Modal Reduction Method Research Articles

Dynamic modeling and trajectory optimization for the rigid-flexible coupled spacecraft with the free-floating manipulator and solar panels

The rigid-flexible coupled spacecraft, composed of flexible solar panels and a multilink manipulator, has gained prominence in on-orbit servicing due to rapid advancements in space technology. However, the intricate effects of rigid-flexible coupling pose significant challenges for dynamic modeling, trajectory planning, and control. This paper aims to develop general dynamic approaches for modeling and trajectory planning in such spacecraft, considering large deformations. The main distinguishing feature is the use of the referenced nodal coordinate formulation to accurately describe the large-deformed solar panels rather than directly treating them as disturbance for the free-floating system. Additionally, the common recursive model for the multilink manipulator is integrated into the same framework. The modal reduction method with modal derivatives techniques is employed to address geometric nonlinearity resulting from large deformations. Polynomial trajectory parameters with different performance characteristics are obtained by defining various optimization objectives. The coupling analysis is conducted based on an accurate reduced-order dynamic model, the results of which can be used for designing manipulator tasks. Coupling values are defined as the objective for trajectory optimization, offering advantages such as insensitivity to dynamic model accuracy and a fast optimization process. After validating the accuracy of the proposed dynamic model through simulations, the performances of different optimized trajectories are demonstrated using numerical examples.

Efficient Forced Response Minimization Using a Full-Viscosity Discrete Adjoint Harmonic Balance Method

A forced response induced by inlet distortion and blade row interactions can lead to high-cycle fatigue failure, especially when the unsteady excitation frequency is close to the natural vibration frequency of a blade. This paper presents the study of forced response sensitivity analysis and minimization using a full-viscosity unsteady discrete adjoint method. The goal is to improve the aeroelastic performances of turbomachinery blades and simultaneously constrain/improve aerodynamic performances. The decoupled modal reduction method is used to compute the forced response, which is considered the objective function in adjoint-based design optimization. To analyze aeroforcing and aerodamping flow and adjoint fields efficiently, the harmonic balance method and its adjoint counterpart developed by an automatic differentiation tool are applied. Two cases—the NASA Rotor 67 with inlet distortion and a three-row configuration with multiple fundamental frequencies in the second row—are used to demonstrate the effectiveness of the aerodynamic and aeroelastic coupled design optimization system developed in this work. For the latter case, the Fourier-transform-based method that first decomposes and then matches the time and space modes at two sides of an interface is used for interface coupling. The almost-periodic Fourier transform method is used to determine the time instances for cases with multiple fundamental frequencies.

Reduced-order Model Parameterization for Uncertain Lti Siso Systems

Abstract The primary goal of this paper is to develop a formal foundation to design an adaptive output feedback predictor for a class of unknown systems where parameters and order are unknown or high-dimensional. We present a reduced-order adaptive output-predictor scheme based on modal reduction and Lyapunov's method. Moreover, the credibility of the proposed reduced-order adaptive output-predictor scheme is validated by mathematical proof and numerical studies.

A hybrid finite element and extended transfer matrix method for the dynamic modeling of fluid-conveying pipeline with breathing cracks

In this study, a hybrid finite element and extended transfer matrix method (FEM-ETMM) is formulated to simulate the breathing effect and fluid disturbances of a fluid-conveying cracked pipeline (FCCP). Furthermore, the frequency spectra of structural intensity (SI) responses and the normalized displacement-structural intensity (D-SI) curves are used to detect cracks. In this FEM-ETMM, the finite element method (FEM) is adopted to determine the contact state and establish the equivalent contact model, and the extended transfer matrix method (ETMM) is obtained by constructing the state transfer matrix in the crack region. Based on ETMM, the time-varying fluid disturbances are obtained. Furthermore, the model modal reduction and parametric modeling methods are used to improve solution efficiency. Finally, the effects of the crack parameters on vibration responses with and without considering fluid disturbances are analyzed. The results show that SI responses can amplify the odd-mode harmonics, thereby accurately detecting cracks at different depths and locations. In addition, the fluid disturbances change the contact states and enhance the damping effect. The proposed modeling and analysis method will provide valuable insights for crack detection of pipelines.

Data-Driven Reconstruction of Dynamical Forces and Responses through Bayesian Expectation-Maximization and Modal Reduction Methods

This paper presents a new Bayesian approach for estimating input and state in linear structures using response-only measurements. The proposed approach benefits from a modally reduced state-space model, which circumvents the dimensionality of dynamical responses in complex structures through a low-dimensional subspace. It also substitutes unknown physical forces with a set of equivalent modal forces, which is beneficial when the magnitude and location of input forces are unknown. In this work, these forces are characterized through the conventional random walk model and a class of stationary Gaussian processes. Subsequently, an augmented state-space model is constructed to describe modal states and input loads. Based on this model, a Bayesian expectation-maximization (BEM) methodology is developed to identify the input, state, and noise parameters. This noise identification perspective activates uncertainty quantification in joint input-state estimation problems and enables quantifying the degree of confidence in the estimated quantities. When the proposed method is tested using numerical and experimental examples, accurate estimations and reasonable uncertainty bounds are acquired for the dynamical state and input forces. Although the literature reports the superiority of the Gaussian process latent force model over the random walk model without using a unified noise calibration strategy, this study, to our best knowledge, is the first effort to compare and interpret the results on a consistent basis where the noise and input characteristics are all identified from the data through BEM.

The simulation of the rigid-flexible coupling model of the tank firing on the move based on fully flexible body modeling method

Generally, the rigid-flexible coupling model based on the modal reduction method cannot realize the contact between the rigid body and the flexible body. Based on the multi-body systems dynamics and fully flexible body modeling method, the rigid-flexible coupling dynamics model of the tank gun system was constructed. Then by combining with the PID control model, the mechanical/control co-simulation model of the tank gun on the move was constructed. The influences of the control system and vehicle speed on the muzzle vibration during the shooting process were respectively analyzed. It has been shown that the control system model can effectively simulate the response characteristics of the muzzle during the move of the tank. Under the same road surface and the same control law, the frequency and amplitude of the muzzle angular displacement in the low frequency range increase with the increase of the vehicle speed. The constructed model and the simulation results are of great significance to the analysis and optimization of the shooting accuracy of the tank gun on the move.

Discrete material optimization of composite structures subjected to initial excitation for minimum residual vibration

This work studies design optimization of laminated structures subjected to initial excitation for minimum residual vibration. The Discrete Material Optimization (DMO) method with solid isotropic material with penalization (SIMP) model is employed to perform the design optimization of stacking sequence, fiber angles, and selection of material for the composite structures. The transient vibration analysis of the composite structures excited by initial excitation can be performed via the finite element method, where the residual vibration is identified. The integration of squared displacement of residual vibration can be selected as the objective function, which can be greatly simplified by Lyapunov equation to reduce the computational time. A modal reduction method is employed to further improve the computational efficiency of vibration analysis. Since many linear constraints are included in the optimization formulation for discrete material optimization, sequential linear programming (SLP) algorithm with a trust region approach is used to conduct the optimization iteration with the help of the design sensitivities obtained by an adjoint method. Numerical examples are presented to illustrate the effectiveness of the proposed method. The influences of the distribution and magnitude of several initial excitations on the optimized solutions are discussed.

Modal reduction-based finite element method for nonlinear FG piezoelectric energy harvesters

In this study, harvesting energy from a nonlinear functionally graded (FG) piezoelectric cantilever beam under harmonic excitation is investigated. The material properties of the piezoelectric are assumed to be a combination of piezo-ceramic and piezo-polymer materials for high performance and flexibility. The neutral axis is obtained in order to eliminate bending–stretching coupling. The geometrical nonlinearity and electromechanical coupling are incorporated in the coupled nonlinear equations that are derived using the generalized Hamilton’s principle and solved using the combination of modal reduction and finite element methods. The shooting method is employed to obtain steady-state periodic response of an FG nonlinear harvester with appropriate initial conditions. Also it is shown that at least two-mode approximation is required for accurate estimation of nonlinear response and harvested power. Using the method of nonlinear modal reduction, the unstable branches for frequency domain solution are estimated and the computational time is reduced considerably compared to full finite element method. A case study is also accomplished in detail to analyze the effects of base amplitude values and material distribution on harvested power and bandwidth.

Damage identification of structures using incomplete mode shape and improved TLBO-PSO with self-controlled multi-stage strategy

The present work presents an efficient multi-stage optimization method-based damage detection method for truss and frame structures equipped with a limited number of sensors. In this approach, a Finite Element (FE) model is developed to simulate the response of the actual structure. The limited sensor condition for this FE model is achieved by the modal reduction method. A comparison study among three well-established modal reduction methods has been performed, and Iterated Improved Reduction System (IIRS) approach has been selected for the present study. Next, the damage identification problem is defined as an unconstrained optimization problem. The objective function of the optimization problem is formulated using the weighted linear combination of the frequencies and mode shapes obtained from the actual damaged structure and reduced FE model. This objective function is minimized by the improved version of hybrid Teaching Learning Based Optimization - Particle Swarm Optimization (ITLBO-PSO) utilizing a self-controlled multi-stage (SCMS) strategy. In this method, the SCMS strategy automatically reduces the search dimension of the optimization problem in each stage. Four examples with different damage scenarios from the relevant literature are considered in the present study to demonstrate the efficacy of the proposed method. The proposed method results for both with noise and without noise are compared with existing literature and nine other well-established algorithms. The results show that the proposed ITLBO-PSO with SCMS strategy identifies damages with adequate precision and outperforms the other algorithms regarding the accuracy and computational cost.

Research Progress of Synthesis and Modification Methods Based on Dynamic Substructures

In the process of model modification of large and complex structures, substructure synthesis method and modal reduction method have been widely used, but there are still some difficulties in precision control and engineering application in the process of model updating. In order to better study the dynamic response of the vibration substructure, the synthesis and correction method of the classical dynamic substructure is described in this paper, which provides a new idea for further engineering development. In the aspect of substructure synthesis method, the modal reduction of substructures, two methods of classical substructure synthesis, mechanical impedance method, singular value decomposition method, rigidity-flexibility equivalence, and transformation of degree of freedom are analyzed. The advantages and disadvantages of the above methods are discussed. In terms of substructure modification, the reference datum method, function dynamic modification method, neural network model modification, and frequency response function modification are analyzed, and the shortcomings of the dynamic substructure modification method are summarized. Finally, the development trend of dynamic substructure synthesis and modification algorithm is proposed.

Generalized modal reduction method for the dynamic analysis ofrotating mechanical systems

The paper proposes modal reduction method of the dynamic systems composed of linear nonconservative sub-systems coupled by nonlinear discrete couplings. Classical approach to the modal reduction is based on thetransformation of the generalized coordinates by the real modal submatrix of the linear conservative part of thewhole system. In case of modal synthesis method, transformation matrices are the real modal submatrices of theconservative part of mutually isolated subsystems. Rotating mechanical systems contain gyroscopic effects andother influences of rotation and damping. The paper introduces a generalized modal reduction method based on thecomplex modal values of the whole system or the isolated subsystems. Their complex eigenvalues and eigenvectorsare used for transformation of the generalized coordinates and reduction of the number of degrees of freedom. Thepresented method is focused on vibrating rotating systems with gyroscopic and dissipative effects and nonlinearinternal couplings.

基于自由度缩聚的最小秩修正损伤诊断

To solve the problem that the measured modal parameters and the finite element analysis modal parameters do not match in damage identification, the residual force vector formula and the minimum rank update formula based on the modal reduction method were derived. Through the calculation of the residual force vectors of damaged and undamaged structures, the change rate of the residual force vector was obtained, and the absolute values of the residual force change rate vector elements were defined as the improved residual force vector. With the improved residual force vector, the correct number of modes used in the minimum rank update method was determined, and the minimum rank update formula based on modal reduction was used to calculate the damage extent. The results show that, under noise interference, the presented method could well locate the damage with incomplete modal data. The damage extent identification based on the derived minimum rank update formula is reliable. The method is robust and accurate in structural damage identification for measured incompleteDOF structures.

Reduced-Order Modeling for Stability and Steady-State Response Analysis of Asymmetric Rotor Using Three-Dimensional Finite Element Model

A generalized and efficient technique of reduced-order model (ROM) is proposed in this paper for stability and steady-state response analysis of an asymmetric rotor based on three-dimensional (3D) finite element model. The equations of motion of the asymmetric rotor-bearing system are established in the rotating frame. Therefore, the periodic time-variant coefficients only exist at a tiny minority of degrees-of-freedom (DOFs) of bearings. During the model reduction process, the asymmetric rotor-bearing system is divided into rotor and bearings. Only the rotor was reduced. And the physical coordinates of bearings are kept in the reduced model during reduction. Then, the relationship between the rotor and bearings is established by inserting periodic time-variant stiffness and damping matrix of bearings into the reduced model of rotor. There is no reduction to the matrices of bearings, which guarantees the accuracy of the calculation. This technique combined with fixed-interface component mode synthesis (CMS) and free-interface CMS is compared with other existing modal reduction method on an off-center asymmetric rotor and shows good performance.

Estimation of heat flux by using reduced model and the adjoint method. Application to a brake disc rotating

In this study, an approach for inverse problem is proposed to estimate a time dependent heat flux on a braking system. This approach combines modal reduction and conjugate gradient method with adjoint method. A brake disc with variable velocity and thermal solicitations is treated. In a first step, Branch Eigenmodes Reduction Method is used to build a reduced model which allows reduction of computing time while conserving a satisfying precision on temperature field. In a second step, an inverse algorithm based on adjoint method is applied to recover time dependent heat flux from temperature calculation. Compared to Beck method, adjoint method gives accurate results without any regularization. The proposed approach enables to perform a quasi on-line estimation.

Exact model reduction by a slow–fast decomposition of nonlinear mechanical systems

We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the softer degrees of freedom. This slow–fast decomposition (SFD) enslaves exponentially fast the stiffer degrees of freedom to the softer ones as all oscillations converge to the reduced model defined on a slow manifold. We obtain an expression for the domain boundary beyond which the reduced model ceases to be relevant due to a generic loss of stability of the slow manifold. We also find that near equilibria, the SFD gives a mathematical justification for two modal reduction methods used in structural dynamics: static condensation and modal derivatives. These formal reduction procedures, however, are also found to return incorrect results when the SFD conditions do not hold. We illustrate all these results on mechanical examples.

Numerical Modeling of Piston Lubrication with Body Deformation through Modal Reduction Method

An analysis for operating characteristics of piston lubrication system is performed based on the numerical model in this study. Dynamic piston lubrication characteristics such as oil film pressure and thickness distribution can be analyzed through a numerical model with an integration of elastohydrodynamics and multi-flexible-body dynamics (MFBD). In particular, the oil film thickness variation by elastic deformation is considered in the elastohydrodynamic analysis by using the modal reduction method in MFBD system. And this effect is reflected on the fluid governing equations to evaluate the oil film pressure in the lubrication region. A series of process proposed in this study is available for the analysis of realistic elastohydrodynamic lubrication phenomenon. A numerical example for the piston lubrication system is also demonstrated.

Reduction Methods Applied to Aircraft Brake Squeal Prediction and Simulation

Aircraft braking systems may be subjected to friction-induced vibrations, during which the brake is unstable and behaves as a source of mechanical vibrations. This is an issue for aircraft brake manufacturers because it may jeopardize structural integrity due to accelerated fatigue life or generate discomfort for the aircraft crew and passengers. Hence, there is a need for instability simulation and prediction methods that can be used as early as the design stage. At this stage, finite element models are available, but their sizes are prohibitive as far as vibration-level prediction is concerned, thus requiring model reduction. This paper presents a comparative study between several modal reduction methods, among them the well-known Craig–Bampton method. The reference model to be reduced is a full brake system finite element model. The system stability is assessed by means of the classical complex eigenvalue analysis. Convergence of the implemented reduction strategies is studied. Double modal synthesis is implemented, and an improved version is introduced. They appear to be efficient model reduction strategies for application to squeal prediction. By reducing all the system’s degrees of freedom on several sets of generalized coordinates, an excellent compromise between system size and precision can be found.

Model Order Reduction of Large-Scale Finite Element Systems in an MPI Parallelized Environment for Usage in Multibody Simulation

Abstract The use of elastic bodies within a multibody simulation became more and more important within the last years. To include the elastic bodies, described as a finite element model in multibody simulations, the dimension of the system of ordinary differential equations must be reduced by projection. For this purpose, in this work, the modal reduction method, a component mode synthesis based method and a moment-matching method are used. Due to the always increasing size of the non-reduced systems, the calculation of the projection matrix leads to a large demand of computational resources and cannot be done on usual serial computers with available memory. In this paper, the model reduction software Morembs++ is presented using a parallelization concept based on the message passing interface to satisfy the need of memory and reduce the runtime of the model reduction process. Additionally, the behaviour of the Block-Krylov-Schur eigensolver, implemented in the Anasazi package of the Trilinos project, is analysed with regard to the choice of the size of the Krylov base, the block size and the number of blocks. Besides, an iterative solver is considered within the CMS-based method.

Numerical methods for parametric model reduction in the simulation of disk brake squeal

We present numerical methods for model reduction in the numerical simulation of disk brake squeal. Automotive disk brake squeal is a high frequency noise phenomenon based on self excited vibrations. Our method is based on a variation of the proper orthogonal decomposition method and involves the solution of a large scale, parametric eigenvalue problem. Several important challenges arise, some of which can be traced back to the finite element modeling stage. Compared to the current industrial standard our new approach is more accurate in vibration prediction and achieves a better reduction in model size. This comes at the price of an increased computational cost, but it still gives useful results when the classical modal reduction method fails to do so. We illustrate the results with several numerical experiments, some from real industrial models, some from simpler academic models. These results indicate where improvements of the current black box industrial codes are advisable.

Nonlinear dynamic response of functionally graded shallow shells under harmonic excitation in thermal environment using finite element method

The nonlinear dynamic response of functionally graded material (FGM) shallow shells subjected to thermal and harmonic loads is studied using finite element method. The material properties vary continuously in the thickness direction based on a simple power law distribution. The equations of motion are obtained using modal reduction method based on the third order shear deformation theory. The shooting method is used to obtain appropriate initial conditions for having only steady state response. The effects of thickness ratio and radii of curvature on the dynamic response of FGM shallow shells are studied. Buckled equilibrium positions (BEPs) are obtained for thermally loaded FGM shallow shells with immovable middle or physical neutral surfaces. It is shown that there exist one stable and two unstable BEPs. Then the behavior of an FGM shallow shell subjected to harmonic excitation in thermal environment is investigated. It is seen that the thermally loaded FGM shallow shell has different responses for the same frequency and amplitude of excitation depending on its initial conditions.