Joshua H. Gordis

A general theory for frequency domain structural synthesis

Abstract Fundamental issues in frequency domain structural synthesis are addressed. A new derivation of the operative synthesis equation which is based on a congruent transformation of the pre-synthesis frequency response matrix is developed. That this transformation is intrinsic to the theory is evident in the form of this operative equation found in virtually all prior references to the theory. The structure of the operative equation facilitates its straightforward incorporation into a finite element analysis environment. The new formulation provides an exact and orderly treatment of both “coupling”, the joining of previously uncoupled structures, and “modification”, the creation of redundant load paths in a structure, and makes plain their equivalence with respect to the derivation and application of the theory. The new formulation encompasses previously addressed capabilities of the theory, and provides new capabilities. A primary feature of the new formulation is the provision for general indirect synthesis, the inclusion of intermediate or interconnecting structural elements of any type. Lumped elements as well as distributed (finite) elements can be included, either between substructures in a coupling operation, or as redundant load paths in a single structure, in a modification. The application of graph theory to structural synthesis gives rise to Boolean and non-Boolean mapping matrices which organize the coupling force sign conventions. The new theory provides a thorough accounting for the role of graph theory in structural synthesis, and a fundamental understanding of the relationship between a given structural element and its mapping matrix is achieved.

Efficient transient analysis for large locally nonlinear structures

gardless of the size of the (linear) portion of the model, the motivation exists for the development of a solution method which isolates the nonlinearities, thereby preserving the linearity of the bulk of the model for an efficient linear solution prior to (repeated) nonlinear design analyses. It is also desirable that such a solution procedure not require all model degrees-of-freedom (DOF) to be retained in the analysis; the solution procedure should require only those DOF directly associated with the nonlinear portions of the model.

Structural Synthesis in the Frequency Domain: A General Formulation

A general formulation for frequency domain structural synthesis is presented. The formulation is based on a physical coordinate transformation of the presynthesis frequency response system model and addresses modification, general coupling, and constraint imposition. Notable features of the formulation includes the ability to directly synthesize response quantities such as displacements, stresses, and strains, and the accommodation of Boolean matrices that organize the connectivity in the synthesis of complex systems. The theory is shown to be a highly efficient and exact means of doing static and complex dynamic reanalysis.

Integral equation formulation for transient structural synthesis

An exact formulation for time domain structural synthesis is developed. Volterra integral equations are derived from the convolution integral which address substructure coupling and structural modification. The theory is cast in physical coordinates and, therefore, no transformation or mode truncation is required to achieve model reduction. As a minimum, only those coordinates directly involved in the synthesis need be retained, although synthesized transient response can be found for all coordinates, if so desired. The formulation makes use of transient response data and impulse response functions at the retained physical coordinates. Modified or coupled transient response is directly calculated ; no synthesized system model is assembled. The formulation exactly synthesizes system damping, regardless of the uncoupled system damping models used. The numerical solution of the integral equations involves the solution of a lower triangular linear system only ; no matrix factorization or eigensolution is required. Simple yet representative numerical examples are included.

Analysis of stress due to fastener tolerance in assembled components

The location of fasteners in a manufactured component are commonly specified with an allowed deviation from nominal location, known as tolerance. The assembly of such components can generate stress due to the accumulation of these tolerances. A highly efficient and exact method for the linear static and complex dynamic analysis of assembly stress is presented. An exact reduced representation of the assembled components is generated using frequency domain structural synthesis. An alternative coordinate system is employed in the synthesis that allows the direct application of fastener tolerances to the component assembly model and the rapid calculation of the resulting displacements, strains, and stresses. The method provides an efficient means of rapidly assessing the effects of proposed (maximum allowable) tolerance limits for each component, thereby aiding in the design process and minimizing manufacturing costs.

Dynamic Mechanical Properties of Spirally Wound Paper Tubes

The use of experimental modal analysis to obtain the dynamic mechanical properties of spirally wound paper tubes is investigated. Based on experimentally measured natural frequencies in the free-free mode of transverse vibration, tube flexural stiffness properties are predicted using three beam theories: Euler-Bernoulli beam theory, Timoshenko beam theory for isotropic materials, and Timoshenko beam theory for anisotropic materials.

Frequency domain structural synthesis applied to quasi-static crack growth modeling

Quasi-static crack growth in a composite beam was modeled using the structural synthesis technique along with a finite element model. The considered crack was an interface crack in the shear mode (i.e. mode II), which occurs frequently in the scarf joint of composite structures. The analysis model was a composite beam with an edge crack at the midplane of the beam subjected to a three-point bending load. In the finite element model, beam finite elements with translational degrees of freedom only were used to model the crack conveniently. Then, frequency domain structural synthesis (substructure coupling) was applied to reduce the computational time associated with a repeated finite element calculation with crack growth. The quasi-static interface crack growth in a composite beam was predicted using the developed computational technique, and its result was compared to experimental data. The computational and experimental results agree well. In addition, the substructure-based synthesis technique showed the significantly improved computational efficiency when compared to the conventional full analysis.

Optimal Selection of Artificial Boundary Conditions for Model Update and Damage Detection – Part 1: Theory

Sensitivity-based model error localization and damage detection is hindered by the relative differences in modal sensitivity magnitude among updating parameters. The method of artificial boundary conditions is shown to directly address this limitation, resulting in the increase of the number of updating parameters at which errors can be accurately localized. Using a single set of FRF data collected from a modal test, the artificial boundary conditions (ABC) method identifies experimentally the natural frequencies of a structure under test for a variety of different boundary conditions, without having to physically apply the boundary conditions, hence the term “artificial.” The parameter-specific optimal ABC sets applied to the finite element model will produce increased sensitivities in the updating parameter, yielding accurate error localization and damage detection solutions. A method is developed for identifying the parameter-specific optimal ABC sets for updating or damage detection, and is based on the QR decomposition with column pivoting. Updating solution residuals, such as magnitude error and false error location, are shown to be minimized when the updating parameter set is limited to those corresponding to the QR pivot columns. The existence of an optimal ABC set for a given updating parameter is shown to be dependent on the number of modes used, and hence the method developed provides a systematic determination of the minimum number of modes required for localization in a given updating parameter. These various concepts are demonstrated on a simple model with simulated test data.

Optimal Selection of Artificial Boundary Conditions for Model Update and Damage Detection – Part 2: Experiment

Sensitivity-based model error localization and damage detection is hindered by the relative differences in modal sensitivity magnitude among updating parameters. The method of artificial boundary conditions is shown to directly address this limitation, resulting in the increase of the number of updating parameters at which errors can be accurately localized. Using a single set of FRF data collected from a modal test, the artificial boundary conditions (ABC) method identifies experimentally the natural frequencies of a structure under test for a variety of different boundary conditions, without having to physically apply the boundary conditions, hence the term “artificial.” The parameter-specific optimal ABC sets applied to the finite element model will produce increased sensitivities in the updating parameter, yielding accurate error localization and damage detection solutions. A method is developed for identifying the parameter-specific optimal ABC sets for updating or damage detection, and is based on the QR decomposition with column pivoting. Frequency response data collected from a simple laboratory experiment is used, along with the corresponding finite element-generated data, to demonstrate the effectiveness of the ABC-QR method.