In this manuscript, the dynamic stability and bifurcation occurrence for three famous types of plates including orthotropic sigmoid, power-law and exponential functionally graded plates under lateral stochastic loads are studied. Due to randomness, the behavior and analysis are not conventional deterministic investigation. So, the dynamic stability zone and border curves of bifurcation are evaluated using probability density function of the response. The latter is computed from a completely exact solution of the Fokker Planck Kolmogorov equation. The three dimensional dynamic stable zone and the border surfaces of bifurcation are obtained as a function of material parameter, in-plane forces and the mean value of lateral load. To generalize the results, all the parameters are transformed to some proper non-dimensional variables and then the effects of all prescribed parameters on the dynamic stability are completely discussed and compared. The comparison is done between the plates with themselves and also the corresponding homogenous plate. Finally the results are validated by the bifurcation diagrams of non dimensional deflection of plates that are obtained directly and numerically from the governing equations of plates.
AbediMAsnafiA (2015a) Instability and Bifurcation Behavior of Orthotropic S-FGM Plates Under Lateral Stochastic Loads Considering Variation of Material Properties. International Journal of Structural Stability and Dynamics. 1450108.
2.
AbediMAsnafiA (2015b) To reduce the instability region in the nonlinear transverse vibration of randomly excited plates using orthotropic P-FG material. Nonlinear Dynamics80(3): 1413–1430.
AmabiliM (2008) Nonlinear vibrations and stability of shells and plates, Cambridge: Cambridge University Press.
5.
AmabiliMPaıdoussisMP (2003) Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Applied Mechanics Reviews56: 349–381.
6.
AsnafiAAbediM (2015) A complete analogical study on the dynamic stability analysis of isotropic functionally graded plates subjected to lateral stochastic loads. Acta Mechanica. 1–17.
7.
Asnafi A and Mahzoon M (2005) Using FPK Equation to Analyze Nonlinear Behavior of Some Oscillators under Random Forcing Function. In: 13th Annual International Mechanical Engineering Conference, Isfahan University of Technology, Isfahan, Iran.
8.
BolotinVV (1984) Random Vibration in Elastic Systems, New York: Martinus Nijhoff.
9.
CaiGQLinYK (1998) On exact stationary solutions of equivalent non-linear stochastic systems. International Journal of Non-Linear Mechanics23: 315–325.
10.
CaugheyTKMaF (1982) The exact steady-state solution of a class of non-linear stochastic systems. International Journal of Non-Linear Mechanics17: 137–142.
11.
ChakrabartiASheikhAH (2010) Dynamic instability of imperfect laminated sandwich plates with in-plane partial edge load. Latin American Journal of Solids and Structures7: 457–474.
12.
ChenCSChenCWChenWR (2013) Dynamic Stability Characteristics of Functionally Graded Plates under Arbitrary Periodic Loads. International Journal of Structural Stability and Dynamics13: 98–118.
13.
ChibaRSuganoY (2007) Stochastic thermoelastic problem of a functionally graded plate under random temperature load. Archive of Applied Mechanics77: 215–227.
14.
DagS (2006) Thermal fracture analysis of orthotropic functionally graded materials using an equivalent domain approach. Eng. Fract. Mech73: 2802–2828.
15.
DagSYildirimBSarikayaD (2007) Mixed-mode fracture analysis of orthotropic functionally graded materials under mechanical and thermal loads. International Journal of Solids and Structures44: 7816–7840.
16.
DeyPSinghaMK (2006) Dynamic stability analysis of composite skew plates subjected to periodic in-plane load. Thin-Walled Structures44: 937–942.
17.
DowellEH (1970) Panel flutter-A review of the aeroelastic stability of plates and shells. AIAA Journal8: 385–399.
18.
DimentbergMF (1982) An exact solution to a certain non-linear random vibration problem. International Journal of Non-Linear Mechanics17: 231–236.
HasheminejadSKeshvariMAshoryM (2014) Dynamic Stability of Superelliptical Plates Resting on Elastic Foundations under Periodic In-Plane Loads. J. Eng. Mech140: 172–181.
21.
HuttJMSalamAE (1971) Dynamic Stability of Plates by Finite Elements. Journal of the Engineering Mechanics Division97: 879–899.
22.
JagtapKRLalASinghBN (2012) Stochastic nonlinear bending response of functionally graded material plate with random system properties in thermal environment. International Journal of Mechanics and Materials in Design8: 149–167.
KimJHKimHS (2000) A study on the dynamic stability of plates under a follower force. Computers & Structures74: 351–363.
25.
KitipornchaiSYangJLiewKM (2006) Random vibration of the functionally graded laminates in thermal environments. Computer Methods in Applied Mechanics and Engineering195: 1075–1095.
26.
LalAJagtapKRSinghBN (2013) Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties. Applied Mathematical Modelling37: 2900–2920.
27.
LanheW (2004) Thermal buckling of a simply supported moderately thick rectangular FGM plate. Composite Structures64: 211–218.
28.
LanheWHongjunWDaobinW (2007) Dynamic stability analysis of FGM plates by the moving least squares differential quadrature method. Composite Structures77: 383–394.
29.
LiJJChengCJZhangNH (2002) Dynamic stability of viscoelastic plates with finite deformation and shear effects. Journal of Shanghai University (English Edition)6: 115–124.
30.
LinYKCaiG (1988) Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II. J. Appl. Mech55: 702–705.
31.
RahmaniHMahmoudiSHAshoriA (2014) Mechanical performance of epoxy/carbon fiber laminated composites. Journal of Reinforced Plastics and Composites33(8): 733–740.
32.
RaoSS (2010) Mechanical Vibration, New York: Prentice Hall.
33.
RaoBNRahmanS (2005) A continuum shape sensitivity method for fracture analysis of orthotropic functionally graded materials. Mechanics of Materials37: 1007–1025.
34.
ReddyDVRRatnamBSRaoGV (2014) Prediction of dynamic stability behavior of thin square plates subjected to constant compressive and periodic including constant compressive loads on perpendicular edges. International Journal of Current Engineering and Technology2: 620–624.
35.
RiskenH (1984) Fokker-Planck Equation, Berlin Heidelberg: Springer, pp. 63–95.
36.
SahaKNKarRCDattaPK (1997) Dynamic stability of a rectangular plate on non-homogeneous Winkler foundation. Computers & structures63: 1213–1222.
37.
SahuSKDattaPK (2007) Research advances in the dynamic stability behavior of plates and shells: 1987–2005—part I: conservative systems. Applied mechanics reviews60: 65–75.
38.
SinghaMKDaripaR (2009) Nonlinear vibration and dynamic stability analysis of composite plates. Journal of Sound and Vibration328: 541–554.
39.
SladekJSladekVHellmichCHEberhardsteinerJ (2007) Heat conduction analysis of 3-D axisymmetric and anisotropic FGM bodies by mesh-less local Petrov–Galerkin method. Comput. Mech39: 323–333.
TimoshenkoSWoinowsky-KriegerS (1959) Theory of plates and shells, New York: McGraw–Hill.
42.
TouatiDCederbaumG (1995) Influence of large deflections on the dynamic stability of nonlinear viscoelastic plates. Acta Mechanica113: 215–231.
43.
XuHYaoXFengXHisenY (2008) Dynamic stress intensity factors of a semi-infinite crack in an orthotropic functionally graded material. Mechanics of Materials40: 7–47.
44.
YangJLiewKMKitipornchaiS (2004) Dynamic stability of laminated FGM plates based on higher-order shear deformation theory. Computational Mechanics33: 305–315.
45.
YongYLinYK (1987) Exact stationary-response solution for second order nonlinear systems under parametric and external white-noise excitations. Journal of Applied Mechanics54: 414–418.
46.
ZhangNHChengCJ (2003) Two-mode Galerkin approach in dynamic stability analysis of viscoelastic plates. Applied Mathematics & Mechanics24: 247–255.
