Matthew Robert Brake

Variability and Repeatability of Jointed Structures with Frictional Interfaces

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.

The Effects of Boundary Conditions, Measurement Techniques, and Excitation Type on Measurements of the Properties of Mechanical Joints

This paper investigates how the responses of mechanical joints are influenced by using different experimental setups. The experiments are conducted on both a monolithic beam and a bolted beam, and the beams are excited by hammer tests and a shaker. Multiple boundary conditions are also studied. It is found that the hammer tests performed on the “free” boundary condition monolithic beam (for multiple bungee lengths and positions) had a negligible influence on the system in terms of damping ratio and frequency variation. Multiple sensors attached to the monolithic beam are studied; the effect of multiple accelerometers manifests as a significant shift of frequency and damping due to the additional mass. In the case of the jointed beam, both mirror-like and rough interfaces are used. Several sets of different interface pairs, bolt torques, bolt preloads, excitation frequency sweep rates and bolt tightening orders are considered in this study. The time varying changes in stiffening and damping are measured by testing multiple combinations of the experimental setup at different levels of excitation. The results showed that the mirror-like surface finish for the interface has higher damping values compared to the rough surface across multiple bolt torque scenarios (such as preload and tightening order) and modes of vibration. Guidelines for a more reliable measurement of the properties of a mechanical joint are made based on the results of this research.

Evaluating Convergence of Reduced Order Models Using Nonlinear Normal Modes

It is often prohibitively expensive to integrate the response of a high order nonlinear system, such as a finite element model of a nonlinear structure, so a set of linear eigenvectors is often used as a basis in order to create a reduced order model (ROM). By augmenting the linear basis with a small set of discontinuous basis functions, ROMs of systems with local nonlinearities have been shown to compare well with the corresponding full order models. When evaluating the quality of a ROM, it is common to compare the time response of the model to that of the full order system, but the time response is a complicated function that depends on a predetermined set of initial conditions or external force. This is difficult to use as a metric to measure convergence of a ROM, particularly for systems with strong, non-smooth nonlinearities, for two reasons: (1) the accuracy of the response depends directly on the amplitude of the load/initial conditions, and (2) small differences between two signals can become large over time. Here, a validation metric is proposed that is based solely on the ROM’s equations of motion. The nonlinear normal modes (NNMs) of the ROMs are computed and tracked as modes are added to the basis set. The NNMs are expected to converge to the true NNMs of the full order system with a sufficient set of basis vectors. This comparison captures the effect of the nonlinearity through a range of amplitudes of the system, and is akin to comparing natural frequencies and mode shapes for a linear structure. In this research, the convergence metric is evaluated on a simply supported beam with a contacting nonlinearity modeled as a unilateral piecewise-linear function. Various time responses are compared to show that the NNMs provide a good measure of the accuracy of the ROM. The results suggest the feasibility of using NNMs as a convergence metric for reduced order modeling of systems with various types of nonlinearities.

Modeling and Measurement of a Bistable Beam in a Microelectromechanical System

Design and fabrication of microelectromechanical systems (MEMS) can be costly, time consuming, and necessitating accurate models for their behavior. Current theoretical models of bistable beams in MEMS devices are limited to numerical or small deformation models and current measurement techniques are unable to fully characterize these devices as they only determine thresholds or have resolutions that are too coarse to adequately explore the force-deflection relationship of bistable mechanisms. Two analytical models are developed: a stepped Euler-Bernoulli beam and a large deformation model. To validate these models, a new technique for measuring in-plane mechanical properties of MEMS devices is introduced that measures normal and lateral forces against a probe tip, while electrostatic actuation and a force-feedback loop maintain the desired tip position. This allows true displacement-controlled measurements along two axes and facilitates automated positioning. Measurements validate the large deformation model and show that Euler-Bernoulli beam theory is inadequate for modeling the mechanisms bistable behavior. A parameter study in edge width using the large deformation model accounts for discrepancies between predicted and measured forces. The models utility is further demonstrated by an optimization study that redesigns the mechanism to be less sensitive to the edge width variation introduced in the manufacturing process.

Experimental Determination of Frictional Interface Models

The focus of this paper is on continuing the experimental/modeling investigation of the Brake-Reus beam which was initiated a year ago as part of the NOMAD program at Sandia National Labs. The ultimate goal of the overall effort is to (1) determine the parameters of joint models, in particular the Iwan model in its modal form, from well delineated tests and (2) extend this approach to identify statistical distributions of the model parameters to account for joint uncertainty. The present effort focused on free response of the beam resulting from an impact test. The use of this data in conjunction with the Hilbert transform is shown to provide a straightforward framework for the identification of the joint model parameters at the contrary of the forced response data used earlier. The resulting frequency and damping vs. amplitude curves are particularly conducive to a Iwan-type modeling which is demonstrated. The curves also show the effect of the bolt torque on the joint behavior, i.e., increase in natural frequency, linear limit, and macroslip threshold. Macroslip is shown to have occurred in some of the tests and it is concluded from ensuing testing that this event changed the nature of the jointed beams. Specifically, the linear natural frequency (observed under very low level impact test) shifted permanently by 20 Hz and, in one case, the linear natural frequency was observed to decrease with increasing bolt torque level in opposition to other beams and physical expectations. An analysis of the joint surface strongly suggest that a significant plastic zone developed during the macroslip phase which induced the above unusual behaviors.

A Numerical Round Robin for the Prediction of the Dynamics of Jointed Structures

Motivated by the current demands in high-performance structural analysis, and by a desire to better model systems with localized nonlinearities, analysts have developed a number of different approaches for modelling and simulating the dynamics of a bolted-joint structure. However, the types of conditions that make one approach more effective than the others remains poorly understood due to the fact that these approaches are developed from fundamentally and phenomenologically different concepts. To better grasp their similarities and differences, this research presents a numerical round robin that assesses how well three different approaches predict and simulate a mechanical joint. These approaches are applied to analyze a system comprised of two linear beam structures with a bolted joint interface, and their strengths and shortcomings are assessed in order to determine the optimal conditions for their use.

Nonlinear Model Reduction of von Kármán Plates Under Quasi-Steady Fluid Flow

A reduced-order model for von Karman plates, using a method of quadratic components, is presented in this paper. The method of quadratic components postulates the full kinematics of the plate as consisting of a combination of linear and quadratic components. This reduced model is then discretized in space with a Galerkin approximation and in time with an implicit integration scheme. The plate is coupled with a fluid flowing over it at a supersonic speed using a quasi-steady pressure model commonly referred to as piston theory. A static loading example is used to validate the model reduction of the plate with respect to other numerical and approximate solutions. The limit cycles of the plate coupled with piston theory are calculated via a cyclic method, and the results from parameter studies are compared to classical results. A previously unexplored regime of the limit cycle amplitudes is investigated; a second, coexisting, limit cycle is found that yields a discontinuity in the limit cycle amplitudes at a critical dynamic pressure.

Modelling localized nonlinearities in continuous systems via the method of augmentation by non-smooth basis functions

Existing solutions for continuous systems with localized, non-smooth nonlinearities (such as impacts) focus on exact methods for satisfying the nonlinear constitutive equations. Exact methods often require that the non-smooth nonlinearities be expressed as piecewise-linear functions, which results in a series of mapping equations between each linear regime of the nonlinearities. This necessitates exact transition times between each linear regime of the nonlinearities, significantly increasing computational time, and limits the analysis to only considering a small number of nonlinearities. A new method is proposed in which the exact, nonlinear constitutive equations are satisfied by augmenting the systems primary basis functions with a set of non-smooth basis functions. Two consequences are that precise contact times are not needed, enabling greater computational efficiency than exact methods, and localized nonlinearities are not limited to piecewise-linear functions. Since each nonlinearity requires only a few non-smooth basis functions, this method is easily expanded to handle large numbers of nonlinearities throughout the domain. To illustrate the application of this method, a pinned–pinned beam example is presented. Results demonstrate that this method requires significantly fewer basis functions to achieve convergence, compared with linear and exact methods, and that this method is orders of magnitude faster than exact methods.

Experimental Assessment of the Influence of Interface Geometries on Structural Dynamic Response

Jointed interfaces are sources of the greatest amount of uncertainty in the dynamics of a structural assembly. In practice, jointed connections introduce nonlinearity into a system, which is often manifested as a softening response in frequency response, exhibiting amplitude dependent damping and stiffness. Additionally, standard joints are highly susceptible to unrepeatability and variability that make meaningful prediction of the performance of a system prohibitively difficult. This high degree of uncertainty in joint structure predictions is partly due to the physical design of the interface. This paper experimentally assesses the influence of the interface geometry on both the nonlinear and uncertain aspects of jointed connections. The considered structure is the Brake-Reus beam, which possesses a lap joint with three bolted connections, and can exhibit several different interface configurations. Five configurations with different contact areas are tested, identified, and compared, namely joints with complete contact in the interface, contact only under the pressure cones, contact under an area twice that of the pressure cones, contact only away from the pressure cones and Hertzian contact. The contact only under the pressure cone and Hertzian contact are found to behave linearly in the range of excitation used in this work. The contact area twice that of the pressure cone behaves between the complete contact and contact only under the pressure cone cases.

Nonlinear Model Reduction of von Karman Plates Under Linearized Compressible Fluid Flow.

Areduced ordermodel (ROM) of linearized compressiblefluidflowcoupledwith a vonKarmanplate is developed. SeparateROMs for both thefluid and structure are derived and are coupled through a solidwall boundary condition at the interface boundary. The structural ROM is formulated using the method of quadratic components, which postulates that the full kinematics of the plate can be represented using linear and quadratic terms. The fluidROM is constructed via a proper orthogonal decomposition method. Both ROMs are further reduced via a Galerkin discretization, and the coupled system is implicitly integrated in time. The coupled model is subsequently compared with previously validated models that used either a linearized plate model with the linearized compressible fluid model or a quasi-static fluidmodel with the vonKarman platemodel. The comparisons are conducted in high-Machnumber regimeswhere the comparison is admissible, and excellent agreement is found. Analysis of the resulting limit cycles shows two sets of discontinuities in the limit cycle amplitudes. These discontinuities have two salient features: a snap through phenomenon in phase space is observed in the higher modes, and the coalescing of several of the lower modes is observed below the critical dynamic pressures followed by a sudden separation above the critical dynamic pressures.