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The dynamic response of a Timoshenko beam with immovable ends resting on a nonlinear viscoelastic foundation and subjected to motion of a traveling mass moving with a constant velocity is studied. Primarily, the beam’s nonlinear governing coupled PDEs of motion for the lateral and longitudinal displacements as well as the beam’s cross-sectional rotation are derived using Hamilton’s principle. On deriving these nonlinear coupled PDEs the stretching effect of the beam’s neutral axis due to the beam’s fixed end conditions in conjunction with the von-Karman strain-displacement relations is considered. To obtain the dynamic responses of the beam under the act of a moving mass, derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic responses are obtained using mode summation technique. Furthermore, after verification of our results with other sources in the literature a parametric study on the dynamic response of the beam is conducted by changing the velocity of the moving mass, damping coefficient, and stiffnesses of the foundation including linear and cubic nonlinear parts, respectively. It is observed that the inclusion of geometrical and foundation stiffness nonlinearities into the system in presence of the foundation damping will produce significant effect in the beam’s dynamic response.

The topic of vibration study of structural elements such as strings, beams, plates, and shells under the act of a moving mass is of great interest and importance in the field of structural dynamics. It should be noted that the review of numerous reported studies related to the dynamic behavior of mechanical/structural systems discloses that almost linear behavior of such systems is considered. Indeed, in reality such systems inherently and naturally have nonlinear behavior, for example, due to the geometrical nonlinearity or when they are subjected to external loadings comparatively large enough. As we will see later on in the modeling of the problem, the stretching of the beam’s neutral axis due to fixed ends condition adds another nonlinearity to the dynamical behavior of the system. In addition, there are some other external distinct mechanical elements having nonlinear behavior attached to such structures like shock energy absorbers or viscoelastic foundations which will add further other nonlinearities in the model analysis. From mechanical point of view, any beam structure can be modeled as a thin or thick beam for which different theories usually can be implemented. In extending the issue of the linear analysis of a Timoshenko beam resting on a viscoelastic foundation and under the act of a moving mass, certainly including the nonlinear behavior in the analysis will well provide more reliable and accurate results specifically in cases where the energy damping or any restraint on the beam’s amplitude vibrations plays an important role in this filed.

Some examples of actual cases can be addressed as a bridge when traveled by moving vehicles, moving trains on railway tracks, simulation of high axial speed machining processes in milling operation, and internal fluid flow in piping systems resting on a soil foundation. Moreover, shafts in the rotating machinery resting on the elastic supports (journal bearings) and floating in an industrial lubricant can be modeled as beams on viscoelastic foundation. This approach can be viewed in modeling of rotating blades in a helicopter or in rotating blades of turbomachineries.

In [

Based on this presumption and on the same line as the other studies this work is initiated. In the present study for the first time three nonlinear governing coupled PDEs of motion for a Timoshenko beam resting on a nonlinear viscoelastic foundation and under the act of a moving mass are derived. Then by applying Galerkin’s method, these three obtained governing nonlinear coupled second order ODEs are solved numerically using

In this study, the dynamic responses of a pinned-pinned Timoshenko beam resting on a nonlinear viscoelastic foundation and under the act of a moving mass are the main subject in our analysis. In fact our analysis focused on the study of the effects of the nonlinearities introduced due to large deflection of the beam (geometric nonlinearity) and the nonlinear stiffness of the foundation on the dynamic behaviour of above mentioned Timoshenko beam.

Consider an isotropic and homogenous Timoshenko beam with beam density

Timoshenko beam under the motion of a concentrated point mass resting on a viscoelastic foundation. (a) Timoshenko beam with no stretching effect on

According to von-Karman’s theory, the kinematics relations for the beam shown in Figure

Force relation in

Force relation in

Moment relation about

In this study Galerkin’s method is chosen as a powerful mathematical tool to analyze the vibrations of a Timoshenko beam. Based on the separation of variables technique, the response of the Timoshenko beam in terms of the linear free-oscillation modes can be assumed as follows [

In the next step primarily we substitute (

As mentioned in the introduction at the moment no specific results exist for the problem under consideration in the literature. Therefore, to verify the validity of the results obtained in this study we primarily consider some special cases by which our results can be compared with those existing in the literature.

In the first attempt we set ^{2}, ^{2}, ^{3}, ^{2}, ^{2 }, and

Based on the above data, the computer code was run for this case, and the normalized instantaneous lateral displacement (

Instantaneous normalized lateral displacement ^{2},

A simply supported Timoshenko beam resting on a Pasternak-type viscoelastic and a shear foundation and traveled by a constant velocity moving force as shown in Figure

A simply supported Timoshenko beam resting on a nonlinear Pasternak viscoelastic foundation and subjected to a moving force

To check on the validity of our solutions based on (^{4}, ^{2}, ^{2}, ^{2}, ^{3}, ^{2}, ^{4}, ^{2},

Lateral displacement

After being satisfied with the validity of solution technique, the beam’s instantaneous dynamic lateral deflection is calculated in the next step. In obtaining these results the following data [

It should be mentioned that all deflection variations versus moving mass instantaneous positions are given in a nondimensional form, that is,

The effect of variation of nonlinear part of foundation stiffness, ^{2}, ^{2}, and ^{4}. Furthermore, when the mass velocity increases, the maximum value of the beam’s dynamic response increases up to ^{21} N/m^{4}, by increasing the value of ^{4} and

Variation of dimensionless instantaneous dynamic response, ^{2}, ^{2}, and

Figure ^{2}, ^{4}, ^{2}, and

Variation of dimensionless instantaneous dynamic lateral deflection, ^{2}, ^{2}, and ^{4}.

Using linear (M-1 model) and nonlinear (M-2 and M-3 models) Timoshenko beam models, Figure ^{22} N/m^{4}, ^{2}, and ^{4} are the same and always lower than the one obtained from the M-1 model, where the maximum difference between M-1 and M-2 or M-3 models always happens at ^{2}, this difference becomes negligible. Furthermore, by comparing Figure

Variation of dimensionless instantaneous dynamic lateral deflection, ^{2}, and ^{4}.

Figure ^{2}, ^{4}, and ^{2} this difference becomes negligible. Furthermore, by comparing Figure ^{2} the obtained beam’s dynamic response at

Variation of dimensionless instantaneous dynamic lateral deflection, ^{2}, and ^{4}.

Figure ^{2}, ^{4}, and ^{2} are considered. It can be seen from this figure that the maximum value of dynamical response,

Variation of dimensionless instantaneous dynamic lateral deflection, ^{2}, ^{4}, and ^{2}; (- - - - -) M-1 model and (—) M-2 or M-3 model.

The longitudinal and lateral deflections as well as the rotation of warped cross section of three different Timoshenko beams of M-1, M-2, and M-3 models resting on viscoelastic foundation and subjected to a moving mass with constant velocity are all considered by including the nonlinear nature in the beam’s geometry and the stiffness of viscoelastic foundation. The outcome results are as the following.

It is concluded that by including geometrical and foundation nonlinearities into the reference M-1 model, as the size of nonlinearity increases the difference of the beam’s dynamic response in the nonlinear models (M-2 or M-3) and M-1 model increases. This difference becomes more pronounced for the comparatively large enough values of nonlinear stiffness part of viscoelastic foundation.

It is observed that the beam’s dynamic response results for the fully nonlinear beam model, that is, M-3 model, are always lower than the one obtained from the linear beam model, that is, M-1 model.

It is concluded that when the value of linear stiffnesspart of foundation increases, the difference between M-1 and M-2 or M-3 models decreases accordingly, where forthe large values of linear stiffness part of foundation this difference becomes negligible. Moreover, when the value of foundation damping increases the beam’s dynamical response difference between M-1 and M-2 or M-3 models decreases accordingly.

Because both stretching effect and nonlinear stiffness part of viscoelastic foundation in conjunction with moving mass condition and the other values of the foundation’s parameters are significant factors in the nonlinear dynamic behavior of the beam, with changing the beam’s slenderness ratio

The definition of different matrices used in calculation of the nonlinear coupled ODEs of modal relations (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper is dedicated to Ahmad Mamandi’s dearest Professor Dr. Mohammad Hossein Kargarnovin, passed away on Monday, November 4, 2013. He was a fine, kind, decent, sincere, good tempered, and well-liked professor who will be sadly missed by all who knew him. May his soul rest in peace.