A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods

El-Sayed, T.A.; Farghaly, S.H.

doi:10.1155/2017/8186976

Cited by 11 publications

(8 citation statements)

References 40 publications

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“…Anyway, this method's application is limited to multi-span [11] or multi-stepped beams [12] and can, therefore, not be applied on plane frame structures. Also, this method has been extended to crack identification of stepped beams with multiple edge cracks and different boundary conditions [13].…”

Section: Introductionmentioning

confidence: 99%

The Numerical Assembly Technique for arbitrary planar systems based on an alternative homogeneous solution

Krämer¹,

Gfrerer²

2022

Preprint

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The Numerical Assembly Technique is extended to investigate arbitrary planar frame structures with the focus on the computation of natural frequencies. This allows us to obtain highly accurate results without resorting to spatial discretization. To this end, we systematically introduce a frame structure as a set of nodes, beams, bearings, springs, and external loads and formulate the corresponding boundary and interface conditions. As the underlying homogeneous solution of the governing equations, we use a novel approach recently presented in the literature. This greatly improves the numerical stability and allows the stable computation of very high natural frequencies accurately. We show this numerically at two frame structures by investigation of the condition number of the system matrix and also by the use of variable precision arithmetic.

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Section: Introductionmentioning

confidence: 99%

The Numerical Assembly Technique for arbitrary planar systems based on an alternative homogeneous solution

Krämer¹,

Gfrerer²

2022

Preprint

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“…Then these equations are assembled. There are several methods to assemble the segments equations but the most common methods are numerical assembly technique (NAT) [30,31], transfer matrix method (TMM) [32,33] and DSM [34][35][36]. In the current manuscript, DSM method is adopted.…”

Section: Introductionmentioning

confidence: 99%

Longitudinal Vibration Analysis of a Stepped Nonlocal Rod Embedded in Several Elastic Media

Taima

El-Sayed

Farghaly

2022

J. Vib. Eng. Technol.

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Purpose Mechanical properties of 1D nanostructures are of great importance in nanoelectromechanical systems (NEMS) applications. The free vibration analysis is a non-destructive technique for evaluating Young's modulus of nanorods and for detecting defects in nanorods. Therefore, this paper aims to study the longitudinal free vibration of a stepped nanorod embedded in several elastic media. Methods The analysis is based on Eringen’s nonlocal theory of elasticity. The governing equation is obtained using Hamilton’s principle and then transformed into the nonlocal analysis. The dynamic stiffness matrix (DSM) method is used to assemble the rod segments equations. The case of a two-segment nanorod embedded in two elastic media is then deeply investigated. Results The effect of changing the elastic media stiffness, the segments stiffness ratio, boundary conditions and the nonlocal parameter are examined. The nano-rod spectrum and dispersion relations are also investigated. Conclusion The results show that increasing the elastic media stiffness and the segment stiffness ratio increases the natural frequencies. Furthermore, increasing the nonlocal parameter reduces natural frequencies slightly at lower modes and significantly at higher modes.

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“…Wu and Chen (2000) developed an analytical and numerical–combined method (ANCM) to determine the natural frequencies of a uniform clamped–free beam with several inspan 1-DOF spring–mass–damper subsystems. For more complicated problems such as multi-span beams and beams that carry higher-order discrete subsystems, the frequency equation is obtained by solving the boundary conditions for each span and then applying one of the assembly methods such as the numerical assembly technique (El-Sayed and Farghaly, 2016a), dynamic stiffness matrix (Elsawaf et al, 2020), or transfer matrix method (TMM) (El-Sayed and Farghaly, 2017). El-Sayed and Farghaly (2016b) studied the free vibration of a single-span and multi-span Timoshenko beams, each mounted on a lumped parameter subsystem located at its ends and represented by a two-degree-of-freedom (2-DOF) spring–mass system.…”

Section: Introductionmentioning

confidence: 99%

A new numeric–symbolic procedure for variational iteration method with application to the free vibration of generalized multi-span Timoshenko beam

El-Sayed

El-Mongy

2021

Journal of Vibration and Control

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In this article, a novel approach is introduced for the free vibration analysis of beams based upon the variational iteration method. The new approach uses a numeric–symbolic procedure that tackles the problem of increased execution time involved in symbolic integrations. This drawback is usually encountered in solving complicated free vibration problems such as stepped beams connected to lumped parameter subsystems. The proposed procedure is applied for free vibration analysis of a generalized multi-span Timoshenko beam connected to multiple lumped subsystems. Each subsystem is represented by a two-degree-of-freedom spring–mass–damper system. Several verification examples are presented where the results of the proposed numeric–symbolic variational iteration method are compared with the conventional symbolic approach symbolic variational iteration method in terms of execution time. Special attention is given to the verification of the new results against finite element modeling results and exact solutions where possible. Based on the presented results, it is shown that the new numeric–symbolic variational iteration method procedure efficiently reduces the time required for solving the free vibration problem while maintaining the high accuracy and robustness of the variational iteration method. The new procedure presented here may facilitate solving some engineering problems in which the conventional symbolic approach usually fails to solve owing to extensive memory requirements. The study contributes toward further improvements of the variational iteration method and its application to sophisticated dynamic systems.

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A Normalized Transfer Matrix Method for the Free Vibration of Stepped Beams: Comparison with Experimental and FE(3D) Methods

Cited by 11 publications

References 40 publications

The Numerical Assembly Technique for arbitrary planar systems based on an alternative homogeneous solution

The Numerical Assembly Technique for arbitrary planar systems based on an alternative homogeneous solution

Longitudinal Vibration Analysis of a Stepped Nonlocal Rod Embedded in Several Elastic Media

A new numeric–symbolic procedure for variational iteration method with application to the free vibration of generalized multi-span Timoshenko beam

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