Oscillations

Focus is now shifted to the oscillations that are observed behind the primary peak of these pulsed waves. The origin of these oscillations is still uncertain. The typical frequency of the oscillations is

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 20 40 60 80 100 120 140

Oscillation mean pressure (kPa)

Oscillation frequency (Hz)

Figure 4.7: Frequency of the oscillations after the passing of the initial peak plotted as a function of the mean pressure during the oscillations. Measurements were made in PVC (◦) and 3 mm glass (2) particles.

in the range 25 to 80 Hz. These frequencies are far too low to be attributed to the natural frequency of the transducer diaphragm. They are more likely associated with some structure on a larger length scale. One possible explanation is that these oscillations constitute a natural frequency of the force chains through which the waves travel. In this scenario, one would expect the frequency to increase with the confining stress on the chain. Thus, ringing about an elevated pressure level should occur at a higher frequency than ringing about the static pressure.

Two different types of ringing were observed. Short-duration oscillations were seen at the elevated pressure due to the push of the piston (see Figure 4.3: 41 and 65 ms widths) and just after the release of the piston about the static pressure (see Figure 4.4: 6 ms width). These oscillations were characterized by a relatively smooth, sinusoidal shape. They were limited in duration to a maximum of three periods. Long-duration oscillations were observed behind the wave after any short-duration oscillations (see Figure 4.9 after 0.2 s) and could last for as long as twenty periods. Their structure was comparatively noisy, but their frequency was consistently 20 Hz. In contrast, the short-duration oscillations could take on various frequencies in the range 25-80 Hz.

sure on the particles. The degree of confining pressure was determined as the mean pressure about which the oscillations occurred. The frequencies of the short-duration oscillations were determined by windowing two to three periods of the ringing and calculating the frequency by inspection. The use of Fourier spectra proved ineffective due to the limited number of periods of the oscillation. No more than three periods were used as the character of the oscillations appeared to change to that of the long-duration oscillations after this period of time.

The results of these calculations are shown in Figure 4.7. There is an apparent linear dependence of the frequency on the mean pressure despite the scatter in the data. Note that the gap in the mean pressure for both the PVC and glass constitutes the boundary between oscillations about the static pressure and those about the elevated pressure. Ringing about the elevated pressure consistently occurs at a higher frequency than that about the static pressure. This result provides at least some indication that the frequency is dependent on some variable property of the bed and is consistent with the hypothesis that the oscillations arise from a natural frequency of the force chains.

To estimate the natural frequency expected from theory, a 1-D chain of identical particles as depicted in Figure 1.3 is considered. The system is modeled as a series of masses for the particles connected by nonlinear springs with a Hertzian relationship between the force,F, and the overlap, δbetween particles such that

F =K2δ3/2. (4.1)

It is assumed that the chain of particles is subjected to an initial force,F0, which leads to an initial

particle overlap, δ0, between each pair of particles. The nonlinear force-overlap relation is then

linearized about this initial overlap. The linearized stiffness about this initial overlap will be

k=dF dδ =

3K2δ10/2

2 . (4.2)

Written in terms of the initial confining force, this stiffness will be

k= 3 2K 2/3 2 F 1/3 0 . (4.3)

The force amplitude of the vibrations is assumed to be much less than the confining force. Otherwise, the particles would come out of contact due to the vibrations or the linearization would no longer hold.

The system is now a series of point masses with this linearized stiffness between them. The equation of motion is written for each particle in the chain. The resulting system of equations is solved to find the natural frequencies of the chain of particles. The primary frequency, fnat, will have the form

fnat =G

r

k

where k is the linearized stiffness between particles, m is the mass of a particle and Gis a factor that depends on the number of particles in the chain and the boundary conditions at the end of the chain. In general,G decreases as the number of particles increases. Substituting fork and m (= 1

6πρD3), the frequency becomes

fnat =G3 2/3 π1/2 1 ρ1/2_D4/3 µ E 1−ν2 ¶1/3 F₀1/6. (4.5)

For a chain of 100 particles with its ends fixed, Gwill be on the order of 1×10−3_{. The value}

of

q

k

m will be on the order of 1×105 Hz for glass spheres if the initial compression force is taken as the weight of particles occupying a height of 80 mm above the point of measurement. The result will be a frequency on the order of 100 Hz which is in the range of the measurements of Figure 4.7. The data in Figure 4.7 may also scale with the pressure to the 1/6 power as the theory predicts the frequency to scale with confining force, but more data is needed to draw a firm conclusion on this dependence.