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19 pages, 1249 KiB

Open AccessArticle

Dynamic Stiffness for a Levinson Beam Embedded Within a Pasternak Medium Subjected to Axial Load at Both Ends

by Zhijiang Chen, Qian Cheng, Xiaoqing Jin and Feodor M. Borodich

Buildings 2024, 14(12), 4008; https://doi.org/10.3390/buildings14124008 - 17 Dec 2024

Viewed by 1094

This work presents accurate values for the dynamic stiffness matrix coefficients of Levinson beams under axial loading embedded in a Winkler–Pasternak elastic foundation. Levinson’s theory accounts for greater shear deformation than the Euler–Bernoulli or Timoshenko theories. Using the dynamic stiffness approach, an explicit algebraic expression is derived from the homogeneous solution of the governing equations. The dynamic stiffness matrix links forces and displacements at the beam’s ends. The Wittrick–Williams algorithm solves the eigenvalue problem for the free vibration and buckling of uniform cross-section parts. Numerical results are validated against published data, and reliability is confirmed through consistency tests. Parametric studies explore the effects of aspect ratio, boundary conditions, elastic medium parameters, and axial force on beam vibration properties. The relative deviation for the fundamental frequency is almost

6.89 %

for a cantilever beam embedded in the Pasternak foundation,

5.16 %

for a fully clamped beam, and

4.79 %

for a clamped–hinged beam. Therefore, Levinson beam theory can be used for calculations relevant to loads with short durations that generate transient responses, such as impulsive loads from high-speed railways, using the mode superposition method. Full article

(This article belongs to the Special Issue Advanced Numerical and Computer Methods in Civil Engineering—2nd Edition)

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21 pages, 4027 KiB

Open AccessArticle

Closed-Form Exact Solution for Free Vibration Analysis of Symmetric Functionally Graded Beams

by Lorenzo Ledda, Annalisa Greco, Ilaria Fiore and Ivo Caliò

Symmetry 2024, 16(9), 1206; https://doi.org/10.3390/sym16091206 - 13 Sep 2024

Cited by 1 | Viewed by 1697

Abstract

The dynamic stiffness method is developed to analyze the natural vibration characteristics of functionally graded beams, where material properties change continuously across the beam thickness following a symmetric law distribution. The governing equations of motion and associated natural boundary conditions for free vibration analysis are derived using Hamilton’s principle and closed-form exact solutions are obtained for both Euler–Bernoulli and Timoshenko beam models. The dynamic stiffness matrix, which governs the relationship between force and displacements at the beam ends, is determined. Using the Wittrick–Williams algorithm, the dynamic stiffness matrix is employed to compute natural frequencies and mode shapes. The proposed procedure is validated by comparing the obtained frequencies with those given by approximated well-known formulas. Finally, a parametric investigation is conducted by varying the geometry of the structure and the characteristic mechanical parameters of the functionally graded material. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Partial Differential Equations)

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18 pages, 2680 KiB

Open AccessArticle

Research on the Dynamic Characteristics of the Double Slings System with Elastic Connection Considering Boundary Conditions

by Xia Liao, Danhui Dan, Fei Han and Rui Zhao

Mathematics 2022, 10(17), 3129; https://doi.org/10.3390/math10173129 - 1 Sep 2022

Cited by 1 | Viewed by 1770

Abstract

As the length of single sling increases, double slings with transverse connections are gradually becoming one of the effective measures to control the undesirable vibration of single slings. In the analysis of the dynamic characteristics of the double sling system, this paper firstly deduces the dynamic stiffness matrix of the elastically connected double sling system by the dynamic stiffness method (DSM), solves the frequency equation evolved from the dynamic stiffness matrix by using the Wittrick-Williams (W-W) algorithm, and obtains the systematic analysis and calculation of the dynamic characteristics of the double sling system under arbitrary boundary conditions. Secondly, a complete and accurate analysis method of the dynamic characteristics of the double sling system is obtained by comprehensively considering the bending stiffness and boundary conditions of the sling, and the accuracy of the calculation can be verified by the actual measurement data. Finally, the best installation position and quantity of transverse sling clamps in the double sling system are obtained by the parametric analysis of transverse sling clamps. The analysis of this paper will provide a theoretical basis for the design and optimization of slings, and further promote the wide application of the double sling system. Full article

(This article belongs to the Special Issue Modeling and Analysis in Dynamical Systems and Bistability)

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21 pages, 1713 KiB

Open AccessArticle

Development of the Dynamic Stiffness Method for the Out-of-Plane Natural Vibration of an Orthotropic Plate

by Manish Chauhan, Pawan Mishra, Sarvagya Dwivedi, Minvydas Ragulskis, Rafał Burdzik and Vinayak Ranjan

Appl. Sci. 2022, 12(11), 5733; https://doi.org/10.3390/app12115733 - 5 Jun 2022

Cited by 4 | Viewed by 2466

Abstract

In this present paper, the dynamic stiffness method (DSM) was formulated to investigate the out-of-plane natural vibration of a thin orthotropic plate using the classical plate theory (CPT). Hamilton’s principle was implemented to derive the governing differential equation of motion for free vibration of the orthotropic plate for Levy-type boundary conditions. The Wittrick–Williams (W–W) algorithm was used as a solution technique to compute the natural frequencies of a thin orthotropic plate for different boundary conditions, aspect ratios, thickness ratios, and modulus ratios. The obtained results are compared with the results by the finite element method using commercial software (ANSYS and those available) in the published literature. The presented results by the dynamic stiffness method can be used as a benchmark solution to compare the natural frequencies of orthotropic plates. Full article

(This article belongs to the Special Issue Advances in Vibroacoustics and Aeroacustics of Marine, Aerospace and Automotive Systems)

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16 pages, 2991 KiB

Open AccessArticle

Dynamic Stiffness Matrix Approach to Free Vibration Analysis of Functionally Graded Rotor Bearing System Subjected to Thermal Gradients

by Bharath Obalareddy, Prabhakar Sathujoda and Roberto Citarella

Materials 2022, 15(4), 1540; https://doi.org/10.3390/ma15041540 - 18 Feb 2022

Cited by 7 | Viewed by 2705

Abstract

The dynamic stiffness matrix (DSM) method, an analytical method that provides exact solutions, has been used for the first time for the free vibration analysis of a functionally graded (FG) rotor bearing system subjected to temperature gradients and to investigate its application to FG rotors. The material gradation occurs based on the power law between the inner metal core and the outer ceramic rich layer of the FG rotor. The temperature gradation follows the Fourier law of heat conduction which leads to non-linear temperature distribution (NLTD) in the radial direction of the FG rotor. The development of the DSM formulations for Timoshenko FG rotor elements using the governing equations derived from translational and rotational equilibrium conditions is the novelty of the present work. The DSM of the FG rotor elements, rigid disk and linear isotropic bearings are assembled to obtain the global dynamic stiffness matrix of the FG rotor bearing system. The natural whirl frequencies are computed from the global DSM using the Wittrick–William algorithm as a root searching technique. The natural and whirl frequencies are validated with the results available in the literature and the exactness of the DSM method has been exemplified. Full article

(This article belongs to the Special Issue On the Residual Strength and Damage Identification of Damaged Composite Structure)

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23 pages, 2068 KiB

Open AccessArticle

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

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

Mathematics 2022, 10(1), 57; https://doi.org/10.3390/math10010057 - 24 Dec 2021

Cited by 9 | Viewed by 4202

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. Full article

(This article belongs to the Special Issue Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences)

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