Harmonic Forced Response Research Articles

Rainbow gyroscopic disk metastructures for broadband vibration attenuation in rotors

In the last decade, the study of metastructures has gained increasing attention and its engineering applications for vibration control have been gaining maturity for industrial applications. Remarkably, there is a gap on the application of the former approach for rotating machines yet to be bridged. In this work, we present an investigation on the use of locally resonant metastructures for vibration attenuation on rotating machines. A simple rotor structure with periodically attached, compliantly mounted disk resonators is modelled and its dynamical behaviour is investigated. Both forced harmonic response and modal analyses are explored and a thorough evaluation on the interactions between the resonators and main structure is presented for varying conditions of rotating speed and forward and backward excitations. Periodic and rainbow type of resonators with a gradient spatial profile are considered for translational and rotational resonators. It is shown that the former does not present any gyroscopic effect while the attenuation performance of latter depends on the interaction of the forward and backward resonating modes with the excitation. In addition, the rainbow disk metastructure presented attenuation bands up to 60% larger than the periodic case with the same added mass. The results suggest that the proposed gyroscopic periodic structures can be effectively applied for new alternatives for vibration attenuation in rotors paving the way for further innovative engineering solutions on vibration control in lightweight and high speed hermetic rotating machinery.

An Efficient Approach for Predicting Resonant Response with the Utilization of the Time Transformation Method and the Harmonic Forced Response Method

Resonant response of turbomachinery blades can lead to high cycle fatigue (HCF) if the vibration amplitudes are significant. Therefore, the dangerousness assessment of the resonance crossing is important. It requires accurate predictions of the aerodynamic excitation, damping, and response, which will consume immense computational costs. The novel aspect of this study is the development of an efficient approach, which incorporates the time transformation (TT) method to predict the aerodynamic excitations and the harmonic forced response method to obtain the response levels. The efficiency and accuracy of this method were evaluated by comparing with traditional methods for the resonance crossing excited by upstream wake in a 1.5 multistage compressor. For the aerodynamic excitation, discrepancies of ±2% at the mean pressure and ±25% at the harmonic pressure in most areas expect for the blade root were observed, but the calculation time required by the TT method was only 5% of that by the time-marching method. Moreover, response levels with the same aerodynamic forces were compared between the harmonic forced-response and transient dynamic methods. Small differences in the displacement and stress variables were observed; the relative deviation was smaller than 2% with only 1% computing time compared with the transient method, indicating the high accuracy and efficiency of the efficient approach.

Aeroelastic assessment of a highly loaded high pressure compressor exposed to pressure gain combustion disturbances

A numerical aeroelastic assessment of a highly loaded high pressure compressor exposed to flow disturbances is presented in this paper. The disturbances originate from novel, inherently unsteady, pressure gain combustion processes, such as pulse detonation, shockless explosion, wave rotor or piston topping composite cycles. All these arrangements promise to reduce substantially the specific fuel consumption of present-day aeronautical engines and stationary gas turbines. However, their unsteady behavior must be further investigated to ensure the thermodynamic efficiency gain is not hindered by stage performance losses. Furthermore, blade excessive vibration (leading to high cycle fatigue) must be avoided, especially under the additional excitations frequencies from waves traveling upstream of the combustor. Two main numerical analyses are presented, contrasting undisturbed with disturbed operation of a typical industrial core compressor. The first part of the paper evaluates performance parameters for a representative blisk stage with high-accuracy 3D unsteady Reynolds-averaged Navier-Stokes computations. Isentropic efficiency as well as pressure and temperature unsteady damping are determined for a broad range of disturbances. The nonlinear harmonic balance method is used to determine the aerodynamic damping. The second part provides the aeroelastic harmonic forced response of the rotor blades, with aerodynamic damping and forcing obtained from the unsteady calculations in the first part. The influence of blade mode shapes, nodal diameters and forcing frequency matching is also examined.

Transient Forced Response Analysis of Mistuned Steam Turbine Blades During Startup and Coastdown

It is well known that the vibrational behavior of a mistuned bladed disk differs strongly from that of a tuned bladed disk. A large number of publications dealing with the dynamics of mistuned bladed disks are available in the literature. The vibrational phenomena analyzed in these publications are either forced vibrations or self-excited flutter vibrations. Nearly, all published literature on the forced vibrations of mistuned blades disks considers harmonic, i.e., steady-state, vibrations, whereas the self-excited flutter vibrations are analyzed by the evaluation of the margin against instabilities by means of a modal, or rather than eigenvalue, analysis. The transient forced response of mistuned bladed disk is not analyzed in detail so far. In this paper, a computationally efficient mechanical model of a mistuned bladed disk to compute the transient forced response is presented. This model is based on the well-known fundamental model of mistuning (FMM). With this model, the statistics of the transient forced response of a mistuned bladed disk is analyzed and compared to the results of harmonic forced response analysis.

Forced harmonic response of viscoelastic sandwich beams by a reduction method

ABSTRACTThe aim of this article is to develop a reduction method to determine the forced harmonic response of viscoelastic sandwich structures at a reasonable computational cost. The numerical resolution is based on the asymptotic numerical method. This type of problem is complex, and its number of degrees of freedom is double the number of the undamped structure, leading to a high computational time. To address the problem, three reduction methods are evaluated, which differ from the projection operator. Numerical tests have been performed in the case of cantilever sandwich beams. The comparison of the results obtained by the reduction order resolution with those given by the full order resolution shows both a good agreement and a significant reduction in computational cost.

Investigation on the forced response of a radial turbine under aerodynamic excitations

Rotor blades in a radial turbine with nozzle guide vanes typically experience harmonic aerodynamic excitations due to the rotor stator interaction. Dynamic stresses induced by the harmonic excitations can result in high cycle fatigue (HCF) of the blades. A reliable prediction method for forced response issue is essential to avoid the HCF problem. In this work, the forced response mechanisms were investigated based on a fluid structure interaction (FSI) method. Aerodynamic excitations were obtained by three-dimensional unsteady computational fluid dynamics (CFD) simulation with phase shifted periodic boundary conditions. The first two harmonic pressures were determined as the primary components of the excitation and applied to finite element (FE) model to conduct the computational structural dynamics (CSD) simulation. The computed results from the harmonic forced response analysis show good agreement with the predictions of Singh's advanced frequency evaluation (SAFE) diagram. Moreover, the mode superposition method used in FE simulation offers an efficient way to provide quantitative assessments of mode response levels and resonant strength.

Effects of Forward/Backward Whirl Mechanism on Nonlinear Normal Modes of a Rotor/Stator Rubbing System

In this paper, the effects of forward and backward whirl mechanism on the existence and the stability of multiple nonlinear normal modes (NNMs) in a four degree-of-freedom (DOF) rotor/stator rubbing system with cross-coupling stiffness and dry friction are investigated analytically. The NNMs may possess either positive or negative modal frequencies, corresponding, respectively, to the inherent motions of forward or backward whirl, and can be either stable or unstable. The relationship between the NNMs, regarding to their stability, and the forced system responses of the system is of great interest. It is found that a stable NNM corresponds to a forced harmonic response with the same whirl direction and frequency as the NNM, and an unstable NNM may still influence some forced system responses by contributing a frequency component equal to the modal frequency to the response spectrum.

New advances in the forced response computation of periodic structures using the wave finite element (WFE) method

The wave finite element (WFE) method is investigated to describe the harmonic forced response of one-dimensional periodic structures like those composed of complex substructures and encountered in engineering applications. The dynamic behavior of these periodic structures is analyzed over wide frequency bands where complex spatial dynamics, inside the substructures, are likely to occur. Within the WFE framework, the dynamic behavior of periodic structures is described in terms of numerical wave modes. Their computation follows from the consideration of the finite element model of a substructure that involves a large number of internal degrees of freedom. Some rules of thumb of the WFE method are highlighted and discussed to circumvent numerical issues like ill-conditioning and instabilities. It is shown for instance that an exact analytic relation needs to be considered to enforce the coherence between positive-going and negative-going wave modes. Besides, a strategy is proposed to interpolate the frequency response functions of periodic structures at a reduced number of discrete frequencies. This strategy is proposed to tackle the problem of large CPU times involved when the wave modes are to be computed many times. An error indicator is formulated which provides a good estimation of the level of accuracy of the interpolated solutions at intermediate points. Adaptive refinement is carried out to ensure that this error indicator remains below a certain tolerance threshold. Numerical experiments highlight the relevance of the proposed approaches.

A wave finite element‐based formulation for computing the forced response of structures involving rectangular flat shells

SUMMARYThe harmonic forced response of structures involving several noncoplanar rectangular flat shells is investigated by using the Wave Finite Element method. Such flat shells are connected along parallel edges where external excitation sources as well as mechanical impedances are likely to occur. Also, they can be connected to one or several coupling elements whose shapes and dynamics can be complex. The dynamic behavior of the connected shells is described by means of numerical wave modes traveling towards and away from the coupling interfaces. Also, the coupling elements are modeled by using the conventional finite element (FE) method. A FE mesh tying procedure between shells having incompatible meshes is considered, which uses Lagrange multipliers for expressing the coupling conditions in wave‐based form. A global wave‐based matrix formulation is proposed for computing the amplitudes of the wave modes traveling along the shells. The resulting displacement solutions are obtained by using a wave mode expansion procedure. The accuracy of the wave‐based matrix formulation is highlighted in comparison with the conventional FE method through three test cases of variable complexities. The relevance of the method for saving large CPU times is emphasized. Its efficiency is also highlighted in comparison with the component mode synthesis technique. Copyright © 2013 John Wiley & Sons, Ltd.

Effect of damping on the nonlinear modal characteristics of a piecewice-smooth system through harmonic forced response

Several engineering systems present piecewise-linear characteristics, among them, the damaged beams with breathing cracks are of particular interest. The dynamics of such systems exhibits bifurcations at internal resonances characterized by the onset of superabundant nonlinear normal modes with their individual modal shapes. In this paper, a 2-DOF system with piecewise linear stiffness, representative of a damaged system with a breathing crack, is analyzed by means of a numerical and theoretical investigation. The oscillator is forced by a harmonic base excitation and the role of damping on the modification of the nonlinear modal characteristics is investigated. The outcomes are compared with the reference behavior of the undamped system which allows for semi-analytical solution. It is found that the damping, on one hand, softens the abrupt transitions from one behavior to another, typical of undamped systems, on the other, affects the bifurcations of the nonlinear modes causing some of them to completely disappear and leaving others largely unaffected.

Assessment of third order smeared and zigzag theories for buckling and vibration of flat angle-ply hybrid piezoelectric panels

A recently developed improved third order theory (ITOT) for angle-ply hybrid piezoelectric plates in cylindrical bending is extended to include geometric non-linearity in the Von Karman sense. The transverse deflection is approximated non-uniformly to explicitly account for the transverse strain due to temperature and electric potential. The coupled non-linear equations of motion and the boundary conditions are derived using the extended Hamilton’s principle. The non-linear theory is used to obtain the buckling and free vibration response of symmetrically laminated hybrid angle-ply panels under inplane electro-thermomechanical loading. This theory and the third order zigzag theory with additional layerwise terms for inplane displacements are assessed in direct comparison with the exact 2D piezothermoelasticity solutions for forced harmonic response, buckling and free vibration response under initial inplane electro-thermomechanical loading. The comparison establishes the accuracy of the results of the zigzag theory and its superiority over the ITOT for the dynamic and buckling response of angle-ply hybrid panels.

Identification of noise modes for automobile operational modal analysis

In rotating machinery, the input excitation does not meet the requirement of operational modal analysis (OMA). In this article, an effective and improved method of distinguishing forced harmonic response from measured vibration signal is presented based on the differences of the statistical properties of a stochastic signal and a harmonic signal. It makes the OMA applicable to the situation of existing periodic excitation. In order to prove the effectiveness of the proposed method, a modal experiment of a steel plate excited by a motor is performed. Then, an OMA is performed to solve the vehicle interior noise problem. Combining the projection channel technique with the advanced method, most noise modes, especially harmonic modes are successfully separated. Precise modal parameters of the car are obtained in typical working conditions which provide guidance of the dynamic modification and show that the harmonic identification method has great practical value.

Analysis of Piezoelectric Ultrasonic Transducers Attached to Waveguides Using Waveguide Finite Elements

A finite-element modeling procedure for computing the frequency response of piezoelectric transducers attached to infinite constant cross-section waveguides, as encountered in guided wave ultrasonic inspection, is presented. Two-dimensional waveguide finite elements are used to model the waveguide. Conventional three-dimensional finite elements are used to model the piezoelectric transducer. The harmonic forced response of the waveguide is used to obtain a dynamic stiffness matrix (complex and frequency dependent), which represents the waveguide in the transducer model. The electrical and mechanical frequency response of the transducer, attached to the waveguide, can then be computed. The forces applied to the waveguide are calculated and are used to determine the amplitude of each mode excited in the waveguide. The method is highly efficient compared to time integration of a conventional finite-element model of a length of waveguide. In addition, the method provides information about each mode that is excited in the waveguide. The method is demonstrated by modeling a sandwich piezoelectric transducer exciting a waveguide of rectangular cross section, although it could be applied to more complex situations. It is expected that the modeling method will be useful during the optimization of piezoelectric transducers for exciting specific wave propagation modes in waveguides.

2D exact solutions for flat hybrid piezoelectric and magnetoelastic angle-ply panels underharmonic load

A benchmark two-dimensional (2D) exact piezoelasticity solution is presented for thesteady-state forced harmonic response of simply supported angle-ply laminated flat panelswith embedded or surface-bonded piezoelectric layers, in cylindrical bending, underelectromechanical and magneto-mechanical harmonic excitation with damping. Thevariables for each layer are expanded in Fourier series to satisfy the boundary conditions atthe simply supported ends. The governing equations reduce to ordinary differentialequations in thickness coordinates with constant coefficients. Their general solution isobtained using a state-space approach. A transfer matrix approach is presented to relateeight primary variables at the top and bottom using the electromechanical andmagneto-mechanical boundary conditions at the top and bottom of the panel, theconditions of prescribed potentials and the conditions of continuity/jump at the layerinterfaces. Results for the amplitude and phase lag of the central deflection arepresented for a hybrid piezoelastic highly inhomogeneous test panel, composite andsandwich panels, and a hybrid magnetoelastic panel. The present benchmarksolution would help assess 1D panel theories for damped response under harmonicloads.

INCLUSION OF MEASURED FREQUENCY- AND AMPLITUDE-DEPENDENT MOUNT PROPERTIES IN VEHICLE OR MACHINERY MODELS

This article proposes several new or refined analytical methods for vehicle or machinery system models that include measured dynamic stiffness of vibration isolators or mounts. Complications arising due to the spectrally varying and/or amplitude-dependent parameters are categorized, and the associated eigenvalue and frequency response problems are defined. First, the real and complex eigenvalue problems that include both viscous and visco-elastic damping models are critically examined and illustrated via examples. Second, a non-linear eigenvalue problem is formulated and the resulting eigensolutions are determined for a two-degree-of-freedom system with frequency-dependent elastic and dissipative parameters. Several approximate methods, including the modal expansion method, are also proposed to calculate the forced harmonic response, and their solution errors are assessed. Third, a quasi-linear method is applied to a 1/2 car model, using measured data of a typical hydraulic engine mount, to see the effect of excitation amplitude-dependent dynamic stiffnesses. Finally, a refined non-linear, frequency domain synthesis method is proposed that includes local non-linearities in the form of measured dynamic stiffness data. The forced harmonic response of the overall system is obtained, and comparing to the corresponding time domain method for a specific 1/4 car vehicle model validates it.

Acoustic radiation from a cylindrical shell with an internal plate

The presence of a substructure in a submerged cylindrical shell gives rise to dynamic interactions which influence the radiated acoustic field. The lowest modes of the substructure may resonate vigorously with a higher order mode of the shell. In this paper, a Lagrange's equations formulation is used to analyze the forced harmonic response of a submerged shell/substructure system. An infinitely long cylindrical shell is considered and the substructure is a plate with various types of connections to the shell. Harmonic forcing is applied to the structure and expressions are developed for the far-field radiated pressure. The frequency window method is used to simplify the analysis and to provide approximate solutions in a frequency range of interest.

A Direct Two Response Approach for Updating Analytical Dynamic Models of Structures With Emphasis on Uniqueness

An approach, based on utilizing only two sets of structural responses and the enforcement of the conditions for a unique solution, is presented for the updating of Finite Element Models. The responses required can be any two identified normal modes, any two identified complex modes, or two forced harmonic response vectors in the neighborhood of any two natural frequencies of the structure under test. The mass, stiffness, and damping matrices are interactively and simultaneously corrected in a direct noniterative procedure. A uniqueness factor is automatically computed in the procedure to indicate the correctability of the Finite Element Model under consideration. The number of measurement locations is assumed to be less than the number of degrees of freedom of the analytical model. Provisions for completing and smoothing the measured or identified responses are included to reduce the effects of measurement noise and identification error. Preliminary results on simple models are presented in support of the proposed technique.

A receptance formula for general second‐degree square lambda matrices

AbstractA computational algorithm utilizing the free vibration modes of a structure is presented for calculating receptances. The usual eigensystem computed for large structural models is incomplete; hence the receptances are approximate. The formulae developed here increased receptance accuracy compared to classical spectral representations. Receptances are used extensively in eigensolution reanalysis, design and synthesis and also for forced harmonic response studies. In these areas receptance approaches offer a popular alternative to Rayleigh–Ritz subspace type methods. Structural models represented by nonsymmetric mass, damping and stiffness matrices, which occur frequently in rotating structures, may be treated using the receptance formulae presented. The receptance matrix derived is applicable to general, second‐degree, square lambda matrices. This generalizes the receptance matrix commonly associated with matrix pencils.