Classical Component Mode Synthesis Research Articles

Evaluation of free interface-based reduction techniques for nonlinear forced response analysis of shrouded blades

The efficient dynamic stress assessment of turbine blades is of prime importance in turbomachinery design. An accurate prediction of forced response level of shrouded blades requires a very detailed finite element model in addition to a nonlinear solver. In order to perform nonlinear forced response analysis of blades at an affordable computational cost, applying a model order reduction technique is essential. The appeal for component mode synthesis methods in dimension reduction of structures with friction contacts is due to the possibility of retaining a subset of physical degrees of freedom (e.g. the contact degrees of freedom) in the set of generalized coordinates. In this paper, a reduction method recently developed for nonlinear forced response analysis of structures with local nonlinearity is evaluated and compared with two classical component mode synthesis reduction techniques. All three methods have the same projection basis, which includes residual flexibility attachment modes and free interface modes, but different implementation. The response is computed in the frequency domain using multiharmonic balance method and periodic contact forces are modeled with a node-to-node 3D friction contact model. In order to demonstrate the efficiency of the three formulations, a rod and a simplified shrouded turbine blade are considered as case studies.

Dynamic Analysis and Reduction of a Cyclic Symmetric System Subjected to Geometric Nonlinearities

The search for ever lighter weight has become a major goal in the aeronautical industry as it has a direct impact on fuel consumption. It also implies the design of increasingly thin structures made of sophisticated and flexible materials. This may result in nonlinear behaviors due to large structural displacements. Stator vanes can be affected by such phenomena, and as they are a critical part of turbojets, it is crucial to predict these behaviors during the design process in order to eliminate them. This paper presents a reduced order modeling process suited for the study of geometric nonlinearities. The method is derived from a classical component mode synthesis (CMS) with fixed interfaces, in which the reduced nonlinear terms are obtained through a stiffness evaluation procedure (STEP) procedure using an adapted basis composed of linear modes completed by modal derivatives (MD). The whole system is solved using a harmonic balance procedure and a classic iterative nonlinear solver. The application is implemented on a schematic stator vane model composed of nonlinear Euler–Bernoulli beams under von Kàrmàn assumptions.

Simulation of Moving Loads in Elastic Multibody Systems With Parametric Model Reduction Techniques

Abstract In elastic multibody systems, one considers large nonlinear rigid body motion and small elastic deformations. In a rising number of applications, e.g. automotive engineering, turning and milling processes, the position of acting forces on the elastic body varies. The necessary model order reduction to enable efficient simulations requires the determination of ansatz functions, which depend on the moving force position. For a large number of possible interaction points, the size of the reduced system would increase drastically in the classical Component Mode Synthesis framework. If many nodes are potentially loaded, or the contact area is not known a-priori and only a small number of nodes is loaded simultaneously, the system is described in this contribution with the parameter-dependent force position. This enables the application of parametric model order reduction methods. Here, two techniques based on matrix interpolation are described which transform individually reduced systems and allow the interpolation of the reduced system matrices to determine reduced systems for any force position. The online-offline decomposition and description of the force distribution onto the reduced elastic body are presented in this contribution. The proposed framework enables the simulation of elastic multibody systems with moving loads efficiently because it solely depends on the size of the reduced system. Results in frequency and time domain for the simulation of a thin-walled cylinder with a moving load illustrate the applicability of the proposed method.

Component Mode Synthesis Using Undeformed Interface Coupling Modes to Connect Soft and Stiff Substructures

Classical component mode synthesis methods for reduction are usually limited by the size and compatibility of the coupling interfaces. A component mode synthesis approach with constrained coupling interfaces is presented for vibro-acoustic modelling. The coupling interfaces are constrained to six displacement degrees of freedom. These degrees of freedom represent rigid interface translations and rotations respectively, retaining an undeformed interface shape. This formulation is proposed for structures with coupling between softer and stiffer substructures in which the displacement is chiefly governed by the stiffer substructure. Such may be the case for the rubber-bushing/linking arm assembly in a vehicle suspension system. The presented approach has the potential to significantly reduce the modelling size of such structures, compared with classical component mode synthesis which would be limited by the modelling size of the interfaces. The approach also eliminates problems of nonconforming meshes in the interfaces since only translation directions, rotation axes and the rotation point need to be common for the coupled substructures. Simulation results show that the approach can be used for modelling of systems that resemble a vehicle suspension. It is shown for a test case that adequate engineering accuracy can be achieved when the stiffness properties of the connecting parts are within the expected range of rubber connected to steel.

Component Mode Synthesis Using Undeformed Interface Coupling Modes to Connect Soft and Stiff Substructures

Classical component mode synthesis methods for reduction are usually limited by the size and compatibility of the coupling interfaces. A component mode synthesis approach with constrained coupling interfaces is presented for vibro-acoustic modelling. The coupling interfaces are constrained to six displacement degrees of freedom. These degrees of freedom represent rigid interface translations and rotations respectively, retaining an undeformed interface shape. This formulation is proposed for structures with coupling between softer and stiffer substructures in which the displacement is chiefly governed by the stiffer substructure. Such may be the case for the rubber-bushing/linking arm assembly in a vehicle suspension system. The presented approach has the potential to significantly reduce the modelling size of such structures, compared with classical component mode synthesis which would be limited by the modelling size of the interfaces. The approach also eliminates problems of nonconforming meshes in the interfaces since only translation directions, rotation axes and the rotation point need to be common for the coupled substructures. Simulation results show that the approach can be used for modelling of systems that resemble a vehicle suspension. It is shown for a test case that adequate engineering accuracy can be achieved when the stiffness properties of the connecting parts are within the expected range of rubber connected to steel.

Component mode synthesis methods using partial interface modes: Application to tuned and mistuned structures with cyclic symmetry

We present new component mode synthesis methods using partial interface modes which are the structure normal modes resulting from the static condensation of the structure to the interface between the substructures and which are possibly clamped at a part of this interface. These methods are the generalization of the classical component mode synthesis methods and those using the interface modes. These methods allow to reduce the number of the interface coordinates and at the same time to keep some of the physical interface displacements. These methods are applied to a structure with cyclic symmetry in both tuned and mistuned cases.

Intelligent interfaces for sound insulation: Numerical and experimental optimisation

Sound insulation is actually achieved thanks to passive multilayers. These ones are classically made of elastic, acoustic and poroelastic media. Good results are obtained in medium and high frequencies, but low frequencies disturbances still remain a problem. Indeed, local and global resonant behaviours of the panel, such as plate bending mode or the double-leaf phenomenon, occur and lower the performances. An active concept has already been proposed and validated. Piezoelectric components are added to the interface and act a secondary source aiming at minimising the sound transmitted. In this contribution, a double-plate system is studied. A reduced finite element model of an elementary cell is presented. This one relies on modal projection and classical component mode synthesis (CMS) procedures. This model is first updated and validated thanks to measurements performed on a dedicated test bench. The use of active control allows a 15 dB enhancement of sound insulation at low resonant frequencies. Numerical experiments are designed. Response surfaces are obtained thanks to kriging procedures. Finally, the configuration of the optimal interface is discussed in a physical point of view.

Proper orthogonal decomposition and component mode synthesis in macromodel generation for the dynamic simulation of a complex MEMS device

In this paper, we develop a novel method for the macromodel generation for the dynamic simulation and analysis of a structurally complex MEMS device, by making use of proper orthogonal decomposition (POD), also known as the Karhunen–Loève decomposition and classical component mode synthesis. The complex microelectromechanical systems (MEMS) device is divided into interconnected components and each of these components is treated separately using POD to extract its proper orthogonal modes (POMs) and their corresponding proper orthogonal values. The separate component responses are then expressed in generalized coordinates that are defined by the POMs. The requirements of the displacement and force compatibility at the interface of components serve as constraint equations among the component coordinates, and are used to construct a transformation relating the component coordinates to system coordinates. This transformation is used to formulate the low-order macromodel to determine system dynamic responses. Numerical results obtained from the simulation of pull-in dynamics of a non-uniform microbeam MEMS device subjected to electrostatic actuation force with squeezed gas-film damping effect show that the macromodel generated this way can dramatically reduce the computation time while capturing the device behaviour faithfully.

Component mode synthesis methods using interface modes. Application to structures with cyclic symmetry

Classical component mode synthesis methods can generate coupled systems with large size due to the important number of degrees of freedom at the interface. A method is proposed to reduce the number of interface coordinates in the component mode synthesis methods. This procedure, already available for the fixed interface method, is extended to the free and hybrid interface methods. It is based on the use of a truncated basis of interface modes in place of the constraint modes, which are introduced in the free and hybrid interface methods. Classical and new methods are applied to structures with cyclic symmetry in combination with the wave propagation properties. The eigensolutions and the frequency response obtained with these methods are compared on a numerical example.

A substructure synthesis method for non-symmetric viscoelastic systems

Based on a new variational principle in the time domain, a substructure synthesis method is developed for non-symmetric viscoelastic systems without precluding a symmetric formulation. The viscoelastic damping is assumed to be generally non-proportional and in an hereditary integral form. The traditional state space formulation is avoided by the proposed substructure synthesis method so that the approach is independent of viscoelastic models. On the basis of the Rayleigh-Ritz approach, each substructure is approximated by a set of admissible trial vectors. In accordance with substructure order reduction, the geometric compatibility conditions can be approximated by a weighted residual method. The method is independent of the viscoelastic damping models and the effort on the reduction process is similar to that of classic methods for undamped systems. If all the coefficient matrices are symmetric, the formulation preserves the symmetry in the reduced systems by choosing the same trial vectors and weighting vectors for both the original and adjoint systems. In the absence of damping, the method is further reduced to classical component mode synthesis if undamped substructure modes are used as trial vectors.

Selection of component modes for flexible multibody simulation

The assumed-modes method in multibody dynamics allows the elastic deformation of each component in the system to be approximated by a sum of products of spatial and temporal functions commonly known as modes and modal coordinates, respectively. This paper focuses on the choice of component modes used to model articulating and nonarticulating flexible multibody systems. Attention is directed toward three classical component mode synthesis methods whereby component normal modes are generated by treating the component interface as either fixed, free, or loaded with mass and stiffness contributions from the remaining components. The fixed and free interface normal modes are augmented by static shape functions termed constraint and residual modes, respectively. In this paper, a mode selection procedure is outlined whereby component modes are selected from the Craig-Bampton (fixed interface plus constraint), MacNeal-Rubin (free interface plus residual), or Benfield-Hruda (loaded interface) mode sets in accordance with a modal ordering scheme derived from balanced realization theory. The success of the approach is judged by comparing the actuator-to-sensor frequency response of the reduced-order system with that of the full-order system over the frequency range of interest. A finite element model of the Galileo spacecraft serves as an example in demonstrating the effectiveness of the proposed mode selection method.