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The dynamic stability behavior of simply supported beams on the Pasternak foundation, with emphasis on the effect of the transition foundation parameter, subjected to a concentrated end axial periodic load, is presented in this paper. The energy method, using a single term exact trigonometric admissible function to represent the lateral deflection, is employed to obtain the dynamic instability regions. These regions are presented in the graphical form, for the first and second foundation parameters of the Pasternak foundation. The effect of the transition foundation parameter, which can be exactly evaluated for the simply supported beams on the Pasternak foundation, on the dynamic instability behavior of the simply supported beams, is investigated. The existence of the master dynamic instability curves is established in the present study by using the specific non-dimensional parameters for the applied periodic load and its radian frequency.

Nomenclature G = system geometric stiffness matrix K = system stiffness matrix M = system mass matrix N = axial or in-plane compressive periodic load N cr = buckling load N s = constant part of N N t = periodic part of N = N s =N cr = N t =N cr fg = as given in Eq. (7) f 1 g = eigenvector of the dynamic stability problem f 2 g = eigenvector of the free vibration problem f 3 g = eigenvector of the buckling problem = radian frequency of N t = as defined in Eq. (12) = =! ! = natural radian frequency

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