Complex Mode Shapes Research Articles

Inference of dominant modes for linear stochastic processes.

For dynamical systems that can be modelled as asymptotically stable linear systems forced by Gaussian noise, this paper develops methods to infer (estimate) their dominant modes from observations in real time. The modes can be real or complex. For a real mode (monotone decay), the goal is to infer its damping rate and mode shape. For a complex mode (oscillatory decay), the goal is to infer its frequency, damping rate and (complex) mode shape. Their amplitudes and correlations are encoded in a mode covariance matrix that is also to be inferred. The work is motivated and illustrated by the problem of detection of oscillations in power flow in AC electrical networks. Suggestions of some other applications are given.

Output only system identification using complex wavelet modified second order blind identification method - A time-frequency domain approach

This paper reviewed a few output-only system identification algorithms and identified the shortcomings of those popular blind source separation methods. To address the issues such as less sensors than the targeted modal modes (under-determinate problem), repeated natural frequencies as well as systems with complex mode shapes, this paper proposed a complex wavelet modified second order blind identification method (CWMSOBI) by transforming the time domain problem into time-frequency domain. The wavelet coefficients with different dominant frequencies can be used to address the under-determinate problem, while complex mode shapes are addressed by introducing the complex wavelet transformation. Numerical simulations with both high and low signal-to-noise ratios validate that CWMSOBI can overcome the above-mentioned issues while obtaining more accurate identified results than other blind identification methods.

Uncertainty quantification for the Modal Phase Collinearity of complex mode shapes

The Modal Phase Collinearity (MPC) is a modal indicator designed to decide whether the mode shape used in its computation is a real or complex-valued vector. Its estimate inherits the statistical properties of the corresponding mode shape estimate. While the statistical framework for the uncertainty quantification of modal parameters is well-known and developed in the context of subspace-based system identification methods, uncertainty quantification for the MPC estimate has not been carried out yet. In this paper, the uncertainty quantification of the MPC estimates is developed when the corresponding mode shapes are complex-valued vectors. In this case, the theoretical value of the MPC is strictly lower than 1 and it is shown that the distribution of the MPC estimate can be approximated as Gaussian. The computation of its variance and the resulting confidence intervals of the MPC estimate are developed. The proposed framework is validated in Monte Carlo simulations and illustrated on experimental data of an offshore structure.

A coupled efficient layerwise finite element model for free vibration analysis of smart piezo-bonded laminated shells featuring delaminations and transducer debonding

This paper presents a computationally efficient multifield finite element (FE) model for accurate analysis of smart composite and sandwich shells equipped with piezoelectric patch actuators and sensors, featuring multiple delaminations and transducer debonding. It employs a facet shell element with four physical nodes and one electric node, based on the third order zigzag theory approximations for displacements and a piecewise quadratic through-thickness variation for the electric potential. A hybrid point-least squares method recently proposed by the authors is extended to impose the condition of continuity of the nonlinear displacement field at the delamination, debonding, and patch fronts. The methodology is generic for multiple patch transducers, delaminations, and debonding, occurring at any arbitrary interface and in-plane locations. The model shows excellent accuracy in comparison with the coupled full-field 3D FE solutions, for the natural frequencies and complex global, local and hybrid mode shapes of delaminated smart composite shells and soft-core sandwich plates, with partially debonded actuators and sensors. The developed model will be of immense use for the design of active vibration control and structural health monitoring strategies for smart laminated shell-type structures featuring such damages.

Dynamic damage identification using complex mode shapes

Dynamic damage identification using complex mode shapes

High Frequency Modal Testing of the Multiblade Packets Using a Noncontact Measurement and Excitation System

High cycle failure of blades and vanes caused by the vibration is one of the major causes reducing the lifetime of turbomachines. For multiblade packets, the failure may occur at vibrations with high frequencies that can reach up to tens of kHz. The experimental modal testing of blades is crucial for the validation of numerical models and for the optimization of turbomachine design. In this paper, the test rig and procedure for measurements of dynamic characteristics of lightweight multiblade packets in wide and high frequency ranges are developed. The measurements are based on a noncontact excitation and noncontact measurement method, which allows the determination of the modal characteristics of the packets with high accuracy in wide frequency ranges. The responses of the multiblade packets are measured using a Scanning Laser Doppler Vibrometry (SLDV), while vibrations are excited by the acoustic excitation technique. Modal tests of the blade packet comprising 18 vane blades connected by shrouds are performed. The measurements are performed within the high frequency range of 0–30 kHz, and the natural frequencies and mode shapes are obtained for first 97 modes. To capture the complex high frequency blade mode shapes, each blade in the packet is scanned over 25 reference points uniformly distributed over the blade concave surface. In order to obtain the high frequency resolution, the frequency range used for the measurements is split into several frequency intervals accordingly to the number of spectral lines available in the used data acquisition system, and for each such interval, the test is performed separately. The finite model of the packet is created, and the numerical modal analysis is performed to compare the calculated natural frequencies and mode shapes with the experimental measurements. The comparison shows the satisfactory with those from finite element analysis. It illustrates the measurement method described in this work is effective and reliable.

Payload Parameter Identification of a Flexible Space Manipulator System via Complex Eigenvalue Estimation

Manipulator systems are widely used in payload capture and movement in the ground/space operation due to their dexterous manipulation capability. In this study, a method for identifying the payload parameters of a flexible space manipulator using the estimated system of complex eigenvalue matrix is proposed. The original nonlinear dynamic model of the manipulator is linearized at a selected working point. Subsequently, the system state-space model and corresponding complex eigenvalue parameters are determined by the observer/Kalman filter identification algorithm using the torque input signal of the motor and the vibration output signals of the link. Therefore, the inertia parameters of the payload, that is, the mass and the moment of inertia, can be derived from the identified complex eigenvalue system and mode shapes by solving a least-squares problem. In numerical simulations, the proposed parameter identification method is implemented and compared with the classical recursive least-squares and affine projection sign algorithms. Numerical results demonstrate that the proposed method can effectively estimate the payload parameters with satisfactory accuracy.

A Modal Perturbation Method for Eigenvalue Problem of Non-Proportionally Damped System

The non-proportionally damped system is very common in practical engineering structures. The dynamic equations for these systems, in which the damping matrices are coupled, are very time consuming to solve. In this paper, a modal perturbation method is proposed, which only requires the first few lower real mode shapes of a corresponding undamped system to obtain the complex mode shapes of non-proportionally damped system. In this method, an equivalent proportionally damped system is constructed by taking the real mode shapes of a corresponding undamped system and then transforming the characteristic equation of state space into a set of nonlinear algebraic equations by using the vibration modes of an equivalent proportionally damped system. Two numerical examples are used to illustrate the validity and accuracy of the proposed modal perturbation method. The numerical results show that: (1) with the increase of vibration modes of the corresponding undamped system, the eigenvalues and eigenvectors monotonically converge to exact solutions; (2) the accuracy of the proposed method is significantly higher than the first-order perturbation method and proportional damping method. The calculation time of the proposed method is shorter than the state space method; (3) the method is particularly suitable for finding a few individual orders of frequency and mode of a system with highly non-proportional damping.

A Bayesian model updating with incomplete complex modal data

Abstract Finite element (FE) model updating using measured dynamic data has gained much attention in view of its contribution towards important areas like model-correction, structural health monitoring. Usually model updating is observed to be performed using measured modal data like modal frequencies and mode shapes – indeed very few works are observed to provide scope for using damping measurements in FE model updating. The present work proposes a Bayesian probabilistic FE model updating using multiple sets of complex modal data (viz. complex modal frequencies and complex mode shapes) and thereby enabling the scope for incorporating damping measurements in updating. Moreover there remains the opportunity for using incomplete complex modal data and avoiding the requirement of mode matching. In this paper, a computationally efficient model updating approach based on maximum a posteriori (MAP) is formulated where all the mass, stiffness and damping parameters are updated in a sequential iterative manner. Detailed formulations are obtained considering damping to be contributed by classical damping (Rayleigh damping) and non-classical viscous damping. Besides, formulations for uncertainty estimation and probabilistic damage detection are also developed. The proposed approach is validated using two numerical examples while utilizing incomplete complex modal data. Performance in updating is evaluated for both undamaged and damaged cases. The proposed approach is observed to be successful for updating of FE model as well as for detection of changes/damages in probabilistic manner, while being computationally efficient as well.

Analysis of Aeolian Vibrations of Transmission Line Conductors and Extraction of Damper Optimal Placement with a Comprehensive Methodology

Energy balance method is an effective and simple method which is used in the amplitude calculation of Aeolian vibration in transmission lines with Stockbridge damper. However, the accuracy of the results obtained by this method, heavily depends on the assumed mode shapes of the conductor vibration. In this study, by considering an appropriate model for the conductor vibration, a comprehensive methodology is presented to calculate the steady-state vibration amplitude of a conductor with arbitrary number of dampers. In this proposed method, the effects of traveling waves, variations of amplitude and phase, boundary conditions (finite length of the conductor), as well as the effect of number, location and impedance of the dampers are taken into account. Natural frequencies, damping rates and complex mode shapes are also obtained from forming and solving the nonlinear eigenvalue problem. Using this method, the effects of the damper placement on the vibration amplitude and bending strain are examined to achieve an optimum damper location. The comparison of the obtained values shows that considering the above parameters has a significant effect on the accuracy of the results.

Coupled Bending–Bending–Axial–Torsional Vibrations of Rotating Blades

In this paper, the coupled bending–bending–axial–torsional free vibrations of rotating blades are investigated based on the Euler–Bernoulli beam model. The coupled partial differential equations governing flapwise, edgewise, axial and torsional motions are derived by the Hamilton’s principle, wherein three types of velocity-dependent terms, namely static centrifugal terms, dynamic centrifugal terms and gyroscopic coupling terms, are focused. The ordinary differential equations are acquired by the Galerkin truncation, and the natural frequencies in all directions and complex mode shapes of the rotating blades are analyzed in detail. It is revealed that the three types of velocity-dependent terms have different effects on the natural frequencies. The natural frequencies are noticeably dependent on the rotating speed and preset angle, except for the axial vibration, which is almost immune to the preset angle. The complex modal motions are displayed by a series of positions of the central line and free-end cross section for different time instants, showing the coupled vibrations among different directions.

Vibration response of multi-span fluid-conveying pipe with multiple accessories under complex boundary conditions

Realistic multi-span fluid-conveying pipe may contain various accessories such as valves, clamps, flanges, elastic supports and vibration absorbers under complex boundary conditions in engineering applications. The dynamic response of the multi-span pipe may be affected by the presence of accessories, giving rise to complex mode shapes. Simplified and reliable methods for multi-span mode shapes calculation from eigenvector of the characteristic equation are widely applied in the pipeline engineering community. However, current methods are not valid when it comes to the amplitude of the eigenvector with a resonance frequency. Consequently, corresponding stresses cannot be further evaluated exactly. To address the above mentioned issues, a novel Frequency Response Function (FRF)-based method is proposed in this paper for the calculation of the mode shapes exactly. Furthermore, a method combining the Spectral Analysis Method (SAM) and Transfer Matrix Method (TMM) is first proposed in this paper to obtain the natural frequencies and transient response for the cascaded pipeline. The results calculated by the present method are validated by comparing them with those obtained from existing literature and conventional Finite Element Method (FEM). The effects of the accessories on the vibration characteristics of the multi-span pipes are further analyzed.

A Benchmark Study of Modeling Lamb Wave Scattering by a Through Hole Using a Time-Domain Spectral Element Method

Ultrasonic guided waves (GWs) are being extensively investigated and applied to nondestructive evaluation and structural health monitoring. Guided waves are, under most circumstances, excited in a frequency range up to several hundred kilohertz or megahertz for detecting defect/damage effectively. In this regard, numerical simulation using finite element analysis (FEA) offers a powerful tool to study the interaction between wave and defect/damage. Nevertheless, the simulation, based on linear/quadratic interpolation, may be inaccurate to depict the complex wave mode shape. Moreover, the mass lumping technique used in FEA for diagonalizing mass matrix in the explicit time integration may also undermine the calculation accuracy. In recognition of this, a time domain spectral element method (SEM)—a high-order FEA with Gauss–Lobatto–Legendre (GLL) node distribution and Lobatto quadrature algorithm—is studied to accurately model wave propagation. To start with, a simplified two-dimensional (2D) plane strain model of Lamb wave propagation is developed using SEM. The group velocity of the fundamental antisymmetric mode (A0) is extracted as indicator of accuracy, where SEM exhibits a trend of quick convergence rate and high calculation accuracy (0.03% error). A benchmark study of calculation accuracy and efficiency using SEM is accomplished. To further extend SEM-based simulation to interpret wave propagation in structures of complex geometry, a three-dimensional (3D) SEM model with arbitrary in-plane geometry is developed. Three-dimensional numerical simulation is conducted in which the scattering of A0 mode by a through hole is interrogated, showing a good match with experimental and analytical results.

Galloping Analysis of a Stay Cable with an Attached Viscous Damper Considering Complex Modes

AbstractThe use of viscous dampers to mitigate cable vibrations on cable-stayed bridges is very popular. A viscous damper attached to a stay cable results in complex mode shapes. Such complexity co...

Transverse vibration and instability of axially travelling web subjected to non-homogeneous tension

In many engineering applications of roll-to-roll manufacturing, the inhomogeneity in the tension profile applied at both the ends of web is apparent, which cannot be completely ignored in the modeling so as to accurately predict the dynamic behaviors of the travelling web. In this paper, transverse vibration and instability of axially travelling web subjected to parabolic tension profile are investigated. The governing equations of motion are derived by employing the Hamilton principle. The Galerkin method is used to compute the complex natural frequencies and mode shapes. The instability of travelling web versus different axial velocities is studied. This paper focuses on the influence of tension inhomogeneity on the critical velocity and mode shape. The sensitivity of aspect ratio and average tension to the tension inhomogeneity are also discussed. The critical velocities of the travelling web may be greatly decreased because of the tension inhomogeneity and a small tension inhomogeneity is enough to change the mode shapes completely.

A new Gibbs sampling based algorithm for Bayesian model updating with incomplete complex modal data

Model updating using measured system dynamic response has a wide range of applications in system response evaluation and control, health monitoring, or reliability and risk assessment. In this paper, we are interested in model updating of a linear dynamic system with non-classical damping based on incomplete modal data including modal frequencies, damping ratios and partial complex mode shapes of some of the dominant modes. In the proposed algorithm, the identification model is based on a linear structural model where the mass and stiffness matrix are represented as a linear sum of contribution of the corresponding mass and stiffness matrices from the individual prescribed substructures, and the damping matrix is represented as a sum of individual substructures in the case of viscous damping, in terms of mass and stiffness matrices in the case of Rayleigh damping or a combination of the former. To quantify the uncertainties and plausibility of the model parameters, a Bayesian approach is developed. A new Gibbs-sampling based algorithm is proposed that allows for an efficient update of the probability distribution of the model parameters. In addition to the model parameters, the probability distribution of complete mode shapes is also updated. Convergence issues and numerical issues arising in the case of high-dimensionality of the problem are addressed and solutions to tackle these problems are proposed. The effectiveness and efficiency of the proposed method are illustrated by numerical examples with complex modes.

Interpretation and generalization of complexity pursuit for the blind separation of modal contributions

Complexity pursuit (CP) has recently been proposed as an elegant and simple solution to blindly (i.e. without measuring the inputs) separate the modal contributions in the vibration responses of a structure. This potentially finds considerable interest in operational modal analysis and related applications. This paper analyses the theoretical ins and outs of the method. It also revises its physical interpretation in the modal analysis context. CP is found to separate components which are the least dispersive (i.e. invariant under linear filtering), a property that well characterizes the modal responses of lightly damped systems. However, it is also found to suffer from the same limitations as other blind source separation methods used in the state-of-the-art, namely the difficulty to separate strongly coupled modes and to identify complex mode shapes. Finally a generalization of CP is proposed which intends to widen its applicability. Interestingly, the generalized CreP happens to include the well-known SOBI algorithm as a particular case.

Research of complex modal parameters extraction of a multi-degree-of-freedom structure based on similarity search

We present a method for the extraction of modal parameters that could be useful for the parameter identification of complex structural dynamic modes of multi degree of freedom systems that may be subject to environmental load excitations. The approach is linear and in this sense it operates on the basis of the principles suggested by the so-called Rayleigh superposition method. This atoms which have the form of free vibration are constructed. The method combined with MP (Matching Putsuit) and GA (Genetic Algorithm). MP is borrowed to extract every most similar atom to the random decrement of the structure response to identify modal parameters. GA is used to speed up this process. The method is applied on the analysis of model experiments for an FPSO single point mooring system and results are compared against real monitoring data. It is shown that the method is capable of extracting successfully modal characteristics such as complex modal frequency, damping ratio and mode shapes.

Complex modes in damped sandwich beams using beam and elasticity theories

We investigated complex damped modes in beams in the presence of a viscoelastic layer sandwiched between two elastic layers. The problem was solved using two approaches, (1) Rayleigh beam theory and analyzed using the Ritz method, and (2) by using 2D plane stress elasticity based finite-element method. The damping in the layers was modeled using the complex modulus. Simply-supported, cantilever, and viscously supported boundary conditions were considered in this study. Simple trigonometric functions were used as admissible functions in the Ritz method. The key idea behind sandwich structure is to increase damping in a beam as affected by the presence of a highly-damped core layer vibrating mainly in shear. Different assumptions are utilized in the literature, to model shear deformation in the core layer. In this manuscript, we used FEM without any kinematic assumptions for the transverse shear in both the core and elastic layers. Moreover, numerical examples were studied, where the base and constraining layers were also damped. The loss factor was calculated by modal strain energy method, and by solving a complex eigenvalue problem. The efficiency of the modal strain energy method was tested for different loss factors in the core layer. Complex mode shapes of the beam were also examined in the study, and a comparison was made between viscoelastically and viscously damped structures. The numerical results were compared with those available in the literature, and the results were found to be satisfactory.

Sound Radiation Properties of Complex Modes in Rectangular Plates: A Numerical Study

The sound radiation properties of real normal modes are often used to characterise acoustic sources such as vibrating plates. The assumption of homogeneously distributed structural damping is usually made and in consequence the eigenmodes are real. With the increasing use of high-performance composite structures and the demanding requirements for acoustic comfort, damping treatments such as constrained layer damping or embedded elastomer layers may be distributed locally at some critical locations of the vibrating structure. This violates the assumption of homogeneously distributed structural damping and results in complex mode shapes. In this case, dynamics of the vibrating systems within the resonances is no longer dominated by pure standing waves, but also by a superposition with travelling waves. This paper presents the results of a numerical study on sound radiation properties of this phenom-enon, generally known as complex modes of vibration. The finite element model of an inhomoge-neously damped plate is used to generate and investigate complex vibration patterns. The level of modal complexity of the generated complex modes is characterized by the modal collinearity index and the spatial distribution of the standing wave ratio. The fluid structure interaction is implemented by a elemental radiator approach and used for the acoustic characterisation of vibration patterns in terms of far-field radiation efficiencies and spatial distributions of sound intensity. It is shown that the eigenvector complexity can seriously affect the sound radiation properties of structural modes.