Publications

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Structural dynamic finite element models typically use multipoint constraints (MPC) to condense the degrees of freedom (DOF) near bolted joints down to a single node, which can then be joined to neighboring structures with linear springs or nonlinear elements. Scalability becomes an issue when multiple joints are present in a system, because each requires its own model to capture the nonlinear behavior. While this increases the computational cost, the larger problem is that the parameters of the joint models are not known, and so one must solve a nonlinear model updating problem with potentially hundreds of unknown variables to fit the model to measurements. Furthermore, traditional MPC approaches are limited in how the flexibility of the interface is treated (i.e. with rigid bar elements the interface has no flexibility). To resolve this shortcoming, this work presents an alternative approach where the contact interface is reduced to a set of modal DOF which retain the flexibility of the interface and are capable of modeling multiple joints simultaneously. Specifically, system-level characteristic constraint (S-CC) reduction is used to reduce the motion at the contact interface to a small number of shapes. To capture the hysteresis and energy dissipation that is present during microslip of joints, a hysteretic element is applied to a small number of the S-CC Shapes. This method is compared against a traditional MPC method (using rigid bar elements) on a two-dimensional finite element model of a cantilever beam with a single joint near the free end. For all methods, a four-parameter Iwan element is applied to the interface DOF to capture how the amplitude dependent modal frequency and damping change with vibration amplitude.

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Lacayo et al. (Mechanical Systems and Signal Processing, 118: 133–157, 2019) recently proposed a fast model updating approach for finite element models that include Iwan models to represent mechanical joints. The joints are defined by using RBE3 averaging constraints or RBAR rigid constraints to tie the contact surface nodes to a single node on each side, and these nodes are then connected with discrete Iwan elements to capture tangential frictional forces that contribute to the nonlinear behavior of the mechanical interfaces between bolted joints. Linear spring elements are used in the remaining directions to capture the joint stiffness. The finite element model is reduced using a Hurty/Craig-Bampton approach such that the physical interface nodes are preserved, and the Quasi-Static Modal Analysis approach is used to quickly predict the effective natural frequency and damping ratio as a function of vibration amplitude for each mode of interest. Model updating is then used to iteratively update the model such that it reproduces the correct natural frequency and damping at each amplitude level of interest. In this paper, Lacayo’s updating approach is applied to the S4 Beam (Singh et al., IMAC XXXVI, 2018) giving special attention to the size and type of the multi-point constraints used to connect the structures, and their effect on the linear and nonlinear modal characteristics.

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Transient simulations of linear viscoelastically damped structures require excessive computational resources to directly integrate the full-order finite element model with time-stepping algorithms. Traditional modal reduction techniques are not directly applicable to these systems since viscoelastic materials depend on time and frequency. A more appropriate reduction basis is obtained from the nonlinear, complex eigenvalue problem, whose eigenvectors capture the appropriate kinematics and enable frequency-based mode selection; unfortunately, the computational cost is prohibitive for computing these modes from large-scale engineering models. To address this shortcoming, this work proposes a novel two-tier reduction procedure to reduce the upfront cost of solving the complex, nonlinear eigenvalue problem. The first reduction step reduces the full-order model with real mode shapes linearized about various centering frequencies to capture the kinematics over a full range of viscoelastic material behavior (glassy, rubbery, and glass-transition zones). This tier-one reduction preserves time-temperature superposition and allows the equations to depend parametrically on operating temperature. The second-level reduction then solves the complex, nonlinear eigenmode solutions in the tier-one reduced space about a fixed temperature to further reduce the equations-of-motion. The method is demonstrated on a cantilevered sandwich plate to showcase its accuracy and efficiency in comparison to full-order model predictions.

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In computational structural dynamics problems, the ability to calibrate numerical models to physical test data often depends on determining the correct constraints within a structure with mechanical interfaces. These interfaces are defined as the locations within a built-up assembly where two or more disjointed structures are connected. In reality, the normal and tangential forces arising from friction and contact, respectively, are the only means of transferring loads between structures. In linear structural dynamics, a typical modeling approach is to linearize the interface using springs and dampers to connect the disjoint structures, then tune the coefficients to obtain sufficient accuracy between numerically predicted and experimentally measured results. This work explores the use of a numerical inverse method to predict the area of the contact patch located within a bolted interface by defining multi-point constraints. The presented model updating procedure assigns contact definitions (fully stuck, slipping, or no contact) in a finite element model of a jointed structure as a function of contact pressure computed from a nonlinear static analysis. The contact definitions are adjusted until the computed modes agree with experimental test data. The methodology is demonstrated on a C-shape beam system with two bolted interfaces, and the calibrated model predicts modal frequencies with <3% total error summed across the first six elastic modes.

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Contact in structures with mechanical interfaces has the ability to significantly influence the system dynamics, such that the energy dissipation and resonant frequencies vary as a function of the response amplitude. Finite element analysis is commonly used to study the physics of such problems, particularly when examining the local behavior at the interfaces. These high fidelity, nonlinear models are computationally expensive to run with time-stepping solvers due to their large mesh densities at the interface, and because of the high expense required to update the tangent operators. Hurty/Craig-Bampton substructuring and interface reduction techniques are commonly utilized to reduce computation time for jointed structures. In the past, these methods have only been applied to substructures rigidly attached to one another, resulting in a linear model. The present work explores the performance of a particular interface reduction technique (system-level characteristic constraint modes) on a nonlinear model with node-to-node contact for a benchmark structure consisting of two c-shape beams bolted together at each end.

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.

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