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Mechanical Considerations for Dynamic Operation of PZTs

Resonant Frequency
In general, the resonant frequency of any spring/mass system is a function of its stiffness and effective mass. The resonant frequency given in the technical data tables always refers to the unloaded actuators, rigidly mounted on one end.

Effective mass of an actuator fixed on one end.

f₀ = (1/2p)*Ö(k_T/m_eff)       (4-10)

Resonant frequency of an ideal spring/mass system.

where
f₀ = resonant frequency [Hz]

k_T = actuator stiffness [N/m]

m_eff = effective mass (about 1/3 of the mass of the ceramic stack plus any installed end pieces) [kg]

Note:
Due to the non-ideal spring behavior of Piezo ceramics, the theoretical result of the above equation does not necessarily match the real-world behavior of a Piezo system.

When adding a mass to the actuator the resonant frequency drops according to the following equation:

f₀'=f₀ Ö(m_eff/(m_eff+M))       (4-11)

Resonant frequency with additional mass M.

The above equations show that increasing the mass on the actuator by a factor of 4 will reduce the response (resonant frequency) by a factor of 2. Increasing the spring preload on the actuator does not significantly affect its resonant frequency.

The phase response of a Piezo system can be approximated by a second order system and is described by

j » 2 * arc tan (f/f₀)       (4-12)

j = phase angle [deg]

f₀ = resonant frequency [Hz]

f = operating frequency [Hz]

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