1. Introduction
In the industrial fields, lots of machines and structures had experienced the troubles caused by vibration and unbalance phenomena. Numerous studies have been done to control the vibrations of the machines in operation in the past decades.
Currently it is known well that the fast and the accurate motions in the machining duration becomes one of the common trends for the optimal design. In the high speed operation, the vibration which may occurs at each parts of machine should be controlled actively or passively for the stability of the machine structure.
The vibration problems of machines and structures had been studied by lots of researchers. The vibration energy flow and the dynamic response of a beam, plate or some compound system have been analyzed. By using a model of an elastic beam, the vibrational intensity and control skills had been presented^{[14]}. The structure analysis and the control of the vibrations of the machine tools had been introduced ^{[510]}.
It is known that a body or part of machinery or robots in the industrial fields experience a few types of dynamic response being induced by a bouncing movement, a rotation and/or the constant reciprocal motions. So to control the above mentioned responses, the dynamic absorber is simply applied to the machine. The Dynamic Absorber (DA)^{[11]} is known to be effective over narrow band of frequencies and is designed to make the natural frequencies of the main system be away from the forcing frequency. Hence in this study a machine which is subjected to a harmonically excited force is modelled theoretically as a rigid body in a two dimension and employed to be controlled passively by a Dynamic Absorber which consists of a mass and a spring. Two supports of a body are considered as linear springs having only the elastic property. Based on the equation of motion, two motions (bounce and rotation about the center of gravity (CG)) of a rigid body will be discussed.
To define the optimal design parameters of a DA, the cost function which is the ratio of the dynamic characteristics of a body without DA to that of a body with DA will be evaluated and discussed. Fig. 1 shows the theoretical model of a rigid body which is subjected to a harmonically excited force F(t) and has a Dynamic Absorber (DA) which is located at La being distanced from the center of gravity (CG).
2. The Governing Equations
The governing equations for the motions of a rigid body which has a dynamic absorber(DA), can be written in matrix form as:
where x and θ mean the bouncing and rotating displacements of a rigid body, respectively and y is the displacement of a Dynamic Absorber(DA), J_{G} is a mass moment of inertia of a rigid body. The elements of a matrix are given as
In the above equations, the displacements of a rigid body and a DA are expressed into the amplitude of a spatial function and the time harmonic function. Then by suppressing the time term in Eq. (1), it can be rearranged in terms of amplitudes as,
where $\begin{array}{l}{z}_{11}={k}_{1}+{k}_{2}+{k}_{a}{\omega}^{2},{z}_{12}={k}_{a}={z}_{21},\\ {z}_{13}={k}_{2}{L}_{2}{k}_{1}{L}_{1}={z}_{31},{z}_{22}={k}_{a}{m}_{a}{\omega}^{2},\\ {z}_{23}=\mp {k}_{a}{X}_{a}={z}_{32},\\ {z}_{33}={k}_{1}{L}_{1}^{2}+{k}_{2}{L}_{2}^{2}+{k}_{a}{X}_{a}^{2}J{\omega}^{2}\end{array}$
Here the amplitudes can be obtained by multiplying both sides of Eq.(3) by [Z]^{1} which is the inverse of a thirdorder matrix [Z] in Eq.(3). Finally the amplitudes become as follows,
where ‘Δ’ represents the determinant of a thirdorder matrix [Z] and is expressed as
$\begin{array}{c}\Delta ={z}_{11}{z}_{22}{z}_{33}{z}_{12}{z}_{21}{z}_{33}+{z}_{13}{z}_{21}{z}_{32}{z}_{11}{z}_{23}{z}_{32}\\ +{z}_{12}{z}_{23}{z}_{31}{z}_{13}{z}_{22}{z}_{31}\end{array}$
The determinant will be used for determining the resonant frequencies to control the vibration of a rigid body. All values obtained in this study have been expressed into the dimensionless forms which are based on a certain quantity. The dimensionless mass (m_{a}) of a DA which is divided by a mass of rigid body (m) is given as 0.1 ~ 0.3 and the mass moment of inertia (J_{G}) is kept a constant as 0.1 based on the quantity of mL^{2}.
The center of a gravity (CG) of a lot machines are observed to deviate away from a geometric center of a body. So the dimensionless lengths (L_{1}, L_{2}) of both supports from CG are given as 0.4 and 0.6, respectively. In Table 1, the dimensionless properties and the base quantity are introduced as
3. Resonant frequency coefficients (α)
In Eq.(4), the denominator part vanishes for certain specific values of the frequency ratio(α). The roots of denominator which are expressed in α are called the resonance frequency coefficient for the rigid body with a DA. For three mass ratios (ma = 0.1, 0.2 and 0.3) of a DA stiffness ratio (k_{a}=0.1), the variations of three resonance frequency coefficients (α1, α2, α3) versus location of a DA (Xa) are plotted in Fig. 2(a), (b) and (c), respectively. Based on the CG, the left direction is marked negative sign (0 ~ 0.4) and the positive sign (0 ~ 0.6) is given for the right direction. As the mass is increasing, the values of the coefficients are getting higher. But the differences of the value is getting smaller as be higher the order of the coefficient. It is confirmed that as the total mass is increasing, is getting lower the value of the resonant frequencies. Fig. 1
In Fig. 2(a), (b) and (c), the variations of the three frequency coefficients are plotted versus the location of a DA for three stiffness ratios (ka = 0.1, 0.3 and 0.5) of a mass ratio (ma=0.2), respectively. It is noted that the higher stiffness produces the higher coefficient. The variation of each coefficient along the length of a body is found to show the similar trend for the same order of coefficient.
4. Control of Dynamic Characteristics
4.1 Reduction of amplitudes
In order to control passively the dynamic responses of a rigid body, In Eq. (4), two amplitudes for the bounce (X) and the rotation (Θ) are employed for the responses of a rigid body. The reduction level of the dynamic response (RD) is evaluated in comparison with two cases  a response without DA and a response with DA. The given external force ratio (f) is set to be a unity and the forcing frequency (α_{f}) is randomly entered. The ratio of a response without DA to one with DA is expressed in terms of decibel [dB] as follows;
In Fig.3, the reduction of two amplitudes (bounce and rotation) for ma = 0.2 and α_{f} = 2, are plotted along the DA stiffness ratio (ka) for three locations of a DA  Xa = 0.1 (the left side from CG), Xa = 0.0 (positioned at CG) and Xa = 1.0 (positioned at the location of the external force (Lf)).
In case of bouncing motion, the huge reduction is observed near or at the value of ka = 0.8 for three locations and otherwise the reduction is found to be small or worse.
The large reduction of rotating case occurs at the value of ka = 0.8 only for the case of Xa = 0.1 which is the same place of the external force. Here the value of ka = 0.8 matches the optimal stiffness of a DA being reported previously^{[11]} such that the natural frequency of DA should be set equal to the forcing frequency of a main system. So the optimal value of the stiffness ratio (ka) can be given as;
In Fig.4, the value of ka being estimated by Eq. (6) gives the large reduction level compared to those of other values of ka. It is proved that the optimal value of ka in Eq. (6) is still effective to control two motions of a rigid body (bounce and rotation).
4.2 Reduction of transmitted power
The Vibrational energy which is induced by the harmonic motion of the external force is known to transmit to the main and the neighbour systems. In practice, a lot machines and frames are known to be troubled by excessive energy which is transmitted to the machine body and to the surrounding parts. Hence power which is the energy rated by time is evaluated and is expressed as follows;
where P[m], P[L] and P[R] represent the power be transmitted to a rigid body, to the left support and to the right support, respectively. The reduction level of power (RP) is expressed in terms of decibel [dB] as;
For three parts of a system (a rigid body, a left and right supports), Fig.5 shows the reductions of the transmitted power (RP) of three DA locations (Xa= 0.1, 0.1 and 0.2) versus the forcing frequency coefficient (α_{f}) for the optimal value of a DA stiffness ratio (ka=Eq. (6)). As shown in Fig.5, reduction level of the location of Xa = 0.1 is exclusively high compared to other locations. In Fig.6, the reductions of the transmitted power of a rigid body for three cases of a force location (Lf= 0.1, 0.1 and 0.3) are plotted against the location of a DA (Xa). It is observed that the powers be reduced largely near the same location of a DA and a external force. It is surely noted that the optimal location of a DA be near or at the place of an external force.
5. Conclusion
In this paper, the passive control of a vibrating machine with a Dynamic Absorber is studied to determine the optimal design parameters of a DA. On the bases of the analyses in this study, the conclusions are obtained as follows,

1) A Dynamic Absorber is observed to have the satisfactory results for the reduction of the dynamic characteristics, which are the dynamic responses and the power be transmitted to a rigid body and two supports.

2) The location of a DA should be positioned near or at the place of a external force for the control of two motions of a rigid body  bounce and rotation.

3) The variation of a mass of a DA might be a helpful tool to avoid the resonant phenomena by changing the range of the natural frequencies.

4) It is proved that the optimal stiffness of a DA which is determined by Eq. (6) is still conservative as the design parameter of a DA.