A Unified Formulation for Free Vibration of Spherical Cap Based on the Ritz Method

Table 2

Convergence study of the frequency parameters with respect to the section number of spherical cap (N_p).

n	m	Number of the segments (Np)									Ref [2]	Ref [39]	FEM
		2	3	4	5	6	7	8	9	10
1	1	0.8229	0.8229	0.8228	0.8228	0.8228	0.8228	0.8228	0.8228	0.8228	0.8247	0.8228	0.8226
	2	0.9147	0.9146	0.9145	0.9145	0.9145	0.9145	0.9145	0.9145	0.9145	0.9184	0.9145	0.9144
	3	0.9448	0.9443	0.9445	0.9445	0.9445	0.9445	0.9445	0.9445	0.9445	0.9485	0.9446	0.9445
	4	0.9708	0.9707	0.9703	0.9703	0.9703	0.9703	0.9703	0.9703	0.9703	0.9772	0.9705	0.9704
	5	1.0101	1.0099	1.0098	1.0098	1.0098	1.0098	1.0098	1.0098	1.0098	1.0228	1.0099	1.0098
2	1	0.9195	0.9194	0.9193	0.9193	0.9193	0.9193	0.9193	0.9193	0.9193	0.9215	0.9192	0.9193
	2	0.9508	0.9507	0.9505	0.9505	0.9505	0.9505	0.9505	0.9505	0.9505	0.9528	0.9504	0.9505
	3	0.9791	0.979	0.9787	0.9787	0.9787	0.9787	0.9787	0.9787	0.9787	0.9819	0.9788	0.9788
	4	1.0169	1.0167	1.0165	1.0165	1.0165	1.0165	1.0165	1.0165	1.0165	1.0184	1.0164	1.0164
	5	1.0542	1.0541	1.0541	1.0541	1.0541	1.0541	1.0541	1.0541	1.0541	1.0685	1.0538	1.0539
3	1	0.9350	0.9349	0.9348	0.9348	0.9348	0.9348	0.9348	0.9348	0.9348	0.9369	0.9346	0.9347
	2	0.9624	0.9623	0.9621	0.9621	0.9621	0.9621	0.9621	0.9621	0.9621	0.9652	0.9618	0.9620
	3	0.9965	0.9964	0.9963	0.9963	0.9963	0.9963	0.9963	0.9963	0.9963	1.0007	0.9961	0.9962
	4	1.0443	1.0442	1.0441	1.0441	1.0441	1.0441	1.0441	1.0441	1.0441	1.0474	1.0438	1.044
	5	1.0938	1.0935	1.0932	1.0932	1.0932	1.0932	1.0932	1.0932	1.0932	1.1110	1.0931	1.0931