Reduced Order Modeling Based on Complex Nonlinear Modal Analysis and Its Application to Bladed Disks With Shroud Contact

2013 ◽  
Vol 135 (10) ◽  
Author(s):  
Malte Krack ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek ◽  
Christian Siewert ◽  
Andreas Hartung
Keyword(s):  
Limit Cycle ◽  
Large Scale ◽  
Computational Effort ◽  
Reduced Order Modeling ◽  
Forced Response ◽  
Reduced Order Model ◽  
Limit Cycle Oscillations ◽  
Order Model ◽  
Reduced Order ◽  
Modal Properties

The design of bladed disks with contact interfaces typically requires analyses of the resonant forced response and flutter-induced limit cycle oscillations. The steady-state vibration behavior can efficiently be calculated using the multiharmonic balance method. The dimension of the arising algebraic systems of equations is essentially proportional to the number of harmonics and the number of degrees of freedom (DOFs) retained in the model. Extensive parametric studies necessary, e.g., for robust design optimization are often not possible in practice due to the resulting computational effort. In this paper, a two-step nonlinear reduced order modeling approach is proposed. First, the autonomous nonlinear system is analyzed using the generalized Fourier-Galerkin method. In order to efficiently study localized nonlinearities in large-scale systems, an exact condensation approach as well as analytically calculated gradients are employed. Moreover, a continuation method is employed in order to predict nonlinear modal interactions. Modal properties such as eigenfrequency and modal damping are directly calculated with respect to the kinetic energy in the system. In a second step, a reduced order model is built based on the single nonlinear resonant mode theory. It is shown that linear damping and harmonic forcing can be superimposed. Moreover, similarity properties can be exploited to vary normal preload or gap values in contact interfaces. Thus, a large parameter space can be covered without the need for recomputation of nonlinear modal properties. The computational effort for evaluating the reduced order model is almost negligible since it contains a single DOF only, independent of the original system. The methodology is applied to both a simplified and a large-scale model of a bladed disk with shroud contact interfaces. Forced response functions, backbone curves for varying normal preload, and excitation level as well as flutter-induced limit cycle oscillations are analyzed and compared to conventional methods. The limits of the proposed methodology are indicated and discussed.

Reduced Order Modeling Based on Complex Nonlinear Modal Analysis and its Application to Bladed Disks With Shroud Contact

2013 ◽  
Author(s):  
Malte Krack ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek ◽  
Christian Siewert ◽  
Andreas Hartung
Keyword(s):  
Modal Analysis ◽  
Limit Cycle ◽  
Large Scale ◽  
Computational Effort ◽  
Reduced Order Modeling ◽  
Forced Response ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Nonlinear Modal Analysis

The design of bladed disks with contact interfaces typically requires analyses of the resonant forced response and flutter-induced limit cycle oscillations. The steady-state vibration behavior can efficiently be calculated using the Multi-Harmonic Balance method. The dimension of the arising algebraic systems of equations is essentially proportional to the number of harmonics and the number of degrees of freedom (DOFs) retained in the model. Extensive parametric studies necessary e.g. for robust design optimization are often not possible in practice due to the resulting computational effort. In this paper, a two-step nonlinear reduced order modeling approach is proposed. First, the autonomous nonlinear system is analyzed using a Complex Nonlinear Modal Analysis technique based on the work of Laxalde and Thouverez [1]. The methodology in [1] was refined by an exact condensation approach as well as analytical calculation of gradients in order to efficiently study localized nonlinearities in large-scale systems. Moreover, a continuation method was employed in order to predict nonlinear modal interactions. Modal properties such as eigenfrequency and modal damping are directly calculated with respect to the kinetic energy in the system. In a second step, a reduced order model is built based on the Single Nonlinear Resonant Mode theory. It is shown that linear damping and harmonic forcing can be superimposed. Moreover, similarity properties can be exploited to vary normal preload or gap values in contact interfaces. Thus, a large parameter space can be covered without the need for re-computation of nonlinear modal properties. The computational effort for evaluating the reduced order model is almost negligible since it contains a single DOF only, independent of the original system. The methodology is applied to both a simplified and a large-scale model of a bladed disk with shroud contact interfaces. In contrast to [1], the contact constraints account for variable normal load and lift-off in addition to dry friction. Forced response functions, backbone curves for varying normal preload and excitation level as well as flutter-induced limit cycle oscillations are analysed and compared to conventional methods. The limits of the proposed methodology are indicated and discussed.

Model Order Reduction of Transmission Line Model

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Transmission Line ◽  
Large Scale ◽  
Original System ◽  
Reduced Order Model ◽  
Benchmark Problems ◽  
Transmission Line Model ◽  
Order Model ◽  
Line Model ◽  
Reduced Order ◽  
Numerical Effort

Transmission Line model are an important role in the electrical power supply. Modeling of such system remains a challenge for simulations are necessary for designing and controlling modern power systems.In order to analyze the numerical approach for a benchmark collection Comprehensive of some needful real-world examples, which can be utilized to evaluate and compare mathematical approaches for model reduction. The approach is based on retaining the dominant modes of the system and truncation comparatively the less significant once.as the reduced order model has been derived from retaining the dominate modes of the large-scale stable system, the reduction preserves the stability. The strong demerit of the many MOR methods is that, the steady state values of the reduced order model does not match with the higher order systems. This drawback has been try to eliminated through the Different MOR method using sssMOR tools. This makes it possible for a new assessment of the error system Offered that the Observability Gramian of the original system has as soon as been thought about, an H∞ and H2 error bound can be calculated with minimal numerical effort for any minimized model attributable to The reduced order model (ROM) of a large-scale dynamical system is essential to effortlessness the study of the system utilizing approximation Algorithms. The response evaluation is considered in terms of response constraints and graphical assessments. the application of Approximation methods is offered for arising ROM of the large-scale LTI systems which consist of benchmark problems. The time response of approximated system, assessed by the proposed method, is also shown which is excellent matching of the response of original system when compared to the response of other existing approaches .

scholarly journals Order Reduction based on Coulomb’s and Franklin’s laws Algorithm

2021 ◽  
Author(s):  
Ram Kumar ◽  
Afzal Sikander
Keyword(s):  
Large Scale ◽  
Linear Time ◽  
Order Reduction ◽  
Absolute Error ◽  
Search Space ◽  
Engineering Problem ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Linear Time Invariant

Abstract The Coulomb and Franklin laws (CFL) algorithm is used to construct a lower order model of higher-order continuous time linear time-invariant (LTI) systems in this study. CFL is quite easy to implement in obtaining reduced order model of large scale system in control engineering problem as it employs the combined effect of Coulomb’s and Franklin’s laws to find the best values in search space. The unknown coefficients are obtained using the CFLA methodology, which minimises the integral square error (ISE) between the original and proposed ROMs. To achieve the reduced order model, five practical systems of different orders are considered. Finally, multiple performance indicators such as the ISE, integral of absolute error (IAE), and integral of time multiplied by absolute error were calculated to determine the efficacy of the proposed methodology. The simulation results were compared to previously published well-known research.

Accuracies of Reduced Order Models of a Bladed Rotor With Geometric Mistuning

2013 ◽  
Vol 136 (7) ◽  
Author(s):  
Yasharth Bhartiya ◽  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Domain Analysis ◽  
Forced Response ◽  
Reduced Order Model ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
Model Based ◽  
Bladed Rotor

The results from a reduced order model based on frequency mistuning are compared with those from recently developed modified modal domain analysis (MMDA). For the academic bladed rotor considered in this paper, the frequency mistuning analysis is unable to capture the effects of geometric mistuning, whereas MMDA provides accurate estimates of natural frequencies, mode shapes, and forced response.

Order Reduction of Parametrically Excited Nonlinear Systems Subjected to External Periodic Excitations

2009 ◽  
Author(s):  
Sangram Redkar ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Fundamental Solution ◽  
Invariant Manifold ◽  
Large Scale ◽  
Order Reduction ◽  
Reduced Order Model ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
And Control

In this work, some techniques for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and/or nonlinearity takes the form of quasiperiodic functions. The techniques proposed here; construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system. Specifically, two methods are outlined to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’, the second novel technique proposed here, utilizes the concept of ‘invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on invariant manifold technique yields unique ‘reducibility conditions’. If these ‘reducibility conditions’ are satisfied only then an accurate order reduction via ‘invariant manifold’ is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. One can also recover ‘resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handing systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. These methodologies are applied to a typical problem and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.

Vibration Control for Integrally Bladed Disks Using Friction Ring Dampers

2007 ◽  
Author(s):  
Denis Laxalde ◽  
Fabrice Thouverez ◽  
Jean-Pierre Lombard
Keyword(s):  
Frequency Domain ◽  
Forced Response ◽  
Reduced Order Model ◽  
Order Model ◽  
Bladed Disks ◽  
Reduced Order ◽  
Design And Optimization ◽  
Beam Elements ◽  
Parametric Studies ◽  
Frequency Domain Approach

A damping strategy for integrally bladed disks (blisks) is discussed in this paper; this involves the use of friction rings located underside the wheel of bladed disks. The forced response of the blisk with friction rings is derived in the frequency domain using a frequency domain approach known as Dynamic Lagrangian Frequency-Time method. The blisk is modeled using a reduced-order model and the rings are modeled using beam elements. The results of some numerical simulations and parametric studies are presented. The range of application of this damping device is discussed. Parametric studies are presented and allow to understand the dissipation phenomena. Finally some design and optimization guidelines are given.

LIMIT CYCLE OSCILLATION PREDICTION AND CONTROL DESIGN METHOD FOR AEROELASTIC SYSTEM BASED ON NEW NONLINEAR REDUCED ORDER MODEL

2011 ◽  
Vol 08 (01) ◽  
pp. 77-90 ◽  
Author(s):  
GANG CHEN ◽  
YUE-MING LI ◽  
GUI-RONG YAN
Keyword(s):  
Limit Cycle ◽  
Active Control ◽  
Reduced Order Model ◽  
Nonlinear Flow ◽  
Limit Cycle Oscillation ◽  
Control Law ◽  
Order Model ◽  
Reduced Order ◽  
Large Perturbation ◽  
Cycle Oscillation

When the amplitude of the oscillation of the unsteady flow is large or there is large perturbation relative to the mean background flow, the traditional proper orthogonal decomposition/reduced order model (POD/ROM) based on linearized time or frequency domain small disturbance solvers cannot capture the main nonlinear features well such as limit cycle oscillation (LCO), which is very dangerous for the structure. Therefore, the traditional linear ROMs are not good enough for limit cycles prediction and active control law design. A new nonlinear ROM based on dynamically nonlinear flow equation NPOD/ROM was investigated. The nonlinear second-order snapshot equation in time domain for POD basis construction is obtained from the Taylor series expansion of the flow solver. The simulation results indicate that the NPOD/ROM can capture LCO very well and is also very convenient for active control law design, while the traditional POD/ROM lose effectiveness.

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