NORMAS DE CARÁCTER GENERAL:

For these experiments, boundary conditions were designed that could be decoupled from the system response and simply modeled, while applying the static load and allowing the necessary measurements and actuation. For the actuator boundary, a steel block was designed that would act as a rigid wall. Holes were milled to allow the granular crystal alignment rods to be adjusted in length (similar to the polycar- bonate guide plate pattern), without moving the actuator mount. The dimensions (the actuator mounting block was a cube of 8.9 cm per side, and the block at the other end of the crystal was a cube of 7.6 cm per side) were designed based on the estimate that the frequency of the first resonant mode of the block should be greater than the frequencies of the system (approximately 23 kHz). This is critical to ensure that the “rigid” wall does not begin to vibrate on its own, otherwise vibrations aside from what is calculated by the applied voltage, and measured by the strain gage will be applied to the granular crystal. Additionally, a resonant frequency within the range of the granular crystal response would cause the boundary to interact with the response of the crystal – creating a nontrivial boundary condition to model. The resonant frequency of the first mode was estimated where fb = _2Lb1

q

Eb

ρb, where Lb is the length of the block, Eb is its elastic modulus, and ρb is its density. The resonant frequency was experimentally checked by applying a impulse excitation to the block and measuring the frequency spectrum of the response with accelerometer bolted to the opposite side. As the actuator mount is rigid, and the actuator (as previously described) of high stiffness compared to the granular crystal, in numerics, the front of the actuator is modeled as a moving wall.

Opposite the actuator “rigid wall”, a static load was applied to statically com- press the granular crystal. Two methods were used for this. Initially (for the discrete breathers experiments, see figure3.1), a lever–hanging mass system was used to apply the static load, where the static load applied was calculated based on the lever geom- etry and calibrated with a static load cell which was then removed. This method was

difficult to model, and not fully decoupled from the system. Because this mechanism was built from steel (stiff at the contact), but allowed to pivot around the lever, it acted like a large mass with applied force boundary condition. However, this did not greatly effect the experiment as the dissipation was high enough and the chain long enough that the dynamic effect of the boundary could not be seen at the beginning of the chain (where the relevant phenomena was occuring). Modifications were at- tempted, such as adding additional mass to the lever or adding dissipative elements; however in all cases, using an accelerometer measurements, it was found that there was significant movement of the boundary.

Following this, an attempt was made to make a fixed boundary at the other end (similar to the actuator boundary) which would also apply the static load. Though this method was never actually used, it is important for understanding the design of the final boundary condition. A steel block of similar dimensions was fabricated where its position could be adjusted and then fixed to the optical table some dis- tance with respect to the actuator mount – thus statically compressing the crystal. There were several challenges with this method. The first is the measurement of the static load. With the tools then available, the static load could be measured by the displacement of the actuator (based of the embedded strain gage measurement) or with a static load cell. The static load cells used were soft elements (compared to the stiffness of the granular crystal contacts), which create a stiffness defect in the granular crystal or at the boundary, so these were not used in this configuration. The second major challenge was due to the stiffness of the granular crystal as a whole. Under a fixed-fixed condition, any small buckling of the chain, or actuator hysterisis caused a significant change in the effective static load.

Following these attempts, a “free” boundary condition with applied static load was designed (see figures. 1.3, 2.1, and 5.1). A soft stainless steel linear spring (stiffness 1.24 kN/m) was placed in between the moveable steel cube. Thus when the moveable steel block was positioned and fixed with respect to the actuator mount, the spring is compressed and a static load applied to the granular crystal. Because the linear spring is so much softer than the contact stiffness between the particles in the

granular crystal, the boundary is modelled as a free boundary condition. This was confirmed by placing a piezoelectric sensor at the last particle in the chain, applying an impulsive excitation, and measuring the frequency response. The frequency matched closely with that predicted by a free-boundary condition surface mode. Furthermore, because of the low stiffness of the compression spring, anything placed behind the spring (with respect to the granular crystal) is effectively decoupled from the system. The static load cell is thus placed between the spring and the rigid boundary, where the static load cell is mounted in a dissipative teflon holder. This configuration allows the static load to be measured without affecting the response of the chain, and allows for small deviations in the granular crystal realignment and actuator hysterisis without significantly affecting the static load applied to the crystal.