Attenuation of short strongly nonlinear stress pulses in dissipative granular chains

Attenuation of short, strongly nonlinear stress pulses in chains of spheres and cylinders was investigated experimentally and numerically for two ratios of their masses keeping their contacts identical. The chain with mass ratio 0.98 supports solitary waves and another one (with mass ratio 0.55) supports nonstationary pulses, which preserve their identity only on relatively short distances, but attenuate on longer distances because of radiation of small amplitude tails generated by oscillating small mass particles. Pulse attenuation in experiments in the chain with mass ratio 0.55 was faster at the same number of the particles from the entrance than in the chain with mass ratio 0.98. It is in quantitative agreement with results of numerical calculations with effective damping coefficient 6 kg/s. This level of damping was critical for eliminating the gap openings between particles in the system with mass ratio 0.55 present at lower or no damping. With increase of dissipation numerical results show that the chain with mass ratio 0.98 provides faster attenuation than the chain with mass ratio 0.55 due to the fact that the former system supports the narrower pulse with the larger difference between velocities of neighboring particles. The investigated chains demonstrated similar behavior at large damping coefficient 100 kg/s.

Figure 1

Experimental setup. (a) Cylinders and spheres aligned in 1D chain inside the holder and (b) cross-sectional view of the assembly. Four aluminum rods hold the particles in aligned chain. In the numerical calculations the particles inside the chain with even numbers are spheres and particles with odd numbers are cylinders, $i=2$, 3, 4, …, N/2. The impactor is a separate particle outside of chain with number 1.

Figure 2

Experimental results. Single pulse propagating through a chain with the mass ratio of 0.98. (a) Signals correspond to sensors embedded into the fourth and fifth cylinder. (b) Signals correspond to sensors embedded into the fourth and 21st cylinder. Pulses were excited by the impact of a spherical steel particle. The vertical scale in both figures is 20 N and zero time is arbitrary, the curves are offset for visual clarity.

Figure 3

Comparison of experimental results and numerical calculations of the pulse propagating through the system, which has a mass ratio of 0.98. Sensors are placed in the fourth, fifth, and 21st cylinders. (a) Experimental results, (b), (c) numerical calculations (without dissipation) related to the forces in the corresponding cylinders. The vertical scale is 20 N and the curves are offset for visual clarity and zero time is arbitrary.

Figure 4

Experimental results and numerical calculations of the pulse propagating through the system with mass ratio of 0.98. (a) Experimental results, sensors are placed in the fourth and 21st cylinders, (b) Numerical calculation with damping coefficient 6 kg/s related to the forces in the corresponding cylinders. The vertical scale in (a), (b) is 20 N. The curves are offset for visual clarity and zero time is arbitrary.

Figure 5

Relative amplitude of signal at $i\mathrm{th}$ cylinders (corresponding to different depths) with respect to the amplitude of the reference pulse detected by the sensor in the fourth cylinder in the chain with mass ratio 0.98, experimental data and numerical results with different damping coefficients.

Figure 6

Attenuation in experiments (blue) and in numerical calculations with damping coefficient 6 kg/s (red) comparing with exponential decay.

Figure 7

The formation of two-wave structure (solitary wave followed by shock wave) in one mass chain impacted by sphere with velocity 1.45 m/s, damping coefficient 6 kg/s. The $y$ axes scale for all curves representing forces in corresponding particles (all of them are cylinders) are offset by 10 N for clarity.

Figure 8

Dependence of the force in the 21st cylinder normalized with respect to the force at the entrance (at fourth cylinder) on mass ratio in nondissipative and dissipative chains with fixed contacts. Mass of sphere was kept the same and mass of cylinder changed keeping fixed contacts.

Figure 9

Experimental results. Stress pulses propagating through a chain with the mass ratio 0.55, mass of cyliders is larger than mass of spheres. (a) Signals correspond to sensors embedded into the fourth and fifth cylinder. (b) Signals correspond to sensors embedded into the fourth and 21st cylinder. Pulses were excited by the impact of a spherical bead the same as the spherical particles in the chain. The vertical scale in both figures is 20 N and zero time is arbitrary, the curves are offset for visual clarity.

Figure 10

Comparison of the experimental results and numerical calculations (without dissipation) of the pulse propagating through the dimer system with mass ratio 0.55, mass of cylinders is larger than mass of spheres. (a) Experimental results; (b), (c) numerical calculations (without dissipation) related to the forces in corresponding cylinders. The vertical scale is 20 N and the curves are offset for visual clarity and zero time is arbitrary.

Figure 11

Results of experiments and numerical calculations of the pulse propagating through the dimer system, mass of cylinders is larger than mass of spheres. (a) experimental results (sensors are placed in the fourth and 21st cylinders), (b) and (c) results of numerical calculations related to the forces in fourth and 21st cylinders with μ = 4 kg/s and μ = 6 kg/s, correspondingly. The vertical scale is 20 N and the curves are offset for visual clarity, zero time is arbitrary.

Figure 12

The comparison of the experimental results and data from the numerical calculations with different values of damping coefficient in the dimer chain with mass ratio 0.55. Relative amplitude of leading stress pulse at different depths (number of cylinders are shown on the horizontal axes) is calculated with respect to the amplitude of the reference pulse in the fourth cylinder. Mass of cylinders is larger than mass of spheres.

Figure 13

Attenuation of relative amplitude in experiments (blue) and in numerical calculations with damping coefficient 6 kg/s (red) comparing with exponential decay. Relative amplitude is calculated with respect to the amplitude of the reference pulse in the fourth cylinder. Mass of cylinders is larger than mass of spheres.

Figure 14

Comparison of the forces inside 21st cylinder and out of phase displacements of neighboring spheres in numerical calculations: (a) damping coefficient equal to zero, opening of gaps on both sides of 21st cylinder is evident at moments corresponding to zero forces and (b) no openings of gaps at damping coefficient equal to 6 kg/s. In both cases chain with mass ratio 0.55 impacted by sphere with velocity 1.45 m/s.

Figure 15

The attenuation of the relative pulse amplitude (with respect to the amplitude of the reference pulse in the fourth cylinder, mass ratios 0.98 and 0.55) and change of decay efficiency in two systems with increased damping coefficient at the same contact number in the chains at corresponding values of damping coefficients (a) 0, (b) 6 kg/s, (c) 10 kg/s, (d) 15 kg/s, and (e) 100 kg/s. Mass of cylinders is larger than mass of spheres (2.085 g).

Figure 16

The change of particle velocities profiles in both sytems with increase of damping coefficient. The maximum of pulse amplitude corresponds to 42nd particle (21st cylinder), odd numbers are related to spheres and even to cylinders. Numerical calculations with damping coefficients (a) 0, (b) 6 kg/s, (c) 10 kg/s, (d) 15 kg/s, and (e) 100 kg/s. Particle velocities in system with mass ratio 0.98 (blue, open circles) and particle velocities in system with mass ratio 0.55 (red, solid circles). Mass of cylinders is larger than mass of spheres (2.085 g).

Figure 17

Relative amplitude (with respect to the amplitude of the reference pulse at the fourth cylinder) after propagation through the chain with the same mass at different values of damping coefficient in (a) experiments and in numerical calculations with different damping coefficients: (b) 0, (c) 6 kg/s, (d) 10 kg/s, (e) 15 kg/s, and (f) 100 kg/s. Mass of cylinders is larger than mass of spheres (2.085 g).

Figure 18

The attenuation of the relative pulse amplitude with respect to the amplitude of the reference pulse in the fourth cylinder, mass ratios 0.98 and 0.55, mass of cylinders is smaller than mass of spheres (2.085 g) and change of decay efficiency in two systems with increased damping coefficient. Relative amplitudes in numerical calculations at the same position in the chains at corresponding values of damping coefficients (a) 6 kg/s, (b) 10 kg/s, (c) 15 kg/s, and (d) 100 kg/s.

Figure 19

Relative amplitude (with respect to the amplitude of the reference pulse at the fourth cylinder) after propagation through the chain with the same mass at different values of damping coefficient in in numerical calculations with different damping coefficients: (a) 6 kg/s, (b) 10 kg/s, (c) 15 kg/s, and (d) 100 kg/s. Mass of cylinders is smaller than mass of spheres (2.085 g), mass ratio is 0.55.