Dynamic Analysis of Encastre Beams by Modification of the System’s Stiffness Distribution

##plugins.themes.bootstrap3.article.main##

  •   V. O. Okonkwo

  •   C. H. Aginam

  •   C. M. O. Nwaiwu

Abstract

Numerical and energy methods are used to dynamically analyze beams and complex structures. Hamilton’s principle gives exact results but cannot be easily applied in frames and complex structures. Lagrange’s equations can easily be applied in complex structures by lumping the continuous masses at selected nodes. However, this would alter the mass distribution of the system, thus introducing errors in the results of the dynamic analysis. This error can be corrected by making a corresponding modification in the systems’ stiffness matrix. This was achieved by simulating a beam with uniformly distributed mass with the force equilibrium equations. The lumped mass structures were simulated with the equations of motion. The continuous systems were analyzed using the Hamilton’s principle and the vector of nodal forces {P} causing vibration obtained. The nodal forces and displacements were then substituted into the equations of motion to obtain the modified stiffness values as functions of a set of stiffness modification factors. When the stiffness distribution of the system was modified by means of these stiffness modification factors, it was possible to predict accurately the natural fundamental frequencies of the lumped mass encastre beam irrespective of the position or number of lumped masses.


Keywords: Lagrange’s equations; mass lumping; encastre beam; natural frequency; stiffness matrix, finite element analysis.

References

Choi W. S, Park G. J, “Transformation ofdynamic loads into equivalent static loads based on modal analysis”, International Journalfor Numerical Methods in Engineering. Vol 46, Iss. 1, pp 29–43. 1999 https://doi.org/10.1002/(SICI)1097-207(19990910)46:1<29:AID NME661>3.0.CO;2-D.

Choi W. S, Park K. B., Park G. J,“Calculation of Equivalent Static Loads and its Application”, Transactions, SMiRT 16, WashingtonDC. 2001. https://repository.lib.ncsu.edu/bitstream/handle/1840.20/30457/B1111.pdf?sequence=1&isAllowed=y.

Kim H., Kim E., Cho M., “Transformation of Dynamic Loads inEquivalent Static Load based on the stress constraint conditions”,Journal of the Computational Structural Engineering, Vol 26 (2013). DOI:10.7734/COSEIK.2013.26.2.165

Kim H., Kim E., Cho M., “Study on the Structural Optimization based on Equivalent Static Load under Dynamic Load”, Journal of theComputational Structural Engineering, Vol 26. (2014) pp 421-427.DOI:10.7734/COSEIK.2014.27.5.421

Park K. J., Lee J. N., Park G. J., “Structural Shape Optimization using equivalent static loads transformed from dynamic loads”, International Journal of Numerical Methods in Engineering, Vol. 63, Issue 4 pp 589-602, 2005, DOI:10.1002/nme.1295.

Rajasekaran, S., Structural Dynamics of EathquakeEngineering:Theory and Application using Mathematica and Matlab, Woodhead Publishing Limited Cambridge, 2009.

Ezeokpube, G. C.,“Dynamic Response of Frames with stiffened joints”, M.Eng Thesis University of Nigeria Nsukka, 2002.

De Basabe J. D., Sen M. K., “A comparison of finite difference and spectral-element methods for elastic wave propagation in media with a fluid-solid interface”, Geophysical Journal International, vol. 200, Issue, 1 pp 278 – 298, 2015. https://doi.org/10.1093/gji/ggu389.

Perumal P., “A review on Polygonal/Polyhedral finite element methods”, Mathematical Problems in Engineering, vol. 2018, pp 1 – 23, 2018. https://doi.org/10.1155/2018/5792372.

Beaurepaire, P and Schueller, G. I., “Modelling of the Variability of fatigue Crack growth using cohesive zone element”, Engineering Fracture Mechanics Vol 78 Issue 12 pp 2399-2413, 2011. Elsevier. DOI: 10.1016/j.engfracmech.2011.05.011.

Tornabene, F., Nicholas, Fanluzzi, Uberlini, F., Erasmo, V., “Strong Formulation Finite Element Method based on Differential Quadrature: A Survey”, Applied Mechanics Review Vol 67 pp 1-50, 2015, ASME. https://doi.org/10.1115/1.4028859.

Tauchert, T. R., Energy Principles in Structural Mechanics, International Student Edition, McGraw-Hill Kogakusha Ltd Tokyo, 1974.

Ahmad, Z. and Campbell, J., “Development of Two-dimensional Solver Code for Hybrid Model of Energy absorbing system”, International Journal of Physical Sciences, vol 8 (13), pp 510 – 525, 2013. DOI: 10.5897/IJPS12.584.

Ashithamol S., Yedu K. M., Nithin W., “Study on mass lumping methods of framed structures”, International Journal of Engineering Research and Technology, vol. 5, Issue 9, pp 253 – 257, 2016. http://dx.doi.org/10.17577/IJERTV5IS090271.

Lin HY., Tsai YC, “Free Vibration Analysis of uniform multi-span beam carrying multi spring-mass systems”, Journal of Sound and Vibration, Vol 302, Issue 3, pp 442 – 456, 2007. https://doi.org/10.1016/j.jsv.2006.06.080.

Zhai, W., Cai, Z., “Dynamic Interaction between a lumped mass vehicle and a discretely supported continuous rail track”, Computers and Structures Vol. 63, Issue 5. pp 987 – 997. 1997.https://doi.org/10.1016/S0045-7949(96)00401-4.

Reichi, K. K. and Iman, D. J., “Lumped mass model of a 1-dimensional metastructure for vibration suppression with no additional mass”, Journal of Sound and Vibration, Vol. 403. pp 75- 89. 2017.https://doi.org/10.1016/j.jsv.2017.05.026.

Torkian, B. B., Chandran, P., Ratnagaran B. J., Miller R. and Lu S., “Validation of lumped mass stick models for surface founded structures”, Transactions, SMiRT-22, San Francisco, California, USA.pp 1 – 11. 2013.

Iyer, R.,1993, Error Estimation and Adaptive Refinements for Finite ElementAnalysis of Structures, PhD Thesis, Indian Institute of Science, Bangalore.

Ma L., Liu J., Jia X., Yan Y., “Lumped mass matrix of three-node beam element”, Advanced Materials Research, vol. 616-618,pp 1969 – 1973. 2012.DOI: 10.4028/www.scientific.net/AMR.616-618.1969.

Cohen G., Fauqueux S., “Mixed finite elements with mass-lumping for the transient wave equation”, Journal of Computational Acoustics, vol 8(1), (2000), pp 171-188. DOI: 10.1016/S0218-396X(00)00011-X.

Iyer R., Pilani, G. S. and Rao, T. V., “Influence of Mass Representation Schemeson Vibration Characteristics of Structures”, Technical Journal: Aerospace Engineering, The Institute of Engineers Vol 84, pp 19 – 26, 2003.

Nandi, S.K. and Bosu, S., “Effect of Mass Matrix Formulation Schemes on Dynamics of Structures”,International ANSYS Conference Proceeding, www.ansys.com, paper 50, 2004.

Ferro R. M., Ferreira W. G, Calenzani A. F. G., “Dynamic Analysis of support frame structures of rotating machinery”, Global Journal of Research in Engineering, vol. 14, Issue 5, pp 27-31. 2014 https://globaljournals.org/GJRE_Volume14/4-Dynamic-Analysis-of-Support.pdf.

Sule, S., “Approximate Method for the determination of natural frequencies of a Multi-degree of freedom beam system”, Nigerian Journal of Technology, Vol 30, No 2, pp 1 – 6. 2011https://www.ajol.info/index.php/njt/article/view/123519.

Osadebe, N. N., “An Improved MDOF Model Simulating some Systems with distributed mass”, Journal of University of Science and Technology Kumasi, Vol. 19, pp. 56-61, 1999.

Ross R. G., “Synthesis of stiffness and mass matrices from experimental vibration modes”,SAE Transactions, vol. 80, section 4, (1971) pp 2627 – 2635. https://doi.org/10.4271/710787.

Cha P. D., De Pillis L. G., “Model Updating by adding known masses, International Journal for Numerical Methods in Engineering, vol. 50, pp 2547 – 2571. 2001https://doi.org/10.1002/nme.136.

Torabi K, Afshari H, Zafari E., “Transverse Vibration of Non-uniform.

Euler-Bernoulli Beam using Differential Transform Method (DTM)”, Applied Mechanics and Materials, vols 110 – 116 (2012) pp 2400 –2405, DOI:10.4028/www.scientific.net/AMM.110-116.2400.

Thomson, W. T. and Dahleh M. D., Theory of Vibrations with Applications, 5th Edition, Prentice Hall New Jersey, 1998.

Gao, W.,”Natural frequency and mode shape analyses of structures with uncertainty”, Mechanical Systems and Signal Processing, vol. 21, issue 1, pp 24 – 39, 2007, Elsevier. DOI: 10.1016/j.ymssp.2006.05.007.

Kumar A., Jaiswal H., Jain R., Patil P. P., “Free vibration and material mechanical properties influence based frequency and mode shape analysis of transmission gearbox casing”, 12th Global Congress on Manufacturing and Management, Procedia Engineering 97,(2014) pp 1097 – 1106. https://doi.org/10.1016/j.proeng.2014.12.388.

Hibbeler R. C., Structural Analysis, 8th Edition, Person, 2012.

Downloads

Download data is not yet available.

##plugins.themes.bootstrap3.article.details##

How to Cite
[1]
Okonkwo, V.O., Aginam, C.H. and Nwaiwu, C.M.O. 2020. Dynamic Analysis of Encastre Beams by Modification of the System’s Stiffness Distribution. European Journal of Engineering Research and Science. 5, 11 (Nov. 2020), 1334-1342. DOI:https://doi.org/10.24018/ejers.2020.5.11.2220.