Sharif Digital Repository

Natural frequencies are important dynamic characteristics of a structure where they are required for the forced vibration analysis and solution of resonant response. Therefore, the exact solution to free vibration of elastically restrained Timoshenko beam on an arbitrary variable elastic foundation using Green Function is presented in this paper. An accurate and direct modeling technique is introduced for modeling uniform Timoshenko beam with arbitrary boundary conditions. The applied method is based on the Green Function. Thus, the effect of the translational along with rotational support flexibilities, as well as, the elastic coefficient of Winkler foundation and other parameters are... 

In this paper the free vibration of single walled carbon nanopeapods encapsulating C60 molecules is considered. The nanopeapod is embedded in an elastic medium and clamped at both ends. The Euler-Bernoulli beam model is used for the carbon nanotube and the C60 molecules are considered as lumped masses attached to the beam. Based on the nonlocal elasticity theory the governing equation of motion is derived and the resonance frequencies of the nanopeapod are obtained. The effects of small scale, foundation stiffness and ratio of the fullerenes' mass to the nanotube's mass on the frequencies are studied and some conclusions are drawn  

An exact solution is established pertaining to the problem of undamped free vibration of a thin circular plate resting on a Winkler foundation with variable subgrade modulus. The solution is performed by applying the infinite power series method. Moreover, the solution procedure is demonstrated through an illustrative example, wherein the general frequency equation is derived for two different boundary conditions. The correctness of the solution is also verified using results available in the literature. Finally, it is shown that the proposed method of solution is directly applicable to the more-general problem of circular plates on a variable-modulus Pasternak-type foundation  

In this paper, the whirling frequencies of simply supported and clamped rotating cylindrical shells surrounded by an elastic foundation are investigated. The Love's shell theory is used along with the Winkler foundation to obtain the governing equations of motion. An exact power series solution is obtained for arbitrary boundary conditions and the results are verified with the literature. Several case studies are performed, and the effect of spinning speed, foundation stiffness, and geometrical dimensions of the cylinder on the whirling frequencies are investigated  

An exact solution is established pertaining to the dynamic response of an Euler–Bernoulli beam resting on a Winkler foundation with variable subgrade modulus. The solution is performed by employing the infinite power series method. Moreover, using the Frobenius theorem, the proposed method is extended in order to solve the problems wherein the variation of the modulus is not an analytic function. The solution procedure is demonstrated through several illustrative examples, and the correctness of the results has been ascertained through comparison with recognized solutions in the literature. Finally, it is shown that the proposed method of solution is directly applicable to the more general... 

This paper examines the effects of foundation uplift on the dynamic response of steel frames supported by shallow foundations. In order to achieve this goal, the dynamic responses of a number of two-dimensional concentrically braced steel frames with fixed base, flexible base preventing uplift, and flexible base allowing uplift, are compared. The steel frames are modeled by considering material nonlinearities and P-Delta effects, while the soil-foundation system is modeled through the Winkler foundation approach. The dynamic analyses are conducted using 15 far-fault ground-motions with various intensities and different response parameters are monitored. According to the achieved responses,... 

Plates on elastic foundations have attracted the attention of many researchers. Some elementary models have been introduced to consider interactions between the plate and its foundation. Other improved models have been proposed to develop basic models. In this work, a model based on the Winkler-foundation theory is proposed, while the constant parameter of Winkler is assumed to be variable; such as non-uniform springs with the functionality of the domain position, along with the plate and beam span in order to consider the non-uniform behavior of the foundation. The governing equation on the system is solved by using the Galerkin method and effects such as the presence of rigid points in the... 

In this paper, the vibration of a moderately thick plate to a moving mass is investigated. Pasternak foundation with a variable subgrade modulus is considered to tackle the shortcomings of Winkler model, and an analytical-numerical solution is proposed based on the eigenfunction expansion method. Parametric studies by using both CPT (Classical Plate Theory) and FSDT (First-Order Shear Deformation Plate Theory) are carried out, and, the differences between them are also highlighted. The obtained results reveal that utilizing FSDT without considering the rotary inertia leads to a smaller deflection in comparison with CPT pertaining to a thin plate, while it demonstrates a greater response for... 

This study pertains to the free vibration problem of beams on an elastic foundation of the Winkler type, which is distributed over a particular length of the beam. Closed form solutions are developed by solving the governing differential equations of beams. Moreover, an innovative mathematical approach is proposed to find the precise analytical solution of the free vibration of beams with mixed boundary conditions. Results are discussed in detail through verification studies. Ultimately, it was concluded that the proposed mathematical method could successfully obtain the exact solution to the free vibration problem of beams on partial elastic foundations under mixed boundary conditions  

Natural frequencies are important dynamic characteristics of a structure. Therefore, the exact solution pertaining to free vibration of stepped circular plate elastically restrained against rotation, translation, and internal elastic ring support resting on an arbitrary variable elastic foundation using Green Function is presented in this paper. Thus, an accurate and direct modeling technique is introduced for modeling stepped circular plate on an arbitrary variable elastic foundation with arbitrary boundary conditions and internal elastic ring support. The effect of the translational along with rotational support flexibilities, as well as, the elastic coefficient of Winkler foundation and... 

In the present study, free vibrations of single walled carbon nanotubes (SWCNT) on an elastic foundation is investigated by nonlocal theory of elasticity with both beam and shell models. The nonlocal boundary conditions are derived explicitly and effectiveness of nonlocal parameter appearing in nonlocal boundary conditions is studied. Also it is demonstrated that the beam model is comparatively incapable of capturing size effects while shell model captures size effects more precisely. Moreover, the effects of some parameters such as mechanical properties, foundation stiffness, length and radius ratios on the natural frequencies are studied and some conclusions are drawn