Indian Journal of Science and Technology Vol. 5 No. 9 (Sep. 2012) ISSN: 0974- 6846 Research article “Thermal gradient” A.Khanna et al. Indian Society for Education and Environment (iSee) http://www.indjst.org Indian J.Sci.Technol. 3263 Effect of varying poisson ratio on thermally induced vibrations of non-homogeneous rectangular plate Anupam Khanna 1 *, Narinder Kaur 2 , Ashish Kumar Sharma 3 1 *Department of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana, India 2 Department of Mathematics, Maharishi Markandeshwar University, Mullana, Ambala, Haryana, India 3 Department of Mathematics, ManavBharti University, Solan (H.P.), India 1 *anupam_rajie@yahoo.co.in, 2 narinder89.kaur@gmail.com, 3 ashishk482@gmail.com Abstract A mathematical model will help the engineers in science and technology applications. In this paper, authors studied a temperature-thickness coupling problem of a non-homogeneous rectangular plate in which temperature varies bi- linearly & thickness varies linearly in x-direction. Due to non-homogeneity, it is considered that poisson ratio varies exponentially in x direction. Ray-Leigh Ritz method has been adopted to calculate the time period for fixed two modes of vibration for different values of aspect ratio, thermal gradient and taper constants. All results are shown in graphs. Keywords: Vibration, Thermal gradient, Taper constant, Aspect Ratio, Time Period Introduction There has been amazing discoveries in the field of vibration research, drawing the attention of scientists and design engineers to study tremendous effect of vibrational behaviour in engineering and modern technology. In the field of mechanical engineering, new discoveries can’t be possible without considering the effect of vibration as almost all machines and engineering structures experience. Study of vibration is not just confined to science but also our day to day life. From its constructive aspects in aircrafts engineering, space technology, etc., to the destructive aspects e.g. earthquake, satellite; nothing is untouched by vibration effects. Plates of variable thickness are commonly used in many engineering applications like spacecrafts, submarine, nuclear reactor ships etc. So it is the need of the hour to get deeper knowledge of the plate’s behaviour and their characteristics, which would in turn help to perceive their potential in many fields. In the modern technology, an interest towards the effect of high temperatures on non-homogeneous rectangular plates of variable thickness is developed due to applications in various engineering branches such as nuclear, power plants, aeronautical, chemical etc. where metals and their alloys exhibits visco-elastic behaviour. The reason for these is that during heating up periods, structures are exposed to high intensity heat fluxes and material properties undergo significant changes; in particular thermal effect cannot be negligible. A lot of literature is available in one dimensional variation in temperature with thickness variation of plates, but negligible work is found in two dimensional temperature variations. Cheung and Zhou (1999) have studied several aspects concerning the formulation and the solution of amplitude equations for free vibration of systems with cubic non-linearity. Dhotarad and Ganesan (1978) analysed the dynamic free response of thin rectangular plates subjected to one and two dimensional steady state temperature distributions. Gupta et al. (2007) had analysed the effect of non-homogeneity on thermally induced vibration of orthotropic visco-elastic rectangular plate of linearly varying thickness. Gupta et al. (2009) had evaluated time period and deflection for the first two modes of vibration of visco-elastic rectangular plate and for various bi-linearly thickness variation. Jain and Soni (1973) had studied the free vibrations of rectangular plates with thickness varying parabolically. Kumar and Sanjay (2003) had analysed the vibration of visco-elastic isotropic rectangular plate with varying thickness in two directions i.e. linearly in one and parabolically in other direction. Khanna and Anupam (2005) had studied the vibrations on visco-elastic rectangular plate with the variable thickness without considering thermal effects. Lal Roshan and Dhanpati (2009) had analysed the vibrations of non-homogeneous orthotropic rectangular plates of varying thickness with two opposite simply supported edges and resting on two-parameter foundation. Laura et al. (1979) had studied the transverse vibrations of rectangular plates with linear variation of the thickness in the x and y directions. Leissa (1987) had helped the many researchers by collecting the various research papers which show the latest researches done in the field of vibration of plates without considered two dimensional thermal effects. Leissa (1987) had evaluated the effect of thermal gradient on the vibration of parallelogram plate with linearly varying thickness in both direction and thermal effect in linear form only. Singh and Saxena V (1996) had studied the transverse vibrations of a rectangular plate of variable thickness with different combinations of boundary conditions at the four edges. Tomar and Gupta (1983) had evaluated the thermal gradient effect on the vibration of a rectangular plate having bi-directional variation thickness. Tomar and Gupta (1985) had studied the effect of thermal gradient on frequencies of an orthotropic rectangular plate whose thickness varies in two directions. Li (2005) gave an analysis on modal characteristics on vibrations of rectangular plate with general elastic supports along its edges.