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2. Numerical analysis of the friction-induced oscillator of Duffing's type with modified LuGre friction model.
- Author
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Pikunov, Danylo and Stefanski, Andrzej
- Subjects
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LYAPUNOV exponents , *DIFFERENTIAL equations , *CALCULUS , *MATHEMATICAL physics , *ADJOINT differential equations - Abstract
Abstract This paper focuses on analysis of a friction-induced mechanical oscillator with cubic nonlinearity and an applied dynamical model of dry friction. The studied model is based on the classical LuGre approach to friction force modelling. Lyapunov exponents spectrum of the frictional oscillator is calculated by means of a recently implemented method of their estimation for non-smooth systems. Results of the numerical research, observations of the nature of the oscillator's response and interaction of the applied dynamic friction model with the oscillator are reported. Highlights • The dynamics of stick-slip oscillator with cubic nonlinearity is demonstrated. • Modified LuGre friction model is applied. • The stability of the system is proven by conditional Lyapunov exponents. • The correlation (synchrony) between oscillator response and dynamics of friction force is confirmed. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
3. On the reflected wave superposition method for a travelling string with mixed boundary supports.
- Author
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Chen, E.W., Zhang, K., Ferguson, N.S., Wang, J., and Lu, Y.M.
- Subjects
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BOUNDARY value problems , *COMPLEX variables , *DIFFERENTIAL equations , *DOMAIN decomposition methods , *MATHEMATICAL physics - Abstract
Abstract An analytical vibration response in the time domain for an axially translating and laterally vibrating string with mixed boundary conditions is considered in this paper. The domain of the string is a constant, dependent upon the general initial conditions. The translating tensioned strings possess different types of mixed boundary conditions, such as fixed_dashpot, fixed_spring-dashpot, fixed_mass-spring-dashpot. An analytical solution using a reflected wave superposition method is presented for a finite translating string. Firstly, the cycle of boundary reflection for strings is provided, which is dependent upon the string length. Each cycle is divided into three time intervals according to the travelling speed and direction of the string. Applying D'Alembert's principle and the reflection properties, expressions for the reflected waves under three different non-classical boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals by using a reflected wave superposition method. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark- β method solution. The energy expressions for a travelling string with a fixed_dashpot boundary condition is obtained and the time domain curves for the total energy and the change of energy at the boundaries are given. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
4. A new condition for the concavity method of blow-up solutions to p-Laplacian parabolic equations.
- Author
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Chung, Soon-Yeong and Choi, Min-Jun
- Subjects
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BOUNDARY value problems , *DEGENERATE parabolic equations , *COMPLEX variables , *DIFFERENTIAL equations , *MATHEMATICAL physics - Abstract
Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
5. Asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition–diffusion system.
- Author
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Du, Li-Jun, Li, Wan-Tong, and Wang, Jia-Bing
- Subjects
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FLIP-flop circuits , *DIFFERENTIAL equations , *MATHEMATICAL physics , *BERNOULLI equation , *BESSEL functions - Abstract
Abstract This paper is concerned with a time periodic competition–diffusion system { u t = u x x + u (r 1 (t) − a 1 (t) u − b 1 (t) v) , t > 0 , x ∈ R , v t = d v x x + v (r 2 (t) − a 2 (t) u − b 2 (t) v) , t > 0 , x ∈ R , where u (t , x) and v (t , x) denote the densities of two competing species, d > 0 is some constant, r i (t) , a i (t) and b i (t) are T -periodic continuous functions. Under suitable conditions, it has been confirmed by Bao and Wang (2013) [2] that this system admits periodic traveling fronts connecting two stable semi-trivial T -periodic solutions (p (t) , 0) and (0 , q (t)) associated to the corresponding kinetic system. In the present work, we first investigate the asymptotic behavior of periodic bistable traveling fronts with non-zero speeds at infinity by a dynamical approach combined with the two-sided Laplace transform method. With these asymptotic properties, we then obtain some key estimates. As a result, by applying the super- and subsolutions techniques as well as the comparison principle, we establish the existence and various qualitative properties of the so-called entire solutions defined for all time and the whole space, which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
6. The existence of solitary wave solutions of delayed Camassa–Holm equation via a geometric approach.
- Author
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Du, Zengji, Li, Ji, and Li, Xiaowan
- Subjects
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SOLITONS , *PERTURBATION theory , *APPROXIMATION theory , *DIFFERENTIAL equations , *MATHEMATICAL physics - Abstract
This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
7. On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime.
- Author
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Heitman, Zhu, Bremer, James, and Rokhlin, Vladimir
- Subjects
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OSCILLATING chemical reactions , *DIFFERENTIAL equations , *NUMERICAL analysis , *CALCULUS , *MATHEMATICAL physics - Abstract
We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate several implications of this fact. For example, that many special functions of great interest — such as the Bessel functions J ν and Y ν — can be evaluated accurately using a number of operations which is O ( 1 ) in the order ν . The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
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