Parametric instability of exponentially tapered, pre-twisted sandwich beam on a variable Pasternak foundation with viscoelastic end supports in temperature gradient
P.R. Dash * C R Nayak
The purpose of this paper is to study about the dynamic stability of exponentially tapered pre-twisted sandwich beam on a variable Pasternak foundation which is propped at ends by viscoelastic supports subjected to an axial periodic force and temperature environment. A set of equations of motion are developed by using Lagrange method and instability regions are plotted using Hsu’s method. The effects of pre-twisted angle, temperature parameter, taper parameter, modulus of shear layer of foundation, stiffness of Pasternak foundation, and elastic foundation parameter on the regions of parametric instability are plotted.
Keywords: sandwich beams, viscoelastic core, pre-twisted tapered beams, Pasternak foundations, elastic foundation parameter, coreloss factor.
Lists of symbols
The inflating calls for high strength and light-weight structures in different fields of engineering claims the use of sandwich beams with some modifications. To reduce instability of these sandwich beams due to vibration during their practical applications, a viscoelastic core is used at the mid-plane of the beam. For convenience to analysis generally the sandwich structures (e.g.: plates, beams etc.) have been constructed, examined and analyzed for frequency response and coreloss factor. But in many practical applications, the constructions of real structures are not of that simple. Due to so many assumptions and design errors there be pre-twisting, un-symmetricity etc. will present in structures. So, researchers are interested for doing analysis on pre-twisted sandwich beams to study the effect of pre-twist and to use advantages of pre-twisting. The objectives of this paper are to study the effect of pre-twist angle, taper parameters etc. on the frequency plots by considering the Pasternak foundation and viscoelastic supports at ends.
The dynamic stability of pre-twisted sandwich beams was examined by different researchers during past time. Ray, et al 1 have examined for parametric instability of sandwich beam for various boundary conditions and concluded that loss factor increases with increase in static load parameter which also increase system stability. Parametric instability of asymmetric sandwich beam considering thermal gradient for different boundary conditions was investigated by Dash, et al 2 with a new conclusion that the thermal gradient parameter has very least effect on system stability. Very important results i.e. increase in taper parameters and stiffness of foundation increase the natural frequencies but decrease the regions of instability were given by Kar, et al 3 also suggested that increasing the modulus of shear layer will reduce the natural frequencies. Pradhan, et al 4 have analyzed on static and dynamic stability of asymmetric sandwich beam on variable Pasternak foundation under thermal gradient and concluded that static buckling load decreases with increase in thermal gradient and not effected by the modulus ratio. M.pradhan, et al 5 have also studied for analysis of asymmetric tapered sandwich beam under thermal gradient and same author 6 have also analyzed about free vibration of beam on variable Pasternak foundation. D.K Rao 7 have given data’s about loss factors and frequencies of different modes using various boundary conditions for sandwich beams. Study on the transverse vibration of pre-twisted sandwich beams is also done by same author 8. Here author has used Hamilton’s principle to get equations of motion and gave idea about change of loss factor with change in pre-twist angle. There were also some researchers T Rout, et al 9 who have studied on sigmoid Timoshenko beam resting on variable Pasternak foundation. Authors have considered different stiffness functions (like; sinusoidal, linear and parabolic). Saito and Otomi 10 have studied theoretically and experimentally the effect of viscoelastic supports on stability plot in a beam. Kar and Ray 11 have not only developed equations of motion using Galerkin method but also plotted dynamic stability graphs using Hsu’s method and concluded that with increase in pre-twist angle, stability decreases. Wang, et al 12 have analyzed about natural frequencies of vibration of beam on Pasternak foundation considering the rotary inertia effect. The solutions for parametrically excited multi degree of freedom dynamic system have been explained by C.S Hsu 13. Different dynamic plots were analyzed by Nayak, et al 14 for rotating sandwich beam by changing core material with magnetorheological elastomer material.
In this paper, authors study the dynamic analysis of viscoelastically supported, exponentially tapered, pre-twisted sandwich beam on Pasternak foundation placed in temperature environment. Authors are also interested to know the combined effect of both pre-twist angle and taper parameter along with temperature parameter on the regions of instability for the system. The similar analyses being carry out for the system by changing different parameters of Pasternak foundation, via modulus of shear layer, elastic foundation parameter and spring stiffness value.