Fixed-interface Component Mode Synthesis Research Articles

Dynamic characteristics investigation of a dual rotor-casing system considering the comprehensive stiffness of the intermediate bearing

This paper focuses on the vibrational properties of a dual rotor-casing coupled system with nonlinear clearance in the intermediate bearing, considering both the oil film stiffness and Hertzian contact stiffness between rolling elements and raceways. A comprehensive stiffness model for the intermediate bearing has been developed based on elastohydrodynamic lubrication (EHL) and Hertz contact theory. Subsequently, a dynamical model of the dual rotor-bearing-casing system is established using the finite element method, thereby verifying the feasibility of simulating the casing with beam elements. The dimensionality of the system equation is reduced using the fixed-interface component mode synthesis (CMS) method, and the resulting reduced-dimensionality equation is solved using the improved Newmark-β method. The effects of casing support, rotational speed, and radial clearance on the vibration characteristics of the system are investigated. Then, the comprehensive stiffness and coupled vibration characteristics of the system are analyzed to elucidate its intrinsic mechanism. Finally, the contact characteristics within the intermediate bearing under dynamic load are also studied. The results indicate that the support stiffness of the casing has a more significant impact on the second resonance caused by both the inner and outer rotors. The system is prone to experiencing varying compliance (VC) resonance at low rotational speeds. The excessive bearing clearance leads to complex vibrational characteristics of the system, making it challenging to predict its motion. The impact of minor changes in stiffness on system vibration is small when the comprehensive stiffness is large. At this time, the limited effect of oil film stiffness allows it to be disregarded in terms of comprehensive stiffness. Furthermore, the speed significantly influences the contact characteristics within the intermediate bearing under the unbalanced force of the rotor and gravity.

Study on dynamic characteristics of the gear-dual-rotor system with multi-position rubbing

To reveal the mechanism of gear meshing induced rubbing and investigate the nonlinear dynamic response characteristics of a gear-dual-rotor system with multi-position rubbing, a new finite element model (FEM) of the gear-dual-rotor system considering the gear meshing excitation is established based on the in-house beam element code. Then, the fixed interface component mode synthesis method (FICMSM) is used to establish the reduced gear-dual-rotor dynamic model. A rotor-rotor rubbing model between the inner and outer rotors (IORs) and a rotor-stator rubbing model are built through vectors considering whirling velocities. In addition to the general concern about the low-frequency components around rotating frequencies and transversal responses with rubbing, more attention is given to the high-frequency components around the gear meshing frequency and the torsional responses. Finally, some interesting phenomena are found as follows: gear meshing is a new cause of partial rubbing. Some combined frequencies between the rotating frequencies of IORs and modulation frequencies between the meshing frequency and rotating frequencies are found in the transversal and torsional responses.

Mobile Interface CMS Method for Vibration Characteristics Prediction of Mistuned Bladed Disk with Different Coupling Degrees

The prediction of vibration characteristics was studied in the mistuned bladed disk by the mobile interface prestressed component mode synthesis (CMS) superelement method. When the strongly, generally, and weakly coupling in the mistuned bladed disk, according to the results of the direct FEM method, the prediction accuracy of this method was verified and compared with the fixed-interface CMS method by using the relative error of dynamic frequency, vibration mode matching function, and dimensionless root mean square error of vibration amplitudes. It is pointed that for mistuned bladed disk in the strong coupling, the prediction accuracy of dynamic frequency and vibration amplitudes are higher by the mobile interface CMS method and the vibration modes are matched with the direct method. In weak coupling, the results of dynamic frequency and vibration modes predicted by the mobile interface CMS method and the fixed-interface CMS method are consistent with the direct method, but the vibration amplitudes’ prediction error of the mobile interface CMS method is lower than that of the fixed-interface CMS method. In general coupling, the mobile interface CMS method has higher dynamic frequency prediction accuracy at low order, and the two methods have comparable dynamic frequency prediction accuracy at high order. The vibration modes predicted by the two methods are matched with the direct FEM method, and the prediction accuracy of vibration amplitude by the mobile interface CMS method is better than that of the fixed-interface CMS method. The results indicate that the mobile interface CMS method could more accurately predict vibration characteristics of the mistuned bladed disk with different coupling degrees and could be an effective measurement for studying the vibration characteristics of the mistuned bladed disk system.

A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures

Based on the fixed interface component mode synthesis, a multiple and multi-level substructure method for the modeling of complex structures is proposed in this paper. Firstly, the residual structure is selected according to the structural characteristics of the assembled complex structure. Secondly, according to the assembly relationship, the parts assembled with the residual structure are divided into a group of substructures, which are named the first-level substructure, the parts assembled with the first-level substructure are divided into a second-level substructure, and consequently the multi-level substructure model is established. Next, the substructures are dynamically condensed and assembled on the boundary of the residual structure. Finally, the substructure system matrix, which is replicated from the matrix of repeated physical geometry, is obtained by preserving the main modes and the constrained modes and the system matrix of the last level of the substructure is assembled to the upper level of the substructure, one level up, until it is assembled in the residual structure. In this paper, an assembly structure with three panels and a gear box is adopted to verify the method by simulation and a rotor is used to experimentally verify the method. The results show that the proposed multiple and multi-level substructure modeling method is not unique to the selection of residual structures, and different classification methods do not affect the calculation accuracy. The selection of 50% external nodes can further improve the analysis efficiency while ensuring the calculation accuracy.

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.

A developed component mode synthesis for parametric response analysis of large-scale asymmetric rotor

A generalized and efficient method is proposed in this paper for parametric response analysis of large-scale asymmetric rotor. If the bearings are anisotropic in asymmetric rotor, the motion of the rotor-bearing system will be governed by ordinary differential equations with periodically time-variant coefficients in both fixed and rotating frame. Response analysis of such parametric system can be attempted by assuming solution based on Hill’s method. If the number of degree of freedoms (DOFs) of the original model is N and the number of terms considered in the truncated assumed solution is m, for response analysis one has to solve a set of equations with (2m+1)N unknowns. Calculating this kind of equations requires a lot of computational costs, and the application of 3D finite element model will lead to greater computational costs. To avoid this costly approach, the fixed-interface component mode synthesis (CMS) is employed to form a reduced-order model (ROM). Although conventional CMS is adequate enough to make response analysis of largescale structures, an optimized scheme still needs to be developed for the response analysis of large-scale asymmetric rotor to achieve high efficiency and accuracy. The optimized scheme concerning a single substructure and multiple substructures are established respectively. Finally, the developed approach is validated through a simple model and a large-scale realistic two-pole generator rotor. And factors that influence the parametric response are investigated.

Stabilized Solution to Spurious Mode Problem and Ill-Conditioning in Interface Force Based Substructure Coupling Method

There are two major types of substructure mode synthesis methods, i.e., the fixed-interface component mode synthesis and free-interface component mode synthesis. There are two coupling methods, the interface degrees of freedom based coupling method and the interface force based coupling method, the former one is referred to as the primary assembly method, and the latter is referred to as the dual assembly method. However, the dual assembly method is theoretically shown to be unstable in this research, such reduced stiffness matrix is indefinite, this fatal weakness can be conquered by further interface reduction, and the interface compatibility is therefore rigorously enforced. Unfortunately, Craig’s method leads to another numerical instability when inverting a submatrix of residual flexibility on the interface degrees of freedom, this problem is neglectable in small dimensional matrix problems, but it is prominent in large models when the number of interface degrees of freedom is large, this ill-conditioning problem may be circumvented by truncated singular value decomposition technique; here, a more efficient strategy is proposed, the substructure reduction is modified, this modification proves to be numerically stable, and it can be even more accurate than the prevailing Craig-Bampton method; the numerical examples validate the suggestion.

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.

Seismic analysis of the APR 1400 reactor vessel internals using the model reduction method

ABSTRACTTo secure reliability of the seismic design of the reactor vessel internals (RVIs) through the finite element analysis, it is important to develop the accurate analysis model which can represent the geometric complexity of the RVIs. However, the seismic analysis requires too large computation cost to solve the complex equations; thus, it needs to reduce the overall size of the analysis model. Here, we apply a model reduction method based on the fixed-interface component mode synthesis (CMS) method to practical RVIs to solve complex numerical problems efficiently. To verify the model reduction method, several cases of the RVIs with different conditions are analyzed for the static and dynamic problems. Finally, the seismic analysis was performed with the suggested reduced model with the CMS method. The time history analysis is performed to extract important seismic responses at the specified locations, and the stress analysis is also performed to identify that the RVIs satisfy the seismic design. In the last part of the paper, an example of the design modification is suggested to reduce the stress intensity at the support locations.

Comparison of Common Methods in Dynamic Response Predictions of Rotor Systems with Malfunctions

The efficiency and accuracy of common time and frequency domain methods that are used to simulate the response of a rotor system with malfunctions are compared and analyzed. The Newmark method and the incremental harmonic balance method are selected as typical representatives of time and frequency domain methods, respectively. To improve the simulation efficiency, the fixed interface component mode synthesis approach is combined with the Newmark method and the receptance approach is combined with the incremental harmonic balance method. Numerical simulations are performed for rotor systems with single and double frequency excitations. The inherent characteristic that determines the efficiency of the two methods is analyzed. The results of the analysis indicated that frequency domain methods are suitable single and double frequency excitation rotor systems, whereas time domain methods are more suitable for multifrequency excitation rotor systems.

Dynamic Analysis of Satellite Antenna System with Joint Clearance and Reflector Flexibility

AbstractA methodology for modeling and analysis of satellite antenna systems is presented considering the effects of the joint clearance and reflector flexibility in the system. The joint clearance model is established based on a thorough geometric description of the eccentricity vector. The contact-impact forces are evaluated based on a Hertz contact theory and a modified Coulomb’s friction law. The fixed-interface component-mode synthesis method and Lagrange’s equations are used to achieve a lower-order dynamic model by modal truncation. Furthermore, in the process of dynamic modeling, the changes of topology were expressed as constraint variations; namely, adding or deleting the variable structure constraint in the dynamic model. Finally, the multibody dynamics of satellite antenna are analyzed by considering a flexible reflector with ideal and nonideal joints. Thus, the simulation results can predict the dynamic characteristics of satellite attitude and antenna pointing with the effects of clearance a...

Transient non-linear simulation with component mode synthesis

A method is presented to incorporate non-linear material behavior in a transient simulation with component mode synthesis reduction. This method is used in conjunction with rigid body solution techniques to expand the useful range of rigid body design tools for the development of complex mechanisms. A fixed interface component mode synthesis technique is enhanced with an algorithm to approximate the effect of plastic deformation during a dynamic simulation. The plastic strain is determined from the elastic modal response using classical plasticity theory and applied to the modal solution by projecting an effective nodal force vector on the modal coordinates to induce the necessary plastic deformation. This method can be used during the design process to approximate the non-linear dynamic response of complex mechanisms and offer significant computational savings over a full fidelity, non-linear dynamic solution.

Component Mode Synthesis Using Nonlinear Normal Modes

This paper describes a methodology for developing reduced-order dynamic models of structural systems that are composed of an assembly of nonlinear component structures. The approach is a nonlinear extension of the fixed-interface component mode synthesis (CMS) technique developed for linear structures by Hurty and modified by Craig and Bampton. Specifically, the case of nonlinear substructures is handled by using fixed-interface nonlinear normal modes (NNMs). These normal modes are constructed for the various substructures using an invariant manifold approach, and are then coupled through the traditional linear constraint modes (i.e., the static deformation shapes produced by unit interface displacements). A class of systems is used to demonstrate the concept and show the effectiveness of the proposed procedure. Simulation results show that the reduced-order model (ROM) obtained from the proposed procedure outperforms the ROM obtained from the classical fixed-interface linear CMS approach as applied to a nonlinear structure. The proposed method is readily applicable to large-scale nonlinear structural systems that are based on finite-element models.

Approximation of Power Flow Between Two Coupled Beams Using Statistical Energy Methods

In this work, an investigation is performed into developing a general framework for predicting the power flow between coupled component structures with uncertain system parameters. A specific example of two coupled beams is considered, in which a torsional spring is attached at the coupling point to adjust the coupling strength. The power flow in the nominal system is formulated using component mode synthesis (CMS). First, the parameter-based statistical energy method, which employs free-interface component modes, is applied to obtain approximations for the ensemble-averaged power flow with each beam length having a uniformly-distributed random perturbation. Then, using fixed-interface component modes and constraint modes, the Craig-Bampton method of CMS is employed to formulate the nominal power flow equation in terms of the constraint-mode degrees of freedom. This fixed-interface CMS method is seen to provide a systematic and efficient platform for power flow analysis. Using this CMS basis, a general approximation for the ensemble-averaged power flow is formulated regardless of the probability distribution of the random parameters or the coupling strengths between the substructures. This approximation is derived using Galerkin’s method, in which each modal response is expanded in locally linear interpolation functions in the random system parameters. The proposed general framework is numerically validated by comparisons with wave approximations from the literature for this two-coupled-beam system.

AN ALGORITHM FOR RESPONSE AND STABILITY OF LARGE ORDER NON-LINEAR SYSTEMS — APPLICATION TO ROTOR SYSTEMS

A numerical algorithm to calculate the periodic response, stability and bifurcations of a periodically excited non-conservative, Multi-Degree of Freedom (MDOF) system with strong local non-linearities is presented. First, the given large order system is reduced using a fixed-interface component mode synthesis procedure (CMS) in which the degrees of freedom associated with non-linear elements are retained in the physical co-ordinates while all others, whose number far exceeds the number of non-linear DOF, are transformed to modal co-ordinates and reduced using real mode CMS. A shooting and continuation method is then applied to the reduced system to solve for the periodic response. Floquet stability theory is used to calculate stability and bifurcations of the periodic response. The algorithm is applied to study the response to imbalance, stability, and bifurcations of a 24 DOF flexible rotor supported on journal bearings. The results indicate that the proposed algorithm, though approximate, can yield very accurate information about dynamic behavior of large order non-linear systems, even with few numbers of retained component modes. The algorithm, which imposes less demand on computer time and memory, is believed to be of considerable potential in analyzing a variety of practical problems.