Dual Craig-Bampton Research Articles

Upgraded Reduction Technique for Dynamic Analysis of Structures with Friction Contact

A recent reduction technique for the nonlinear forced response analysis of structures with contact interfaces is upgraded. The evaluations of the existing method (Dual formulation), which is based on dual Craig–Bampton method, show its considerable off-line computational time saving because of no matrix manipulation, whereas the resulting reduced model is less accurate than Rubin’s in approximating the full finite element model. The accuracy of this formulation is significantly improved with a low additional computational cost in the proposed upgraded formulation. That is obtained by consistent projection of all structural matrices on a reduction basis made up of two sets of vectors: free-interface normal modes and residual flexibility attachment modes. Assembly approach can be considered as dual because the interface forces are kept as generalized coordinates in opposition to primal methods, where interface displacements are retained. To reduce the computational cost, the nonlinear governing equations are also solved in frequency domain using the multiharmonic balance method and alternating frequency time method. Contact elements are introduced between adjacent interface nodes to find the nonlinear contact forces. The performance of the new formulation is demonstrated through two numerical models with friction contact.

A strategy to stabilize the transient analysis and increase the approximation accuracy of dual Craig-Bampton reduced systems

We consider reduced systems obtained by a dual Craig-Bampton reduction, which are always unstable, and propose a modification of such systems allowing for stable transient analysis. Additionally, we use modal truncation augmentation to improve the approximation accuracy of such systems.The dual Craig-Bampton method for the reduction and successive coupling of dynamic systems employs free interface vibration modes, attachment modes, and rigid body modes to build the reduction bases of the substructures, but assembles the substructures using interface forces. Thereby, the interface kinematic conditions are transformed, enforcing only weak interface compatibility between the substructures and allowing for incompatibilities associated with the interface degrees of freedom. As a result, the dual Craig-Bampton reduced system has always as many negative eigenvalues as interface coupling conditions. The reduced system is unstable, rendering a straightforward time integration of the dual Craig-Bampton reduced system impossible.The feasibility of a reliable time integration of dual Craig-Bampton reduced systems is demonstrated and investigated in detail. The unstable behavior when time integrating such systems without further modifications is illustrated and an approach to overcome this instability is suggested: A modal analysis of the reduced system is performed as a subsequent step to the dual Craig-Bampton reduction. Only modes corresponding to positive eigenvalues are kept for transient analysis. This leads to a stable reduced systems and allows for stable time integration. The accuracy using this approach is demonstrated by two examples with either different initial conditions or varying external periodic excitations. Results are compared to the behavior of a classical Craig-Bampton reduction. Additionally, modal truncation augmentation is used to improve the approximation accuracy by capturing the spatial distribution of the external loading not captured by the retained modes of the dual Craig-Bampton reduction basis.

Numerical Assessment of Reduced Order Modeling Techniques for Dynamic Analysis of Jointed Structures With Contact Nonlinearities

This work presents an assessment of classical and state of the art reduced order modeling (ROM) techniques to enhance the computational efficiency for dynamic analysis of jointed structures with local contact nonlinearities. These ROM methods include classical free interface method (Rubin method, MacNeal method), fixed interface method Craig-Bampton (CB), Dual Craig-Bampton (DCB) method and also recently developed joint interface mode (JIM) and trial vector derivative (TVD) approaches. A finite element (FE) jointed beam model is considered as the test case taking into account two different setups: one with a linearized spring joint and the other with a nonlinear macroslip contact friction joint. Using these ROM techniques, the accuracy of dynamic behaviors and their computational expense are compared separately. We also studied the effect of excitation levels, joint region size, and number of modes on the performance of these ROM methods.

Dual Craig-Bampton component mode synthesis method for model order reduction of nonclassically damped linear systems

The original dual Craig-Bampton method for reducing and successively coupling undamped substructured systems is extended to the case of arbitrary viscous damping. The reduction is based on the equations of motion in state-space representation and uses complex free interface normal modes, residual flexibility modes, and state-space rigid body modes. To couple the substructures in state-space representation, a dual coupling procedure based on the interface forces between adjacent substructures is used, which is novel compared to other methods commonly applying primal coupling procedures in state-space representation. The very good arbitrary viscous damping of the dual Craig-Bampton approach is demonstrated on a beam structure with localized dampers. The results are compared to a classical Craig-Bampton approach for damped systems showing the potential of the proposed method.

Improving the accuracy of the dual Craig-Bampton method

The objective of the work reported in this paper is to improve the well-known dual Craig-Bampton (DCB) method. The original transformation matrix of the DCB method is enhanced by considering the higher-order effect of residual substructural modes through residual flexibility. Using the new transformation matrix, original finite element models can be more accurately approximated in the reduced models. Herein, additional generalized coordinates are newly defined for considering the 2nd order residual flexibility. Additional coordinates related to the interface boundary can be eliminated by applying the concept of SEREP (the system equivalent reduction expansion process). The formulation of the improved DCB method is presented in detail, and its accuracy is investigated through numerical examples.

A mixed-contact formulation for a dynamics simulation of flexible systems: An integration with model-reduction techniques

A new numerical procedure for efficient dynamics simulations of linear-elastic systems with unilateral contacts is proposed. The method is based on the event-driven integration of a contact problem with a combination of single- and set-valued force laws together with classical model-reduction techniques. According to the contact state, the developed event-driven integration enables the formulation of reduced system matrices. Moreover, to enable the transition among different reduced spaces the formulation of the initial conditions is also presented. The method has been developed separately for each of the four most popular model-reduction techniques (Craig–Bampton, MacNeal, Rubin and dual Craig–Bampton). The applicability of the newly presented method is demonstrated on a simple clamped-beam structure with a unilateral contact, which is excited with a harmonic force at the free end.

On the dual Craig–Bampton method for the forced response of structures with contact interfaces

Assembled structures are characterized by contact interfaces that introduce a local non-linearity and affect the dynamics of the assembly in terms of resonance frequencies and vibration levels. To assess the forced response levels of the assemblies during the design, nonlinear dynamic analyses are performed and, in order to reduce the computation time, spatial and temporal reductions of the governing equations must be used. A classical way to achieve temporal reduction is to implement the harmonic balance method to turn the time-domain differential governing equations into frequency-domain algebraic equations. Due to the local nature of contact interfaces, which usually involve a subset of degrees of freedom (dofs) of the structure, a common strategy to achieve spatial reduction is to use component mode synthesis (CMS), by retaining the contact dofs as master dofs. In this paper, a recent CMS approach, named dual Craig–Bampton method (Rixen in J Comput Appl Math, 2004. doi: 10.1016/j.cam.2003.12.014 ), is applied to the nonlinear forced response of structures with contact interfaces. The spectral orthogonality of the two subsets of mode shapes used as a projection basis is exploited to write a set of algebraic equations of the contact dofs in the frequency domain, with no need to compute the reduced matrices of the system. Different formulations of the governing equations are proposed for different configurations (i.e., outer contacts, inner contacts and structures with floating components), and two academic numerical test cases are used to demonstrate the method.

Evaluation of Substructure Reduction Techniques with Fixed and Free Interfaces

Substructure reduction techniques are efficient methods to reduce the size of large models used to ana-lyze the dynamical behavior of complex structures. The most popular approach is a fixed interface method, the Craig-Bampton method (1968), which is based on fixed interface vibration modes and inter-face constraint modes. In contrast, free interface methods employing free interface vibration modes together with attachment modes are also used, e.g. MacNeal's method (1971) and Rubin's method (1975). The methods mentioned so far assemble the substructures using interface displacements (primal assembly). The dual Craig-Bampton method (2004) uses the same ingredients as the two aforementioned free interface methods, but assembles the substructures using interface forces (dual assembly). This method enforces only weak interface compatibility between the substructures, thereby avoiding inter-face locking problems as sometimes experienced in the primal assembly approaches using free interface modes. The dual Craig-Bampton method leads to simpler reduced matrices compared to other free inter-face methods and the reduced matrices are sparse, similar to the classical Craig-Bampton matrices. In this contribution we evaluate the primal (classical) formulation of the Craig-Bampton method, the MacNeal method, the Rubin method and the dual formulation of the Craig-Bampton method. The pre-sented theory and the comparison between the four substructuring methods will be illustrated on the Benfield truss, on a three-dimensional beam frame and on a two-dimensional solid plane stress problem.

Generalized Methodology for Assembly and Reduction of Component Models for Dynamic Substructuring

A classic issue in component mode synthesis methods is the choice for fixed or free boundary conditions at the interface degrees of freedom and the associated vibration modes in the component's reduction basis. Closely related is the aspect of substructure assembly. In this paper a general approach to mixed boundary component mode synthesis problems is proposed, by considering component reduction and assembly separately. First, a general framework for substructure assembly is presented, enabling assembly of all types of reduced component models, regardless of their interface conditions. Thereafter, a general mixed boundary component mode synthesis method called the mixed Craig―Bampton method is introduced. The method is derived by dividing the substructure degrees of freedom into a set of internal degrees of freedom, free interface degrees of freedom, and fixed interface degrees of freedom using a simple but effective selection scheme. Based on this selection a reduction basis is computed consisting of vibration modes, constraint modes, and residual flexibility modes, which leads to relatively sparse reduced matrices. As such, the mixed Craig―Bampton method forms a natural generalization of the classic Craig―Bampton and more recent dual Craig―Bampton methods. Finally, the method is applied to a test problem, revealing that the proposed method has comparable or better accuracy and superior versatility with respect to existing methods.

A general mixed boundary model reduction method for component mode synthesis

A classic issue in component mode synthesis (CMS) methods is the choice for fixed or free boundary conditions at the interface degrees of freedom (DoF) and the associated vibration modes in the components reduction base. In this paper, a novel mixed boundary CMS method called the "Mixed Craig-Bampton" method is proposed. The method is derived by dividing the substructure DoF into a set of internal DoF, free interface DoF and fixed interface DoF. To this end a simple but effective scheme is introduced that, for every pair of interface DoF, selects a free or fixed boundary condition for each DoF individually. Based on this selection a reduction basis is computed consisting of vibration modes, static constraint modes and static residual flexibility modes. In order to assemble the reduced substructures a novel mixed assembly procedure is developed. It is shown that this approach leads to relatively sparse reduced matrices, whereas other mixed boundary methods often lead to full matrices. As such, the Mixed Craig-Bampton method forms a natural generalization of the classic Craig-Bampton and more recent Dual Craig-Bampton methods. Finally, the method is applied to a finite element test model. Analysis reveals that the proposed method has comparable or better accuracy and superior versatility with respect to the existing methods.

A dual Craig–Bampton method for dynamic substructuring

A novel component mode synthesis method for dynamic analysis of structures is presented. It is based on free interface vibration modes and residual flexibility components. Although the ingredients are the same as in previously published procedures (e.g. MacNeal or Rubin), our method is fundamentally different in that it assembles the substructures using interface forces (dual assembly) and enforces only weak interface compatibility. The new formulation is based on a fully consistent reduction approach and the reduced matrices so-obtained are exactly dual to the Craig–Bampton reduced matrices. The new free interface substructuring method proposed here is thus more natural then classical free interface synthesis procedures and leads to simpler reduced matrices. We illustrate the efficiency of the dual Craig–Bampton approach for reducing the dynamical representation on a three-dimensional frame.