مدل سازی و تجزیه و تحلیل حساسیت یک سیستم جداسازی ارتعاش پنوماتیک با دو محفظه هوا

This paper aims at accurate modeling and sensitivity analysis for a pneumatic vibration isolation system (PVIS) as a foundation for practical design. Even though the PVIS is widely used for its effective performance in vibration isolation, its design has depended largely on trial-and-error methods. In previous studies, nonlinear characteristics of the diaphragm and the air flow restrictor, which significantly affect the performance of a PVIS, have been investigated. However, several hurdles, such as the absence of a mathematical model for the diaphragm, still remain with regard to the model-based prediction of performance. Therefore, a fractional derivative model for the diaphragm and a quadratic damping model for the air flow restrictor are newly developed based on the careful examination of previous studies. Then, sensitivities of vibration isolation performance indices with regard to major design variables are analyzed and new approximation formulas are created based on the dynamic characteristics of the PVIS. Our models with a transmissibility-computing algorithm are verified by comparison with experimental data. The sensitivity analyses and approximation formulas are expected to be useful for practical PVIS design owing to their simplicity and accuracy.

As high precision industries such as semiconductor production, precision metrology, optics, and microbiology continue to grow, higher performance vibration isolation systems are needed to meet the corresponding vibration tolerance requirements [1], [2] and [3]. To achieve vibration isolation for local precision equipment, a pneumatic vibration isolation system (PVIS) is widely used because it needs no energy supply and no control unit, and performs stable and effective vibration attenuation across a wide frequency range. Even though a PVIS is very useful, its design for better vibration isolation has depended largely on trial and error methods. Vibration isolation performance enhancement of the PVIS has been attempted by a variety of ways such as reshaping of elastomeric diaphragm [4], usage of an air flow restrictor of porous media [5], energy dissipation by a gimbal piston in oil chamber [6], parallelization with a negative-stiffness device [7] and adoption of active control schemes [8], [9], [10] and [11]. In all those attempts, the basic work is the modeling of components of the PVIS since effects of design variables on vibration isolation performance can be predicted only with accurate mathematical models and corresponding computational techniques. In this paper, our goal is accurate modeling and vibration isolation performance evaluation of a PVIS such that our results may be used to predict the performance for practical PVIS design. We examined previous studies and determined that three additional efforts are required. The first is to make a mathematical model of the diaphragm. Some studies pointed out that the diaphragm has an important role in the elastic and damping characteristic of a PVIS [4], [12] and [13]. However, its nonlinear properties have been thus far neglected in many cases, since the diaphragm have been treated as a linear element in most analyses [14], [15], [16] and [17] or the properties of the diaphragm have taken the form of lookup table even in some studies that regarded it as a nonlinear element [12] and [13]. The mathematical modeling of the diaphragm is essential since it makes transfer functions to be analytic so that the transfer functions might be used not only for transmissibility calculation but also for sensitivity analysis and performance optimization. The second is to consider the air flow restrictor as a nonlinear damper, and the air volume through the restrictor as a simultaneous independent variable. Some previous studies regarded the flow restrictor as a linear damper, in which case the transfer function of the PVIS is easily formulated [12], [14] and [18]. However, the nonlinearity of the damping characteristics of the flow restrictor was demonstrated in many studies [11], [19], [20] and [21]. Lee and Kim [13] advanced the analysis of a PVIS by considering the flow restrictor as a nonlinear damper. They made the approximation that the air volume passing through the restrictor is proportional to tabletop displacement. However, since tabletop displacement is not proportional to base displacement of vibration, it is easily inferred that the air volume coupled with both of them is not proportional to either one of them. The last is to analyze sensitivities of vibration isolation performance indices to design variables based on nonlinear dynamic characteristics of the PVIS. Many design strategies were developed using the assumption that nonlinear models can be approximated as linear models [5], [6], [14], [18] and [22]. However, the design strategies could not cope with the nonlinear characteristics of a PVIS. Therefore, the sensitivity analyses using nonlinear models are required for accurate performance prediction and for efficient design of a PVIS. Consideration of the first two issues, which is presented in Section 2, results in the transfer functions of a PVIS consisting of two simultaneous nonlinear complex equations. To derive and calculate the transfer functions, the equal energy dissipation method is adopted, and an equivalent mechanical system and a recursive numerical method are devised in Section 3. The models and the PVIS transfer functions are verified by experimental data in Section 4, in which the discrepancies between the previous air flow restrictor models and experimental data lead to an adjustment of the model. Based on the dynamic characteristics of the PVIS, sensitivities of PVIS performance indices to design variables are analyzed and approximation formulas are created in Section 5. Our concluding remarks are given in Section 6. In exploring the modeling of the PVIS, this research will be limited to considering only vertical rigid-body mode vibration of the tabletop because horizontal vibration and tabletop flexural modes can be controlled by very different mechanisms, which should be subjects of distinct research [23].

The nonlinear properties of the diaphragm of a PVIS have been described using look-up tables in previous studies since it is very difficult to model the stiffness and damping, which depend on the amplitude and frequency of deformation. In this paper, a fractional derivative model for the diaphragm was newly developed and was shown to agree very well with experimental data. The capillary tube in the PVIS, referred to as an air flow restrictor, has previously been regarded as a linear damper or as a nonlinear damper with its air flow volume being proportional to the tabletop displacement. We have described the limitations of these assumptions of air flow restrictors and improved the model for the air flow restrictor. In our model, the damping force is proportional to the square of the flow rate and the air volume through the capillary tube is defined as an independent variable. Based on comparisons with experimental data, the new air flow restrictor model has proved to be more accurate than previous models. The PVIS transfer functions were derived from our new models by the equal energy dissipation method and the conversion to equivalent dynamic variables. To calculate the obtained transfer functions that cannot be calculated in one step due to the nonlinearity, fixed-point iteration with under-relaxation was applied. The computational algorithm was proven to be valid by comparison of its results with experimental findings. The algorithm enabled sensitivity analysis as well as transmissibility calculations. The equivalent mechanical system, which was newly proposed in the formulation of the transfer functions, clarified the physical meanings of components of the PVIS and was useful in adjusting the air flow restrictor model. The dynamic characteristic of the PVIS, namely, that the transmissibility surface should be divided into two characteristic regions according to the ranges of the frequency and amplitude, could be explained using the equivalent mechanical system. For the two characteristic regions, the sensitivities of vibration isolation performance indices to design variables were analyzed using the PVIS transfer functions and the golden-section search method. In order to facilitate efficient design, approximation formulas were created based on the dynamic characteristics of the PVIS. The results agreed very well with the results of sensitivity analyses through the tuned coefficients.