R-3 Repository :: Browsing by Author "Brake, Matthew R.W."

Browsing by Author "Brake, Matthew R.W."

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A quantitative assessment of the model form error of friction models across different interface representations for jointed structures
(Elsevier, 2022) Porter, Justin H.; Balaji, Nidish Narayanaa; Little, Clayton R.; Brake, Matthew R.W.
Hysteretic models are widely used to model frictional interactions in joints to recreate experimental behavior. However, it is unclear which models are best suited for fitting or predicting the responses of structures. The present study evaluates 26 friction model/interface representation combinations to quantify the model form error. A Quasi-Static Modal Analysis approach (termed Rayleigh Quotient Nonlinear Modal Analysis) is adopted to calculate the nonlinear system response, and a Multi-Objective Optimization is solved to fit experimental data of the first mode of the Brake-Reuß Beam. Optimized parameters from the first mode are applied to the second and third bending modes to quantify the predictive ability of the models. Formulations for both tracing full hysteresis loops and recreating hysteresis loops from a single loading curve (Masing assumptions) are considered. Smoothly varying models applied to a five patch representation showed the highest flexibility (for fitting mode 1) and good predictive potential (for modes 2 and 3). For a second formulation, which uses 152 frictional elements to represent the interface, the physically motivated spring in series with a Coulomb slip model (elastic dry friction) has high error for fitting mode 1 and performs near the middle for predicting higher modes. For both interface representation, the best fit models are not the most physical, but rather the ones with the most parameters (as expected); however, the more physical models perform somewhat better for predicting the higher modes.
A Review of Damping Models for Structures With Mechanical Joints
(ASME, 2020) Mathis, Allen T.; Balaji, Nidish N.; Kuether, Robert J.; Brink, Adam R.; Brake, Matthew R.W.; Quinn, D. Dane
In a standard design practice, it is often necessary to assemble engineered structures from individually manufactured parts. Ideally, the assembled system should perform as if the connections between the components were perfect, that is, as if the system were a single monolithic piece. However, the fasteners used in those connections, such as mechanical lap joints, are imperfect and highly nonlinear. In particular, these jointed connections dissipate energy, often through friction over highly localized microscale regions near connection points, and are known to exhibit history dependent, or hysteretic behavior. As a result, while mechanical joints are one of the most common elements in structural dynamics problems, their presence implies that assembled structural systems are difficult to model and analyze. Through rigorous experimental, analytical, and numerical work over the past century, researchers from several different disciplines have developed numerous damping models that give rise to the dynamical behavior attributed to joints. This work seeks to review, compare, and contrast several linear and nonlinear damping models that are known to be relevant to modeling assembled structural systems. These models are presented and categorized to place them in the proper historical and mathematical context as well as presenting numerous examples of their applications. General properties of hysteretic friction damping models are also studied and compared analytically. Connections are drawn between the different models so as to not only identify differences between models, but also highlight commonalities not normally seen to be in association.
Comparison of nonlinear system identification methods for free decay measurements with application to jointed structures
(Elsevier, 2019) Jin, Mengshi; Brake, Matthew R.W.; Song, Hanwen
Assembled structures are nonlinear. The sources of this nonlinearity could include the jointed interfaces, damage and wear, non-idealized boundary conditions, or other features inherent in real parts. To study these systems and to ascertain if they will be operating in a regime in which the nonlinearity is prominent,ﾠnonlinear system identificationﾠtechniques are needed to assess and characterize the nature of the nonlinearity in the structure. Significant progress over the last few years has focused on using nonlinear system identification to identify damage and other deviations from idealized structures. This research reviews nine different methods for nonlinear system identification (restoring force surface,ﾠHilbert transform, directﾠquadrature, zero-crossing,ﾠshort-time Fourier transform, Gaborﾠwavelet, Morlet wavelet, Morse wavelet, and a neural network-based algorithm) in order to assess their accuracy. The methods are compared by identifying characteristics of two systems: a singleﾠdegree of freedomﾠmodel of a Duffing oscillator and measured data from a jointed structure. Asﾠneural networksﾠare not commonly used for system identification, multiple variations of the method are investigated to study its effectiveness.ﾠPerturbationﾠanalysis is conducted to see the efficacy of the different methods for identifying parameters across a large range of design spaces, and the advantages and disadvantages of each method are discussed. The primary contribution of this paper is a comparison on both analytical and experimental data of multiple widely used system identification methods, and an assessment of when each method is most and least applicable, specifically in the context of jointed structures.
Constitutive Modeling of Friction in Bolted Connections
(2021-11-16) Porter, Justin H.; Brake, Matthew R.W.
Bolted joints are ubiquitous in mechanical engineering, requiring accurate models to optimize designs. However, the exact nature of frictional contact between components is unknown and poses a significant challenge to modeling the nonlinear vibration of assemblies. This thesis applies empirical and physics-based modeling approaches to identify improvements to current models and a potential path towards predictive models of friction in bolted joints. The empirical modeling approach solves a multi-objective optimization to fit 26 friction model/interface representation combinations to experimental data and quantify the model form error. While the empirical models are not physical, the optimized results highlight the benefits of using smooth friction models and the limitations of a common physically motivated model. The physics-based model formulates the frictional force based on contact interactions of surface features and derives parameters from surface scans. While the physics-based model is not completely predictive, results show promising agreement with experiments.
Determining the Influence of Experimental Setup and Inputs on Nonlinear Systems
(2020-04-15) Smith, Scott A; Brake, Matthew R.W.
In the automotive and aerospace industries, every component experiences dynamic loading. The complex, multi-axial loads and surrounding system are the environmental conditions. Due to the lack of understanding of nonlinear system dynamics, the components and structures have been over-designed to handle these environmental conditions and reduce nonlinear influences. Over-designing leads to an increase in weight, which in turn reduces fuel efficiency. In recent years, there has been a drive to improve fuel economy resulting in the redesign of these over-designed systems. Reducing the component conservativeness increases nonlinear influences on the dynamic response of the system. Research is ongoing to characterize these new systems in laboratory settings. However, there is a disconnect between the environmental conditions and laboratory experiments. These disconnects are a result of improper model simplifications and input forces, costing billions of dollars in inappropriate/ additional qualification tests. Currently, researchers are using varying levels of fidelity to model the salient physics of the nonlinear systems. Unfortunately, they are still unable to match the response of systems with strong nonlinearities fully. This thesis is motivated by the need to improve and develop experimental methods to more accurately update models for environmental conditions. The specific issues addressed and significant contributions include: 1. A refinement of a recently formalized method proposed by D.J. Ewins, which consists of ten steps to perform model validation of nonlinear structures. This work details through a series of experimental studies that many standard test setup assumptions and properties of nonlinearities are invalid. This invalidation is due to both known and unrecognized nonlinear properties. A review of current methods for characterizing nonlinear and gaps in the approaches. 2. A series of tests to determine the influence of multi-axial excitation on system responses. Due to the complex environmental loads, a component will not experience excitation from only one direction. Current laboratory tests qualify the system one axis at a time, under the assumption that a nonlinearity has no dependence on the load direction. 3. A design methodology to create a test fixture to emulate the environmental boundary condition. The methodology truncates boundaries to a more reasonable size for laboratory testing by utilizing spring-mass systems to account for the loss of mass and stiffness. The method is validated by comparing stress states of an attached component of the full system and truncated one.
Experimental Investigation of Local Dynamics in a Bolted Lap Joint Using Digital Image Correlation
(ASME, 2020) Brøns, Marie; Kasper, Thomas A.; Chauda, Gaurav; Klaassen, Steven W.B.; Schwingshackl, Christoph W.; Brake, Matthew R.W.
The dynamics of structures with joints commonly show nonlinearity in their responses. This nonlinear behavior can arise from the local dynamics of the contact interfaces. The nonlinear mechanisms at an interface are complicated to study due to the lack of observability within the contact interface itself. In this work, digital image correlation (DIC) is used in combination with a high-speed camera to observe the local motion at the edge of the interface of a bolted lap joint. Results demonstrate that it is possible to use this technique to monitor the localized motion of an interface successfully. It is observed that the two beam parts of the studied lap joint separate when undergoing bending vibrations and that there is a clear asymmetry in the response of the left and the right end of the interface. Profilometry indicates that the asymmetry in the response is due to the mesoscale topography of the contact interface, highlighting the importance of accounting for surface features in order to model the nonlinearities of a contact interface accurately.
Finite element analysis of screw fixation durability under multiple boundary and loading conditions for a custom pelvic implant
(Elsevier, 2023) Zhu, Yuhui; Babazadeh-Naseri, Ata; Dunbar, Nicholas J.; Brake, Matthew R.W.; Zandiyeh, Payam; Li, Geng; Leardini, Alberto; Spazzoli, Benedetta; Fregly, Benjamin J.
Despite showing promising functional outcomes for pelvic reconstruction after sarcoma resection, custom-made pelvic implants continue to exhibit high complication rates due to fixation failures. Patient-specific finite element models have been utilized by researchers to evaluate implant durability. However, the effect of assumed boundary and loading conditions on failure analysis results of fixation screws remains unknown. In this study, the postoperative stress distributions in the fixation screws of a state-of-the-art custom-made pelvic implant were simulated, and the risk of failure was estimated under various combinations of two bone-implant interaction models (tied vs. frictional contact) and four load cases from level-ground walking and stair activities. The study found that the average weighted peak von Mises stress could increase by 22-fold when the bone-implant interactions were modeled with a frictional contact model instead of a tied model, and the likelihood of fatigue and pullout failure for each screw could change dramatically when different combinations of boundary and loading conditions were used. The inclusion of additional boundary and loading conditions led to a more reliable analysis of fixation durability. These findings demonstrated the importance of simulating multiple boundary conditions and load cases for comprehensive implant design evaluation using finite element analysis.
Identification of Instantaneous Frequency and Damping From Transient Decay Data
(ASME, 2020) Jin, Mengshi; Chen, Wei; Brake, Matthew R.W.; Song, Hanwen
Jointed interfaces, damage, wear, or non-idealized boundary conditions often introduce nonlinear characteristics to assembled structures. Consequently, extensive research has been carried out regarding nonlinear system identification. The development of nonlinear system identification is also enabling the intentional application of nonlinearities towards practical fields such as vibration control and energy harvesting. This research proposes a nonlinear identification procedure that consists of two steps: first, the raw data is filtered by the Double Reverse Multimodal Decomposition method that involves system reconstruction, expansion, and filtering twice. Second, the Peak Finding and Fitting method is applied to the filtered signal to extract the instantaneous amplitude and frequency. The identification procedure is applied to the measured responses from a jointed structure to assess its efficacy. The results are compared with those obtained from other well-known methods—the Hilbert transform and zero-crossing methods. The comparison results indicate that the Peaking Finding and Fitting method extracts the amplitude of the response signal more accurately. Consequently, this yields a higher signal-to-noise ratio in the extracted damping values. As a recommended last step, uncertainty assessment is conducted to calculate the 95% confidence intervals of the nonlinear properties of the system.
Interface reduction for Hurty/Craig-Bampton substructured models: Review and improvements
(Elsevier, 2019) Krattiger, Dimitri; Wu, Long; Zacharczuk, Martin; Buck, Martin; Kuether, Robert J.; Allen, Matthew S.; Tiso, Paolo; Brake, Matthew R.W.
The Hurty/Craig-Bampton method in structural dynamics represents the interior dynamics of each subcomponent in a substructured system with a truncated set of normal modes and retains all of the physical degrees of freedom at the substructure interfaces. This makes the assembly of substructures into a reduced-order system model relatively simple, but means that the reduced-order assembly will have as many interface degrees of freedom as the full model. When the full-model mesh is highly refined, and/or when the system is divided into many subcomponents, this can lead to an unacceptably large system of equations of motion. To overcome this, interface reduction methods aim to reduce the size of the Hurty/Craig-Bampton model by reducing the number of interface degrees of freedom. This research presents a survey of interface reduction methods for Hurty/Craig-Bampton models, and proposes improvements and generalizations to some of the methods. Some of these interface reductions operate on the assembled system-level matrices while others perform reduction locally by considering the uncoupled substructures. The advantages and disadvantages of these methods are highlighted and assessed through comparisons of results obtained from a variety of representative linear FE models.
Measurement and identification of the nonlinear dynamics of a jointed structure using full-field data, Part I: Measurement of nonlinear dynamics
(Elsevier, 2022) Chen, Wei; Jana, Debasish; Singh, Aryan; Jin, Mengshi; Cenedese, Mattia; Kosova, Giancarlo; Brake, Matthew R.W.; Schwingshackl, Christoph W.; Nagarajaiah, Satish; Moore, Keegan J.; Noël, Jean-Philippe
Jointed structures are ubiquitous constituents of engineering systems; however, their dynamic properties (e.g., natural frequencies and damping ratios) are challenging to identify correctly due to the complex, nonlinear nature of interfaces. This research seeks to extend the efficacy of traditional experimental methods for linear system identification (such as impact testing, shaker ringdown testing, random excitation, and force or amplitude-control stepped sine testing) on nonlinear jointed systems, e.g., the half Brake–Reuß beam, by augmenting them with full-field data collected by high-speed videography. The full-field response is acquired using high-speed cameras combined with Digital Image Correlation (DIC), which enables studying the spatial–temporal dynamic characteristics of the system. As this is a video-based experiment, additional constraints are attached to the beam at the node points to remove the rigid body motion, which ensures that the beam is in the view of the camera during the entire test. The use of a video-based method introduces new sources of experimental error, such as noise from the high-speed camera’s fan and electrical noise, and so the measurement accuracy of DIC is validated using accelerometer data. After validating the DIC data, the measurements are recorded for several types of excitation, including hammer testing, shaker ringdown testing, fixed sine testing, and stepped sine testing. Using the DIC data to augment standard nonlinear system identification techniques, modal coupling and the mode shapes’ evolution are investigated. The suitability of videography methods for nonlinear system identification of nonlinear beams is explored for the first time in this paper, and recommendations for techniques to facilitate this process are made. This article focuses on developing an accurate data collection methodology as well as recommendations for nonlinear testing with DIC, which paves the way for video-based investigation of nonlinear system identification. In Part-II (Jin et al., 2021) of this work, the same data set is used for a rigorous assessment of nonlinear system identification with full-field DIC data.
Measurement and identification of the nonlinear dynamics of a jointed structure using full-field data; Part II - Nonlinear system identification
(Elsevier, 2022) Jin, Mengshi; Kosova, Giancarlo; Cenedese, Mattia; Chen, Wei; Singh, Aryan; Jana, Debasish; Brake, Matthew R.W.; Schwingshackl, Christoph W.; Nagarajaiah, Satish; Moore, Keegan J.; Noël, Jean-Philippe
The dynamic responses of assembled structures are greatly affected by the mechanical joints, which are often the cause of nonlinear behavior. To better understand and, in the future, tailor the nonlinearities, accurate methods are needed to characterize the dynamic properties of jointed structures. In this paper, the nonlinear characteristics of a jointed beam is studied with the help of multiple identification methods, including the Hilbert Transform method, Peak Finding and Fitting method, Dynamic Mode Decomposition method, State-Space Spectral Submanifold, and Wavelet-Bounded Empirical Mode Decomposition method. The nonlinearities are identified by the responses that are measured via accelerometers in a series of experiments that consist of hammer testing, shaker ringdown testing, and response/force-control stepped sine testing. In addition to accelerometers, two high-speed cameras are used to capture the motion of the whole structure during the shaker ringdown testing. Digital Image Correlation (DIC) is then adopted to obtain the displacement responses and used to determine the mode shapes of the jointed beam. The accuracy of the DIC data is validated by the comparison between the identification results of acceleration and displacement signals. As enabled by full-field data, the energy-dependent characteristics of the structure are also presented. The setup of the different experiments is described in detail in Part I (Chen et al., 2021) of this research. The focus of this paper is to compare nonlinear system identification methods applied to different measurement techniques and to exploit the use of high spatial resolution data.
On the Characterization of Nonlinearities in Assembled Structures
(ASME, 2020) Smith, Scott A.; Brake, Matthew R.W.; Schwingshackl, Christoph W.
This work refines a recently formalized methodology proposed by D.J. Ewins consisting of ten steps for model validation of nonlinear structures. This work details, through a series of experimental studies, that many standard test setup assumptions that are made when performing dynamic testing are invalid and need to be evaluated for each structure. The invalidation of the standard assumptions is due to the presence of nonlinearities, both known and unrecognized in the system. Complicating measurements, many nonlinearities are currently characterized as constant properties instead of variables that exhibit dependency on system hysteresis and actuation amplitude. This study reviews current methods for characterizing nonlinearities and outlines gaps in the approaches. A brief update to the CONCERTO method, based on the accelerance of a system, is derived for characterizing a system’s nonlinearities. Finally, this study ends with an updated methodology for model validation and the ramifications for modeling assemblies with nonlinearities are discussed.
Reduced order modeling for the dynamics of jointed structures through hyper-reduced interface representation
(Elsevier, 2021) Balaji, Nidish Narayanaa; Dreher, Tobias; Krack, Malte; Brake, Matthew R.W.
One strategy to develop both accurate and computationally tractable models of jointed structures is reduced order modeling through hyper-reduced representations of the interfaces in contact. Hyper reduction refers to reduction techniques that result in a Reduced Order Model (ROM) that is complete by itself, i.e., all displacements and forces are fully described in the ROM coordinates directly. Focusing primarily on applications involving small relative displacement contacts, two fundamentally different approaches are formulated and compared for merits and limitations in applicability. The first is an adaptation of the stiffness-preserving RBE3 constraint elements, and the second is an interpolation approach based on remeshing the interface. Although RBE3 is extensively used in the literature, the current formulation derives stiffness preserving elements that are specifically useful for contact dynamics applications. Transformations are developed to express force–displacement relationships in the ROM coordinates that are congruent (in the sense of using the same contact models) as well as consistent (in the sense of being faithful to the quantities involved) with a high-fidelity model of the same structure. These approaches are applied to study a three-bolted lap-joint structure (the Brake-Reuß Beam (BRB) benchmark) that has been observed to demonstrate characteristic contact non-linearities. Multiple strategies for the hyper reduction are evaluated, including graph partitioning, finite element coarsening, and homogenization of field objectives, some of which involve an extra step of remeshing/choosing patches based on a field objective (e.g., contact pressure). The performances of the ROMs are assessed by conducting nonlinear modal analysis and computing a posteriori error metrics.
Strain Hardening for Elastic-Perfectly Plastic to Perfectly Elastic Flattening Single Asperity Contact
(ASME, 2018) Ghaednia, Hamid; Brake, Matthew R.W.; Berryhill, Michael; Jackson, Robert L.
For elastic contact, an exact analytical solution for the stresses and strains within two contacting bodies has been known since the 1880s. Despite this, there is no similar solution for elastic-plastic contact due to the integral nature of plastic deformations, and the few models that do exist develop approximate solutions for the elastic-perfectly plastic material model. In this work, the full transition from elastic-perfectly plastic to elastic materials in contact is studied using a bilinear material model in a finite element environment for a frictionless dry flattening contact. Even though the contact is considered flattening, elastic deformations are allowed to happen on the flat. The real contact radius is found to converge to the elastic contact limit at a tangent modulus of elasticity around 20 %. For the contact force, results show a different trend in which there is a continual variation in forces across the entire range of material models studied. A new formulation has been developed based on finite element results to predict the deformations, the real contact area, and contact force. A second approach has been introduced to calculate the contact force based on the approximation of the Hertzian solution for the elastic deformations on the flat. The proposed formulation is verified for five different materials sets.
Strain Hardening From Elastic-Perfectly Plastic to Perfectly Elastic Indentation Single Asperity Contact
(Frontiers Media S.A., 2020) Ghaednia, Hamid; Mifflin, Gregory; Lunia, Priyansh; O'Neill, Eoghan O.; Brake, Matthew R.W.
Indentation measurements are a crucial technique for measuring mechanical properties. Although several contact models have been developed to relate force-displacement measurements with the mechanical properties, they all consider simplifying assumptions, such as no strain hardening, which significantly affects the predictions. In this study, the effect of bilinear strain hardening on the contact parameters for indentations is investigated. Simulations show that even 1% strain hardening causes significant changes in the contact parameters and contact profile. Pile-up behavior is observed for elastic-perfectly plastic materials, while for strain hardening values greater than 6%, only sink-in (i.e., no pile-up) is seen. These results are used to derive a new, predictive formulation to account for the bilinear strain hardening from elastic-perfectly plastic to purely elastic materials.
The Influence of Additively Manufactured Nonlinearities on the Dynamic Response of Assembled Structures
(ASME, 2020) Shu, Hang; Smith, Scott A.; Brake, Matthew R.W.; Tribomechadynamics Laboratory
Structural dynamic techniques have been proven accurate at predicting the vibrations of single parts (i.e., monolithic specimens), which are widely used in industrial applications. However, vibration analysis of such assemblies often exhibits high variability or nonrepeatability due to jointed interfaces. Inspired by advances in additive manufacturing (AM) and nonlinear vibration absorber theory, this research seeks to redesign jointed structures in an attempt to reduce the nonlinear effects introduced by the jointed interfaces. First, the nonlinear dynamics of a conventionally manufactured beam and an AM beam are measured in both a traditional (flat) lap joint assembly and also a “linearized” lap joint configuration (termed the small pad). Second, the internal structure of the AM beam is varied by printing specimens with internal vibration absorbers. With the two interface geometries studied in this experiment, the flat interface is found to be predominantly nonlinear, and introducing a vibration absorber fails to reduce the nonlinearities from the jointed interface. The small-pad responses are relatively linear in the range of excitation used in the analysis, and the nonlinear effects are further reduced with the presence of a center vibration absorber. Overall, the energy dissipation at the interface is highly dependent on the design of the contact interface and the internal vibration absorber. Adding a nonlinear vibration absorber alone is insufficient to negate the interfacial nonlinearity from the assembly|| therefore, future work is needed to study the shape, location, and material for the design and fabrication of nonlinear vibration absorbers.
The Variability of Strains in Bolts and the Effect on Preload in Jointed Structure
(2019-04-19) Ruan, Mianmian; Brake, Matthew R.W.
Torque wrenches are the most common method used for tightening bolts; however, this method can cause a large amount of uncertainty in the resulting bolt preload. With the inability to determine the preload in bolted structures precisely, overdesign is necessary for bolts to meet safety requirements and prevent jointed structures from looseness failure. A series of comparative experiments are completed to quantify the influence of bolt strain variation based on strain gauges, force transducers, and accelerators. Moreover, this research uses the Peak Finding and Fitting (PFF) PFF algorithm to analyze nonlinear dynamics of the experimental system, which demonstrates the ramifications of the experimental results. Measured strains in bolts showed significant variabilities, which indicates larger the expected uncertainty in preload. Impact test results also indicate the advantages of pre-strain method for tightening bolts compared to torque wrenches.
Towards a predictive, physics-based friction model for the dynamics of jointed structures
(Elsevier, 2023) Porter, Justin H.; Brake, Matthew R.W.
Bolted connections are ubiquitous in mechanical designs and pose a significant challenge to understanding and predicting the vibration response of assembled structures. The present paper develops a physics-based rough contact model of the frictional interactions within a joint. This model sums over the probable interactions of asperities – defined as locally maximum surface features – to determine the contact forces. Here, the tangential contact forces vary smoothly between sticking and slipping and allow the model to better capture the qualitative trends of experimental amplitude dependent frequency and damping than previous studies. Furthermore, the novel model is generalized to allow for arbitrarily varying normal pressure including potential separation to better represent the interfacial dynamics. This includes developing a new, computationally tractable approximation to the analytical Mindlin partial slip solution for tangential loading of contacting spheres. The results highlight the importance of accurately characterizing the as-built topology of the interface, the plastic behavior of the contacting asperities, the relevant length scale of asperities, and the eccentricity of asperities. A predictive friction coefficient based on plasticity provides a poor match to experiments, so fitting the friction coefficient is also considered. Numerical results are compared to experiments on the Brake-Reuß Beam to assess the predictive potential of the models. While blind predictions over-predict the slip limit, the current model presents a significant improvement in physics-based modeling and highlights areas for ongoing research.
Wave-based analysis of jointed elastic bars: nonlinear periodic response
(Springer Nature, 2022) Balaji, Nidish Narayanaa; Brake, Matthew R.W.; Leamy, Michael J.
In this paper, we develop two wave-based approaches for predicting the nonlinear periodic response of jointed elastic bars. First, we present a nonlinear wave-based vibration approach (WBVA) for studying jointed systems informed by re-usable, perturbation-derived scattering functions. This analytical approach can be used to predict the steady-state, forced response of jointed elastic bar structures incorporating any number and variety of nonlinear joints. As a second method, we present a nonlinear Plane-Wave Expansion (PWE) approach for analyzing periodic response in the same jointed bar structures. Both wave-based approaches have advantages and disadvantages when compared side-by-side. The WBVA results in a minimal set of equations and is re-usable following determination of the reflection and transmission functions, while the PWE formulation can be easily applied to new joint models and maintains solution accuracy to higher levels of nonlinearity. For example cases of two and three bars connected by linearly damped joints with linear and cubic stiffness, the two wave-based approaches accurately predict the expected Duffing-like behavior in which multiple periodic responses occur in the near-resonant regime, in close agreement with reference finite element simulations. Lastly, we discuss extensions of the work to jointed structures composed of beam-like members, and propose follow-on studies addressing opportunities identified in the application of the methods presented.
Wave-based analysis of jointed elastic bars: stability of nonlinear solutions
(Springer Nature, 2022) Balaji, Nidish Narayanaa; Brake, Matthew R.W.; Leamy, Michael J.
In this paper we develop two new approaches for directly assessing stability of nonlinear wave-based solutions, with application to jointed elastic bars. In the first stability approach, we strain a stiffness parameter and construct analytical stability boundaries using a wave-based method. Not only does this accurately determine stability of the periodic solutions found in the example case of two bars connected by a nonlinear joint, but it directly governs the response and stability of parametrically forced continuous systems without resorting to discretization, a new development in of itself. In the second stability approach, we pose a perturbation eigenproblem residue (PER) and show that changes in the sign of the PER locate critical points where stability changes from stable to unstable, and vice-versa. Lastly, we discuss follow-on research using the developed stability approaches. In particular, we identify an opportunity to study stability around internal resonance, and then identify a need to further develop and interpret the PER approach to directly predict stability.

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