Linear Free Vibration Analysis of Tapered Rectangular Cantilevered Timoshenko Beams using Energy Field Method

Here in this paper we discuss about Energy field method. There are many methods to evaluate the natural frequency of the structures but in this research paper the authors developed a method called “Energy field method” which reduces computational efforts compared with the other methods and which is successfully applied for the Cantilever boundary condition of a tapered (rectangular cross section) Timoshenko beam and calculated the fundamental frequency parameter values and compared the results with existing literature. To confirm the precision, coherence and adaptability of the model these resultant values are also compared with modal structural analysis values in Ansys10 software.


Introduction
The cantilever beam is a constructional piece in which one end is fixed supported and other end is free. The connection of beam with support is usually perpendicular and beam is vertical. For a normal cantilever beam we can consider only shear modulus but in case if Timoshenko cantilever beam we can consider both normal and rotational effects at a time.
Vibrations are the mechanical structures experience whereby swinging backwards and forwards occurs about the equipoise spot. When a mechanical structures is in swinging by its own weight that vibrations are called free vibrations. The frequencies of those vibrations are called as natural frequencies of the structure. Vibration motion of a mechanical structure could understand by conservation of energy.
In the present method the concept of Energy Field Method is presented for the vibration analysis of tapered Timoshenko cantilevered beams, where the transverse displacement is expressed in terms of total rotation and its derivatives depending on the transverse shear deformation theory, in which the number of un known coefficients is brought down to exactly hk/l in the coupled displacement field method. To demonstrate the efficiency of the proposed method, the expressions for fundamental frequency are obtained for tapered shear flexible Timoshenko beams with Cantilever type of boundary condition. The numerical results presented in this paper are matching very closely with the existing literature.
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Coupling equation
According to law conversation of energy total energy of the system at any state is constant applying the same principle we can develop the present concept. The concept of coupled displacement field is explained in detail. In the Energy Field Method with the single term admissible function for Eita, the displacement function for w is derived using the coupling equation.
While the beam is in stationary position the sum of strain energy and potential energy is constant (U+W= constant). When the beam is free to vibrate by its own weight the sum of strain energy and kinetic energy is constant (U+K.E= constant) this is the basic principle of energy field method. For derivation of coupling and frequency function we can use the principle of minimization of potential energy from FEM equations.

Simulation results
The concept of Energy field method is used to determine the fundamental frequencies of tapered shear flexible beams, with most practically used boundary condition. The boundary conditions of the beams considered is Cantilever beam, one with axially immovable ends. For more accuracy of the results these values are checked by using ANSYS 10 software. The procedure of the simulation is explained below.
1. Open the ANSYS workbench tool.
2. Click on structural analysisone window will be appear.
3. Give engineering data related your applied material.
6. Click on modelsupportsselect fixed supportclick on one face of the geometry.
7. Click on model -Analysis settingsgive max number of modessolve.
We can note the frequencies values which will be appear on the screen.

Conclusion and future scope
This method is developed for linear free vibration analysis of tapered Timoshenko cantilever beams. Moreover the tapered functions of breadth and height of the cross section of the beam have been taken into account for deriving the natural frequency function with cantilever beam boundary conditions. These values are compared with existing method called coupled displacement field method and Rayleigh-Ritz method for The convergency, accuracy, efficiency, and versatility of the method these frequencies values are also compared with Ansys software. The accuracy of the present method results is very rapid. This present research work can be extend for hinged-hinged beam and simply supported beams.