1.3. Aspectos físicos en radiografía computarizada
1.3.3. Parámetros que afectan la dosis del paciente y la calidad de la imagen
In order to increase the maximum measurable duration in a SHPB test, wave separation methods have been developed which are mainly categorised into two groups: The two-point method and the one-point method. The two-point method developed by Lundberg and Henchoz [82] separates the forward and backward moving waves in the time domain using strain signals from two different locations on the bar. Similarly, Yanagihara [83] independently proposed the two-point method using a similar numerical method as the one given by Lundberg and Henchoz [82] to measure impact forces. The one-point method proposed by Park and Zhou [84] replaced the second strain measurement by the condition of the zero strain at the free end. However, the one-point method only extends the test duration by a factor of two compared with the conventional SHPB duration [85]. One dimensional wave propagation in elastic bars was assumed for both of these methods, i.e. both dispersion and attenuation effects were assumed negligible.
Zhao and Gary [75] proposed a new method for separating waves propagating in opposite directions, which utilises the two-point measurement technique. The method takes into account both dispersion and attenuation effects and can be applied to both elastic and viscoelastic bars. With this technique wave shifting is performed
Chapter 5: Wave Separation Techniques
in the frequency domain and subsequently all the calculations regarding the separation of waves at a strain gauge location are then performed in the time domain. The authors [75] used a pressure bar with two strain gauges cemented at two
different locations (e.g. SG2 (xB =d1) and SG3 (xC =d2) in Figure 5.1), in order to
record the strain histories at these points. Both strain measurements were divided
into equal time intervals ∆tBC, which is twice the time needed for waves to travel the
distance between the two gauges. The first measured incident wave from the first strain gauge (SG2 (xB =d1), see Figure 5.1) is contained in the first time interval.
Then the magnitude of the first incident wave at position xC =d2 (Figure 5.1) is
predicted by time shifting the measured strain from point xB =d1 (Figure 5.1) using
a Fourier transform. Since the strain at every cross section of the bar is equal to the sum of the stains due to both incident and reflected waves, the reflected wave at
point xC =d2 (Figure 5.1) can be calculated.
The above method leads to the evaluation of all waves propagating in both directions for all the time intervals by means of an iterative process. This method allows tests to be performed on a SHPB with increased observation time. In order to obtain more accurate results, dispersion effects were taken into account using the generalised Pochhammer-Chree wave equation [26]. This technique was used for both elastic and viscoelastic SHPB arrangements for the determination of the dynamic behaviour of several materials including metallic tubes, polymeric foams [75] and aluminium honeycombs [31]. It was shown that the measured duration can be increased by a factor of 100 compared with the conventional SHPB set up [75] due to repeated loading of the specimen.
The method proposed above can be applied only if the first incident wave is fully measured before the reflected wave has reached the first strain gauge. Hence, the duration of the incident wave is limited. This is important especially in cases where a viscoelastic projectile is used, since as it has been reported the duration of the incident pulse is extended compared with the duration produced when using an elastic projectile [27]. Furthermore, the time intervals ∆tBC are approximate, since
Chapter 5: Wave Separation Techniques
Zhao and Gary [75] first discussed the possibility of wave separation in the frequency domain and stated the problems involved when adopting this approach.
The forward wave at position xA in the frequency domain is given by:
( )
( )
( ) ( ) ( ) ( ) (2 ) ( ) B A A B A x x A B x x x e P e e γ ω γ ω γ ωε ω ε ω
ω
= − − − − − − − % % % . (5.12)This equation cannot be defined when the denominator is zero, which occurs at certain frequencies for both elastic and viscoelastic bars, as discussed in Section 2.7. Also, errors are caused when the recorded strain signals do not attenuate completely at the end of the test duration due to the finite time limits of the integration involved when performing a Fourier transform [75, 74, 86, 87]. To overcome this different testing procedures [86, 87, 88, 89, 90] and different approaches to solving Equation (5.12) have been proposed [85, 74, 86].
Bacon extended the two-point method for separating waves for both elastic and viscoelastic bars, where dispersion and attenuation effects are encountered [74]. The proposed technique involved an iterative calculation in the time domain when Equation (5.12) was not defined or could not be used i.e. when the strain signals were truncated. Details of this approach are provided in Section 2.7.
The separation of waves in large diameter elastic bars was investigated by Zhao and Lok [85]. Dispersion effects were taken into account by solving numerically the
Pochhammer-Chree frequency equation. For frequencies (
ω
=ω
0) where Equation(5.12) could not be defined, Equation (5.12) becomes an undefined fraction of type 0/0 and was calculated using L’Hospital’s rule. Outside these frequencies Equation (5.12) was used directly to determine the forward moving wave. The problem is that the application of L’Hospital’s rule involves the determination of the spectral derivatives of the measured strain signals and the wave number. In particular, the calculation of the wave number’s derivative was considered rather complex as pointed out by the authors [85]. Furthermore, it is not clear if problems due to truncation of the signals are overcome with this method. Also, the case of viscoelastic pressure bars, where waves attenuate and disperse, was not examined.
Chapter 5: Wave Separation Techniques
However, the method was validated both numerically and experimentally for 75 mm diameter steel SHPB.
A modified SHPB set-up was introduced by Meng and Li [90] where an iterative algorithm was performed in the time domain to separate the overlapping waves using measurements from two strain gauges. Based on the fact that the dispersion and attenuation effects become important only if the wave travels over a considerable length of the bar, two strain gauges having a small distance between them were placed near the bar/specimen interface. Therefore, both effects could be disregarded. A detailed finite element analysis revealed that a minimum distance of 1.5D (where D is the bar diameter) ensures a uniform strain state on the cross-section of the elastic bar. Hence, a reduction in the time shift of the waves was introduced and the shape of the pulses was considered unchanged. The accuracy of the method was confirmed using both numerical and experimental examples. However, an error analysis needed to be employed in order to improve the accuracy of the results. Also, although the presented method has the advantage of minimising the time shift and the dispersion and attenuation effects in elastic bars, further investigation is needed before the technique can be employed for viscoelastic SHPB arrangements.
Velocity measurements have been utilised by many researchers when separating waves in SHPB set-ups [86, 88]. The main advantage is that the use of direct velocity measurements minimises errors due to noise, which are important when using strain signals [75, 86, 89, 87]. Casem et al. [88] proposed a method to separate overlapping waves and could be applied to both elastic and viscoelastic bars where dispersion and attenuation effects are important. A strain gauge is employed together with an electromagnetic velocity gauge to measure both the axial strain and velocity at a single point. The forward and backward waves were separated in the frequency domain by solving a system of two simultaneous equations. It was suggested that by this method unlimited test duration is achieved and there is no restriction for the position of the gauge stations. The accuracy of the technique was validated by evaluating the stress at the impact and free ends of a 19.1 mm diameter polycarbonate bar from an impact test of the bar with an aluminium projectile.
Chapter 5: Wave Separation Techniques
that the signals have to attenuate completely in order to avoid truncation errors. Also, questions arise in whether the magnetic field induced by the velocity gauges will affect the operation of the strain gauges. Lastly limitations exist when using velocity gauges on magnetic materials.
A detailed work on the factors that influence the accuracy of the separated strain signals has been performed by Bussac et al. [86]. It was suggested that when using the two-point method in the frequency domain, the presence of noise in the measured strain signals, the imprecise knowledge of the amplifier gain, the inaccurate measurements of the strain gauge positions and the incorrect null strain, will introduce errors when calculating forces and displacements at any cross-section of the bar. Using the Maximum Likelihood Principle both the forward and backward waves were expressed as functions of the strain signals and the bar dispersion relation. It was concluded that only a set-up consisting of three strain gauges and two velocity stations will be able to minimise any source of errors. Although, this solution is optimal regarding the elimination of errors, the mathematics involved are rather complex as pointed out by other researchers [87]. Furthermore, the fact that each pressure bar in a SHPB set-up requires to be implemented with five different measuring stations makes this technique rather expensive.