Modeling and Analyzing the Free Vibration of Simply Supported Functionally Graded Beam

Neamah, Raghad Azeez; Nassar, Ameen Ahmed; Alansari, Luay Sadiq

doi:10.1590/jatm.v14.1257

Raghad Azeez Neamah ^**Correspondence author: ragada.deibel@uokufa.edu.iq

University of Basrah – College of Engineering – Mechanical Engineering Department – Basrah – Iraq.

University of Kufa – Faculty of Engineering – Mechanical Engineering Department – Najaf – Iraq.

http://orcid.org/0000-0003-1780-5035

Ameen Ahmed Nassar

University of Basrah – College of Engineering – Mechanical Engineering Department – Basrah – Iraq.

http://orcid.org/0000-0001-7230-4394

Luay Sadiq Alansari

University of Kufa – Faculty of Engineering – Mechanical Engineering Department – Najaf – Iraq.

http://orcid.org/0000-0002-2989-8614

*Correspondence author: ragada.deibel@uokufa.edu.iq

Section editor: Renato Rebouças de Medeiros

AUTHORS’ CONTRIBUTION

Conceptualization: Neamah RA, Nassar AA and Alansari LS; Methodology: Neamah RA, Nassar AA and Alansari LS; Investigation: Neamah RA; Writing – Original Draft: Neamah RA; Writing – Review and Editing: Neamah RA; Funding Acquisition: Neamah RA; Resources: Neamah RA; Supervision: Nassar AA and Alansari LS.

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Figure 1
The geometry of FGB.

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Figure 2
Mesh of 1 and 10 layers of FGB.

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Figure 2
Mesh of 1 and 10 layers of FGB.

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Figure 3
The first mode shape of simply supported FGB at L/h 5.

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Figure 4
The first mode shape of simply supported FGB at L/h 10.

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Figure 5
The first mode shape of simply supported FGB at L/h 20.

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Figure 6
The first mode shape of simply supported FGB at L/h 50.

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Figure 7
The first mode shape of simply supported FGB at L/h 100.

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Figure 8
Dimensionless frequency of FGB when E ratio = 0.25 by CBT.

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Figure 9
Dimensionless frequency of FGB when E ratio=0.5 by CBT.

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Figure 10
Dimensionless frequency of FGB when E ratio = 1 by CBT.

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Figure 11
Dimensionless frequency of FGB when E ratio = 2 by CBT.

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Figure 12
Dimensionless frequency of FGB when E ratio = 4 by CBT.

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Figure 13
Dimensionless frequency of FGB when E ratio = 0.25 by FSDT.

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Figure 14
Dimensionless frequency of FGB when E ratio = 0.5 by FSDT.

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Figure 15
Dimensionless frequency of FGB when E ratio = 1 by FSDT.

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Figure 16
Dimensionless frequency of FGB when E ratio = 2 by FSDT.

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Figure 17
Dimensionless frequency of FGB when E ratio = 4 by FSDT.

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Figure 18
Dimensionless frequency of FGB when E ratio = 0.25 by HSDT.

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Figure 19
Dimensionless frequency of FGB when E ratio = 0.5 by HSDT.

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Figure 20
Dimensionless frequency of FGB when E ratio = 1 by HSDT.

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Figure 21
Dimensionless frequency of FGB when E ratio = 2 by HSDT.

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Figure 22
Dimensionless frequency of FGB when E ratio = 4 by HSDT.

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Figure 23
Dimensionless frequency of FGB when E ratio = 0.25 by Ansys.

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Figure 24
Dimensionless frequency of FGB when E ratio = 0.5 by Ansys.

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Figure 25
Dimensionless frequency of FGB when E ratio = 1 by Ansys.

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Figure 26
Dimensionless frequency of FGB when E ratio = 2 by Ansys.

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Figure 27
Dimensionless frequency of FGB when E ratio = 4 by Ansys.

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Figure 28
Dimensionless frequency of FGB when L/h = 5 by CBT.

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Figure 29
Dimensionless frequency of FGB when L/h = 5 by CBT.

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Figure 30
Dimensionless frequency of FGB when L/h = 5 by FSDT.

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Figure 31
Dimensionless frequency of FGB when L/h = 5 by FSDT.

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Figure 32
Dimensionless frequency of FGB when L/h = 5 by HSDT.

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Figure 33
Dimensionless frequency of FGB when L/h = 5 by HSDT.

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Figure 34
Dimensionless frequency of FGB when L/h = 5 by Ansys.

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Figure 35
Dimensionless frequency of FGB when L/h = 5 by Ansys.

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Figure 36
Dimensionless frequency of FGB when L/h = 5 and E ratio = 0.25 for all theories.

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Figure 37
Dimensionless frequency of FGB when L/h = 5 and E ratio = 0.333 for all theories.

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Figure 38
Dimensionless frequency of FGB when L/h = 5 and E ratio = 0.5 for all theories.

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Figure 39
Dimensionless frequency of FGB when L/h = 5 and E ratio = 0.75 for all theories.

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Figure 40
Dimensionless frequency of FGB when L/h = 5 and E ratio = 1 for all theories.

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Figure 41
Dimensionless frequency of FGB when L/h = 5, E ratio = 1.333 for all theories.

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Figure 42
Dimensionless frequency of FGB when L/h = 5, E ratio = 2 for all theories.

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Figure 43
Dimensionless frequency of FGB when L/h = 5, E ratio = 3 for all theories.

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Figure 44
Dimensionless frequency of FGB when L/h = 5 E ratio = 4 for all theories.

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Table 1
The material properties used.

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Table 2
Validation of dimension less natural frequency for s-s FGB when L/H = 20.

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Table 3
Validation of dimension less natural frequency for s-s FGB when L/H = 100.

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Property	Steel	Al₂O₃
Modulus of elasticity (E) GPas	210	390
Poison ratio (ν)	0.3	0.3
Density (ρ) kg·m^–3	7800	3960

ER	Author	Index
ER	Author	0.1	0.2	0.5	1	2	5	10
0.25	Present Ansys – Shell Model	2.3374	2.414	2.571	2.715	2.846	2.947	3.005
	Present CBT	2.3733	2.460	2.596	2.702	2.804	2.928	3.006
	Present FSDT	2.3697	2.456	2.592	2.698	2.800	2.924	3.002
	Present HSDT	2.3726	2.459	2.595	2.701	2.803	2.927	3.006
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	2.1571	2.232	2.386	2.528	2.657	2.765	2.802
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	2.3739	2.460	--	2.703	2.805	--	3.008
0.5	Present Ansys – Shell Model	2.6526	2.689	2.770	2.850	2.927	3.005	3.029
	Present CBT	2.7093	2.756	2.834	2.893	2.944	3.009	3.054
	Present FSDT	2.7052	2.751	2.830	2.888	2.940	3.005	3.050
	Present HSDT	2.7085	2.755	2.833	2.892	2.943	3.008	3.053
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	2.4544	2.489	2.567	2.647	2.724	2.792	2.816
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	2.7104	2.757	--	2.894	2.945	--	3.056
1	Present Ansys – Shell Model	3.0669	3.066	3.066	3.066	3.066	3.066	3.066
	Present CBT	3.1383	3.138	3.138	3.138	3.138	3.138	3.138
	Present FSDT	3.1336	3.133	3.133	3.133	3.133	3.133	3.133
	Present HSDT	3.1374	3.137	3.137	3.137	3.137	3.137	3.137
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	2.8442	2.844	2.844	2.844	2.844	2.844	2.844
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	3.1399	3.139	3.139	3.139	3.139	3.139	3.139
2	Present Ansys – Shell Model	3.589	3.558	3.479	3.389	3.289	3.182	3.135
	Present CBT	3.675	3.628	3.527	3.440	3.374	3.317	3.270
	Present FSDT	3.6699	3.622	3.522	3.435	3.369	3.312	3.265
	Present HSDT	3.6742	3.627	3.526	3.439	3.373	3.316	3.269
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	3.335	3.306	3.233	3.146	3.045	2.936	2.894
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	3.6775	3.630	--	3.442	3.376	-	3.272
4	Present Ansys – Shell Model	4.2329	4.174	4.023	3.840	3.620	3.370	3.251
	Present CBT	4.3344	4.243	4.0324	3.821	3.647	3.530	3.4529
	Present FSDT	4.3280	4.237	4.0269	3.816	3.642	3.524	3.447
	Present HSDT	4.3331	4.242	4.0312	3.820	3.646	3.529	3.4519
	(Marzoq and Al-Ansari 2021Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1... )	3.9361	3.882	3.745	3.572	3.353	3.089	2.981
	(Şimşek and Kocatürk 2009Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200... )	4.337	4.245	-	3.823	3.648	-	3.454

ER	Author	Index
ER	Author	0.1	0.2	0.5	1	2	5	10

0.25	Present Ansys Shell Model	2.482	2.546	2.6717	2.775	2.861	2.952	3.0108
	Present CBT	2.374	2.461	2.5980	2.703	2.805	2.930	3.008
	Present FSDT	2.374	2.461	2.5976	2.703	2.805	2.930	3.008
	Present HSDT	2.373	2.460	2.5976	2.703	2.805	2.930	3.008
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	2.504	2.569	2.6976	2.805	2.898	2.995	3.053
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	2.375	2.462	--	2.705	2.807	--	3.01
0.5	Present Ansys Shell Model	2.758	2.791	2.8593	2.917	2.966	3.020	3.056
	Present CBT	2.710	2.757	2.8361	2.894	2.946	3.010	3.056
	Present FSDT	2.710	2.757	2.8360	2.894	2.945	3.010	3.056
	Present HSDT	2.710	2.757	2.8362	2.894	2.946	3.010	3.056
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	2.776	2.810	2.8782	2.937	2.989	3.044	3.080
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	2.711	2.758	--	2.896	2.947	--	3.057
1	Present Ansys Shell Model	3.140	3.140	3.1405	3.140	3.140	3.140	3.140
	Present CBT	3.140	3.140	3.1400	3.140	3.140	3.140	3.140
	Present FSDT	3.139	3.139	3.1397	3.139	3.139	3.139	3.139
	Present HSDT	3.139	3.139	3.1398	3.139	3.139	3.139	3.139
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	3.163	3.163	3.1632	3.163	3.163	3.163	3.163
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	3.141	3.141	3.1415	3.141	3.141	3.141	3.141
2	Present Ansys Shell Model	3.640	3.608	3.5381	3.469	3.407	3.339	3.287
	Present CBT	3.677	3.629	3.5298	3.442	3.376	3.319	3.272
	Present FSDT	3.676	3.629	3.5293	3.442	3.376	3.319	3.272
	Present HSDT	3.676	3.629	3.5292	3.442	3.376	3.319	3.272
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	3.679	3.636	3.5644	3.493	3.456	3.383	3.332
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	3.679	3.632	--	3.444	3.378	--	3.274
4	Present Ansys Shell Model	4.268	4.208	4.0693	3.924	3.787	3.637	3.523
	Present CBT	4.336	4.245	4.0345	3.823	3.649	3.532	3.454
	Present FSDT	4.336	4.245	4.0342	3.823	3.649	3.532	3.454
	Present HSDT	4.335	4.244	4.0338	3.823	3.648	3.532	3.454
	Marzoq and Al-Ansari (2021)Marzoq Z, Al-Ansari LS (2021) Calculating the fundamental frequency of power law functionally graded beam using Ansys software. IOP Conf Ser: Mater Sci Eng 1090:012014. https://doi.org/10.1088/1757-899X/1090/1/012014 https://doi.org/10.1088/1757-899X/1090/1...	4.345	4.284	4.1394	3.984	3.831	3.669	3.557
	Şimşek and Kocatürk (2009)Şimşek M, Kocatürk T (2009) Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Compos Struct 90(4):465-473. https://doi.org/10.1016/j.compstruct.2009.04.024 https://doi.org/10.1016/j.compstruct.200...	4.339	4.248	--	3.825	3.651	-	3.456

\begin{aligned} δ K = \int ρ (z) \times [((\frac{\partial u}{\partial t}) - (z \times \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) + (f (z) \times (\frac{\partial u_{1}}{\partial t})) \times (δ (\frac{\partial u}{\partial t}) - \\ (z \times δ \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) + (f (z) \times δ (\frac{\partial u_{1}}{\partial t})) + ((\frac{\partial w}{\partial t}) \times δ (\frac{\partial w}{\partial t}))] d v = \int_{0}^{l} [δ (\frac{\partial u}{\partial t}) ((m_{00} (\frac{\partial u}{\partial t})) - \\ (m_{11} \times \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) + (m_{f} \times (\frac{\partial u_{1}}{\partial t}))) + δ \frac{\partial}{\partial x} (\frac{\partial w}{\partial t}) ((- m_{11} \times (\frac{\partial u}{\partial t})) + (m_{22} \times \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) - \\ (m_{f z} \times (\frac{\partial u_{1}}{\partial t}))) + δ (\frac{\partial u_{1}}{\partial t}) ((m_{f} \times (\frac{\partial u}{\partial t})) - (m_{f z} \times \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) + (m_{f^{2}} \times (\frac{\partial u_{1}}{\partial t}))) + \\ (δ (\frac{\partial w}{\partial t}) \times (\frac{\partial w}{\partial t}))] d x \end{aligned}

\begin{matrix} δ (U_{i n t}) = \int [(σ_{x x} \times δ ε_{xx}) + (σ_{x z} \times δ γ_{x z})] d V = \int [σ_{x x} \times (δ \frac{d u}{d x} - z δ \frac{d^{2} w}{d x^{2}} + f (z) δ \frac{d u_{1}}{d x}) \\ + σ_{x z} \times (\frac{d f (z)}{d z} \times δ u_{1})] d V = \int_{0}^{l} [N \times δ \frac{d u}{d x} - M \times δ \frac{d^{2} w}{d x^{2}} + M^{s} \times δ \frac{d u_{1}}{d x} + Q^{s} \times δ u_{1}] d x \end{matrix}

\begin{matrix} \int_{t 1}^{t 2} \int_{0}^{l} [δ (\frac{\partial u}{\partial t}) ((m_{00} (\frac{\partial u}{\partial t})) - (m_{11} \times \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) + (m_{f} \times (\frac{\partial u_{1}}{\partial t}))) + δ \frac{\partial}{\partial x} (\frac{\partial w}{\partial t}) ((- m_{11} \times (\frac{\partial u}{\partial t})) + \\ (m_{22} \times \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) - (m_{f z} \times (\frac{\partial u_{1}}{\partial t}))) + δ (\frac{\partial u_{1}}{\partial t}) ((m_{f} \times (\frac{\partial u}{\partial t})) - (m_{f z} \times \frac{\partial}{\partial x} (\frac{\partial w}{\partial t})) + \\ (m_{f^{2}} \times (\frac{\partial u_{1}}{\partial t}))) + (m_{00} \times δ (\frac{\partial w}{\partial t}) \times (\frac{\partial w}{\partial t})) - (N \times δ \frac{d u}{d x} - M \frac{d^{2} w}{d x^{2}} + M^{s} \times δ \frac{d u_{1}}{d x} + Q^{s} \times δ u_{1})] dx d t = 0 \end{matrix}

((\frac{d^{2} M}{d x^{2}})) = (m_{11} \times \frac{\partial}{\partial x} (\frac{\partial^{2} u}{\partial t^{2}})) - (m_{22} \times \frac{\partial^{2}}{\partial x^{2}} (\frac{\partial^{2} w}{\partial t^{2}})) + (m_{f z} \times \frac{\partial}{\partial x} (\frac{\partial^{2} u_{1}}{\partial t^{2}})) + (m_{00} \times (\frac{\partial^{2} w}{\partial t^{2}}))

\begin{aligned} (A_{11} \times \frac{d^{2} u}{d x^{2}}) - (B_{11} \times \frac{d^{3} w}{d x^{3}}) + (E_{11} \times \frac{d^{2} u_{1}}{d x^{2}}) = (m_{00} (\frac{\partial^{2} u}{\partial t^{2}})) - (m_{11} \times \frac{\partial}{\partial x} (\frac{\partial^{2} w}{\partial t^{2}})) + \\ (m_{f} \times (\frac{\partial^{2} u_{1}}{\partial t^{2}})) \end{aligned}

\begin{matrix} (B_{11} \times \frac{d^{3} u}{d x^{3}}) - (D_{11} \times \frac{d^{4} w}{d x^{4}}) + (F_{11} \times \frac{d^{3} u_{1}}{d x^{3}}) = (m_{11} \times \frac{\partial}{\partial x} (\frac{\partial^{2} u}{\partial t^{2}})) - \\ (m_{22} \times \frac{\partial^{2}}{\partial x^{2}} (\frac{\partial^{2} w}{\partial t^{2}})) + (m_{f z} \times \frac{\partial}{\partial x} (\frac{\partial^{2} u_{1}}{\partial t^{2}})) + (m_{00} \times (\frac{\partial^{2} w}{\partial t^{2}})) \end{matrix}

\begin{matrix} (E_{11} \times \frac{d^{2} u}{d x^{2}}) - (F_{11} \times \frac{d^{3} w}{d x^{3}}) + (H_{11} \times \frac{d^{2} u_{1}}{d x^{2}}) - (A_{55} \times u_{1}) = (m_{f} \times (\frac{\partial^{2} u}{\partial t^{2}})) - \\ (m_{f z} \times \frac{\partial}{\partial x} (\frac{\partial^{2} w}{\partial t^{2}})) + (m_{f^{2}} \times (\frac{\partial^{2} u_{1}}{\partial t^{2}})) \end{matrix}

\begin{aligned} ([\begin{array}{ccc} (- α^{2} \times A_{11}) & (α^{3} \times B_{11}) & (- α^{2} \times E_{11}) \\ (α^{3} \times B_{11}) & (- α^{4} \times D_{11}) & (F_{11} \times α^{3}) \\ (- α^{2} \times E_{11}) & (F_{11} \times α^{3}) & (- α^{2} \times H_{11} - A_{S S} \times K_{S}) \end{array}] - \\ {ω_{n}}^{2} [\begin{array}{ccc} m_{00} & - α \times m_{11} & m_{f} \\ - α \times m_{11} & m_{22} \times α^{2} + m_{00} & - m_{f z} \times α \\ m_{f} & - m_{f z} \times α & m_{f^{2}} \end{array}]) [\begin{matrix} U_{n} \\ W_{n} \\ G_{n} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] \end{aligned}

[\begin{array}{cc} (- α^{2} \times A_{11}) & (α^{3} \times B_{11}) \\ (α^{3} \times B_{11}) & (- α^{4} \times D_{11} + α^{2} \times P) \end{array}] - {ω_{n}}^{2} [\begin{array}{cc} m_{00} & - m_{11} \times α \\ - m_{11} \times α & m_{22} \times α^{2} \end{array}] [\begin{array}{l} U_{n} \\ W_{n} \end{array}] = [\begin{array}{l} 0 \\ 0 \end{array}]

[1] *Correspondence author: ragada.deibel@uokufa.edu.iq

Brasil

Brasil

Modeling and Analyzing the Free Vibration of Simply Supported Functionally Graded Beam

ABSTRACT