• Victor Okonkwo Nnamdi Azikiwe University
  • Chukwurah Aginam Nnamdi Azikiwe University Awka
  • Charles Nwaiwu



Lagrage equations, Stiffness matrix, Inertia matrix, Lumped mass


Continuous systems are sometimes analysed as lumped masses connected by massless elements. This reduces the structure’s degree of freedom and therefore simplifies the analysis. However this over simplification introduces an error in the analysis and the results are therefore approximate. In this work sections of the vibrating beam were isolated and the equations of the forces causing vibration obtained using the Hamilton’s principle. These forces were applied to the nodes of an equivalent lumped mass beam and the stiffness modification needed for it to behave as a continuous beam obtained. The beam’s stiffness was modified using a set of stiffness modification factors to . It was observed that by applying these factors in the dynamic analysis of the beam using the Lagrange’s equation, we obtain the exact values of the fundamental frequency irrespective of the way the mass of the beam was lumped. From this work we observed that in order to obtain an accurate dynamic response from a lumped mass beam there is need to modify the stiffness composition of the system and no linear modification of the stiffness distribution of lumped mass beams can cause them to be dynamically equivalent to the continuous beams. This is so because the values of the modification factors obtained for each beam segment were not equal. The stiffness modification factors were obtained for elements at different sections of the beam


A. A. Shabana, Vibration of Continuous Systems. In:Theory of Vibration, Mechanical Engineering Series, Springer, New York, NY pp 175-251, 1991

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

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

J. Humar, Dynamics of structures, 3rd Edition, CRC Press, London, 2012

M. Simsek, “Vibration analysis of a functionally graded beam under a moving mass by using different beam theories”, Composite Structures vol. 92, no. 4, pp 904-917, Elsevier, 2010,

Chung J., Yoo H. H., “Dynamic Analysis of a rotating cantilever beam by using the finite element method”, Journal of Sound and Vibration,vol. 249, Issue 1, pp 147-164, Elsevier, 2002,

L. Li, Y. Hu, X.Li, “Longitudinal vibration of size dependent rods via non-local strain gradient theory”, International Journal of Mechanical Science, vols. 115-116, pp 135-144, Elsevier, 2016,

Y. Saad and H. A. V. Vorst, “Iterative Solution of Linear Systems in the 20th Century”, Journal of Computational and Applied Mathematics Vol 123, Issue 1-2 pp 1-33, 2000,

N. J. Fergusson, W. D. Pilkey, “Frequency-dependent element matrices”, Journal of Applied Matrices, vol.59. issue 1, pp 1-10, 1992,

Downs B., “Vibration Analysis of Continuous systems by Dynamic discretization”,Journal of Mechanical Design, vol. 102, Issue 2. 1980.

A. Houmat, Nonlinear free vibration of a shear deformable laminated composite annular elliptical plate, ActaMechanica. Springer 208:281.

P. Beaurepaire and G. I. Schueller, “Modelling of the Variability of fatigue Crack growth using cohesive zone element”, Engineering Fracture Mechanics vol. 78 Issue 12 pp 2399-2413 Elsevier. 2011.

F. Tornabene, F. Nicholas, F. Uberlini, V. Erasmo, “Strong Formulation Finite Element Method based on Differential Quadrature: A Survey”, Applied Mechanics Review vol. 67 pp 1-50 ASME 2015.

G. Cocchetti, M. Pagani, U. Perego, “Selective mass scaling for distorted solid –shell elements in explicit dynamics:Optimal scaling factor and stable time step estimate”, International Journal for Numerical Methods in Engineering, vol. 101, no. 9, pp 1-30 2013.

X. Hua, T. C. Lin, T. Peng, W. E. Wali, “Dynamic Analysis of spiral level geared rotor systems applying finite elements and enhanced lumped parameters”, International Journal of Automotive Technology, vol.13. no. 1, pp 97-107, 2012. DOI: 10.1007/s12239-012-0009-4

G. Kouroussis, G. Gazetas, I. Anastasopoulos, C. Conti, O. Verlinden, “Lumped mass model of vertical dynamic coupling of a railway track on elastic homogenous or layered halfspace”, Proceedings of the 8th International Conference on Structural Dynamics EURODYN Belgium, pp 676-683, 2011

M. Blundell, D. Harty, The multibody systems approach to vehicle dynamics, Butterworth-Heinemann Uk, pp 185-334, 2015

G. Kouroussis, O. Verlinden, “Prediction of railway ground vibrations: Accuracy of a coupling lumped mass model for representing the track/soil interaction”, Soil Dynamics and Earthquake Engineering, vol. 69, pp. 220-226, Elsevier, 2015.

K. K. Reichl, D. J. Inman, Lumped mass model of a 1D metastructure for vibration suppression with no additional mass, Journal of Sound and Vibration, vol. 403, pp. 75-89, Elsevier, 2017.

H. Ahmadian, M. I. Friswell, J. E. Mottershead, “Minimization of the discretization error in mass and stiffness formulations by an inverse method”. International Journal for Numerical Methods in Engineering, vol. 41, no. 2, pp 371-387, 1998. 0207(19980130)41:2<371::AID-NME288>3.0.CO;2-

C. Hager, B. I. Wohlmuth, “Analysis of a space-time discretization for dynamic elasticity problems based on mass-free surface elements”, SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp 1863-1885, 2009.

D. Wang, W. Liu, H. Zhang, “Superconvergent Isogeometric free vibration analysis of Euler-Bernoulli Beams and Kirchhoff plates with higher order mass matrices”, Computer Methods in Applied Mechanics and Engineering, vol. 286, pp. 230-267, 2015.

T. M. Ericson, R. G. Parker, “Planetary gear modal vibration experiments and correlation against lumped-parameter and finite element models”, Journal of Sound and Vibration, vol. 332, no.9, pp 2350- 2375 Elservier, 2013.

I. O. Onyeyili, C. H. Aginam, V. O. Okonkwo, “Analysis of a propped cantilever under longitudinal vibration by a modification of the system’s stiffness distribution”, IOSR Journal of Mechanical and Civil Engineering, vol. 13, Iss. 5, pp 65-78, 2016

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

J. Buskiewicz, “A Dynamic Analysis of a coupled beam/slider system”, Applied Mathematical Modelling vol. 32, Issue 10, pp 1941 – 1955, 2008.

V. O. Okonkwo, Dynamic Analysis of frames by the modification of the system’s stiffness distribution, Unpublished PhD Thesis, Civil Eng. NnamdiAzikwe University, Awka

V. O. Okonkwo, Analysis of Multi-storey steel frames, Unpublished MEng Thesis, Civil Eng. Nnamdi Azikwe University, Awka

R. L. Bisplinghoff, H. Ashley, R. L. Halman, Aeroelasticity, Cambridge Mass Addison-Wesley Publishing Company Inc. pp 90, 1955




How to Cite

Okonkwo, V., Aginam, C., & Nwaiwu, C. (2021). IMPROVED ANALYSIS OF A PROPPED CANTILEVER UNDER LATERAL VIBRATION . Stavební Obzor - Civil Engineering Journal, 30(4).