ENG  Vol.13 No.8 , August 2021
Hermite Finite Element Method for a Class of Viscoelastic Beam Vibration Problem
Abstract: Beam equation can describe the deformation of beams and reflect various bending problems and it has been widely used in large engineering projects, bridge construction, aerospace and other fields. It has important engineering practice value and scientific significance for the design of numerical schemes. In this paper, a scheme for vibration equation of viscoelastic beam is developed by using the Hermite finite element. Based on an elliptic projection, the errors of semi-discrete scheme and fully discrete scheme are analyzed respectively, and the optimal L2-norm error estimates are obtained. Finally, a numerical example is given to verify the theoretical predictions and the validity of the scheme.
Cite this paper: Tang, Y. and Yin, Z. (2021) Hermite Finite Element Method for a Class of Viscoelastic Beam Vibration Problem. Engineering, 13, 463-471. doi: 10.4236/eng.2021.138033.

[1]   Putrevu, H., Subramanian, H. and Mulay, S.S. (2021) On the Viscoelastic Dynamic Beam Modelling. International Journal of Advances in Engineering Sciences and Applied Mathematics, 13, 18-32.

[2]   Pierro, E. (2020) Damping Control in Viscoelastic Beam Dynamics. Journal of Vibration and Control, 0, 1-12.

[3]   Huang, Z.C., Wang, X.G., Wu, N.X., Chu, F.L. and Luo, J. (2019) A Finite Element Model for the Vibration Analysis of Sandwich Beam with Frequency-Dependent Viscoelastic Material Core. Materials, 12, No. 20.

[4]   Kouami, K., Foudil, M., Erasmo, C. and Mostafa, D.E. (2021) A Finite Element Approach for the Static and Vibration Analyses of Functionally Graded Material Viscoelastic Sandwich Beams with Nonlinear Material Behavior. Composite Structures, 274, Article ID: 114315.

[5]   Snchez, E.D., Nallim, L.G., Bellomo, F.J. and Oller, S.H. (2019) Generalized Viscoelastic Model for Laminated Beams Using Hierarchical Finite Elements. Composite Structures, 235, Article ID: 111794.

[6]   Lekdim, B. and Khemmoudj, A. (2021) Existence and Energy Decay of Solution to a Nonlinear Viscoelastic Two-Dimensional Beam with a Delay. Multidimensional Systems and Signal Processing, 32, 915-931.

[7]   Wu, Q.L. and Qi, G.Y. (2020) Viscoelastic String-Beam Coupled Vibro-Impact System: Modeling and Dynamic Analysis. European Journal of Mechanics-A/Solids, 82, Article ID: 104012.

[8]   Elhuni, H. and Basu, D. (2019) Dynamic Soil Structure Interaction Model for Beams on Viscoelastic Foundations Subjected to Oscillatory and Moving Loads. Computers and Geotechnics, 115, Article ID: 103157.

[9]   Yu, C., Zhang, J., Chen, Y., Feng, Y.J. and Yang, A.M. (2019) A Numerical Method for Solving Fractional-Order Viscoelastic Euler-Bernoulli Beams. Chaos, Solitons & Fractals, 128, 275-279.

[10]   Li, R.H. and Liu, B. (2009) Numerical Solution of Differential Equation. Higher Education Press, Beijing.

[11]   Ciarlet, P. (1978) The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam.