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Free Vibrations of a Thin Elastic Orthotropic Cylindrical Panel with Free Еdges

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Mechanics of Composite Materials Aims and scope

Using a system of equations corresponding to the classical theory of orthotropic cylindrical shells, the free vibrations of a thin elastic orthotropic cylindrical panel with free edges is investigated. To calculate its natural frequencies and to identify the respective vibration modes, the generalized Kantorovich–Vlasov method of reduction to ordinary differential equations is employed. To find the natural frequencies of possible types of vibrations, dispersion equations are derived. An asymptotic relation between the dispersion equations of the problem in hand and of an analogous problem for a rectangular plate with free sides is established. Determined is also a relation between the dispersion equations of the problem and of the boundary-value problem for a semi-infinite orthotropic nonclosed circular cylindrical shell with three free edges. With the example of an orthotropic cylindrical panel, the values of dimensionless characteristics of its natural frequencies are derived.

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Authors and Affiliations

  1. Kh. Abovyan ASPU, Yerevan, Armenia

    G. R. Ghulghazaryan & L. G. Ghulghazaryan

  2. Institute of Mechanics, NAS of Armenia, Yerevan, Armenia

    L. G. Ghulghazaryan

  3. University of Kettering, Flint, USA

    I. I. Kudish

Authors
  1. G. R. Ghulghazaryan
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  2. L. G. Ghulghazaryan
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  3. I. I. Kudish
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Correspondence to L. G. Ghulghazaryan.

Additional information

Translated from Mekhanika Kompozitnykh Materialov, Vol. 55, No. 5, pp. 813-836, September-October, 2019

Appendix

Appendix

Here, the analytical expressions for mij are presented:

$$ {\displaystyle \begin{array}{c}{m}_{11}=H{\chi}_1^4+{d}_1{\chi}_1^2+{d}_2,\kern0.36em {m}_{12}=H{\overline{\overline{f}}}_3+{d}_1{\overline{\overline{f}}}_1\kern0.36em {m}_{13}=H{\overline{f}}_2+{d}_1,{m}_{14}= Hf,\\ {}{m}_{21}=T{\chi}_1^5+{d}_3{\chi}_1^3+{d}_4{\chi}_1,{m}_{22}=T{\overline{\overline{f}}}_4+{d}_3{\overline{\overline{f}}}_2+{d}_4,{m}_{23}=T{\overline{f}}_3+{d}_3{\overline{f}}_1,{m}_{24}=T{f}_2+{d}_3,\\ {}{m}_{31}={\delta}_m{\chi}_1^6+{d}_5{\chi}_1^4+{d}_6{\chi}_1^2+{d}_7,{m}_{32}={\delta}_m{\overline{\overline{f}}}_5+{d}_5{\overline{\overline{f}}}_3+{d}_6{\overline{\overline{f}}}_1,\\ {}{m}_{33}={\delta}_m{\overline{f}}_4+{d}_5{\overline{f}}_2+{d}_6,{m}_{34}={\delta}_m{f}_3+{d}_5{f}_1,\\ {}{m}_{41}={\delta}_m{\chi}_1^7+{d}_8{\chi}_1^5+{d}_9{\chi}_1^3+{d}_{10}{\chi}_1,{m}_{42}={\delta}_m{\overline{\overline{f}}}_6+{d}_8{\overline{\overline{f}}}_4+{d}_9{\overline{\overline{f}}}_2+{d}_{10},\\ {}{m}_{43}={\delta}_m{\overline{f}}_5+{d}_8{\overline{f}}_3+{d}_9{\overline{f}}_1,{m}_{44}={\delta}_m{f}_4+{d}_8{f}_2+{d}_9,{\delta}_m=1+4{a}^2+{\varepsilon}_m^2,\\ {}{m}_{i5}={\left(-1\right)}^{i+1}{m}_{i1}\exp {z}_1,{m}_{i6}={\left(-1\right)}^{i+1}\left({m}_{i2}\exp {z}_2+{m}_{i1}\left[{z}_1{z}_2\right]\right),\\ {}{m}_{i7}={\left(-1\right)}^{i+1}\left({m}_{i3} ep{z}_3+{m}_{i2}\left[{z}_2{z}_3\right]+{m}_{i1}\left[{z}_1{z}_2{z}_3\right]\right),\\ {}{m}_{i8}={\left(-1\right)}^{i+1}\left({m}_{i4}\exp {z}_4+{m}_{i3}\left[{z}_3{z}_4\right]+{m}_{i2}\left[{z}_2{z}_3{z}_4\right]+{m}_{i1}\left[{z}_1{z}_2{z}_3{z}_4\right]\right),i=\overline{1,4,}\end{array}} $$

where

$$ {\displaystyle \begin{array}{c}H=-{a}^2\frac{B_{12}+4{B}_{66}}{B_{66}}{\beta}_m^{\hbox{'}},\kern0.6em T=-\frac{B_{12}}{B_{66}}{a}^2{\delta}_m{\beta}_m^{\hbox{'}},\kern0.48em {\delta}_m=1+4{a}^2{\varepsilon}_m^2,\\ {}{d}_1=\frac{B_{11}{B}_{22}-{B}_{12}^2}{B_{11}^2}{\beta}_m^{\hbox{'}}-\frac{B_{12}{B}_{66}}{B_{11}^2}{\eta}_{1m}^2+4{a}^2{\varepsilon}_m^2\frac{B_{12}{B}_{66}}{B_{11}^2}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\\ {}+{a}^2{\beta}_m^{\hbox{'}}\left[\frac{B_{12}}{B_{11}}{\beta}_m^{\hbox{'}\hbox{'}}-\frac{B_{12}\left({B}_{12}+4{B}_{66}\right)}{B_{11}^2}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\right],\\ {}{d}_2=-\frac{B_{12}}{B_{11}^2}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\left({B}_{66}{\eta}_{2m}^2+{B}_{22}\left({\beta}_m^{\hbox{'}}-{\beta}_m^{\hbox{'}\hbox{'}}\right)+{a}^2{\beta}_m^{\hbox{'}\hbox{'}}{B}_{22}\left({\beta}_m^{\hbox{'}}-{\varepsilon}_m^2\right)\right),\\ {}{d}_3=\frac{B_{11}{B}_{22}-{B}_{12}^2}{B_{11}^2}{\delta}_m{\beta}_m^{\hbox{'}}+{a}^2{\beta}_m^{\hbox{'}}\left(4{\eta}_{2m}^2-3{B}_2-2\frac{B_{12}}{B_{11}}{\beta}_m^{\hbox{'}}-\frac{B_{12}}{B_{11}}{\eta}_{1m}^2\right)+4{a}^4{\varepsilon}_m^2\left[\frac{B_{12}}{B_{11}}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\right],\\ {}{d}_4=\left(\frac{B_{12}}{B_{11}}{\eta}_{1m}^2+\frac{B_{12}}{B_{11}}{\eta}_{2m}^2\right)\kern0.24em {\beta}_m^{\hbox{'}}+{a}^2{\beta}_m^{\hbox{'}}\left[\frac{B_{12}\left({B}_{12}+4{B}_{66}\right)}{B_{11}{B}_{66}}{\beta}_m^{\hbox{'}}{\beta}_m^{\hbox{'}\hbox{'}}-3\frac{B_{12}}{B_{11}}{\beta}_m^{\hbox{'}\hbox{'}}-4\frac{B_{66}}{B_{11}}{\eta}_{2m}^2\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\right]\\ {}-{a}^2{\varepsilon}_m^2{\beta}_m^{\hbox{'}}\frac{B_{12}}{B_{11}}\left(\frac{\left({B}_{12}{\beta}^{\hbox{'}\hbox{'}}+4a{B}_{66}{\beta}_m^{\hbox{'}}\right)}{B_{66}}-4{\eta}_{1m}^2\right)+\frac{B_{12}{B}_{22}}{B_{11}{B}_{66}}{\beta}_m^{\hbox{'}}\left({\beta}_m^{\hbox{'}}-{\beta}_m^{\hbox{'}\hbox{'}}\right),\\ {}{d}_5=\frac{B_{66}}{B_{11}}{\eta}_{1m}^2+{\eta}_{2m}^2-\frac{B_{11}{B}_{22}{\beta}_m^{\hbox{'}\hbox{'}}-{B}_{12}^2{\beta}_m^{\hbox{'}}-{B}_{12}{B}_{66}{\beta}_m^{\hbox{'}}}{B_{11}{B}_{66}}\\ {}-{a}^2{\varepsilon}_m^2\left[\frac{B_{11}{B}_{22}{\beta}_m^{\hbox{'}\hbox{'}}-{B}_{12}^2{\beta}_m^{\hbox{'}}}{B_{11}{B}_{66}}+\frac{4{B}_{66}}{B_{11}}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\right],\\ {}{d}_6=\frac{B_{12}}{B_{11}}{B}_2+\frac{B_{22}}{B_{11}}{\beta}_m^{\hbox{'}}{\beta}_m^{\hbox{'}\hbox{'}}-\left(\frac{B_{11}{B}_{22}{\beta}^{\hbox{'}\hbox{'}}+{B}_{12}{B}_{66}{\beta}_m^{\hbox{'}}}{B_{11}^2}{\eta}_{1m}^2+\frac{B_{12}{B}_{66}}{B_{11}}{\beta}_m^{\hbox{'}}{\eta}_{2m}^2\right)\\ {}+\frac{B_{66}}{B_{11}}{\eta}_{1m}^2{\eta}_{2m}^2+{\varepsilon}_m^2\left[{a}^2\frac{{}_{11}{B}_{22}{\beta}^{\hbox{'}\hbox{'}}-{B}_{12}^2{\beta}_m^{\hbox{'}}}{B_{12}^2}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)-\frac{B_{12}}{B_{11}}{B}_1{\beta}_m^{\hbox{'}}\right],\\ {}{d}_7=\frac{B_{12}}{B_{11}}{\beta}_m^{\hbox{'}}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\left(\frac{B_{22}}{B_{11}}{\varepsilon}_m^2-\frac{B_{22}}{B_{11}}{\beta}_m^{\hbox{'}\hbox{'}}+\frac{B_{66}}{B_{11}}{\eta}_{2m}^2\right),\\ {}{d}_8=\frac{B_{66}}{B_{11}}{\eta}_{1m}^2+{\eta}_{2m}^2-\frac{B_{11}{B}_{22}{\beta}_m^{\hbox{'}\hbox{'}}-{B}_{12}^2{\beta}_m^{\hbox{'}}-{B}_{12}{B}_{66}{\beta}_m^{\hbox{'}}+4{B}_{66}^2{\beta}_m^{\hbox{'}}}{B_{11}{B}_{66}}\\ {}-{a}^2{\varepsilon}_m^2\left({B}_2+\frac{4{B}_{66}-2{B}_{12}}{B_{11}}{\beta}_m^{\hbox{'}}-\frac{4{B}_{66}}{B_{11}}{\eta}_{1m}^2\right),\\ {}{d}_9=\frac{B_{66}}{B_{11}}{\eta}_{1m}^2{\eta}_{2m}^2-\frac{B_{11}{B}_{22}{\beta}_m^{\hbox{'}\hbox{'}}-{B}_{12}^2{\beta}_m^{\hbox{'}}-{B}_{12}{B}_{66}{\beta}_m^{\hbox{'}}+4{B}_{66}^2{\beta}_m^{\hbox{'}}}{B_{11}{B}_{66}}{\eta}_{1m}^2\\ {}+\frac{B_{12}5{B}_{66}}{B_{11}}{\beta}_m^{\hbox{'}}{\eta}_{2m}^2+\frac{\left({B}_{12}+4{B}_{66}\right)}{B_{11}}{B}_2{\beta}_m^{\hbox{'}}+\frac{B_{22}}{B_{11}}{\beta}_m^{\hbox{'}}{\beta}_m^{\hbox{'}\hbox{'}}\\ {}+{\varepsilon}_m^2\left[{a}^2\frac{B_{11}{B}_{22}{\beta}_m^{\hbox{'}\hbox{'}}-{B}_{12}^2{\beta}_m^{\hbox{'}}-4{B}_{12}{B}_{66}{\beta}_m^{\hbox{'}}}{B_{12}^2}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)-\frac{\left({B}_{12}+4{B}_{66}\right)}{B_{11}}{B}_1{\beta}_m^{\hbox{'}}\right],\\ {}{d}_{10}=\frac{B_{12}+4{B}_{66}}{B_{11}}{\beta}_m^{\hbox{'}}\left({\beta}_m^{\hbox{'}}-{\eta}_{1m}^2\right)\left(\frac{B_{22}}{B_{11}}{\varepsilon}_m^2-\frac{B_{22}}{B_{11}}{\beta}_m^{\hbox{'}\hbox{'}}+\frac{B_{66}}{B_{11}}{\eta}_{2m}^2\right).\end{array}} $$

This research was performed at a partial support of the Erasmus + ICM program within the framework of cooperation with Keele University, UK.

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Ghulghazaryan, G.R., Ghulghazaryan, L.G. & Kudish, I.I. Free Vibrations of a Thin Elastic Orthotropic Cylindrical Panel with Free Еdges. Mech Compos Mater 55, 557–574 (2019). https://doi.org/10.1007/s11029-019-09834-9

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