| 
                                    [1]
                                 | 
                                
                                    Advances in Structural Integrity for Mechanical, Civil, and Aerospace Applications                                     
                                    Lecture Notes in Mechanical Engineering,      
                                    2025 
                                    DOI:10.1007/978-981-97-6367-2_8
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [2]
                                 | 
                                
                                    Nano‐Enhanced Functionally Graded Epoxy Composites and their future potential: Exploring their synthesis, properties, and applications                                     
                                    Progress in Engineering Science,      
                                    2025 
                                    DOI:10.1016/j.pes.2024.100041
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [3]
                                 | 
                                
                                    Solving coupled differential eigenvalue problems using the differential transformation method numerical example: Dynamic analysis of multi-span beams                                     
                                    Mechanics Research Communications,      
                                    2025 
                                    DOI:10.1016/j.mechrescom.2024.104366
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [4]
                                 | 
                                
                                    Free Vibration of Bi-Directional Functionally Graded Nanobeams Resting on Winkler–Pasternak Foundations                                     
                                    Journal of Vibration Engineering & Technologies,      
                                    2024 
                                    DOI:10.1007/s42417-024-01511-z
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [5]
                                 | 
                                
                                    Dynamic analysis of double cracked bi-directional functionally graded nanobeam using the differential quadrature method                                     
                                    Acta Mechanica,      
                                    2024 
                                    DOI:10.1007/s00707-023-03797-8
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [6]
                                 | 
                                
                                    Investigating the free vibration of viscoelastic FGM Timoshenko nanobeams resting on viscoelastic foundations with the shear correction factor using finite element method                                     
                                    Mechanics Based Design of Structures and Machines,      
                                    2024 
                                    DOI:10.1080/15397734.2022.2141777
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [7]
                                 | 
                                
                                    Dynamic analysis of double cracked bi-directional functionally graded nanobeam using the differential quadrature method                                     
                                    Acta Mechanica,      
                                    2024 
                                    DOI:10.1007/s00707-023-03797-8
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [8]
                                 | 
                                
                                    Dynamic analysis of double cracked bi-directional functionally graded nanobeam using the differential quadrature method                                     
                                    Acta Mechanica,      
                                    2024 
                                    DOI:10.1007/s00707-023-03797-8
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [9]
                                 | 
                                
                                    Dynamic examination of closed cylindrical shells utilizing the differential transform method                                     
                                    Scientific Reports,      
                                    2024 
                                    DOI:10.1038/s41598-024-66095-w
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [10]
                                 | 
                                
                                    Free Vibration of Bi-Directional Functionally Graded Nanobeams Resting on Winkler–Pasternak Foundations                                     
                                    Journal of Vibration Engineering & Technologies,      
                                    2024 
                                    DOI:10.1007/s42417-024-01511-z
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [11]
                                 | 
                                
                                    Vibration characteristics of a tapered- pre-twisted-rotating blade made of Ti-6Al-4 V and ZrO2 type of functionally graded material                                     
                                    International Journal on Interactive Design and Manufacturing (IJIDeM),      
                                    2024 
                                    DOI:10.1007/s12008-024-02064-x
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [12]
                                 | 
                                
                                    Free vibration analysis of functionally graded beam based on different beam theories                                     
                                    4TH INTERNATIONAL SCIENTIFIC CONFERENCE OF ENGINEERING SCIENCES AND ADVANCES TECHNOLOGIES,      
                                    2023 
                                    DOI:10.1063/5.0156796
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [13]
                                 | 
                                
                                    Dynamic characteristics of cracked simply supported bidirectional functionally graded Rayleigh beam                                     
                                    Results in Materials,      
                                    2023 
                                    DOI:10.1016/j.rinma.2023.100441
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [14]
                                 | 
                                
                                    Free vibration of axially or transversely graded beams using finite-element and artificial intelligence                                     
                                    Alexandria Engineering Journal,      
                                    2022 
                                    DOI:10.1016/j.aej.2021.07.004
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [15]
                                 | 
                                
                                    Modeling and Analyzing the Free Vibration of Simply Supported Functionally Graded Beam                                     
                                    Journal of Aerospace Technology and Management,      
                                    2022 
                                    DOI:10.1590/jatm.v14.1257
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [16]
                                 | 
                                
                                    Investigating the free vibration of viscoelastic FGM Timoshenko nanobeams resting on viscoelastic foundations with the shear correction factor using finite element method                                     
                                    Mechanics Based Design of Structures and Machines,      
                                    2022 
                                    DOI:10.1080/15397734.2022.2141777
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [17]
                                 | 
                                
                                    Calculating the fundamental frequency of power law functionally graded beam using ANSYS software                                     
                                    IOP Conference Series: Materials Science and Engineering,      
                                    2021 
                                    DOI:10.1088/1757-899X/1090/1/012014
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [18]
                                 | 
                                
                                    A Comprehensive Review on vibration analysis of functionally graded beams                                     
                                    International Journal of Structural Stability and Dynamics,      
                                    2020 
                                    DOI:10.1142/S0219455420300025
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [19]
                                 | 
                                
                                    DYNAMIC ANALYSIS OF UNIFORM AND NON-UNIFORM CROSS-SECTION CANTILEVER SANDWICH BEAMS                                     
                                    European Journal of Technic,      
                                    2019 
                                    DOI:10.36222/ejt.632784
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [20]
                                 | 
                                
                                    Assessing the Effects of Porosity on the Bending, Buckling, and Vibrations of Functionally Graded Beams Resting on an Elastic Foundation by Using a New Refined Quasi-3D Theory                                     
                                    Mechanics of Composite Materials,      
                                    2019 
                                    DOI:10.1007/s11029-019-09805-0
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [21]
                                 | 
                                
                                    Vibration Analysis of Functionally Graded Timoshenko Beams                                     
                                    International Journal of Structural Stability and Dynamics,      
                                    2018 
                                    DOI:10.1142/S0219455418500074
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [22]
                                 | 
                                
                                    Modeling and analysis of functionally graded sandwich beams: A review                                     
                                    Mechanics of Advanced Materials and Structures,      
                                    2018 
                                    DOI:10.1080/15376494.2018.1447178
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [23]
                                 | 
                                
                                    Free vibration of functionally graded beams resting on Winkler-Pasternak foundation                                     
                                    Arabian Journal of Geosciences,      
                                    2018 
                                    DOI:10.1007/s12517-018-3579-2
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [24]
                                 | 
                                
                                    Effect of Taper on Free Vibration of Functionally Graded Rotating Beam by Mori-Tanaka Method                                     
                                    Journal of The Institution of Engineers (India): Series C,      
                                    2018 
                                    DOI:10.1007/s40032-018-0477-z
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [25]
                                 | 
                                
                                    The isogeometric free vibration and transient response of functionally graded piezoelectric curved beam with elastic restraints                                     
                                    Results in Physics,      
                                    2018 
                                    DOI:10.1016/j.rinp.2018.10.019
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [26]
                                 | 
                                
                                    NATURAL FREQUENCY ANALYSIS OF LAYERED FUNCTIONALLY GRADED BEAMS                                     
                                    ANADOLU UNIVERSITY JOURNAL OF SCIENCE AND TECHNOLOGY A - Applied Sciences and Engineering,      
                                    2018 
                                    DOI:10.18038/aubtda.345541
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [27]
                                 | 
                                
                                    Mode Shape Analysis of Multiple Cracked Functionally Graded Timoshenko Beams                                     
                                    Latin American Journal of Solids and Structures,      
                                    2017 
                                    DOI:10.1590/1679-78253496
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [28]
                                 | 
                                
                                    Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method                                     
                                    KSCE Journal of Civil Engineering,      
                                    2017 
                                    DOI:10.1007/s12205-016-0149-6
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [29]
                                 | 
                                
                                    Elastostatic analysis of two-directional functionally graded beams using various beam theories and Symmetric Smoothed Particle Hydrodynamics method                                     
                                    Composite Structures,      
                                    2017 
                                    DOI:10.1016/j.compstruct.2016.10.065
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [30]
                                 | 
                                
                                    Bending behaviour of two directional functionally graded sandwich beams by using a quasi-3d shear deformation theory                                     
                                    Composite Structures,      
                                    2017 
                                    DOI:10.1016/j.compstruct.2017.04.046
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [31]
                                 | 
                                
                                    Free vibration analysis of functionally graded beams using complementary functions method                                     
                                    Archive of Applied Mechanics,      
                                    2017 
                                    DOI:10.1007/s00419-017-1338-6
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [32]
                                 | 
                                
                                    Free vibration of functionally Euler-Bernoulli and Timoshenko graded porous beams using the transfer matrix method                                     
                                    KSCE Journal of Civil Engineering,      
                                    2017 
                                    DOI:10.1007/s12205-016-0149-6
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [33]
                                 | 
                                
                                    On vibration behavior of rotating functionally graded double-tapered beam with the effect of porosities                                     
                                    Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering,      
                                    2016 
                                    DOI:10.1177/0954410015619647
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [34]
                                 | 
                                
                                    Finite Element Modelling for Static and Free Vibration Response of Functionally Graded Beam                                     
                                    Latin American Journal of Solids and Structures,      
                                    2016 
                                    DOI:10.1590/1679-78252159
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [35]
                                 | 
                                
                                    Free vibration of functionally graded thin beams made of saturated porous materials                                     
                                    Steel and Composite Structures,      
                                    2016 
                                    DOI:10.12989/scs.2016.21.5.999
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [36]
                                 | 
                                
                                    Active vibration control of functionally graded beams with piezoelectric layers based on higher order shear deformation theory                                     
                                    Earthquake Engineering and Engineering Vibration,      
                                    2016 
                                    DOI:10.1007/s11803-016-0352-y
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [37]
                                 | 
                                
                                    Static and free vibration analysis of functionally graded beams by combination Timoshenko theory and finite volume method                                     
                                    Composite Structures,      
                                    2016 
                                    DOI:10.1016/j.compstruct.2015.11.027
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [38]
                                 | 
                                
                                    On vibration behavior of rotating functionally graded double-tapered beam with the effect of porosities                                     
                                    Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering,      
                                    2016 
                                    DOI:10.1177/0954410015619647
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [39]
                                 | 
                                
                                    Analysis of Bending Deflections of Functionally Graded Beams by Using Different Beam Theories and Symmetric Smoothed Particle Hydrodynamics                                     
                                    International Journal of Engineering Technologies IJET,      
                                    2016 
                                    DOI:10.19072/ijet.259394
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 | 
                            
                        
                            
                                | 
                                    [40]
                                 | 
                                
                                    Exact solutions for static and dynamic analyses of carbon nanotube-reinforced composite plates with Pasternak elastic foundation                                     
                                    Applied Mathematical Modelling,      
                                    2015 
                                    DOI:10.1016/j.apm.2014.12.058
                                 | 
                            
                            
                                | 
                                 | 
                                
                                 |