A reduction method for nonlinear structural dynamic analysis

A computational algorithm for predicting the nonlinear dynamic response of a structure is presented. The nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a Rayleigh-Ritz analysis. The basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing. Only a few basis updatings are required during the whole time integration.The truncation error introduced at every change of basis is pointed out as the cause for a divergence-type behaviour, and some means for eliminating it are discussed.Results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations.     

Nonlinear dynamics by mode superposition

A mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward-difference operators. The initial natural frequencies and mode shapes are found by inverse power iteration with the trial vectors for successively higher modes being swept by Gram—Schmidt orthonormalization at each iteration. The subsequent modal spectrum for nonlinear states is based upon the tangent stiffness of the structure and is calculated by a subspace iteration procedure that involves matrix multiplication only, using the most recently computed spectrum as an initial estimate. Then, a precise time integration algorithm that has no artificial damping or phase velocity error for linear problems is applied to the uncoupled modal equations of motion. Squared-frequency extrapolation is examined for nonlinear problems as a means by which these qualities of accuracy and precision can be maintained when the state of the system (and, thus, the modal spectrum) is changing rapidly.The results indicate that a number of important advantages accrue to nonlinear mode superposition: (a) there is no significant difference in total solution time between mode superposition and implicit direct integration analyses for problems having narrow matrix half-bandwidth (in fact, as bandwidth increases, mode superposition becomes more economical), (b) solution accuracy is under better control since the analyst has ready access to modal participation factors and the ratios of time step size to modal period, and (c) physical understanding of nonlinear dynamic response is improved since the analyst is able to observe the changes in the modal spectrum as deformation proceeds.     

含间隙弹簧振动系统的非线性模态特性

针对含间隙的两自由度弹簧-质量分段振动系统的非线性模态开展了研究.首先,解析确定了分段保守自治系统发生同相和反相模态运动的初始位移,并采用加权平均方法确定了分段振动系统的模态频率,及其在位形空间模态曲线.然后,采用数值方法求解了系统的非线性模态曲线和模态频率,与本文获得的解析模态频率比较,说明本文的结果较等效模态频率有更好的精度.研究结果表明:在一定的参数条件下,系统的非线性模态个数会高于系统的自由度数目,系统可能发生内共振,而产生多余模态.多余模态运动是两振子同向振动中含有异向振动,说明多余模态是在同相模态运动和反相模态运动之间转换的模态.     

Dynamic analysis of flexible multi-body systems using mixed modal and tangent coordinates

This paper presents a mixed modal and tangent coordinate technique for computer aided analysis of flexible mechanical systems whose components undergo large translations and large rotations. In this model the configuration of a flexible component is identified by using two sets of generalized coordinates, namely rigid body and elastic coordinates. The rigid body coordinates define the location and orientation of a body axis, whereas the elastic coordinates define the displacement field of a component with respect to its body axis. The elastic coordinates are introduced by using finite element discretization to model flexible components with complex geometries. A modal analysis technique is used to identify the elastic mode shapes and to eliminate insignificant higher frequency modes. An orthonormalization of constraint Jacobian matrix associated with rigid body coordinates is used to identify the rigid body tangent coordinates. The resulting modal and tangent coordinates are used to develop an automated numerical integration scheme to solve the system differential and algebraic equations. Two numerical examples are considered to show the feasibility of dynamic analysis of flexible mechanical systems using this scheme.     

Flexible multibody dynamics using coordinate reduction improved by dynamic correction

This paper is concerned with the efficient dynamic analysis of flexible multibody systems using a robust coordinate reduction technique. Unlike conventional static correction, the formulation is derived by dynamic correction that considers the inertia effect. In this formulation, the constraint and fixed-interface normal modes, which are representative modes in the typical coordinate reduction, are corrected by considering the truncated modal effect with the residual flexibility. Therefore, the proposed method can offer a more precise reduced system without increasing the dimension, which consequently leads to a more accurate and efficient flexible multibody simulation. We implement here the proposed method under augmented formulations of the floating reference frame approach, and test its performance with numerical examples.     

Modal synthesis method of lateral vibration analysis for rotor-bearing system

Rigid coupling and flexible connection made up of elastic coupling units are widely applied to rotor-bearing system with multi-branched shafting system. This paper proposes a modal synthesis method of lateral vibration analysis for such kind of rotor-bearing system. When the proposed approach is developed, the elastic coupling unit is defined as “flexible substructure” which is treated individually and the other parts are partitioned into some substructures which are analyzed by finite element method. The lower-frequency normal modes of the substructures are retained; whereas the higher-frequency normal modes are neglected by a frequency truncation criterion and the residual flexibility of those omitted modes is considered. The lower-frequency normal modes and residual flexibility are considered to be the assumed modes of Rayleigh–Ritz analysis of whole structure. The approach to the treatment of higher-frequency modes leads to a great reduction in the calculation time and a significant improvement in the efficiency of modal synthesis. Examples show that the proposed approach is very effective for the vibration analysis of rotor-bearing system with multi-branched shafting system and the results derived from this approach are in a good agreement with those from transfer matrix method.     

Incremental Deformation Subspace Reconstruction

Recalculating the subspace basis of a deformable body is a mandatory procedure for subspace simulation, after the body gets modified by interactive applications. However, using linear modal analysis to calculate the basis from scratch is known to be computationally expensive. In the paper, we show that the subspace of a modified body can be efficiently obtained from the subspace of its original version, if mesh changes are small. Our basic idea is to approximate the stiffness matrix by its low‐frequency component, so we can calculate new linear deformation modes by solving an incremental eigenvalue decomposition problem. To further handle nonlinear deformations in the subspace, we present a hybrid approach to calculate modal derivatives from both new and original linear modes. Finally, we demonstrate that the cubature samples trained for the original mesh can be reused in fast reduced force and stiffness matrix evaluation, and we explore the use of our techniques in various simulation problems. Our experiment shows that the updated subspace basis still allows a simulator to generate visual plausible deformation effects. The whole system is efficient and it is compatible with other subspace construction approaches.     

Time-varying system identification using a newly improved HHT algorithm

A newly improved Hilbert–Huang transform (HHT) algorithm is presented for identification of time-varying systems and analysis of nonlinear structural response with closely spaced modes. In this improved HHT, the auto-correlation function of the structural response is taken as a substitute of input to reduce noise effect. A band-pass filter scheme and an effective intrinsic mode function (IMF) selection principle are combined to overcome the modal perturbation problem. Based on these, a time-varying system identification procedure is developed. Its robustness and effectiveness are verified by both numerically simulated and laboratory measured vibration data on a scaled concrete-steel composite beam model.     

Model predictive control of axial dispersion chemical reactor

This paper discusses the development of model predictive control algorithm which accounts for the input and state constraints applied to the parabolic partial differential equations (PDEs) system describing the axial dispersion chemical reactor. Spatially varying terms arising from the nonlinear PDEs model are accounted for in model development. Finite-dimensional modal representation capturing the dominant dynamics of the PDEs system is derived for controller design through Galerkin's method and modal decomposition technique. Tustin's discretization and Cayley transform are used to obtain infinite-dimensional discrete-time dynamic modal representations which are used in subsequent constrained controller design. The proposed discrete-time constrained model predictive control synthesis is constructed in a way that the objective function is only based on the low-order modal representation of the PDEs system, while higher-order modes are utilized only in the constraints of the PDEs state. Finally, the MPC formulations are successfully applied, via simulation results, to the PDEs system with input and state constraints.     

工程结构多刚度尺度分析与模态理论

当前工程结构设计大多基于成熟的线性振动理论,未综合考虑非线性、多尺度等特性的影响,导致各类工程振动问题频发,减振措施失效.首先,本文以质量弹簧系统为例,对系统刚度比和质量比等关键参数开展分析,指出刚度比对系统模态具有显著影响;再者,简述大跨桥梁动力学研究现状,从系统全局动力学角度,根据非线性动力学分析和有限元分析,提出工程结构多刚度尺度概念,分析并指出多刚度尺度耦合系统的全局模态、局部模态和混合模态基于不同刚度尺度的定义.为建立桥梁全局动力学模型和理论,桥梁非线性动力学研究奠定基础.     

Cyclical Parthenogenesis Algorithm for guided modal strain energy based structural damage detection

In this paper, a newly developed multi-agent meta-heuristic method, named Cyclical Parthenogenesis Algorithm (CPA), is incorporated into a guided modal strain energy based structural damage detection technique. A modal strain energy based index is used to guide the structural damage identification process, which is formulated as an inverse optimization problem. Generalized Flexibility Matrix (GFM) of the structure is used to define the objective function of the optimization problem. Three numerical examples are provided in order to examine the viability of the proposed method. The results indicate that the proposed method is capable of locating and quantifying structural damage using only the first few modes of the structure. The results are also compared with those of three other meta-heuristic algorithms in order to show the efficiency of CPA in solving the problem.     

On the applicability of static modes switching in gear contact applications

Contact modeling is an active research area in the field of multibody dynamics. Despite the important research effort of the last decades, two main challenging issues, namely accuracy and speed, are far from being jointly solved. One main issue remains the current lack of model order reduction schemes capable of efficiently treat systems where multiple, a priori unknown, input-output locations are present. This work discusses how a methodology named static modes switching can be extended for its use in gear contact simulation. The method proposes an on-line strategy for the selection of residual attachment modes for accurate local deformation prediction. The applicability of the method is discussed by means of several numerical experiments. The result is an indication that the static modes switching underlying idea can be applied also in case of impulsive phenomena like gear impacts. Moreover, a numerical study on penalty factor values and number of eigenmodes and residual attachment modes shows an interesting relation between these three quantities, the dynamic behavior of the system, and the solution accuracy. Finally, some simulations are performed to compare the adopted model order reduction strategy with a reference nonlinear finite element simulation.     

Structural archetypes for coherency: A framework for comparing power system equivalents

The power system model is divided into a study system (SS) and an external group (EG). Three structural archetypes for coherency (SAC) are defined. Each SAC is a set of structural conditions on the power system which when satisfied cause the EG to remain coherent for disturbances confined to the SS. The validity of each SAC for both nonlinear and linear models of the power system is determined. The SAC provide a framework for comparing the three principal methods of determining reduced-order dynamic equivalents of power systems: coherency analysis, modal analysis and the singular perturbation decomposition (SPD) based on the separation of high and low frequency modes. It is shown that satisfaction of any of the three SAC causes the coherency and modal equivalents to be identical. Satisfaction of one particular SAC, termed strict synchronizing coherency, causes the coherency, modal and SPD equivalents to all be identical. The concept of a hierarchy of equivalents, based on the SAC is introduced and briefly discussed.     

Nonlinear dynamic response structural optimization using equivalent static loads

It is well known that nonlinear dynamic response optimization using a conventional optimization algorithm is fairly difficult and expensive for the gradient or non-gradient based optimization methods because many nonlinear dynamic analyses are required. Therefore, it is quite difficult to find practical large scale examples with many design variables and constraints for nonlinear dynamic response structural optimization. The equivalent static loads (ESLs) method is newly proposed and applied to nonlinear dynamic response optimization. The equivalent static loads are defined as the linear static load sets which generate the same response field in linear static analysis as that from nonlinear dynamic analysis. The ESLs are made from the results of nonlinear dynamic analysis and used as external forces in linear static response optimization. Then the same response from nonlinear dynamic analysis can be considered throughout linear static response optimization. The updated design from linear response optimization is used again in nonlinear dynamic analysis and the process proceeds in a cyclic manner until the convergence criteria are satisfied. Several examples are solved to validate the method. The results are compared to those of the conventional method with sensitivity analysis using the finite difference method.     

System reduction in multibody dynamics of wind turbines

A system reduction scheme is devised related to a multibody formulation from which the dynamic response of a wind turbine is determined. In this formulation, each substructure is described in its own frame of reference, which is moving freely in the vicinity of the moving substructure. The Ritz bases spanning the reduced system comprises of rigid body modes and some dynamic low-frequency elastic eigenmodes compatible to the kinematic constraints of the related substructure. The high-frequency elastic modes are presumed to cause merely quasi-static displacements, and thus are included in the expansion via a quasi-static correction. The results show that by using the derived reduction scheme it is only necessary with 2 dynamical modes for the blade substructure when the remaining modes are treated as quasi-static. Moreover, it is shown that it has little to none effect if the gyroscopic stiffness matrix during a stopped situation or under nominal operational conditions is used to derive the functional basis of the modal expansion.     

A practical algorithm for the efficient computation of eigenvector sensitivities

Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. In this paper, a practical algorithm has been developed for efficiently computing eigenvector derivatives of generalized symmetric eigenvalue problems. For eigenvector derivative of a separate mode, the computation only requires the knowledge of eigenvalue and eigenvector of the mode itself and an inverse of system matrix accounts for most computation cost involved. In the case of two close modes, the modal information of both modes is required and the eigenvector derivatives can be accurately determined simultaneously at minor additional computational cost. Further, the proposed method has been extended to the case of practical structural design where structural modifications are made locally and the eigenderivatives of the modes concerned before are still of interest. By combining the proposed algorithm together with the proposed inverse iteration technique and singular value decomposition theory, eigenproperties and their derivatives can be very efficiently computed. Numerical results from a practical finite element model have demonstrated the practicality of the proposed method. The proposed method can be easily incorporated into commercial finite element packages to improve the computational efficiency of eigenderivatives needed for practical applications.     

An accurate modal truncation method for eigenvector derivatives

This paper deals with obtaining accurate eigenvector derivatives in damped vibration systems with modal truncation method only by system matrices and a few available lower modes. A numerical example shows that the method of this paper is correct and efficient.     

Feedback control of unstable thermoacoustic modes in an annular Rijke tube

Simulation and experimental results from an annular Rijke tube are presented. This system is a thermoacoustic surrogate system of an annular gas turbine combustor which, despite its simplicity, possesses the basic mechanisms to feature unstable azimuthal modes. A thermoacoustic network model is set up and used to derive low-order models for modal control of the system. The derived controllers are successfully applied in simulation and experiment. With the modal controllers, all unstable acoustic modes can be eliminated individually. A simultaneous use of all controllers results in a complete stabilization of the system.