Igor A. Sergey M. Vladimir I. Ranjan Ganguli. Gevorg Baghdasaryan. Iosif Vulfson. Agnesa Kovaleva. Jan Awrejcewicz. Boris F. Nikita Morozov.
Vladimir Palmov. Oleg V. Grigori Muravskii. Home Contact us Help Free delivery worldwide. Free delivery worldwide. Bestselling Series. It is very difficult to study the nonlinear dynamic response of a shallow conical shell in an alternating magnetic field and subjected to mechanical loads.
The study on the nonlinear dynamic response of the shell is very rare.
Thermo-viscoelastic analysis of composite plates and shells using a triangular flat shell element
Based on the heat conduction equation and the heat balance equation, temperature field equations are derived. Based on nonlinear equations of classical plates and shells, considering the coupling effect of the Lorentz force and temperature stress, nonlinear magneto-elastic heat equations of shallow conical shell are deduced. Applying Galerkin method, the solution of thermomagneto-elastic equation, the rule of temperature, magnetic field and displacement varying with time under the coupling effect of the applied magnetic field and surface uniform stable mechanical loads are obtained.
H r is a function of the coordinate z and time t, and P z is constant. Ignoring the mechanical electric effect and considering the axial symmetry, the electrodynamics Equation 1 of shallow conical shell can be simplified as.
Figure 1. The diagram of shallow conical shell. As electromagnetic field varying with time induces current in shallow conical shell, which formats Joule heating effect, that is induction current loss. As to shallow conical shell, it can approximately assume that current distribute uniformly in shallow conical shell because of a low frequency of the applied magnetic field and current, then the current loss per unit time per volume can be calculated by the following formula:. According to Fourier heat transfer law and energy conservation law, the control equation of heat conduction is established, which means that the transient temperature field T z , t of shallow conical shell should satisfy the followed equation:.
In the axial symmetry condition, the control equation of heat conduction of shallow conical shell can be simplified to. Thus, the boundary condition is. The shallow conical shell in the time dependent electromagnetic field also suffer the temperature stress induced by Joule heat and Lorentz force in addition to the external mechanical loads P, The Lorentz force can be expressed as. When discussing the thermo-magneto-elastic nonlinear problem of a shallow conical shell, the Kichhoff-love straight normal hypothesis is adopted, that the normal section perpendicular to the neutral surface before deformation of the shell remains a straight line after deformation and perpendicular to the neutral plane after deformation, and its length remains unchanged.
Based on this assumption, the radial, toroidal and normal displacements of any point in a shell with a distance of z from the neutral plane under axial symmetry conditions can be expressed as:. Among them, u and w are the radial displacement and deflection at any point on the neutral surface of the shell respectively. Considering the Von-Karman type large deflection geometric relation of conical shell, the strain-displacement relation at any point in the shell is obtained:.
The physical equation of shallow conical shell considering temperature change can be calculated according to follow formula. In the axial symmetry condition, considering the coupling effect of Lorentz force, temperature stress and mechanical load, according to the classical theory of plates and shells, the control equation of shallow conical shell can be derived as follows:.
F z and N T are the resultant forces of Lorentz force in the transverse direction and the film force produced by thermal stress respectively. Consider the following boundary conditions and initial conditions of electromagnetic field, temperature field and elastic field. Equation 14 is solved by separation of variables. Substituting Formula 24 in to Equation 14 , boundary conditions and initial Condition 19 can be written as follows.
Assume that the solution of Equation 25 satisfy the Conditions 26 and 27 is following series form. Based on the separation of variables, assume that the solution of Equation 16 which satisfies the boundary Condition 20 shows the following form. Both sides of Equation 12 multiply. Applying Galerkin method, the initial boundary value problem 17a , 17b and 21 - Substituting expression 37 into Equation 17a , the solution of equation 17a satisfying with boundary Condition 21 and 22 is.
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Numerical solution of the dynamic response Equation 40 is obtained by Runge-Kutta method, the response relation of deflection and time is obtained. The shallow conical thin shell is shown in Figure 1 , which is made of aluminum and subjected to mechanical load and time-varying magnetic field.
Thermo-Dynamics of Plates and Shells
The physical parameters of the shallow conical thin shell are as follows. According to the above analysis, numerical calculation is obtained by Matlab, the results is shown in Figures Figure 2. Figure 3.
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