Torque load measurements play an important part in various
engineering applications, as for automotive industry, in which the drive
torque of a motor has to be determined. A widely used measuring method are
strain gauges. A thin flexible foil, which supports a metallic pattern, is
glued to the surface of the object the torque is being applied to. In case of
a deformation due to the torque load, the change in the electrical resistance
is measured. With the combination of constitutive equations the applied
torque load is determined by the change of electrical resistance. The creep
of the glue and the foil material, together with the temperature and humidity
dependence, may become an obstacle for some applications

The state of the art method for the measuring of mechanical torque loads are
strain gauges. These are thin flexible foils glued to the surfaces of an
object, that change the electrical resistance due to deformation. Their
application for engineering purposes is cheap, simple and well established.
However, the measurement accuracy depends on the humidity and the surrounding
temperature. Also, the creep of the foil and the glue contributes to the
measurement error

As shown in Fig.

Measuring setup consisting of the beam, electret and the ideal conducting plate.

In order to model the behavior of an electrostatic field, it is important to
understand the model geometry and to define the mechanical assumptions for a
beam with a torque load

Assuming the shear angle

As shown in Fig.

Torsional deformation of the beam. Figures adapted from

Foregoing the derivation of the St. Vénant torsion theory, the torsion rate
at the end is defined as

Using the described geometry and the St. Vénant torsion theory, it is
possible to formulate a simple geometrical model of the small deformations as
shown in Fig.

Simple geometrical model of the charge holding particles. For a
linear approximation the twist rate follows from Eqs. (

The presence of a static charge holding particle results in a physical
electrostatic field

Based on the experiments, it is known that the forces on the particle are
central forces. Thus, the curl of the described fields has to be equally
zero,

Furthermore, the divergence of the electric displacement field is used to
derive an elliptic partial differential equation

In the case of homogeneous potential field problems, in which the only
boundary condition is postulated by the decrease of the field over the
distance and eventually vanishing at an infinite distance, there are known
analytic solutions to the elliptic partial differential, e.g. Kirchhoff's
integral theorem based on Greens identities

Considering the measuring setup in Fig.

However, there is a possibility to substitute the current setup with an
alternative setup just for the computation purpose. The first point to
consider, is the fact that the potential

Inhomogeneous electrostatic field problem to solve, with a constant potential boundary condition at the plate.

Furthermore, the electret consists of a thin foil, so it is possible to
simplify the integration in Eq. (

Homogeneous electrostatic field problem, with a constant potential at the plane described by the plate.

Although, an analytic solution for the sensor model has been derived, there still exists the problem of integrating the charge over a complex geometry. For the modeling purpose it is easier to use a discretization of the electret and translate the integration into a sum.

For the discretization, the electret is divided into regular small part-areas

This restriction to the solution is based on the fact, that in reality the space below the ideal conducting plate is shielded from the influence of the electric charge by the plate itself.

Regular discretization of the electret.

Potential planes

As described in Sect. 2 the induced charge

In Fig.

Figure

The quantity of the induced charge is calculated as described in Sect.

Charge density

The characteristic curve of the sensor fitted by a polynomial
function of 2nd order

As one can see in Fig.

Furthermore, an expected sensitivity deficiency is being observed. This is bound to the small quasi-permanent charge an electret holds and the distance to the ideal conducting plate.

Interestingly, the polynomial function of 2nd order

In this contribution an analysis of an electrostatic sensor for torque load measurements is presented. Due to the necessity of describing the position of the charge in the physical space, a geometrical and mechanical model based on the St. Vénant torsion theory is described. In order to solve the well known Poisson equation for this problem, the Kirchhoff's integral theorem and an alternated model are used. Further, a discretization is used to solve the integral and also to calculate the induced charge numerically. As shown, the proposed sensor has only a low sensitivity. Considering the dimensions and the sensitivity issues of the measuring setup, the practical application is not optimal. Nevertheless, in the further research an improvement of the sensitivity will be considered.

No data sets were used in this article.

The authors declare that they have no conflict of interest.