# Dynamic Operation

## Resonant Frequency

The resonant frequencies specified for longitudinal stack actuators apply to operation when not clamped on both sides. In an arrangement with unilateral clamping, the value has to be divided in half.

The reducing influence of an additional load on the resonant frequency can be estimated with the following equation (for the calculation of meff see fig. 1):

In positioning applications, piezo actuators are operated considerably below the resonant frequency in order to keep the phase shift between the control signal and the displacement low. The phase response of a piezo system is approximately that of a second order system:

 f0' Resonant frequency with load [Hz] f0 Resonant frequency without load [Hz] meff Effective mass of the piezo stack actuator [kg] meff' Effective mass of the piezo stack actuator with load [kg] φ Phase angle [degrees] f Control frequency [Hz] m Mass of the piezo actuator M Additional load Tmin Time in which the piezo actuator can reach its nominal displacement.

## Response Behavior: How Fast Can a Piezo Actuator Expand?

Fast response behavior is a characteristic feature of piezo actuators. A fast change in the operating voltage causes a fast position change. This behavior is desired especially in dynamic applications, such as scanning microscopy, image stabilization, valve controls, generating shock waves, or active vibration damping. When the control voltage suddenly increases, a piezo actuator can reach its nominal displacement in approximately one third of the period of its resonant frequency f0 (fig. 2):

In this case, a strong overshoot occurs. With the corresponding control technology, this can partially be compensated for.

Example: A unilaterally clamped piezo actuator with a resonant frequency of f0 = 10 kHz can reach its nominal displacement in 30 μs.

## Dynamic Forces

With suitable drive electronics, piezo actuators can generate high accelerations of several ten thousand m/s². As a result of the inertia of possible coupled masses and of the actuators themselves, dynamic pull forces occur, which have to be compensated for with >> Mechanical Preloads. In sinusoidal operation, the maximum forces can be estimated as follows:

 Fdyn Max. dynamic force meff' Effective mass of the piezo stack actuator [kg] ΔL Displacement (peak-to-peak) [m] f Control frequency

Example: The dynamic forces at 1,000 Hz, 2 μm of displacement (peak-to-peak) and 3 kg mass are approximately ±40 N.

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