Using linear model reduction to investigate the dynamics of structures with local non-linearities

doi:10.1006/mssp.1995.0026

Mechanical Systems and Signal Processing

Volume 9, Issue 3, May 1995, Pages 317-328

https://doi.org/10.1006/mssp.1995.0026 Get rights and content

Abstract

This paper considers the application of model reduction methods, which are popular for linear systems, to systems with local non-linearities, modeled using finite element analysis. In particular these methods are demonstrated by obtaining the receptance of a discrete and a continuous system with cubic stiffening discrete springs, using the harmonic balance method. Time simulation, using the reduced model, is also demonstrated by computing the Poincaré map of a pinned-pinned beam with backlash. The methods provide satisfactory results providing sufficient degrees of freedom are retained in the reduced model and these retained degrees of freedom are chosen with some care.

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Cited by (53)

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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.
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In this study, an improved model order reduction method is developed for the dynamic analysis of large-scale structures with local nonlinearities. In this method, the reduced system associated with the elements in nonlinear state is reconstructed adaptively based on the expressions of the element generalized deformation, which is defined by the eigen-decomposition of the element stiffness matrix. With the introduction of a simplified pre-conditioning method, a novel Newton iteration approach with Krylov subspace methods is proposed to solve the reduced system efficiently. Thus, dynamic analysis can be conducted efficiently for the entire structure. To expand the scope of application, an alternating iterative solution algorithm based on the model order reduction method is further developed for the structural systems simulated with both pseudo-static models and parameterized dynamic models. According to the results of numerical investigation, the novel Newton iteration approach significantly improves the solution efficiency of dynamic analysis for the large-scale structures with local nonlinearities. Meanwhile, the implementation of the alternating iterative solution algorithm contributes a new idea to the fast solution of structural systems with various types of nonlinear models.
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Citation Excerpt :
Nonlinear interface reduction (NLIR) methods seek to develop deformation shapes that can approximate the essential features of the interface with fewer degrees of freedom than needed with the discretized mesh. Early efforts in NLIR extended straightforward linear model reduction strategies to problems with localized contact nonlinearities [15–17]. Gaul and Becker [3] used SCC modes and a free-interface reduction approach to estimate structural damping in a beam-damper assembly.
Virtual prototyping in engineering design rely on modern numerical models of contacting structures with accurate resolution of interface mechanics, which strongly affect the system-level stiffness and energy dissipation due to frictional losses. High-fidelity modeling within the localized interfaces is required to resolve local quantities of interest that may drive design decisions. The high-resolution finite element meshes necessary to resolve inter-component stresses tend to be computationally expensive, particularly when the analyst is interested in response time histories. The Hurty/Craig-Bampton (HCB) transformation is a widely used method in structural dynamics for reducing the interior portion of a finite element model while having the ability to retain all nonlinear contact degrees of freedom (DOF) in physical coordinates. These models may still require many DOF to adequately resolve the kinematics of the interface, leading to inadequate reduction and computational savings. This study proposes a novel interface reduction method to overcome these challenges by means of system-level characteristic constraint (SCC) modes and properly orthogonal interface modal derivatives (POIMDs) for transient dynamic analyses. Both SCC modes and POIMDs are computed using the reduced HCB mass and stiffness matrices, which can be directly computed from many commercial finite element analysis software. Comparison of time history responses to an impulse-type load in a mechanical beam assembly indicate that the interface-reduced model correlates well with the HCB truth model. Localized features like slip and contact area are well-represented in the time domain when the beam assembly is loaded with a broadband excitation. The proposed method also yields reduced-order models with greater critical timestep lengths for explicit integration schemes.
An improved nonlinear dynamic reduction method for complex jointed structures with local hysteresis model
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Citation Excerpt :
In the literature, two categories of nonlinear dynamic reduction methods have been widely used to decrease the dimensions of iteration vector and Jacobian matrix: the local nonlinearity transformation reduction and generalized modal superposition reduction. As for the former one, the nonlinear solutions are obtained by performing iteration only related to the DOFs of the nonlinear joints, such as the Guyan method, dynamics reduction, improved dynamics reduction, and equivalent expansion reduction [28–31]. However, these reduction methods require the inversion of dynamic stiffness matrix at each discrete harmonic frequency, which is impractical and very difficult for large-scale engineering structures.
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Regular ArticleUsing linear model reduction to investigate the dynamics of structures with local non-linearities

Abstract

Regular Article
Using linear model reduction to investigate the dynamics of structures with local non-linearities