scholarly journals Key Features of the Transient Amplification of Mistuned Systems

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.

scholarly journals Identification of the Essential Features of the Transient Amplification of Mistuned Systems

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.

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

2011 ◽  
Vol 22 (6) ◽  
pp. 613-629 ◽  
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.

Bicameralism in a Unitary State

2021 ◽  
pp. 329-348
Author(s):  
Mary C. Murphy
Keyword(s):  
Unitary State ◽  
Lower House ◽  
Key Features ◽  
Complex Manner

This chapter describes, explains, and analyses the Irish Senate (officially Seanad Éireann). It details the birth and evolution of the bicameral principle in Ireland and provides an account of its key features, including its vocational character and complex electoral arrangements. The chapter also outlines the role, powers, and functions of Seanad Éireann and identifies the key criticisms of this body. These include the institution’s subservience to the lower house, the complex manner in which members are elected and appointed, and the limited nature of vocational representation in the Seanad. Campaigns for reform and the 2013 abolition referendum are discussed with reference to the several reform proposals of varying depth and substance that appeared over the decades.

Shaft Instabilities in Two-Stage Gear Systems

2013 ◽  
Vol 275-277 ◽  
pp. 930-935
Author(s):  
Zhe Rao ◽  
Chun Yan Zhou
Keyword(s):  
Multiple Scale ◽  
Gear System ◽  
Dynamic Equations ◽  
Time Varying ◽  
Kutta Method ◽  
Two Stage ◽  
Scale Method ◽  
System A ◽  
Runge Kutta Method ◽  
Intermediate Shaft

The present paper is focused on the torsional instabilities of the intermediate shaft in a two stage gear system. A theoretical model is established taking account in the torsional flexibility of the intermediate shaft and the meshing time-varying stiffness of the gears. Multiple scale method is applied to analysis the instability areas of the gear system for which the generalized modal coordinate is adopted. The result is certificated by numerical integrals of the dynamic equations by Runge-Kutta Method.

Natural Frequencies and Damping of a Full-Scale Pipe Loop in Air and Water

2017 ◽  
Author(s):  
Hugh Goyder
Keyword(s):  
Oil And Gas ◽  
Natural Frequencies ◽  
Damping Ratio ◽  
Full Scale ◽  
Surrounding Water ◽  
Sea Floor ◽  
Dry Conditions ◽  
Vibration Tests ◽  
Complex Manner ◽  
Submerged Conditions

A full scale pipework system, typical of oil and gas installations located on the sea floor, was subjected to vibration tests in both dry and submerged conditions. The frequency range examined covered 10 Hz to 500 Hz. The objective of the tests was to provide experimental data so that computer simulations could be developed and validated. The method used to determine the vibration properties was that of an experimental modal analysis using an impact hammer. The hammer was modified for underwater use. In dry conditions the damping was found to be very small (damping ratio less than 0.0002) despite the construction being typical. When submerged the effect of the surrounding water was significant. The changes in the natural frequencies from the dry case to the wet case occurred in such a complex manner that it was not possible to identify a simple shift between wet and dry vibration modes. It was necessary to include appropriate added mass coefficients in the computer simulation for both the pipe and the support system. The effect of the surrounding water on the damping was measured but found to be insignificant. It was concluded that immersion in water does not add significant damping to oil and gas pipework.

QUANTUM QUASI-SOLITONS IN THE TWO-DIMENSIONAL FERROMAGNETIC LATTICE WITH AN EXTERNAL FIELD

2013 ◽  
Vol 27 (09) ◽  
pp. 1350058 ◽  
Author(s):  
DE-JUN LI ◽  
BING TANG ◽  
KE HU ◽  
YI TANG
Keyword(s):  
Magnetic Field ◽  
Quantum Theory ◽  
External Magnetic Field ◽  
External Field ◽  
Multiple Scale ◽  
Two Dimensional ◽  
Scale Method ◽  
Nonlinear Excitations ◽  
Localized Mode ◽  
Intrinsic Localized Mode

Based on the quantum theory and a simplified version of the multiple-scale method, the nonlinear excitations in a two-dimensional ferromagnetic lattice with an external magnetic field are analytically investigated. The standard two-dimensional nonlinear Schrödinger equation is obtained. Results show that the quantum quasi-soliton can exist in the two-dimensional ferromagnetic lattice. In addition, when the group velocity is equal to zero, at the boundary of the Brillouin zone, the quantum quasi-soliton becomes the quantum intrinsic localized mode.

scholarly journals A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-26 ◽  
Author(s):  
M. Ilić ◽  
I. W. Turner ◽  
V. Anh
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Poisson Equation ◽  
Matrix Function ◽  
Fractional Power ◽  
Analytic Solutions ◽  
Linear System Of Equations ◽  
Exact Analytic Solutions ◽  
The Matrix ◽  
Symmetric Positive Definite Matrices

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

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