(Equation 12) Note: In positioning applications, piezo actuators are operated well below their resonant fre- quencies. Due to the non-ideal spring behavior of piezoceram-ics, the theoretical result fromthe above equation does not necessarily match the real- world behavior of the piezo actuator system under large signal conditions. When adding a mass M to the actua- tor, the resonant frequencydrops according to the follow-ing equation: Resonant frequency withadded mass.m´ (Equation 13) (Equation 11) where: Resonant Frequency Resonant frequency of an idealspring/mass system. In general, the resonant fre-quency of any spring/mass sys- tem is a function of its stiffness and effective mass. Unless otherwise stated, the resonant frequency given in the tech- nical data tables for actuators always refer to the unloadedactuator with one end rigidlyattached. For piezo positioning systems, the data refers to the unloaded system firmly attached to a significantly larg- er mass. where:f

Piezo · Nano · Positioning

= phase angle [deg]F
max = resonant frequency [Hz]f = operating frequency[Hz]
eff = additional mass M + m
O = resonant frequency ofunloaded actuator [Hz]k
eff .The above equations show thatto double the resonant fre- quency of a spring-mass sys- tem, it is necessary to either increase the stiffness by a fac-tor of 4 or decrease the effec-tive mass to 25% of its original value. As long as the resonant frequency of a preload spring is well above that of the actua-tor, forces it introduces do not significantly affect the actua- tor’s resonant frequency.The phase response of a piezoactuator system can be approx-imated by a second order sys-tem and is described by the fol- lowing equation:
T = piezo actuator stiffness[N/m]m
eff = effective mass (about1/3 of the mass of the ceramic stack plus any installed end pieces) [kg]
Effective mass of an actuator fixed at one end 49