Finite element models can be used to model and predict the hysteresis and energy dissipation exhibited by nonlinear joints in structures. As a result of the nonlinearity, the frequency and damping of a mode is dependent on excitation amplitude, and when the modes remain uncoupled, quasi-static modal analysis has been shown to efficiently predict this behavior. However, in some cases the modes have been observed to couple such that the frequency and damping of one mode is dependent on the amplitude of other modes. To model the interactions between modes, one must integrate the dynamic equations in time, which is several orders of magnitude more expensive than quasi-static analysis. This work explores an alternative where quasi-static forces are applied in the shapes of two or more modes of vibration simultaneously, and the resulting load–displacement curves are used to deduce the effect of other modes on the effective frequency and damping of the mode in question. This methodology is demonstrated on a simple 2D cantilever beam structure with a single bolted joint which exhibits micro-slip nonlinearity over a range of vibration amplitudes. The predicted frequency and damping are compared with those extracted from a few expensive dynamic simulations of the structure, showing that the quasi-static approach produces reasonable albeit highly conservative bounds on the observed dynamics. This framework is also demonstrated on a 3D structure where dynamic simulations are infeasible.
Finite element models can be used to model and predict the hysteresis and energy dissipation exhibited by nonlinear joints in structures. As a result of the nonlinearity, the frequency and damping of a mode is dependent on excitation amplitude, and when the modes remain uncoupled, quasi-static modal analysis has been shown to efficiently predict this behavior. However, in some cases the modes have been observed to couple such that the frequency and damping of one mode is dependent on the amplitude of other modes. To model the interactions between modes, one must integrate the dynamic equations in time, which is several orders of magnitude more expensive than quasi-static analysis. This work explores an alternative where quasi-static forces are applied in the shapes of two or more modes of vibration simultaneously, and the resulting load–displacement curves are used to deduce the effect of other modes on the effective frequency and damping of the mode in question. This methodology is demonstrated on a simple 2D cantilever beam structure with a single bolted joint which exhibits micro-slip nonlinearity over a range of vibration amplitudes. The predicted frequency and damping are compared with those extracted from a few expensive dynamic simulations of the structure, showing that the quasi-static approach produces reasonable albeit highly conservative bounds on the observed dynamics. This framework is also demonstrated on a 3D structure where dynamic simulations are infeasible.
In the context of experimental vibration data, strain gauges can obtain linear and nonlinear dynamic measurements. However, measuring strain can be disincentivizing and expensive due to the complexity of data acquisition systems, lack of portability, and high costs. This research introduces the use of a low-cost efficient wireless intelligent sensor for strain (LEWIS-S) that is based on a portable-sensor-design platform that streamlines strain sensing. Additionally, the softening behavior of a cantilever beam with geometric and inertial nonlinearities is characterized by the LEWIS-S based on high force level inputs. Two experiments were performed on a nonlinear cantilever beam with measurements obtained by the LEWIS-S sensor and an accelerometer. First, a sine sweep test was performed through the fundamental resonance of the system, then a ring-down test was performed from a large initial static deformation. Good agreement was revealed in quantities of interest such as frequency response functions, the continuous wavelet transforms, and softening behavior in the backbone curves.
In this work, we study how a contact/impact nonlinearity interacts with a geometric cubic nonlinearity in an oscillator system. Specific focus is shown to the effects on bifurcation behavior and secondary resonances (i.e., super- and sub-harmonic resonances). The effects of the individual nonlinearities are first explored for comparison, and then the influences of the combined nonlinearities, varying one parameter at a time, are analyzed and discussed. Nonlinear characterization is then performed on an arbitrary system configuration to study super- and sub-harmonic resonances and grazing contacts or bifurcations. Both the cubic and contact nonlinearities cause a drop in amplitude and shift up in frequency for the primary resonance, and they activate high-amplitude subharmonic resonance regions. The nonlinearities seem to never destructively interfere. The contact nonlinearity generally affects the system's superharmonic resonance behavior more, particularly with regard to the occurrence of grazing contacts and the activation of many bifurcations in the system's response. The subharmonic resonance behavior is more strongly affected by the cubic nonlinearity and is prone to multistable behavior. Perturbation theory proved useful for determining when the cubic nonlinearity would be dominant compared to the contact nonlinearity. The limiting behaviors of the contact stiffness and freeplay gap size indicate the cubic nonlinearity is dominant overall. It is demonstrated that the presence of contact may result in the activation of several bifurcations. In addition, it is proved that the system's subharmonic resonance region is prone to multistable dynamical responses having distinct magnitudes.
Dynamical systems subject to intermittent contact are often modeled with piecewise-smooth contact forces. However, the discontinuous nature of the contact can cause inaccuracies in numerical results or failure in numerical solvers. Representing the piecewise contact force with a continuous and smooth function can mitigate these problems, but not all continuous representations may be appropriate for this use. In this work, five representations used by previous researchers (polynomial, rational polynomial, hyperbolic tangent, arctangent, and logarithm-arctangent functions) are studied to determine which ones most accurately capture nonlinear behaviors including super- and subharmonic resonances, multiple solutions, and chaos. The test case is a single-DOF forced Duffing oscillator with freeplay nonlinearity, solved using direct time integration. This work intends to expand on past studies by determining the limits of applicability for each representation and what numerical problems may occur.
Conference Proceedings of the Society for Experimental Mechanics Series
Saunders, Brian E.; Vasconcellos, Rui M.G.; Kuether, Robert J.; Abdelkefi, Abdessattar
Dynamical systems containing contact/impact between parts can be modeled as piecewise-smooth reduced-order models. The most common example is freeplay, which can manifest as a loose support, worn hinges, or backlash. Freeplay causes very complex, nonlinear responses in a system that range from isolated resonances to grazing bifurcations to chaos. This can be an issue because classical solution methods, such as direct time integration (e.g., Runge-Kutta) or harmonic balance methods, can fail to accurately detect some of the nonlinear behavior or fail to run altogether. To deal with this limitation, researchers often approximate piecewise freeplay terms in the equations of motion using continuous, fully smooth functions. While this strategy can be convenient, it may not always be appropriate for use. For example, past investigation on freeplay in an aeroelastic control surface showed that, compared to the exact piecewise representation, some approximations are not as effective at capturing freeplay behavior as other ones. Another potential issue is the effectiveness of continuous representations at capturing grazing contacts and grazing-type bifurcations. These can cause the system to transition to high-amplitude responses with frequent contact/impact and be particularly damaging. In this work, a bifurcation study is performed on a model of a forced Duffing oscillator with freeplay nonlinearity. Various representations are used to approximate the freeplay including polynomial, absolute value, and hyperbolic tangent representations. Bifurcation analysis results for each type are compared to results using the exact piecewise-smooth representation computed using MATLAB® Event Location. The effectiveness of each representation is compared and ranked in terms of numerical accuracy, ability to capture multiple response types, ability to predict chaos, and computation time.
The goal of this paper is to present a set of measurements from a benchmark structure containing two bolted joints to support future efforts to predict the damping due to the joints and to model nonlinear coupling between the first two elastic modes. Bolted joints introduce nonlinearities in structures, typically causing a softening in the natural frequency and an increase in damping because of frictional slip between the contact interfaces within the joint. These nonlinearities pose significant challenges when characterizing the response of the structure under a large range of load amplitudes, especially when the modal responses become coupled, causing the effective damping and natural frequency to not only depend on the excitation amplitude of the targeted mode, but also the relative amplitudes of other modes. In this work, two nominally identical benchmark structures, known in some prior works as the S4 beam, are tested to characterize their nonlinear properties for the first two elastic modes. Detailed surface measurements are presented and validated through finite element analysis and reveal distinct contact interactions between the two sets of beams. The free-free test structures are excited with an impact hammer and the transient response is analyzed to extract the damping and frequency backbone curves. A range of impact amplitudes and drive points are used to isolate a single mode or to excite both modes simultaneously. Differences in the nonlinear response correlate with the relative strength of the modes that are excited, allowing one to characterize mode coupling. Each of the beams shows different nonlinear properties for each mode, which is attributed to the different contact pressure distributions between the parts, although the mode coupling relationship is found to be consistent between the two. The test data key finding are presented in this paper and the supporting data is available on a public repository for interested researchers.
Conference Proceedings of the Society for Experimental Mechanics Series
Singh, Aabhas; Wielgus, Kayla M.; Dimino, Ignazio; Kuether, Robert J.; Allen, Matthew S.
Morphing wings have great potential to dramatically improve the efficiency of future generations of aircraft and to reduce noise and emissions. Among many camber morphing wing concepts, shape changing fingerlike mechanisms consist of components, such as torsion bars, bushings, bearings, and joints, all of which exhibit damping and stiffness nonlinearities that are dependent on excitation amplitude. These nonlinearities make the dynamic response difficult to model accurately with traditional simulation approaches. As a result, at high excitation levels, linear finite element models may be inaccurate, and a nonlinear modeling approach is required to capture the necessary physics. This work seeks to better understand the influence of nonlinearity on the effective damping and natural frequency of the morphing wing through the use of quasi-static modal analysis and model reduction techniques that employ multipoint constraints (i.e., spider elements). With over 500,000 elements and 39 frictional contact surfaces, this represents one of the most complicated models to which these methods have been applied to date. The results to date are summarized and lessons learned are highlighted.
Nonlinear force appropriation is an extension of its linear counterpart where sinusoidal excitation is applied to a structure with a modal shaker and phase quadrature is achieved between the excitation and response. While a standard practice in modal testing, modal shaker excitation has the potential to alter the dynamics of the structure under test. Previous studies have been conducted to address several concerns, but this work specifically focuses on a shaker-structure interaction phenomenon which arises during the force appropriation testing of a nonlinear structure. Under pure-tone sinusoidal forcing, a nonlinear structure may respond not only at the fundamental harmonic but also potentially at sub- or superharmonics, or it can even produce aperiodic and chaotic motion in certain cases. Shaker-structure interaction occurs when the response physically pushes back against the shaker attachment, producing non-fundamental harmonic content in the force measured by the load cell, even for pure tone voltage input to the shaker. This work develops a model to replicate these physics and investigates their influence on the response of a nonlinear normal mode of the structure. Experimental evidence is first provided that demonstrates the generation of harmonic content in the measured load cell force during a force appropriation test. This interaction is replicated by developing an electromechanical model of a modal shaker attached to a nonlinear, three-mass dynamical system. Several simulated experiments are conducted both with and without the shaker model in order to identify which effects are specifically due to the presence of the shaker. The results of these simulations are then compared to the undamped nonlinear normal modes of the structure under test to evaluate the influence of shaker-structure interaction on the identified system's dynamics.
A proper understanding of the complex physics associated with nonlinear dynamics can improve the accuracy of predictive engineering models and provide a foundation for understanding nonlinear response during environmental testing. Several researchers and studies have previously shown how localized nonlinearities can influence the global vibration modes of a system. This current work builds upon the study of a demonstration aluminum aircraft with a mock pylon with an intentionally designed, localized nonlinearity. In an effort to simplify the identification of the localized nonlinearity, previous work has developed a simplified experimental setup to collect experimental data for the isolated pylon mounted to a stiff fixture. This study builds on these test results by correlating a multi-degree-of-freedom model of the pylon to identify the appropriate model form and parameters of the nonlinear element. The experimentally measured backbone curves are correlated with a nonlinear Hurty/Craig-Bampton (HCB) reduced order model (ROM) using the calculated nonlinear normal modes (NNMs). Following the calibration, the nonlinear HCB ROM of the pylon is attached to a linear HCB ROM of the wing to predict the NNMs of the next-level wing-pylon assembly as a pre-test analysis to better understand the significance of the localized nonlinearity on the global modes of the wing structure.
Conference Proceedings of the Society for Experimental Mechanics Series
Saunders, Brian E.; Vasconcellos, Rui M.G.; Kuether, Robert J.; Abdelkefi, Abdessattar
Physical systems that are subject to intermittent contact/impact are often studied using piecewise-smooth models. Freeplay is a common type of piecewise-smooth system and has been studied extensively for gear systems (backlash) and aeroelastic systems (control surfaces like ailerons and rudders). These systems can experience complex nonlinear behavior including isolated resonance, chaos, and discontinuity-induced bifurcations. This behavior can lead to undesired damaging responses in the system. In this work, bifurcation analysis is performed for a forced Duffing oscillator with freeplay. The freeplay nonlinearity in this system is dependent on the contact stiffness, the size of the freeplay region, and the symmetry/asymmetry of the freeplay region with respect to the system’s equilibrium. Past work on this system has shown that a rich variety of nonlinear behaviors is present. Modern methods of nonlinear dynamics are used to characterize the transitions in system response including phase portraits, frequency spectra, and Poincaré maps. Different freeplay contact stiffnesses are studied including soft, medium, and hard in order to determine how the system response changes as the freeplay transitions from soft contact to near-impact. Particular focus is given to the effects of different initial conditions on the activation of secondary- and isolated-resonance responses. Preliminary results show isolated resonances to occur only for softer-contact cases, regions of superharmonic resonances are more prevalent for harder-contact cases, and more nonlinear behavior occurs for higher initial conditions.
This paper presents a set of tests on a bolted benchmark structure called the S4 beam with a focus on evaluating coupling between the first two modes due to nonlinearity. Bolted joints are of interest in dynamically loaded structures because frictional slipping at the contact interface can introduce amplitude-dependent nonlinearities into the system, where the frequency of the structure decreases, and the damping increases. The challenge to model this phenomenon is even more difficult if the modes of the structure become coupled, violating a common assumption of mode orthogonality. This work presents a detailed set of measurements in which the nonlinearities of a bolted structure are highly coupled for the first two modes. Two nominally identical bolted structures are excited using an impact hammer test. The nonlinear damping curves for each beam are calculated using the Hilbert transform. Although the two structures have different frequency and damping characteristics, the mode coupling relationship between the first two modes of the structures is shown to be consistent and significant. The data is intended as a challenge problem for interested researchers; all data from these tests are available upon request.
The 2020 Nonlinear Mechanics and Dynamics (NOMAD) Research Institute was successfully held from June 15 to July 30, 2020. NOMAD brings together participants with diverse technical backgrounds to work in small teams to cultivate new ideas and approaches in engineering mechanics and dynamics research. NOMAD provides an opportunity for researchers – especially early career researchers - to develop lasting collaborations that go beyond what can be established from the limited interactions at their institutions or at annual conferences. A total of 11 students participated in the seven-week long program held virtually due to the COVID-19 health pandemic. The students collaborated on one of four research projects that were developed by various mentors from Sandia National Laboratories, the University of New Mexico, and other academic and research institutions. In addition to the research activities, the students attended weekly technical seminars, various virtual tours, and socialized at virtual gatherings. At the end of the summer, the students gave a final technical presentation on their research findings. Many of the research discoveries made at NOMAD 2020 are published as proceedings at technical conferences and have direct alignment with the critical mission work performed at Sandia.
This work studies the different types of behavior and inaccuracies that can occur when contact is not adequately accounted for in a dynamical system with freeplay, as the strength of the contact stiffness increases. The MATLAB® ode45 time integration solver, with the built-in Event Location capability, is first validated using past experimental data from a forced Duffing oscillator with freeplay. Next, numerical results utilizing event location are compared to results neglecting event location in order to highlight possible numerical errors and effects on multistable dynamical responses. Inaccuracies tend to occur in two different ways. First, neglecting event location can affect the boundaries between basins of attraction. Second, neglecting event location has little effect on the behaviors of the attractor solutions themselves besides merely resembling poorly converged solutions. Errors are less pronounced at the limits of soft or hard contact stiffness. This study shows the importance of accurately solving piecewise-smooth systems and the existing correlation between the strength of the contact force and possible numerical inaccuracies.
Virtual prototyping in engineering design rely on modern numerical models of contacting structures with accurate resolution of interface mechanics, which strongly affect the system-level stiffness and energy dissipation due to frictional losses. High-fidelity modeling within the localized interfaces is required to resolve local quantities of interest that may drive design decisions. The high-resolution finite element meshes necessary to resolve inter-component stresses tend to be computationally expensive, particularly when the analyst is interested in response time histories. The Hurty/Craig-Bampton (HCB) transformation is a widely used method in structural dynamics for reducing the interior portion of a finite element model while having the ability to retain all nonlinear contact degrees of freedom (DOF) in physical coordinates. These models may still require many DOF to adequately resolve the kinematics of the interface, leading to inadequate reduction and computational savings. This study proposes a novel interface reduction method to overcome these challenges by means of system-level characteristic constraint (SCC) modes and properly orthogonal interface modal derivatives (POIMDs) for transient dynamic analyses. Both SCC modes and POIMDs are computed using the reduced HCB mass and stiffness matrices, which can be directly computed from many commercial finite element analysis software. Comparison of time history responses to an impulse-type load in a mechanical beam assembly indicate that the interface-reduced model correlates well with the HCB truth model. Localized features like slip and contact area are well-represented in the time domain when the beam assembly is loaded with a broadband excitation. The proposed method also yields reduced-order models with greater critical timestep lengths for explicit integration schemes.