Reduced-order Structural Models Research Articles

Reduced order modeling for real-time monitoring of structural displacements due to electromagnetic forces in large scale tokamaks

Abstract The real-time monitoring of the structural displacement of the vacuum vessel of thermonuclear fusion devices caused by electromagnetic loads is of great interest. In this paper, model order reduction is applied to the integral equation methods and the finite elements method to develop electromagnetic and structural reduced order models (ROMs) compatible with real-time execution which allows for the real-time monitoring of strain and displacement in critical positions of Tokamaks machines. Low-rank compression techniques based on hierarchical matrices are applied to reduce the computational cost during the offline stage when the ROMs are constructed. Numerical results show the accuracy of the approach and demonstrate the compatibility with real-time execution in standard hardware.

High fidelity adaptive mirror simulations with reduced order models

In the design process of large adaptive mirrors numerical simulations represent the first step to evaluate the system design compliance in terms of performance, stability and robustness. For the next generation of Extremely Large Telescopes increased system dimensions and bandwidths lead to the need of modeling not only the deformable mirror alone, but also all the system supporting structure or even the full telescope. The capability to perform the simulations with an acceptable amount of time and computational resources is highly dependent on finding appropriate methods to reduce the size of the resulting dynamic models. In this paper we present a framework developed together with the company Microgate to create a reduced order structural model of a large adaptive mirror as a preprocessing step to the control system simulations. The reduced dynamic model is then combined with the remaining system components allowing to simulate the full adaptive mirror in a computationally efficient way. We analyze the feasibility of our reduced models for Microgate’s prototype of the adaptive mirror of the Giant Magellan Telescope.

Aeroelastic Influence of Blade-Shaft Coupling in a 1 1/2-Stage Axial Compressor

Abstract The design process of turbomachinery, is often constraint by aeroelastic phenomena. The design choices are limited by possible structural failure, which can be caused by high vibration amplitudes. In particular, damping has an important impact on these phenomena. In the absence of friction, damping is mainly created by aerodynamics. In this paper, additional damping created by the shaft will be investigated. This becomes relevant when blade vibrations with nodal diameters 1 and -1 couple structurally with shaft vibrations. To investigate the blade-shaft coupling, a simulation process is set up based on a full structural dynamic model of the blisk-shaft assembly and a harmonic balance CFD model to account for the aeroelastic effects. In addition, mistuning identification is performed based on an experimental modal analysis at standstill. All results are incorporated into a structural reduced order model that calculates the vibrational behavior of the blading. These results are compared to damping determined during operation using an acoustic excitation system and measured forced frequency responses. The numerical results agree well with the experimental results, i.e. within the measurement uncertainty. Furthermore, the blade-shaft coupling results in significant changes of the eigenfrequencies and damping. As a consequence, damping increases by up to twelve times due to the coupling. This reduces amplitudes by a factor of nine for the mistuned blade responses. Consequently, higher structural safety factors can be achieved by taking the blade-shaft coupling into account so that the remaining potentials in the aerodynamic design could be better exploited.

Bayesian calibration of hysteretic reduced order structural models for earthquake engineering applications

Seismic risk assessment entails frequently a large number of nonlinear time-history analyses. The use of high-fidelity Finite Element Models (FEMs) in this context, though supports high accuracy predictions, involves a substantial computational burden. This has incentivized researchers to explore the use of calibrated Reduced Order Models (ROMs) to replace the computational expensive FEMs, with the calibration process established so that the ROM can match the desired FEM characteristics. Recent studies have explicitly examined this calibration for hysteretic, multi-degree-of-freedom ROMs utilizing FEM time-history response data. This calibration has been posed as a least-squares optimization problem for the configurable ROM parameters, considering the discrepancy between the ROM and FEM responses for a selected number of excitations, for different output quantities of interest (QoIs). This work extends these efforts to examine a Bayesian calibration framework of ROMs for earthquake engineering applications. The entire posterior distribution of the ROM parameters is identified, instead of solely the point estimate that the current least-squares optimization provides. This facilitates a quantification of the uncertainty in the calibrated ROM parameters. For formulating the likelihood function different assumptions are discussed, impacting, ultimately, how the different earthquake excitations and QoIs are effectively combined. Comparisons to the objective function utilized in the established least-squares approach are drawn. Furthermore, a model class selection is advocated to compare different candidate ROMs in order to choose the most appropriate one given the available data. Finally, a hierarchical Bayesian inference approach is examined, treating independently the ROM calibration for each earthquake excitation, but unifying the ROM parameters under a common prior, updated using data from all excitations. Different challenges are discussed for implementing such a multilevel Bayesian framework for the current application. The illustrative implementation considers two different structures, corresponding to different materials and heights, with the FEMs developed in OpenSees. The different calibration approaches are evaluated by utilizing the calibrated ROMs for seismic risk assessment for different seismicity scenarios.

Aeroelastic behaviour of UAV wings due to morphing

PurposeThe purpose of this paper is to analyse the effects of morphing on the aeroelastic behaviour of unmanned aerial vehicle (UAV) wings to make an emphasis on the required aeroelastic tailoring starting from the conceptual design of the morphing mechanisms.Design/methodology/approachIn this study, flutter and divergence characteristics of a fully morphing wing design were discussed to show the dilapidating effect of morphing on the related parameters. The morphing wings were intended to achieve a high efficiency at different flight phases; thus, various morphing concepts were integrated into a UAV wing structure. Although it is considered beneficial to have the morphing capabilities to avoid the failure due to a possible wear out in flutter and divergence parameters; it is necessary to include the aeroelastic analyses at the early design phases. This study utilizes a combination of a reduced order structural model and Theodorsen unsteady aerodynamic model as primary analyses tools for flutter and divergence. The analyses were conducted by using an in-house developed pk-algorithm coupled with a commercial finite element analysis (FEA) tool. This approach yielded a fast solution capacity because of the state-space form used.FindingsAnalyses conducted showed that transition between take-off, climb, cruise and loiter phases yield a change in the flutter and divergence speeds as high as 138 and 305 per cent, respectively.Practical implicationsThe research showed that an extensive aeroelastic investigation was required for morphing wing designs to achieve a failure safe design.Originality/valueThe research intends to highlight the possible deteriorating effects on structural design of morphing UAV wings by focusing on the aeroelastic characteristics. In addition to that, fundamental morphing concepts are compared in terms of the order of magnitude of their deteriorating effects.

Estimation of the dynamic response of a slender suspension bridge using measured acceleration data

Suspension bridges with very long spans and slender designs are susceptible to large-amplitude dynamic excitation. Monitoring systems installed on bridges can provide measurement data (e.g. accelerations) and therewith valuable information on the true dynamic behaviour. This pilot study examines the possible use of recently developed methods for real-time response estimation at unmeasured locations. The methodology for response estimation is tested in a case study on the Hardanger Bridge, a 1310 m long suspension bridge in Norway, which has a network of twenty accelerometers. Two techniques, a joint input-state estimation algorithm (JIS) and a dual Kalman filter (DKF), are used to estimate the full-field dynamic response using data measured at the bridge and a reduced order structural model. The results show that the DKF is able to estimate accelerations fairly accurately. The JIS estimate, however, suffer from ill-conditioning and consequently show severe errors. Possible reasons for this ill-conditioning are briefly discussed.

The determination and enhancement of compliant modes for actuation in structural assemblies

Linear algebra methods for determining modes of kinematic and static indeterminacy in jointed frames are extended to reveal modes of compliance in otherwise rigid assemblies. These modes are extracted from a structural model, based on finite elements, via a singular value decomposition and yield the ways in which a structure can be most easily deformed. This modal approach also allows for the formulation of a reduced-order structural model, whereby relevant modes are selected and used as the basis for the optimisation of a complaint structure. The method detailed is shown to be a useful design tool, demonstrated by its application to a structure based on the Kagome lattice geometry. For certain frameworks, first order effects produce tightening under actuation. As a result, a scheme to adjust the modes to take nonlinear effects into account is also given.

Assessment of a rolling isolation system using reduced order structural models

This paper examines the performances of lightly- and heavily-damped rolling isolation systems (RISs) located within earthquake-excited structures. Six steel structures of varying height and stiffness are selected so as to represent a range of potential RIS installations. Computation models of representative frames from each of the six structures are reduced through dynamic condensation and assembled with models for biaxial isotropic hysteretic behavior within each floor. A novel reduced order modeling approach is presented in this paper. The method combines a dynamic condensation of a linear-elastic frame with the inelastic-push over curve for a detailed elastic–plastic frame model and a novel bi-axial hysteretic model for the net inter-story inelastic behavior. The reduced inelastic model combines stiffness and mass matrices from the reduced linear model with the bi-axial inelastic floor model, and is subsequently fit to push-over curves from the detailed hysteretic model. The resulting reduced order model has three coordinates per floor and provides a much simpler model for simulating the floor responses of inelastic structures. The resulting inelastic structural models are isotropic in plan and uniform along the height. Suites of recorded ground motions representative of near-fault and far-field hazards are scaled and inputted into these hysteretic reduced models. The bidirectional floor responses at varying heights are then applied to experimentally-validated models of lightly- and heavily-damped RISs. Empirical cumulative distribution functions of peak isolator responses (relative displacement and total acceleration) for the two systems are compared, from which installation guidelines are presented.

Reduced order structural models for the calculation of wet contact forces due to impacts in hydraulic valves

Impacts in hydraulic valves generate structure-borne sound. Numerical methods are applied to calculate the contact forces, which are related to the acoustic noise generation. The fluid between the moving valve stem and the end stop creates a squeeze film force. A model order reduction technique in two steps is applied to capture the flexibility of the impact bodies in a multi-physics system simulation. The first reduction step is based on modal truncation using normal and attachment modes. The second step aims to select the important states or modes within the model bandwidth based on Hankel singular values, where two selection procedures are compared. The squeeze film effect is represented by an analytical formula which is closely incorporated to the compliant contact model. This contact deformation relation for the wet impact has to account for the transition between the pure squeeze motion and the structural collision. Measurements are used to support the calculation results. The coupled calculation of elastic and hydraulic effects leads to improved models for predicting the noise design and the functionality of hydraulic valves.

Bi-linear reduced-order models of structures with friction intermittent contacts

The dynamic analysis of structures with localized nonlinearities, such as intermittent contacts of cracked structures, is a computationally demanding task because of the large size of the models involved. Thus, high-resolution finite element models are often reduced using a variety of specialized techniques which exploit spatial coherences in the dynamics. In addition, when a steady-state forced response analysis is performed, direct time integration can be replaced with multi-harmonic balance methods. Recently, a technique based on bi-linear normal modes has been successfully applied to piecewise-linear oscillators. The key idea of that approach is to represent the spatial coherences in the system dynamics with two sets of normal modes with special boundary conditions, referred to as bi-linear modes. In this paper, the bi-linear modal representation is extended to the case of intermittent contacts with friction. Furthermore, a novel reduced order modeling method is developed for the 0th order harmonic used in multi-harmonic balance methods. The forced response of a cracked structure is used to demonstrate the proposed methods.

243 セミアクティブ制御におけるモデルと目標制御則の最適化

Various semi-active control methods have been proposed for vibration control of civil structures. In contrast to active vibration control systems, all semi-active control systems are essentially asymptotically stable because of the stability of the structural systems (with structural damping) themselves and the energy dissipating nature of the semi-active control law. In this study, by utilizing the above property on the stability of semi-active control systems, a reduced-order structural model and a semi-active control law are simultaneously obtained so that the performance of the resulting semi-active control system becomes good. A two degree of freedom active control law is adopted as the targeted semi-active control in the present study. Optimal design parameters are obtained with Genetic Algorithm (GA). Simulation results show the effectiveness of the proposed approach.

443 制御性能向上のための低次構造モデルとセミアクティブ制御則の同時設計

Various semi-active control methods have been proposed for vibration control of civil structures. In contrast to active vibration control systems, all semi-active control systems are essentially asymptotically stable because of the stability of the structural systems (with structural damping) themselves and the energy dissipating nature of the semi-active control law. In this study, by utilizing the above property on the stability of semi-active control systems, a reduced-order structural model and a semi-active control law are simultaneously obtained so that the performance of the resulting semi-active control system becomes good. Based on the above fact any semi-active control laws derived from some models stabilize all real-existing structural systems that have structural damping. It means that the difference of dynamic behaviors between the real structural system and the reduced-order mathematical model in the sense of the open-loop response is no longer an important issue. In other words, we do not have to consider the closed-loop stability, which is one of the most important constraints in active control, in the process of the reduced-order structural modeling and the semi-active control design. We can only focus on the control performance of the closed-loop system with the real structure with the (model-based) semi-active control law in obtaining the reduced-order model. The semi-active control law in the present study is based on the one step ahead prediction of the structural response. The Genetic Algorithm (GA) is adopted to obtain the reduced-order model and the semi-active control law based on the reduced order model.

Vascular stents: Coupling full 3-D with reduced-order structural models

Self-expanding nitinol stents are used to treat peripheral arterial disease. The peripheral arteries are subjected to a combination of mechanical forces such as compression, torsion, bending, and contraction. Most commercially available peripheral self-expanding stents are composed of a series of sub-millimeter V-shaped struts, which are laser-cut from a nitinol tube and surface-treated for better fatigue performance. The numerical stent models must accurately predict location and distribution of local stresses and strains caused by large arterial deformations. Full 3-D finite element non-linear analysis of an entire stent is computationally expensive to the point of being prohibitive, especially for longer stents. Reduced-order models based on beam or shell elements are fairly accurate in capturing global deformations, but are not very helpful in predicting stent failure. We propose a mixed approach that combines the full 3-D model and reduced-order models. Several global-local, full 3-D/reduced-order finite element models of a peripheral self-expanding stent were validated and compared with experimental data. The kinematic constraint method used to couple various elements together was found to be very efficient and easily applicable to commercial FEA codes. The proposed mixed models can be used to accurately predict stent failure based on realistic (patient-specific), non-linear kinematic behavior of peripheral arteries.

Numerical and experimental validation of out-of-plane resonance closed formulas for MEMS suspended plates with square holes

A compact analytical model for out-of-plane resonance evaluation is proposed for large diffuse MEMS vibrating plates with squared holes. A closed form expression was developed from a structural reduced order model with equivalent concentrated mass and stiffness parameters. Results were validated through FEM models and experimental modal analysis. Numerical FEM simulations were performed by 1D, 2D and 3D FEM models; experiments on polysilicon test structures were conducted using an interferometric microscope. Specimens were designed ad hoc to highlight the sensitivity of proposed formulas to main structural dimensional parameters of vibrating plates. Experiments and models helped investigate the sensitivity of the proposed reduced linearized model when exposed to electrostatic non-linear coupling effects and fluidic damping coupling. Both numerical results and experiments are in good agreement with analytical results predicted by closed formulas.

Time Simulations of the Response of Maneuvering Flexible Aircraft

This work is concerned with time simulations of the response of flying flexible aircraft to initial conditions and external excitations, such as gusts. The simulations are based on a unified formulation of the problem of dynamics and control of maneuvering flexible aircraft developed recently by the authors. The formulation represents a seamless integration of pertinent material from the areas of analytical dynamics, structural dynamics, aerodynamics, and controls. It incorporates both the aircraft rigid-body motions and elastic deformations and it includes the aerodynamic, propulsion, gravity, and control forces, all in a rigorous manner. The formulation lends itself to efficient computer processing of the aircraft response. The formulation contains improvements over the earlier formulation by the authors in the form of a reduced-order structural model and newly designed, easier-to-implement controller and state observer. The corresponding time simulations are entirely new, and no similar results have been obtained by other investigators. The developments should help with the design of the new generation of unmanned aerial vehicles (UAVs) and in particular with the design of autonomous UAVs controlled by autopilots.

Stochastic optimal control of coupled structures

The stochastic optimal nonlinear control of coupled adjacent building structures is studied based on the stochastic dynamical programming principle and the stochastic averaging method. The coupled structures with control devices under random seismic excitation are first condensed to form a reduced-order structural model for the control analysis. The stochastic averaging method is applied to the reduced model to yield stochastic differential equations for structural modal energies as controlled diffusion processes. Then a dynamical programming equation for the energy processes is established based on the stochastic dynamical programming principle, and solved to determine the optimal nonlinear control law. The seismic response mitigation of the coupled structures is achieved through the structural energy control and the dimension of the optimal control problem is reduced. The seismic excitation spectrum is taken into account according to the stochastic dynamical programming principle. Finally, the nonlinear controlled structural response is predicted by using the stochastic averaging method and compared with the uncontrolled structural response to evaluate the control efficacy. Numerical results are given to demonstrate the response mitigation capabilities of the proposed stochastic optimal control method for coupled adjacent building structures.

Application of the Robust Pole Assignment Technique for Vibration Control of Structures

This paper describes robust vibration framed structure control with active damping consisting of a displacement sensor and an active mass actuator. An H∞ controller based on a reduced order structural model is designed to suppress spillover destabilization. Model error is considered multiplicative uncertainty between the reduced order model and the high precision order model. Robust pole assignment is used to alleviate difficulty in controller design. This technique assigns the poles of a closed loop system in a desirable region on a complex plane by introducing affine transformation into the framework of the H∞ control theory. Experiments of the initial response and frequency response are performed to verify that the controller has robust stability and better control performance.

Reduced-Order Models of Structures with Viscoelastic Components

A useful method of modeling viscoelastic effects in structures for transient response analysis is to treat the frequency dependence of the usual damping model through the introduction of extra dissipation coordinates or internal variables. These new coordinates require a curve e t to the material loss factor data over a range of frequencies. Such a method has the disadvantage of introducing a large number of extra degrees of freedom that make response calculations and controller design very computationally intensive. Methods are examined to reduce the model size and, hence, ease this burden. Eigensystem truncation and balanced realizations are used to successfully reduce the full model. It is demonstrated that methods where the reduction transformation is based on the undamped model produce poor results. An iterative method to calculate the full eigensystem is introduced, using the frequency-dependent material modulus. In an alternative scheme, the model based on the physical degrees of freedom is reduced before the extra dissipation coordinates for the Golla ‐Hughes‐McTavish method are introduced. Also considered is the possibility of reducing the model at the element level, before assembly, although this approach has dife culty retaining the damping characteristics of the full model.

Reduced-order models of structures with viscoelastic components

Reduced-order models of structures with viscoelastic components

Theory for the estimation of structural vibration parameters from incomplete data

N system identification and modal analysis, the investigator is often faced with the problem of having to construct a model of finite order to represent a continuous vibrating structure. In other words, he is charged with obtaining a reduced-order model which, by definition, must be limited in its representation of the continuous system to a finite range of frequencies. The quality of the reduced-order model will be contingent upon the completeness of the measured data: a common cause of incompletenes s occurs when the number of modes excited in the test is fewer than the number of measurement stations (i.e., less than the order of the identified model). When it is required to construct a structural model (in terms of mass, stiffness, and damping parameters) from measured data, it is vital that such a model should in some sense be physically meaningful. The problems associated with the construction of reduced-order structural models from incomplete data have been highlighted by Berman and Flannelly1 and Herman.2 It is obvious that a reduced-order model cannot possibly reproduce the behavior of a continuous structure across an infinite range of frequencies. In the real structure the mass is distributed, whereas in the model the mass is discretized. In the finite-element method, consistent mass and stiffness matrices may be formulated by the minimization of an energy functional and, for this reason, such matrices may be considered to possess physical meaning. In reality, finite-element models do not exactly represent the dynamics of vibrating structures because they are incapable of dealing with boundary conditions such as imperfect hinges or flexibility in welded joints. It is then necessary to modify the finite-element model slightly in order that it should replicate the observed dynamics of the vibrating structure in a required range of frequencies. Methods have been developed for the estimation of structural parameters based on measured modal data. Chen and Garba3 presented a least-squares technique that may be applied iteratively together with matrix perturbation (for computation of the Jacobian matrix and reanalysis of the eigenvalues and eigenvectors) until the measured and computed eigendata are sufficiently in agreement. However, modal methods are known to have difficulty in treating closely spaced modes and can only deal with out-of-range modes by the addition of residual flexibilities. In this Note, a method is presented that does not depend on a modal decomposition but uses measured frequency responses to construct a structural model that varies minimally from an initial finite-element representation. A model constructed by this method is thus physically meaningful in the sense that a finite-element model is meaningful. A simple problem is contrived to illustrate an extreme case of incomplete data, which is solved using a continuous-fre quency- domain, least-squares filter.4 Using this approach, the Jacobian matrix is a constant matrix. Other problems associated with measure