Nonlinear dynamics of mistuned bladed disks with ring dampers

In turbomachinery applications, rotating bladed disks (blisks) are often subject to high levels of dynamic loading, such as traveling wave excitations, which result in large response amplitudes at resonance. To prevent premature high cycle fatigue, various dry friction dampers are designed for blisk systems to reduce the forced responses. Ring dampers are located in the disk, underneath the blades, and are held in contact with the blisk by centrifugal loading. Energy is dissipated by nonlinear friction forces when relative motions between the ring damper and the blisk take place. To investigate the dynamic responses of blisk–damper systems in the presence of the nonlinear frictional contacts, conventional methods based on numerical time integration are not suitable since they are computationally expensive. This paper presents a reduced-order modeling technique to efficiently capture the nonlinear dynamic responses of the blisk–damper systems. Craig–Bampton component mode synthesis (CB-CMS) serves as the first model reduction step. A novel mode basis that mimics the contact behavior under sliding and sticking conditions is developed to further reduce the CB-CMS model while maintaining its accuracy. The resulting reduced nonlinear equations of motion are solved by a hybrid frequency/time domain (HFT) method. In the HFT method, the contact status and friction forces are determined in the time domain by a three-dimensional contact model at each contact point, whereas the reduced equations of motion are solved in the frequency domain according to a harmonic balance formulation. Moreover, to investigate the effects of blade mistuning, which can lead to drastic increase of forced responses, an extension of the reduced-order models (ROMs) is developed based on component mode mistuning. Forced responses computed by the proposed ROMs are validated for both tuned and mistuned systems. A statistical analysis is performed to study the effectiveness of ring dampers under random blade mistuning patterns.     

Convergence predictions for aeroelastic calculations of tuned and mistuned bladed disks

Mistuning changes the dynamics of bladed disks significantly. Frequency domain methods for predicting the dynamics of mistuned bladed disks are typically based on iterative aeroelastic calculations. Converged aerodynamic stiffness matrices are required for accurate aeroelastic results of eigenvalue and forced response problems. The tremendous computation time needed for each aerodynamic iteration would greatly benefit from a fast method of predicting the number of iterations needed for converged results. A new hybrid technique is proposed to predict the convergence history based on several critical ratios and by approximating as linear the relation between the aerodynamic force and the complex frequencies (eigenvalues) of the system. The new technique is hybrid in that it uses a combined theoretical and stochastic/computational approach. The dynamics of an industrial bladed disk is investigated, and the predicted convergence histories are shown to match the actual results very well. Monte Carlo simulations using the new hybrid technique show that the aerodynamic ratio and the aerodynamic gradient ratio are the two most important factors affecting the convergence history.     

The worst response of mistuned bladed disk system using neural network and genetic algorithm

The objective of this paper is to develop an integrated approach using artificial neural networks (ANN) and genetic algorithms (GA) for predicting the worst response of mistuned bladed disk. ANN is used to predict the responses of bladed disk system which are used further in evaluation of fitness and constraint violation in GA process. A multilayer back-propagation neural network is trained with the results obtained from finite element model for different bladed disk configurations. Subsequently, GA is employed for arriving at optimum configuration of the bladed disk system by maximizing the blade responses. By integrating ANN with GA, the computational time required for obtaining optimal solution could be reduced substantially. The efficacy of this approach is demonstrated by carrying out studies on mistuned bladed disk systems for different sets of mistuning parameters, namely mistuning in modulus of elasticity and length of blades. Finally, the effect of adding shroud at the tip of blades in reducing the maximum response of the bladed disk system was investigated.     

The effect of underplatform dampers on the forced response of bladed disks by a coupled static/dynamic harmonic balance method

Friction contacts are often used in turbomachinery design as passive damping systems. In particular, underplatform dampers are mechanical devices used to decrease the vibration amplitudes of bladed disks.Numerical codes are used to optimize during designing the underplatform damper effectiveness in order to limit the resonant stress level of the blades. In such codes, the contact model plays the most relevant role in calculation of the dissipated energy at friction interfaces. One of the most important contact parameters to consider in order to calculate the forced response of blades assembly is the static normal load acting at the contact, since its value strongly affects the area of the hysteresis loop of the tangential force, and therefore the amount of dissipation.A common procedure to estimate the static normal loads acting on underplatform dampers consists in decoupling the static and the dynamic balance of the damper. A preliminary static analysis of the contact is performed in order to get the static contact/gap status to use in the calculation, assuming that it does not change when vibration occurs.In this paper, a novel approach is proposed. The static and the dynamic displacements of the system (bladed disk+underplatform dampers) are coupled together during the forced response calculation. Static loads acting at the contacts follow from static displacements and no preliminary static analysis of the system is necessary.The proposed method is applied to a numerical test case representing a simplified bladed disk with underplatform dampers. Results are compared with those obtained with the classical approach.     

An enriched multiple scales method for harmonically forced nonlinear systems

This article explores enrichment to the method of Multiple Scales, in some cases extending its applicability to periodic solutions of harmonically forced, strongly nonlinear systems. The enrichment follows from an introduced homotopy parameter in the system governing equation, which transitions it from linear to nonlinear behavior as the value varies from zero to one. This same parameter serves as a perturbation quantity in both the asymptotic expansion and the multiple time scales assumed solution form. Two prototypical nonlinear systems are explored. The first considered is a classical forced Duffing oscillator for which periodic solutions near primary resonance are analyzed, and their stability is assessed, as the strengths of the cubic term, the forcing, and a system scaling factor are increased. The second is a classical forced van der Pol oscillator for which quasiperiodic and subharmonic solutions are analyzed. For both systems, comparisons are made between solutions generated using (a) the enriched Multiple Scales approach, (b) the conventional Multiple Scales approach, and (c) numerical simulations. For the Duffing system, important qualitative and quantitative differences are noted between solutions predicted by the enriched and conventional Multiple Scales. For the van der Pol system, increased solution flexibility is noted with the enriched Multiple Scales approach, including the ability to seek subharmonic (and superharmonic) solutions not necessarily close to the linear natural frequency. In both nonlinear systems, comparisons to numerical simulations show strong agreement with results from the enriched technique, and for the Duffing case in particular, even when the system is strongly nonlinear.     

Boundary-layer receptivity to external disturbances using multiple scales

Non-homogeneous multiple scales are introduced to solve the resonant problem of non-parallel boundary-layer receptivity originating from the quadratic mixing of environmental disturbances. The resulting algorithm is computationally inexpensive and can be efficiently included in industrial codes for transition prediction. The mutual interactions between acoustic wave, vorticity wave, wall vibration and wall roughness are discussed in detail and the receptivity coefficient, which relates the amplitude of the excited wave to the amplitude of the exciting sources, is computed. The largest effect is found for the interaction between acoustic waves and wall roughness perturbations. Other coupling mechanisms are less effective. By comparing parallel and non-parallel results, it is found that flow non-parallelism can play a non-negligible role even in Blasius’ boundary layer, although the largest effects are evident for the three-dimensional boundary layer over an infinite swept wing. For the particular case of wall roughness—wall vibration mixing, the velocity disturbance is shown to be exactly equal to the velocity perturbation induced by wall roughness alone on a wall vibrating in the normal direction.     

Analysis of damped guided waves using the method of multiple scales

We analytically investigate the influence of damping on Lamb waves, which are a specific type of guided wave in two-dimensional plates. Considering material attenuation, we suppose that Lamé constants are complex numbers. This leads to the associated wavenumbers being complex, with the imaginary part of the wavenumber being associated with effect of attenuation of the guided wave. In this paper, we show how dispersion curves and attenuation coefficients can be obtained using the self-adjointness and the method of multiple scales (MMS), which is a type of perturbation method. Using the self-adjointness and the MMS, we can calculate the frequency- and wavenumber-dependent attenuation coefficients from the integral values and boundary values of a corresponding eigenfunction with respect to each propagation mode. This analytical method can yield not only dispersion curves but also mode-by-mode attenuation coefficients regardless of the numerical initial values, unlike numerical approaches using the Newton method. Thus, the proposed method can more easily calculate the attenuation coefficients with respect to a particular mode than conventional methods. Furthermore, the results obtained by proposed method were in good agreement with those obtained by the semi-analytical finite element (SAFE) method, which validates the proposed method.     

Vortex-induced vibration of pipes conveying fluid using the method of multiple scales

The nonlinear dynamics of supported pipes conveying fluid subjected to vortex-induced vibration is evaluated using the method of multiple scales. Frequency response portraits for different internal fluid velocities under lock-in conditions are obtained and the stability of steady-state responses is discussed. Results show that the internal fluid velocity has a prominent effect on the oscillation amplitude and that the steady-state responses incorporating unstable solutions in the lock-in region are also obtained. In addition, the effects of two kinds of fluctuating lift coefficients on the steady-state responses are compared with each other.     

Comparison of PIV data using multiple configurations and processing techniques

Particle image velocimetry (PIV) data have been acquired using three different experimental configurations in the far-field of the interaction created by a transverse supersonic jet exhausting from a flat plate into a transonic crossflow. The configurations included two-component PIV in the centerline streamwise plane at two overlapping stations, as well as stereoscopic PIV in both the same streamwise plane and in the crossplane. All measurement planes intersected at a common line. Data from both two-component measurement stations and the stereoscopic streamwise configuration agreed to within the estimated uncertainty, but data from the crossplane exhibited reduced velocity and turbulent stress magnitudes by a small but significant degree. Subsequent reprocessing of the data in nominally the same manner using a newer software package brought all values into close agreement with each other, but produced substantially higher turbulent stresses. The error source associated with the choice of software was traced to the use of image deformation in the newer software to treat velocity gradients, which is shown by synthetic PIV tests to yield a more accurate result for turbulence measurements even for gradients within the recommended limits for classical PIV. These detailed comparisons of replicate data suggest that routine methods of uncertainty quantification used for a turbulent PIV experiment may not fully capture the actual error sources.     

The forced response of the lamped dynamic systems

This paper deals with the forced harmonie responses of the discrete or continuous damped systems to harmonic excitation, where the viscous damping matrix cannot be diagonalized. The explicit expressions of the response solutions are given. Hence, with these expressions, some general and analytical study of some phenomena in vibration is made. For example, the fixed amplitude point phenomena in the single damped systems have been demonstrated generally. The conditions, under which all forces that exert on the system and possess common phase, which will excite common phase responses, are also discussed.The solutions deduced here only involve the inverse matrices of lower order. Thus, in the numerical computation for digital computers, the method is more simple, economical, and accurate than others.The method described here can be used in the analysis of unbalance responses of the rotor systems.     

On the transient response of forced nonlinear oscillators

We consider the transient response of a prototypical nonlinear oscillator modeled by the Duffing equation subjected to near resonant harmonic excitation. Of interest here is the overshoot problem that arises when the system is undergoing free motion and is suddenly subjected to harmonic excitation with a near resonant frequency, which leads to a beating type of transient response during the transition to steady state. In some design applications, it is valuable to know the peak value of this response and the manner in which it depends on system parameters, input parameters, and initial conditions. This nonlinear overshoot problem is addressed by considering the well-known averaged equations that describe the slowly varying amplitude and phase for both transient and steady state responses. For the undamped system, we show how the problem can be reduced to a single parameter χ that combines the frequency detuning, force amplitude, and strength of nonlinearity. We derive an explicit expression for the overshoot in terms of χ, describe how one can estimate corrections for light damping, and verify the results by simulations. For zero damping, the overshoot approximation is given by a root of a quartic equation that depends solely on χ, yielding a simple bound for the overshoot of lightly damped systems.     

On forced response of polygonal plates

Summary An approximate solution is presented for the linear isothermal response of viscously damped thin homogeneous polygonal plates subjected to uniform dynamic loading. Normal modes of vibration are determined with the aid of a boundary collocation technique and forced solutions are calculated by employing a numerical integration procedure. The solution is applicable to elastically built-in plates having edge conditions ranging from the simply supported to the clamped case. Numerical solutions are given for the normal modes of vibration of some simply supported and clamped plates and theoretical and experimental transverse displacements and stress couples are compared for a plate subjected to blast loading.

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Übersicht Es wird eine Näherungslösung für die Bewegung von homogenen, polygonalen Platten unter gleichförmiger dynamischer Belastung angegeben. Dabei wird Linearität, viskose Dämpfung und isothermes Verhalten vorausgesetzt. Die Eigenschwingungsformen werden mit einer Kollokationsmethode berechnet, während die erzwungenen Schwingungen durch numerische Integration bestimmt werden. Die Lösungen können für elastisch eingespannte Platten verwendet werden, deren Randbedingungen von der einfachen Auflage bis zur festen Einspannung reichen. Für einige aufgelagerte und eingespannte Platten werden numerische Lösungen für die Eigenschwingungsformen angegeben. Für eine Platte mit Stoßbelastung werden theoretische und experimentelle Werte für Verschiebungen und Momente verglichen.

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This work was carried out at the Defence Research Establishment Suffield, Alberta as well as at the University of Alberta, Edmonton, Alberta and permission to publish the results is gratefully acknowledged.     

Bounds on the forced response of damped linear systems

Bounds are obtained on the magnitude of the displacement vector representing the response of a damped linear mechanical system to an oscillatory forcing function. Several restrictions used in previous treatments are avoided, particularly to the effect that the system modes must satisfy a decoupling condition.     

Dynamic responses of functionally graded magneto-electro-elastic shells with closed-circuit surface conditions using the method of multiple scales

A three-dimensional (3D) free vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with closed-circuit surface conditions is presented using the method of perturbation. By means of the direct elimination, we firstly reduce the twenty-nine basic equations of 3D magneto-electro-elasticity to ten differential equations in terms of ten primary variables of magnetic, electric and elastic fields. The method of multiple scales is introduced to eliminate the secular terms in various order problems of the present formulation so that the present asymptotic expansion to the primary field variables leads to be uniform and feasible. Through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of governing equations for various order problems. The coupled classical shell theory (CST) is derived as a first-order approximation to the 3D magneto-electro-elasticity. Higher-order modifications can be further determined by considering the solvability and orthonormality conditions in a systematic and consistent way. Some benchmark solutions for the free vibration analysis of FG elastic and piezoelectric plates are used to validate the performance of the present asymptotic formulation. The influence of the material-property gradient index on the natural frequencies and corresponding modal field variables of the FG shells is mainly concerned.     

On the multiple scales perturbation method for difference equations

In the classical multiple scales perturbation method for ordinary difference equations (O Δ Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Δ E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt–Miranker method indeed reflect the behavior of the exact solutions of the O Δ Es. For that reason, a version of the multiple scales perturbation method for O Δ Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Δ Es and to singularly perturbed, linear O Δ Es.     

Particular aspects of the forced response of finite-dimensional modular structures

Summary The paper analyzes, from an engineering point of view, a problem strictly connected to the Large Space Structures dynamics, precisely the dynamics of modular structures of arbitrary length, arbitrarily constrained at either end and arbitrarily subject to external loads at the boundaries of each repetitive element. In particular the attention is focused on the relation between the propagation characteristics of an indefinite structure and the forced response of a finite-dimensional one; this relation is established in closed form for a periodic beam in transverse vibration. A detailed numerical investigation on the same structure confirms the analytical results and shows the peculiar dynamic properties of finite-dimensional modular structures.

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Sommario Viene presentata una analisi ingegneristica di un problema estremamente attuale delle grandi strutture spaziali, ovvero lo studio della dinamica di strutture modulari di lunghezza arbitraria, soggette ad arbitrary vincoli alle estremità e ad arbitrari carichi esterni applicati in corrispondenza delle giunzioni tra due moduli adiacenti. In particolare l'interesse del lavoro è rivolto alla determinazione di una relazione tra le caratteristiche di propagazione di una struttura indefinita e la risposta forzata della omologa struttura di lunghezza finita; tale relazione viene ricavata in forma chiusa per una trave periodica. Una serie di simulazioni numeriche sulla medesima struttura conferma i risultati analitici, mostrando le peculiarità del comportamento dinamico di strutture periodiche di lunghezza finita.

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Paper presented as Particular Aspects of the Dynamics of Finite-Dimensional Modular Structures at the ESA Workshop on Modal Representation of Flexible Structures by Continuum Methods, ESTEC, Noordwijk, The Netherlands, June 1989.     

Free vibration and stability of an axially moving thin circular cylindrical shell using multiple scales method

This paper investigates the linear free vibration of axially moving simply supported thin circular cylindrical shells with constant and time-dependent velocity considering the effect of viscous structure damping. Classical shell theory is employed to express strain-displacement relation. Linear elasticity theory is used to write stress–strain relation considering Hook’s Law. Governing equations in cylindrical coordinates are derived using the Hamilton principle. Equilibrium equations are rewritten with the help of Donnell–Mushtari shell theory simplification assumptions. Motion equations for displacements in axial and circumferential directions are solved analytically concerning to displacement in the radial direction.As the displacement in the radial direction is the combination of driven and companion modes, the third motion equation is discretized using the Galerkin method. The set of ordinary differential equation obtained from the Galerkin method is solved using the steady-state method, which in practice leads to the prediction of the exact frequencies of vibration. By employing multiple scale method the critical speed values of a circular cylindrical shell and several types of instabilities are discussed. The numerical results show that by increasing the mean velocity, the system always loses stability by the divergence instability in different modes, and the critical speed values of lower modes are higher than those of higher modes. As well as the unstable regions for the resonances between velocity function fluctuation frequencies and the linear combination of natural frequencies is gained from the solvability condition of second order multiple scale method. The accuracy of the method is checked against the available data.

     

An application of the method of multiple scales to magnetohydrodynamic flow

In this paper we study the flow of an incompressible conducting fluid along an elliptic duct imposed in an uniform magnetic field. In case Hartmann number of the flow is sufficiently large, the method of multiple scales is adopted for constructing the asymptotic approximation of solution up to any order. Our method can also be applied to study the magnetohydrodynamic flow along any duct whose cross section has smooth boundary.     

Analysis of guided waves with a nonlinear boundary condition caused by internal resonance using the method of multiple scales

We theoretically investigated the cumulative nonlinear guided waves caused by internal resonance, using the method of multiple scales (MMS), which can construct better approximations to the solutions of perturbation problems. In this study, we consider nonlinearity only on the boundary instead of material nonlinearity or geometric nonlinearity. We showed nonlinear effects on the amplitudes of a lower mode and a higher mode depending on the propagation length. Also, we examined effects of wavenumber detuning from a phase matching condition of the two modes. If the wavenumber detuning is exactly equal to zero, the mechanical energy of the lower mode is transferred through nonlinear coupling to the energy of the higher mode, unilaterally. However, if a wavenumber detuning is not equal to zero, amplitude of the two modes change in a cyclic fashion during wave propagation. The amount of this amplitude variation and its cycle length are determined by the eigenfunctions of the two modes, the nonlinear parameter and the wavenumber detuning.