Displacement and External Forces

Like any other actuator, a piezo actuator is compressed when a force is applied. Two cases must be considered when operating a piezo actuator with a load:

a) The load remains constant during the motion process.

b) The load changes during the motion process.

Note

To keep down the loss of travel, the stiffness of the preload spring should be under 1/10 that of the piezo actuator stiffness. If the preload stiffness were equal to the piezo actuator stiffness, the travel would be reduced by 50 %. For primarily dynamic applications, the resonant frequency of the preload must be above that of the piezo actuator.

a) Constant Force

Zero-point is offset A mass is installed on the piezo actuator which applies a force F = M · g (M is the mass, g the acceleration due to gravity).
The zero-point will be shifted by ΔL_N ˜ F/kT, where kT is the stiffness of the actuator. If this force is below the specified load limit (see product technical data), full displacement can be obtained at full operating voltage.

(Equation 6)

Zero-point offset with constant force

where:

ΔL_N = zero-point offset [m] F = force (mass x acceleration due to gravity) [N] k_T = piezo actuator stiffness [N/m]

Example

How large is the zero-point offset of a 30 µm piezo actuator with a stiffness of 100 N/µm if a load of 20 kg is applied, and what is the maximum displacement with this load?

The load of 20 kg generates a force of 20 kg x 9.81 m/s₂ = 196 N. With a stiffness of 100 N/µm, the piezo actuator is compressed slightly less than 2 µm. The maximum displacement of 30 µm is not reduced by this constant force.

b) Changing Force

Displacement is reduced For piezo actuator operation against an elastic load different rules apply. Part of the displacement generated by the piezo effect is lost due to the elasticity of the piezo element. The total available displacement can be related to the spring stiffness by the following equations:

(Equation 7)

Maximum displacement of a piezo actuator acting against a spring load.

(Equation 8)

Maximum loss of displacement due to external spring force. In the case where the restraint is infinitely rigid (ks = 8), the piezo actuator can produce no displacement but acts only as a force generator.

where:

ΔL = displacement with external spring load [m]
ΔL₀ = nominal displacement without external force or restraint [m]
ΔL_R = lost displacement caused by the external spring [m]
k_s = spring stiffness [N/m] k_T = piezo actuator stiffness [N/m]

Example

Q: What is the maximum displacement of a 15 µm piezo translator with a stiffness of 50 N/µm, mounted in an elastic restraint with a spring constant kS (stiffness) of 100 N/µm?

A: Equation 7 shows that the displacement is reduced in an elastic restraint. The spring constant of the external restraint is twice the value of the piezo translator. The achievable displacement is therefore limited to 5 µm (1/3 of the nominal travel).