Abstract
In practice, due to the fact that the phenomenon of drawing self-excited vibration can be deemed as one of the hunting phenomena of the mechanical system, this study focuses on investigating the drawing self-excited vibration process through proposing the fractional differential equation model of hunting phenomenon of the mechanical system. The fractional Legendre functions together with their fractional differential operational matrices are used to numerically solve the model. In this way, the numerical solutions of vibration displacement of the model are obtained. At the end, the proposed model and algorithm are proved to be effective via analyzing the numerical results and phase position.
Highlights
In order to more accurately describe the whole process of drawing self-excited vibration, a fractional differential equation model is proposed, and the fractionalorder Legendre functions are utilized to obtain the numerical vibration displacement solutions
Where Dαt denotes the fractional differential operator defined in the Caputo sense [26, 27], m is the quality of the slider, c is the fractional damping coefficient, k is the spring stiffness coefficient, u0 is the displacement at drive end, u is the slider displacement, and φ(u_) is the frictional force
Suppose y(x) ∈ L2[0, h], it can be expanded in terms of the generalized fractional-order Legendre functions (GFLFs) as follows [29]:
Summary
E fractional differential equation of mechanical system hunting phenomenon is given as follows: mu€ + c Dαt u − Dαt u0 − φ(u_) + k u − u0 0, 0 < α ≤ 1, (1). V ξ √c , 2 km the sliding block motion equation can be simplified as x′′ + 2ξDαx + x −. E dimensionless dynamic friction coefficient f and dimensionless static-dynamic friction drop d are defined as follows:. Mω0v e sliding block motion equation is simplified again as x′′ + 2ξDαx + x − fsgn x′ + 1,. From equations (12) and (13), the general form of fractional differential equation model of mechanical system hunting phenomenon is established: y′′(x) + 2ξDαxy(x) + y(x) f(x), 0 < α ≤ 1, (14)
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