Homogenization by Multiple Scale Asymptotic Expansions

2020 ◽

Author(s):

Luigi Carassale ◽

Vincent Denoël ◽

Carlos Martel ◽

Lars Panning-von Scheidt

Keyword(s):

Linear System ◽

Numerical Simulations ◽

Mechanical System ◽

Asymptotic Expansions ◽

Damping Ratio ◽

Multiple Scale ◽

Analytic Solutions ◽

Sweep Rate ◽

Scale Method ◽

Complex Manner

Abstract The dynamic behavior of bladed disks in resonance crossing has been intensively investigated in the community of turbomachinery, addressing the attention to (1) the transienttype response that appear when the resonance is crossed with a finite sweep rate and (2) the localization of the vibration in the disk due to the blade mistuning. In real conditions, the two mentioned effects coexist and can interact in a complex manner. This paper investigates the problem by means of analytic solutions obtained through asymptotic expansions, as well as numerical simulations. The mechanical system is assumed as simple as possible: a 2-dof linear system defined through the three parameters: damping ratio ξ, frequency mistuning Δ, rotor acceleration Ω˙. The analytic solutions are calculated through the multiple-scale method.

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A Survey in Mathematics for Industry: Two-timing and matched asymptotic expansions for singular perturbation problems

European Journal of Applied Mathematics ◽

10.1017/s0956792511000325 ◽

2011 ◽

Vol 22 (6) ◽

pp. 613-629 ◽

Cited By ~ 6

Author(s):

R. E. O'MALLEY ◽

E. KIRKINIS

Keyword(s):

Asymptotic Expansions ◽

Multiple Scale ◽

Singularly Perturbed ◽

Matched Asymptotic Expansions ◽

Amplitude Equations ◽

Singularly Perturbed Problems ◽

Multi Scale ◽

Scale Method ◽

Perturbation Problems ◽

Appl Math

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems.Stud. Appl. Math.124, 383–410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations.

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Key Features of the Transient Amplification of Mistuned Systems

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.4049501 ◽

2021 ◽

Author(s):

Luigi Carassale ◽

Vincent Denoel ◽

Carlos Martel ◽

Lars Panning-von Scheidt

Keyword(s):

Linear System ◽

Asymptotic Expansions ◽

Damping Ratio ◽

Multiple Scale ◽

Analytic Solutions ◽

Sweep Rate ◽

Scale Method ◽

Key Features ◽

Type Response ◽

Complex Manner

Abstract The dynamic behavior of bladed disks in resonance crossing has been intensively investigated in the community of turbomachinery, addressing the attention to (1) the transient-type response that appear when the resonance is crossed with a finite sweep rate and (2) the localization of the vibration in the disk due to the blade mistuning. In real conditions, the two mentioned effects coexist and can interact in a complex manner. This paper investigates the problem by means of analytic solutions obtained through asymptotic expansions, as well as numerical simulations. The mechanical system is assumed as simple as possible: a 2-dof linear system defined through the three parameters: damping ratio ?, frequency mistuning ?, rotor acceleration . The analytic solutions are calculated through the multiple-scale method.

Download Full-text