Sessile liquid drops damp vibrating structures

Abstract

In this study, we explore the vibration damping characteristics of singular liquid drops of varying viscosity and surface tension resting on a millimetric cantilever. Cantilevers are displaced 0.6 mm at their free end, 6% their length, and allowed to vibrate freely. Such ringdown vibration causes drops to deform, or slosh, which dissipates kinetic energy via viscous dissipation within the drop and through contact line friction. Damping by drop sloshing is dependent on viscosity, surface tension, drop size, and drop location. A solid weight with the same mass as experimental drops is used to compare against the damping imposed by liquids, thereby accounting for other damping sources. Neither the most viscous nor least viscous drops studied imposed the greatest damping on cantilever motion. Instead, drops of intermediate viscosity strike the most effective balance of sloshing and internal dissipative capacity. Very thin cantilevers with sloshing drops express more than one dominant frequency and vibrate erratically, often shifting phase, presenting a challenge for quantification of damping. Finally, we introduce a new dimensionless group aimed at incorporating all salient variables of our cantilever-drop system.

Coupled Dynamics of Droplet Impact on a Flexible, Hydrophilic Cantilever Beam

“…This is explained by the fact that the damping coefficient of the equivalent spring-mass system for the coupled dynamics scales with viscosity is as follows: c ∼ μ d 0 . This result is also consistent with the findings of a previous study, which reported that the stiffest beam damps by a sessile droplet with larger viscosity. As m c increases, ζ decreases for all three droplets, as plotted in Figure b.…”

“…Now, the energy balance between point t 1 and t (Figure a) is given as follows false| E normalk + E normals false| t 1 = false| E normalk + E normals + E normalb false| t + false| E normald normali normals normals false| t 1 − t false| E normald normali normals normals false| t 1 − t = false| E normalk + E normals false| t − false| E normalk + E normals + E normalb false| t where E diss includes energy dissipation due to viscosity, contact line friction, and air damping . The bending energy ( E b ) of the cantilever beam is E b = 3 EI δ 2 /2 L 3 (ref ). The time-averaged bending energy ( E b,avg ) is expressed as follows E normalb , a v g = 1 t f − t 0 ∫ t 0 t normalf 3 E I 2 L 3 δ 2...…”

“…Now, the energy balance between point t 1 and t (Figure a) is given as follows where E diss includes energy dissipation due to viscosity, contact line friction, and air damping . The bending energy ( E b ) of the cantilever beam is E b = 3 EI δ 2 /2 L 3 (ref ). The time-averaged bending energy ( E b,avg ) is expressed as follows where t 0 is the time at the moment of droplet impact, and t f is the time required for the cantilever to undergo N cycles.…”

“…A theoretical model based on the energy balance was used to explain the measurements. The role of droplet viscosity in damping of cantilever motion was quantified by Alam and Dickerson . They showed that a viscous droplet at rest (sessile) dampens the cantilever beam oscillations, and the maximum damping corresponds to an optimum value of viscosity that depends on the substrate’s stiffness.…”

“…In addition, previous studies did not consider the effect of viscosity on the coupled dynamics of the droplet and substrate, and most of the earlier studies employed pure water droplets in this context. One exception is the study of Alam and Dickerson, which showed that the sessile viscous droplet (zero impact velocity) damps the cantilever beam oscillations. However, it is unclear how the time scales of droplets and substrate in the case of an impacting droplet influence the lock-in condition.…”