In order to use the split Hopkinson bar setup in the intermediate strain rate range, several wave separation methods have been proposed in the literature. In this work, three wave separation methods are compared using numerical simulation: Casem-Fourney-Chang (CFC), Jacquelin Hamelin (JH) and Bussac-Collet-Gary-Othman (BCGO) methods. It is observed that the CFC and JH methods are more accurate than the two-strain and three-strain wave separation methods. The fourstrain wave separation method does better than CFC and JH methods, where also, JH method is slightly better than the CFC method. Moreover, the BCGO method error decreases with increasing number of strain measurements.

1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 114-119 © IAEME
114
COMPARISON OF THREE METHODS TO SEPARATE
WAVES IN THE PROCESSING OF LONG-TIME
HOPKINSON BAR EXPERIMENTS
Ramzi Othman
Mechanical Engineering Department, Faculty of Engineering,
King Abdulaziz University, P.O. Box 80248, Jeddah 21589, Saudi Arabia
ABSTRACT
In order to use the split Hopkinson bar setup in the intermediate strain rate range, several
wave separation methods have been proposed in the literature. In this work, three wave separation
methods are compared using numerical simulation: Casem-Fourney-Chang (CFC), Jacquelin-
Hamelin (JH) and Bussac-Collet-Gary-Othman (BCGO) methods. It is observed that the CFC and JH
methods are more accurate than the two-strain and three-strain wave separation methods. The four-
strain wave separation method does better than CFC and JH methods, where also, JH method is
slightly better than the CFC method. Moreover, the BCGO method error decreases with increasing
number of strain measurements.
Keywords: Hopkinson bar, Intermediate strain rate, Kolsky bars, Wave dispersion, Wave separation.
I. INTRODUCTION
Intermediate strain-rate (1-200/s) mehanical testing of materials is a main concern in several
engineering applications [1-4]. The Hopkinson-Kolsky bar technique is mostly used at high strain-
rates (500-5000/s), because of the limitation on the test duration. This limitation is induced by the
superposition of waves propagating in opposite directions.
Since the pioneer works of Lundberg & Henchoz [5] and Yanagihara [6], several wave
separation techniques with no limit on the test duration have been proposed [7-17]. The reader is
referred to Othman et al. [7] for a critical review of these techniques. Mostly, wave separation
methods were applied to extend Hopkinson-Kolsky bar machine to the intermediate strain-rate range
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 114-119 © IAEME
115
[8-11]. Othman et al. [7] have also applied a wave separation method to an elastic bar replacing the
piezo-electric force sensor in a servo-hydraulic machine.
Wave separation techniques applied to elastic and viscoelastic bars can be considered as long-
duration dynamic force measurement. Hence, they have multiple applications in impact engineering.
From the critical review of Othman et al. [7], it seems that three methods are less sensitive to noise
than the others. These methods are: CFC method [12], JH method [13] and BCGO method [14-15].
In this paper, we aim at comparing these three wave separation techniques by using numerical
simulations.
II. METHODOLOGY
The goal of this work is to compare three CFC, JH and BCGO wave separation methods. To
this purpose, we consider a long viscoelastic bar (Fig. 1). The bar left end is impacted by a striker
whereas the right end is free. The three wave separation methods use strain and/or velocity
measurements.
As a first step, strain and velocity measurements are simulated assuming 1D wave
propagation in the bar. However, 3D geometrical effects on wave dispersion are taken into account
[18]. The measurement positions are chosen randomly. One strain and one velocity measurements, in
the same cross-section, are considered for the CFC method. For the JH method, three strain
measurements are simulated.
Fig. 1: Schematic of a long bar impacted by a striker
The BCGO method can use an infinite number of strain measurements. In this work, we study
the BCGO method using 2, 3, 4, 5, 10, 20, 50 and 100 strain measurements. Numerical simulations
are carried out during a period of time T equal to 10 ms. Firstly, the Fourier transforms of strain and
velocity are calculated. Then the inverse Fourier transform is used to deduce strain and velocity in
terms of time. More precisely, the Fourier transform of the strain is given by
ߝ̃ሺ‫,ݔ‬ ߱ሻ = ∑ ߆෨ሺ߱ሻ ݁ି௜కሺఠሻሾሺଶ௠ାଵሻ௅ା௫ሿெబ
௠ୀ଴ + ∑ −߆෨ሺ߱ሻ ݁ି௜కሺఠሻሾሺଶ௠ାଵሻ௅ି௫ሿெబ
௠ୀ଴ , (1)
where ߆෨ሺ߱ሻ is the Fourier transform of the wave generate by the striker impact at the left bar end,
ߦሺ߱ሻ is the wave dispersion relation, ‫ܮ‬ is the bar length, ‫ݔ‬ is the strain measurement position, ߱ is
the angular frequency and ‫ܯ‬଴ is the number of wave round-trips during the time T. Similarly, the
Fourier transform of the velocity reads
ܸ෨ሺ‫,ݔ‬ ߱ሻ =
ఠ
కሺఠሻ
൫∑ ߆෨ሺ߱ሻ ݁ି௜కሺఠሻሾሺଶ௠ାଵሻ௅ା௫ሿெబ
௠ୀ଴ + ∑ ߆෨ሺ߱ሻ ݁ି௜కሺఠሻሾሺଶ௠ାଵሻ௅ି௫ሿெబ
௠ୀ଴ ൯. (2)

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 114-119 © IAEME
116
In order to simulate experimental noise, a numerically build Gaussian noise is added to each strain or
velocity signal. These noises are constructed separately. Therefore, they can be assumed two-by-two
independent.
In this work, we consider that the bar is 4-m long and 40 mm in diameter. It is supposed to be
made of aluminum (Young's modulus ‫ܧ‬ =70GPa, Poisson's ratio ߥ =0.34 and density ߩ =2800
kg/m3
). The wave dispersion equation is obtained by solving the Pochhammer-Chree equation
[19,20]. The calculated strain and velocity signals are sampled at a sampling frequency of 100 MHz.
The second step consists in calculating the force at the free end, which should be equal to
zero, by applying the three different wave separation methods to the appropriate strain and/or
velocity measurements. The reader is referred to Refs. [12-14] for more details on how these
methods work.
Let ‫ܨ‬ఓ
௥
= ‫ܨ‬ఓሺ‫ݐ߂ ݎ‬ሻ be the force at the free bar end at the time ‫ݐ߂ ݎ‬ (߂‫ݐ‬ is the sampling step).
The subscript ఓ refer to the used method, thus, ఓ can be replaced by ஼ி஼, ௃ு, or ஻஼ீை಻
for
CFC method, JH method and J-strain measurements BCGO method, respectively. The error of the
method ఓ is defined by
ߞఓ
௔
=
ฮிഋฮ
ೌ
ிబ
, (3)
where ‖. ‖௔ is a norm and ‫ܨ‬଴ is a reference force. Two norms are considered here the Euclidian norm
‖. ‖ଶ and the max norm ‖. ‖∞. For a vector ܻ = ሺ‫ݕ‬௥ሻ௥ୀ଴...ோ, the two norms read
‖ܻ‖ଶ = ඥ∑ ሺ‫ݕ‬௥ሻଶ௥ୀோ
௥ୀ଴ , (4)
and
‖ܻ‖∞ = max௥ሺ‫ݕ‬௥ሻ. (5)
The reference force ‫ܨ‬଴ is chosen to be the maximum force induced by the striker impact at the
left bar end.
By running the first and second step, we obtain six error values, two for each method. The
results depend on the considered strain and/or velocity simulated measurement positions which are
chosen randomly. It depends also on the numerical noise added to the perfect strain and velocity
measurements. Therefore, steps 1 and 2 are run for one thousand times. Each time six error values
are obtained. At the end of the run, an average value is calculated: 〈ߞఓ
௔〉. 〈ߞఓ
௔〉 is the average error of
the method ߤ calculated using the norm ‖. ‖௔.
III. RESULTS
The errors obtained by the CFC, JH and BCGOJ methods are depicted in Fig. 2. For the
BCGOJ method, eight values of J are considered: J = 2; 3; 4; 5; 10; 25; 50; 100. Even though, it is
hard to cement 10 strain measurements or more on a bar, these values are only considered to check
the J-convergence of the BCGOJ method. Moreover, we can imagine the use of full-field strain
measurement techniques. In that case a high number of strain measurements can be recorded.
Excepting the BCGO2, all methods give quite accurate results as they have average maximum
errors lower than 8% (Fig. 2 (b)) and average ‖. ‖ଶ errors below 2% (Fig. 2(a)). The CFC and JH
methods give better results than the two- and three-strain BCGO wave separation method. The JH
method is slightly better than the CFC. The BCGO method is more accurate for J≥4. The errors of

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 114-119 © IAEME
117
the BCGO method decrease with increasing J. These errors have an exponential behavior for
important values of strain measurements: J≥5. Precisely,
〈ߞ஻஼ீை಻
௔ 〉 ∝ ݁ି௃
. (6)
It is worth noticing that BCGO2 gives highly dispersive results, following in decreasing
order: BCGO3, BCGO4, JH, CFC than BCGO୎ஹହ. The dispersion is most probably due to the variation
of the measurements positions. These positions can influence the poles in the solutions of the BCGO
wave separation methods. As errors are amplified in the neighborhood of these poles, this can
explain why the BCGO method gives more dispersive results. Even though, the BCGO4 is more
dispersive, it gives, in average, better results than JH and CFC methods.
IV. CONCLUSION
Assuming one-dimensional wave propagation and considering the 3D wave dispersion
effects, strain and velocity measurements, induced by a striker impact, were simulated in a free-
ended elastic bar. Subsequently, a numerically Gaussian noise was added to the strain and velocity
measurements to be close to experimental situation. The noise measurements were then processed by
three wave separation methods in order to recover the force at the free end, which should be equal
zero. Thus, we were able to define and calculate an error for each wave separation method.
It was observed that the CFC and JH wave separation methods lead to more accurate results
than the two-strain and three-strain BCGO method. However, the four-strain wave separation
method gives better accuracy than the CFC and JH methods. Comparing these last two, JH method is
slightly better than CFC method.
It is also reported that the increase of the number of strain measurement improves the
accuracy of the BCGO wave separation method. Actually, increasing the number of measurements
increases the redundancy.

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 114-119 © IAEME
118
Fig. 2: Comparison of errors obtained by the three wave separation method: (a) 〈ߞఓ
ଶ〉 and (b) 〈ߞఓ
∞〉
REFERENCES
[1] J.C. Petiteau, R. Othman, P. Guégan, H. Le Sourne, E. Verron. Dynamic uniaxial extension
of elastomers at constant true strain rate. Polymer Testing, 32, 2013, 391-401.
[2] J.C. Petiteau, R. Othman, P. Guégan, H. Le Sourne, E. Verron. A drop-bar setup for the
compressive testing of rubber-like materials in the intermediate strain rate range. Strain, 50,
2014, 555-562.
[3] Z. El-Qoubaa, R. Othman. Volume change in polyetheretherketone under compression loads
over wide ranges of strain rate and temperature. Journal Strain Analysis for Engineering
Design, 49, 2014, 315-324.
[4] V. Delhaye, A.H. Clausen, F. Moussy, et al. Influence of stress state and strain rate on the
behaviour of a rubberparticle reinforced polypropylene. International Journal of Impact
Engineering, 38, 2011, 208–218.
[5] B. Lundberg, A. Henchoz. Analysis of Elastic Waves from Two-Point Strain Measurement.
Experimental Mechanics, 17, 1977, 213-218.
[6] N. Yanagihara. New Measuring Method of Impact Force. Bulletin of Japanese Society of
Mechanical Engineering, 21, 1978, 1085-1088.
[7] R. Othman, P. Guégan, G. Challita, et al. A Modified Servo-Hydraulic Machine for Testing at
Intermediate Strain-rates. International Journal of Impact Engineering, 36, 2009, 460-467.
[8] H. Zhao, G. Gary. A new method for the separation of waves. Application to the SHPB
technique for an unlimited measuring duration. Journal of Mechanics and Physics of Solids,
45, 1997, 1185-1202.
[9] R. Othman, M.N. Bussac, P. Collet, G. Gary. Testing with SHPB from quasi-static to
dynamic strain rates. Journal de Physique IV, 110, 2003, 397-404.

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 11, November (2014), pp. 114-119 © IAEME
119
[10] H. Zhao, S. Abdennadher, R. Othman. An experimental study of square tube crushing under
impact loading using a modified large scale SHPB. International Journal of Impact
Engineering, 32, 2006, 1174-1189.
[11] R. Othman, G. Gary. Testing aluminum alloy from quasi-static to dynamic strain-rates with a
modified Split Hopkinson bar method. Experimental Mechanics, 47, 2007, 295-299.
[12] D.T. Casem, W. Fourney, P. Chang. Wave separation in viscoelastic pressure bar using single
point measurements of strain and velocity. Polymer Testing, 22, 2003, 155-164.
[13] E. Jacquelin, P. Hamelin. Force recovered from three recorded strains. International Journal
of Solids and Structures, 40, 2003, 73-88.
[14] R. Othman, M.N. Bussac, P. Collet, G. Gary. Séparation et reconstruction des ondes dans les
barres élastiques et viscoélastiques partir de mesures Redondantes. Comptes Rendus de
l’Academie des Sciences Serie IIb, 329, 2001, 369-376.
[15] M.N. Bussac, P. Collet, G. Gary, R. Othman. An optimisation method for separating and
rebuilding one-dimensional dispersive waves from multi-point measurements. Application to
elastic or viscoelastic bars. Journal of Mechanics and Physics of Solids, 50, 2002, 321-350.
[16] J. Shim, D. Mohr. Using split Hopkinson pressure bars to perform large strain compression
tests on polyurea at low, intermediate and high strain rates. International Journal of Impact
Engineering, 36, 2009, 1116-1127.
[17] R. Othman. Wave separation in non-uniform Hopkinson bars using redundant measurements.
Journal de Physique IV, 134, 2006, 571-576.
[18] R. Othman, R.H. Blanc, M.N. Bussac, P. Collet, G. Gary. Identification of the dispersion
relation in rods. Comptes Rendus de Mecanique, 330, 2002, 849–855.
[19] L. Pochhammer. Uber die Fortpanzungsgeschwindigkeinten kleiner Schwingungen in einem
unbergrenzten isotropen Kreiszylinder. Journal fur die Reine Angewandte Mathematik, 81,
1876, 324-336.
[20] C. Chree. The equations of an isotropic elastic solid in polar and cylindrical co-ords, their
solutions and applications. Cambridge Philosophical Society Transactions, 14, 1889, 250-
369.