Dynamic Stability Analysis of FGM Beams Based on the Nonlinear Timoshenko Model

Malakzadeh Fard, Keramat; Pourmoayed, Alireza

doi:10.47176/jcme.42.1.8602

Dynamic Stability Analysis of FGM Beams Based on the Nonlinear Timoshenko Model

Document Type : Original Article

Authors

10.47176/jcme.42.1.8602

Abstract

Various types of dynamic instabilities in mechanical systems are one of the most important disruptive factors in such structures. Therefore, an accurate study of dynamic instability in beams, as one of the fundamental engineering structures, is of great importance. In this paper, dynamic instability problem of beams made of Functionally Graded Materials (FGM) is investigated. For this purpose, the first-order shear deformation (or the Timoshenko) beam theory with the effects of geometric nonlinearity is considered. Thus, the proposed model has the ability to determine mechanical behavior of thin and thick beams. By considering the energy functions of the system, and implementing the Hamilton’s principle, the governing equations are obtained along with different types of common boundary conditions. The Differential Quadrature Method (DQM), as one of the best-known numerical methods, is used. The nonlinear partial differential equations are written in the form of equivalent ordinary differential equations. Then, considering the harmonic responses for the system, the differential equations are converted to a set of nonlinear algebraic equations. Finally, in order to study the important parameters, various numerical examples are provided. The obtained numerical results are compared with the literature and thus, the validity of the presented formulation and solution methodology is revealed. Also, a comparative study between linear and nonlinear kinematic models shows that the importance of geometric nonlinearity of the model is quite significant.

Keywords

20.1001.1.22287698.1402.42.1.4.3

References

1. Fu, Y., Wang, J., and Mao, Y., “Nonlinear Analysis of Buckling, Free Vibration and Dynamic Stability for the Piezoelectric Functionally Graded Beams in Thermal Environment”, Applied Mathematical Modelling, Vol. 36, No. 9, pp. 4324-4340, 2012.
1. Chen, L. W., Lin, C. Y., and Wang, C. C., “Dynamic Stability Analysis and Control of a Composite Beam with Piezoelectric Layers”, Composite Structures, Vol . 56, No. 1, pp. 97-109, 2002.
2. Yang, W. P., Chen, L. W., and Wang, C. C., “Vibration and Dynamic Stability of a Traveling Sandwich Beam”, Journal of Sound and Vibration, Vol. 285, No. 3, pp. 597-614, 2005.
3. Lee, H., “Dynamic Stability of a Tapered Cantilever Beam on an Elastic Foundation Subjected to a Follower Force”, International Journal of Solids and Structures, Vol. 33, No. 10, pp. 1409-1424, 1996.
4. Zheng, X., Zhang, J., and Zhou, Y., “Dynamic Stability of a Cantilever Conductive Plate in Transverse Impulsive Magnetic Field”, International Journal of Solids and Structures, Vol. 42, No. 8, pp. 2417-2430, 2005.
5. Shahmohammadi, M. A., Azhari, M., Saadatpour, M. M., Salehipour, H., and Civalek, Ö., “Dynamic Instability Analysis of General Shells Reinforced with Polymeric Matrix and Carbon Fibers Using a Coupled IG-SFSM Formulation”, Composite Structures, Vol. 263, pp.113720, 2021.
6. Pölöskei, T., and Szekrényes, A., “Dynamic Stability of a Structurally Damped Delaminated Beam Using Higher Order Theory”, Mathematical Problems in Engineering, Vol. 2018, pp. 1-15, 2018.
[8] Dario Aristizabal-Ochoa, J., “Static and Dynamic Stability of Uniform Shear Beam-Columns under Generalized Boundary Conditions”, Journal of Sound and Vibration, Vol. 307, No. 1-2, pp. 69-88, 2007.
1. Salehipour, H., Emadi, S., Tayebikhorami, S., and Shahmohammadi, M.A., “A Semi-Analytical Solution for Dynamic Stability Analysis of Nanocomposite/Fibre-Reinforced Doubly-Curved Panels Resting on the Elastic Foundation in Thermal Environment”, The European Physical Journal Plus, Vol. 137, No. 1, pp.1-36, 2022.
2. Malekzadeh, K., Khalili, M., and Mittal, R., “Local and Global Damped Vibrations of Plates with a Viscoelastic Soft Flexible Core: An Improved High-Order Approach”, Journal of Sandwich Structures & Materials, Vol. 7, No. 5, pp. 431-456, 2005.
3. Banerjee ,J., Cheung, C., Morishima, R., Perera, M., and Njuguna, J., “Free Vibration of a Three-Layered Sandwich Beam Using the Dynamic Stiffness Method and Experiment”, International Journal of Solids and Structures, Vol. 44, No. 22-23, pp. 7543-7563, 2007.
4. Karaagac, C., ÖZTÜRK, H., Sabuncu, M., “Lateral Dynamic Stability Analysis of a Cantilever Laminated Composite Beam with an Elastic Support”, International Journal of Structural Stability and Dynamics, Vol. 7, No. 3, pp. 377-402, 2007.
5. Pradhan, M., Dash, P., and Pradhan, P., “Static and Dynamic Stability Analysis of an Asymmetric Sandwich Beam Resting on a Variable Pasternak Foundation Subjected to Thermal Gradient”, Meccanica, Vol. 51, No. 3, pp. 725-739, 2016.
6. Wang, J., Shen, H., Zhang, B., and Liu, J., “Studies on the Dynamic Stability of an Axially Moving Nanobeam Based on the Nonlocal Strain Gradient Theory”, Modern Physics Letters B, Vol. 32, No. 16, pp. 1850167, 2018.
7. Frostig,Y., Baruch, M., Vilnay, O., and Sheinman, I., “High-Order Theory for Sandwich-Beam Behavior with Transversely Flexible Core”, Journal of Engineering Mechanics, Vol. 118, No. 5, pp. 1026-1043, 1992.
8. Yeh, J. Y., Chen, L. W., and Wang, C. C., “Dynamic Stability of a Sandwich Beam with a Constrained Layer and Electrorheological Fluid Core”, Composite Structures, Vol. 64, No. 1, pp. 47-54, 2004.
9. Bozhevolnaya, E., and Frostig, Y., “Free Vibrations of Curved Sandwich Beams with a Transversely Flexible Core”, Journal of Sandwich Structures & Materials, Vol. 3, No. 4, pp. 311-342, 2001.
10. Sahmani, S., Ansari, R., Gholami R., and Darvizeh, A., “Dynamic Stability Analysis of Functionally Graded Higher-order Shear Deformable Microshells Based on the Modified Couple Stress Elasticity Theory”, Composites Part B: Engineering, Vol. 51, pp. 44-53, 2013.
11. Shahmohammadi, M. A., Mirfatah, S. M., Salehipour, H., Azhari, M., and Civalek, Ö., “Free Vibration and Stability of Hybrid Nanocomposite-Reinforced Shallow Toroidal Shells Using an Extended Closed-Form Formula Based on the Galerkin Method”, Mechanics of Advanced Materials and Structures, Vol. 29, No. 26, pp. 5284-5300.
12. Shahmohammadi, M.A., Mirfatah, S.M., Salehipour, H., Azhari, F. and Civalek, Ö., “Dynamic Stability of Hybrid Fiber/Nanocomposite-Reinforced Toroidal Shells Subjected to the Periodic Axial and Pressure Loadings”, Mechanics of Advanced Materials and Structures, Vol. 30, No. 8, pp.1547-1590, 2023.
13. Ansari, R., and Gholami, R., “Dynamic Stability of Embedded Single Walled Carbon Nanotubes Including Thermal Effects”, Iranian Journal of Science and Technology Transactions of Mechanical Engineering, Vol. 39, pp. 153-161, 2015.
14. Ansari, R., Gholami, R., Sahmani, S., Norouzzadeh, A., and Bazdid-Vahdati, M., “Dynamic Stability Analysis of Embedded Multi-Walled Carbon Nanotubes in Thermal Environment”, Acta Mechanica Solida Sinica, Vol. 28, No. 6, pp. 659-667, 2015.
15. Kolahchi, R., and Bidgoli, A. M., “Size-Dependent Sinusoidal Beam Model for Dynamic Instability of Single-Walled Carbon Nanotubes”, Applied Mathematics and Mechanics, Vol. 37, No. 2, pp. 265-274, 2016.
16. Ke, L. L., and Wang, Y. S., “Size Effect on Dynamic Stability of Functionally Graded Microbeams Based on a Modified Couple Stress Theory”, Composite Structures, Vol. 93, No. 2, pp. 342-350, 2011.
25. Reddy, J., and Chin, C., “Thermomechanical Analysis of Functionally Graded Cylinders and Plates”, Journal of Thermal Stresses, Vol. 21, No. 6, pp. 593-626, 1998.