Linear and non-linear vibration analysis of moderately thick isosceles triangular FGPs using a triangular finite p-element
- SA. Belalia^{1}Email author
https://doi.org/10.1186/s40759-017-0018-0
© The Author(s). 2017
Received: 2 September 2016
Accepted: 12 January 2017
Published: 1 February 2017
Abstract
Background
The geometrically non-linear formulation based on Von-Karman’s hypothesis is used to study the free vibration isosceles triangular plates by using four types of mixtures of functionally graded materials (FGMs - AL/AL_{2}O_{3}, SUS304/Si_{3}N_{4}, Ti- AL-4V/Aluminum oxide, AL/ZrO_{2}). Material properties are assumed to be temperature dependent and graded in the thickness direction according to power law distribution.
Methods
A hierarchical finite element based on triangular p-element is employed to define the model, taking into account the hypotheses of first-order shear deformation theory. The equations of non-linear free motion are derived from Lagrange's equation in combination with the harmonic balance method and solved iteratively using the linearized updated mode method.
Results
Results for the linear and nonlinear frequencies parameters of clamped isosceles triangular plates are obtained. The accuracy of the present results are established through convergence studies and comparison with results of literature for metallic plates. The results of the linear vibration of clamped FGMs isosceles triangular plates are also presented in this study.
Conclusion
The effects of apex angle, thickness ratio, volume fraction exponent and mixtures of FGMs on the backbone curves and mode shape of clamped isosceles triangular plates are studied. The results obtained in this work reveal that the physical and geometrical parameters have a important effect on the non-linear vibration of FGMs triangular plates.
Keywords
The mixtures effect of Ceramic-Metal Linear and Non-linear vibration Moderately thick FGM plates p-version of finite element methodBackground
In recent years, the geometrically non-linear vibration of functionally graded Materials (FGMs) for different structures has acquired great interest in many researches. In 1984, The concept of the FGMs was introduced in Japan by scientific researchers (Koizumi 1993; Koizumi 1997). FGMs are composite materials which are microscopically inhomogeneous. The mechanical properties of FGMs are expressed with mathematical functions, and assumed to vary continuously from one surface to the other.
Since the variation of mechanical properties of FGM is nonlinear, therefore, studies based on the nonlinear deformation theory is required for these type of materials. Many works have studied the static and dynamic nonlinear behavior of functionally graded plates with various shapes. The group of researchers headed by (Reddy and Chin 1998; Reddy et al. 1999; Reddy 2000) have done a lot of numerical and theoretical work on FG plates under several effects (thermoelastic response, axisymmetric bending and stretching, finite element models, FSDT-plate and TSDT-plate). Woo & Meguid (2001) analyzed the nonlinear behavior of functionally graded shallow shells and thin plates under temperature effects and mechanical loads. The analysis of nonlinear bending of FG simply supported rectangular plates submissive to thermal and mechanical loading was studied by (Shen 2002). (Huang & Shen 2004) applied the perturbation technique to nonlinear vibration and dynamic response of FG plates in a thermal environment. Chen (2005) investigated the large amplitude vibration of FG plate with arbitrary initial stresses based on FSDT. An analytical solution was proposed by Woo et al. 2006 to analyzed the nonlinear vibration of functionally graded plates using classic plate theory. Allahverdizadeh et al. (2008a, 2008b) have studied the non-linear forced and free vibration analysis of circular functionally graded plate in thermal environment. The p-version of the FEM has been applied to investigate the non-linear free vibration of elliptic sector plates and functionally graded sector plates by (Belalia & Houmat 2010; 2012). Hao et al. 2011 analyzed the non-linear vibration of a cantilever functionally graded plate based on TSDT of plate and asymptotic analysis and perturbation method. Duc & Cong 2013 analyzed the non-linear dynamic response of imperfect symmetric thin sandwich FGM plate on elastic foundation. Yin et al. 2015 proposed a novel approach based on isogeometric analysis (IGA) for the geometrically nonlinear analysis of functionally graded plates (FGPs). the same approach (IGA) and a simple first-order shear deformation plate theory (S-FSDT) are used by Yu et al. 2015 to investigated geometrically nonlinear analysis of homogeneous and non-homogeneous functionally graded plates. Alinaghizadeh & Shariati 2016, investigated the non-linear bending analysis of variable thickness two-directional FG circular and annular sector plates resting on the non-linear elastic foundation using the generalized differential quadrature (GDQ) and the Newton–Raphson iterative methods.
The p-version FEM has many advantages over the classic finite element method (h-version), which includes the ability to increase the accuracy of the solution without re-defining the mesh (Han & Petyt 1997; Ribeiro 2003). This advantage is suitable in non-linear study because the problem is solved iteratively and the non-linear stiffness matrices are reconstructed throughout each iteration. Using the p-version with higher order polynomials, the structure is modeled by one element while satisfying the exactitude requirement. In p-version, the point where the maximum amplitude is easy to find it as there is a single element, contrary to the h-version this point must be sought in every element of the mesh which is very difficult. The advantages of the p-version mentioned previously, make it more powerful to the nonlinear vibration analysis of plates. So far, no work has been published to the study of linear and nonlinear vibration of FGMs isosceles triangular plate by using the p-version of FEM.
In the present work, the non-linear vibration analysis of moderately thick FGMs isosceles triangular plates was investigated by a triangular finite p-element. The shape functions of triangular finite p-element are obtained by the shifted orthogonal polynomials of Legendre. The effects of rotatory inertia and transverse shear deformations are taken into account (Mindlin 1951). The Von-Karman hypothesis are used in combination with the harmonic balance method (HBM) to obtained the motion equations. The resultant equations of motion are solved iteratively using the linearized updated mode method. The exactitude of the p-element is investigated with a clamped metallic triangular plate. Comparisons are made between current results and those from published results. The effects of thickness ratio, apex angle, exponent of volume fraction and mixtures of FGMs on the backbone curves and mode shape of clamped isosceles triangular plates are also studied.
Methods
where q _{ u }, q _{ v } are the vectors of generalized in-plane displacements, q _{ w }, \( {q}_{\theta_y} \) and \( {q}_{\theta_x} \) are the vectors of generalized transverse displacement and rotations, respectively, N(ξ, η) are the hierarchical shape functions of triangular p-element (Belalia & Houmat 2010).
Material | Properties | ||
---|---|---|---|
E (10^{9} N/m^{2}) | ν | ρ (kg/m^{3}) | |
Aluminium (Al) | 70.00 | 0.30 | 2707 |
Alumina (Al_{2}O_{3}) | 380.00 | 0.30 | 3800 |
Stainless steel SUS304 | 207.78 | 0.3177 | 8166 |
Silicon nitride Si_{3}N_{4} | 322.27 | 0.24 | 2370 |
Ti-6AL-4 V | 105.7 | 0.2981 | 4429 |
Aluminum oxide | 320.24 | 0.26 | 3750 |
Zirconia (ZrO_{2}) | 151.00 | 0.30 | 3000 |
Where M is the out-of-plane inertia matrices, \( \overline{\boldsymbol{K}} \), K and \( \overset{\smile }{\boldsymbol{K}} \) are the extension, bending and coupled extension-rotation linear stiffness matrices, \( \tilde{\boldsymbol{K}} \) and \( \widehat{\boldsymbol{K}} \) represent the nonlinear stiffness matrices. These matrices are given in Appendix A.
The system of equations given in Eq. (23) are solved iteratively using the linearized updated mode method. This method needs two type of amplitudes, the first is the specific amplitude which depends on the plate thickness, the second is the maximum amplitude to be calculated for each iteration. The new system of equations is solved using any known technique with an accuracy of around (e.g.10^{−5}).
Results
Study of convergence and comparison for linear vibration
In this part a convergence and comparison study is made for the linear vibration of clamped metallic isosceles triangular plates to validate the current formulation and methods proposed.
Convergence of the first three linear frequency parameters for clamped metallic isosceles triangular plate (β = 90°)
h/b | Mode | p | |||||
---|---|---|---|---|---|---|---|
6 | 7 | 8 | 9 | 10 | 11 | ||
0.05 | Ω_{1} | 166.3 | 164.6 | 164.4 | 164.3 | 164.3 | 164.3 |
Ω_{2} | 277.2 | 265.7 | 261.9 | 261.1 | 260.9 | 260.9 | |
Ω_{3} | 330.4 | 321.2 | 316.3 | 314.3 | 313.8 | 313.7 | |
0.1 | Ω_{1} | 128.2 | 127.9 | 127.9 | 127.9 | 127.9 | 127.9 |
Ω_{2} | 195.1 | 191.2 | 190.5 | 190.2 | 190.2 | 190.2 | |
Ω_{3} | 227.9 | 225.0 | 223.6 | 223.3 | 223.2 | 223.2 | |
0.15 | Ω_{1} | 100.3 | 100.2 | 100.2 | 100.2 | 100.2 | 100.2 |
Ω_{2} | 146.1 | 144.3 | 144.1 | 144.0 | 144.0 | 144.0 | |
Ω_{3} | 168.7 | 167.4 | 166.8 | 166.7 | 166.7 | 166.7 |
Comparison of the first three linear frequency parameters for clamped metallic isosceles triangular plate
h/b | Mode | β | |||||
---|---|---|---|---|---|---|---|
30° | 60° | 90° | |||||
Present | Liew et al. (1998) | Present | Liew et al. (1998) | Present | Liew et al. (1998) | ||
0.05 | Ω_{1} | 51.55 | 51.55 | 91.86 | 91.86 | 164.3 | 164.4 |
Ω_{2} | 80.60 | 80.61 | 167.5 | 167.5 | 260.9 | 260.9 | |
Ω_{3} | 109.5 | 109.5 | 167.5 | 167.5 | 313.7 | 313.7 | |
0.1 | Ω_{1} | 46.35 | 46.35 | 77.76 | 77.79 | 127.9 | 127.9 |
Ω_{2} | 69.82 | 69.81 | 132.3 | 132.3 | 190.2 | 190.3 | |
Ω_{3} | 92.16 | 92.17 | 132.3 | 132.3 | 223.1 | 223.2 | |
0.15 | Ω_{1} | 40.55 | 40.55 | 64.58 | 64.59 | 100.2 | 100.2 |
Ω_{2} | 59.07 | 59.07 | 104.7 | 104.7 | 143.0 | 144.0 | |
Ω_{3} | 76.12 | 76.15 | 104.7 | 104.7 | 166.7 | 166.7 |
Linear vibration of FGMs isosceles triangular plate
The first three linear frequency parameters of clamped FG AL/AL_{2}O_{3} isosceles triangular plate
β | h/b | Mode | n | |||||||
---|---|---|---|---|---|---|---|---|---|---|
ceramic | 0.1 | 0.5 | 1 | 2 | 5 | 10 | metal | |||
30° | 0.05 | Ω_{L1} | 5.0681 | 4.8827 | 4.3399 | 3.9585 | 3.6367 | 3.4112 | 3.2521 | 2.5773 |
Ω_{L2} | 7.9247 | 7.6385 | 6.8050 | 6.2214 | 5.7244 | 5.3517 | 5.0840 | 4.0300 | ||
Ω_{L3} | 10.768 | 10.382 | 9.2453 | 8.4385 | 7.7440 | 7.2297 | 6.8741 | 5.4761 | ||
0.1 | Ω_{L1} | 9.1144 | 8.7992 | 7.8656 | 7.1908 | 6.5871 | 6.0926 | 5.7606 | 4.6349 | |
Ω_{L2} | 13.729 | 13.267 | 11.904 | 10.916 | 10.009 | 9.1978 | 8.6491 | 6.9818 | ||
Ω_{L3} | 18.122 | 17.522 | 15.727 | 14.401 | 13.158 | 12.049 | 11.330 | 9.2160 | ||
0.15 | Ω_{L1} | 11.960 | 11.571 | 10.404 | 9.5364 | 8.7114 | 7.9458 | 7.4521 | 6.0824 | |
Ω_{L2} | 17.422 | 16.873 | 15.237 | 14.013 | 12.812 | 11.599 | 10.814 | 8.8598 | ||
Ω_{L3} | 22.452 | 21.759 | 19.656 | 18.046 | 16.436 | 14.823 | 13.822 | 11.418 | ||
60° | 0.05 | Ω_{L1} | 9.0322 | 8.7089 | 7.7700 | 7.1137 | 6.5499 | 6.1081 | 5.7894 | 4.5931 |
Ω_{L2} | 16.471 | 15.893 | 14.197 | 12.987 | 11.922 | 11.066 | 10.472 | 8.3760 | ||
Ω_{L3} | 16.471 | 15.895 | 14.220 | 13.039 | 11.998 | 11.121 | 10.501 | 8.3760 | ||
0.1 | Ω_{L1} | 15.290 | 14.785 | 13.290 | 12.200 | 11.179 | 10.225 | 9.5914 | 7.7757 | |
Ω_{L2} | 26.023 | 25.197 | 22.727 | 20.880 | 19.081 | 17.304 | 16.159 | 13.233 | ||
Ω_{L3} | 26.023 | 25.197 | 22.734 | 20.895 | 19.098 | 17.307 | 16.159 | 13.233 | ||
0.15 | Ω_{L1} | 19.048 | 18.460 | 16.696 | 15.361 | 14.020 | 12.629 | 11.751 | 9.6867 | |
Ω_{L2} | 30.889 | 29.973 | 27.193 | 25.036 | 22.785 | 20.357 | 18.876 | 15.708 | ||
Ω_{L3} | 30.889 | 29.974 | 27.208 | 25.070 | 22.840 | 20.408 | 18.904 | 15.708 | ||
90° | 0.05 | Ω_{L1} | 16.158 | 15.597 | 13.957 | 12.793 | 11.760 | 10.885 | 10.273 | 8.2171 |
Ω_{L2} | 25.654 | 24.788 | 22.260 | 20.454 | 18.802 | 17.278 | 16.227 | 13.046 | ||
Ω_{L3} | 30.842 | 29.806 | 26.705 | 24.420 | 22.326 | 20.519 | 19.338 | 15.684 | ||
0.1 | Ω_{L1} | 25.141 | 24.354 | 22.002 | 20.236 | 18.487 | 16.701 | 15.561 | 12.785 | |
Ω_{L2} | 37.407 | 36.275 | 32.868 | 30.265 | 27.597 | 24.749 | 22.981 | 19.022 | ||
Ω_{L3} | 43.879 | 42.565 | 38.556 | 35.441 | 32.237 | 28.877 | 26.830 | 22.314 | ||
0.15 | Ω_{L1} | 29.543 | 28.682 | 26.067 | 24.026 | 21.861 | 19.453 | 17.991 | 15.023 | |
Ω_{L2} | 42.470 | 41.260 | 37.557 | 34.620 | 31.440 | 27.861 | 25.732 | 21.597 | ||
Ω_{L3} | 49.166 | 47.778 | 43.507 | 40.090 | 36.371 | 32.198 | 29.726 | 25.002 |
The first three linear frequency parameters of clamped FG SUS304/Si_{3}N_{4} isosceles triangular plate
β | h/b | Mode | n | |||||||
---|---|---|---|---|---|---|---|---|---|---|
ceramic | 0.1 | 0.5 | 1 | 2 | 5 | 10 | metal | |||
30° | 0.05 | Ω_{L1} | 5.8366 | 5.1789 | 4.0506 | 3.5610 | 3.2026 | 2.9134 | 2.7785 | 2.5747 |
Ω_{L2} | 9.1371 | 8.1064 | 6.3395 | 5.5721 | 5.0090 | 4.5535 | 4.3414 | 4.0244 | ||
Ω_{L3} | 12.427 | 11.024 | 8.6199 | 7.5746 | 6.8066 | 6.1846 | 5.8959 | 5.4669 | ||
0.1 | Ω_{L1} | 10.564 | 9.3705 | 7.3225 | 6.4269 | 5.7626 | 5.2220 | 4.9759 | 4.6207 | |
Ω_{L2} | 15.956 | 14.149 | 11.053 | 9.6966 | 8.6850 | 7.8582 | 7.4845 | 6.9544 | ||
Ω_{L3} | 21.102 | 18.708 | 14.609 | 12.811 | 11.465 | 10.363 | 9.8682 | 9.1743 | ||
0.15 | Ω_{L1} | 13.954 | 12.374 | 9.6609 | 8.4647 | 7.5657 | 6.8293 | 6.5019 | 6.0512 | |
Ω_{L2} | 20.390 | 18.075 | 14.106 | 12.353 | 11.028 | 9.9374 | 9.4558 | 8.8055 | ||
Ω_{L3} | 26.332 | 23.336 | 18.2050 | 15.934 | 14.212 | 12.793 | 12.170 | 11.341 | ||
60° | 0.05 | Ω_{L1} | 10.423 | 9.2477 | 7.2314 | 6.3544 | 5.7096 | 5.1871 | 4.9454 | 4.5855 |
Ω_{L2} | 19.056 | 16.902 | 13.210 | 11.601 | 10.413 | 9.4482 | 9.0049 | 8.3552 | ||
Ω_{L3} | 19.056 | 16.902 | 13.211 | 11.603 | 10.415 | 9.4492 | 9.0053 | 8.3552 | ||
0.1 | Ω_{L1} | 17.805 | 15.790 | 12.331 | 10.810 | 9.6719 | 8.7402 | 8.3229 | 7.7404 | |
Ω_{L2} | 30.433 | 26.975 | 21.052 | 18.440 | 16.469 | 14.849 | 14.131 | 13.156 | ||
Ω_{L3} | 30.433 | 26.975 | 21.053 | 18.441 | 16.470 | 14.850 | 14.132 | 13.156 | ||
0.15 | Ω_{L1} | 22.345 | 19.812 | 15.456 | 13.522 | 12.053 | 10.845 | 10.318 | 9.6205 | |
Ω_{L2} | 36.377 | 32.233 | 25.130 | 21.971 | 19.558 | 17.564 | 16.700 | 15.581 | ||
Ω_{L3} | 36.377 | 32.234 | 25.132 | 21.975 | 19.562 | 17.566 | 16.701 | 15.581 | ||
90° | 0.05 | Ω_{L1} | 18.711 | 16.598 | 12.972 | 11.388 | 10.216 | 9.2627 | 8.8272 | 8.1943 |
Ω_{L2} | 29.792 | 26.422 | 20.641 | 18.112 | 16.227 | 14.689 | 13.992 | 12.998 | ||
Ω_{L3} | 35.858 | 31.808 | 24.843 | 21.792 | 19.508 | 17.650 | 16.8101 | 15.621 | ||
0.1 | Ω_{L1} | 29.448 | 26.111 | 20.374 | 17.831 | 15.906 | 14.324 | 13.631 | 12.704 | |
Ω_{L2} | 43.964 | 38.964 | 30.385 | 26.577 | 23.679 | 21.289 | 20.248 | 18.882 | ||
Ω_{L3} | 51.630 | 45.755 | 35.676 | 31.196 | 27.781 | 24.961 | 23.736 | 22.141 | ||
0.15 | Ω_{L1} | 34.858 | 30.906 | 24.089 | 21.036 | 18.693 | 16.762 | 15.937 | 14.894 | |
Ω_{L2} | 50.223 | 44.496 | 34.666 | 30.273 | 26.895 | 24.098 | 22.903 | 21.397 | ||
Ω_{L3} | 58.180 | 51.535 | 40.150 | 35.064 | 31.148 | 27.898 | 26.509 | 24.766 |
The first three linear frequency parameters of clamped FG Ti-6AL-4 V/Aluminum oxide isosceles triangular plate
β | h/b | Mode | n | |||||||
---|---|---|---|---|---|---|---|---|---|---|
ceramic | 0.1 | 0.5 | 1 | 2 | 5 | 10 | metal | |||
30° | 0.05 | Ω_{L1} | 4.8290 | 4.6023 | 4.0178 | 3.6640 | 3.3821 | 3.1477 | 2.9915 | 2.5775 |
Ω_{L2} | 7.5568 | 7.2037 | 6.2958 | 5.7453 | 5.3021 | 4.9231 | 4.6728 | 4.0305 | ||
Ω_{L3} | 10.275 | 9.7962 | 8.5595 | 7.8050 | 7.1942 | 6.6742 | 6.3363 | 5.4770 | ||
0.1 | Ω_{L1} | 8.7224 | 8.3224 | 7.2851 | 6.6421 | 6.1042 | 5.6259 | 5.3293 | 4.6364 | |
Ω_{L2} | 13.162 | 12.565 | 11.018 | 10.053 | 9.2288 | 8.4668 | 8.0039 | 6.9847 | ||
Ω_{L3} | 17.397 | 16.613 | 14.569 | 13.281 | 12.167 | 11.137 | 10.527 | 9.2205 | ||
0.15 | Ω_{L1} | 11.497 | 10.982 | 9.6405 | 8.7886 | 8.0410 | 7.3389 | 6.9297 | 6.0858 | |
Ω_{L2} | 16.783 | 16.041 | 14.109 | 12.873 | 11.761 | 10.679 | 10.060 | 8.8657 | ||
Ω_{L3} | 21.659 | 20.709 | 18.216 | 16.602 | 15.134 | 13.708 | 12.913 | 11.426 | ||
60° | 0.05 | Ω_{L1} | 8.6182 | 8.2172 | 7.1865 | 6.5606 | 6.0523 | 5.6101 | 5.3205 | 4.5939 |
Ω_{L2} | 15.743 | 15.016 | 13.139 | 11.986 | 11.033 | 10.193 | 9.6614 | 8.3781 | ||
Ω_{L3} | 15.743 | 15.017 | 13.149 | 12.006 | 11.058 | 10.209 | 9.6685 | 8.3781 | ||
0.1 | Ω_{L1} | 14.679 | 14.018 | 12.303 | 11.225 | 10.290 | 9.4121 | 8.8890 | 7.7795 | |
Ω_{L2} | 25.055 | 23.944 | 21.047 | 19.198 | 17.544 | 15.951 | 15.037 | 13.242 | ||
Ω_{L3} | 25.055 | 23.944 | 21.049 | 19.201 | 17.547 | 15.951 | 15.037 | 13.242 | ||
0.15 | Ω_{L1} | 18.378 | 17.573 | 15.468 | 14.107 | 12.863 | 11.641 | 10.959 | 9.6939 | |
Ω_{L2} | 29.880 | 28.591 | 25.201 | 22.974 | 20.887 | 18.802 | 17.675 | 15.721 | ||
Ω_{L3} | 29.880 | 28.591 | 25.209 | 22.991 | 20.911 | 18.818 | 17.682 | 15.721 | ||
90° | 0.05 | Ω_{L1} | 15.453 | 14.743 | 12.911 | 11.784 | 10.845 | 10.002 | 9.4702 | 8.2194 |
Ω_{L2} | 24.582 | 23.465 | 20.582 | 18.795 | 17.269 | 15.851 | 14.981 | 13.051 | ||
Ω_{L3} | 29.577 | 28.236 | 24.742 | 22.547 | 20.676 | 18.974 | 17.953 | 15.691 | ||
0.1 | Ω_{L1} | 24.232 | 23.165 | 20.377 | 18.586 | 16.965 | 15.385 | 14.493 | 12.794 | |
Ω_{L2} | 36.137 | 34.565 | 30.445 | 27.765 | 25.284 | 22.822 | 21.468 | 19.038 | ||
Ω_{L3} | 42.422 | 40.584 | 35.747 | 32.574 | 29.626 | 26.711 | 25.127 | 22.333 | ||
0.15 | Ω_{L1} | 28.615 | 27.392 | 24.165 | 22.028 | 19.998 | 17.948 | 16.857 | 15.037 | |
Ω_{L2} | 41.198 | 39.447 | 34.821 | 31.734 | 28.767 | 25.752 | 24.171 | 21.619 | ||
Ω_{L3} | 47.715 | 45.694 | 40.348 | 36.768 | 33.310 | 29.790 | 27.951 | 25.028 |
The first three linear frequency parameters of clamped FG AL/ZrO_{2} isosceles triangular plate
β | h/b | Mode | n | |||||||
---|---|---|---|---|---|---|---|---|---|---|
ceramic | 0.1 | 0.5 | 1 | 2 | 5 | 10 | metal | |||
30° | 0.05 | Ω_{L1} | 3.0021 | 2.9572 | 2.8416 | 2.7888 | 2.7883 | 2.8317 | 2.8112 | 2.5773 |
Ω_{L2} | 4.6942 | 4.6257 | 4.4505 | 4.3711 | 4.3692 | 4.4266 | 4.3896 | 4.0299 | ||
Ω_{L3} | 6.3787 | 6.2869 | 6.0482 | 5.9351 | 5.9251 | 5.9981 | 5.9503 | 5.4760 | ||
0.1 | Ω_{L1} | 5.3989 | 5.3268 | 5.1359 | 5.0398 | 5.0174 | 5.0496 | 5.0015 | 4.6348 | |
Ω_{L2} | 8.1325 | 8.0301 | 7.7590 | 7.6194 | 7.5751 | 7.5901 | 7.5050 | 6.9816 | ||
Ω_{L3} | 10.735 | 10.604 | 10.250 | 10.056 | 9.9768 | 9.9757 | 9.8654 | 9.2158 | ||
0.15 | Ω_{L1} | 7.0850 | 7.0016 | 6.7739 | 6.6473 | 6.5896 | 6.5742 | 6.4964 | 6.0823 | |
Ω_{L2} | 10.320 | 10.207 | 9.8999 | 9.7224 | 9.6219 | 9.5516 | 9.4211 | 8.8596 | ||
Ω_{L3} | 13.300 | 13.161 | 12.769 | 12.526 | 12.368 | 12.251 | 12.086 | 11.418 | ||
60° | 0.05 | Ω_{L1} | 5.3502 | 5.2734 | 5.0781 | 4.9894 | 4.9857 | 5.0425 | 4.9975 | 4.5931 |
Ω_{L2} | 9.7566 | 9.6227 | 9.2739 | 9.1052 | 9.0780 | 9.1532 | 9.0675 | 8.3758 | ||
Ω_{L3} | 9.7566 | 9.6232 | 9.2816 | 9.1224 | 9.0985 | 9.1668 | 9.0741 | 8.3758 | ||
0.1 | Ω_{L1} | 9.0573 | 8.9471 | 8.6541 | 8.4990 | 8.4407 | 8.4359 | 8.3353 | 7.7755 | |
Ω_{L2} | 15.415 | 15.244 | 14.774 | 14.500 | 14.358 | 14.268 | 14.081 | 13.233 | ||
Ω_{L3} | 15.415 | 15.244 | 14.777 | 14.509 | 14.362 | 14.270 | 14.082 | 13.233 | ||
0.15 | Ω_{L1} | 11.283 | 11.165 | 10.838 | 10.642 | 10.517 | 10.414 | 10.266 | 9.6865 | |
Ω_{L2} | 18.297 | 18.125 | 17.627 | 17.299 | 17.040 | 16.784 | 16.532 | 15.707 | ||
Ω_{L3} | 18.297 | 18.125 | 17.632 | 17.310 | 17.056 | 16.797 | 16.538 | 15.707 | ||
90° | 0.05 | Ω_{L1} | 9.5715 | 9.4422 | 9.1085 | 8.9487 | 8.9212 | 8.9816 | 8.8897 | 8.2169 |
Ω_{L2} | 15.196 | 15.003 | 14.502 | 14.254 | 14.185 | 14.215 | 14.050 | 13.045 | ||
Ω_{L3} | 18.269 | 18.040 | 17.419 | 17.086 | 16.970 | 17.004 | 16.829 | 15.684 | ||
0.1 | Ω_{L1} | 14.892 | 14.732 | 14.291 | 14.033 | 13.882 | 13.769 | 13.580 | 12.785 | |
Ω_{L2} | 22.158 | 21.938 | 21.319 | 20.930 | 20.650 | 20.390 | 20.091 | 19.022 | ||
Ω_{L3} | 25.992 | 25.741 | 25.014 | 24.537 | 24.177 | 23.847 | 23.505 | 22.313 | ||
0.15 | Ω_{L1} | 17.500 | 17.340 | 16.880 | 16.571 | 16.314 | 16.039 | 15.785 | 15.023 | |
Ω_{L2} | 25.157 | 24.943 | 24.305 | 23.848 | 23.428 | 22.966 | 22.596 | 21.596 | ||
Ω_{L3} | 29.123 | 28.883 | 28.152 | 27.614 | 27.104 | 26.542 | 26.113 | 25.002 |
Non-linear vibration of isosceles triangular FG-plate
The boundary conditions effects on the fundamental backbone curves for FG AL/AL_{2}O_{3} isosceles triangular plate are investigated in Fig. 5. Four different boundary conditions are considered in this part of study SSS, CSS, SCC and CCC (S: simply supported edge and C : clamped edge). The volume fraction exponent, thickness ratio and the apex angle of FG isosceles triangular plate are taken respectively as n =1, h/b = 0.05 and β = 90°. The figure clearly show that the FG plate with simply supported boundary conditions presents a more accentuated hardening behavior than the other boundary conditions. It is noted that the hardening effect increases when the plate becomes more free (SSS) and decreases as the plate becomes more fixed (CCC), this difference in the results is due to the rotation of the edges.
Conclusions
The non-linear free vibration of moderately thick FGMs clamped isosceles triangular plates was analyzed by a triangular p-element. The material properties of the functionally graded triangular plate assumed to be graded only in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents. The shape functions of triangular finite p-element are obtained by the shifted orthogonal polynomials of Legendre. The components of stiffness and mass matrices were calculated using numerical integration of Gauss-Legendre. The equations of motion are obtained from Lagrange's equation in combination with the harmonic balance method (HBM). Results for linear and non-linear frequency for the lowest three modes of FGMs clamped isosceles triangular plates were obtained. The parametric studies show that the boundary conditions have a great influence on the shape of the backbone curves, the hardening spring effect decreases for clamped FG plate. For simply supported FG plate and by increasing thickness ratio and sector angle of FG plates the hardening spring effect increases. A increase in the volume fraction exponent produces a variation in the hardening spring effect with an increasing part and another decreasing part, the peak in the curves of the nonlinear-to-linear fundamental frequency ratio FG triangular plate is obtained around of n = 1 at which the hardening behavior is maximum, and is obtained for AL/AL_{2}O_{3} FG plate. This value of volume fraction exponent corresponds to equal mixtures of metal and ceramic in the composition of the FG plate. Not only the hardening behavior is influenced by this mixture but the non-linear mode shape of FG isosceles triangular plate is also influenced.
Appendix A
Declarations
Competing interests
The author declare no significant competing financial, professional or personal interests that might have influenced the performance or presentation of the work described in this manuscript.
Author details
Laboratory of Computational Mechanics, Faculty of Technology, Department of Mechanical Engineering, University of Tlemcen, B.P. 230, Tlemcen 13000, Algeria.
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