Mass and Stiffness Matrices and Frequencies of Simple Beam Elements Based on Real Shape Functions

الموضوعات :

Pedram Abouzari ¹ (Department of Civil Engineering, Faculty of Civil & Earth Resources Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran)
Karen Khanlari ² (Department of Civil Engineering, Faculty of Civil & Earth Resources Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran)
Reza Esmaeilabadi ³ (Department of Civil Engineering, Roudehen Branch, Islamic Azad University, Roudehen, Iran)

تاريخ الإرسال : 22 الثلاثاء , جمادى الثانية, 1443 تاريخ التأكيد : 14 الجمعة , رمضان, 1443 تاريخ الإصدار : 02 الأربعاء , ذو القعدة, 1443

الکلمات المفتاحية: Beam Element, Real and classic shape functions, high natural frequency, mass and stiffness matrices,

ملخص المقالة :

In this research, we investigate and compare the natural frequencies of simple beams and their mass and stiffness matrices of the two methods: classic shape functions and real shape functions. To this end, we solve the beam motion Equation and apply boundary conditions. This article shows that the coefficients of the real shape functions, and consequently, the real shape functions, become harmonic and hyperbolic and also, they are dependent on the natural frequency value of the element. As a result, the real mass and the real stiffness matrix of each element are also dependent on the element frequency. The frequency values obtained from these two methods are compared with the exact frequency values of two simple beam types with different support conditions. In this way, we determine which method leads to more accurate and acceptable frequencies for these beams. Based on the obtained results, the percentage of frequency error obtained by the classical method is relatively high in the sample beams. Hence, the natural frequency value of the beams studied using exact shape functions shows a small error compared to the classical method in terms of the exact frequency value of these beams. It is of note that the frequency error obtained from the classical method is greater in the elements with a higher natural frequency. Overall, obtaining the exact natural frequency of an element will result in accurate dynamic responses and more appropriate analyses and designs.

المصادر:

• Rahbar, A., Abdollah, S., Free Vibrations of Reinforced Sheets of Sheep Structures, 17th Marine Industry Conference, Kish Island, Iranian Marine Engineering Association, 2015.


• Azimi-Zawareie, S., Exact Dynamic Analysis of Truss Structures, Iranian Journal of International Institute of Seismology and Earthquake Engineering, No. 1, 1994.


• Rana, H. K., Dynamic Analysis of Fixed-Fixed Beams, 2012.


• Satpathy, S. M., Dash, P., Dynamic Analysis of Cantilever Beam and Its Experimental Validation, 2014.


• Sawant, S., Experimental Verification of Transverse Vibration of Free-Free Beam, International Journal of Advanced Research in Electrical Electronics and iInstrumentation Engineering, 2013.


• Tatar, İ., Vibration Characteristics of Portal Frames, İzmir Institute of Technology, 2013.


• AL, G., Kumawat, S. C., World Research Journal of Civil Engineering, 2011.


• Chao, X., Dong, W., Experimental Investigation of Nonlinear Interface Effects in a Jointed Beam, Vibroengineering Procedia, Vol. 4, 2014, pp. 107-112.


• Delhez, E., Experimental and Numerical Modal Analyses of a Pre-Stressed Steel Strip, ULiège-Université de Liège, 2017.


• Esfandiari, A., Rahai, A., Sanayei, M., and Bakhtiari-Nejad, F., Model Updating of a Concrete Beam With Extensive Distributed Damage Using Experimental Frequency Response Function, Journal of Bridge Engineering, Vol. 21, No. 4, 2016, pp. 04015081.


• Gandomkar, F., Wan Badaruzzaman, W., Osman, S., and Ismail, A., Experimental and Numerical Investigation of The Natural Frequencies of The Composite Profiled Steel Sheet Dry Board (PSSDB) System, Journal of the South African Institution of Civil Engineering= Joernaal van die Suid-Afrikaanse Instituut van Siviele Ingenieurswese, Vol. 55, No. 1, 2013, pp. 11-21.


• Joubaneh, E. F., Barry, O. R., and Tanbour, H. E., Analytical and Experimental Vibration of Sandwich Beams Having Various Boundary Conditions, Shock and Vibration, Vol. 2018, 2018.


• Krishnaraju, A., Ramkumar, R., and Lenin, V., Experimental Investigation of Natural Frequency of a Cantilever Composite Beam using Vibration Absorber, 2016.


• Trišović, N., Eigenvalue Sensitivity Analysis in Structural Dynamics, FME Transactions, Vol. 35, No. 3, 2007, pp. 149-156.


• Yang, C., Zhang, Z., Nong, S., and Zhu, C., Analytical and Experimental Investigation on Eigenfrequency-Based Damage Diagnosis of Cantilever Beam, Journal of Vibroengineering, Vol. 18, No. 8, 2016, pp. 5114-5126.


• Chen, J., Bao, H., Wang, T., Desbrun, M., and Huang, J., Numerical Coarsening using Discontinuous Shape Functions, ACM Transactions on Graphics (TOG), Vol. 37, No. 4, 2018, pp. 1-12.


• Mousavi, Z., Ettefagh, M. M., Sadeghi, M. H., and Razavi, S. N., Developing Deep Neural Network for Damage Detection of Beam-Like Structures Using Dynamic Response Based on FE Model and Real Healthy State, Applied Acoustics, Vol. 168, 2020, pp. 107402.


• Adhikari, S., Karličić, D., and Liu, X., Dynamic Stiffness of Nonlocal Damped Nano-Beams on Elastic Foundation, European Journal of Mechanics-A/Solids, Vol. 86, 2021, pp. 104144.


• Corrêa, R. M., Arndt, M., and Machado, R. D., Free In-Plane Vibration Analysis of Curved Beams by The Generalized/Extended Finite Element Method, European Journal of Mechanics-A/Solids, Vol. 88, 2021, pp. 104244.


• Liu, X., Chang, L., Banerjee, J. R., and Dan, H. C., Closed-Form Dynamic Stiffness Formulation for Exact Modal Analysis of Tapered and Functionally Graded Beams and Their Assemblies, International Journal of Mechanical Sciences, Vol. 214, 2022, pp. 106887.


• Banerjee, J. R., Ananthapuvirajah, A., An Exact Dynamic Stiffness Matrix for a Beam Incorporating Rayleigh-Love and Timoshenko Theories, International Journal of Mechanical Sciences, Vol. 150, 2019, pp. 337-347.


• Talukdar, S., Vibration of Continuous Systems, Profesor, Department of Civil Engineering. Indian Institute of Technology Guwahati-781039, 2016.