Mode Acceleration Method Research Articles

Wind-Induced Response and Equivalent Static Wind Load of Long-Span Roof Structures by Combined Ritz-Proper Orthogonal Decomposition Method

AbstractThe contribution of higher vibration modes is important to the wind-induced response of long-span roofs, which makes it difficult to estimate the wind-induced response and equivalent static wind loads accurately. The authors have studied this topic for a number of years, and some research results are presented. The Ritz-proper orthogonal decomposition method for choosing dominant modes of the wind-induced response is proposed. Eigenvectors of fluctuating wind pressure obtained from the proper orthogonal decomposition are regarded as the spatial distribution of the load to generate Ritz vectors. The background and resonant response are redefined on the basis of the modal acceleration method, and a method is proposed to calculate the correlation coefficient between the background and the resonant responses. Target responses are expressed as the linear combination of basis vectors, which are the dominant eigenmodes of wind pressure and structural vibration modes representing the background and the re...

A new methodology for analysis of equivalent static wind loads on super-large cooling towers

For tall buildings and typical long-span spatial structures, the background and resonant components of wind-induced fluctuating response should be taken into account by different calculation theories. The total fluctuating response is obtained through the square root of sum of squares (hereafter referred to as SRSS) combination (hereafter referred to as tri-component method). However, this method cannot consider the modal coupling effects of the background and resonant components, nor the coupling effects of the resonant component for super-large cooling towers. This paper presents a new approach for analyzing wind-induced responses and corresponding equivalent static wind loads (hereafter referred to as ESWLs) by a consistent coupling method (hereafter referred to as CCM) based on structural random vibration theory. Firstly, the refined definition of the cross term between background and resonant component is explained based on a mode-acceleration method, and covariance matrices of coupled elastic restoring force and resonant elastic restoring force are proposed. Secondly, based on covariance matrix theory, CCM is proposed for calculating the background and resonant components and for compensating the cross term between background and resonant components, and the ESWLs of all components are derived by load response correlation theory. Finally, calculation of wind-induced responses and ESWLs for a super-large cooling tower 215m high demonstrates the superiority and effectiveness of the present approach, the characteristics of ESWL distributions of background, resonant and cross term between background and resonant component, and the wind-induced coefficients for super-large cooling towers are extracted.

An inertia-capacitance beam substructure formulation based on the bond graph method with application to rotating beams

In this paper, a novel inertia-capacitance (IC) beam substructure formulation based on the IC-field presentation from the bond graph method is developed. The IC beam provides a modular, systematic and graphical approach to beam modeling. These features allow the modeler to focus more on the modeling and less on the mathematics. As such, the IC beam is proposed as an alternative to the many existing types of beam models available in the literature. The IC beam is formulated in the center of mass body fixed coordinate system allowing for easy interfacing in a multibody system setting. This floating frame approach is also computationally cheap. Elastic deformations in the IC beam are assumed to be small and described by modal superposition. The formulation couples rigid body and elastic deformations in a nonlinear fashion. The formulation is also compact and efficient. Detailed derivations for a two-dimensional planar IC beam with bending modes are presented. A modal acceleration method based on the decoupling of bending modes is proposed for use in the IC beam. The rotating beam spin-up maneuver problem is solved. The Karnopp–Margolis method is applied to ensure complete integral causality for an efficient numerical system. Geometric substructuring technique is applied to model large deflections. The IC beam is shown to be capable of solving the rotating beam problem accurately and efficiently.

A new modal correction method for linear structures subjected to deterministic and random loadings

In the general framework of linear structural dynamics, modal corrections methods allow improving the accuracy of the response evaluated with a reduced number of modes. Although very often neglected by researchers and practitioners, this correction is particularly important when strains and stresses are computed. Aimed at overcoming the main limitations of existing techniques, a novel dynamic modal acceleration method (DyMAM) is presented and numerically validated. The proposed correction involves a set of additional dummy oscillators, one for each dynamic loading, and can be applied, with a modest computational effort, to discrete and continuous systems under deterministic and random inputs.

Stochastic analysis of large structural systems under fully non-stationary input

Dynamic analysis of large linear systems is usually performed adopting the well-known modal analysis along with modal truncation of higher modes. However, in the case in which the contribution of higher modes is not negligible, modal correction methods have to be introduced to improve the accuracy of the dynamic response either in the case of deterministic or stochastic excitation. Aim of this paper is to propose a new computationally competitive method for the stochastic analysis of large linear system vibrating under fully non-stationary Gaussian excitations. The method is based on the extension of the mode-acceleration method and its variant, the stochastic mode-acceleration method, for the evaluation of the non-geometric spectral moments of the non-stationary response. Numerical results from the study of a large MDoF structure show the accuracy and the efficiency of the proposed technique.

Comparison on numerical solutions for mid-frequency response analysis of finite element linear systems

Mid-frequency response analysis often faces computational difficulties when a conventional modal approach is used for a finite element linear system. In this paper, the computational burden is relieved by frequency sweep algorithm or mode acceleration method for a reduced-order system constructed by algebraic substructuring, a variant of model order reduction. The two methods are compared with the help of numerical experiments and their computational complexity. As demonstrated by the finite element simulations, in which proportional damping is assumed, of a turbo-prop aircraft and a ring resonator, the frequency sweep algorithm for reduced-order systems shows the best performance among all considered numerical solutions, including the conventional approach.

A sparse preconditioned iterative method for vibration analysis of geometrically mistuned bladed disks

In this paper, the Static Mode Compensation method to predict geometrical mistuning effects on the response of bladed disks is reviewed and its limitations are analyzed. This method proved to be effective only for narrow clusters of modes under localized low rank perturbation. A new algorithm is introduced to address its deficiencies that draws on optimal preconditioned methods for generalized eigenvalue problem featuring sparse matrix vector multiplications, being more attractive under limited memory constraints of multi-millon DOF FEM models. The central idea of the SMC is to correct nominal eigenspace using modal acceleration method. It has been extended here by replacing the quasi-static set of modes with inexact solutions of the linear Jacobi–Davidson correction equations. Some heuristic strategies are discussed to lower the computational effort given the block-circulant structure of the nominal system. Numerical experiments on an industrial scale bladed disk model show that this leads to a very competitive tool. Computational performance and accuracy of both methods is compared in two areas of spectrum. The study demonstrates low accuracy of SMC method in the modal interaction zone, while validating efficiency and accuracy of the new algorithm in both areas.

A comparative study of “missing mass” correction methods for response spectrum method of seismic analysis

The techniques that have been developed to account for the effect of “missing mass” contained in the uncalculated high frequency modes for the response spectrum method of seismic analysis, viz, static correction method, mode acceleration method and residual mode method are reviewed in detail. The rationale for using these methods is examined and the merits and demerits are discussed. The methods are compared, with the help of numerical examples by evaluating the response of structures for El Centro (1940) earthquake. “Missing mass” correction using residual mode method is found to be superior to other methods for the seismic analysis of structures using response spectrum method.

Passive control of wind turbine vibrations including blade/tower interaction and rotationally sampled turbulence

AbstractThis paper investigates the use of a passive control device, namely, a tuned mass damper (TMD), for the mitigation of vibrations due to the along‐wind forced vibration response of a simplified wind turbine. The wind turbine assembly consists of three rotating uniform rotor blades connected to the top of a flexible uniform annular tower, constituting a multi‐body dynamic system. First, the free vibration properties of the tower and rotating blades are each obtained separately using a discrete parameter approach, with those of the tower including the presence of a rigid mass at the top, representing the nacelle, and those of the blade including the effects of centrifugal stiffening due to blade rotation and self‐weight. Drag‐based loading is assumed to act on the rotating blades, in which the phenomenon of rotationally sampled wind turbulence is included. Blade response time histories are obtained using the mode acceleration method, allowing base shear forces due to flapping motion for the three blades to be calculated. The resultant base shear is imparted into the top of the tower. Wind drag loading on the tower is also considered, and includes Davenport‐type spatial coherence information. The tower/nacelle is then coupled with the rotating blades by combining their equations of motion. A TMD is placed at the top of the tower, and when added to the formulation, a Fourier transform approach allows for the solution of the displacement at the top of the tower under compatibility of response conditions. An inverse Fourier transform of this frequency domain response yields the response time history of the coupled blades/tower/damper system. A numerical example is included to qualitatively investigate the influence of the damper. Copyright © 2007 John Wiley & Sons, Ltd.

Adaptive mode superposition and acceleration technique with application to frequency response function and its sensitivity

An adaptive mode superposition and acceleration technique (AMSAT) is proposed and implemented into the computation of frequency response functions (FRFs) and their sensitivities. Based on the mode superposition and mode acceleration methods for the FRFs, m-version, s-version, and ms-version adaptive schemes are presented. In these schemes, the error resulted from the mode truncation and/or series truncation is, at first, estimated at every specific frequency, respectively. Then, one more mode (called m-version), or one more level of the series (called s-version), or the combination (called ms-version) is included in the computation of the FRF when its error is greater than the error tolerance. The new FRF is recalculated and its error is re-evaluated. This procedure is repeated until all the errors fall below the specified value. Although only the implementation of FRFs and their sensitivities is demonstrated in this paper, the proposed adaptive technique may be applied to the computation of dynamic responses in time domain and their sensitivities, sensitivity of eigenpairs, modal energy, etc. One numerical example is included to demonstrate the application of the proposed adaptive schemes. The results show that the present schemes work very well. The s- and ms-version adaptive schemes are much more efficient than m-version scheme. Since the intention of this paper is to propose these new procedures, the damping, particularly the non-classical damping, is not included due to the complexity.

Design-Oriented Structural Model Order Reduction for Strain-Actuated Flight Vehicle Structures

This paper assesses the accuracy of a number of structural-model order-reduction techniques as applied to strain-actuated (smart) flight-vehicle structures modeled with finite elements (FE). Model order reduction is an essential facet when working in a design-oriented optimization environment. Therefore, to eventually build a design-oriented multidisciplinary optimization (MDO) capability for large-scale strain-actuated structures (e.g., aircraft, spacecraft), insight into the affects of model order reduction on the accuracy of design constraints and sensitivities must first be gained. In this paper reduced-order models are solved using mode displacement (MD) and mode acceleration (MA) methods. Furthermore, the effect of the chosen reduction bases on model accuracy is also investigated. Specifically, the total number of basis modes used is varied. Fictitious masses (FMs) may be added to areas of the structure where local effects predominate, and mass-normalized static solution vectors (Ritz vectors) of important actuation modes may be appended to the bases. By examining the convergence of design constraints and their sensitivities for a variety of model order-reduction approaches (solution methods and reduction bases), valuable insight is gained into the technical challenges that must be addressed when creating a robust MDO capability for strain-actuated structures.

Along-wind response of a wind turbine tower with blade coupling subjected to rotationally sampled wind loading

This paper proposes an approach to investigate the along-wind forced vibration response of a wind turbine tower and rotating blades assembly subjected to rotationally sampled stationary wind loading. The wind turbine assembly consists of three rotating rotor blades connected to the top of a flexible annular tower, constituting a multi-body dynamic entity. The tower and rotating blades are each modelled as discretized multi-degree-of-freedom (MDOF) entities, allowing the free vibration characteristics of each to be obtained using a discrete parameter approach. The free vibration properties of the tower include the effect of a rigid mass at the top, representing the nacelle, and those of the blade include the effects of centrifugal stiffening due to rotation and blade gravity loadings. The blades are excited by drag force time-histories derived from discrete Fourier transform (DFT) representations of rotationally sampled wind turbulence spectra. Blade response time-histories are obtained using the mode acceleration method, which allows for the quantification of base shear forces due to flapping for the three blades to be obtained. This resultant base shear is imparted into the top of the tower. Wind drag loading on the tower is also considered, with a series of spatially correlated nodal force time-histories being derived using DFTs of wind force spectra. The tower/nacelle is then coupled with the rotating blades by combining their equations of motion and solving for the displacement at the top of the tower under compatibility conditions in the frequency domain. An inverse Fourier transform of the frequency domain response yields the response time-history of the coupled system. The response of an equivalent system that does not consider the blade/tower interaction is also investigated, and the results are compared.

Non-stationary response of large, non-linear finite element systems under stochastic loading

A very efficient and straightforward numerical procedure for the computation of statistical second moment characteristics of large, non-linear finite element systems under stochastic loading is presented. For the modeling of both the loading and the response of the system an orthogonal series expansion of the corresponding covariance matrix, the so-called Karhunen–Loève expansion is applied, allowing to incorporate potentially available statistical data of an excitation process directly into the analysis. The non-linear equation of motion is linearized by the method of equivalent statistical linearization. According to the present capabilities of this linearization technique, one-dimensional hysteretic elements are used for modeling the non-linear system behavior. The mode acceleration method is applied in order to reduce significantly the size of the system equation and thus increasing the computational efficiency of the proposed procedure. Contrary to methodologies based on state space formulations, this procedure relies on deterministic step by step integration, implying that the dimension of the system equation is the same as in a purely deterministic analysis.

Data Recovery Methodologies for Reduced Dynamic Substructure Models with Internal Loads

An important step in the design and verification process of spacecraft structures is the coupled dynamic analysis with the launch vehicle in the low-frequency domain. To reduce the costs of computation, the spacecraft and launcher models are dynamically reduced. The recovery of physical responses from the reduced system solution by means of the mode displacement method can be improved by using the mode acceleration method, which accounts for the effect of modal truncation. A further improvement can be obtained by the application of the modal truncation augmentation method, which augments the reduction transformation matrix with a set of pseudoeigenvectors. A drawback of this method is the high number of pseudoeigenvectors for substructure models with a high number of degrees of freedom at the interface. To overcome this problem a solution is provided to reduce the number of pseudoeigenvectors. The aforementioned recovery methods will be compared with regards to the obtained accuracy, specifically for substructure models with applied loads at the internal degrees of freedom (internal loads). For this purpose, the modal truncation augmentation method is further refined by using additional pseudoeigenvectors associated with the internal loads. It is shown that the accuracy obtained with the modal truncation augmentation method is superior to the other recovery methods, whereas the recovery equations remain as simple as possible. Nomenclature B = reduced number of pseudoeigenvectors associated with interface excitation dy = y displacement dz = z displacement F = matrix of internal load vectors F =e xcitation force vector, internal load vector G =fl exibility matrix I = total number of internal degrees of freedom (DOF) J = total number of interface DOF K = stiffness matrix, number of static internal load vectors M = mass matrix N = total number of pseudoeigenvectors, J + K P = matrix of total internal load vectors, total number of

Mode Acceleration Based Random Gust Stresses in Aeroservoelastic Optimization

The present paper describes a method for obtaining accurate design-oriented stress and stress-sensitivity information from reduced-order linear-time-invariant state-space models of integrated aeroservoelastic systems, using Lyapunov’s Equation for calculating covariance matrices of the displacement and stress responses. A complete formulation of the reduced-order stress gust response problem for aeroservoelastic design synthesis, tailored toward integration with control-system design techniques based on modern control, is presented. It includes an adaptation of the mode-acceleration method, reduced-order analytic sensitivities of stress covariances, and efficient approximations to be used in a nonlinear programming/approximation concepts approach to design optimization. Ar ealistic aeroservoelastic model of a typical passenger airplane is used as a test case, and the paper includes results of convergence studies for the assessment of order-reduction effects on the accuracy of the integrated structure/aerodynamic/control models.

A convex optimization approach to the mode acceleration problem

The purpose of this note is to introduce an alternative procedure to the mode acceleration method when the underlying structure model includes damping. We will show that the problem can be cast as a convex optimization problem that can be solved via linear matrix inequalities.

모우드 가속도법의 수학적 정리(定理)

모우드 중첩법은 구조 동역학 문제의 선형 거동해를 위해서 가장 일반적으로 사용되고 있다. 이러한 모우드 중첩법의 큰 장점은 보통 저차 모우드의 작은 수 만으로 구조물의 거동해석이 충분하다는 것이다. 그러나 많은 수의 자유도를 갖는 거대 구조물에서는 수렴속도가 느리고, 정확한 모우드 중첩법이 되기 위해서는 많은 수의 모우드 수가 필요하게 된다. 모우드 중첩법의 부정확성은 사용되는 모우드 수의 절삭에 의해서 발생된다. 이러한 단점들은 모우드 가속도법에 의해서 극복될 수 있다. 조화하중을 받는 단순보에 대하여 예제해석을 수행하였으며, 두 방법에 의해서 절점 변위들의 수렴성을 비교하였다. 비교적 낮은 주파수를 갖는 하중에 대하여 모우드 가속도법은 저차 모우드 1개만으로도 좋은 결과를 얻을 수 있었으며, 이 방법은 수치해석에 있어서 더 경제적이며 또한 정확한 해가 된다. Mode superposition method(MSM) is the most commonly used for solving linear response problems of structural dynamics. The major advantage of MSM is that usually a small number of lower mode is sufficient to analysis the response. However, the convergence is slow and many modes would be needed to give an accurate MSM in large structure with many degrees of freedom. The inaccuracies of MSM are caused by mode truncation in the solution. These demerits can be overcome by use of the mode acceleration method(MAM). Example analyses are carried out in simple beam subjected to harmonic loadings and compared the convergence of the joint displacements by the two methods. For relatively low frequency loadings, a good results was obtained by the lowest one mode in MAM, so the method is more economic in numerical analysis on an accurate solution.

A CONVEX OPTIMIZATION APPROACH TO THE MODE ACCELERATION PROBLEM

The purpose of this paper is to introduce an alternative procedure to the mode acceleration method when the underlying structure model includes damping. We will show that the problem can be cast as a convex optimization problem that can be solved via linear matrix inequalities.

Quasi-static data recovery for dynamic analyses of structural systems

Data recovery is a post-processing procedure employed in dynamic analyses to recover the physical response from the modal responses obtained using the mode-displacement method. An enhanced data recovery process can improve accuracy by compensating for the modal truncation errors. The mode-acceleration method is a well-known data recovery technique. However, its domain of application is restricted to problems having a low-frequency band of interest. In this paper, new data recovery methods, the quasi-static data recovery methods, are presented. These new methods are based upon a quasi-static compensation technique that is applicable for any banded frequency domain of interest, with both high- and low-frequency modes truncated. Numerical examples are given to demonstrate the improvements on accuracy and applicability of the proposed method compared with traditional data recovery.

Improved dynamic correction method in seismic analysis of both classically and non‐classically damped structures

AbstractA step‐by‐step approximate procedure taking into consideration high‐frequency modes, usually neglected in the modal analysis of both classically and non‐classically damped structures, is presented. This procedure can be considered as an extension of traditional modal correction methods, like the mode‐acceleration method and the dynamic correction method, which are very effective for structural systems subjected to forcing functions described by analytical laws. The proposed procedure, herein called improved dynamic correction method, requires two steps. In the first step, the number of differential equations of motion are reduced and consequently solved by using the first few undamped mode‐shapes. In the second step, the errors due to modal truncation are reduced by correcting the dynamic response and solving a new set of differential equations, formally similar to the original differential equations of motion. The difference between the two groups of differential equations lies in the forcing vector, which is evaluated in such a way as to correct the effects of modal truncation on applied loads. Copyright © 2001 John Wiley & Sons, Ltd.