Reflected waves analysis in case of the deformable bodies' impact using longitudinal wave's propagation theory.

Hule, Voichita Ionela ; Tarca, Ioan Constantin ; Blaga, Florin Sandu 等

Abstract: This paper deals with coaxial percussion study of two cylindrical steel bars, based on longitudinal waves propagation phenomenon. The aim of the paper is to reveal reflected waves from free ends of the bars that are in percussion. It has been made a correlation between analytical and experimental results using an approximation calculus method of percussion parameters and diagram drawing for these waves.

Key words: wave propagation, data acquisition.

1. INTRODUCTION

In the paper are studied both the theoretical and experimental study of the coaxial impact of two elastic bodies having the longitudinal dimensions much greater than those transversals (such as light gauge rods). These kinds of forces appear during mining or geo-thermal soil log upon drilling string elements (Aarrestad & Kyllingstad, 1994).

The impact calculus of the elastic systems assumes the quantification of the stresses and strains that appears in the bodies after their impact. These data may be studied by means of the one-dimensional wave propagation theory in elastic bodies using the longitudinal wave equation theory (Benatar et al., 2003). This is a second order differential equation which allows a large family of solutions, sometimes difficult to achieve through analytical methods (Harrison & Nettleton, 1991), (Graff, 1991). Therefore an approximate calculus method of the impact systems was used, without the necessity of the wave equation solving (Brindeu et al., 2003). This method may be applied for the case of cylindrical long rods coaxial impact. The theory is based on the following hypothesis: at the collision moment all the points situated on the impact surface have the velocity v and the force N. These data (v, N) are transmitted to the adjacent surfaces of the contact surface with the finite velocity (c), as a wave named "impact wave".

The dynamics fundamental laws and the main principles of the plane collision theory are used to primary solve the equation system developed for the plane collision. Solving this system of equations, speeds and impact forces for each transversal section through which the deforming wave passes at a certain moment of time can be computed. Other data can be computed using these two parameters: rod stresses, contact surface displacement, transmitted energy and impact power.

To illustrate the wave's propagation process and also their time succession on the boundary limits, a wave propagation diagram can be drawn. Based on the information it offers the system equations used for the impact parameters calculus can be obtained. Also, the contact time can be calculated. For the case of the finite length rods, an experimental setup has been created. Its purpose was to relieve the phenomena that appear during collision, following the incidental wave and the reflected wave propagation through rod and also the computation of the contact time between rods.

2. THE CALCULUS OF THE PERCUSSION SYSTEM COMPOSED OF TWO FINITE LENGTH RODS

[FIGURE 1 OMITTED]

The percussion system is made of two rods: the percussion rod having the l1 length, and the stricken rod, having the [l.sub.2] length. Fig. 1 shows the percussion system and the corresponding wave diagram. The following notations were made: [S.sub.1], [S.sub.2] = cross-sectional areas of the rods; [[rho].sub.1], [[rho].sub.2] = rods materials densities; [c.sub.1], [c.sub.2] = longitudinal waves propagation velocity; v0 = initial velocity of the percussion rod; [v.sub.i] = instant velocity of the contact surface points; [N.sub.i] = mutual force of the rods during impact; [K.sub.1] = [S.sub.1][[rho].sub.1][c.sub.1];[K.sub.2] = [S.sub.2][[rho].sub.2][c.sub.2] = the impact impedance of the bodies; [T.sub.1] = 2[l.sub.1] / [c.sub.1]= the time necessary for the reflected wave generated at the free end of the percussion rod (C) to reach the contact surface (B); [T.sub.2] = 2[l.sub.2] / [c.sub.2] = the time necessary for the reflected wave generated at the free end of the stricken rod (A) to reach the contact surface (B).

After a period of time (which depends on the percussion rod length and the wave propagation velocity), at the boundary limits new waves occur. These waves are defined by the impact parameters: [N.sub.i] =impact force, and [v.sub.i] = impact speed. For each "i" period of time the corresponding parameters of the impact waves (that is force [N.sub.i] and speed [v.sub.i]) are computed (Brindeu et al., 2003). The particular case [T.sub.2] = 3[T.sub.1] was studied. The expressions for the impact parameters were achieved as function of the time period number, i, in which they occurs. Thus at every moment of time the force and the velocity of the points for a certain transversal section of the rod subjected to longitudinal impact can be computed. The impact force and velocity expressions are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Observations: If [K.sub.2]>[K.sub.1], after t = 2[l.sub.1]/[c.sub.1] period of time the percussion rod recoils from the stricken rod. If [K.sub.1]>[K.sub.2] the impact continues; the impact process ends after t = 2[l.sub.1]/[c.sub.2] period of time from the arrival of the reflected wave from the (A) free end of the stricken rod.

From the wave diagram (Fig. 1) one can see that the percussion rod waves having the [l.sub.1] / [c.sub.1] duration alternates with the stricken rod waves with a 2[l.sub.2] / [c.sub.2] periodicity. The number of these waves can be computed with the relation:

n = [l.sub.2][c.sub.1] / [l.sub.1][c.sub.2] (4)

For case n=3, at the moment when the reflected wave from the free end of the rod having the parameters [N".sub.1] = 0; [v".sub.1] reaches the B surface, the third wave reflected from the free end of the percussion rod having the parameters [N'.sub.3] = 0; [v'.sub.3] reaches that surface too.

For the impact system consisted in two rod of different finite lengths presented in Fig. 1, for a certain case the impact parameters values and the impact time were computed. Also the force and the velocity that occurs on the impact surfaces from the moment of impact till the recoil moment were computed. The impact parameters were computed using software realized in MATLAB.

([v.sub.0] = 2m/s; [d.sub.1] = 0,04m; [d.sub.2] = 0,035m; [l.sub.1] = 2m; [l.sub.2] = 6m)

3. EXPERIMENTAL RESULTS

The theoretical results achieved for the case of two different finite length rods impact were verified through experimental results using an experimental setup consisting in two suspended OLC45 cylindrical rods having diameters of 40mm and 35mm, and the length of 2m and 6m. On each rod strain gauges (T1, ..., T8) were mounted (Fig. 2). The signal is transmitted via a Wheatstone bridge to the data acquisition board inside a PC. Experimental data acquired refers to the experimental verification of the propagation theory and the reflection of the waves in free metallic rods as a result of their impact at low speeds and also to the impact time verification in these conditions. The signal measured on the first strain gauge mounted on the stricken rod (T1, Fig. 2) shows amplitude variations that can be explained by the reflected wave generated at the free end of the stricken rod influence upon it. This phenomenon can be noticed on the comparative diagram shown in Fig. 3. In Fig. 3.a the theoretical impact diagram is presented (similar to that in Fig. 1) and in Fig. 3.b the T1 strain gauge measured signal diagram is presented. Measured signal amplitude changes correspond to the apparition of the percussion rod reflected wave ([t.sub.B], [t.sub.D], [t.sub.F] points). Calculated values for the moments where the reflected waves from the free end of the percussion rod reach the contact surface are:

[t.sub.B] = [T.sub.1] = 0,771ms; [t.sub.D] = 2[T.sub.1] = (5)

In [t.sub.B] and [t.sub.D] points a decrease of the signal amplitude can be noticed (diagram in Fig. 3.b). This decrease corresponds to the moments where the first and the second reflected waves from the free end of the percussion rod reach the contact surface (see Fig. 3.a).

[FIGURE 2 OMITTED]

Fig.2. Strain gauge mounting schema on the percussion rod (1) and the stricken rod (2)

[FIGURE 3 OMITTED]

The signal amplitude change in [t.sub.F] point corresponds to the simultaneous apparition on the contact surface of the third reflected wave from the free end of the percussion rod and the reflected wave from the free end of the stricken rod. Also this moment coincides with the end of the contact period between rods. The impact time experimentally measured corresponds to that achieved through computation.

4. CONCLUSIONS

The correlations between the wave's propagation diagrams inside rods were realized: one of them was created through theoretical methods and the other through experiment (diagrams in Fig. 3), relieving a clear correspondence between measured and calculated results.

For the phenomena analysis which constituted the subject of the experimental studies, software data acquisition in Visual C++ along with computing software in MATLAB needed for experimental data computation and conditioning were made. This software can be use for further researches.

5. REFERENCES

Aarrestad, T. V.& Kyllingstad, A. (1994), Loads on drill-pipe during jarring operations, SPE Drilling Completion, December, 1994, pp. 271-275

Benatar, A.; Rittel, D. & Yarin, A. L. (2003), Theoretical and experimental analysis of longitudinal wave propagation in cylindrical viscoelastic rods, Journal of the Mechanics and Physics of Solids, Vol. 51, Issue 8, August, 2003, pp 1413-1431

Brindeu, L.; Hule, V.& Petcovici, O. (2003), Dynamic model of impact, considering the propagation of the stress waves in the deformable body, Scientific Bulletin of "Politehnica" University of Timisoara, Transaction of Mechanics, Tom 48, Fasc. 1, 2003, ISSN 1224-6077

Graff, K.F. (1991), Wave Motion in Elastic Solids, New York, Dover, ISBN 0-486-66745-6

Harrison, H. R. & Nettleton, T. (1991), Advanced Engineering Dynamics, John Wiley & Sons Inc., New York, ISBN 0-340-64571-7