In this paper, the Equivalent Linearization Method (ELM) with a weighted averaging, which is proposed by Anh (Anh, 2015), is applied to analyze some vibrating systems with nonlinearities. The strongly nonlinear Duffing oscillator with third, fifth, and seventh powers of the amplitude, the other strongly nonlinear oscillators and the cubic Duffing with discontinuity are considered. The results obtained via this method are compared with the ones achieved by the Min-Max Approach (MMA), the Modified Lindstedt -Poincare Method (MLPM), the Parameter -Expansion Method (PEM), the Homotopy Perturbation Method (HPM) and 4 th order RungeKutta method. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully exerted to a lot of practical engineering and physical problems.

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This work was carried out in collaboration between both authors. Author DVH designed the study, performed the statistical analysis, wrote the protocol, wrote the first draft of the manuscript, and managed the analyses of the study. Author NQH managed the literature searches. Both authors read and approved the final manuscript.

Die sterisch überladenen Disiloxane (III) bzw. (VI) werden erhalten bei der Suche nach einfacheren Synthesewegen für das Silandiol (II) bzw. Silanol (V) durch Reaktion der Silane (I) bzw. (IV) mit AgNO3 im Überschuß.

The overcrowded molecules 1,1,3,3-tetra-tert-butyldisiloxane-1,3-diol (2) and penta-tertbutyldisiloxanol (4) have been obtained during the attempted synthesis of tri-tert-butylsilanol and di-tert-butylsilanediol by reaction of tri-tert-butylsilane and di-tert-butylchlorosilane, resp., with silver nitrate. Compounds 2 and 4 were also formed either by condensation of two molecules of di-tert-butylsilanediol or by co-condensation of the silanediol with tri-tert-butylsilanol in the presence of jo-toluenesulfonic acid. Reactions of tri-tert-butylsilanol with silicon tetrachloride yielded 1,1,1,5,5,5-hexa-tert-butyl-3,3-dichlorotrisiloxane.