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This paper focuses on the vibration periodic property of Duffing oscillator with generalized initial conditions. Firstly, the undamped case is solved by Ji-Huan He’s frequency formulation; Secondly, the formulation is extended to the damped case. Numerical verification shows that the frequency formulation is mathematically simple and physically insightful and practically applicable. This paper paves a novel way for engineers to use the formulation to study nonlinear vibration system with ease and reliability. [ABSTRACT FROM AUTHOR]

In this study, chitosan-based silicon nanoparticles (Chsi-NPs) are prepared that primarily consists of C (57.9%), O (31.3%), N (5.6%), and Si (3.5%) and are 10–180 nm in size. We then explore the effect on the foliage applied on rice planted on soil contaminated with 104 mg·kg−1 arsenic (As); low (3 mg·L−1)and high (15 mg·L−1) doses of the foliar Chsi-NPs are administered during the rice grain filling stage. The results showed that the higher dose foliar Chsi-NPs treatment reduced the As concentration in the grain by 61.2% but increased As concentration in the leaves by 47.1% compared to the control treatment. The foliar spraying of the Chsi-NPs inhibited As transport to the grain by facilitating the attachment of As to the cell wall, with higher doses of the foliar Chsi-NPs treatment increased by 8.7%. The foliar spraying of Chsi-NPs increased the malondialdehyde levels by 18.4%, the catalase activity by 49.0%, and the glutathione activity by 99.0%. These results indicated that the foliar Chsi-NPs application was effective for alleviating As toxicity and accumulation in rice. This study provides a novel method for effectively alleviating As accumulation in rice. [Display omitted] • Foliar Chsi-NPs were used to improve the As tolerance of rice in soils. • Foliar Chsi-NPs application promotes the fixation of arsenic in the leaf cell wall. • Foliar Chsi-NPs treatments reduced the As concentration in the grain by 61.2%. [ABSTRACT FROM AUTHOR]