Static and quasi-static operation

The constitutive equations can be specifically applied to an actuator.

In this case, the important characteristics are the mechanical stiffness (from the “sE.T” term) and the piezoelectric free strain (from the “dt.E” term). 

All actuators can thus be described in a first approximation by linear equations. The common representation for this behaviour is a force-displacement diagram.

If the actuator operates freely, it will generate a displacement equal to its free displacement. If it is operating against a spring, it will generate less displacement, but more force, according to the stiffness of the opposing spring. If it is operating against an infinitely stiff spring, it will generate a force equal to its blocking force, but no displacement.

The maximum mechanical power is reached at the middle of this graph (half the free displacement, half the blocking force), i.e. when the opposing spring and the actuator have the same stiffness.
 

Example: An actuator needs to generate a displacement of 0,1 mm against a spring of 10.000 N/mm. The most optimised actuator will have a free stroke of 0,2 mm and a blocking force of 2.000 N.

Characteristics of Noliac standard linear actuators

This situation is realized when placing a constant load M on the actuator. Then the dimensions of the piezoelectric actuators will be initially reduced by an amount ∆LN=M x g x k^-1, where g is the gravitational constant and k the stiffness of the actuator. Except for this initial offset, its capability to produce displacements will remain roughly unaffected and full displacement will be obtained at full operating voltage.

2. The load changes during the motion process

When the opposite force is proportional to the displacement with a spring constant K_o, the final displacement is reduced by ∆LR= K_o x L_o x (k+K_o) ^-1, where k is the stiffness of the actuator and L_o its full stroke.
 

Although non-linear by nature, benders can also be approximated by this free displacement – blocking force linear model.

Characteristics of Noliac standard benders