Free Vibration Analysis of Multiphase Magneto-Electro-Elastic Composite Conical Shells

Objectives: In the present article, free vibration of multiphase Magneto-Electro-Elastic (MEE) conical shell having a uniform thickness is examined for Clamped-Free (C-F) boundary condition. Method: The study is carried out using a semi-analytical approach for diﬀerent volume fractions (V f ) 0, 0.2, 0.6 and 1.0 of BaTiO 3 in BaTiO 3 -CoFe 2 O 4 smart composite conical shell for three diﬀerent semi-vertex angles 20 o ,35 o and 50 o . The piezoelectric (P e ) and piezomagnetic (P m ) phase on natural frequencies of MEE truncated conical shells are discussed for diﬀerent circumferential modes. Findings: The parametric study indicates that natural frequency decrease with an increase in V f of BaTiO 3 in magneto-electro-elastic truncated conical shells. Novelty: Studies on MEE constant thickness truncated conical shell using BaTiO 3 and CoFe 2 O 4 as (P e ) & (P m ) smart composite for clamped-free boundary condition to analyse the eﬀect of the frequency with diﬀerent semi-vertex angle and cone heights. Present commercial FEA software tools are limited to 2 coupling ﬁelds. In this research, coupling between 3 ﬁelds considered for MEE material. Hence, a computer code is developed to study the inﬂuence coupling between electric, elastic and magnetic ﬁelds, which can be used for any combinations of boundary conditions and volume fractions. varying thickness (3) . The frequency characteristics of the truncated conical shell are studied having different geometric parameters (4) . Free vibration of MEE cylindrical shell analysed using governing equations (5) . kernel particle (kp) functions are used to study the thin conical shell (6) . A study is conducted to know the piezomagnetic effect on multiphase MEE cylindrical shell (7) . Shell and plates behaviour is analysed using the differential quadrature method (8) . Novozhilov theory is used to investigated thick cones with smaller height for different vertex angles and end conditions (9) . The study is conducted on the conical shell using closed-form auxiliary functions along with the Rayleigh-Ritz procedure (10) . The truncated composite conical shell is analysed for different volume fraction using third-order shear deformation theory (11) . A study is conducted on free vibration conical shell using FSDT with six degrees of freedom in the direction of thickness (12) . Conical shell is analysed using Flügge thin shell theory with different boundary conditions (13) . Multi-layered conical shell analysed using coupled differential equations and spline approximation method (14) . Truncated conical shell behaviour studied using arbitrary boundary conditions employing Hamilton’s principle (15) . Free vibration of MEE plates analysed using condensation technique (16) . Magneto-Electro-Elastic plates studied using the Hamiltons principle (17) . Review is made on different techniques to analyse the behaviour of MEE materials. (18) . Research on conical shell vibration under various parameters effect like boundary conditions, cone angle, thickness, radius to height ratio, length to radius ratio has attracted attention in engineering (19,20) . With the literature view, numerous works are carried out of free vibration on the conical shell using standard numerical methods. Meanwhile, a very less study on MEE truncated conical shell using BaTiO 3 and CoFe 2 O 4 as (P e ) & (P m ) smart composite for clamped-free boundary condition. At present, a parametric study has been performed to analyse the effect frequency with different and cone heights.


Introduction
Smart materials and smart structures are important as they have wide varieties of applications in engineering such as sensors, actuators and especially in vibration and noise control. Several materials and technologies have been proposed and investigated. Axisymmetric conical shells are shown their applications in aerospace and shipbuilding. A numerical approach is used to obtain the frequency for conical shell for different boundary conditions (1) . The vibration behaviour of the axisymmetric conical shell was analysed using FEM and optimization studies were carried out (2) . A study is conducted for the vibration of the conical shell having different cone angles with constant and https://www.indjst.org/ varying thickness (3) . The frequency characteristics of the truncated conical shell are studied having different geometric parameters (4) . Free vibration of MEE cylindrical shell analysed using governing equations (5) . kernel particle (kp) functions are used to study the thin conical shell (6) . A study is conducted to know the piezomagnetic effect on multiphase MEE cylindrical shell (7) . Shell and plates behaviour is analysed using the differential quadrature method (8) . Novozhilov theory is used to investigated thick cones with smaller height for different vertex angles and end conditions (9) . The study is conducted on the conical shell using closed-form auxiliary functions along with the Rayleigh-Ritz procedure (10) . The truncated composite conical shell is analysed for different volume fraction using third-order shear deformation theory (11) . A study is conducted on free vibration conical shell using FSDT with six degrees of freedom in the direction of thickness (12) . Conical shell is analysed using Flügge thin shell theory with different boundary conditions (13) . Multi-layered conical shell analysed using coupled differential equations and spline approximation method (14) . Truncated conical shell behaviour studied using arbitrary boundary conditions employing Hamilton's principle (15) . Free vibration of MEE plates analysed using condensation technique (16) . Magneto-Electro-Elastic plates studied using the Hamiltons principle (17) . Review is made on different techniques to analyse the behaviour of MEE materials. (18) . Research on conical shell vibration under various parameters effect like boundary conditions, cone angle, thickness, radius to height ratio, length to radius ratio has attracted attention in engineering (19,20) .
With the literature view, numerous works are carried out of free vibration on the conical shell using standard numerical methods. Meanwhile, a very less study on MEE truncated conical shell using BaTiO 3 and CoFe 2 O 4 as (P e ) & (P m ) smart composite for clamped-free boundary condition. At present, a parametric study has been performed to analyse the effect of the frequency with different semi-vertex angle and cone heights.

Formulation
The governing equations of MEE are referred for deriving the FE model in r, θ , z coordinate system such that material properties and geometry remains same in 'q' direction (5) (7) .
Free vibration studies for different V f of MEE conical shell is conducted for Clamped-Free boundary condition. The conical shell structure is modelled using three noded triangular elements with three degrees of freedom (DOF) for each node.
The equations used in the finite element model are; Where u r , u q and u z are mechanical displacements. The relation between electric field vector (E)& electric potential (ϕ ) is.
The relation between magnetic field (H) & magnetic potential (ψ) is.
σ r , σ θ , σ z τ zr are Stress components; D r , D z are elecrtical displacements; B r , B z are magnetic displacements; ε r , ε θ , ε Z , γ zr are strains; E r , E z indicate electric fields; H r , H z represents magnetic fields. C i j , ε i j and µ i j are elastic, dielectric and magnetic permeability constants and e i j , q i j and m i j are the P e , P m and magnetoelectric material constants respectively. https://www.indjst.org/ For free vibration studies (5) (7) , [ [K MEE ] = Stiffness matrix for fully coupled magneto-electro-elastic material.
[K uu ] = Structural stiffness matrix considering elastic constants [K u ] = Structural stiffness matrix considering coupled elastic − electric potential material.
The component matrices of equation The distribution of {ϕ } and {ψ} are

Volume Fraction (V f ) P e phase in MEE Conical Shell
The analysis is conducted for V f (0, 0.2, 0.6, and 1.0) of Pe phase in MEE smart materials. The density for MEE material is 5730 kg m -3 (7) .

Validation
The developed computer code for MEE conical shell finite element analysis is validated (3) and the results of the validation show good agreement. The dimensions of the isotropic conical shell are R/H= 0.3, base radius (R=1.25m), constant thickness (h=0.0625m), density is 2410 kg m -3 , Young's modulus (E=30x10 9 Nm -2 ). https://www.indjst.org/

Analysis of MEE conical shell for Clamped-Free boundary condition
Here free vibration studies on three MEE conical shells with different semi vertex angle, viz.,

Analysis of MEE conical shell for V f = 0 2
The study conducted on Clamped-Free boundary condition for V f = 0.2 and Tables 5 and 6 shows the results of the study. Table 5 shows the frequency for top-end clamped and bottom end free conical shell and Table 6 shows results for top-end free and bottom end clamped. As the semi-vertex angle increases the frequency values in Table 5 is low compared with the frequency values in Table 6.