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As described before the electric field cause mechanical stress which induce a
deformation of the actuator body. The generative force F_{g} is the product of actuator area A and the piezoelectric stress s:

The generated stress s is proportional to the piezoelectric constant d, which is a material property, and the applied electrical voltage. The maximum generative force is provided in the actuator specification. The proper actuator size can be selected by choosing the actuator base area.

It is important to consider that piezo actuators are solid bodies and feature much higher mechanical stiffness than any other
actuator of same size. In mechanical equilibrium the generative force is equal the sum of the useful load force and the internal elastic force which is associated with the actuator deformation.
This means that the useful force reduces linearly with displacement X. X_{0} denotes the free displacement (no lad situation).

An illustration of the force versus displacement output of a piezoelectric actuator with various applied voltages is shown in Figure 3.

The linear constitutive equation describes a linear elastic material with the superimposed effect of electrically induced force, which is linear to the applied electrical field. Second, a piezo actuator represents a capacitor with the superimposed effect of induced charge linear with the applied force. In conclusion, piezo actuation is based on electrically controlled deformation of a solid body. When a voltage is applied to a piezoelelectric actuator piezoelectric forces are generated instantaneously inside the solid body, which deform it and move the load to reach mechanical equilibrium.

From the above noted equations, **important load cases** can be examined:

The displacement X of a piezo actuator X is equal piezoelectric (charge) constant d x voltage U.

Multilayer actuators use a multitude of thin layers and the total achievable displacement is the value of the individual layer multiplied by the number of layers. The layer thickness is in the order of 100 um, voltages are 150 V and field strength is 1 – 2 kV/mm.

T = constant. The "Domain Orientation"-Effect contribute an additional deformation effect which will increases the normal piezoelectric deformation. It occurs when the piezo is compressed or tensional forces are applied. The compression or tension change the domain orientation in the piezo material. When the electrical voltage across the piezo tack element changes, the normal piezoelectric displacement X of the actuator is modified by the interaction of the applied field with the domains. (Field -Dipole Interaction). The electrics field lead to re-orientation of the domains. What happen is, that compression of the material result from reducing the quasi ideal alignment along the 3-axis and the body get shorter than one would expect from an elastic body with a given Young's modulus. The orientation of the domains couples mechanical and electric constitution of the material. An actuator material with high d33-constant is "softer" than a "hard" piezoelectric material which domain structure is not so moveable. The electrical induced displacement is still equal piezoelectric (charge) constant d33 x U. When the voltage is zero for example the compression of an actuator material result in larger displacement then one would imagine form an elastic body with a given Young's modulus. Domain re-orientation represents a resilience that can be influenced electrically. The change in the domain structure is subject to hysteresis and increases both the piezoelectric expansion when a field is applied in the 3-direction, since the alignment of the domains along the 3-axis causes an increase in length. This change in length adds to the known linear piezoelectric effect. In addition, the flexibility of the domain structure (de-orientation reduces the observable continuity of the body. This acts in addition to the well-known elastic effect, whereby the Young's modulus generally describes the linear elastic behavior.The mass force statically deforms the actuator.

The free displacement of the actuator is reduced by the ratio of stiffness of the load spring k_{load} and actuator k_{a}.

Displacement under load of spring is equal to the

free displacement times the factor** k _{a}/(k_{a}+k_{load}**)

with free displacement X_{0} = d U

The generated force is equal base area x piezoelectric constant d x electric field divided by compliance s.

If the voltage source is abruptly switched on, the actuator will experience a step function excitation. The electric behaviour of the actuator is that of a
capacitor and will pull large currents I form the amplifier to ramp up the voltage during rise time DT.

If the amplifier is capable to provide high currents, the actuator will overshoot in this situation and internal tensile stress results. The actuator stack is in danger to be damaged if not sufficient measures are taken:

- First, current limiting is reliable method to reduce rise time and to avoid overshoot.

- Second, a pre-stress mechanism can be installed which compensate transient tensile stress amplitude.

In dynamic actuation mode periodic (sine, square waves) or non-periodic waveforms (e.g. compensation of disturbances in a feedback control loop) are applied to the actuator. In modern car engines high-pressure fuel injectors are used to spray very precisely fuel into the combustion chamber. Piezo is the superior actuator principle to control the injection process and superseded electromagnets. The piezo in a fuel injector is used to generate several fast pulses during each combustion cycle.

The dynamic response of a piezo actuator and connected mechanical load to the electrically controlled piezoelectric force is determined by masses, stiffness’s,
and damping rates. The actuator itself represents a spring mass system with low damping rate. The low frequency response of that basic actuator system is given by the free stroke. At higher
frequencies the stroke is limited by the inertia of the effective actuator mass. The realizable displacement of an actuator in sinusoidal operation is given by the equilibrium of the
piezoelectric force and the force needed to accelerate the effective mass. The following equations illustrates the mechanical response quite below and beyond the mechanical resonance frequency
f_{r}:

a) f < f_{r} X = d U

b) f > f_{r} X = F_{b} /
(m_{load} (2p f)^{2})

Highly dynamic operation of a piezoelectric actuator results in high levels of mechanical (force x speed) and electrical power (voltage x current). Damping is effective in piezoelectric materials and in dynamic operation losses occurs with resulting heating of the material. Continuous dynamic operation can generate high losses and the piezo stack may rapidly heat up. Sufficient measures has be taken to avoid excessive temperatures:

· limit the amplitude (travel, voltage),

· limit frequency,

· limit the duration of operation, and

· adequate cooling.

Contact PIEZOTECHNICS if high power levels are required. PIEZOTECHNICS is experienced in highly dynamic actuation.

Performance made of Passion.

**Piezotechnics** by

Piezotechnics GmbH

Germany