Matthew R. W. Brake

Finite element analysis of complex geometries is limited by several factors, including the time required to develop solid geometries and meshes, and the computational time required to perform the simulation of a system with, potentially, millions of degrees of freedom. As a result, the analysis of complex systems is often relegated to deterministic approaches in which geometries other than the idealized geometry rarely are considered. In order to robustly design a system, though, and to quantify the margins of uncertainty that a manufactured realization might yield, a stochastic sampling of the permissible geometries is needed. For complex geometries, a stochastic sampling can involve dozens, if not hundreds, of dimensions that can vary. Exploring a large sample space on even a modestly large model quickly becomes prohibitively expensive— and nearly impossible when a mesh change is required for each sample point. This paper presents a new methodology for utilizing reduced order models in which the number of necessary full degree of freedom models (and corresponding meshes) is minimized. The benefits of parameterizing the high fidelity model at an elemental level, the reduced order model at a system level, and the eigen-space representation are discussed. An application of the method on a model with a non-trivial number of degrees of freedom is presented.

Bolted joints are found in almost every assembled system. Damping due to the friction in the interface of the bolted joints dominates the overall damping in these systems. Therefore, in order to accurately model assembled systems, the correct amount of damping as well as the nonlinear characteristics of the bolted joint must be appropriately accounted for. The level of damping, however, is sensitive to many factors, such as the interface condition and the residual stresses. The formulation of the equations of motion hereby has to involve the local properties of the interfacial damping. In this contribution, two different approaches are applied to a two-beam structure coupled by three bolted joint connections: the discontinuous basis function method and a frequency based substructuring formulation. Measurements of three related systems are used to assess the two different modeling approaches: a monolithic beam, a monolithic beam with three interfaces, and a jointed beam with three bolted joints. The FRFs of the three systems are measured in order to quantify the effect of the bolted interfaces, and future work will investigate the ability of the models to predict the FRFs.

In order to assess the predicted performance of a manufactured system, analysts must typically consider random variations (both geometric and material) in the development of a finite element model, instead of a single deterministic model of an idealized geometry. The incorporation of random variations, however, could potentially require the development of thousands of nearly identical solid geometries that must be meshed and separately analyzed, which would require an impractical number of man-hours to complete. This paper proposes a new approach to uncertainty quantification by developing parameterized reduced order models. These parameterizations are based upon Taylor series expansions of the system’s matrices about the ideal geometry, and a component mode synthesis representation for each linear substructure is used to form an efficient basis with which to study the system. The numerical derivatives required for the Taylor series expansions are obtained efficiently using hyper dual numbers, which enable the derivatives to be calculated precisely to within machine precision. The theory is applied to a stepped beam system in order to demonstrate proof of concept. The accuracy and efficiency of the method, as well as the level at which the parameterization is introduced, are discussed. Hyper dual numbers can be used to construct parameterized models both efficiently and accurately and constitute an appropriate methodology to account for perturbations in a structural system.

Friction in built-up structures has a nonlinear impact on the overall dynamics and is difficult to predict due to the different phenomena occurring at various length scales. A major challenge is observing and quantifying the interfacial behavior in-situ. One solution that has long been studied is the use of ultrasound to measure the interfacial contact stiffness. However, existing research requires complete immersion of the setup in water or permanently attaching the transducer to the test article using epoxy to ensure good, consistent coupling between the transducer and metal. Immersion would change the interface conditions whereas epoxy-based coupling would mean that an array of transducers is required to measure larger contact surfaces. This work proposes a cost-effective, roving-transducer method that uses a conformable delay-line to couple the transducer to the metal surface. The accuracy of the proposed method is tested against static measurements of contact stiffness from a universal testing machine. Furthermore, the contact stiffness across the interface of two bolted stainless-steel blocks is measured using the proposed approach.

Modern engineering structures exhibit nonlinear vibration behavior as designs are pushed to reduce weight and energy consumption. Of specific interest here, joints in assembled structures introduce friction, hysteresis, and unilateral contact resulting in nonlinear vibration effects. In many cases, it is impractical to remove jointed connections necessitating, the understanding of these behaviors. This work focuses on superharmonic and internal resonances in hysteretic and jointed systems. Superharmonic resonances occur when a nonlinear system is forced at an integer fraction of a natural frequency resulting in a large (locally maximal) response at an integer multiple of the forcing frequency. When a second vibration mode simultaneously responds in resonance at the forcing frequency, the combined phenomena is termed an internal resonance. First, variable phase resonance nonlinear modes (VPRNM) is extended to track superharmonic resonances in multiple degree of freedom systems exhibiting hysteresis. Then a novel reduced order model based on VPRNM (VPRNM ROM) is proposed to reconstruct frequency response curves faster than utilizing the harmonic balance method (HBM). The VPRNM ROM is demonstrated for a 3 degree of freedom system with a 3:1 internal resonance and for the jointed Half Brake-Reuss Beam (HBRB), which exhibits a 7:1 internal resonance. For the HBRB, new experimental results are used to validate the modeling approaches, and a previously developed physics-based friction model is further validated, achieving frequency predictions within 3%. For the considered cases, VPRNM ROM construction is up to 4 times faster than HBM, and the evaluation of the VPRNM ROM is up to 780,000 times faster than HBM. The modeling shows that both tangential slipping and normal direction clapping of the joint play important roles in exciting the superharmonic resonances in the HBRB.

Custom implants used for pelvic reconstruction in pelvic sarcoma surgery face a high complication rate due to mechanical failures of fixation screws. Consequently, patient-specific finite element (FE) models have been employed to analyze custom pelvic implant durability. However, muscle forces have often been omitted from FE studies of the post-operative pelvis with a custom implant, despite the lack of evidence that this omission has minimal impact on predicted bone, implant, and fixation screw stress distributions. This study investigated the influence of muscle forces on FE predictions of fixation screw pullout and fatigue failure in a custom pelvic implant. Specifically, FE analyses were conducted using a patient-specific FE model loaded with seven sets of personalized muscle and hip joint contact force loading conditions estimated using a personalized neuromusculoskeletal (NMS) model. Predictions of fixation screw pullout and fatigue failure—quantified by simulated screw axial forces and von Mises stresses, respectively—were compared between analyses with and without personalized muscle forces. The study found that muscle forces had a considerable influence on predicted screw pullout but not fatigue failure. However, it remains unclear whether including or excluding muscle forces would yield more conservative predictions of screw failures. Furthermore, while the effect of muscle forces on predicted screw failures was location-dependent for cortical screws, no clear location dependency was observed for cancellous screws. These findings support the combined use of patient-specific FE and NMS models, including loading from muscle forces, when predicting screw pullout but not fatigue failure in custom pelvic implants.

The present article summarizes the submissions to the Tribomechadynamics Research Challenge announced in 2021. The task was a blind prediction of the vibration behavior of a system comprising a thin plate clamped on two sides via bolted joints. Both geometric and frictional contact nonlinearities are expected to be relevant. Provided were the CAD models and technical drawings of all parts as well as assembly instructions. The main objective was to predict the frequency and damping ratio of the lowest-frequency mode as function of the amplitude. Many different prediction approaches were pursued, ranging from well-known methods to very recently developed ones. After the submission deadline, the system has been fabricated and tested. The aim of this article is to evaluate the current state of the art in modeling and vibration prediction, and to provide directions for future methodological advancements.

Introduction: Three-dimensional (3D)-printed custom pelvic implants have become a clinically viable option for patients undergoing pelvic cancer surgery with resection of the hip joint. However, increased clinical utilization has also necessitated improved implant durability, especially with regard to the compression screws used to secure the implant to remaining pelvic bone. This study evaluated six different finite element (FE) screw modeling methods for predicting compression screw pullout and fatigue failure in a custom pelvic implant secured to bone using nine compression screws. Methods: Three modeling methods (tied constraints (TIE), bolt load with constant force (BL-CF), and bolt load with constant length (BL-CL)) generated screw axial forces using functionality built into Abaqus FE software; while the remaining three modeling methods (isotropic pseudo-thermal field (ISO), orthotropic pseudo-thermal field (ORT), and equal-and-opposite force field (FOR)) generated screw axial forces using iterative physics-based relationships that can be implemented in any FE software. The ability of all six modeling methods to match specified screw pretension forces and predict screw pullout and fatigue failure was evaluated using an FE model of a custom pelvic implant with total hip replacement. The applied hip contact forces in the FE model were estimated at two locations in a gait cycle. For each of the nine screws in the custom implant FE model, likelihood of screw pullout failure was predicted using maximum screw axial force, while likelihood of screw fatigue failure was predicted using maximum von Mises stress. Results: The three iterative physics-based modeling methods and the non-iterative Abaqus BL-CL method produced nearly identical predictions for likelihood of screw pullout and fatigue failure, while the other two built-in Abaqus modeling methods yielded vastly different predictions. However, the Abaqus BL-CL method required the least computation time, largely because an iterative process was not needed to induce specified screw pretension forces. Of the three iterative methods, FOR required the fewest iterations and thus the least computation time. Discussion: These findings suggest that the BL-CL screw modeling method is the best option when Abaqus is used for predicting screw pullout and fatigue failure in custom pelvis prostheses, while the iterative physics-based FOR method is the best option if FE software other than Abaqus is used.

Modern engineering structures exhibit nonlinear vibration behavior as designs are pushed to reduce weight and energy consumption. Of specific interest here, joints in assembled structures introduce friction, hysteresis, and unilateral contact resulting in nonlinear vibration effects. In many cases, it is impractical to remove jointed connections necessitating, the understanding of these behaviors. This work focuses on superharmonic and internal resonances in hysteretic and jointed systems. Superharmonic resonances occur when a nonlinear system is forced at an integer fraction of a natural frequency resulting in a large (locally maximal) response at an integer multiple of the forcing frequency. When a second vibration mode simultaneously responds in resonance at the forcing frequency, the combined phenomena is termed an internal resonance. First, variable phase resonance nonlinear modes (VPRNM) is extended to track superharmonic resonances in multiple degree of freedom systems exhibiting hysteresis. Then a novel reduced order model based on VPRNM (VPRNM ROM) is proposed to reconstruct frequency response curves faster than utilizing the harmonic balance method (HBM). The VPRNM ROM is demonstrated for a 3 degree of freedom system with a 3:1 internal resonance and for the jointed Half Brake-Reuss Beam (HBRB), which exhibits a 7:1 internal resonance. For the HBRB, new experimental results are used to validate the modeling approaches, and a previously developed physics-based friction model is further validated, achieving frequency predictions within 3%. For the considered cases, VPRNM ROM construction is up to 4 times faster than HBM, and the evaluation of the VPRNM ROM is up to 780,000 times faster than HBM. The modeling shows that both tangential slipping and normal direction clapping of the joint play important roles in exciting the superharmonic resonances in the HBRB.

Many modern engineering structures exhibit nonlinear vibration. Characterizing such vibrations efficiently is critical to optimizing designs for reliability and performance. For linear systems, steady-state vibration occurs only at the forcing frequencies. However, nonlinearities (e.g., contact, friction, large deformation, etc.) can result in nonlinear vibration behavior including superharmonics - responses at integer multiples of the forcing frequency. When the forcing frequency is near an integer fraction of the natural frequency, superharmonic resonance occurs, and the magnitude of the superharmonics can exceed that of the fundamental harmonic that is externally forced. Characterizing such superharmonic resonances is critical to improving engineering designs. The present work extends the concept of phase resonance nonlinear modes (PRNM) to be applicable to general nonlinearities, and is demonstrated for eight different nonlinear forces. The considered forces include stiffening, softening, contact, damping, and frictional nonlinearities that have not been previously considered with PRNM. The proposed variable phase resonance nonlinear modes (VPRNM) method can accurately track superharmonic resonances for hysteretic nonlinearities that exhibit amplitude dependent phase resonance conditions that cannot be captured by PRNM. The proposed method allows for characterization of superharmonic resonances without constructing a full frequency response curve at every force level with the harmonic balance method. Thus, the present method allows for analysis of potential failures due to large amplitudes near the superharmonic resonance with reduced computational cost. The consideration of single degree of freedom systems in the present paper provides insights into superharmonic resonances and a basis for understanding internal resonances for multiple degree of freedom systems.

Many modern engineering structures exhibit nonlinear vibration. Characterizing such vibrations efficiently is critical to optimizing designs for reliability and performance. For linear systems, steady-state vibration occurs only at the forcing frequencies. However, nonlinearities (e.g., contact, friction, large deformation, etc.) can result in nonlinear vibration behavior including superharmonics - responses at integer multiples of the forcing frequency. When the forcing frequency is near an integer fraction of the natural frequency, superharmonic resonance occurs, and the magnitude of the superharmonics can exceed that of the fundamental harmonic that is externally forced. Characterizing such superharmonic resonances is critical to improving engineering designs. The present work extends the concept of phase resonance nonlinear modes (PRNM) to be applicable to general nonlinearities, and is demonstrated for eight different nonlinear forces. The considered forces include stiffening, softening, contact, damping, and frictional nonlinearities that have not been previously considered with PRNM. The proposed variable phase resonance nonlinear modes (VPRNM) method can accurately track superharmonic resonances for hysteretic nonlinearities that exhibit amplitude dependent phase resonance conditions that cannot be captured by PRNM. The proposed method allows for characterization of superharmonic resonances without constructing a full frequency response curve at every force level with the harmonic balance method. Thus, the present method allows for analysis of potential failures due to large amplitudes near the superharmonic resonance with reduced computational cost. The consideration of single degree of freedom systems in the present paper provides insights into superharmonic resonances and a basis for understanding internal resonances for multiple degree of freedom systems.

Experiments and physics-based modeling efforts both show that the features within a jointed interface can have an outsized influence on the nonlinear dynamics of a large-scale structure. The interfacial features, including asperities and meso-scale topology, are often six to ten orders of magnitude smaller in scale than the structure itself, yet can significantly change the natural frequencies and damping of a structure and can lead to the premature failure due to wear if not properly designed. A significant amount of recent research has been invested in understanding and predicting the nonlinear dynamics of structures with jointed interfaces; however, there are many challenges that still remain before accurate predictions of a jointed structure's nonlinear dynamics and wear properties becomes accessible to design engineers. This article is a reflection of the outcomes of the 2023 International Workshop on the Mechanics of Jointed Structures in which the state of the art of joints modeling was assessed and future directions for research on jointed structures were identified. As such, this paper makes several recommendations for new research thrusts to improve the understanding of jointed structures in addition to highlighting the current state of the art and recent advances in modeling and experimentally characterizing jointed structures.

Abstract The effect of three different friction interface models on an elastic half-space is presented. Three constitutive friction models are studied: Coulomb, soil–concrete interface, and Bouc–Wen, using a computational mechanics framework that can represent the contact patch's material response to static and dynamic surface tractions. This response is observed as strains and stresses present from reciprocating sliding using an elastoplastic friction (EPF) algorithm that also captures energy dissipation and hysteresis due to friction sliding. Additionally, the use of the four-parameter Bouc–Wen model represents a new development in contact mechanics that allows micro-slip of the contact interface to be modeled. Hysteresis loops are generated for the three friction models based on a quasi-static assumption. This algorithm is built into a mesoscale finite element method solver that is able to simulate different loading conditions and provide insight into how the friction models respond to load conditions and inform on experimental data. The energy dissipation from reciprocating friction sliding will be generated for each friction model as a metric that captures surface wear and potential material damage.

The Masing conditions establish a criterion to relate the loading curve of a hysteretic system (e.g., systems with friction or plasticity) to its complete hysteresis loop. For the field of joint mechanics, where hysteretic models are often used to describe the dissipative, tangential behavior within an interface, the Masing conditions allow for significant computational savings when the normal load is constant. In practice, though, jointed systems experience time varying normal forces that modify the tangential behavior of the system. Consequently, the hysteretic behavior of jointed structures do not adhere to the Masing conditions. In this work, this discrepancy between the Masing conditions and behavior exhibited by jointed structures is explored, and it is hypothesized that if the Masing conditions accounted for variations in normal force, then they would more accurately represent jointed structures. A new set of conditions is introduced to the original set of Masing conditions, yielding a « Masing manifold » that spans the tangential displacement-tangential force-normal force space. Both a simple harmonic oscillator and a built-up structure are investigated for the case of elastic dry friction, and the results show that the hysteresis of both of these systems conforms to the three dimensional Masing manifold exactly, provided that a set of constraints are satisfied, even though the hysteresis does not conform with the original Masing conditions.