Publications

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The Third International Workshop on Jointed Structures was held from August 16th to 17th, 2012, in Chicago Illinois, following the ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Thirty two researchers from both the United States and international locations convened to discuss the recent progress of mechanical joints related research and associated efforts in addition to developing a roadmap for the challenges to be addressed over the next five to ten years. These proceedings from the workshop include the minutes of the discussions and follow up from the 2009 workshop [1], presentations, and outcomes of the workshop. Specifically, twelve challenges were formulated from the discussions at the workshop, which focus on developing a better understanding of uncertainty and variability in jointed structures, incorporating high fidelity models of joints in simulations that are tractable/efficient, motivating a new generation of researchers and funding agents as to the importance of joint mechanics research, and developing new insights into the physical phenomena that give rise to energy dissipation in jointed structures. The ultimate goal of these research efforts is to develop a predictive model of joint mechanics.

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Random vibration under preload is important in multiple endeavors, including those involving launch and re-entry. There are some methods in the literature to begin to address this problem, but there is nothing that accommodates the existence of preloads and the necessity of making probabilistic statements about the stress levels likely to be encountered. An approach to achieve to this goal is presented along with several simple illustrations.

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A systematic approach to defining margin in a manner that incorporates statistical information and accommodates data uncertainty, but does not require assumptions about specific forms of the tails of distributions is developed. This approach extends to calculations underlying validation assessment and quantitatively conservative predictions.

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Correctly incorporating the influence of mechanical joints in built-up mechanical systems is a critical element for model development for structural dynamics predictions. Quality experimental data are often difficult to obtain and is rarely sufficient to determine fully parameters for relevant mathematical models. On the other hand, fine-mesh finite element (FMFE) modeling facilitates innumerable numerical experiments at modest cost. Detailed FMFE analysis of built-up structures with frictional interfaces reproduces trends among problem parameters found experimentally, but there are qualitative differences. Those differences are currently ascribed to the very approximate nature of the friction model available in most finite element codes. Though numerical simulations are insufficient to produce qualitatively correct behavior of joints, some relations, developed here through observations of a multitude of numerical experiments, suggest interesting relationships among joint properties measured under different loading conditions. These relationships can be generalized into forms consistent with data from physical experiments. One such relationship, developed here, expresses the rate of energy dissipation per cycle within the joint under various combinations of extensional and clamping load in terms of dissipation under other load conditions. The use of this relationship - though not exact - is demonstrated for the purpose of extrapolating a representative set of experimental data to span the range of variability observed from real data. © 2011 American Society of Mechanical Engineers.

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The SNL/AWE joint mechanics workshop, held in Dartington Hall, Totnes, Devon, UK 26-29 April 2009 was a follow up to another international joints workshop held in Arlington, Virginia, in October 2006. The preceding workshop focused on identifying what length scales and interactions would be necessary to provide a scientific basis for analyzing and understanding joint mechanics from the atomistic scale on upward. In contrast, the workshop discussed in this report, focused much more on identification and development of methods at longer length scales that can have a nearer term impact on engineering analysis, design, and prediction of the dynamics of jointed structures. Also, the 2009 meeting employed less technical presentation and more break out sessions for developing focused strategies than was the case with the early workshop. Several 'challenges' were identified and assignments were made to teams to develop approaches to address those challenges.

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The nonlinear behavior of mechanical joints is a confounding element in modeling the dynamic response of structures. Though there has been some progress in recent years in modeling individual joints, modeling the full structure with myriad frictional interfaces has remained an obstinate challenge. A strategy is suggested for structural dynamics modeling that can account for the combined effect of interface friction distributed spatially about the structure. This approach accommodates the following observations: (1) At small to modest amplitudes, the nonlinearity of jointed structures is manifest primarily in the energy dissipation - visible as vibration damping; (2) Correspondingly, measured vibration modes do not change significantly with amplitude; and (3) Significant coupling among the modes does not appear to result at modest amplitudes. The mathematical approach presented here postulates the preservation of linear modes and invests all the nonlinearity in the evolution of the modal coordinates. The constitutive form selected is one that works well in modeling spatially discrete joints. When compared against a mathematical truth model, the distributed dissipation approximation performs well.

This report describes work performed from October 2007 through September 2009 under the Sandia Laboratory Directed Research and Development project titled 'Reduced Order Modeling of Fluid/Structure Interaction.' This project addresses fundamental aspects of techniques for construction of predictive Reduced Order Models (ROMs). A ROM is defined as a model, derived from a sequence of high-fidelity simulations, that preserves the essential physics and predictive capability of the original simulations but at a much lower computational cost. Techniques are developed for construction of provably stable linear Galerkin projection ROMs for compressible fluid flow, including a method for enforcing boundary conditions that preserves numerical stability. A convergence proof and error estimates are given for this class of ROM, and the method is demonstrated on a series of model problems. A reduced order method, based on the method of quadratic components, for solving the von Karman nonlinear plate equations is developed and tested. This method is applied to the problem of nonlinear limit cycle oscillations encountered when the plate interacts with an adjacent supersonic flow. A stability-preserving method for coupling the linear fluid ROM with the structural dynamics model for the elastic plate is constructed and tested. Methods for constructing efficient ROMs for nonlinear fluid equations are developed and tested on a one-dimensional convection-diffusion-reaction equation. These methods are combined with a symmetrization approach to construct a ROM technique for application to the compressible Navier-Stokes equations.

Advanced computing hardware and software written to exploit massively parallel architectures greatly facilitate the computation of extremely large problems. On the other hand, these tools, though enabling higher fidelity models, have often resulted in much longer run-times and turn-around-times in providing answers to engineering problems. The impediments include smaller elements and consequently smaller time steps, much larger systems of equations to solve, and the inclusion of nonlinearities that had been ignored in days when lower fidelity models were the norm. The research effort reported focuses on the accelerating the analysis process for structural dynamics though combinations of model reduction and mitigation of some factors that lead to over-meshing.

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The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transformation leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using a piecewise-smooth finite element basis. Stability of the reduced order model using this approach is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation. © 2008 Elsevier Inc.

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Energy dissipation in mechanical joints occurs as a result of micro-slip motion between contacting rough surfaces. An account of this phenomenon is especially challenging due to the vast differences in the length and time scale differences between the macro-mechanical structure and the micron-scale events at the joint interface. This paper considers the contact between two nominally flat surfaces containing micron-scale roughness. The rough surface interaction is viewed as a multi-sphere elastic interaction subject to a periodic tangential force. It combines the Mindlin's formulation [1,2] for the elastic interaction of two spheres with the Greenwood and Williamson's [3] statistical approach for the contact of two nominally flat rough surfaces so as to develop a model for multi-sphere problem in which sphere radii, contact load and the number of spheres in contact can only be known in a statistical sense and not deterministically. Copyright © 2007 by ASME.

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The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transform leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Coupling with a linearized structural dynamics model is made possible through the solid wall boundary condition. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using piecewise-smooth finite element bases. Stability of the coupled fluid/structure system is examined for the case of uniform flow past a thin plate. Stability of the reduced order model for the fluid is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation.

The NSF/SNL joint mechanics workshop, held in Arlington, Virginia, 16-18 October, 2006, attempted to assess the current state of the art for modeling joint mechanics for the purpose of structural dynamics calculation, to identify the underlying physics issues that must be addressed to advance the field, and to propose a path forward. Distinguished participants from several countries representing research communities that focus on very different length and time scales identified multiple challenges in bridging those scales. Additionally, two complementary points of view were developed for addressing those challenges. The first approach - the 'bottom-up' perspective - attempts to bridge scales by starting from the smallest length scale and working up. The other approach starts at the length scale of application and attempts to deduce mechanics at smaller length scales through reconciliation with laboratory observation. Because interface physics is a limiting element of predictive simulation in defense and transportation, this issue will be of continuing importance for the foreseeable future.

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Using fine material meshes in structural dynamics analysis is often impractical due to time step considerations. Unfortunately, fine meshes are typically required to capture the inherent physics in jointed connections. This is especially true in threaded connections which feature numerous contact interfaces and stress singularities. A systematic method is presented here for representing the threaded volume by a continuous, homogeneous, linear elastic, anisotropic equivalent material. The parameters of that equivalent material depend on thread geometry and the assumed contact condition between adjacent threads and are derived from detailed finite element simulations of a characteristic thread-pair unit cell. Numerical simulations using the equivalent material closely match the local stiffness through the load path calculated from the finely meshed thread models and also reproduce classical theoretical and experimental results from the literature. Copyright © 2007 by ASME.

A relatively new concept in the field of mechanical shock analysis has been introduced whereby an analysis is made on the work done on structures by the excitation force. The energy imparted to a structure by the excitation can then be divided into various storage and loss mechanisms within the structure. These energies can be used to both evaluate shock response severity and characterize the underlying excitation. Previous work has illustrated the many advantages of the energy methods over traditional shock response spectrum techniques. This work will show that the energy delivered to a MDOF system is uncoupled between modes. Therefore, the total deformational energy delivered to a MDOF system is a weighted sum of the uncoupled modal contributions. This leads to the ability to compute input energy on a modal basis using uncoupled, SDOF calculations. Further, the internal storage and loss energies are also uncoupled. When the input excitation is broadband, the energy input into a MDOF structure by ground motion is dominated by that mode with the largest fraction of participating mass, often the fundamental mode of the system. This leads to the justification for treating complex structures as SDOF oscillators when using energy methods to evaluate both the underlying excitation and the structural response.

It is shown that for any material or structural model expressible as a Masing model, there exists a unique parallel-series (displacement-based) Iwan system that characterizes that model as a function of displacement history. This poses advantages both in terms of more convenient force evaluation in arbitrary deformation histories as well as in terms of model inversion. Characterization as an Iwan system is demonstrated through the inversion of the Ramberg-Osgood model, a force(stress)-based material model that is not explicitly invertible. An implication of the inversion process is that direct, rigorous comparisons of different Masing models, regardless of the ability to invert their constitutive relationship, can be achieved through the comparison of their associated Iwan distribution densities.

A technique published in SAND Report 2006-1789 ''Model Reduction of Systems with Localized Nonlinearities'' is illustrated in two problems of finite element structural dynamics. That technique, called here the Method of Locally Discontinuous Basis Vectors (LDBV), was devised to address the peculiar difficulties of model reduction of systems having spatially localized nonlinearities. It's illustration here is on two problems of different geometric and dynamic complexity, but each containing localized interface nonlinearities represented by constitutive models for bolted joint behavior. As illustrated on simple problems in the earlier SAND report, the LDBV Method not only affords reduction in size of the nonlinear systems of equations that must be solved, but it also facilitates the use of much larger time steps on problems of joint macro-slip than would be possible otherwise. These benefits are more dramatic for the larger problems illustrated here. The work of both the original SAND report and this one were funded by the LDRD program at Sandia National Laboratories.

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An LDRD funded approach to development of reduced order models for systems with local nonlinearities is presented. This method is particularly useful for problems of structural dynamics, but has potential application in other fields. The key elements of this approach are (1) employment of eigen modes of a reference linear system, (2) incorporation of basis functions with an appropriate discontinuity at the location of the nonlinearity. Galerkin solution using the above combination of basis functions appears to capture the dynamics of the system with a small basis set. For problems involving small amplitude dynamics, the addition of discontinuous (joint) modes appears to capture the nonlinear mechanics correctly while preserving the modal form of the predictions. For problems involving large amplitude dynamics of realistic joint models (macro-slip), the use of appropriate joint modes along with sufficient basis eigen modes to capture the frequencies of the system greatly enhances convergence, though the modal nature the result is lost. Also observed is that when joint modes are used in conjunction with a small number of elastic eigen modes in problems of macro-slip of realistic joint models, the resulting predictions are very similar to those of the full solution when seen through a low pass filter. This has significance both in terms of greatly reducing the number of degrees of freedom of the problem and in terms of facilitating the use of much larger time steps.

Structural assemblies often include bolted connections that are a primary mechanism for energy dissipation and nonlinear response at elevated load levels. Typically these connections are idealized within a structural dynamics finite element model as linear elastic springs. The spring stiffness is generally tuned to reproduce modal test data taken on a prototype. In conventional practice, modal test data is also used to estimate nominal values of modal damping that could be used in applications with load amplitudes comparable to those employed in the modal tests. Although this simplification of joint mechanics provides a convenient modeling approach with the advantages of reduced complexity and solution requirements, it often leads to poor predicted responses for load regimes associated with nonlinear system behavior. In this document we present an alternative approach using the concept of a "whole-joint" or "whole-interface" model [1]. We discuss the nature of the constitutive model, the manner in which model parameters are deduced, and comparison of structural dynamic prediction with results for experimental hardware subjected to a series of transient excitations beginning at low levels and increasing to levels that produced macro-slip in the joint. Further comparison is performed with a traditional "tuned" linear model. The ability of the whole-interface model to predict the onset of macro-slip as well as the vast improvement of the response levels in relation to those given by the linear model is made evident. Additionally, comparison between prediction and high amplitude experiments suggests areas for further work.

The presence of mechanical joints--typified by the lap joint--in otherwise linear structures has been accommodated in structural dynamics via ad hoc methods for a century. The methods range from tuning linear models to approximate non-linear behavior in restricted load ranges to various methods which introduce joint dissipation in a post-processing stage. Other methods, employing constitutive models for the joints are being developed and their routine use is on the horizon.

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The generalized momentum balance (GMB) methods, explored chiefly by Shabana and his co-workers, treat slap or collision in linear structures as sequences of impulses, thereby maintaining the linearity of the structures throughout. Further, such linear analysis is facilitated by modal representation of the structures. These methods are discussed here and extended. Simulations on a simple two-rod problem demonstrate how this modal impulse approximation affects the system both directly after each impulse as well as over the entire collision. Furthermore, these simulations illustrate how the GMB results differ from the exact solution and how mitigation of these artifacts is achieved. Another modal method discussed in this paper is the idea of imposing piecewise constant forces over short, yet finite, time intervals during contact. The derivation of this method is substantially different than that of the GMB method, yet the numerical results show similar behavior, adding credence to both models. Finally, a novel method combining these two approaches is introduced. The new method produces physically reasonable results that are numerically very close to the exact solution of the collision of two rods. This approach avoids most of the non physical, numerical artifacts of interpenetration or chatter present in the first two methods.

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The Lubkin solution for two spheres pressed together and then subjected to a monotonically increasing axial couple is examined numerically. The Deresiewicz asymptotic solution is compared to the full solution and its utility is evaluated. Alternative approximations for the Lubkin solution are suggested and compared. One approximation is a Pade rational function which matches the analytic solution over all rotations. The other is an exponential approximation that reproduces the asymptotic values of the analytic solution at infinitesimal and infinite rotations. Finally, finite element solutions for the Lubkin problem are compared with the exact and approximate solutions.

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The constitutive behavior of mechanical joints is largely responsible for the energy dissipation and vibration damping in built-up structures. For reasons arising from the dramatically different length scales associated with those dissipative mechanisms and the length scales characteristic of the overall structure, this physics cannot be captured through direct numerical simulation (DNS) of the contact mechanics within a structural dynamics analysis. The difficulties of DNS manifest themselves either in terms of Courant times that are orders of magnitude smaller than that necessary for structural dynamics analysis or as intractable conditioning problems. The only practical method for accommodating the nonlinear nature of joint mechanisms within structural dynamic analysis is through constitutive models employing degrees of freedom natural to the scale of structural dynamics. In this way, development of constitutive models for joint response is a prerequisite for a predictive structural dynamics capability. A four-parameter model, built on a framework developed by Iwan, is used to reproduce the qualitative and quantitative properties of lap-type joints. In the development presented here, the parameters are deduced by matching joint stiffness under low load, the force necessary to initiate macroslip, and experimental values of energy dissipation in harmonic loading. All the necessary experiments can be performed on real hardware or virtually via fine-resolution, nonlinear quasistatic finite elements. The resulting constitutive model can then be used to predict the force/displacement results from arbitrary load histories.

The constitutive behavior of mechanical joints is largely responsible for the energy dissipation and vibration damping in weapons systems. For reasons arising from the dramatically different length scales associated with those dissipative mechanisms and the length scales characteristic of the overall structure, this physics cannot be captured adequately through direct simulation of the contact mechanics within a structural dynamics analysis. The only practical method for accommodating the nonlinear nature of joint mechanisms within structural dynamic analysis is through constitutive models employing degrees of freedom natural to the scale of structural dynamics. This document discusses a road-map for developing such constitutive models.

The constitutive behavior of mechanical joints is largely responsible for the energy dissipation and vibration damping in weapons systems. For reasons arising from the dramatically different length scales associated with those dissipative mechanisms and the length scales characteristic of the overall structure, this physics cannot be captured through direct numerical simulation (DNS) of the contact mechanics within a structural dynamics analysis. The difficulties of DNS manifest themselves either in terms of Courant times that are orders of magnitude smaller than that necessary for structural dynamics analysis or as intractable conditioning problems. The only practical method for accommodating the nonlinear nature of joint mechanisms within structural dynamic analysis is through constitutive models employing degrees of freedom natural to the scale of structural dynamics. In this way, development of constitutive models for joint response is a prerequisite for a predictive structural dynamics capability. A four-parameter model, built on a framework developed by Iwan, is used to reproduce the qualitative and quantitative properties of lap-type joints. In the development presented here, the parameters are deduced by matching experimental values of energy dissipation in harmonic loading and values of the force necessary to initiate macro-slip. (These experiments can be performed on real hardware or virtually via fine-resolution, nonlinear quasi-static finite elements.) The resulting constitutive model can then be used to predict the force/displacement results from arbitrary load histories.

Frictional energy dissipation in joints is an issue of long-standing interest in the effort to predict damping of built up structures. Even obtaining a qualitative understanding of how energy dissipation depends on applied loads has not yet been accomplished. Goodman postulated that in harmonic loading, the energy dissipation per cycle would go as the cube of the amplitude of loading. Though experiment does support a power-law relationship, the exponent tends to be lower than Goodman predicted. Recent calculations discussed here suggest that the cause of that deviation has to do with reshaping of the contact patch over each loading period.

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The structural dynamics modeling of engineering structures must accommodate the energy dissipation due to microslip in mechanical joints. Given the nature of current hardware and software environments, this will require the development of constitutive models for joints that both adequately reproduce the important physics and lend themselves to efficient computational processes. The exploration of the properties of mechanical joints--either through fine resolution finite element modeling or through experiment--is itself an area of research, but some qualitative behavior appears to be established. The work presented here is the presentation of a formulation of idealized elements due to Iwan, that appears capable of reproducing the important joint properties as they are now understood. Further, methods for selecting parameters for that model by joining the results from experiments in regimes of small and large load are developed. The significance of this work is that a reduced order model is presented that is capable of reproducing the important qualitative properties of mechanical joints using only a small number of parameters.

The von Mises stress is often used as the metric for evaluating design margins, particularly for structures made of ductile materials. While computing the von Mises stress distribution in a structural system due to a deterministic load condition may be straightforward, difficulties arise when considering random vibration environments. As a result, alternate methods are used in practice. One such method involves resolving the random vibration environment to an equivalent static load. This technique, however, is only appropriate for a very small class of problems and can easily be used incorrectly. Monte Carlo sampling of numerical realizations that reproduce the second order statistics of the input is another method used to address this problem. This technique proves computationally inefficient and provides no insight as to the character of the distribution of von Mises stress. This tutorial describes a new methodology to investigate the design reliability of structural systems in a random vibration environment. The method provides analytic expressions for root mean square (RMS) von Mises stress and for the probability distributions of von Mises stress which can be evaluated efficiently and with good numerical precision. Further, this new approach has the important advantage of providing the asymptotic properties of the probability distribution. A brief overview of the theoretical development of the methodology is presented, followed by detailed instructions on how to implement the technique on engineering applications. As an example, the method is applied to a complex finite element model of a Global Positioning Satellite (GPS) system. This tutorial presents an efficient and accurate methodology for correctly applying the von Mises stress criterion to complex computational models. The von Mises criterion is the traditional method for determination of structural reliability issues in industry.