Consider the scalar nonlinear delay differential equationddt[x(t)-f(t,x(t-τ))]+g(t,x(t-δ))=0,tt 0,where τ,δ(0,∞),f,gC([t 0,∞)×R,R)and xg(t,x)0 for tt 0,x∈R.The author obtains sufficient conditions for the ...Consider the scalar nonlinear delay differential equationddt[x(t)-f(t,x(t-τ))]+g(t,x(t-δ))=0,tt 0,where τ,δ(0,∞),f,gC([t 0,∞)×R,R)and xg(t,x)0 for tt 0,x∈R.The author obtains sufficient conditions for the zero solution of this equation to be uniformly stable as well as asymptotically stable.展开更多
This paper focuses on the development of an efficient semi-analytical solution of chatter stability in milling based on the spectral method for integral equations. The time-periodic dynamics of the milling process tak...This paper focuses on the development of an efficient semi-analytical solution of chatter stability in milling based on the spectral method for integral equations. The time-periodic dynamics of the milling process taking the regenerative effect into account is formulated as a delayed differential equation with time-periodic coefficients, and then reformulated as a form of integral equation. On the basis of one tooth period being divided into a series of subintervals, the barycentric Lagrange interpolation polynomials are employed to approximate the state term and the delay term in the integral equation, respectively, while the Gaussian quadrature method is utilized to approximate the integral tenn. Thereafter, the Floquet transition matrix within the tooth period is constructed to predict the chatter stability according to Floquet theory. Experimental-validated one-degree-of-freedom and two-degree-of-freedom milling examples are used to verify the proposed algorithm, and compared with existing algorithms, it has the advantages of high rate of convergence and high computational efficiency.展开更多
文摘Consider the scalar nonlinear delay differential equationddt[x(t)-f(t,x(t-τ))]+g(t,x(t-δ))=0,tt 0,where τ,δ(0,∞),f,gC([t 0,∞)×R,R)and xg(t,x)0 for tt 0,x∈R.The author obtains sufficient conditions for the zero solution of this equation to be uniformly stable as well as asymptotically stable.
基金supported by the National Key Basic Research Program (Grant No. 2011CB706804)the Science & Technology Commission of Shanghai Municipality (Grant Nos. 09QH1401500 and 10JC1408000)
文摘This paper focuses on the development of an efficient semi-analytical solution of chatter stability in milling based on the spectral method for integral equations. The time-periodic dynamics of the milling process taking the regenerative effect into account is formulated as a delayed differential equation with time-periodic coefficients, and then reformulated as a form of integral equation. On the basis of one tooth period being divided into a series of subintervals, the barycentric Lagrange interpolation polynomials are employed to approximate the state term and the delay term in the integral equation, respectively, while the Gaussian quadrature method is utilized to approximate the integral tenn. Thereafter, the Floquet transition matrix within the tooth period is constructed to predict the chatter stability according to Floquet theory. Experimental-validated one-degree-of-freedom and two-degree-of-freedom milling examples are used to verify the proposed algorithm, and compared with existing algorithms, it has the advantages of high rate of convergence and high computational efficiency.
