In this article, two-dimensional functional materials have been used using the power law in them, which is a good measure to obtain the properties of a composite material of metal and ceramic. At first, the equations of motion were obtained using Hamiltons method and solved by the GDQ method, and finally, the accuracy of the obtained answers was compared with the existing articles. In the following, the dynamic model of the sheet with two piezoelectric actuator layers at the top and bottom was investigated and the obtained equations were solved using the Ritz method.
Validation of two-dimensional functional sheet vibration with existing articles to validate the results of this research, we first run the written program in the special case of one-dimensional graded material and without piezoelectric. Then we compare the obtained results with references 7, 8 and 11. In these references, the function sheet one dimension is made of aluminum metal and aluminum oxide ceramic with the following specifications.
Table 1: specifications of one-dimensional functional grade material (AL/AL2O3).
material
70 e9 2702
380 e9 3800
The results of this review are as follows
Table 2: Comparison of natural four-frequency with dimension of present research with existing articles for one-dimensional functional material SSSS و ( ) و( ) و ( (
Present 4492 11092 11128 16880
X. Zhao[11] 4347.4 10416 10416 15936
Error (%) 3.326126 6.490015 6.835637 5.923695
H. Matsunaga[7] 4427 10630 10630 16200
Error (%) 1.468263 4.34619 4.684854 4.197531
Table 2 Comparison of the main natural frequency with the dimension of the current research with existing articles for the one-dimensional functional material AL/AL2O3 for boundary conditions SSSS و ( ) و( )و( (
0 0.5 1 4 10
Present 5950 5225 4492 4070 3855
H. Matsunaga [7] 5777 4917 4427 3811 3642
error 2.994634 6.263982 1.468263 6.796117 5.848435
Sh. H.Hashemi [8] 5769 4920 4454 3825 3627
error 3.137459 6.199187 8.53166 6.405229 6.286187
X. Zhao [11] 5676.3 4820.9 4347 - 3592.3
error 4.821803 8.382252 3.335634 - 7.312864
Investigation of two-dimensional functional sheet vibration (without piezoelectric)
Table 3: Specifications of two-dimensional graded material.
material
70 2702
200 7800
380 3800
205 8900
Table 4: Checking the vibration of two-dimensional graded material with support conditions SSSS.
ssss
0 0.5 1 4 10 0 0.5 1 4 10 0 0.5 1 4 10
10 2973 3229.3 3344.1 3536.3 3605.6 3021.9 3415.2 3590 3880.06 3980.7 3251.4 3915.3 4190 4640.2 4790.5
20 1495.
623 1624.56 1682.312 1779.002 1813.865 1520.223 1718.08 1806.017 1951.937 2002.566 1635.678 1969.665 2107.858 2334.339 2409.951
30 998.
064 1084.
185 1122.
905 1186.995 1209.694 1014.754 1146.939 1205.688 1303.158 1336.538 1092.196 1314.507 1406.636 1558.181 1608.252
40 748.548 813.1388 842.1788 890.2463 907.2705 761.0655 860.2043 904.266 977.3685 1002.404 819.147 985.8803 1054.977 1168.636 1206.189
50 598.
8384 650.
511 673.
743 712.
197 725.
8164 608.8524 688.1634 723.4128 781.8948 801.
9228 655.3176 788.7042 843.9816 934.9086 964.9512
80 374.
274 406.5694 421.
0894 445.1231 453.6353 380.5328 430.1021 452.133 488.6843 501.2018 409.5735 492.9401 527.4885 584.3179 603.0945
100 299.
4192 325.
2555 336.8715 356.0985 362.9082 304.4262 344.0817 361.7064 390.9474 400.9614 327.6588 394.3521 421.9908 467.4543 482.4756
The resulting graphs from this table are as follows
Fig. 1: The original natural frequency with 2D functional sheet dimension with SSSS boundary conditions.
Fig. 2: The original natural frequency with the dimension of the two-dimensional functional sheet with boundary SSSSو ( ) conditions.
According to the graph (1), it can be seen that with the increase of the ratio or the decrease of the thickness, the natural frequency of the two-dimensional functional sheet decreases. 2- According to the diagram (2), it can be seen that it increases with increasing (constant) frequency. 3- According to the graph (2), it can be seen that it increases with the increase (constant) of the frequency. This increase is faster for values less than one, and for values greater than one, this increase is slower. 4- According to the diagram (1) by comparing the two cases, it can be seen that the increase has a greater effect on the frequency increase than the increase. 5- According to the diagram (1), it can be seen that the frequency reduction is faster before and after this ratio, the frequency reduction speed becomes lower.
Table 5: Checking the vibration of two-dimensional graded material with SCSC support conditions of the main natural frequency with the dimension of the two-dimensional functional sheet with SCSC boundary conditions.
scsc
0 0.5 1 4 10 0 0.5 1 4 10 0 0.5 1 4 10
10 4428.9 4796.3 4975 5295.2 5385.4 4500.3 5070.1 5345 5820 5955.7 4835.1 5805 6240 6975.3 7175
20 2228 2412.824 2502.721 2663.8 2709.176 2263.918 2550.562 2688.853 2927.806 2996.071 2432.343 2920.26 3139.091 3508.84 3609.452
30 1486.324 1609.622 1669.594 1777.052 1807.322 1510.286 1701.509 1793.764 1953.173 1998.713 1622.644 1948.139 2094.123 2340.787 2407.906
40 1117.621 1210.333 1255.428 1336.23 1358.991 1135.639 1279.426 1348.796 1468.661 1502.905 1220.125 1464.876 1574.647 1760.123 1810.592
50 894.1684 968.3442 1004.423 1069.069 1087.28 908.5836 1023.623 1079.123 1175.023 1202.42 976.1777 1171.995 1259.819 1408.211 1448.589
80 558.8902 605.253 627.8034 668.21 679.5925 567.9002 639.8042 674.4943 734.4353 751.5596 610.1492 732.5425 787.4358 880.1867 905.425
100 447.1479 484.2411 502.2829 534.6107 543.7175 454.3565 511.8843 539.6386 587.5953 601.2957 488.1584 586.0809 629.9991 704.2057 724.398
The resulting graphs from this table are as follows
Fig. 3: The original natural frequency with the dimension of the two-dimensional functional sheet with boundary SCSC conditions.
Fig. 4: Checking the vibration of two-dimensional graded material with CCCS support conditions along with piezoelectric.
Table 6: main natural frequencies of two-dimensional functional sheet with thickness ( ) under the effect of piezoelectric layer with thickness ( ) with CCCS boundary conditions.
CCCS(h=0.1)
0 1 4 1 4
0 4945 5840 6365 6800 7610
0.01 4830 5530 5990 6210 6910
0.02 4840 5425 5815 5955 6550
0.03 4940 5455 5780 5890 6395
0.04 5110 5570 5845 5950 6370
0.05 5325 5745 5970 6085 6435
0.06 5575 5960 6150 6270 6565
0.07 5845 6205 6335 6495 6735
0.08 6130 6470 6600 6735 6940
0.09 6425 6745 6855 6995 7170
0.1 6725 7025 7115 7265 7410
The graph resulting from this table is as follows
Fig. 5: The main natural frequencies of the two-dimensional functional sheet with thickness ( ) under the effect of piezoelectric layer with thickness ( ) with CCCS boundary conditions.
In the case of increasing the piezoelectric thickness ( ), the frequency decreases until the thickness and the frequency increases from this thickness. 2- In the case of ( ) increasing the piezoelectric thickness ( ), the frequency decreases ( ) to the thickness ( ) and the frequency increases from this thickness. 3- In the case of increasing the piezoelectric thickness, the frequency decreases to the thickness ( ) and the frequency increases ( ) from this thickness. 4- In the case of increasing the piezoelectric thickness, the frequency decreases to the thickness and the frequency increases from this thickness. 5- In the case of increasing the piezoelectric thickness ( ), the frequency decreases to the thickness ( ) and the frequency increases from this thickness. 6- By increasing the thickness of the piezo ( ), all the graphs are closer to each other and finally one be. 7- As the effect of piezoelectric increases in the initial reduction of frequencies, it increases. For example, in the case of frequency reduction, it is equal to 5.3% and for the case of equal to 8.6% is. 8- As the effect of piezoelectric increases in the initial reduction of frequencies, it increases. For example, in the mode of reducing the frequency in the mode equal to 5.3% and for the mode equal to 5.8% is. 9- The effect of piezoelectric in the initial reduction of frequencies is greater than For example, in the mode of reducing the frequency in the mode equal to 5.8% and for the mode equal to 8.6% is 6-3-5- Investigating the vibration of two-dimensional graded material with CCCC support conditions along with piezoelectric.
Table 7: main natural frequencies of two-dimensional functional sheet with thickness ( ) under the effect of piezoelectric layer with thickness ( ) with CCCC boundary conditions.
CCCC(h=0.1)
0 1 4 1 4
0 5610 6655 7155 7750 8520
0.01 5475 6305 6745 7080 7760
0.02 5488 6180 6560 6790 7370
0.03 5600 6205 6525 6715 7205
0.04 5790 6330 6600 6775 7185
0.05 6035 6525 6750 6920 7265
0.06 6315 6765 6950 7125 7415
0.07 6620 7040 7195 7370 7610
0.08 6945 7335 7460 7640 7845
0.09 7275 7640 7745 7930 8095
0.1 7610 7955 8040 8225 8365
The graph resulting from this table is as follows
Fig. 6: The main natural frequencies of the two-dimensional functional sheet with thickness ( ) under the effect of piezoelectric layer with thickness ( ) with CCCC boundary conditions.
Comparing the vibration of two-dimensional graded material with two piezoelectric layers with SSSS, SCSC, SSSC, SSCC, SCCC and CCCC boundary conditions.
Fig. 7: The original natural frequency with the dimension of a two-dimensional functional sheet with two piezoelectric layers with different boundary conditions in the state.
According to the diagram above, in a specific state (here for), the percent increase and decrease in frequency caused by changes in the piezoelectric thickness is almost constant and uniform. So that all six lines in the above diagram move parallel to each other. Investigating the causes of errors 1- The classical theory of sheets is only valid for thin sheets. In this theory, shear force and rotational inertia are zero. 2- To consider zero and to simplify the equations. 3- Using the numerical method (GDQ) to solve equations: the causes of errors in this part include two parts becomes: a: The smaller the number of points in the grid of the sheet, the greater the error. b: In this research, the method is used to apply the boundary conditions. In this method, we apply the boundary conditions regardless of the delta distance from the edges.
Fig. 8: Boundary condition in the state.
According to the tables and charts, it was observed that: For all boundary conditions and different power coefficients, natural frequency decreases with increa-sing ratio ( ). They increase with increasing and frequencies. It should be noted that the frequency changes depend on the arrangement of metals and ceramics in the two-dimensional graduated functional material. As expected, in a certain case, the thickness of the functional sheet and the power coefficients are ordered from higher to lower according to the boun-dary conditions as follows: from CCCC, CCCS, SCSC, CCSS, SSSC and SSSS. The effect of the presence of two piezoelectric layers on the top and bottom of the sheet in the (close-circuit) mode is such that with the increase in thickness, the piezoelectric frequencies decrease up to a certain thickness and then increase. The piezo effect in a certain state of plate thickness and power coefficients is completely similar for different boundary conditions. This means that the frequencies decrease and then increase at a certain rate.
We are grateful to all the dear professors for providing their information regarding this research.
Conflicts of interest are declared obviously in the manuscript. Authors also state separately that they have all read the manuscript and have no conflict of interest.
Academic Editor
Dr. Sonjoy Bishwas, Executive, Universe Publishing Group (UniversePG), California, USA.
Sharafi M. (2023). Vibration analysis of two-dimensional functionally graded plate with piezoelectric layers using the classical theory of plates, Br. J. Arts Humanit., 5(5), 243-252. https://doi.org/10.34104/bjah.02302430252