Plasma Beams Free Vibration Ivestigation using the Boubaker Polynomials Expansion Scheme

Problem statement: In the present study, plasma beams free vibration frequencies alteration has been investigated through an origina l protocol. Approach: The Boubaker Polynomials Expansion scheme BPES has been performed for deduci ng and ranging the dimensionless frequencies of the beam vibrations. Results: Natural frequencies of the plasma beam have been e stimated for given parameters values. Conclusion: Yielded results have been compared and discussed. It was found that a good agreement as the values determined with experi mental results determined by previous researchers.


INTRODUCTION
In the two last decade, plasma beams have been given interesting electric and fire properties (Dohnalik et al., 2011;Vo et al., 2010;Ghanouchi et al., 2008;Chen 2003;Qindeel et al., 2007;Khare et al., 2004;Della and Shu, 2005;Kisa et al., 1998;Meirovitch, 1986). Sollier et al. (2011) studied plasma beems vibration patterns in terms of peak irradiance, pulse width and duration for energy pulses transmitted through the breakdown plasmas generated in fluids. Recorded data for 25 ns-1064 nm beam pulses outlined the roles of avalanche ionization and multiphoton ionization. Vo et al. (2010) studied the Metastability Exchange Optical Pumping (MEOP) performed at elevated 3He gas pressures using an annular beam and elaborated an accurate vibration model profiling. Many analytical methods of analysis have been used to study the vibration of plates, shells and beams (Khare et al., 2004;Della and Shu, 2005;Kisa et al., 1998;Meirovitch, 1986;Sollier et al., 2011;.
In this study, a model on the vibration analysis of plasma beam has been developed and studied using a confirmed and tested scheme under some presumptions.

MATERIALS AND METHODS
According to the model presented in Fig. 1, The relationship between normal stress and bending moment, according to Bernoulli-Euler hypotheses, is given by Eq. 1: Where: ρ = Beam curvature. H = Beam radius. N = The number of layers Z k = k th layer outer face distance to neutral plane E x = Elasticity modulus Consequently, the relationship between the bending moment and the curvature can be written as Eq. 2 follows: Where: E ef = Effective elasticity modulus. I yy = Beam cross-sectional inertia moment. W = Lateral deflection x-dependent mode profile Plasma beam flexural motion is described by the Eq. 3: With a trivial solution Eq. 4: where, ϖ n is the frequency.
By introducing the expression (4) in Eq. 3, the following time-independent non-variable characteristic value problem is deduced Eq. 5: where, f n is the dimensionless frequency of the beam vibrations given by Eq. 6: Along with the boundary conditions, as per Eq. 7 Shu (2000): The solution of the system referring to Eq. 4 and 6 is obtained using the Boubaker Polynomials Expansion Scheme BPES (Awojoyogbe and Boubaker, 2009;Labiadh and Boubaker, 2007;Slama et al., 2008;Hossein et al., 2009;Fridjine and Amlouk, 2009;Belhadj et al., 2009a;2009b), through establishing the expression Eq. 8: where, B k4 are the 4k-order Boubaker polynomials, x∈]0,L[ is the normalized distance, r k are B k4 minimal positive roots, N 0 is a prefixed integer and 0 k k 1,..N = λ are unknown pondering real coefficients. Consequently, it comes for Eq. 5 that Eq. 9: The related boundary conditions expressed through Eq. 6. The Boubaker Polynomials Expansion Scheme (BPES) protocol ensures their validity regardless main equation features. In fact, thanks to Boubaker polynomials first derivatives properties are Eq. 10 and 11: Boundary conditions are inherently verified Eq. 12: The BPES solution is obtained by integrating, for a given value of N 0 , the whole expression given by Eq. 8 And finally deducing, for increasing values of N 0 , the corresponding frequency using Eq. 6.

RESULTS
Plasma beam frequencies estimated using the boubaker polynomials expansion scheme PBES are gathered in Table 1.

DISCUSSION
The natural frequency changes as a direct result of the change in the stacking sequence causes resonance if the changed frequency becomes closer to the working frequency. This feature is in good agreements with the results published by Aly et al. (2010); Alsultanny (2006); Mardani (2008) and Hashemi et al. (2007).
With relation to the deviations of the numerical results in relation to those recorded in the related literature, some possible measurement errors can be pointed out such as measurement noise or nonuniformity in the beam properties (voids, variations in radius, non uniform outer shape ..). Such factors are not taken into account during the numerical analysis, since the model considers a strictly homogeneous beam, what rarely occurs in practice.

CONCLUSION
This study presents a protocol for the calculation of natural frequency of the plasma beam. Calculations performed by means of Boubaker Polynomials Expansion Scheme (PBES) resulted in coherent similar results. Changes in the stacking sequence as well as damping effects, which can have a large influence on the path of the beam, are the subject of coming investigations.