Guyan Reduction Research Articles

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system. Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘ invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘ reducibility conditions’. If these ‘ reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘ primary’ and ‘ secondary resonances’ present in the system. One can also recover ‘ resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.

An interpolation scheme for the approximation of dynamical systems

The demand of many industries for highly accurate predictions of the structural response of dynamical systems leads to a rapidly growing complexity of the established numerical models. Due to the thereby arising large computational efforts, the use of model reduction techniques becomes an essential task. Model reduction procedures in the context of deterministic structural analysis are already well developed and widely used, such as Guyan reduction and component mode synthesis techniques. However, the assumption of deterministically known structural properties is in most cases not realistic. Hence, the rational propagation of uncertainties becomes an indispensable task in the robust prediction of the structural response. The associated multiple simulation runs of complex models bring the topic of model reduction to the foreground of technical and scientific interest. In this work, a novel method which approximates eigenfrequencies and mode shape vectors needed for the evaluation of frequency response functions, will be presented. This meta-model of structural matrices is established by employing interpolation between the matrices evaluated at some supporting points. The technique has no difficulties with closely spaced eigenfrequencies and mode switching and it can be employed over the entire frequency range on the full model, but also on the substructures and it is hence applicable together with deterministic model reduction techniques.

Finite element modelling for a piezoelectric ultrasonic system

In this paper, a three-dimension (3-D) mechanical element with an extra electrical degree of freedom is employed to simulate the dynamic vibration modes of the linear piezoelectric, piezoelectro-mechanic and mechanical behaviors of a metal disc structure embedded with a piezoelectric actuator. In piezoelectric finite element formulation, a discretized equation of motion is developed and solved by using the integration scheme to explain why an adaptive boundary condition, a simple support condition with three non-equal-triangular (120°–90°–150°) fixed points near the edge, which is the asymmetric disc used as the stator of the studied ultrasonic motor, for the mechanical design of an asymmetric disc-type piezoelectric ultrasonic stator, is defined so that a lateral elliptical motion of the contact point between stator and rotor can be realized for driving the rotor. It starts from hybrid elements with displacement and electric potential as the nodal d.o.f.s model and uses Guyan reduction and Householder-Bisection inverse iteration to find the displacement profile and displacement vector flow of the stator under frequency driving. The standing wave existence is also proven by the displacement patterns of the finite element theoretical model.

モーダル差分構造法とGuyanの静縮約による区分モード合成法の一般化(機械力学,計測,自動制御)

モーダル差分構造法とGuyanの静縮約による区分モード合成法の一般化(機械力学,計測,自動制御)

Thermal post-buckling and limit-cycle oscillation of functionally graded panel with structural damping in supersonic airflow

Abstract This paper investigates the supersonic aero-thermo post-buckling behaviors and limit-cycle oscillations of Functionally Graded Material (FGM) panels. In order to model the panel, the first-order shear deformation theory (FSDT) of plate is applied, and structural damping effect is considered in the supersonic airflow. Material properties are assumed to be temperature dependent, and to vary in the thickness direction of the panel according to a simple power law distribution. Also, the von-Karman strain–displacement relations are adopted to consider the geometric nonlinearity due to large deformation of the structure. Rayleigh damping coefficient is used to consider the structural damping, and the first-order piston theory is applied to model the supersonic aerodynamic loads. Newmark’s time integration method is adopted, and Guyan reduction technique is employed to get the computational efficiency. Numerical results are compared with the previous works, and the effects of structural damping on the limit-cycle oscillation of FGM panels under supersonic airflow are studied in detail.

Standard input data for FEM–MBS coupling: importing alternative model reduction methods into SIMPACK

Various research areas in the field of vehicle modelling, structural mechanics, engine dynamics, microelectromechanical systems (MEMS), etc. require the utilization of both multibody system formalism (MBS) and finite element method (FEM) in order to sufficiently capture the model's dynamics. The FEM–MBS coupling is accomplished by reducing the dimension of the FE-modelled part and then importing it into an MBS-code for further simulation. When using commercial FEM (Nastran, ANSYS, etc.) as well as MBS (SIMPACK) software packages the necessary standard input data (SID) file is needed for the coupling procedure (FEMBS interface). A problem arises by the restriction that both commercial FEM and MBS codes support only two condensation methods (Guyan reduction and component mode synthesis (CMS)), thus disabling the direct application of any other reduction approach (e.g. from the field of control theory) that actually could be better. In this article, the theoretical background of an implemented FEM–MBS interface (MORPACK) is presented allowing the application of any kind of reduction method for FE-modelled structures and furthermore their import (Ritz approximation) into SIMPACK via the SID file generation. A benchmark problem (UIC60-rail) is used in order to capture in SIMPACK the discrepancy between the standardized CMS and the Krylov subspace method (KSM), as one of the alternatives offered by the interface.

Frequency Domain Identification of the Active Beam Model for the Vibration Control System

An identification procedure of the active vibration damping system with the piezo-elements glued to the structure is presented in the paper. Such identification can be carried out in analytical way (Guyan reduction) by using of low-band pass amplifiers. First, we have identified the open loop system in the frequency domain by using experimental setup with real amplifier and beam equipped in piezo-strips working as actuators and sensor, respectively. For the frequency range 1 Hz to 400 Hz we have obtained Bode plots with helped of Dynamic Signal Analyzer which were base to further calculations. Second, mathematical model have been adjusted to the experimental results by the least square sum approach and also by the consideration of a system which was modeled by finite elements. A three first resonance frequencies and three first anti-resonance frequencies have obtained.

Employing incomplete complex modes for model updating and damage detection of damped structures

In the study of finite element model updating or damage detection, most papers are devoted to undamped systems. Thus, their objective has been exclusively restricted to the correction of the mass and stiffness matrices. In contrast, this paper performs the model updating and damage detection for damped structures. A theoretical contribution of this paper is to extend the cross-model cross-mode (CMCM) method to simultaneously update the mass, damping and stiffness matrices of a finite element model when only few spatially incomplete, complex-valued modes are available. Numerical studies are conducted for a 30-DOF (degree-of-freedom) cantilever beam with multiple damaged elements, as the measured modes are synthesized from finite element models. The numerical results reveal that applying the CMCM method, together with an iterative Guyan reduction scheme, can yield good damage detection in general. When the measured modes utilized in the CMCM method are corrupted with irregular errors, assessing damage at the location that possesses larger modal strain energy is less sensitive to the corrupted modes.

Using incomplete modal data for damage detection in offshore jacket structures

The development of robust techniques for early damage detection for offshore structures is crucial to avoid the possible catastrophe caused by structural failures. This article applies the cross-model cross-mode (CMCM) method for damage detection that is capable of identifying the damage to individual members of offshore jacket platforms, when limited, spatially incomplete modal data is available. Basically, the CMCM method is classified as a direct, physical property adjustment model updating method. Implementing this method requires only a few modes measured from the damaged structure. In dealing with spatial incompleteness, this paper investigates both model reduction and modal expansion techniques. Specifically, either Guyan (static condensation) or SEREP (system equivalent reduction expansion process) transformation matrix, between the master and slave degrees-of-freedom, is employed in the model reduction or modal expansion process. One theoretical development is an iterative procedure to compute the transformation matrix associated with the (unknown) damaged structure. Numerical studies are conducted for a jacket platform with multiple damaged members based on synthetic data generated from finite-element models. The results suggest that (i) Guyan scheme always outperforms SEREP, (ii) model reduction is always better than modal expansion, and (iii) the CMCM method in conjunction with iterative Guyan reduction approach yields the best damage location and severity estimate.

Dedicated finite elements for electrode thin films on quartz resonators

The accuracy of the finite element analysis for thickness shear quartz resonators is a function of the mesh resolution; the finer the mesh resolution, the more accurate the finite element solution. A certain minimum number of elements are required in each direction for the solution to converge. This places a high demand on memory for computation, and often the available memory is insufficient. Typically the thickness of the electrode films is very small compared with the thickness of the resonator itself; as a result, electrode elements have very poor aspect ratios, and this is detrimental to the accuracy of the result. In this paper, we propose special methods to model the electrodes at the crystal interface of an AT cut crystal. This reduces the overall problem size and eliminates electrode elements having poor aspect ratios. First, experimental data are presented to demonstrate the effects of electrode film boundary conditions on the frequency-temperature curves of an AT cut plate. Finite element analysis is performed on a mesh representing the resonator, and the results are compared for testing the accuracy of the analysis itself and thus validating the results of analysis. Approximations such as lumping and Guyan reduction are then used to model the electrode thin films at the electrode interface and their results are studied. In addition, a new approximation called merging is proposed to model electrodes at the electrode interface.

Analytic modeling and driving for an ultrasonic wheel system

In this paper, a 3-D mechanical element with an extra electrical degree of freedom is employed to simulate the dynamic vibration modes of the linear piezoelectric, mechanical, and piezoelectro-mechanic behaviors of an ultrasonic piezoelectric wheel system. In piezoelectric finite element formulation, a discretized equation of motion is developed and solved by using the integration scheme to explain why an adaptive boundary condition, a simple support condition with one fixed point on the wheel central for the mechanical design of ultrasonic directly driving system, is defined so that a lateral elliptical motion of the contact point between wheel and ground can be realized for driving the wheel carrier. Guyan reduction and Householder–Bisection inverse iteration are used as an optimal solver to find the modal frequencies and associated mode shapes. In this research, a speed control scheme is implemented by using commercial DSP technique for an open-loop speed driver.

A Direct Approach to Order Reduction of Nonlinear Systems Subjected to External Periodic Excitations

In this work, the basic problem of order reduction of nonlinear systems subjected to an external periodic excitation is considered. This problem deserves special attention because modes that interact (linearly or nonlinearly) with external excitation dominate the response. These dominant modes are identified and chosen as the “master” modes to be retained in the reduction process. The simplest idea could be to use a linear approach such as the Guyan reduction and choose those modes whose natural frequencies are close to that of external excitation as the master modes. However, this technique does not guarantee accurate results when nonlinear interactions are strong and a nonlinear approach must be adopted. Recently, the invariant manifold technique has been extended to forced problems by “augmenting” the state space, i.e., forcing is treated as an additional state and an invariant manifold is constructed. However, this process does not provide a clear picture of possible resonances and conditions under which an order reduction is possible. In a direct innovative approach suggested here, a nonlinear time-dependent relationship between the dominant and nondominant states is assumed and the dimension of the state space remains the same. This methodology not only yields accurate reduced order models but also explains the consequences of various primary and secondary resonances present in the system. One obtains various reducibility conditions in a closed form, which show interactions among eigenvalues, nonlinearities and the external excitation. One can also recover all “resonance conditions” obtained via perturbation or averaging techniques. The “linear” as well as the “extended invariant manifold” techniques are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control of large-scale externally excited nonlinear systems.

Nonlinear thermal flutter of functionally graded panels under a supersonic flow

Abstract Thermal flutter characteristics of functionally graded (FG) ceramic/metal panels under the thermal and aerodynamic loads are investigated. The volume fractions of the constitutive materials are determined by a simple power-law distribution, and material properties are assumed to be a linear rule of mixture. The panels are considered as rectangular plates based on the first-order shear deformation theory, and the von Karman strain–displacement relations are used to account for the geometric nonlinearity. The first-order piston theory is adopted to represent aerodynamic pressures induced by supersonic airflows. The principle of virtual work is applied to derive equations of motion, and a finite element method is used to obtain numerical solutions. The Newton–Raphson method is adopted to obtain approximate solutions of the nonlinear governing equations. Flutter boundaries are defined by eigenvalue analysis, and the Guyan reduction is used to reduce degree of freedom. Flutter motions of FG panels are investigated using the Newmark method. The effects of volume fraction distributions, boundary conditions, temperature changes and aerodynamic pressures on panel flutter characteristics are analyzed in detail.

Mass-weighting methods for sensor placement using sensor set expansion techniques

Placement of sensors is one of the most important tasks performed during pretest planning. The purpose of this work was to develop and investigate the use of an iterative Guyan expansion for mass weighting of target modes for sensor placement analogous to the common iterative Guyan reduction technique. The goal was to determine the appropriate mass-weighting approach to use in conjunction with effective independence sensor set expansion. In either sensor set expansion, or reduction, mass weighting requires a reduction of the FEM mass matrix to the current sensor set size Test-Analysis-Model (TAM). A general theory is presented for target mode mass weighting that can accommodate any type of reduction technique. The theory predicts that sensor set expansion using static mass weighting will result in sensor configurations that produce poor static TAMs. In contrast, sensor set expansion using modal mass weighting exactly reproduces the correct mass distribution during the expansion process. The results of a numerical example corroborate the theory. The modal mass sensor set expansion process produced significantly more accurate static TAMs than the static mass expansion. The modal expansion process was not quite as accurate as the iterative static reduction approach, but modal expansion was over 1600 times faster.

A General Reduced Representation of One-Dimensional Frictional Interfaces

A physically-motivated, reduced representation of a general one-dimensional frictional interface is developed. Friction is introduced into the system as a state variable and is modeled by nonlinear springs of large but finite stiffness. The set of equations for the interface is reduced in a procedure similar to Guyan reduction by assuming that the system must deform in its quasistatic displacement shapes. The result of this reduction is that the degrees of freedom internal to the interface are removed from the analysis and only the boundary degrees of freedom are retained. The reduced system is then specialized to the case of a bar on a frictional surface. For this problem, a second reduction is made by noting that the time derivative of the friction force on the stuck block nearest the slip zone is much greater than the time derivatives of the friction forces elsewhere. Therefore only the friction force on the stuck block nearest the slip zone needs to be updated at each time step. The reduced representation developed in this paper is compared with a formulation from the literature and it is seen that the two match very closely and that the reduced representation is far less computationally intensive.

Simultaneous Mass, Damping, and Stiffness Updating for Dynamic Systems

different modes associated with the mathematical and experimental models, respectively. Two numerical examples are demonstrated: a 4-degree-of-freedom mass–spring–damper system and a 30-degree-of-freedom finite element model for a cantilever beam. The numerical updating by the cross-model cross-mode method is excellent for all system matrices when the measured modes are spatially complete and noise free. The cross-model cross-mode method, together with the Guyan reduction scheme, also performs reasonably well under a spatial incompleteness situation.

Structural stability of functionally graded panels subjected to aero-thermal loads

Abstract Static and dynamic stabilities of functionally graded (FG) panels which are subjected to combined thermal and aerodynamic loads are investigated. The volume fractions of constituent materials composing the FG panels are assumed to be given by a simple power law distribution. Material properties of the FG panels are obtained by a linear rule of mixture. Panels are considered as rectangular plates which are based on the first-order shear deformation theory. The von Karman strain–displacement relation is used to account for geometric nonlinearity, which is caused by a large deformation. The first-order piston theory is used to simulate supersonic aerodynamic loads acting on the panels. Equations of motion are derived by the principle of virtual work and numerical solutions are obtained by a finite element method. The Newton–Raphson method is applied to get solutions of the nonlinear governing equations. Flutter boundaries are defined by linear flutter analysis and the Guyan reduction is applied to reduce degree of freedom for eigenvalue analysis. The influence of the material constitution, asymmetric characteristics of FG panels on thermal buckling and flutter characteristics are examined. Static and dynamic stability margins of FG panels are defined for various volume fractions.

Dynamic reduction strategies to extend modal analysis approach at higher frequencies

Modal analysis on huge finite element models requires a numerical simplification in terms of total number of degrees of freedom (d.o.f.'s). The above aim is generally achieved by choosing some master d.o.f.'s and condensing the structure matrices on those d.o.f.'s. Static condensation (i.e. stiffness matrix reduction) is theoretically an errorless operation; on the other hand, mass condensation can only be approximate in dynamic applications. In order to accomplish matrix condensation, the technique mainly used is the so-called Guyan reduction. The present paper outlines the limitations of the technique, introducing some significant improvements. These are related to the inertia conservation properties of the reduced mass matrix and the condensed mass matrix assembly by means of fictitious and appropriate stiffness connections that are different from those obtained by the stiffness model. The effectiveness of the modified approach is demonstrated with respect to the modal analysis results obtained by Guyan approach, through three different test cases.

Reduced-Order Models for the Calculation of Thermal Transients of Heat Conduction/Convection FE Models

On-line calculation of temperature and thermal stresses at critical locations of structural components is currently performed in nuclear and aeronautical applications in order to assess fatigue damage accumulation and residual life. Since it is not possible to use full-scale finite element models because of the large calculation times, ad-hoc simplified algorithms are developed and employed. Thermal stresses commonly arise because of temperature gradients within the component due to heat exchange between the solid and the surrounding fluid. Commonly on-line monitoring methods are based on the assumption that time histories of the temperatures of the fluids around the component are known (e.g., measured) and are used as the inputs for the calculation of temperature and thermal stress at the critical locations of the component. If the temperatures of the surrounding fluid can not be measured, they must be numerically computed integrating an FE model of the fluid in the thermal FE model of the component; as a result, the size of the coupled thermal model is further increased. In the present work, a methodology is developed to reduce the size of thermal models including not only the FE model of the component but also the coupled fluid model. In detail, Guyan reduction and component mode synthesis are applied to a coupled thermal FE model, in order to reduce the size of the problem and to make it suitable for on-line calculations. The appropriate mathematical formulation for the reduction of the fluid FE model has been developed. Two reduction methods are proposed and are applied to the axial symmetric FE model of a turbine disk pointing out their capabilities and limitations for on-line temperature calculation.

A variational approach for solving an inverse vibration problem¶

The present investigation is focused on the solution of a dynamic inverse problem which is concerned with the assessment of damage in structures by means of measured vibration data. This inverse problem has been presented as an optimization problem and has been solved through the use of the Variational Approach, i.e. the conjugate gradient method (CGM) coupled with the adjoint equation. The results have been presented in a satisfactory form when a small structure with few degrees of freedom (DOF) is considered, however, when a higher DOF structure is studied, the simple application of the variational approach is not any more satisfactory, being necessary the application of an additional methodology. In order to solve this difficulty, a new approach based on the use of the genetic algorithm (GA) method has been proposed. The GA method is used to generate a primary solution which is employed as the initial guess for the CGM. The application of this new approach has shown that better results can be achieved, although the computational time for the application analyzed here could be increased. The damage estimation has been evaluated using noiseless and noisy synthetic experimental data, and the reported results are concerned with both truss and beam-like structures, which have been modeled through a finite element technique. Moreover, in order to take into account the reduced set of experimental data to be employed in the optimization algorithm, a Guyan reduction technique has been adopted on the finite element formulation.¶Selected paper from Inverse Problems, Design and Optimization Symposium, 2004.