# Undamped Free Vibration Analysis of Functionally Graded Beams: A Dynamic Finite Element Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{t}and P

_{b}are the material properties at the top and bottom surfaces of the beam, respectively. The effective material properties are assumed to vary according to a power law distribution. V

_{t}is the volume fraction of the top constituent of the beam, defined as [9,21,22]:

_{i}and A

_{i}are defined as:

_{int}stands for the internal virtual work, and W

_{ext}is the external virtual work. The underlined terms in expressions (12) and (13) represent the virtual work generated by the load boundary conditions of the system, presented earlier as expressions (9). For the free vibration of a system, the total external work is null, W

_{ext}= 0, and therefore:

_{1,2,3,4}and D

_{1,2}are constants and

_{n}]

_{v}and [P

_{n}]

_{w}are defined as:

**,**which yields a zero determinant, $\left|K\left(\omega \right)\right|=0$, for the global dynamic stiffness matrix [28].

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

- Niino, M.; Hirai, T.; Watanabe, R. Functionally gradient materials. In pursuit of super heat resisting materials for spacecraft. J. Jpn. Soc. Compos. Mater.
**1987**, 13, 257–264. [Google Scholar] [CrossRef] - Mahamood, R.M.; Akinlabi, E.T. Laser metal deposition of functionally graded Ti6Al4V/TiC. Mater. Des.
**2015**, 84, 402–410. [Google Scholar] [CrossRef] - Taminger, K.M.B.; Hafley, R.A. Electron Beam Freeform Fabrication: A Rapid Metal Deposition Process. In Proceedings of the 3rd Annual Automotive Composites Conference, Hampton, VA, USA, 9–10 September 2003. [Google Scholar]
- Li, L.; Syed, W.U.H.; Pinkerton, A.J. Rapid additive manufacturing of functionally graded structures using simultaneous wire and powder laser deposition. Virtual Phys. Prototyp.
**2006**, 1, 217–225. [Google Scholar] [CrossRef] - Akbaş, Ş.D.; Fageehi, Y.A.; Assie, A.E.; Eltaher, M.A. Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load. Eng. Comput.
**2022**, 38, 365–377. [Google Scholar] [CrossRef] - Esen, I.; Koc, M.A.; Cay, Y. Finite element formulation and analysis of a functionally graded Timoshenko beam subjected to an accelerating mass including inertial effects of the mass. Lat. Am. J. Solids Struct.
**2018**, 15, 1–18. [Google Scholar] [CrossRef] - Esen, I.; Koc, M.A.; Eroğlu, M. Dynamic behaviour of functionally graded Timoshenko beams on a four parameter linear elastic foundation due to a high speed travelling mass with variable velocities. J. Smart Syst. Res.
**2021**, 2, 48–75. [Google Scholar] - Koç, M.A. Finite element and numerical vibration analysis of a Timoshenko and Euler–Bernoulli beams traversed by a moving high-speed train. J. Braz. Soc. Mech. Sci. Eng.
**2021**, 43, 165. [Google Scholar] [CrossRef] - Şimşek, M. Vibration analysis of a functionally graded beam under a moving mass by using different beam theories. Compos. Struct.
**2010**, 92, 904–917. [Google Scholar] [CrossRef] - Al-Zahrani, M.A.; Asiri, S.A.; Ahmed, K.I.; Eltaher, M.A. Free Vibration Analysis of 2D Functionally Graded Strip Beam using Finite Element Method. J. Appl. Comput. Mech.
**2022**, 8, 1422–1430. [Google Scholar] - Giunta, G.; Crisafulli, D.; Belouettar, S.; Carrera, E. Hierarchical theories for the free vibration analysis of functionally graded beams. Compos. Struct.
**2011**, 94, 68–74. [Google Scholar] [CrossRef] - Neamah, R.A.; Nassar, A.A.; Alansari, L.S. Modeling and Analyzing the Free Vibration of Simply Supported Functionally Graded Beam. J. Aerosp. Technol. Manag.
**2022**, 14. [Google Scholar] [CrossRef] - Aubad, M.J.; Khafaji, S.O.W.; Hussein, M.T.; Al-Shujairi, M.A. Modal analysis and transient response of axially functionally graded (AFG) beam using finite element method. Mater. Res. Express
**2019**, 6, 1065g4. [Google Scholar] [CrossRef] - Kahya, V.; Turan, M. Finite element model for vibration and buckling of functionally graded beams based on the first-order shear deformation theory. Compos. Part B Eng.
**2017**, 109, 108–115. [Google Scholar] [CrossRef] - Kahya, V.; Turan, M. Vibration and stability analysis of functionally graded sandwich beams by a multi-layer finite element. Compos. Part B Eng.
**2018**, 146, 198–212. [Google Scholar] [CrossRef] - Rahmani, F.; Kamgar, R.; Rahgozar, R. Finite element analysis of functionally graded beams using different beam theories. Civ. Eng. J.
**2020**, 6, 2086–2102. [Google Scholar] [CrossRef] - Yarasca, J.; Mantari, J.L.; Arciniega, R.A. Hermite–Lagrangian finite element formulation to study functionally graded sandwich beams. Compos. Struct.
**2016**, 140, 567–581. [Google Scholar] [CrossRef] - Pradhan, K.K.; Chakraverty, S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Compos. Part B Eng.
**2013**, 51, 175–184. [Google Scholar] [CrossRef] - Wattanasakulpong, N.; Mao, Q. Dynamic response of Timoshenko functionally graded beams with classical and non-classical boundary conditions using Chebyshev collocation method. Compos. Struct.
**2015**, 119, 346–354. [Google Scholar] [CrossRef] - Zhao, Y.; Huang, Y.; Guo, M. A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory. Compos. Struct.
**2017**, 168, 277–284. [Google Scholar] [CrossRef] - Su, H.; Banerjee, J.R.; Cheung, C.W. Dynamic stiffness formulation and free vibration analysis of functionally graded beams. Compos. Struct.
**2013**, 106, 854–862. [Google Scholar] [CrossRef] - Su, H.; Banerjee, J.R. Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams. Comput. Struct.
**2015**, 147, 107–116. [Google Scholar] [CrossRef] - Banerjee, J.R.; Ananthapuvirajah, A. Free vibration of functionally graded beams and frameworks using the dynamic stiffness method. J. Sound Vib.
**2018**, 422, 34–47. [Google Scholar] [CrossRef] - Rajasekaran, S. Buckling and vibration of axially functionally graded nonuniform beams using differential transformation based dynamic stiffness approach. Meccanica
**2013**, 48, 1053–1070. [Google Scholar] [CrossRef] - Deng, H.; Cheng, W. Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams. Compos. Struct.
**2016**, 141, 253–263. [Google Scholar] [CrossRef] - Banerjee, J.R. Dynamic stiffness formulation for structural elements: A general approach. Comput. Struct.
**1997**, 63, 101–103. [Google Scholar] [CrossRef] - Banerjee, J.R.; Sobey, A.J.; Su, H.; Fitch, J.P. Use of computer algebra in Hamiltonian calculations. Adv. Eng. Softw.
**2008**, 39, 521–525. [Google Scholar] [CrossRef] - Hashemi, S.M. Free Vibrational Analysis of Rotating Beam-Like Structures: A Dynamic Finite Element Approach. Ph.D. Dissertation, Department of Mechanical Engineering, Lavel University, Quebec, QC, Canada, 1998. [Google Scholar]
- Hashemi, S.M.; Richard, M.J. A Dynamic Finite Element (DFE) method for free vibrations of bending-torsion coupled beams. Aerosp. Sci. Technol.
**2000**, 4, 41–55. [Google Scholar] [CrossRef] - Hashemi, S.M.; Adique, E.J. A Quasi-Exact Dynamic Finite Element for Free Vibration Analysis of Sandwich Beams. Appl. Compos. Mater.
**2010**, 17, 259–269. [Google Scholar] [CrossRef] - Erdelyi, N.H.; Hashemi, S.M. A Dynamic Stiffness Element for Free Vibration Analysis of Delaminated Layered Beams. Model. Simul. Eng.
**2012**, 2012, 2. [Google Scholar] [CrossRef] [Green Version] - Kashani, M.T.; Jayasinghe, S.; Hashemi, S.M. Dynamic finite element analysis of bending-torsion coupled beams subjected to combined axial load and end moment. Shock. Vib.
**2015**, 2015, 471270. [Google Scholar] [CrossRef] [Green Version] - Erdelyi, N.H.; Hashemi, S.M. On the Finite Element Free Vibration Analysis of Delaminated Layered Beams—A New Assembly Technique. Shock. Vib.
**2016**, 2016, 3707658. [Google Scholar] [CrossRef] [Green Version] - Borneman, S.R.; Hashemi, S.M. Vibration-Based, Nondestructive Methodology for Detecting Multiple Cracks in Bending-Torsion Coupled Laminated Composite Beams. Shock. Vib.
**2018**, 2018, 9628141. [Google Scholar] [CrossRef] - Kashani, M.T.; Hashemi, S.M. A Finite Element Formulation for Bending–Torsion Coupled Vibration Analysis of Delaminated Beams Subjected to Combined Axial Load and End Moment. Shock. Vib.
**2018**, 2018, 1348970, Special Issue on SHMV. [Google Scholar] [CrossRef] [Green Version] - Kashani, M.T.; Hashemi, S.M. Dynamic Finite Element Modelling and Vibration Analysis of Prestressed Layered Bending–Torsion Coupled Beams. Appl. Mech.
**2022**, 3, 103–120. [Google Scholar] [CrossRef] - Tse, F.S.; Morse, I.E.; Hinkle, R.T.; Hinkle, R.T.; Morse, I.E. Mechanical Vibrations: Theory and Applications, 2nd ed.; Tse, F.S., Morse, I.E., Hinkle, R.T., Hinkle, R.T., Morse, I.E., Eds.; Allyn and Bacon: Boston, MA, USA, 1978; ISBN 0-205-05940-6. [Google Scholar]

**Figure 1.**Coordinate system and notation for formulation [22].

**Figure 3.**Convergence study for the 3rd mode of a pure aluminum beam with clamped–free boundary conditions.

**Figure 4.**Convergence study for the 4th mode of a FGM beam $\left(k=1\right)$, with clamped–free boundary conditions.

**Figure 5.**First 4 mode shapes for a FGM beam $\left(k=1\right)$; (

**A**) clamped–free, (

**B**) simply supported, (

**C**) clamped–clamped, and (

**D**) clamped–pinned boundary conditions.

**Table 1.**The first three non-dimensional frequencies of a pure aluminum beam for a range of slenderness ratios and simply supported boundary conditions.

Frequency No. i | L/h | $\mathbf{Non-Dimensional}\text{}\mathbf{Fundamental}\text{}\mathbf{Natural}\text{}\mathbf{Frequency}\text{}\left({\mathit{\lambda}}_{\mathit{i}}\right)$ | |||||
---|---|---|---|---|---|---|---|

Classical Results [27] | DSM Ref [13] | DSM Code | DFE (1 ELE) | DFE (3 ELE) | FEM (10 ELE) | ||

1 | 10 | 9.8696 | 9.8293 | 9.8293 | 9.8293 | 9.8293 | 9.8293 |

30 | 9.8696 | 9.8651 | 9.8651 | 9.8651 | 9.8651 | 9.8652 | |

100 | 9.8696 | 9.8692 | 9.8692 | 9.8692 | 9.8692 | 9.8693 | |

2 | 10 | 39.478 | 38.845 | 38.845 | 38.845 | 38.845 | 38.849 |

30 | 39.478 | 39.406 | 39.406 | 39.406 | 39.406 | 39.411 | |

100 | 39.478 | 39.472 | 39.472 | 39.472 | 39.472 | 39.476 | |

3 | 10 | 88.826 | 85.711 | 85.711 | 85.711 | 85.711 | 85.757 |

30 | 88.826 | 88.463 | 88.463 | 88.463 | 88.462 | 88.511 | |

100 | 88.826 | 88.794 | 88.794 | 88.794 | 88.794 | 88.841 |

**Table 2.**The first three non-dimensional frequencies of a pure aluminum beam for a range of slenderness ratios and clamped–clamped boundary conditions.

Frequency No. i | L/h | $\mathbf{Non-Dimensional}\text{}\mathbf{Fundamental}\text{}\mathbf{Natural}\text{}\mathbf{Frequency}\text{}\left({\mathit{\lambda}}_{\mathit{i}}\right)$ | |||||
---|---|---|---|---|---|---|---|

Classical Results [27] | DSM Ref [12] | DSM Code | DFE (1 ELE) | DFE (3 ELE) | FEM (10 ELE) | ||

1 | 10 | 22.373 | 22.259 | 22.259 | 22.259 | 22.259 | 22.260 |

30 | 22.373 | 22.361 | 22.361 | 22.361 | 22.361 | 22.361 | |

100 | 22.373 | 22.372 | 22.372 | 22.373 | 22.372 | 22.373 | |

2 | 10 | 61.673 | 60.522 | 60.522 | 60.522 | 60.522 | 60.538 |

30 | 61.673 | 61.542 | 61.542 | 61.542 | 61.542 | 61.558 | |

100 | 61.673 | 61.673 | 61.661 | 61.661 | 61.660 | 61.661 | |

3 | 10 | 120.90 | 116.21 | 116.21 | 116.21 | 116.21 | 116.32 |

30 | 120.90 | 120.35 | 120.35 | 120.35 | 120.35 | 120.47 | |

100 | 120.90 | 120.85 | 120.85 | 120.86 | 120.85 | 120.97 |

**Table 3.**The first three non-dimensional frequencies of a pure aluminum beam for a range of slenderness ratios and clamped–free boundary conditions.

Frequency No. i | L/h | $\mathbf{Non-Dimensional}\text{}\mathbf{Fundamental}\text{}\mathbf{Natural}\text{}\mathbf{Frequency}\text{}\left({\mathit{\lambda}}_{\mathit{i}}\right)$ | |||||
---|---|---|---|---|---|---|---|

Classical Results [27] | DSM Ref [12] | DSM Code | DFE (1 ELE) | DFE (3 ELE) | FEM (10 ELE) | ||

1 | 10 | 3.5160 | 3.5092 | 3.5092 | 3.5092 | 3.5092 | 3.5092 |

30 | 3.5160 | 3.5153 | 3.5153 | 3.5153 | 3.5153 | 3.5153 | |

100 | 3.5160 | 3.5160 | 3.5159 | 3.5159 | 3.5159 | 3.5159 | |

2 | 10 | 22.035 | 21.743 | 21.743 | 21.743 | 21.743 | 21.743 |

30 | 22.035 | 22.002 | 22.002 | 22.001 | 22.001 | 22.002 | |

100 | 22.035 | 22.032 | 22.032 | 22.032 | 22.032 | 22.032 | |

3 | 10 | 61.677 | 59.801 | 59.801 | 59.801 | 59.801 | 59.816 |

30 | 61.677 | 61.478 | 61.478 | 61.478 | 61.478 | 61.493 | |

100 | 61.677 | 61.677 | 61.677 | 61.677 | 61.677 | 61.693 |

**Table 4.**The first three non-dimensional frequencies of a pure aluminum beam for a range of slenderness ratios and clamped–pinned boundary conditions.

Frequency No. i | L/h | ||||||
---|---|---|---|---|---|---|---|

Classical Results [27] | DSM Ref [12] | DSM Code | DFE (1 ELE) | DFE (3 ELE) | FEM (10 ELE) | ||

1 | 10 | 15.418 | 15.345 | 15.345 | 15.345 | 15.345 | 15.345 |

30 | 15.418 | 15.410 | 15.410 | 15.410 | 15.410 | 15.410 | |

100 | 15.418 | 15.418 | 15.417 | 15.417 | 15.417 | 15.418 | |

2 | 10 | 49.965 | 49.095 | 49.095 | 49.095 | 49.095 | 49.103 |

30 | 49.965 | 49.866 | 49.866 | 49.866 | 49.866 | 48.875 | |

100 | 49.965 | 49.956 | 49.956 | 49.956 | 49.956 | 49.965 | |

3 | 10 | 104.25 | 100.39 | 100.39 | 100.39 | 100.39 | 100.46 |

30 | 104.25 | 103.80 | 103.80 | 103.80 | 103.80 | 103.87 | |

100 | 104.25 | 104.21 | 104.21 | 104.21 | 104.21 | 104.28 |

**Table 5.**The first (fundamental) non-dimensional frequency of a FGM beam with $k=0.3$ under simply supported boundary conditions.

L/h |
$$\mathbf{Non-Dimensional}\text{}\mathbf{Fundamental}\text{}\mathbf{Natural}\text{}\mathbf{Frequency}\text{}\left({\mathit{\lambda}}_{\mathit{i}}\right)$$
| ||||||
---|---|---|---|---|---|---|---|

Ref [25] | DSM Ref [13] | DSM (1 ELE) | DFE (1 ELE) | DFE (3 ELE) | FEM (3 ELE) | ||

Euler-Bernoulli | Timoshenko | ||||||

10 | 17.329 | 17.138 | 17.614 | 17.328 | 17.378 | 17.350 | 17.365 |

30 | 17.392 | 17.373 | 17.676 | 17.395 | 17.447 | 17.420 | 17.433 |

100 | 17.405 | 17.398 | 17.684 | 17.402 | 17.455 | 17.430 | 17.441 |

**Table 6.**The first 4 non-dimensional frequencies for a FGM beam for (k = 0.1, 1, and 5), under clamped–free boundary conditions.

Frequency No. i | |||||||||
---|---|---|---|---|---|---|---|---|---|

k = 0.1 | k = 1 | k = 5 | |||||||

DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | |

1 | 6.2673 | 6.2673 | 6.2673 | 4.7669 | 4.7672 | 4.7672 | 4.0493 | 4.0495 | 4.0495 |

2 | 39.272 | 39.272 | 39.274 | 29.870 | 29.883 | 29.883 | 25.374 | 25.381 | 25.382 |

3 | 109.92 | 109.95 | 109.97 | 83.599 | 83.703 | 83.722 | 71.014 | 71.085 | 71.102 |

4 | 215.38 | 215.40 | 215.60 | 163.81 | 164.11 | 164.26 | 139.16 | 139.34 | 139.47 |

**Table 7.**The first 4 non-dimensional frequencies for a FGM beam for (k = 0.1, 1, 5), under simply supported boundary conditions.

Frequency No. i | |||||||||
---|---|---|---|---|---|---|---|---|---|

k = 0.1 | k = 1 | k = 5 | |||||||

DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | |

1 | 17.592 | 17.592 | 17.592 | 13.381 | 13.382 | 13.382 | 11.366 | 11.367 | 11.367 |

2 | 70.360 | 70.361 | 70.367 | 53.515 | 53.542 | 53.547 | 45.459 | 45.476 | 45.481 |

3 | 158.28 | 158.28 | 158.37 | 120.38 | 120.51 | 120.58 | 102.26 | 102.34 | 102.40 |

4 | 281.30 | 281.32 | 281.78 | 213.94 | 214.36 | 214.69 | 181.74 | 182.00 | 182.29 |

**Table 8.**The first 4 non-dimensional frequencies for a FGM beam for (k = 0.1, 1, and 5), under clamped–clamped boundary conditions.

Frequency No. i | |||||||||
---|---|---|---|---|---|---|---|---|---|

k = 0.1 | k = 1 | k = 5 | |||||||

DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | |

1 | 39.872 | 39.872 | 39.881 | 30.326 | 30.333 | 30.352 | 25.761 | 25.766 | 25.778 |

2 | 109.89 | 109.88 | 109.95 | 83.576 | 83.627 | 83.734 | 70.997 | 71.027 | 71.103 |

3 | 215.35 | 215.36 | 215.66 | 163.79 | 164.00 | 164.38 | 139.13 | 139.27 | 139.56 |

4 | 355.83 | 355.85 | 356.99 | 270.63 | 271.21 | 272.40 | 229.90 | 230.27 | 231.19 |

**Table 9.**The first 4 non-dimensional frequencies for a FGM beam for (k = 0.1, 1, and 5), under clamped–pinned boundary conditions.

Frequency No. i | |||||||||
---|---|---|---|---|---|---|---|---|---|

k = 0.1 | k = 1 | k = 5 | |||||||

DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | DSM Code (1 ELE) | DFE (10 ELE) | FEM (10 ELE) | |

1 | 27.485 | 27.485 | 27.485 | 20.947 | 20.954 | 20.955 | 17.783 | 17.788 | 17.788 |

2 | 89.050 | 89.053 | 89.069 | 67.770 | 67.832 | 67.842 | 57.560 | 57.597 | 57.607 |

3 | 185.75 | 185.77 | 185.90 | 141.32 | 141.56 | 141.66 | 120.04 | 120.19 | 120.27 |

4 | 317.55 | 317.59 | 318.25 | 241.55 | 242.21 | 242.70 | 205.20 | 205.60 | 206.02 |

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**MDPI and ACS Style**

Gee, A.; Hashemi, S.M.
Undamped Free Vibration Analysis of Functionally Graded Beams: A Dynamic Finite Element Approach. *Appl. Mech.* **2022**, *3*, 1223-1239.
https://doi.org/10.3390/applmech3040070

**AMA Style**

Gee A, Hashemi SM.
Undamped Free Vibration Analysis of Functionally Graded Beams: A Dynamic Finite Element Approach. *Applied Mechanics*. 2022; 3(4):1223-1239.
https://doi.org/10.3390/applmech3040070

**Chicago/Turabian Style**

Gee, Aaron, and Seyed M. Hashemi.
2022. "Undamped Free Vibration Analysis of Functionally Graded Beams: A Dynamic Finite Element Approach" *Applied Mechanics* 3, no. 4: 1223-1239.
https://doi.org/10.3390/applmech3040070