Flutter of a Beam in Supersonic Flow: No Need in Original Timoshenko-Ehrenfest Equations

This paper deals with comparison of consistent Timoshenko-Ehrenfest beam model with the original Timoshenko-Ehrenfest equations in vibration field. Omission of the fourth order derivative term in the governing differential equation of the original equations of Timoshenko and Ehrenfest is shown to be more consistent than retaining it as is done in the original Timoshenko-Ehrenfest set. This also simplifies the analysis considerably.

In this work we study the vibration of thick beam in two main sections: free vibrations and dynamic stability: flutter.

In the first section we shown derivation of both governing differential equations. After that we illustrate for each approach the solution of natural frequencies problem for a cantilever homogeneous beam keeping into account the shear deformation. The end of first section is characterized of a numerical example in which we report the first ten natural frequencies obtained with both approaches compared with each other and with also the results obtained with FEM software Strand7 which implements beam element with shear deformation.

The second section deals with dynamic stability of a beam into gas flow: flutter analysis. The beam, as in the previous section, is a cantilever homogeneous beam with shear deformation. We model the load due the gas flow with the piston theory. This theory represents the aerodynamic load by two terms: one for the aerodynamic damping and one for the velocity of the flow. After that we illustrate for each approach the derivation to the governing differential equation of flutter problem. We solved the problem with the well knowns Galerkin method. We treat the aerodynamic damping and the speed of the flow as parameters in order to obtain for each combination of both the stability boundary with Routh-Hurwitz criteria. After that selecting two values of damping, we analyse the relationship of real and imaginary part of the frequencies obtained with Galerkin method finding the critical velocity.

Conclusion are that in this study we have analysed two Timoshenko-Ehrenfest beam’s theory to understand the differences in terms of results. We applied these theories to free vibrations and flutter problem. We saw that for the natural frequencies the differences are negligible. In the flutter problem, Galerkin method required at least the sixth terms to obtain an acceptable result but with the eighth is better. If we compare the critical velocities obtain by the two theories at the eighth order, we saw that are the same.

As a conclusion, both versions of the Timoshenko-Ehrenfest beam theory perform analogously with consistent version being much easier to implement.