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In the last decade there has been an increasing interest in the use of highly and weakly nonlinear solitary waves in engineering and physics. Nonlinear solitary waves can form and travel in nonlinear systems such as onedimensional chains of particles, where they are conventionally generated by the mechanical impact of a striker and are measured either by using thin transducers embedded in between two halfparticles or by a force sensor placed at the chain's base. These waves have a constant spatial wavelength and their speed, amplitude, and duration can be tuned by modifying the particles' material or size, or the velocity of the striker. In this paper we propose two alternative sensing configurations for the measurements of solitary waves propagating in a chain of spherical particles. One configuration uses piezo rods placed in the chain while the other exploits the magnetostrictive property of ferromagnetic materials. The accuracy of these two sensing systems on the measurement of the solitary wave's characteristics is assessed by comparing experimental data to the numerical prediction of a discrete particle model and to the experimental measurements obtained by means of a conventional transducer. The results show very good agreement and the advantages and limitations of the new sensors are discussed.
In the last fifteen years the numerical and experimental studies on the propagation of nonlinear solitary waves in onedimensional chains of granular media, and in particular of spherical elastic beads, have thrived [
It has been demonstrated that HNSWs propagating in granular crystals have the potential to be used as acoustic lenses [
In the study presented here, we investigated numerically and experimentally two alternative sensing systems to measure the propagation of solitary waves in a 1D chain of metallic particles. The first design replaces the sensor beads with piezo rods having thickness and diameter comparable to the size of the particles composing the chain. The second system considers the use of coils wrapped around a segment of the chain to create a magnetostrictive sensor (MsS). To the best of the authors' knowledge, the use of magnetostriction or piezoelectric cylinders to measure the propagation of HNSWs was never reported in the past. In this paper the working principles of these novel transducers are introduced and the experimental results are compared to the measurements obtained using conventional instrumented beads and to the numerical prediction derived with a discrete particle model.
The paper is organized as follows: the experimental setup is described in Section 2. The principles of the three types of sensors are introduced in Section 3. Section 4 presents the numerical model of wave propagation in a chain of spherical particles. In Section 5, the experimental results are presented. Finally, Section 6 concludes the paper with a discussion on the advantages and disadvantages of the three sensing configurations.
In order to compare the novel sensing systems to the conventional one, a plastic tube with inner diameter of 4.8 mm and outer diameter of 12.7 mm was filled with twenty nine 4.76 mmdiameter, 0.45 gr, low carbon steel beads (McMasterCarr product number 96455K51). An identical bead was used as striker. For convenience, the particles are herein numbered 1 to 30 where particle 1 identifies the striker and particle 30 represents the sphere at the opposite end of the chain. The stroke of the particle 1, equal to 7.2 mm, was governed by an electromagnet mounted on top of the tube and remotely controlled by a switch circuit connected to a National Instruments PXI running in LabVIEW.
Three pairs of sensors were used in this study: bead sensors, rodform piezos, and MsSs. Each bead sensor was assembled by embedding a zirconate titanate based piezogauge (3 mm by 3 mm by 0.5 mm) inside two half steel spheres, as shown in
The capability and the repeatability to measure the amplitude and speed of the HNSWs was evaluated by taking 500 measurements at 10 MHz sampling rate for each sensing configuration.
In a onedimensional chain of spherical particles, the interaction between two adjacent beads is governed by the Hertz's law [
A single pulse is commonly induced by mechanically impacting the first bead of the chain with a striker having the same mass of the particles composing the chain. Ni
The MsS takes advantage of the efficient coupling between the elastic and magnetic states of the ferromagnetic particles and in particular of the magnetostrictive phenomena that convert magnetic energy into mechanical energy and vice versa [
The inverse mechanism can be used for the detection of waves. A pulse propagating in the ferromagnetic material modulates an existing magnetic field by means of the Villari's effect [
In the design of the MsS used in the present study we exploited the Villari effect to detect the propagation of nonlinear solitary waves across the chain. The particles are the magnetostrictive material subjected to a biased magnetic field and are surrounded by a coil. We hypothesized that the change of the magnetic induction is proportional to the change of the dynamic contact force between neighboring particles. The output voltage was proportional to the timederivative of dynamic contact force:
Therefore, the dynamic force associated with the solitary wave propagation is proportional to the integral of the sensor output voltage. Based upon the geometry of the MsS [see
The experimental setup was simulated using a chain of spherical particles in contact with a wall which was considered as a halfinfinite medium, as shown in
Here, the subscripts
The impact of the striker was simulated by setting the initial displacement
The model of the chain with the instrumented beads did not consider the fact that they are slightly heavier than the other spheres. This is because it was demonstrated [
In order to model the presence of the piezo rods, the modeled chain comprised two solid rods having the same geometric and material properties of the piezoelectric cylinders at position 13 and 18. And the mass and contact stiffness (terms
The numerical model was applied to simulate the three setups described in Section 2. To predict the measurements of the three types of transducers, the numerical values of the forcetime profiles at contact points c_{8}, c_{11}, c_{12}, and c_{13} (see the notation in
The numerical results of the force profiles at position 13 for the three sensing systems are shown in
Finally,
Although the dissipation coefficients can be determined empirically by measuring the magnitudes of the dynamic forces at different positions in the chain [
In order to calibrate the numerical model to our experimental setup, the dissipation was taken into account. Because the force amplitudes of both incident and reflected solitary pulses were proportional to the voltageforce conversion factor, their AR was independent upon this conversion factor. We computed the dissipation coefficient by considering the 500 experimental measurements of the amplitude ratios
As is said earlier, when the piezoelectric cylinders and MsSs were used, the sensor beads were replaced by the two piezo rods and two spheres, respectively. Because the coefficient
In applications such as nondestructive testing, voltage measurements are sufficient to correlate the characteristics of the solitary pulses to the properties of the structure or material under inspection. However, other engineering applications may require the quantitative measurement of the dynamic force associated with the traveling pulse. Thus, the relationship between this force and the output voltage from the sensor needs to be known. To establish this relationship, we adopted the following procedure for all sensing configuration. The experimental time profiles, expressed in Volts and collected using the experimental setup described in Section 2 were compared to the force profiles (expressed in Newton) computed with the discrete particle model described in Section 4. The model considered the effect of dissipation.
Here
In the last part of our study, we quantitatively compared the shape of the experimental and numerical force profiles.
Similarly,
Finally,
To assess and compare the capability of the novel transducers to measure the characteristics of the HNSW propagating in a straight chain of particles,
In this paper we investigated numerically and experimentally three sensing systems for the measurements of highly nonlinear solitary pulses propagating in a chain of spherical particles. The transducers were a conventional pair of instrumented beads, and two novel designs based on the utilization of a pair of piezo rods, and a pair of coils. The latter aimed at exploiting the magnetostrictive properties of the particles.
We compared the experimental results to the numerical predictions obtained by means of a discrete particle model. We found that the two novel designs performed equally well when compared to the conventional sensor beads, which require micromachining and must not be allowed to rotate in order to keep their sensitivity constant. From this study the following consideration can be made. Owing to its geometry the piezoelectric cylinder is not prone to rotation and does not require the machining of halfparticles. However it may originate unwanted secondary pulses that trails the incident wave and attenuates the amplitude of the incident pulse. To prevent this problem, the cylindrical sensor should have the same mass of the other particles composing the chain, and the spherecylinder contact stiffness should be the same as the spheresphere contact stiffness. Based on the contact mechanics theory, in order to have same contact stiffness, the elastic modulus of the cylinder should be approximately equal to 55% of the particle material's elastic modulus [
The authors acknowledge the support of the University of Pittsburgh's Mascaro Center for Sustainable Innovation seed grant program, the Federal Railroad Administration under contract DTFR5312C00014 (Leith AlNazer was the Program Manager), and the U.S. National Science Foundation (CMMI 1200259).
Schematic diagram of the experimental setup.
Sensing technologies used in this study. (
Schematic diagram of the onedimensional discrete element model. The
Discrete particle model results showing the temporal force profile for all threesensing configurations at contact points: (
Discrete particle model results showing the temporal force profile at some contact points (dashed lines) and as measured by three sensors (solid lines): (
Typical waveforms measured by the bead sensors.
(
Calibration coefficients adopted in this study.
Bead sensor top  2.072  4.71  N/V 
Bead sensor bottom  2.160  4.31  N/V 
Cylindrical sensor top  0.632  4.00  N/V 
Cylindrical sensor bottom  0.557  4.01  N/V 
MSS top  0.131  1.44  N/V·sec 
MSS bottom  0.156  1.56  N/V·sec 
Experimental and numerical timeofflight and amplitude ratio.

 

Bead  Top  318.7  309.8  2.6  2.79  0.57  0.55  0.04  3.51 
Bottom  227.5  219.9  1.6  3.34  0.68  0.73  0.02  7.35  
Cylindrical  Top  341.0  337.0  5.3  1.17  0.42  0.37  0.03  11.9 
Bottom  234.9  228.4  3.0  2.77  0.68  0.60  0.07  11.8  
MSS  Top  298.8  287.3  0.9  3.85  0.70  0.69  0.02  1.43 
Bottom  217.6  207.0  0.7  4.87  0.79  0.86  0.01  8.86 
Experimental and numerical speed of the incident and reflected HNSW pulse.
Bead  Incident  540.0  547.0  13.1  1.30 
Reflected  505.6  514.1  16.0  1.68  
Cylindrical  Incident  486.3  512.0  14.6  5.28 
Reflected  458.7  422.9  38.6  7.80  
MSS  Incident  596.8  602.1  1.75  0.89 
Reflected  576.6  584.8  3.36  1.42 