All real mechanical system has mass and spring characteristics. In a dynamic analysis the inertia forces must be considered with a mass dependent term. The simplest case is a stack itself in highly dynamic operation. In the next differential equation of motion the Newton’s law are expressed in a lumped parameter model:

F_{p} is the piezoelectric force controlled by the voltage, x is the displacement, m the effective inertial mass
of the piezo, and k is a term for damping.

This equation represents a damped spring-mass system, which will perform damped oscillation when excited by a voltage step function. The following data are exemplarily presented for a Piezotechnik PB5.8 stack:

· f_{res} 76 kHz

· m_{eff} 3,4 grams

· stiffness k 6 000 N/m equivalent to a effective Young’s modulus of 44 GPa

· damping parameter b 150 Ns/m

It is well known that this dynamic system have a resonance effect and will overshoot and ring when excited with a voltage
step function. The actuator has a large initial acceleration a_{0}

In the chosen example the initial acceleration of the stack a_{0} is 250 000 m/s^{2} or 25 500 g. In a practical case
there will be a load mass and resonance frequency would be reduced of course.

Rise time for spring-mass system is usually approximated by
t_{r} = 1/3f_{res}. In the given example t = 5 ms. Please notice, that for faster dynamic operation a dynamic overshoot reaction occurs. An overshoot over
the maximum static displacement cause internal stress in the ceramics in tensile direction. As ceramics is sensitive to tensile stress compensation by pre-stress is required to avoid cracking and
failure of the piezo. Those pre-stress mechanism cause mass effort and will reduce reaction speed of the actuator.

These physics principles has to be taken in consideration when designing a piezo stack actuation mechanism for **strong, fast and precise**
systems.

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