Compliant mechanisms in precision weighing technology are highly sensitive mechanical systems with continuously rising demands for performance in terms of resolution and measurement uncertainty. The systematic combination of adjustment measures represents a promising option for the enhancement of weighing cells which is not yet fully exhausted. A novel adjustment concept for electromagnetic force compensated weighing cells designed for 1 kg mass standards is introduced. The effect on the mechanical behavior is analyzed in detail using a planar compliant mechanism with semi-circular flexure hinges. Design equations for a first layout of the mechanical system are derived from a linearized rigid body model. Existing adjustment concepts for the stiffness characteristic and the sensitivity to quasi-static ground tilt are included. They are extended by the novel approach to attach trim weights to the levers of the linear guide. Based on this concept, an optimal design for the weighing cell is determined. The comparison with a finite element model reveals further effects given by the more precise description of the mechanical behavior. By parametric studies of the adjustment parameters in the mechanical models, it is shown that the stiffness and tilt sensitivity can be reduced significantly compared to the non-adjusted weighing cell. The principal correlation of the trim weights and their effect on the mechanical properties is experimentally verified using a commercially available weighing cell.

Precision weighing technology is a research area of
persisting importance for science and economy. The reference of the unit of
mass in the International System of Units (SI) presently depends on the
performance of mass comparators

Besides the dissemination of the present international definition of the unit
of mass, mass comparators are an integral part of research activities in
preparation for the upcoming redefinition of the kilogram and the revision of
the SI system of units

The mechanical system of mass comparators consists of a mechanism based on
concentrated compliance in the form of flexure hinges, a fixed counterweight
and an electromagnetic force compensation (EMFC; see
Fig.

Monolithic EMFC weighing cell with flexure hinges and typical joint orientation. 1 – base, 2 – lower lever, 3 – upper lever, 4 – load carrier, 5 – lower weighing pan, 6 – upper weighing pan, 7 – coupling element, 8 – transmission lever.

The sensitivity of the properties of the mechanism to manufacturing
tolerances necessitates adjustment measures. This is common practice for all
kinds of precision balances, but in contrast to equal arm beam balances

Extensive work on the dynamic behavior of EMFC-weighing cells and their
optimization is presented in

The present work focuses on the static behavior of the mechanical structure of an EMFC-weighing cell. It aims to advance the understanding of the mechanical properties and their enhancement by means of targeted adjustments. This includes the improvement of the sensitivity of the overall system with a simultaneous reduction of its sensitivity to environmental disturbances.

The general scheme of an EMFC-weighing cell is presented in
Fig.

Employing the principle of electromagnetic force compensation, the mechanism
of EMFC balances is quasi not deflected during operation. Residual
deflections result from deviations of the position controller and elastic
deformations of the structure itself. This excludes deflection-dependent
geometrical nonlinearities and anelastic effects of the material to a large
extent. Besides the intrinsic error sources, major disturbances arise from
the environmental surroundings of the precision weighing device. Factors such
as temperature, humidity, and air pressure are of high relevance, as well as
turbulence and electric and magnetic fields

Quasi-static
ground tilt and ground vibrations slightly move the base of the weighing cell

The modeling of the monolithic mechanism of the EMFC-weighing cell is divided
into two stages which can be differentiated by their modeling assumptions and
applied methods. The first model, the linearized rigid body (RB) model, is
based on strongly simplifying modeling assumptions; see
Fig.

In contrast, the finite
element analysis (FEA) is capable of considering elastic deformations of the
mechanism; thus, non-ideal deflections of the flexure hinges are included.
Thus, the comparison between the RB and FEA models reveals the limitations of
the RB model and the influence of elastic deformations. The adjustment
concept presented in Sect.

Weighing cells with high resolution rely on very thin flexure hinges as
rotational joints to obtain a high sensitivity. The minimum thickness of the
joints is technologically limited to about

Figure

Rigid body model of the deflected weighing cell (

The adjustment measures include the adjustment of the centers of mass of
parts in the kinematic system by displacing small trim weights
(

Two modeling stages are used to
investigate the behavior of the weighing cell structure. The mechanical
models of the weighing cell include all adjustment parameters (see
Fig.

The mechanical model of the monolithic mechanism is simplified based on the
following assumptions: the compliant mechanism has concentrated compliance
(semi-circular flexure hinges). All other parts are modeled as rigid bodies
with lumped masses. The flexure hinges are modeled as perfect rotational
joints with a fixed rotational axis and a constant rotational stiffness. All
flexure hinges in the mechanism are modeled with equal geometric parameters
and rotational stiffness. Frictional losses in the joints are not considered.
The coupling element between the subsystems is modeled as a
deflection-dependent transmission ratio between the deflection angles

Kinematically equivalent mechanism of the weighing
cell in Fig.

Hence,

Overview of the adjustment concept.

^{a}

The tilt sensitivity

The FEA model is created using ANSYS Mechanical
APDL^{®}. The structural parts are meshed
with SOLID186 elements based on quadratic displacement functions. The structure
consists of several components that are linked by flexure hinges.

Finite element model of the weighing cell structure with flexure hinges. The point masses attached
to the structure are displayed as

The volume is assigned with linear elastic material properties of aluminum
alloy, which is commonly in use for precision weighing cells

Model parameters.

^{b}

The setup of the model enables the alteration of several parameters. The most influential parameters for the properties of the structure with respect to its application in precision weighing are the centers of rotation and centers of mass. Their influence on the properties of the total structure is determined by parametric studies.

Figure

The computational results from the modeling approaches are presented and compared. From this juxtaposition, consequences for the applicability ranges of linear model equations are derived.

Investigations on flexure hinges reveal that flexures do not show pure
rotations due to a shift of the rotational axis during their deflection

The joint orientation, which proves to be relevant in other fields

Another aspect is the change in rotational stiffness of the joint due to the
static axial load

The geometrically nonlinear FEA model reveals the limited validity of the
linearized rigid body model. This can be shown in the case of the modeling of
the coupling element. According to the model Eq. (

The destabilizing effect of the

.

Visualization of the angles of the coupling element for a deflected state of the weighing cell.
The adjustment parameter

In Fig.

The compensation of restoring forces of the compliant mechanism can be
equated with the destabilization of the system. The stiffness

Variation of

It involves the downside of being dependent on the mass of the sample weight
that is placed on the weighing pan, as can be observed in
Fig.

Linear load dependency of the stiffness

The parameter variation is limited to the
highlighted adjustment parameters in Fig.

FEA results of the function

adjustment of

parameter variation of

determine minimum:

Adjustments of the FEA model.

For the experimental setup, the weighing cell of commercially available mass
comparator MCM1005 was set up on top of the precision tilt table

Experimental setup for the determination of tilt sensitivities with the EMFC-weighing cell based on the mass comparator MCM1005.

The measurement results of weighing systems are sensitive for tilt. If the
weighing cell is not horizontally aligned, as shown in
Fig.

Tilt influence on mass determination systems.

After calibration of the weighing cell by use of an interferometer and an
E2-calibration weight, the setup is prepared for the measurement of tilt
sensitivities. For a first overview the controller, operating in normal mode,
keeps the transmission lever in its undeflected position while the tilt table
approaches different stages of tilt. Once at the beginning, in the middle and
at the end of each measurement the tilt table is returned to its zero
position (

Measured force during tilt about the

Measured force during tilt about the

Influence of tilt on the weighing result.

The results of the measurements are calculated based on the continuous
measurement signal of the weighing cell during the static states of the tilt
table. The measured tilt reactions are correlated with the respective tilt
angles as presented in Fig.

Correlation between tilt angle and tilt sensitivity of the weighing cell.

With this effective and automated method to measure the tilt dependencies in predefined tilt positions, the influence of adjusting parameters such as additional trim weights on the upper lever (3) or the transmission lever (8) can be investigated in dependence of tilt.

Influence of the trim weight (

The investigation was continued by adding small trim weights on the
transmission lever (as shown in Fig.

Influence of the trim weight (

Influence of the trim weight at the lever (

The influence of trim weights on tilt sensitivity shows a linear relationship
with

Modeling aspects of high-precision monolithic weighing cells based on quasi-static mechanical models are discussed. A linear equation system is introduced, presenting the most relevant mechanical properties of the weighing cell at a glance. The solution of the equation system, involving adjustable parameters, provides a foundation for a preliminary design definition of a weighing cell based on geometry, lumped masses and joint stiffness.

The comparison with a geometric nonlinear FEA model reveals the limitations
of the linear RB model and stresses the need for advanced models to refine
the design. The two models show good correspondence in terms of stiffness in
the relevant range and for a sufficient length of the coupling element
(

All the mentioned effects are covered by the FEA model and, as
Sect.

The underlying measurement data and software codes are not publicly available and can be requested from the authors if required.

Closed vector loop derived from Fig.

The simplified kinematic structure shown in Fig.

The mesh of the thin flexure hinges in the FEA model is crucial for the overall accuracy of the computation covering the entire mechanism. The convergence in terms of bending stress is determined for a single flexure hinge.

Convergence of the maximum bending stress in a single hinge (deflection angle

Figure

MD and MP wrote the initial draft of the paper. MD elaborated the mechanical models, whereas MP conducted the measurements. TF and RT conceptualized the research and raised the funds. TF, RT, RW and LZ were involved in reviewing and editing the paper.

The authors declare that they have no conflict of interest.

The authors would like to thank the German Research Foundation (DFG) for the financial support of the project with grant nos. TH 845/7-1 and FR 2779/6-1. Edited by: Bernhard Jakoby Reviewed by: three anonymous referees