In this research, we propose an analytical method to control the period and the clarity of the beat in a slightly asymmetric ring. The concept of a simple equivalent ring is applied, i.e. a slightly asymmetric ring with known modal data is modeled as an equivalent circular ring with one or multiple point masses, which has the same modal data as that of the original ring. By using the equivalent ring model, the optimal beat tuning is achieved by adding a 2 nd point mass in order to obtain the required period and clarity in the beating vibration. The results obtained from the analytical method are compared with those obtained from finite element analysis(FEA) and the validity of the proposed method is verified. The proposed method can be effectively used to predict the mode pair and control the beat property in belltype structures.

1999 esterification, ester hydrolysis esterification, ester hydrolysis O 0310 -053Synthesis of Five-Membered 2-Heteroaryl 2-Heteroaromatic Carboxylates and Attempted Cyclization to Bisheteroaryl[2,3-b:3',2'd]pyran-2-one.-A method for the preparation of the title carboxylates such as (III) and (V) is described and their spectroscopic data are discussed in detail. All attempts to cyclize the obtained compounds fail. It is noteworthy that the reaction in pyridine only yields anhydrides. -(LEE, C. K.; YU, J. S.; KIM, S. H.; J.

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