Dynamic Stability Investigation of a Sandwich Beam Tapered along width and thickness with Temperature Gradient

The investigation to analyze a sandwich beam's dynamic stabilitywith asymmetric configuration, tapered along the thickness and width, and influenced by an alive axial load with temperature gradient is executed for several boundary conditions employing computational method. Use of Hamilton’s principle results in the equations of motion and related boundary conditions. Hill’s equations are achieved using non-dimensionalized equations of motion with the Galerkin’s method. Then, the influence ofseveral parameterson the dynamic stability for different boundary conditions are attained by applying Saito-Otomi conditions. The impact of differentparameters on the regions of instabilities observed and is showcased in a sequence of graphs using the appropriate MAT LAB program.

Core's complex shear modulus orG 2 (1+j) The ratio of the in-phase shear modulus of theviscoelastic core and the young's modulus of elasticity of the elastic layers.

Introduction
To attain superior characteristics like better stiffness and less weight, sandwich beams are extensively developedfor different engineering structures such as aerospace, helicopter blades, etc. The blades in gas or steam turbines and aerospace applications are exposed to high temperatures, so the temperature gradient's impact must be considered during the design of beams. The beams can be economical with the variation of cross-section configuration. Karand Sujata [1] evaluated the stability of a cantilever type sandwich beam under the effect of periodic load having symmetric configuration. They witnessed that the geometric and shear parameterenhances the system's stability, while the taper parameter had a detrimental effect on stability. The same authors investigated the stability of a nonuniformconfigurationbeam subjected to temperature gradient [2]. They realized that the beam's taper profile, thermal gradient, and elastic foundation stiffness affected the stability. Ray and Kar [3]inspected a sandwich beam with 3-layers and symmetric configuration for several boundary conditions. Theydetected that the core's loss factor, along with the shear parameter, improved the beam's stability. Asnani and Nakra [4] established the equations of motion for a multilayered sandwich beam and acquired the vibration damping features of beams with 15 layersand simply supported at the ends forcases such as constant weight, constant size, and flexural rigidity.The dynamic along with staticstability of a sandwich beamlying on a Pasternak foundation in the influence of temperature environment is investigated by Pradhan et al. [7].Pradhan et al. [8] examined a tapered sandwich beam's stability condition, which is on a variable Pasternak foundation. Chand et al. [9] examinedthe stability of a rotational beam with a parabolic-tapered profile and variable temperature grade.Pradhan and Dash [10] inspected the stability of asandwich beam with non-uniform configuration and viscoelastic supportwith variable temperature gradient.
The literature valuation informs that selective study has been performed for the stability of non-uniform beams with various conditions. Nevertheless, no research has been implemented before to analyzethe dynamic stability investigation of an asymmetric sandwich beam tapered along width and thickness with temperature gradient.This research work investigates the above-suggested configuration.

Figure 1. Asymmetric Tapered Beam Configuration
A taper3-layer sandwich beam with asymmetric configuration and length l with a pulsating load, Z acting axially along the un-deformed axis, is presented in figure 1. The assumptions to formulate equations of motion are same as in [7]. In this, Potential energy is expressed as The Kinetic energy is expressed as U represent the displacements for the top elastic and bottom elastic layer in the axial direction and 2 J represents the shearing strain in the viscoelastic core, represented by The mathematical modeling of the equations areattainedby the Hamilton's principle as presented below.
, , , (2) The dependent parameters for different section are, Here, E represent the thickness parameter. 1 2 and P P represent the density parameters.
The associated boundary conditions at the ends are

Approximation Solution
For (1)-(2), the approximate solutions are assumed to be as follows Where j f and l f , j w and l J are used as the time shape function and co-ordinate shape functions respectively. These are considered in such a manner so that it will satisfy maximum number of boundary conditions and the equations of motion [5]. By using Galerkin method and putting the solutionsachieved from (12)-(13) in dimensionlessform of equation of motion (1)-(2), the consequent equations are obtained in matrix form. As in [3], the shape function is implemented for the required boundary condition.

Regions of instability
The unstable zone boundaries for main and combination type resonances are found using the conditions given by Saito and Otomi [6].

3.Result Analysis
The model of sandwich beam is explored forvarious boundary conditions like fixed-fixed and fixedhinged. Numerical values are determined for various parameters like temperature grade parameter, width taper parameter, and depth taper parameters. The given values of parametersare taken for the considered problem Here we are considering, G represents the temperature gradient for the 1 st and 3 rd layers respectively.   values, the unstable areas shift towards lower frequency, resulting in worsening of the system stability. 3 D , also has the same trend on the system stability, as of 1 D .

Conclusion
The examination of dynamic stability of an asymmetric sandwich beam, tapered along thickness and width, subjectedto an axial pulsatingload with temperature gradient is executed for two different end conditions employing computational method.
Results proved that the upsurge in the temperature grade for the bottom layer increases the system's stiffness, thus improving the system's dynamic stability. An increase in taper parameters' values along with thickness and thermal gradient for the top layer worsen the dynamic stability. The beam's dynamic stability improves with an increase in the values of taper parameters along the width of the beam.