Free-interface Component Mode Synthesis Method Research Articles

Nonlinear aeroelastic analysis of a twin-tail boom UAV with rudder freeplay nonlinearities

The issue of nonlinear structural freeplays in aircraft has always been a significant concern. This study investigates the aeroelastic characteristics of a twin-tail boom Unmanned Aerial Vehicle (UAV) with simultaneous freeplay nonlinearity in its left and right rudders. A comprehensive Limit Cycle Oscillation (LCO) solution route is proposed for complex aircraft with multiple freeplays, which can consider both accuracy and efficiency. For the first time, this study reveals the unique LCO characteristics exhibited by twin-tail boom UAVs with rudder freeplays and provides simulations and explanations of interesting phenomena observed during actual flight. The governing equations are established using the free-interface component mode synthesis method, and the LCOs of the system are mainly solved through the improved time-domain numerical continuation method and frequency-domain numerical continuation method. Furthermore, the study investigates the influence of the left and right rudder freeplay size ratio on the LCO characteristics. The results demonstrate that the twin-tail boom UAV exhibits two stable LCO types: close and differing left and right rudder amplitudes. The proposed method successfully describes the complete LCO behaviors of the system. Overall, this study makes significant contributions to our understanding of the aeroelastic behavior of twin-tail boom UAVs with rudder freeplays.

Extended component mode synthesis method for dynamic analysis of mechanical systems with local nonlinearities

Engineering structures are generally designed based on linear elasticity assumptions. However, it is difficult to describe the connected structures using a simple linear system, due to factors such as sliding and friction at interfaces, and variations in contact areas during vibration processes. Therefore, it is necessary to develop an approach to tackle such systems with interface nonlinearities. In this paper, we proposed an extended free-interface component mode synthesis method. According to the proposed method, the substructures are separated at the nonlinear interfaces. Then the component mode synthesis procedure is carried out. By reasonably neglecting the contributions of higher order modes to the damping and inertia terms, it is demonstrated that the nonlinear interface forces and their time derivatives can be expressed as functions of themselves, as well as modal displacements and velocities. This characteristic facilitates the synthesis of the reduced order models for substructures in the state space. Also owing to this operation, the original properties of nonlinear interface forces can be preserved as much as possible, without any linearization. This method is suitable for non-proportionally damped structures with local nonlinearities such as mechanical structures with nonlinear spring and dashpot connections, rotor-bearing systems and so on. It also may be an alternative for the dynamic analysis of jointed structures with contact nonlinearities. Two numerical examples are presented to demonstrate the capability of the proposed method: (1) an eighteen degrees of freedom spring-dashpot-mass system with cubic spring and cubic dashpot interface connections is studied and (2) a more complex mechanical structure: a dual-rotor system with deep-groove ball bearing inter-shaft support. The numerical simulation results indicate that the dynamic responses of the reduced order model match very well with that of the full model, revealing the high accuracy of the proposed method with low computational costs.

Experimental and numerical flutter analysis of a folding fin with multiple asymmetric free-plays

Experimental folding fin models with an adjustable free-play are tested in a wind tunnel. The fin structure is modeled using the free-interface component mode synthesis method, and its free-play is modeled as four independent nonlinear springs with asymmetric stiffness. A nonplanar unsteady vortex-lattice method considering compressibility is employed to address nonlinear deformation and high subsonic flow. Surface spline interpolation is improved through projection and partition. The aeroelastic characteristics of folding fins with different free-play magnitudes, initial conditions and elastic-axis positions are analyzed using an established time-marching method because of its relatively small computation scale and high precision. The results show good consistency among the presented method, the wind tunnel test and the harmonic balance method. There is a negative correlation between the critical speed of divergent motion and the ratio of the initial condition to the free-play magnitude. If either the free-play magnitude or the initial condition is extreme (tiny or vast), the system nonlinearity degenerates to linearity. Generally, the flutter prevention design of a linear model can be applied to a nonlinear model, such as moving the elastic-axis position aftward. The presented fin configuration exhibits an unstable limit cycle oscillation because the orders of coupled flutter modes do not change with variations in equivalent linear stiffness.

A hybrid Bloch mode synthesis method based on the free interface component mode synthesis method

For band-structure computation, the Bloch mode synthesis (BMS) or the generalized Bloch mode synthesis (GBMS) method is a series of model-reduction computational algorithms based on the Craig-Bampton method. A hybrid Bloch mode synthesis (HBMS) method with a hybrid Floquet-Bloch periodic boundary condition is proposed in this paper. A force-based Floquet-Bloch periodic boundary condition is derived and combined with the default displacement-based Floquet-Bloch periodic boundary condition to form the hybrid Floquet-Bloch periodic boundary condition. The hybrid BMS method has been developed by combining the hybrid Floquet-Bloch periodic boundary conditions with the free-interface component mode synthesis method. The HBMS method considers the influence of higher-order modes on periodic boundary structures so that the hybrid Floquet-Bloch periodic boundary conditions can be applied to the reduced-order modal space without introducing errors, which improves the computational efficiency. According to different truncation errors, the HBMS method has been developed into three algorithms, and they each have a different precision-efficiency ratio. Numerical experiments demonstrate that the developed method outperforms in accuracy and computational efficiency compared to the default BMS/GBMS method.

Nonlinear vibration analysis and stability analysis of rotor systems supported on SFD by combining DQFEM, CMS and IHB methods

In this paper, the differential quadrature finite element method (DQFEM), component mode synthesis (CMS) method and incremental harmonic balance (IHB) method are combined to propose a methodology to perform vibration analysis and stability analysis of rotor systems supported on fluid-film bearings that exhibit localized nonlinearities such as squeeze-film damper (SFD). In multi-degree of freedom systems with SFD, a method for dynamic analysis of the system by reducing the degrees of freedom is first suggested in this paper. DQFEM are used to create the mass matrix, stiffness matrix, and gyroscopic matrix of the system. Then, the system is divided into linear system and nonlinear system, and the motion equation of the linear system reduces the degrees of freedom using the free interface CMS method, and the motion equation of the nonlinear system is written without reducing the degrees of freedom. By synthesizing the equations of motion of each component, the motion equation of the entire system is created. That is, the motion equation of the whole system uses hybrid coordinates. The motion equation of the system is solved using the modified IHB method. To evaluate the effectiveness of the suggested method, the computed solutions are compared with the results presented in the previous literature, and the results are very consistent. Also, stability analysis of the obtained solutions is performed by the Floquet theory. The method suggested in the paper can be effectively used to analyze the dynamic behavior of rotor systems exhibiting localized nonlinearities such as SFD.

Multi-mode vibration attenuation of mistuned bladed disks by frictional tuned mass dampers array

An effective method is presented for multi-mode vibration attenuation of mistuned bladed disk (blisk) by employing ring-like array structure composed of multiple frictional tuned mass dampers (FTMDs). The multi-mode vibration attenuation is realized by arranging multiple series of FTMDs. A novel design scheme of the FTMD array is presented. 3D finite element method is employed for quantitative modelling of the blisk-FTMD array system, and the rotordynamic effects of Coriolis forces, spin softening and centrifugal stiffening are taken into account to reach better tuning in natural frequencies in the rotating conditions. The free-interface component mode synthesis method is used to reduce the order of the model. An efficient technique, which combines the harmonic balance method, the analytical formulation of Jacobian matrix and the parameterized Newton-Raphson method, is employed to solve the nonlinear equations. Finally, a blisk, with its first 4ND and 5ND modes being targeted, is employed as an example to illustrate the effectiveness of the presented method. Numerical results show that the method can obviously mitigate the multi-mode vibration of blisk with wide levels of mistuning and is robust against the small tuning deviations of the oscillators.

Model order reduction of thermo-mechanical coupling flexible multibody dynamics via free-interface component mode synthesis method

A systematical procedure of model order reduction for the thermo-mechanical coupling flexible multibody system (FMS) is proposed. The displacement and temperature field are discretized by unified element mesh based on the absolute nodal coordinate formulation (ANCF). Since the expression of the elastic forces is highly nonlinear and the tangential stiffness matrix of ANCF is time varying, the Taylor expansion is used to linearized the dynamic and heat transfer equations to obtain constant tangential stiffness matrix. The thermo-mechanical coupling system is divided into several substructures and the free-interface component mode synthesis method is used at the interface. The modal truncation for the dynamic and thermal coordinates are performed to reduce the system degrees of freedom (DOF). Therefore, the computational efficiency is greatly improved. The system thermo-mechanical equations are reunited and the solving scheme is given. Numerical results show that the proposed method can significantly improve the simulation efficiency without losing accuracy.

Considering higher-order effects of residual attachment modes in free-interface component mode synthesis method for non-classically damped systems

In this paper, by considering higher-order effects of residual attachment modes, a new free-interface component mode synthesis (CMS) method is developed for non-classically damped systems. Firstly, based on the Neumann expansion, the higher-order effects of the residual attachment modes are accurately expressed by the available lower-order modes and the system matrices. Secondly, by using the residual attachment modes and complex modes, the non-classically damped subcomponents are reduced and assembled in the physical space instead of the frequently-used state-space. Then, the new free-interface CMS method is derived and solved built on a frequency shifting technique and an iteration method. Finally, the performances of the presented method are illustrated by two numerical examples and compared with other first- and second-order approximated CMS methods. The results indicate that the presented CMS method for the non-classically damped systems is more accurate than other methods and suitable for the case when both high- and low-order modes are truncated.

A Novel Gap Element for the Coupling of Incompatible Interface in Component Mode Synthesis Method

The component mode synthesis (CMS) methods are often utilized for modal analysis to investigate the vibration characteristics of the complex structures which are commonly divided into several substructures. However, non-matching finite element meshes may occur at the interfaces between components and virtual gaps are easily produced along the curved interfaces, which limit the application of CMS and lead to larger numerical errors for vibration analysis. To overcome the problem, a novel gap element method (GEM) is employed into a free-interface CMS method in this paper, where both displacements and forces of the nodes on the incompatible interfaces are introduced by two independent Lagrange multipliers to enforce the compatibility conditions. Two-dimensional numerical examples are given to validate the effectiveness of the proposed method. The results of natural frequencies and modal shapes obtained using the proposed method agree very well with the ones obtained using full finite elements model, no matter the gaps along the interface exist or not. The influence of the number of nodes on the non-matching interfaces on the accuracy of frequencies is also discussed.

A model-updating approach based on the component mode synthesis method and perturbation analysis

A model-updating approach based on an improved free-interface component mode synthesis method is proposed. First, the equivalent higher-order matrix of the system is developed using a set of linearly independent vectors to capture the effects of the neglected higher-order modes. Second, the perturbation method is applied to derive the change for both eigenvalues and eigenvectors of each component while implementing a modification to the updated finite element model. The perturbation for both the eigenvalues and eigenvectors of each component are used to formulate the synthesis equations containing the perturbation expression. Next, the model-updating process is conducted using the synthesis model for which the degrees of freedom have been greatly reduced. The reduction in the total degree of freedom speeds up the solution process for the model-updating problem. Finally, the computational efficiency and accuracy of the presented method is demonstrated using a wing structure and by modal testing of a bolted plate.

Dynamic Analysis of an Offshore Platform With Compressor Packages—Application of the Substructure Method

This paper presents the substructure-based dynamic analysis of an offshore platform with compressor packages. Three typical substructure methods, the Guyan condensation method, the fixed-interface component mode synthesis (CMS) method and the free-interface CMS method, were compared to identify the appropriate substructure method for this application. A mode truncation criterion was proposed to ensure the accuracy of the recommended substructure method. The results indicated that the free-interface CMS method could generate almost the same results as the fully coupled method and save more than 50% in calculation time and more than 60% in storage space. When the same amount of time was used, the free-interface CMS method obtained more accurate results than the fixed-interface CMS method and Guyan condensation method; thus, the use of this method for evaluating the dynamics of an offshore platform with compressor packages was recommended. The cutoff frequency of the substructure was suggested to be 1.25 times the highest frequency of interest when conducting a dynamic analysis of an offshore platform with compressor packages using the free-interface CMS method. In addition, the offshore platform is a flexible structure with low and dense mechanical natural frequencies (MNFs), with approximately 4500 orders vibration modes in the frequency range of 0–40 Hz, and the displacement response at the area around the compressor package exceeded the allowable value under the excitation of the compressor package.

A free interface component mode synthesis method for viscoelastically damped systems

The analysis of large, complex systems can be simplified and more computationally efficient if smaller, simpler subcomponents can be treated and assembled. Component Mode Synthesis (CMS) methods allow one to reduce the order of the model at the subcomponent level. Although CMS methods are widely applied in the field of structure dynamics for undamped or viscously damped systems, their counterparts for viscoelastically damped systems seem not well-developed. Motivated by such points, this paper develops a model-order-reduction approach for viscoelastically damped systems based on a free interface CMS method. Due to the fact that the nonviscous modes and higher-order elastic modes are difficult to be obtained accurately for viscoelastic systems, the mode truncation error is often introduced. To reduce the mode truncation error problem, two residual attachment modes of viscoelastically damped systems are proposed in terms of the lower modes and system matrices by using the first one or two terms of Neumann expansion. Finally, three numerical studies are provided to illustrate the accuracy and effectiveness of the derived methods. It is shown that the proposed methods can receive good results and reduce the modal truncation error significantly over the conventional method.

Element-by-element model updating of large-scale structures based on component mode synthesis method

Component mode synthesis (CMS) method is developed and applied to the element-by-element model updating of large-scale structures in this study. Several lowest frequencies and mode shapes of the global structure are obtained with the free interface CMS method by employing the several lowest frequencies and mode shapes of each substructure individually. In this process, the removal of higher modes is compensated by the residual modes. The eigensensitivity of the global structure is then assembled from the eigensensitivities of each substructure to the updating element parameters. Subsequently, the global model is updated using the sensitivity-based optimization technique. The application of the present method to an 11-floor frame structure and to a large-scale structure demonstrates its accuracy and efficiency. The computational time required by the substructuring method to calculate the eigensensitivity matrices is significantly reduced, as compared with that consumed by the conventional global-based approach. Selection of the number of master modes is also proposed.

Identification of Nonlinear Structures by the Conditioned Reverse Path Method

The modeling method and identified method adapted to multi-degree-of-freedom structures with structural nonlinearities are established. The free-interface component mode synthesis method is used to establish the nonlinear governing equations by extending the connected relationships. Based on the modeling method, the conditioned reverse path method is used to identify frequency response functions of the nominal linear system without the influence of nonlinearities, and the parameters of nonlinearities are identified in the frequency domain. Nonlinear analysis and identification of a typical folding wing configuration with three free-play hinges are investigated. The nonlinear governing equation is established based on present methods, and the computational results are checked by finite-element programming. Based on coherence function, the effects of a nonlinear function are evaluated. The frequency response funtion of the nominal linear system and parameters of the nonlinear function are identified by the conditioned reverse path method within the noise signal. A ground vibration test of an all-movable fin is performed. The results show that the extending component mode synthesis method in the present work could be used to establish governing equations with structural nonlinearities. Based on the modeling method, the conditioned reverse path method can be used to identify the structural nonlinearities accurately.

Free-Interface Component Mode Synthesis Method with Link Substructure as Super-Element

According to the definition of link substructure in component mode synthesis (CMS) method, link substructure is essentially a kind of super element whose master degree of freedoms (DOF) is the interface DOF. Based on this, a new method of free-interface CMS, compatible for both displacement and force on interfaces, was proposed, by transforming link substructure into super element with Guyan static condensation or dynamic condensation. The new method not only retains free-interface CMS's advantages of reducing the system DOF efficiently and high accuracy, but also can deal with lumped damping reasonably, thus, it has a widespread application prospect in dynamic analysis of the structures with local non-linearity. Then, the application of the proposed technique was shown by modal and seismic response analysis of a truss bridge in which the girder and brace are lead rubber bearings (LRB) linked, in the automated container terminal. Regarding LRB as super element link substructure, the calculation accuracy and efficiency of Guyan and dynamic condensation methods are compared with finite element method (FEM) or direct integration method. Furthermore, inherent characteristic results of the truss bridge under different LRB disposition forms were obtained.

Inability of Free-Interface Modes in Representing Stress Resultants Near the Free Interface

As the fundamental cause of the slow convergence of a direct free-interface component mode synthesis method, we point out the inability of free-interface normal modes in representing the stress resultants near the free interface of a substructure. Although the effects of the residual flexibility of a substructure on the solution convergence are well known, the present theoretical analysis would help understand more clearly the convergence problems of the direct free-interface substructure coupling method. [S0739-3717(00)01704-9]