“…The 3rd- and 4th-mode frequencies converge to μ3=50.3926 and to μ4=99.0805 after n=10 and n=11. It can be seen from the figure that iteration steps necessitating converged first four frequencies are largely reduced in comparison with other approximate methods by Shahba and Rajasekaran (2012) and by Ghazaryan et al (2018).Figure 3.Convergence of dimensionless natural frequencies (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode (cb=ch=0.2) for example 1 under C–F boundary condition.…”
Section: Numerical Examplementioning
confidence: 87%
“…The 3rd-and 4th-mode frequencies converge to μ 3 ¼ 50:3926 and to μ 4 ¼ 99:0805 after n ¼ 10 and n ¼ 11. It can be seen from the figure that iteration steps necessitating converged first four frequencies are largely reduced in comparison with other approximate methods by Shahba and Rajasekaran (2012) and by Ghazaryan et al (2018).…”
Section: Numerical Example 41 Numerical Examplementioning
confidence: 88%
“…Another contribution was made by Rajasekaran (2013) who investigated free vibration of nonrotating and rotating AFG tapered beams using the differential quadrature element method of lowest order (DQEL) as well as the differential transformation method (DTM). Similarly, the DTM was also used by Ghazaryan et al (2018) to examine the vibrational behavior of AFG tapered beams. Another attempt was made by Liu et al (2016) who used the spline finite point method (SFPM) to investigate the vibrational behavior of AFG beams.…”
Tapered beams constitute functionally graded materials that are widely used in various engineering fields. The vibrational characteristics of tapered beams made of axially varying functionally graded materials are investigated via variational iteration method. Natural frequencies and corresponding mode shapes of axially functionally graded tapered beams are examined, and calculated frequencies are compared with results provided by previous researchers using other approximate methods. The efficiency and accuracy of this technique are demonstrated. The effect of nonhomogeneous material distribution, taper ratio as well as boundary conditions on the dynamical behavior of axially functionally graded tapered beams is thoroughly investigated as well.
“…The 3rd- and 4th-mode frequencies converge to μ3=50.3926 and to μ4=99.0805 after n=10 and n=11. It can be seen from the figure that iteration steps necessitating converged first four frequencies are largely reduced in comparison with other approximate methods by Shahba and Rajasekaran (2012) and by Ghazaryan et al (2018).Figure 3.Convergence of dimensionless natural frequencies (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode (cb=ch=0.2) for example 1 under C–F boundary condition.…”
Section: Numerical Examplementioning
confidence: 87%
“…The 3rd-and 4th-mode frequencies converge to μ 3 ¼ 50:3926 and to μ 4 ¼ 99:0805 after n ¼ 10 and n ¼ 11. It can be seen from the figure that iteration steps necessitating converged first four frequencies are largely reduced in comparison with other approximate methods by Shahba and Rajasekaran (2012) and by Ghazaryan et al (2018).…”
Section: Numerical Example 41 Numerical Examplementioning
confidence: 88%
“…Another contribution was made by Rajasekaran (2013) who investigated free vibration of nonrotating and rotating AFG tapered beams using the differential quadrature element method of lowest order (DQEL) as well as the differential transformation method (DTM). Similarly, the DTM was also used by Ghazaryan et al (2018) to examine the vibrational behavior of AFG tapered beams. Another attempt was made by Liu et al (2016) who used the spline finite point method (SFPM) to investigate the vibrational behavior of AFG beams.…”
Tapered beams constitute functionally graded materials that are widely used in various engineering fields. The vibrational characteristics of tapered beams made of axially varying functionally graded materials are investigated via variational iteration method. Natural frequencies and corresponding mode shapes of axially functionally graded tapered beams are examined, and calculated frequencies are compared with results provided by previous researchers using other approximate methods. The efficiency and accuracy of this technique are demonstrated. The effect of nonhomogeneous material distribution, taper ratio as well as boundary conditions on the dynamical behavior of axially functionally graded tapered beams is thoroughly investigated as well.
“…An approach implementing the differential transform method for free vibration analysis of nonuniform cross-section of functionally graded beams has been presented, e.g. in [41,42].…”
A three-dimensional modelling of free vibrations and static response of functionally graded material (FGM) sandwich plates is presented. Natural frequencies and associated mode shapes as well as displacements and stresses are determined by using the finite element method within the ABAQUS TM code. The three-dimensional (3-D) brick graded finite element is programmed and incorporated into the code via the user-defined material subroutine UMAT. The results of modal and static analyses are demonstrated for square metal-ceramic functionally graded simply supported plates with a power-law through-the-thickness variation of the volume fraction of the ceramic constituent. The through-the-thickness distribution of effective material properties at a point are defined based on the Mori-Tanaka scheme. First, exact values of displacements, stresses and natural frequencies available for FGM sandwich plates in the literature are used to verify the performance and estimate the accuracy of the developed 3-D graded finite element. Then, parametric studies are carried out for the frequency analysis by varying the volume fraction profile and value of the ceramic volume fraction.
“…Usually, such sandwich configuration is achieved by gradually changing the volume fraction of the FGM constituents across the sandwich plate thickness from the bottom face sheet to the top face sheet [23]. This removes the interface stress concentration and allows controlling deformation, dynamic responses, and etc, by customizing the gradation profile [24,25,26]. Another characteristic of FGM sandwich panels, with metal and ceramic material, is that they can be used at elevated service temperatures [27,28].…”
Using the finite element code ABAQUS and the user-defined material utilities UMAT and UMATHT, a solid brick graded finite element is developed for three-dimensional (3D) modeling of free vibrations of thermally loaded functionally gradient material (FGM) sandwich plates. The mechanical and thermal material properties of the FGM sandwich plates are assumed to vary gradually in the thickness direction, according to a power-law fraction distribution. Benchmark problems are firstly considered to assess the performance and accuracy of the proposed 3D graded finite element. Comparisons with the reference solutions revealed high efficiency and good capabilities of the developed element for the 3D simulations of thermomechanical and vibration responses of FGM sandwich plates. Some parametric studies are carried out for the frequency analysis by varying the volume fraction profile and the temperature distribution across the plate thickness.
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