Research Article

Structural-Electrical-Coupled Formulation for the Free Vibration of a Piezoelectric-Laminated Plate Using the Analytical Arbitrary Quadrilateral p Element

Table 5

Natural frequencies (Hz) of the aluminum trapezoidal plates with a top and a bottom PZT-5H layers.

degreeMode 1Mode 2Mode 3Mode 4Mode 5

CFFF

0.410973.253453.325008.238766.1212533.2
16.70*1012.183593.015054.039450.8312505.7
30977.833471.645015.18848.5612546.9

0.420254.191305.316033402.944021.24
8.53*258.771335.071586.973499.123991.47
302285.421149.581605.292970.134028.11

0.610897.682764.034863.967820.3510630.2
11.31*9162805.254907.478118.0510327.9
30856.462670.144742.597284.4911111.1

0.6202291218.281300.723295.313528.89
5.71*231.291238.891289.273389.23460.19
30199.231028.211314.462697.623588.43

FFFF(exclude rigid body)

0.4103751.025081.517227.248855.4312778.2
16.70*4451.764829.616826.499817.2312464.5
303817.435072.357192.898949.3212759.9

0.4201232.022244.23311.344476.956100.56
8.53*1264.082241.913412.14483.826086.82
301046.532225.882814.414341.845257.84

0.6103531.184795.266657.038273.7711532.4
11.31*3757.894763.26469.258660.7811502.7
303137.34788.396705.127939.4311502.7

0.6201229.731884.673329.663938.85896.75
5.71*1245.761880.093396.073915.95860.11
30973.251921.312571.673959.414738.01

CCCC

0.41011972.118409.323380.926724.331304.3
16.70*11527.918338.322439.72702230731.8
30119081839123266.426747.231258.5

0.4208440.9411415.714477.417653.618326.9
8.53*8326.4411312.614397.217678.817994.8
309247.0212159.915084.217999.420069.6

0.6109867.6116105.620005.424434.326701.4
11.31*9695.8616169.719492.525235.825762.5
3010309.616075.821104.623884.727731.9

0.6207346.329395.8711756.914912.516565.9
5.71*7295.949350.0711731.714930.816444.5
308418.0410405.812455.314958.317976.5

*Symmetric trapezoidal plates.