Dynamic Analysis of Functionally Graded Porous (FGP) Elliptic Cylindrical Shell Based on Jacobi Polynomials Ritz Method
Abstract
:1. Introduction
2. Theoretical Formulations
2.1. FGP Elliptic Cylindrical Shell and Panel Model
2.2. Energy Expression
2.3. Admissible Displacements and Solving Process
3. Numerical Results and Discussion
3.1. Dynamic Characteristics of FGP Elliptic Cylindrical Shell under Free Vibration
3.2. Dynamic Characteristics of FGP Elliptic Cylindrical Shell under Forced Vibration
3.2.1. Steady-State Response in Frequency Domain
3.2.2. Transient Response in Time Domain
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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e0 | Mode | FGP Elliptic Cylindrical Panel | FGP Elliptic Cylindrical Shell | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
CCCC | SSSS | CSCF | CC | SS | CF | ||||||||
Guan [21] | Present | Guan [21] | Present | Guan [21] | Present | Guan [21] | Present | Guan [21] | Present | Guan [21] | Present | ||
0.2 | 1 | 0.569 | 0.569 | 0.497 | 0.497 | 0.241 | 0.241 | 0.423 | 0.422 | 0.386 | 0.385 | 0.171 | 0.171 |
2 | 0.695 | 0.695 | 0.890 | 0.590 | 0.262 | 0.262 | 0.440 | 0.440 | 0.406 | 0.406 | 0.247 | 0.246 | |
3 | 0.911 | 0.911 | 0.825 | 0.825 | 0.449 | 0.449 | 0.545 | 0.543 | 0.517 | 0.515 | 0.284 | 0.284 | |
4 | 0.938 | 0.937 | 0.844 | 0.844 | 0.517 | 0.517 | 0.613 | 0.613 | 0.594 | 0.594 | 0.432 | 0.430 | |
0.4 | 1 | 0.547 | 0.547 | 0.480 | 0.480 | 0.232 | 0.232 | 0.406 | 0.406 | 0.368 | 0.367 | 0.165 | 0.165 |
2 | 0.668 | 0.668 | 0.563 | 0.563 | 0.252 | 0.252 | 0.424 | 0.424 | 0.388 | 0.388 | 0.237 | 0.237 | |
3 | 0.877 | 0.877 | 0.797 | 0.797 | 0.433 | 0.432 | 0.523 | 0.521 | 0.494 | 0.492 | 0.274 | 0.274 | |
4 | 0.903 | 0.902 | 0.810 | 0.810 | 0.497 | 0.496 | 0.591 | 0.591 | 0.572 | 0.572 | 0.414 | 0.412 |
Type | e0 | FGP Elliptic Cylindrical Shell | FGP Elliptic Cylindrical Panel | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
CC | CF | CS | E1E2 | E1E3 | E2E3 | CCCC | CFCF | CSCF | E1E2E1E2 | E1E3E1E3 | E2E3E2E3 | ||
Type 1 | 0.1 | 0.118 | 0.046 | 0.116 | 0.104 | 0.076 | 0.090 | 0.135 | 0.073 | 0.078 | 0.112 | 0.098 | 0.119 |
0.2 | 0.116 | 0.046 | 0.114 | 0.103 | 0.078 | 0.090 | 0.134 | 0.072 | 0.078 | 0.111 | 0.098 | 0.119 | |
0.3 | 0.115 | 0.046 | 0.112 | 0.101 | 0.078 | 0.089 | 0.134 | 0.072 | 0.077 | 0.111 | 0.097 | 0.118 | |
0.4 | 0.113 | 0.045 | 0.111 | 0.100 | 0.078 | 0.089 | 0.133 | 0.071 | 0.076 | 0.110 | 0.097 | 0.117 | |
Type 2 | 0.1 | 0.118 | 0.046 | 0.115 | 0.104 | 0.076 | 0.090 | 0.134 | 0.073 | 0.078 | 0.112 | 0.098 | 0.119 |
0.2 | 0.116 | 0.045 | 0.113 | 0.102 | 0.078 | 0.089 | 0.132 | 0.071 | 0.077 | 0.110 | 0.097 | 0.117 | |
0.3 | 0.114 | 0.044 | 0.111 | 0.100 | 0.077 | 0.088 | 0.130 | 0.070 | 0.075 | 0.109 | 0.095 | 0.116 | |
0.4 | 0.112 | 0.044 | 0.109 | 0.098 | 0.076 | 0.087 | 0.127 | 0.069 | 0.074 | 0.107 | 0.094 | 0.114 | |
Type 3 | 0.1 | 0.117 | 0.046 | 0.114 | 0.103 | 0.076 | 0.089 | 0.133 | 0.072 | 0.077 | 0.111 | 0.097 | 0.118 |
0.2 | 0.114 | 0.044 | 0.111 | 0.100 | 0.076 | 0.087 | 0.130 | 0.070 | 0.075 | 0.108 | 0.095 | 0.115 | |
0.3 | 0.110 | 0.043 | 0.108 | 0.097 | 0.075 | 0.086 | 0.126 | 0.068 | 0.073 | 0.105 | 0.093 | 0.112 | |
0.4 | 0.107 | 0.042 | 0.105 | 0.094 | 0.073 | 0.083 | 0.122 | 0.066 | 0.071 | 0.102 | 0.090 | 0.109 |
Type | a/b | FGP Elliptic Cylindrical Shell | FGP Elliptic Cylindrical Panel | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
e0 = 0.2 | e0 = 0.3 | e0 = 0.4 | e0 = 0.5 | e0 = 0.6 | e0 = 0.7 | e0 = 0.2 | e0 = 0.3 | e0 = 0.4 | e0 = 0.5 | e0 = 0.6 | e0 = 0.7 | ||
Type 1 | 0.5 | 0.161 | 0.160 | 0.159 | 0.157 | 0.154 | 0.152 | 0.250 | 0.248 | 0.246 | 0.245 | 0.244 | 0.244 |
1 | 0.159 | 0.157 | 0.155 | 0.154 | 0.152 | 0.151 | 0.249 | 0.247 | 0.244 | 0.241 | 0.239 | 0.237 | |
1.5 | 0.139 | 0.137 | 0.135 | 0.133 | 0.132 | 0.130 | 0.194 | 0.193 | 0.192 | 0.191 | 0.191 | 0.192 | |
2 | 0.114 | 0.112 | 0.111 | 0.109 | 0.108 | 0.107 | 0.134 | 0.134 | 0.133 | 0.132 | 0.132 | 0.132 | |
Type 2 | 0.5 | 0.157 | 0.154 | 0.151 | 0.147 | 0.143 | 0.137 | 0.246 | 0.242 | 0.237 | 0.232 | 0.225 | 0.219 |
1 | 0.157 | 0.154 | 0.151 | 0.147 | 0.143 | 0.139 | 0.247 | 0.243 | 0.238 | 0.233 | 0.227 | 0.221 | |
1.5 | 0.138 | 0.135 | 0.132 | 0.129 | 0.126 | 0.123 | 0.190 | 0.187 | 0.183 | 0.178 | 0.173 | 0.167 | |
2 | 0.113 | 0.111 | 0.109 | 0.106 | 0.104 | 0.101 | 0.132 | 0.130 | 0.127 | 0.124 | 0.121 | 0.117 | |
Type 3 | 0.5 | 0.154 | 0.150 | 0.145 | 0.140 | 0.134 | 0.127 | 0.242 | 0.235 | 0.227 | 0.219 | 0.210 | 0.199 |
1 | 0.155 | 0.150 | 0.145 | 0.140 | 0.134 | 0.128 | 0.242 | 0.235 | 0.228 | 0.219 | 0.210 | 0.200 | |
1.5 | 0.136 | 0.132 | 0.127 | 0.123 | 0.118 | 0.112 | 0.186 | 0.181 | 0.175 | 0.169 | 0.162 | 0.154 | |
2 | 0.111 | 0.108 | 0.105 | 0.101 | 0.097 | 0.092 | 0.130 | 0.126 | 0.122 | 0.117 | 0.112 | 0.107 |
Type | L/a | FGP Elliptic Cylindrical Shell | FGP Elliptic Cylindrical Panel | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
e0 = 0.2 | e0 = 0.3 | e0 = 0.4 | e0 = 0.5 | e0 = 0.6 | e0 = 0.7 | e0 = 0.2 | e0 = 0.3 | e0 = 0.4 | e0 = 0.5 | e0 = 0.6 | e0 = 0.7 | ||
Type 1 | 2 | 0.243 | 0.240 | 0.238 | 0.236 | 0.235 | 0.234 | 0.273 | 0.270 | 0.269 | 0.267 | 0.266 | 0.266 |
3 | 0.173 | 0.171 | 0.169 | 0.168 | 0.167 | 0.166 | 0.189 | 0.187 | 0.185 | 0.184 | 0.183 | 0.183 | |
4 | 0.138 | 0.137 | 0.136 | 0.135 | 0.134 | 0.132 | 0.153 | 0.152 | 0.151 | 0.150 | 0.149 | 0.149 | |
5 | 0.114 | 0.112 | 0.111 | 0.109 | 0.108 | 0.107 | 0.134 | 0.134 | 0.133 | 0.132 | 0.132 | 0.132 | |
Type 2 | 2 | 0.239 | 0.235 | 0.230 | 0.225 | 0.220 | 0.213 | 0.269 | 0.264 | 0.259 | 0.253 | 0.247 | 0.239 |
3 | 0.170 | 0.167 | 0.163 | 0.160 | 0.155 | 0.150 | 0.186 | 0.183 | 0.179 | 0.175 | 0.171 | 0.166 | |
4 | 0.136 | 0.133 | 0.131 | 0.127 | 0.124 | 0.120 | 0.151 | 0.148 | 0.145 | 0.142 | 0.139 | 0.134 | |
5 | 0.113 | 0.111 | 0.109 | 0.106 | 0.104 | 0.101 | 0.132 | 0.130 | 0.127 | 0.124 | 0.121 | 0.117 | |
Type 3 | 2 | 0.235 | 0.228 | 0.221 | 0.213 | 0.204 | 0.194 | 0.263 | 0.256 | 0.247 | 0.238 | 0.228 | 0.217 |
3 | 0.167 | 0.162 | 0.157 | 0.151 | 0.145 | 0.138 | 0.182 | 0.177 | 0.171 | 0.165 | 0.158 | 0.151 | |
4 | 0.134 | 0.130 | 0.126 | 0.121 | 0.116 | 0.110 | 0.148 | 0.144 | 0.139 | 0.134 | 0.129 | 0.122 | |
5 | 0.111 | 0.108 | 0.105 | 0.101 | 0.097 | 0.092 | 0.130 | 0.126 | 0.122 | 0.117 | 0.112 | 0.107 |
Type | θ0 | e0 | ||||||
---|---|---|---|---|---|---|---|---|
0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | ||
Type 1 | π/6 | 4.998 | 4.940 | 4.882 | 4.825 | 4.768 | 4.717 | 4.675 |
π/3 | 1.287 | 1.273 | 1.259 | 1.246 | 1.233 | 1.223 | 1.216 | |
π/2 | 0.267 | 0.265 | 0.264 | 0.263 | 0.263 | 0.263 | 0.264 | |
2π/3 | 0.160 | 0.158 | 0.157 | 0.156 | 0.154 | 0.153 | 0.153 | |
5π/6 | 0.146 | 0.145 | 0.144 | 0.143 | 0.143 | 0.142 | 0.143 | |
Type 2 | π/6 | 4.974 | 4.889 | 4.799 | 4.702 | 4.599 | 4.486 | 4.361 |
π/3 | 1.280 | 1.258 | 1.235 | 1.211 | 1.184 | 1.155 | 1.124 | |
π/2 | 0.264 | 0.260 | 0.255 | 0.250 | 0.244 | 0.237 | 0.228 | |
2π/3 | 0.159 | 0.156 | 0.153 | 0.150 | 0.147 | 0.143 | 0.139 | |
5π/6 | 0.145 | 0.142 | 0.140 | 0.137 | 0.134 | 0.130 | 0.126 | |
Type 3 | π/6 | 4.931 | 4.800 | 4.660 | 4.509 | 4.346 | 4.165 | 3.962 |
π/3 | 1.269 | 1.235 | 1.199 | 1.160 | 1.118 | 1.072 | 1.019 | |
π/2 | 0.261 | 0.254 | 0.247 | 0.239 | 0.230 | 0.221 | 0.210 | |
2π/3 | 0.157 | 0.153 | 0.149 | 0.144 | 0.139 | 0.133 | 0.126 | |
5π/6 | 0.143 | 0.139 | 0.135 | 0.131 | 0.126 | 0.121 | 0.115 |
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Lu, J.; Yang, Q.; Meng, Z.; Yang, K.; Xu, W.; Chiu, C. Dynamic Analysis of Functionally Graded Porous (FGP) Elliptic Cylindrical Shell Based on Jacobi Polynomials Ritz Method. J. Compos. Sci. 2023, 7, 344. https://doi.org/10.3390/jcs7080344
Lu J, Yang Q, Meng Z, Yang K, Xu W, Chiu C. Dynamic Analysis of Functionally Graded Porous (FGP) Elliptic Cylindrical Shell Based on Jacobi Polynomials Ritz Method. Journal of Composites Science. 2023; 7(8):344. https://doi.org/10.3390/jcs7080344
Chicago/Turabian StyleLu, J., Q. Yang, Z. Meng, K. Yang, W. Xu, and C. Chiu. 2023. "Dynamic Analysis of Functionally Graded Porous (FGP) Elliptic Cylindrical Shell Based on Jacobi Polynomials Ritz Method" Journal of Composites Science 7, no. 8: 344. https://doi.org/10.3390/jcs7080344