Updating structural models containing nonlinear Iwan joints using quasi-static modal analysis
Introduction
Much of the uncertainty in finite element models for built-up structures comes from the interfaces (e.g. bolted joints, riveted connections, press-fits, etc.). They often cause the structure to behave nonlinearly, making experiments more challenging, and to model them accurately requires a nonlinear model that can be dramatically more expensive. See [1], [2], [3] for a review of these issues. This work focuses on methods by which joints are represented in a simplified way, so that the structure’s response can be simulated quickly and the model updated to correlate with experimental measurements.
In most industrial applications, the bolts are designed to retain integrity, so most of the joint nonlinearity is attributed to microslip friction in which the joint remains intact yet partial slipping occurs for some material near the edges of the contact region [4], [5]. Commercial finite element codes typically include the ability to simulate the static and dynamic response of flexible structures with Coulomb friction between contacts. For example, Abaqus [6] includes the capability to model friction using Penalty and Lagrange methods. However, once the contact interfaces are meshed with sufficient resolution to capture micro-slip and the penalty stiffness and stabilization are set to acceptable levels, the simulations become very expensive for static response [7] and are completely prohibitive when a dynamic response is desired that spans many cycles of oscillation. To address this, harmonic balance methods have been developed that compute the frequency response directly [8], [9]. Other works have proposed 2-D zero-thickness elements [10], [11] or 3-D thin-layer elements [12], [13], both of which implement a tangential stick–slip condition in a continuum element, and have demonstrated their implementation in joints modeling with frequency-domain solvers. In order to model microslip behavior accurately, such models typically require hundreds of friction elements per interface [14]. The resulting simulations are still quite expensive and may be impractical if the model contains many joints. Furthermore, when updating a model to correlate with measurements, one must typically run many simulations while varying the parameters for the joint, a daunting prospect with current models.
A recent work by Festjens et al. [15] presented an alternative to traditional time and frequency domain methods. They computed the quasi-static response of the structure to a loading representative of a certain mode, and then used the results of that simulation to compute the effective natural frequency and damping for the structure when it vibrates in that mode. Their predictions were validated by comparing them to the transient response computed by integrating a full-order nonlinear finite element model, which included the preload in the fastener and the resulting Coulomb friction between the contacting surfaces. Their methodology was found to provide good estimates of the effective damping and natural frequency, however the costs are still quite high if many iterations on the model must be run such as for model updating applications.
One way of reducing the computational burden associated with joints is to replace the joint region with a simpler element that captures its effective stiffness and damping. Then, instead of enforcing frictional contact between individual nodes or element faces, those degrees of freedom can instead be tied (e.g. by multi-point constraints as in Fig. 2) to two ends of a single nonlinear element whose constitutive formulation is designed to reproduce microslip behavior internally. Such a modeling approach is the premise of Segalman’s “whole-joint” model [16]. The nonlinear element may be a hysteretic model such an Iwan model [17], [18] or a Bouc-Wen [19] model. A whole-joint representation is appealing because joints are often only one small part of a complicated, assembled structure. Various experimental studies have shown that the presence of one or more bolted joints induces only a slight softening in the natural frequencies, accompanied with a large increase in the damping, as the structure is excited to higher response amplitudes [14], [20], [21], [22], [23]. Unfortunately, no methods exist in literature that can predict the parameters of these hysteresis elements based on joint geometry, surface roughness, material, etc. This work instead uses a model updating approach to determine the parameters. An experiment is performed in which the transient free response is measured and then the instantaneous natural frequency and damping for each mode are extracted from those measurements [24], [25], [26], [27]. The model updating approach can then be used to find the joint parameters that bring the model into agreement with the measurements.
Specifically, this work presents a few extensions to the quasi-static method introduced by Festjens et al., creating a highly efficient tool for extracting the amplitude-dependent natural frequency and damping from a finite element model containing discrete (localized) nonlinearities. The structure of interest is reduced to a Hurty/Craig-Bampton model [28], which contains only the nodes at each end of the nonlinear elements representing the joints. Here, Iwan elements are considered for the nonlinear representation, and the approach is optimized for those but it could be extended to other nonlinear elements. Then, a variation of the method by Festjens et al. is presented, here referred to as the method of quasi-static modal analysis (QSMA), which is used to compute and update the amplitude-dependent natural frequency and damping. By iterating on the parameters of the nonlinear Iwan joints, one can obtain a model that reproduces the measurements very well, including damping that varies in a power-law fashion with respect to response amplitude. The updated model is then validated using additional transient measurements taken from the structure, and, in the application presented here, the method showed favorable results.
Section snippets
The method of quasi-static modal analysis
The method of quasi-static modal analysis (QSMA) implemented in this work is a modification on the method in [15]. In that work, Festjens, Chevallier, and Dion solved a quasi-static problem in which a quasi-static, distributed force was applied representing the inertial loading when the structure vibrates in the mode of interest. Then, the nonlinear response of that quasi-static model was used together with Masing’s rules to relate the load–deflection behavior to the dynamic response when the
Validation of QSMA with a test structure
In this section, the method of QSMA is validated by using it to compute the stiffness and damping characteristics of a few modes of a reduced finite element model. Separately, the dynamic response of the system is computed and the modal stiffness and damping are extracted and then compared to the QSMA predictions. The results demonstrate that the amplitude-dependent frequency and damping acquired through the quasi-static analysis are essentially equivalent to those obtained from a dynamic
Model updating
As shown in Section 3.3, the method of quasi-static modal analysis (QSMA) provides an efficient means of accurately estimating the amplitude-dependent natural frequency and damping of the modes of a finite element model containing Iwan elements. The speed at which these properties can be computed make QSMA highly desirable as a tool for iterative model updating. Such capability is demonstrated in this section by updating the parameters of the five Iwan elements in the Brake-Reuß beam FEM so
Conclusion
In this work, the method of quasi-static modal analysis (QSMA) was developed as an efficient tool for extracting the amplitude-dependent natural frequency and damping of each mode of a structure that contains weak nonlinearities due to bolted joints. The premise of QSMA is that these properties can be computed from a modal force–deflection curve, which can be used to reconstruct the hysteresis loop by applying Masing’s rules. The force–deflection curve is found by applying a
Acknowledgments
This work was funded by Sandia National Laboratories. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.
References (37)
- et al.
Modeling the dynamics of mechanical joints
Mech. Syst. Sig. Process.
(2011) - et al.
Investigation of a jointed friction oscillator using the multiharmonic balance method
Mech. Syst. Sig. Process.
(2015) - et al.
A numerical tool for the design of assembled structures under dynamic loads
Int. J. Mech. Sci.
(2013) - et al.
Experimental study of non-linear effects in a typical shear lap joint configuration
J. Sound Vib.
(2004) - et al.
Design of experiments and energy dissipation analysis for a contact mechanics 3D model of frictional bolted lap joints
Adv. Eng. Softw.
(2012) - et al.
Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam elements
J. Sound Vib.
(2004) - et al.
Nonlinear characterization of a bolted, industrial structure using a modal framework
Mech. Syst. Sig. Process.
(2017) - et al.
A numerical study on the limitations of modal Iwan models for impulsive excitations
J. Sound Vib.
(2017) Non-linear system vibration analysis using Hilbert transform–I. free vibration analysis method FREEBIB
Mech. Syst. Sig. Process.
(1994)- et al.
A global, single-input-multi-output (SIMO) implementation of the algorithm of mode isolation and application to analytical and experimental data
Mech. Syst. Sig. Process.
(2006)
The Mechanics of Jointed Structures: Recent Research and Open Challenges for Developing Predictive Models for Structural Dynamics
Concepts and Applications of Tribology
Friction damping and isolation systems
J. Mech. Des.
Microslip and macroslip in bolted joints
Exp. Mech.
Nonlinear dynamics of structures assembled by bolted joints
Acta Mech.
Analytical formulation of friction interface elements for analysis of nonlinear multi-harmonic vibrations of bladed disks
J. Turbomach.
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