Three-Dimensional Exact Free Vibration Analysis of Spherical, Cylindrical, and Flat One-Layered Panels

Table 1

Assessment 1. Simply supported aluminum square plate with Poisson ratio 𝜈=0.3. First six exact natural circular frequencies in no-dimensional form 𝜔=𝜔(𝑎²/ℎ)𝜌/𝐸 for imposed half-wave numbers 𝑚=𝑛=1. Comparison between present three-dimensional analysis and three-dimensional analyses by Vel and Batra [34] and Srinivas et al. [22]. 𝑀 is the number of fictitious layers and 𝑁 is the order of expansion for the exponential matrix.

	Mode
	I	II	III	IV	V	VI
	𝑎/ℎ=10
Present 3D (𝑀=1, 𝑁=3)	5.3370	27.554	46.493	154.39	168.09	268.36
Present 3D (𝑀=1, 𝑁=7)	5.7769	27.554	46.502	192.91	197.98	329.35
Present 3D (𝑀=1, 𝑁=10)	5.7769	27.554	46.502	196.21	201.87	308.34
Present 3D (𝑀=1, 𝑁=15)	5.7769	27.554	46.502	196.77	201.34	358.23
Present 3D (𝑀=1, 𝑁=18)	5.7769	27.554	46.502	196.77	201.34	357.42
Present 3D (𝑀=10, 𝑁=3)	5.7769	27.554	46.502	196.71	201.27	355.40
Present 3D (𝑀=50, 𝑁=3)	5.7769	27.554	46.502	196.77	201.34	351.90
Present 3D (𝑀=100, 𝑁=3)	5.7769	27.554	46.502	196.77	201.34	357.42
3D [34]	5.7769	27.554	46.503	196.77	201.34	357.42
3D [22]	5.7769	27.554	46.502	196.77	201.34	357.42
	𝑎/ℎ=√10
Present 3D (𝑀=1, 𝑁=3)	/	8.7132	14.182	17.512	/	/
Present 3D (𝑀=1, 𝑁=7)	4.6598	8.7132	14.463	20.987	23.999	29.925
Present 3D (𝑀=1, 𝑁=10)	4.6582	8.7132	14.463	21.383	25.009	31.399
Present 3D (𝑀=1, 𝑁=15)	4.6581	8.7132	14.463	21.343	24.830	34.049
Present 3D (𝑀=1, 𝑁=18)	4.6581	8.7132	14.463	21.343	24.830	33.982
Present 3D (𝑀=10, 𝑁=3)	4.6580	8.7132	14.463	21.337	24.821	33.939
Present 3D (𝑀=50, 𝑁=3)	4.6581	8.7132	14.463	21.343	24.830	33.923
Present 3D (𝑀=100, 𝑁=3)	4.6581	8.7132	14.463	21.343	24.830	33.982
3D [34]	4.6582	8.7132	14.463	21.343	24.830	33.982
3D [22]	4.6582	8.7134	14.462	21.343	24.830	33.982