Erratum to: Updated numerical integration method for stability calculation of Mathieu equation with various time delays

Nonlinear Dynamics, Mar 2017

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Erratum to: Updated numerical integration method for stability calculation of Mathieu equation with various time delays

Erratum to: Updated numerical integration method for stability calculation of Mathieu equation with various time delays Xiao Jian Zhang 0 Cai Hua Xiong 0 Ye Ding 0 Han Ding 0 0 Y. Ding State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University , Shanghai 200240 , China It is assumed that A(t ) is a constant matrix on the small interval t ∈ [ti−1, ti ]. For convenience, A(t ) is replaced with A(ti ) as an approximation. For the corresponding discretized time points ti = t1 + (i − 1)h (i = 1, . . . , n + 1), x(ti ) is approximated by NewtonCotes formulas according to Ding et al. [1]. - eA(t)·(t−ξ)B j (ξ ) · x(ξ − τ j )dξ , (3) j=1 ti−1 ·x(ξ − τ j )dξ, t ∈ [ti−1, ti ]. eA(t ) · (t − ξ )B j (ξ ) x(ti ) = eA(ti )·(ti −ti−1)x(ti−1) j=1 + s ti −2ti−1 eA(ti )·(ti −ti−1)B j x(ti−1 − τ j ) Since the Newton–Cotes formula has the local truncation error O(h3), the second term of right-hand side of Eq. (5) has the local truncation error O(h3). Therefore, the discretization error of the proposed method is O(h3), which can also be verified via the convergence of critical eigenvalues. Equation (39) in Sect. 3 can be expressed in the integral form when A(t ) is a constant matrix The period interval T is also equally discretized with a time step h. At each small subinterval [ti−1, ti ], Eq. (40) is represented as It is assumed that A(t ) is a constant matrix on the small interval t ∈ [ti−1, ti ]. For simplicity, A(t ) can be replaced with A(ti ) as an approximation. At the corresponding discretized time points ti = t1 + (i − 1)h (i = 1, . . . , n + 1), we approximate x(ti ) by using Newton–Cotes formulas, leading to x(ti ) = eA(ti )·(ti −ti−1)x(ti−1) + ti−1 B(θ )x(ξ + θ )dθ dξ, t ∈ [ti−1, ti ] t=ti t=ti−1 However, it does not affect the results and conclusions of the paper. The authors would like to apologize for any inconvenience caused to the readers. 1. Ding , Y. , Zhu , L.M. , Zhang , X.J. , Ding , H. : Numerical integration method for prediction of milling stability . J. Manuf. Sci. Eng . 133 ( 3 ), 031005 ( 2011 )


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Erratum to: Updated numerical integration method for stability calculation of Mathieu equation with various time delays, Nonlinear Dynamics, 2017, DOI: 10.1007/s11071-017-3331-6