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

Xiang Liu,Suchao Xie,Shaoqi Qiu,Jnan Ranjan Banerjee

doi:10.3390/math10010057

Xiang Liu, Suchao Xie + Show 2 more

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https://doi.org/10.3390/math10010057

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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.

Highlights

Built-up structures in engineering are often composed of plates, beams and other elements through a variety of imposed constraints or connections [1,2,3]
The new dynamic stiffness (DS) formulations and the corresponding enhanced Wittrick-Williams algorithm have been implemented in a Matlab code
Levy-type plate is a = b = 1 m, the boundary condition for the typical FSFS is shown in Figure 10, Young’s modulus is taken as E pl = 7.2 × 1010 Pa, Poisson ratio νpl = 0.3 and the mesh size of plate finite element method (FEM) elements is chosen to be 5 mm

Summary

Introduction

Built-up structures in engineering are often composed of plates, beams and other elements through a variety of imposed constraints or connections [1,2,3]. Vibration analysis is an essential requirement for the design of these built-up structures. Supports/connections can significantly affect the dynamic behavior of built-up structures, such as natural frequency, transmission path, and vibro-acoustic characteristics. If these constraints or connections are not properly considered in the design, inaccurate dynamic analysis may result in undesirable phenomena such as resonance or structural fatigue. The supports/connections of beam-plate combination of built-up structures can be divided into continuous (linear) supports/connections and discrete (point) supports/connections

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