Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

This paper deals with the existence of ω-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. Lnu(t)=f(t,u(t),u(t−τ)),t∈R, where Lnu(t):=u(n)(t)+∑i=0n−1aiu(i)(t), ai∈R, i=0,1,⋯,n−1, are constants, f(t,x,y):R×E×E⟶E is continuous and ω-periodic with respect to t, τ>0. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.

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

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 (DDEs) 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 DDEs are established with centrifugal stiffness and the impacts of stacking sequences on the blade damping taking into consideration. Semidiscretization 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.

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

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.

Normal form of double-Hopf singularity with 1:1 resonance for delayed differential equations

In this manuscript, we provide a framework for the double-Hopf singularity with 1:1 resonance for general delayed differential equations (DDEs). The corresponding normal form up to the third-order terms is derived. As an application of our framework, a double-Hopf singularity with 1:1 resonance for a van der Pol oscillator with delayed feedback is investigated to illustrate the theoretical results.