## 1. Introduction

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

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

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 of Timoshenko beam due to a moving force and the weight of beam. Recently, (Wu, 2005) studied dynamic analysis of an inclined beam due to moving loads.

Most studies on the dynamic response of a beam carrying moving mass or loads are analytical, but a small number of experimental investigations are recently conducted in parallel. The studies on the dynamic response of a beam caused by moving masses or loads have been also conducted domestically. For the studies on the vibration control of moving masses or loads, (Abdel-Rohman & Leipholz, 1980) applied the active vibration control method to control the beam vibration caused by moving masses. They applied the bending moment produced by tension and compression of an actuator to the beam when the vibration of a simply supported beam carrying a moving mass occurs. With the active control, the passive control approaches have been proposed in many engineering fields.(Kwon et al., 1998) presented an approach to reduce the deflection of a beam under a moving load by means of adjusting the parameters of a conceptually second order damped model attached to a flexible structures.

Recently, the piezoelectric material has been used for the active vibration control. (Ryou et al., 1997) studied the vibration control of a beam by employing the distributed piezoelectric film sensor and the piezoelectric ceramic actuator. They verified their sensor and actuator system by observing the piezoelectric film sensor blocked effectively the signal from the uncontrolled modes.

(Bailey& Hubbard Jr., 1985) conducted the active vibration control on a thin cantilever beam through the distributed piezoelectric-polymer and designed the controller using Lyapunov's second and direct method. (Kwak & Sciulli, 1996) performed experiments on the vibration suppression control of active structures through the positive position feedback(PPF) control using a piezoelectric sensor and a piezoelectric actuator based on the fuzzy logic. (Sung, 2002) presented the modeling and control with piezo-actuators for a simply supported beam under a moving mass. Recently, (Nikkhoo et al., 2007) investigated the efficiency of control algorithm in reducing the dynamic response of beams subjected to a moving mass. After that, (Prabakar et al., 2009) studied optimal semi-active control of the half car vehicle model with magneto-rheological damper. (Pisarski & Bajer, 2010) conducted semi-active control of strings supported by viscous damper under a travelling load.

In this chapter, firstly, dynamic response of a simply supported beam caused by a moving mass is investigated by numerical method and experiments. Secondly, the device of an electromagnetic actuator is designed by using a voice coil motor(VCM) and used for the fuzzy control in order to suppress the vibration of the beam generated by a moving mass. Governing equations for dynamic responses of a beam under a moving mass are derived by Galerkin's mode summation method, and the effect of forces (gravity force, Coliolis force, inertia force caused by the slope of the beam, transverse inertia force of the beam) due to the moving mass on the dynamic response of a beam is discussed. For the active control of dynamic deflection and vibration of a beam under the moving mass, the controller based on fuzzy logic is used and the experiments are conducted by VCM(voice coil motor) actuator to suppress the vibration of a beam.

## 2. Theoretical analyses

### 2.1. Governing equations of a simply supported beam traversed by a moving mass

The mathematical model of a beam traversed by a moving mass is shown in Fig. 1, where

The governing equation of the system can be expressed as,

(1) |

Where

Employing Galerkin's mode summation method, the displacement of a beam,

where the dimensionless displacement,

For the simply supported beam, the shape function,

The eigenvalue

From the following equation,

The static deflection,

The maximum static deflection

Substituting the solution of Eq. (5) into Eq. (1) and performing inner product the shape function of

(12) |

where, the dimensionless variables and parameter are defined as

Eq. (12) can be expressed in matrix form as

where, the matrix components are

The response of Eq. (14) may be analyzed using Runge-Kutta integration method.

### 2.2. Designing fuzzy controller

A general linear control theory is not applicable for the study since it is the time variant system and has a large non-linearity, as presented in Eq. (14). The fuzzy controller is effectively applicable for the study since the controller is designed by considering the characteristics of the vibration produced only, without taking into account the dynamic characteristics of the system.

The fundamental structure of the fuzzy controller applied to the simply supported beam carrying a moving mass is shown in Fig. 2.

The system output measured from the laser displacement sensor is transferred to the fuzzy set element by fuzzification. The decision making rule was properly designed by the controller designer based on the dynamic characteristics of the system to be studied. It produced the control input by defuzzifying the fuzzified output calculated from the measured values of the system.

The designer's experience, the professional's knowledge, and the dynamic characteristics of the system to be controlled are added by the designer in every process. The advantage of the fuzzy control theory using this type of design method is to design the controller easily when the characteristics or trends of the system are known or predictable to a certain extent, but the mathematical modelling for control is very difficult and complicated like the one for the study.

The system output obtained from the standardization process was fuzzified by formulating a fuzzy set. There are various ways to formulate a fuzzy set. A fuzzy set, which divides the range from -1 to 1 into 7 equal sections and has a triangular function as a membership function, was constructed and used for the present study as shown in Fig. 3.

It is desired to utilize the time response curve of a typical second-order system to a step input, as shown in Fig. 4, to determine the fuzzy rule for designing a fuzzy controller. At the point of "a", the control input should be PB(Positive Big) since the error is NB(Negative Big) and the time rate of the error is ZE(zero). At the point of "b", the control input is desired to be NS(Negative Small) if the error is ZE and the time rate of the error is PS(Positive Small). From the same logic, the control input at the point of "c" is needed to be PM(Positive Medium). Such a logical reasoning may be justified from the point of professional.

NB | NM | NS | ZE | PS | PM | PB | ||

NB | PB | PB | PM | PM | PS | PS | ZE | |

NM | PB | PM | PM | PS | PS | ZE | NS | |

NS | PM | PM | PS | PS | ZE | NS | NS | |

ZE | PM | PS | PS | ZE | NS | NS | NM | |

PS | PS | PS | ZE | NS | NS | NM | NM | |

PM | PS | ZE | NS | NS | NM | NM | NB | |

PB | ZE | NS | NS | NM | NM | NB | NB |

By expanding the logic stated above, the fuzzy rule may be constructed as shown in Table 1 for all cases of the fuzzy set consisted of seven elements.

where the errors

The normalized error

The actual control input was obtained by defuzzifying the fuzzified control input that was found based on the fuzzy rule from the normalized error and the time rate of the error obtained through the fuzzification process. For the study, the normalized control input was found by employing the min-max centroid method for the defuzzification.

## 3. Numerical analysis results of dynamic response to a moving mass

In order to produce numerical analysis results of dynamic response of a beam traversed by a moving mass, Runge-Kutta integration method was applied to Eq. (14). The dynamic deflection caused by a moving mass with constant velocity was investigated for the study, even though the dynamic deflection by a moving mass with constant acceleration can also be obtained from Eq. (14).

Figs. 5-7 show the dynamic responses at the dimensionless position

Figs. 8-10 show the dynamic deflection at the dimensionless position of the moving mass for various mass ratios of

Figs. 11 and 12 present the comparison of current results on the dynamic deflection and those from previous studies (references Olsson, M. (1991); Esmailzadeh, E. & Ghorashi, M. (1992); Lin, Y. H. (1997)) that do not consider as many effects of a moving mass as the present study does, for

## 4. Experimental apparatus and experiments

### 4.1. Experimental apparatus

An experimental apparatus was set up to control the dynamic deflection and the vibration of a simply supported uniform beam traversed by a moving mass as shown in Fig. 13.

The test beam has a groove along its length to help various sizes of a moving mass to run smoothly. The details of the test beam are shown in Table 2.

Material | Aluminum 6061 |

Modulus of Elasticity(Gpa) | 7.07e+10 |

Density( | 2700 |

Mass( | 283.0 |

Length ( | 1000.0 |

Width ( | 32.0 |

Thickness ( | 4.0 |

Groove width ( | 10.0 |

Groove depth ( | 2.0 |

Steel balls were chosen as moving masses to reduce the friction with the test beam. The details of the moving masses are shown in Table 3

Photo. 1 depicts the experimental set-up for the study. A non-contact laser displacement sensor, a sensor controller, and a digital memory oscilloscope are installed to measure the dynamic response of the test beam traversed by a moving mass.

Figs 14-15 show schematic diagram of the front view and details for the structure of a voice coil motor(VCM) actuator, respectively. The active actuator was reconstructed by using a commercial speaker. In the structure of the actuator, permanent magnet and voice coil wound in bobbin of the speaker were used, and an attacher and a shaft were produced in order to deliver control force from actuator to the beam. The actuator was built to suppress the dynamic deflection and the vibration of the test beam caused by a moving mass. The actuator is able to generate the control input by using a voice coil motor. The actuator is desired to be installed above and below the test beam to apply the force from the magnetic field to the beam. However, the control input is applied through a slender rod which connects the control actuator to the bottom of a beam, since the upper part of the beam should be free from any bar for a moving mass running. Photo. 2 depicts a test beam, moving masses and an actuator.

### 4.2. Experiments

Experiments are conducted to measure the dynamic response first, and then to control the vibration of a beam traversed by a moving mass. Three different mass ratios and three different velocity ratios of a moving mass were chosen to investigate the dynamic response of the system. The moving mass was released freely at the designated location of the guide beam and traveled the horizontal part of the guide beam of 210

In order to suppress the dynamic deflection and vibration of a beam traversed by a moving mass, the control input is supplied by an actuator as shown in Fig. 13. The control input in accordance with the fuzzy logic is applied to the system through a voice coil motor at the same time when the moving mass enters the test beam. The signal of the dynamic response wasamplified by an amplifier and then displayed on an oscilloscope.

Such experiments on the dynamic deflection and the vibration control were performed in order for the selected masses and velocities of moving masses. The details of the moving mass used are shown in each figure.

#### 4.3. Experimental results and discussion

### 4.3.1. Experimental results of the dynamic response and discussion

Figs. 16-18 present both the experimental results and the analytical results from the numerical simulation on the dynamic response of a simply supported beam traversed by a moving mass. Figs. 16-18 show both analytical and experimental results on the dynamic response for three different velocities and magnitudes of a moving mass (

The analytical results agree well with the experimental results in both the magnitude and the shape of the dynamic response curve for twovelocities(

### 4.3.2. Experimental results of vibration control of a beam

A control using a voice-coil motor was conducted to suppress the amplitude of dynamic response and vibration of a beam traversed by a moving mass. The Matlab simulation was performed to find the best location of the actuator for the control input.

The results are presented in Figs. 19-20. The simulation was performed by investigating the response to the disturbance applied to a certain point for various locations of the actuator. Two typical simulation results are presented in Figs. 19-20. As shown in these figures, a better controlled result was obtained when the actuator is located at 3/10 of the beam length

The experimental results under both uncontrolled and controlled conditions are presented in Figs. 21-23.

The experimental results of the dynamic deflection in 4.3.1 and the test results for the uncontrolled case in 4.3.2 are supposed to be identical for the same mass ratio

Therefore, it is necessary to include the slender rod to simulate the dynamic deflection curve for the uncontrolled cases shown in Figs. 21-23. It is, however, very difficult to develop the mathematical governing equation for such a system. Thus, the present study focuses on the control effect only by applying the control input from the measured dynamic responses of the system with the control actuator.

For three different velocities(

## 5. Conclusion

The following results were obtained from the fuzzy control studies on the dynamic response and the vibration of a simply supported beam traversed by a moving mass with a constant velocity. Firstly, the position of a moving mass at the maximum dynamic deflection moves to the right end of a beam as the mass ratio of a moving mass

Thirdly, the experimental results of the dynamic deflection of a beam traversed by a moving mass agree well with the simulation results.

Fourthly,the dynamic deflection and the residual vibration of a beam traversed by a moving mass were successfully reduced more than 50 % through the fuzzy control.