Periodic solutions of the Nathanson's and the Comb-drive models. (September 2018)

• Record Type: Journal Article
• Authors: Llibre, Jaume; Nuñez, Daniel E.; Rivera, Andrés
• Is Part Of: International journal of non-linear mechanics. Volume 104(2018)
• Page Start: 109
• Page End: 115
• Publication Date: 2018-09
• Subjects: 37G15 -- 37C80 -- 37C30; Nathanson's model -- Comb-drive model -- Periodic solution -- Second-order differential equation -- Averaging theory -- Lower and upper solution method; Nonlinear mechanics -- Periodicals; Mécanique non linéaire -- Périodiques; Nonlinear mechanics; Periodicals; 531
• DOI: 10.1016/j.ijnonlinmec.2018.05.009
• Languages: English
• ISSNs: 0020-7462

Abstract

Abstract: We provide sufficient conditions for the existence and stability properties of periodic solutions of the second-order non-autonomous differential equation of the Nathanson's model x ̈ + x + a x ̇ − b ( v 0 + δ v ( ω t ) ) 2 ( 1 − x ) 2 = 0, and of the Comb-drive model x ̈ + x + a x ̇ − 4 b ( v 0 + δ v ( ω t ) ) 2 x ( 1 − x 2 ) 2 = 0, where x ∈ R, a, b, v 0, δ are positive parameters and v ( ω t ) is a 2 π ∕ ω -periodic function. The results are obtained using the averaging theory and the lower and upper solution method. Highlights: We prove the existence of an unstable periodic orbit for the nathanson model via averaging theory. We give sufficient conditions for the stability of the equilibrium of the comb-drive model. The comb-drive model is studied via the upper and lower solution method.