Multi‐cavity Waveguide Theory

Abstract

In a multi-nozzle piezoelectrically driven print head a large number of miniature valveless pumps are integrated. In order to place these miniature pumps as close as possible to each other the length of the pump cavity must be long compared to its cross-sectional dimensions. This is needed to generate enough volume displacement by the piezo actuator. Each pump out of such a print head consists of a pump chamber, a connecting channel, a nozzle and a restriction (throttle). This design will be referred to as the closed end/closed end arrangement. The restriction communicates with the main supply channel. For the waveguide design it is possible to have an open connection to the main supply channel. This will be referred to as the open end/closed end design. By actuation pressure waves will be induced that travel back and forth through the waveguide, their evolution in time depending on the reflection characteristics at the open end to the main supply channel and at the nozzle. Characteristic of the nozzle/throttle design (closed end/closed end) is the restricted connection to the ink supply. It is in principle a half wave length resonator with an open, but restricted, connection to the ink supply and a small but open restriction (the nozzle) to ambient. Because of the restrictions these acoustic end conditions will be referred to as closed. Characteristic of the open end/closed end design is the open connection to the ink supply. As far as the acoustic properties are concerned it is a quarter wave length resonator with an open connection to the ink supply and a small but open restriction (the nozzle) to ambient. The fact that in a multi-nozzle print head the pumps communicate with each other through the main supply channel adds to the complexity of the modelling of a real linear array waveguide system, because it is prone to acoustic cross-talk. In a waveguide type of pump the acoustic end conditions are governed by the fluid column contained in the nozzle and either the fluid inside the throttle or the open end. The compressibility of the fluid contained in the pump chamber will now be dealt with in terms of compression and expansion waves travelling back and forth through the long pump chamber and connecting ducts, the masses and damping in both the nozzle and throttle for the closed end/closed end arrangement are determining the reflection properties at both ends of the waveguide. A similar argument holds true for the open end/closed end case. A large number of pumps are connected to the main supply channel; therefore its length will be large. To avoid acoustic cross-talk the main supply channel must be acoustically soft; the speed of sound corrected for the compliance of the environment must be low compared to the isentropic value. In order to study the dynamics of a piezo driven multi-nozzle linear array inkjet printer head of the waveguide type, in this chapter the following simplified set-up is considered, namely a small and long cavity connected to the environment via a small hole, the nozzle. The other side of the pump chamber is connected to the ink supply channel through a throttle. For the open end design the throttle is absent. Although the cross-sectional dimensions of throttle and nozzle are usually chosen roughly equal, the length of the throttle is much larger than the length of the nozzle. The nozzles are placed at a constant pitch; the surface area of the cross-section of the supply channel and its length are much larger than the corresponding dimensions of the pumps. In this chapter the acoustic cross-talk will be dealt with by making use of symmetry arguments. The linear array print head is considered to be infinitely long with an infinite number of pumps integrated. By driving alternately one pump active and the next one idling, local symmetry allows for confining the analysis to just two neighboring pumps. This approach will be extended to the driving of one nozzle and either two adjacent ones idling or three idling. The method will be applied to the analysis of a multi-nozzle shared wall shear mode print head.