Vibration of an axially moving string with nonclassical boundary conditions subjected to harmonic excitation based on the method of multiple scales

Wu, Yuanfeng; Chen, Enwei; Zhu, Weidong; He, Yuteng; Lu, Yimin; Chen, Pin

doi:10.1007/s11071-023-09115-0

Nonlinear Dyn

2024

DOI: 10.1007/s11071-023-09115-0

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Vibration of an axially moving string with nonclassical boundary conditions subjected to harmonic excitation based on the method of multiple scales

Yuanfeng Wu,

Enwei Chen,

Weidong Zhu

et al.

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2024

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Cited by 2 publications

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Coupled Vertical–Horizontal Vibration Characteristics of a Cold Rolling Mill Based on an Optimized Karman Differential Equation

Wang,

Zhang,

Zhang

et al. 2024

Iran J Sci Technol Trans Mech Eng

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Coupled Vertical–Horizontal Vibration Characteristics of a Cold Rolling Mill Based on an Optimized Karman Differential Equation

Wang,

Zhang,

Zhang

et al. 2024

Iran J Sci Technol Trans Mech Eng

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Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

Lu,

Guo

2024

Int. J. Bifurcation Chaos

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In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

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Vibration of an axially moving string with nonclassical boundary conditions subjected to harmonic excitation based on the method of multiple scales

Cited by 2 publications

References 38 publications

Coupled Vertical–Horizontal Vibration Characteristics of a Cold Rolling Mill Based on an Optimized Karman Differential Equation

Coupled Vertical–Horizontal Vibration Characteristics of a Cold Rolling Mill Based on an Optimized Karman Differential Equation

Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

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