Beam is one of the common structures in engineering, with the development of technology, homogeneous beams no longer meet the needs of engineering structural design, for this reason, people have researched the non-homogeneous beams. In this paper, we study the mixed finite element method for the vibration problem of non-homogeneous damped beams. The fourth-order differential equations are transformed into a system of low-order partial differential equations by introducing intermediate variables, constructing a semi-discrete extended mixed finite element format, proving the existence and uniqueness of the solution of the format, and utilizing the elliptic projection operator for the error estimation. The time derivative term is discretized by the central difference, and the fully discrete mixed element format is given to prove the stability and convergence of the format. The feasibility and effectiveness of the mixed method are verified by numerical examples, and the effects of different damping coefficients μ on beam vibration are investigated.
References
[1]
Gupta, A.K. (1985) Vibration of Tapered Beams. Journal of Structural Engineering, 111, 19-36. https://doi.org/10.1061/(asce)0733-9445(1985)111:1(19)
[2]
Ece, M.C., Aydogdu, M. and Taskin, V. (2007) Vibration of a Variable Cross-Section Beam. MechanicsResearchCommunications, 34, 78-84. https://doi.org/10.1016/j.mechrescom.2006.06.005
[3]
Lin, Y. and Trethewey, M.W. (1990) Finite Element Analysis of Elastic Beams Subjected to Moving Dynamic Loads. JournalofSoundandVibration, 136, 323-342. https://doi.org/10.1016/0022-460x(90)90860-3
[4]
Ait Atmane, H., Tounsi, A., Meftah, S.A. and Belhadj, H.A. (2010) Free Vibration Behavior of Exponential Functionally Graded Beams with Varying Cross-Section. Journal ofVibrationandControl, 17, 311-318. https://doi.org/10.1177/1077546310370691
[5]
Cao, D., Gao, Y., Wang, J., Yao, M. and Zhang, W. (2019) Analytical Analysis of Free Vibration of Non-Uniform and Non-Homogenous Beams: Asymptotic Perturbation Approach. AppliedMathematicalModelling, 65, 526-534. https://doi.org/10.1016/j.apm.2018.08.026
[6]
Awrejcewicz, J., Krysko, A.V., Mrozowski, J., Saltykova, O.A. and Zhigalov, M.V. (2011) Analysis of Regular and Chaotic Dynamics of the Euler-Bernoulli Beams Using Finite Difference and Finite Element Methods. ActaMechanicaSinica, 27, 36-43. https://doi.org/10.1007/s10409-011-0412-5
[7]
Alotta, G., Failla, G. and Zingales, M. (2017) Finite-Element Formulation of a Nonlocal Hereditary Fractional-Order Timoshenko Beam. Journal of Engineering Mechanics, 143, D4015001. https://doi.org/10.1061/(asce)em.1943-7889.0001035
[8]
Dönmez Demir, D., Bildik, N. and Sınır, B.G. (2013) Linear Dynamical Analysis of Fractionally Damped Beams and Rods. JournalofEngineeringMathematics, 85, 131-147. https://doi.org/10.1007/s10665-013-9642-9
[9]
Wang, T., Jiang, Z. and Yin, Z. (2021) Mixed Finite Volume Element Method for Vibration Equations of Beam with Structural Damping. AmericanJournalofComputationalMathematics, 11, 207-225. https://doi.org/10.4236/ajcm.2021.113014
[10]
Zhang, R., Yin, Z. and Zhu, A. (2023) Numerical Simulations of a Mixed Finite Element Method for Damped Plate Vibration Problems. Mathematical Modelling and Control, 3, 7-22. https://doi.org/10.3934/mmc.2023002
[11]
Yuan, J., Yin, Z. and Zhu, A. (2024) H1-Galerkin Mixed Finite Element Method for the Vibration Equation of Beam with Structural Damping. Computational and AppliedMathematics, 43, Article No. 308. https://doi.org/10.1007/s40314-024-02831-2
[12]
Meng, J. and Mei, L. (2020) A Mixed Virtual Element Method for the Vibration Problem of Clamped Kirchhoff Plate. AdvancesinComputationalMathematics, 46, Article No. 68. https://doi.org/10.1007/s10444-020-09810-1
[13]
Babuška, I. (1973) The Finite Element Method with Lagrangian Multipliers. NumerischeMathematik, 20, 179-192. https://doi.org/10.1007/bf01436561
[14]
Fortin, M. and Brezzi, F. (1991) Mixed and Hybrid Finite Element Methods. Springer-Verlag. https://doi.org/10.1007/978-1-4612-3172-1
[15]
Makridakis, C.G. (1992) On Mixed Finite Element Methods for Linear Elastodynamics. NumerischeMathematik, 61, 235-260. https://doi.org/10.1007/bf01385506
[16]
Burger, M., A. Carrillo, J. and Wolfram, M. (2010) A Mixed Finite Element Method for Nonlinear Diffusion Equations. Kinetic&RelatedModels, 3, 59-83. https://doi.org/10.3934/krm.2010.3.59
[17]
Lamichhane, B.P. (2011) A Stabilized Mixed Finite Element Method for the Biharmonic Equation Based on Biorthogonal Systems. Journal of Computational and Applied Mathematics, 235, 5188-5197. https://doi.org/10.1016/j.cam.2011.05.005
[18]
Liu, X. and Yang, X. (2021) Mixed Finite Element Method for the Nonlinear Time-Fractional Stochastic Fourth-Order Reaction-Diffusion Equation. Computers & Mathematics withApplications, 84, 39-55. https://doi.org/10.1016/j.camwa.2020.12.004
[19]
Meng, J. and Mei, L. (2022) The Optimal Order Convergence for the Lowest Order Mixed Finite Element Method of the Biharmonic Eigenvalue Problem. JournalofComputationalandAppliedMathematics, 402, Article ID: 113783. https://doi.org/10.1016/j.cam.2021.113783
[20]
Huang, C., An, N. and Chen, H. (2022) Local H1-Norm Error Analysis of a Mixed Finite Element Method for a Time-Fractional Biharmonic Equation. Applied Numerical Mathematics, 173, 211-221. https://doi.org/10.1016/j.apnum.2021.12.004
[21]
Huang, C. and Stynes, M. (2020) α-Robust Error Analysis of a Mixed Finite Element Method for a Time-Fractional Biharmonic Equation. Numerical Algorithms, 87, 1749-1766. https://doi.org/10.1007/s11075-020-01036-y
[22]
Cowsat, L.C., Dupont, T.F. and Wheeler, M.F. (1990) A Priori Estimates for Mixed Finite Element Methods for the Wave Equation. Computer Methods in Applied MechanicsandEngineering, 82, 205-222. https://doi.org/10.1016/0045-7825(90)90165-i
[23]
Li, J. (2006) Optimal Convergence Analysis of Mixed Finite Element Methods for Fourth-Order Elliptic and Parabolic Problems. Numerical Methods for Partial Differential Equations, 22, 884-896. https://doi.org/10.1002/num.20127
[24]
He, S., Li, H. and Liu, Y. (2013) Analysis of Mixed Finite Element Methods for Fourth-Order Wave Equations. Computers&MathematicswithApplications, 65, 1-16. https://doi.org/10.1016/j.camwa.2012.10.002
[25]
Ciarlet, P.G. (2002) The Finite Element Method for Elliptic Problems. SIAM Publications Library.
