Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes

Abstract

This paper extends the Wittrick-Williams (W-W) algorithm for hybrid dynamic stiffness (DS) models connecting any combinations of line and point nodes. The principal novelties lie in the development of both the DS formulation and the solution technique in a sufficiently systematic and general manner. The parent structure is considered to be in the form of two dimensional DS elements with line nodes, which can be connected to rigid/spring point supports/connections, rod/beam point supports/connections, and point connections to substructures. This is achieved by proposing a direct constrain method in a strong form which makes the modeling process straightforward. For the solution technique, the W-W algorithm is extended for all of the above hybrid DS models. No matrix inversion is needed in the proposed extension, making the algorithm numerically stable, especially for complex built-up structures. A mathematical proof is provided for the extended W-W algorithm. The proposed DS formulation and the extended W-W algorithm are validated by the FE results computed by ANSYS. This work significantly extends the application scope of the DS formulation and the W-W algorithm in a methodical and reliable manner, providing a powerful eigenvalue analysis tool for beam-plate built-up structures.

Applications of randomly selected sets of exact Voight's solutions for vibration of thin plates

“…In spite of very long history of investigation the vibration of plates is still a topic of primary interest important for various engineering application, for example, in solar plates, circuit plates, robots, aircraft, etc. Sometimes these plates are constrained by others structural elements, which should be considered and modeled withing one comprehensive calculational scheme [1]. Such necessity restricts the efficiency of very popular numerical techniques (finite difference, finite element methods) [1], and promote the development of analytical and semianalytical methods.…”

“…Sometimes these plates are constrained by others structural elements, which should be considered and modeled withing one comprehensive calculational scheme [1]. Such necessity restricts the efficiency of very popular numerical techniques (finite difference, finite element methods) [1], and promote the development of analytical and semianalytical methods. Except for the huge practical significance, this topic is the most efficient educational example for students for the study of partial differential equation eigenvalue problems [2].…”