Piezo-University
Dynamic Operation Fundamentals
Description   Specifications   Downloads   CAD/Video
Drawings & Images   Models/Get Price Quote   Related Info   ?? Ask an Engineer

Dynamic Forces
Every time the piezo drive voltage changes, the piezo element changes its dimensions. Due to the inertia of the piezo actuator mass (plus any additional load), a rapid move will generate a force acting on (pushing or pulling) the piezo. The maximum force that can be generated is equal to the blocked force, described by:

(Equation 8)





Maximum force available to accelerate the piezo mass plus any additional load. Tensile forces must be compensated, for example, by a spring preload.

Where:

Fmax = max. force [N]

DL0 = max. nominal displacement without external force or restraint [m]

kT = piezo actuator stiffness [N/m]

The preload force should be around 20% of the compressive load limit. The preload should be soft compared to the piezo actuator, at most 10% the actuator stiffness.
In sinusoidal operation peak forces can be expressed as:

(Equation 9)





Dynamic forces on a piezo actuator in sinusoidal operation at frequency f.

Where:

Fdyn = dynamic force [N]
meff = effective mass [kg], see p. see link
DL = peak-to-peak displacement [m]
f = frequency [Hz]

The maximum permissible forces must be considered when choosing an operating frequency.

Example:
Dynamic forces at 1000 Hz, 2 µm peak-to-peak and 1 kg load reach approximately ±40 N.

Note
A guiding system (e.g. diaphragm type) is essential when loads which are heavy or large (relative to the piezo actuator diameter) are moved dynamically. Without a guiding system, there is a potential for tilt oscillations that may damage the piezoceramics.


Resonant Frequency
In general, the resonant frequency of any spring/mass system is a function of its stiffness and effective mass (see Fig. 23). Unless otherwise stated, the resonant frequency given in the technical data tables for actuators always refer to the unloaded actuator with one end rigidly attached. For piezo positioning systems, the data refers to the unloaded system firmly attached to a significantly larger mass.

(Equation 10)





Resonant frequency of an ideal spring/mass system.

Where:

fO = resonant frequency of unloaded actuator [Hz]

kT = piezo actuator stiffness [N/m]

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

Note:
In positioning applications, piezo actuators are operated well below their resonant frequencies. Due to the non-ideal spring behavior of piezoceramics, the theoretical result from the 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 actuator, the resonant frequency drops according to the following equation:

(Equation 11)





Resonant frequency with added mass.

m“eff = additional mass M + meff.

The above equations show that to double the resonant frequency of a spring-mass system, it is necessary to either increase the stiffness by a factor of 4 or decrease the effective mass to 25 % of its original value. As long as the resonant frequency of a preload spring is well above that of the actuator, forces it introduces do not significantly affect the actuator"s resonant frequency.

The phase response of a piezo actuator system can be approximated by a second order system and is described by the following equation:

(Equation 12)





Where:

j = phase angle [deg]

Fmax = resonant frequency [Hz]

f = operating frequency [Hz]


How Fast Can a Piezo Actuator Expand?
Fast response is one of the characteristic features of piezo actuators. A rapid drive voltage change results in a rapid position change. This property is especially welcome in dynamic applications such as scanning microscopy, image stabilization, switching of valves/shutters, shock-wave generation, vibration cancellation systems, etc.

A piezo actuator can reach its nominal displacement in approximately 1/3 of the period of the resonant frequency, provided the controller can deliver the necessary current. If not compensated by appropriate measures (e.g. notch filter, InputShaping®, see p. see link) in the servo-loop, such rapid expansion will be accompanied by significant overshoot.

(Equation 13)





Minimum rise time of a piezo actuator (requires an amplifier with sufficient output current and slew rate).

Where:

Tmin = time [s]

f0 = resonant frequency [Hz]]

Example: A piezo translator with a 10 kHz resonant frequency can reach its nominal displacement within 30 µs.

Drawings & Images:
 
Fig. 22. Recommended guiding for large masses.
Fig. 22. Recommended guiding for large masses.

Fig. 23. Effective mass of an actuator fixed at one end.
Fig. 23. Effective mass of an actuator fixed at one end.

Fig. 24. Response of an undamped, lever-amplified piezo actuator (low resonant frequency) to a rapid drive-voltage change. This behavior can be prevented by intelligent control techniques or position servo-control.
Fig. 24. Response of an undamped, lever-amplified piezo actuator (low resonant frequency) to a rapid drive-voltage change. This behavior can be prevented by intelligent control techniques or position servo-control.





© 1996-2013 PI (Physik Instrumente) GmbH & Co. KG. All rights reserved. PI - Moving the NanoWorld. ISO 9001. Specifications subject to change without notice.    nanopositioning Linear Stage PI USA Piezo