Free Vibration Characteristics of the Conical Shells Based on Precise Integration Transfer Matrix Method

Modal number
m	n	Reference ^[13]13 Cong Zhang, Meixia Cheng. (2011). Free vibration of a conical shell in arbitrarily boundary condition. The 13th Academic Conference on Marine Underwater Noise (China).	PITMM	Relative deviation
1	1	29.9129	29.3296	-1.95%
	2	26.8326	26.7380	-0.35%
	3	18.0585	17.8254	-1.29%
	4	12.7885	12.7324	-0.44%
	5	11.0225	11.1634	1.28%
	6	11.2995	11.3225	0.20%
	7	12.8908	12.4141	-3.69%
	8	15.3580	15.4831	0.81%
	9	18.2627	18.0296	-1.28%
	10	21.4599	21.8944	2.02%

Modal number
m	n	Reference ^[13]13 Cong Zhang, Meixia Cheng. (2011). Free vibration of a conical shell in arbitrarily boundary condition. The 13th Academic Conference on Marine Underwater Noise (China).	PITMM	Relative deviation
1	1	37.6501	37.4014	-0.66%
	2	0.4311	0.4375	1.48%
	3	1.4162	1.4732	4.09%
	4	2.5699	2.5873	0.68%
	5	4.0152	4.0197	0.11%
	6	5.7521	5.7113	-0.71%
	7	7.4893	7.3211	-2.24%
	8	9.6646	9.6310	-0.35%
	9	11.9858	11.6183	-3.06%
	10	14.5985	14.1648	-2.97%

Modal number
m	n	Reference ^[13]13 Cong Zhang, Meixia Cheng. (2011). Free vibration of a conical shell in arbitrarily boundary condition. The 13th Academic Conference on Marine Underwater Noise (China).	PITMM	Relative deviation
1	1	37.9421	37.8789	-0.16%
	2	28.2925	27.6930	-2.11%
	3	21.5622	21.6901	0.59%
	4	17.4600	17.3930	-0.38%
	5	15.1097	15.0056	-0.69%
	6	14.5112	14.5282	0.12%
	7	15.2266	15.3239	0.64%
	8	17.2558	17.2338	-0.13%
	9	20.0146	20.6211	3.03%
	10	23.3577	23.4859	0.55%

(N_{x θ}, N_{θ x}, Q_{θ}, S_{x}) = \frac{K}{R^{2}} \sum_{\partial = 0}^{1} \sum_{n} (\bar{N_{x θ}}, \bar{N_{θ x}}, \bar{Q_{θ}}, \bar{S_{x}}) \cos (n θ + \frac{α π}{2})

M_{θ} = \frac{E h^{3}}{12 (1 - ν^{2})} [\frac{1}{s} (- \frac{1}{s \sin^{2} α} \frac{\partial^{2} w}{\partial θ^{2}} - v \frac{\partial^{2} w}{\partial s^{2}}) - v \frac{\partial^{2} w}{\partial s^{2}}]

(N_{s θ}, N_{θ s}, Q_{θ}, S_{s θ}) = \frac{K}{R^{2}} \sum_{α = 0}^{1} \sum_{n} ({\tilde{N}}_{s θ}, {\tilde{N}}_{θ s}, {\tilde{Q}}_{θ}, {\tilde{S}}_{s θ}) \cos (n θ + \frac{α π}{2})

(N_{s}, N_{θ}, Q_{s}, V_{s}) = \frac{K}{R^{2}} \sum_{α = 0}^{1} \sum_{n} ({\tilde{N}}_{s}, {\tilde{N}}_{θ}, {\tilde{Q}}_{s}, {\tilde{V}}_{s}) \sin (n θ + \frac{α π}{2})

{[\begin{matrix} - T_{1} & I & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - T_{2} & I & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - T_{3} & I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & ... & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - T_{i + 1} P & I & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & ... & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - T_{n} & I \end{matrix}]}_{(8 n,8 n + 8)} {\begin{matrix} Z (ξ_{0}) \\ Z (ξ_{1}) \\ Z (ξ_{2}) \\ ⋮ \\ Z (ξ_{i}) \\ ⋮ \\ Z (ξ_{n}) \end{matrix}}_{(8 n + 8,1)} = {\begin{matrix} 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ ⋮ \\ 0 \end{matrix}}

Z (ξ_{0}) = {\begin{matrix} {\bar{u}}^{0} & 0 & 0 & {\bar{ψ}}^{0} & 0 & \bar{V_{x}^{0}} & \bar{S_{x φ}^{0}} & 0 \end{matrix}}^{T}

Z (ξ_{n}) = {\begin{matrix} {\bar{u}}^{n} & 0 & 0 & {\bar{ψ}}^{n} & 0 & \bar{V_{x}^{n}} & \bar{S_{x φ}^{n}} & 0 \end{matrix}}^{T}

Z (ξ_{0}) = {\begin{matrix} 0 & 0 & 0 & 0 & \bar{M_{x}^{0}} & \bar{V_{x}^{0}} & \bar{S_{x φ}^{0}} & \bar{N_{x}^{0}} \end{matrix}}^{T}

Z (ξ_{n}) = {\begin{matrix} 0 & 0 & 0 & 0 & \bar{M_{x}^{n}} & \bar{V_{x}^{n}} & \bar{S_{x φ}^{n}} & \bar{N_{x}^{n}} \end{matrix}}^{T}

Z (ξ_{n}) = {\begin{matrix} {\bar{u}}^{n} & 0 & 0 & {\bar{ψ}}^{n} & 0 & \bar{V_{x}^{n}} & \bar{S_{x φ}^{n}} & 0 \end{matrix}}^{T}

Z (ξ_{n}) = {\begin{matrix} 0 & 0 & 0 & 0 & \bar{M_{x}^{n}} & \bar{V_{x}^{n}} & \bar{S_{x φ}^{n}} & \bar{N_{x}^{n}} \end{matrix}}^{T}

Z (ξ_{0}) = {\begin{matrix} {\bar{u}}^{0} & 0 & 0 & {\bar{ψ}}^{0} & 0 & \bar{V_{x}^{0}} & \bar{S_{x φ}^{0}} & 0 \end{matrix}}^{T}

Z (ξ_{0}) = {\begin{matrix} {\bar{u}}^{0} & 0 & 0 & {\bar{ψ}}^{0} & 0 & \bar{V_{x}^{0}} & \bar{S_{x φ}^{0}} & 0 \end{matrix}}^{T}

Z (ξ_{n}) = {\begin{matrix} 0 & 0 & 0 & 0 & \bar{M_{x}^{n}} & \bar{V_{x}^{n}} & \bar{S_{x φ}^{n}} & \bar{N_{x}^{n}} \end{matrix}}^{T}

Z (ξ_{0}) = {\begin{matrix} 0 & 0 & 0 & 0 & \bar{M_{x}^{0}} & \bar{V_{x}^{0}} & \bar{S_{x φ}^{0}} & \bar{N_{x}^{0}} \end{matrix}}^{T}

Z (ξ_{0}) = {\begin{matrix} 0 & 0 & 0 & 0 & \bar{M_{x}^{0}} & \bar{V_{x}^{0}} & \bar{S_{x φ}^{0}} & \bar{N_{x}^{0}} \end{matrix}}^{T}

Z (ξ_{n}) = {\begin{matrix} {\bar{u}}^{n} & 0 & 0 & {\bar{ψ}}^{n} & 0 & \bar{V_{x}^{n}} & \bar{S_{x φ}^{n}} & 0 \end{matrix}}^{T}

{[\begin{matrix} - \overset{⌢}{T_{1}} & I & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - T_{2} & I & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - T_{3} & I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & ... & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - T_{i + 1} P & I & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & ... & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - T_{n} & \overset{⌢}{I} \end{matrix}]}_{(8 n,8 n)} {\begin{matrix} \overset{⌢}{Z} (ξ_{0}) \\ Z (ξ_{1}) \\ Z (ξ_{2}) \\ ⋮ \\ Z (ξ_{i}) \\ ⋮ \\ \overset{⌢}{Z} (ξ_{n}) \end{matrix}}_{(8 n,1)} = {\begin{matrix} 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ ⋮ \\ 0 \end{matrix}}_{(8 n,1)}

{| \begin{matrix} - \overset{⌢}{T_{1}} & I & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - T_{2} & I & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - T_{3} & I & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & ... & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - T_{i + 1} P & I & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & ... & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - T_{n} & \overset{⌢}{I} \end{matrix} |}_{(8 n,8 n)} = 0

U_{63} = - \frac{12 (1 - ν^{2})}{ξ^{2} h \tan^{2} α} - {2 + \frac{n^{2} (1 + ν^{2})}{\sin^{2} α}} \frac{(1 - ν^{2}) n^{2} h}{ξ^{4} \sin^{2} α} + \frac{12 λ^{2}}{h}

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