A high-order full-discretization method using Hermite interpolation for periodic time-delayed differential equations

2015 ◽  
Vol 31 (3) ◽  
pp. 406-415 ◽  
Author(s):  
Yilong Liu ◽  
Achim Fischer ◽  
Peter Eberhard ◽  
Baohai Wu
Keyword(s):  
Differential Equations ◽  
Hermite Interpolation ◽  
High Order ◽  
Discretization Method ◽  
Full Discretization ◽  
Delayed Differential Equations

scholarly journals Stability Analysis of Multi-Discrete Delay Milling with Helix Effects Using a General Order Full-Discretization Method Updated with a Generalized Integral Quadrature

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1003
Author(s):  
Chigbogu Ozoegwu ◽  
Peter Eberhard
Keyword(s):  
Stability Analysis ◽  
High Speed ◽  
High Order ◽  
Discretization Method ◽  
High Order Methods ◽  
Full Discretization ◽  
Low Speed ◽  
Automated Algorithm ◽  
Stability Lobes ◽  
General Order

A tensor-based general order full-discretization method is enhanced with the capacity to handle multiple discrete delays and helix effects leading to a unique automated algorithm in the stability analysis of milling process chatter. The automated algorithm is then exploited in investigating the effects of interpolation order of chatter states and helix-induced terms on the convergence of milling stability lobes. The enhanced capacity to handle the distributed helix effects is based on a general order formulation of the Newton-Coates integral quadrature method. Application to benchmark milling models showed that high order methods are necessary for convergence of the low speed domain of stability lobes while all the numerically stable orders converge in the high speed domain where the ultra-high order methods are prone to numerical instability. Also, composite numerical integration of the helix-induced integrand beyond the usual zero-th order method leads to higher accuracy of stability lobes especially in the low speed domain.

Analysis of Composite Blade/Casing Rub Stability Through Delayed Differential Equations

2019 ◽  
Author(s):  
Jiaguangyi Xiao ◽  
Yong Chen ◽  
Jie Tian ◽  
Hua Ouyang ◽  
Anjenq Wang
Keyword(s):  
Differential Equations ◽  
Experimental Investigations ◽  
Discretization Method ◽  
Reduced Order ◽  
Composite Blade ◽  
Characteristic Roots ◽  
Engine Industry ◽  
Delayed Differential Equations ◽  
Inner Surface ◽  
Aero Engine

Abstract To improve aerodynamic efficiencies, the clearances between blades and casings are becoming smaller and smaller in the aero-engine industry, which might lead to the interactions between these components. These unexpected interactions are known as the so called blade/casing rubs. Abradable materials are implemented on the inner surface of the casings to reduce the potential damages caused by it. However, failures may still arise from blade/casing rubs according to experimental investigations and actual accidents. In this paper, a reduced-order delayed differential equations are used to simplify the rubbing process between composite blade and casing. It is assumed that the removal of the abradable material in blade/casing rubbing process shares a resemblance with machine tool chatters encountered in machining. The delayed differential equations are established with centrifugal stiffness and the impacts of stacking sequences on the blade damping taking into consideration. Semi-Discretization Method (SDM) is used to study the stabilities of the simplified system, which is verified by Cluster Treatment of Characteristic Roots (CTCR) and direct integrations. The results show that the stacking sequences, rub positions, blade damping and stiffness could have much impact on the relatively dangerous interaction regimes. With the help of this method, one can assist the design processes of the composite blade-casing interface in initial aero-engine structural designs.

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

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