Free-interface Component Mode 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 with multiple localized nonlinearities

This study proposes a methodology to analyze the nonlinear vibration characteristics of rotor systems with multiple localized nonlinearities adopting the Finite Element Method (FEM), free interface Component Mode Synthesis (CMS) method, and modified Incremental Harmonic Balance (IHB) method. The rotor system is supported by squeeze film dampers (SFDs) on both sides, and at the nodes of the SFD arrangement, strong local nonlinearities will appear due to fluid-film forces. The methodology to analyze the nonlinear vibration characteristics of the system by reducing the degree of freedom of the rotating system with multiple local nonlinear factors and combining with the IHB method is proposed for the first time in this paper. The FEM is used to write motion equations in components, and the CMS method is applied to reduce the degrees of freedom of linear components. The IHB method is used to solve the motion equations of the nonlinear system. The system has one linear component and two nonlinear components. For linear components, modal coordinates are used, and for nonlinear components, the original physical coordinate system is used. By synthesizing these three components, the motion equation of the whole system is created. In order to validate the effectiveness of the method, the results obtained by the proposed method are compared with the data in the published literature, and the system responses are considered when specific parameters are changed. The stability analysis of the calculated solutions is carried out using the Floquet theory.

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.

Modal Analysis of Complex Structures via a Sub-Structuring Approach

In this paper, the problems arising from determining the modal properties of large and complex structures are investigated. For this purpose, the free interface component mode synthesis method has been used. In the following, Singular-Value Decomposition (SVD) is applied as a powerful mathematical tool to determine the appropriate coordinates to participate in the coupling process. Also, the effective error resources including modal shear error and the continuous systems overlapping error and their solution are introduced. Initially, a discrete system has been employed to investigate the free interface component mode synthesis method. Eventually, the studied main samples in this research are beam, plate and cylindrical shell. It is worth noting that the application of this method on the cylindrical shell has not been observed in previous researches.

The Vibration of a Transversely Cracked Rotor Supported by Anisotropic Journal Bearings with Speed-Dependent Characteristic

This paper presents the vibration of a transversely cracked rotor supported by anisotropic journal bearings, where the speed-dependent characteristic of bearing is considered. A 3D finite element model and the contact-based approach are employed for the shaft and crack. The governing differential equations of the whole cracked rotor-bearing system were obtained by synthesizing the equations of the cracked shaft, the breathing crack and the journal bearings. In order to solve the computational difficulties caused by the high dimensions of model, the free-interface complex component mode synthesis method (CMS) is employed to reduce the order of the model. On this basis, the eigenvalue and the steady-state forced response of the cracked rotor-bearing system are obtained by the Hill’s method. Finally, the effects of the anisotropic and speed-dependent characteristics of bearings on the vibration of the system are studied. Numerical results show that both the two characteristics can significantly affect the response of the system. The anisotropy in the bearing leads to the split of resonant peaks and influence the amplitudes of the peaks. The speed-dependent characteristic mainly affects the responses at the speeds close to the resonant regions, because the parametric excitation effect of the resonance region is greater than other speeds.

Model-Order Reduction of Flexible Multibody Dynamics Via Free-Interface Component Mode Synthesis Method

Abstract A new method of model-order reduction for the flexible multibody system which undergoes large deformation and rotation is proposed. At first, the flexible multibody system is modeled by absolute nodal coordinate formulation (ANCF), and then, the whole motion process of the system is divided into a series of quasi-static equilibrium configurations according to a given criterion. Afterward, motion equation is locally linearized based on the Taylor expansion. Therefore, the constant tangent stiffness matrix is obtained and does not need to be updated until the next configuration. Based on the locally linearized motion equation, the free-interface component mode synthesis (CMS) method is adopted to reduce the degrees-of-freedom (DOF) of the flexible multibody system molded by ANCF. The generalized-α integrator is used to solve the reduced motion equation. To verify the accuracy and efficiency of the proposed method, three examples including a free-falling pendulum, a flexible spinning beam and a deployable sail arrays are presented. Results show that the proposed method is able to reduce the computing time and maintain high 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.

Formulation of Flexibility-Based Component Mode Synthesis for Transient Analysis

Flexibility-based component mode synthesis (F-CMS) is one of the most popular free-interface component mode synthesis (CMS) methods. In the F-CMS method, localized Lagrange multipliers are employed along the interface boundary as a constraint condition between substructures; thus, it offers various advantages, including the perfect independence of subdomains, the easy treatment of the incompatible interface mesh, and parallelized formulation. However, the use of the Lagrange multipliers requires nonstandard specialized eigenvalue analysis procedures. This, in turn, requires a considerable modification in the standard displacement-based time integrators for transient analysis by using the F-CMS method. In the present study, the localized Lagrange multipliers are condensed by exploiting residual flexibility, and this results in a standard displacement-based reduced-order model. A report is made of the performance of the proposed method as compared to several CMS methods, with favorable features observed for the proposed method.

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.

A Free Interface Component Mode Synthesis Approach for Determining the Micro-End Mill Dynamics

The prediction accuracy of the stability boundary in the machining process depends upon accurate estimation of cutting tool-tip dynamics. Note that the experimental modal analysis using direct impact at miniature end mill (typically 50–500 μm in diameter) is not feasible as it can result in tool failure. Consequently, alternative techniques such as experimental modal analysis using reciprocity theory and frequency-based receptance coupling substructure analysis (RCSA) have been used extensively for determining tool-tip dynamics. The experimental approach based on reciprocity theory assumes that the structure is symmetric (cross frequency response functions (FRFs) are same between two points of interest in a structure). RCSA requires a very fine frequency resolution and matrix inversion, which can lead to computational complexities. In addition, RCSA takes into account the FRFs only at the interface and free end, which can induce errors. Owing to these issues with existing approaches, this paper proposes a free-interface component mode synthesis (CMS) approach for estimation of micro-end mill dynamics. The effect of machine tool compliance including the collet–tool interface has been included for estimation of micro-end mill dynamics via a free-interface CMS approach wherein the experimental and analytical mode shapes are coupled. The predicted micro-end mill dynamics have been compared with RCSA and experimental modal analysis using reciprocity theory. Finally, the stability lobe diagrams for high-speed micromilling of Ti6Al4V has been made using the tool-tip dynamics from CMS, RCSA, and experimental technique using reciprocity theory and validated against experimental measurements for onset of instability.

Effects of unbalance on the nonlinear dynamics of rotors with transverse cracks

Unbalance is an inevitable issue in rotating machineries due to the manufacturing tolerances, the assembly errors, the damage in rotating components, etc. The effects of unbalance on the nonlinear dynamics of cracked rotors are investigated in this paper. In order to overcome the assumption of weight dominance, 3D finite element models are employed to discretize the rotor and several contact pairs are defined on the cracked surfaces to simulate the intermittent breathing behavior of crack. Then, the complex free interface component mode synthesis method (CMS) is employed to reduce the model’s order and to increase the computational efficiency. Finally, the obtained models are used to study the effects of concentrated and distributed unbalances on the nonlinear dynamic response of cracked rotors. The results show that the unbalance can significantly affect the breathing mechanism of crack and consequently dramatically change the rotor’s dynamic behaviors. Moreover, the different levels and orientations of unbalance lead to obviously different results. Besides, the responses of rotors with two asymmetric distributed unbalance are similar to those of rotors with a single concentrated unbalance though the rotors are overall balanced.

An improved component mode synthesis method for interval uncertainty analysis

In order to improve the computational efficiency of the large-scale structures with uncertainty parameters, this paper present a methodological approach for interval uncertainty treatment based on an improved free interface Component Mode Synthesis (CMS) method. Firstly, the structure is divided into substructures and the perturbation method is employed for the eigenvalue analysis of substructures with the interval non-deterministic characteristics. To reduce the mode truncation error, the residual flexibility matrix is considered by constructing a set of weighted orthogonal modal vectors with low-order system modal vectors and system matrices. Different from the previous studies, this method proposed in this paper avoids calculating directly inverse of the stiffness matrix, which makes it easier to get the residual flexibility matrix. Then the synthesis equations including parameters perturbation can be deduced in terms of the interface compatibility conditions. Finally, two examples including with a numerical example as well as an experiment example are given to demonstrate the effectiveness and the efficiency of the proposed method.

A 3D nonlinear finite element method for the dynamic analysis of rotating rotor with a transverse crack

A general and efficient method is presented in this paper for studying the effects of unbalance on the breathing mechanism of crack. Based on 3D finite element models combined with a nonlinear contact approach for crack modeling, the method is free from the assumption of weight-dominance and can be used to gain deep insights into the breathing mechanism of crack. In order to greatly reduce the computational time, a complex free-interface component mode synthesis (CMS) method is employed to reduce the order of the model. Based on the proposed method, the effects of unbalance on the breathing mechanism of crack are discussed. Numerical results show that the unbalance can lead to significant changes in the breathing of crack, even when the unbalance force is about an order of magnitude smaller than the self-weight. Moreover, the level and orientation of the unbalance have also remarkable effects on the breathing behaviors of crack. Besides, a new universal non-steady breathing phenomenon of crack is firstly found in this paper, which denotes that the breathing speed of a crack is fluctuated over one revolution when there exists residual unbalance in the cracked rotor.

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.

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.

A real decoupled method and free interface component mode synthesis methods for generally damped systems

This paper reports on the development of a new transformation method. In contrast to most existing mode transformation methods in which the first-order state-space equation of the damped vibration system is transformed into a decoupled form with complex coefficient matrices, using the decoupled method presented in this paper, the equation of the damped system can be decomposed into a decoupled equation with real coefficient matrices. Two new free interface component mode synthesis methods are also presented. The equivalent full-mode matrix of the damped structure is used to capture the effects of the higher-order modes. Additionally, this work modifies the compatibility conditions at the junctions that are employed in most of the previous component mode synthesis methods for generally damped systems. The first component mode synthesis method is performed in complex space, whereas the second method can be applied in real space. Because the coefficient matrices of the coupled equation constructed by the second component mode synthesis method are all real-valued, the solution of the eigenproblem for this coupled equation can be performed in real space as well. Additionally, numerical examples demonstrate the accuracy and validity of these two component mode synthesis methods.