Sistemas Rotatorios, Revista Iberoamericana de Automática e Informática Industrial. Vol. 7, No.

Carmignani, C.; Forte, P. and Rustighi, E. (2001). Active Control of Rotors by Means of Piezoelectric Actuators, *Proceedings of Design Engineering Technical Conference and Computers and Information in Engineering Conference*, p.757-764, ISBN 0791835413, Vol. 6,

Chong-Won, L. (2006). Mechatronics in Rotating Machinery. 7th IFToMM-Conference on

El-Shafei, A. (2000). Active Control Algorithms for the Control of Rotor Vibrations Using

Fliess, M. and Sira-Ramírez, H. (2003) An algebraic framework for linear identification, *ESAIM: Control, Optimization and Calculus of Variations*, pp. 151-168, Vol. 9, 2003. Green K, Champneys A.R., Friswell M.I. y Muñoz (2008) A.M.Investigation of a multi-ball, automatic dynamic balancing mechanism for eccentric rotors. *Royal Society Publishing*,

Guozhi, Y., Fah, Y.F., Guang, C., Guang, M., Tong, F. and yang, Q., Electro-Rheological Multi-layer Squeeze Film Damper and its Application to Vibration Control of Rotor

Hathout, J. P and El-Shafei, A. (1997). PI Control of HSFDs for Active Control of Rotor-Bearing Systems, *Journal of Engineering for Gas Turbines and Power*, pp. 658-667, ISSN

Hredzak, B. and Guo, G. (2006). Adjustable Balancer With Electromagnetic Release of Balancing Members. *IEEE Transactions on Magnetics*, pp. 1591-1596, Vol. 42, No. 5.

System, *Journal of Vibration and Acoustics*, pp. 7-11, Vol. 122, No. 1, 2000.

Dimarogonas, A. (1996). *Vibration for Engineers*. Prentice Hall, ISBN 978-0134562292**,** 1996. Dyer,, S. W.; Ni, J.; Shi, J. and Shin, K. (2002). Robust Optimal Influence-Coefficient Control of Multiple-Plane Active Rotor Balancing Systems, *Journal of Dynamic Systems*,

Sweden, July 7-10, 2003.

4244-2079-7, Seattle, WA, USA, June 11-13, 2008.

4, (Octubre 2010.) pp. 36-43, ISSN 1697-7912.

pp. 705-728, Vol. 366, No. 1866.

0742-4795, Vol. 119, No. 3.

Pittsburgh, Pennsylvania, USA, September 9-12, 2001.

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HSFDS. Proc. of ASME TURBOEXPO 2000, Munich Germany.

*Universidad de Antioquia*. No. 55, pp. 125-133. ISSN. 0120-6230.

5-7, 2007.

### **Author details**

Andrés Blanco-Ortega, Jorge Colín-Ocampo, Marco Oliver-Salazar and Gerardo Vela-Valdés *Centro Nacional de Investigación y Desarrollo Tecnológico, CENIDET, México* 

Gerardo Silva-Navarro *Centro de Investigación y de Estudios Avanzados del IPN, CINVESTAV, México* 

## **Acknowledgement**

Research reported here was supported by grants of the Dirección General de Educación Superior Tecnológica, DGEST through PROMEP under Project "Vibration control of rotating machinery".

### **7. References**


Blanco, A.; Silva, G. and Gómez, J. C. (2003). Dynamic Stiffness Control and Acceleration Scheduling for the Active Balancing Control of a Jeffcott-Like Rotor System, *Proceedings of The tenth International Congress on Sound and Vibration*, pp. 227-234, Stockholm, Sweden, July 7-10, 2003.

228 Advances on Analysis and Control of Vibrations – Theory and Applications

The active vibration control of rotor-bearing systems using active disks for actively balancing a rotor is addressed. This approach consists of locating a balancing mass at a suitable position. Since this active control scheme requires information of the eccentricity, a novel algebraic identification approach is proposed for the on-line estimation of the eccentricity parameters. This approach is quite promising, in the sense that from a theoretical point of view, the algebraic identification is practically instantaneous and robust with respect to parameter uncertainty, frequency variations, small measurement errors and noise. Thus the algebraic identification is combined with two control schemes to place the balancing mass in the correct position to cancel the unbalance of the rotor. A velocity control is designed to take the rotor velocity to a desired operating point over the first critical speed in order to show the vibration cancellation. The controllers were developed in the context of an off-line prespecified reference trajectory tracking problem. Numerical simulations were included to illustrate the proposed high dynamic performance of the active vibration control

Andrés Blanco-Ortega, Jorge Colín-Ocampo, Marco Oliver-Salazar and Gerardo Vela-Valdés

Research reported here was supported by grants of the Dirección General de Educación Superior Tecnológica, DGEST through PROMEP under Project "Vibration control of rotating

Beltrán F.; Silva, G.; Sira, H. and Quezada, J. (2005). Active vibration control using on-line algebraic identification of harmonic vibrations, *Proceedings of American Control Conference*, pp. 4820 - 4825, ISBN 0-7803-9099-7, Portland, Oregon, USA, June 8-10, 2005. Beltrán, F.; Sira, H. and Silva, G. (2006). Adaptive-like Active Vibration Suppression for a Nonlinear Mechanical System Using On-Line Algebraic Identification, *Proceedings*. *of the Thirteenth International Congress on Sound and Vibration*, Vienna, Austria, July 2-6,

Beltrán F.; Silva, G.; Sira, H. and Blanco, A. (2010). Active Vibration Control Using On-line Algebraic Identification and Sliding Modes. Computación y Sistemas. Vol. 13, No. 3, pp

*Centro Nacional de Investigación y Desarrollo Tecnológico, CENIDET, México* 

*Centro de Investigación y de Estudios Avanzados del IPN, CINVESTAV, México* 

**6. Conclusions** 

scheme proposed.

**Author details** 

Gerardo Silva-Navarro

**Acknowledgement** 

machinery".

**7. References** 

2006.

313-330. ISSN 1405-5546.

	- Krodkiewski, J. M. and Sun L. (1998). Modelling of Multi-Bearing Rotor Systems Incorporating an Active Journal Bearing, 215-229, Vol. 10, No. 2.

**Chapter 11** 

© 2012 Ryu and Kong, licensee InTech. This is an open access chapter 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 Ryu and Kong, 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.

**Dynamic Responses and Active Vibration Control** 

The dynamic deflection and vibration control of an elastic beam structure carrying moving masses or loads have long been an interesting subject to many researchers. This is one of the most important subjects in the areas of structural dynamics and vibration control. Bridges, railway bridges, cranes, cable ways, tunnels, and pipes are the typical structural examples of

While the analytical studies on the dynamic behavior of a structure under moving masses and loads have been actively performed, a small number of experimental studies, especially for the vibration control of the beam structures carrying moving masses and loads, have been conducted. It is, therefore, strongly desired to conduct both analytical and experimental studies in parallel to develop the algorithm that controls effectively the

The dynamic responses and vibrations of structures under moving masses and loads were initially studied by (Stokes, 1849; Ayre et al., 1950) who tried to solve the problem of railway bridges. This type of study has been actively performed by employing the finite element method (Yoshida & Weaver, 1971). (Ryu, 1983) used the finite difference method to study the dynamic response of both the simply supported beam and the continuous beam model carrying a moving mass with constant velocity and acceleration. (Sadiku & Leipholz, 1987) utilized the Green's function to present the difference of the solutions for the moving mass problem without and with including the inertia effect of a mass. (Olsson, 1991) studied the dynamic response of a simply supported beam traversed by a moving object of the constant velocity without considering the inertia effect of moving mass. (Esmailzadeh & Ghorash, 1992) expanded Olsson's study by including the inertia effect of the moving mass. (Lin, 1997) suggested the effects of both centrifugal and Coliolis forces should be taken into account to obtain the dynamic deflection. (Wang & Chou, 1998) conducted nonlinear

vibration and the dynamic response of structures under moving masses and loads.

**of Beam Structures Under a Travelling Mass** 

Bong-Jo Ryu and Yong-Sik Kong

http://dx.doi.org/10.5772/39272

**1. Introduction** 

Additional information is available at the end of the chapter

the structure to be designed to support moving masses and loads.

