Coaxial impact of elastic bodies experimental analysis of the reflected waves.

Hule, Voichita ; Tarca, Ioan ; Blaga, Florin 等

1. INTRODUCTION

The paper presents an experimental study of the coaxial impact of two cylindrical rods having the longitudinal dimensions much greater than those transversals based on the longitudinal wave propagation theory.

An experimental setup for finite length rods was conceived in order to reveal the phenomena that appear during the impact period which are the longitudinal wave's propagation inside rods, analyzing the incident wave and also the reflected wave movements through rods and also the contact time analysis between rods (Brindeu et al., 2003).

For the phenomena analysis which constituted the subject of the experimental studies, software data acquisition in Visual C++ were made. MATLAB software was used for data processing because it has a series of predefined functions useful for the data analysis (FFT and wavelet) (Davis & Nosratinia 1998, Strang 1993, Tsai et al. 2005, Josso et al. 2001).

2. EXPERIMENTAL SETUP DESCRIPTION

A variant with suspended rods was used for the experimental setup. Elastic wires through metallic rings fixed on holders were used to suspend the rods. This solution offers a series of advantages comparing to that one in which the rods are guided. The construction is simple and is suitable for the testing of a wide variety of rods' lengths and sections.

The experimental setup is composed on two suspended cylindrical rods of C45 (OLC45 in Romanian STAS), having 40 mm, respective 35 mm in diameter and 2 m, respectively 6 m in length. Strain gauges were mounted on the rods equally disposed at L =1 m. They transmit the signals through a Wheatstone bridge to a data acquisition board. The study was made in the low velocity impact zone, with [v.sub.0] = 2m/s. At the impact moment compress strain occurs, together with mechanic waves which propagate inside the rod material. Compression strains cause the strain gauges to deform proportionally with their magnitude, thus generating electric signals toward Wheatstone bridge.

3. EXPERIMENTAL ANALYSIS OF THE WAVES' PROPAGATION GENERATED AT THE RODS' IMPACT

For the test rig described earlier some experiments were conducted connecting the strain gauges in full bridge. The voltage generated during rods' impact on each strain gauge was measured. Time correlation of the measurements has been realized with a trigger.

Because of the great length of the stricken rod, is difficult to relieve the reflected wave from it's free end. The main cause of this difficulty is that the wave is significantly absorbed along the rod. Beside its attenuation, a series of other effects occurs (side reflections, bending vibrations of the rods) which significantly diminishes the possibility of detecting the reflected wave.

Two sets of 5 measurements were conducted to detect and analyze the reflected wave. In the first set the signal acquired from the first strain gauge (T1) was measured for an impact realized in the conditions mentioned above. The second measurement set was conducted in the same conditions excepting the fact that at the free end of the stricken rod a vaseline film was applied. This layer of viscous material partly absorbs the reflected wave energy thus the reflected signal being diminished. The average of the 5 unabsorbed signals (without vaseline layer--[ms.sub.fa]) and also the average of the 5 absorbed signals (with vaseline layer--[ms.sub.a]) were computed. Both average diagrams are presented in Fig. 1.

The difference between these signals should be noticed at the moment of time t = 2,313 x [10.sup.-3] sec., on which (based on computed values and also on conducted measurements) both the separation of the rods and the reflected wave return in the section which corresponds to T1 strain gauge position occurs. Because of the reduced damping caused by the vaseline layer, this difference couldn't be easily noticed on the diagram in Fig. 1. Because of this a frequency analyze program created in MATLAB which computes and displays the frequency spectrum of the signal was realized. The software offers the possibility of signal filtration eliminating the insignificant frequency ranges (from the analyzed signal point of view) after which an inverse transformation from the frequency range to the time range is applied. This way noise can be eliminated (for example the 50 Hz frequency component induced by the a.c. net. Fig. 2 shows the frequency spectrum of the [ms.sub.fa] (black) and [ms.sub.a] (grey) signals. Three domains appear on the diagram in Fig. 2 in which significant differences between the two signals type occur. In the A domain a phase delay can be observed while in B and C zones a significant amplitude difference can be noticed.

[FIGURE 1 OMITTED]

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Keeping these frequencies domains and eliminating the others in which differences were not been noticed, and then applying an Inverse Fourier transforming the diagram presented in Fig. 3 is achieved. (The damped signal is [sm.sub.a], the undamped signal is [sm.sub.fa], the difference between them after an Inverse Fourier transformation is [dif.sub.sm]). In this diagram a significant variation of the difference between the signals (point A1) can be noticed, but such variations can be noticed in other time ranges in which the apparition of the reflected wave is not estimated, such as the range (0 ... 1 x [10.sup.3]) seconds.

The limitation of the Fourier transformation can be noticed in this case; this means that it is useful for signal filtration but presents the major disadvantage of loosing information regarding time localization of the phenomena. To localize the reflected wave in time the [dif.sub.sm] signal was analyzed using the MATLAB software. The diagram shown in Fig. 4 was created with this software. The program uses the "rbio 3.1" wavelet (Reverse Biorthogonal wavelets version 3.1). From the time-level diagram it can be noticed that in the range of interest (around 2,4 x [10.sup.-3] seconds) a prominent amplitude of the signal exists on the D1 and D2 detail levels. The reflected signal synthesis was realized through the selection of the "rbio3.1" wavelet transformation coefficients corresponding to D1 and D2 details.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Fig. 5 shows the steps of the transformation, coefficient selection and the resulted time range diagram. Through the synthesis of the reflected wave signal achieved from the [dif.sub.sm] signal using wavelet transformation, its share in the base signal can be estimated and also the shape of the signal together with the frequency range in which the signal appears.

4. CONCLUSIONS

The parameters used for data acquisition were correctly been defined, the analyzed phenomena being properly relieved.

Because the reflected wave at the free end of the stricken rod measured on the T1 strain gauge is difficult to be detected, a viscous layer was applied on the free end thus observing the differences between the damped and normal reflected signals. This technique didn't offer remarkable results either in time analysis or in frequency analysis. This is why the wavelet analysis (in time-scale range) was used thus resulting the signal filtration and the reconstitution of the component due to the reflected wave.

For the phenomena analysis which constituted the subject of the experimental studies, software that can be used for further researches for data acquisition and signal conditioning was made.

5. REFERENCES

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 Polytechnical University Timisoara, Mechanics Series, Vol. 48, No. 1, 2003, ISSN 1224-6077

Davis, G. & Nosratinia, A. (1998). Wavelet-Based Image Coding: An Overview. Applied and Computational Control, Signals, and Circuits, Vol. 1, No. 1, Spring 1998.

Josso, B., Burton, D. R. & Lalor, M. J. (2001). Wavelet Strategy for Surface Roughness Analysis and Characterisation. Computer Methods in Applied Mechanics & Engineering, Liverpool John Moores University, Vol. 191, No. 8-10, pp. 829-842

Strang, G. (1993). Wavelet transforms versus Fourier transforms, Bulletin of the American Mathematical Society, Vol. 28, No. 2, April 1993, pp. 288-305

Tsai, D. M., Wu, S. K., Chen, M. C. (2005). Optimal Gabor filter design for texture segmentation using stochastic optimization, Image and Vision Computing, Elsevier Science B.V.