__Derivations of more precise
mathematical models:__

Accepted mathematical models of
hydraulic spool valves and amplifiers do not provide accurate enough results to
use existing models to predict the dynamics of piezoelectric/hydraulic hybrid
actuators. Therefore, an improvement to the mathematical models is integrated
into this program. Ultimately, the use of improved models will allow for
simulation of actuator designs. This simulation will provide a means of
refinement and optimization before developing any actuator hardware. Refinement
of the existing mathematical models will focus on the inclusion of a 2^{nd}
order function to simulate the observable motion of a piezoelectric stack and a
variable flow-by rate for hydraulic fluid within a hydraulic amplifier. Models
will be developed in accepted state space format and tested in a variety of
formats: Matlab/Simulink, C++, and VTB.

In conjunction with improvements to the actuator models, a mathematical relationship between the piston position and the engine valve position will be established. It is anticipated that an equivalent frequency can be calculated. This equivalent frequency takes into account the interrupted motion of engine valves within an engine cycle. For example, a typical intake valve is only open for 270 degrees out of every two engine revolutions (720 degrees)[1]. Therefore, the actuator must open and close the engine valve during this 270 degree portion of crank rotation. This results in a minimum equivalent frequency that the actuator must use as input for a given engine speed. Shown in Figure 1 is a graphical interpretation of equivalent frequency.

Figure 1 Equivalent Frequency

The development of mathematical
models that address the observable 2^{nd} order motion of piezoelectric
stacks combined with traditional hydraulic actuator relationships will lead to a
more accurate mathematical model of hybrid actuator systems. Such a model will
improve the accuracy of actuator simulations, lead to shorter development time
for future designs, and provide a foundation for optimizing performance of the
actuators being developed.