Investigations on anelastic effects in electrical discharge machined titan grade 2 flexures for the use in precision force instruments

1 Introduction

Monolithic mechanisms composed of flexure hinges have been a crucial part of force measurement systems for a long time in both, strain gauge systems and systems based on electromagnetic force compensation (EMFC) [1]. During the process of the redefinition of the kilogram, Kibble balances were developed that extend the EMFC principle with a dynamic calibration mode. It has been shown that integrating the guiding function for the calibration mode into the balance mechanism of the weighing mode is a good strategy to improve the achievable measurement uncertainty of these instruments [2]. Since the calibration mode requires a large deflection, which is unnecessary in common EMFC systems, new balance mechanisms need to be developed to improve existing Kibble balances or to engineer new compact table top systems [3], [4].

One crucial part of the development of these new mechanisms are investigations on the anelastic effects in the flexure hinges [5], [6], which are typically made of copper-beryllium alloys or aluminum in existing Kibble balances. A promising group of materials, which were proposed recently, are titanium alloys due to their low magnetic permeability and excellent tensile strength to density ratio [7]. The anelastic effects themselves have been the subject of research for many years. Researchers have investigated many materials with regard to their anelastic behavior [8], [9]. However, due to the low deformation at room temperature caused by these, measuring them is highly complex. Therefore, a significant research subject is the investigation of anelastic effects, often called creep, at temperatures around 50 % of the melting temperature [10]. In this article, we measure the anelastic effects at room temperature in a temperature controlled climate chamber. There are two main methods for this, dynamic and static measurement. In the dynamic measurement, a beam is set in vibration, similar to the velocity mode in the Kibble balances. The internal damping of the material causes a phase shift. This phase shift can be evaluated and changes over the time of the vibration. The reason for this is the anelastic effect. In static measurement, on the other hand, a load is induced in the material by applying forces. This leads to an initial deformation, which has relatively big magnitude whose experimental determination is not quite challenging. Over time, the body continues to deform under load. This time-dependent deformation is minimal and, therefore, complex to measure, especially compared to the initial deformation. If one evaluates the data after measuring the displacements, the deformation increases over time, and the anelastic effect is the reason [8].

In Table 1 are different studies listed which examined anelastic effects in different materials with different measuring methods and specimens. Currently, there is no research on the anelastic effects of grade 2 titanium. Besides, most of the research is carried out with dynamic analysis with inverted pendulums or static with the single specimen in either three-point bending tests or with nanowires as bending beams in a clamping.

As previously described, anelastic behavior is very important in weighing technologies. Therefore, it is important to find materials with only slightly anelastic behavior. We aim to measure the anelastic effects in grade 2 titanium using a load cell design in a static measurement setup to obtain direct observations of the time-dependent deformation of the material under sustained loading.

2 Measurement set-up

2.1 Description of the specimen

The specimens are designed similarly to binocular-shaped load cells, commonly used as force transducers for low measurement ranges. In contrast to the classical design, the load cells utilized in this work feature a corner-filleted flexure hinge design. This new design was chosen in order to achieve bending stress in the individual thin sections mainly. In addition, the overall stiffness is reduced by the length of the thin sections l _h , which simplifies the measurement of the deformation. A schematic representation of the geometry used is shown in Figure 1, the associated parameters are listed in Table 2. The total length, height and width are specified by the testing facility used.

Figure 1:

Schematic representation of the load cells used. The associated dimensions are described in Table 2.

Table 2:

Dimensions of the load cell specimens under consideration. The hinge-height h _h is described by h _h ∈ {0.6, 1, 2} mm.

Parameter	L	l h	l m	H	h h	h m	r e	w
Dimension in mm	107	10	16	30	h h	9	2	12.7

This study’s three load cell specimens were manufactured from grade 2 titanium via electric discharge manufacturing (EDM). The load cells were manufactured in the same batch and only differ in the hinge height h _h with values of 0.6 mm, 1 mm and 2 mm. Previous studies showed that titanium has different stiffnesses depending on the orientation with respect to the rolling direction [19]. In this batch, the load cells were manufactured in the same orientation and from the same plate. After manufacturing, the load cells were measured with a digital microscope (KEYENCE VHX-7000) to verify their geometry. The results showed that all hinges were between 10 µm and 30 µm smaller than the specified nominal dimensions, which is well within the ordered tolerance for the thin sections (h _h ± 0.05 mm). An example of that measurement is depicted in Figure 2. The image also shows that the surfaces perpendicular to the wire-cut ones are milled.

Figure 2:

Detailed view on a hinge with a nominal thickness of 1 mm with a measurement probe created by the Keyence VHX analysis software. The measured thickness is 972 µm.

2.2 Description of the testing facility

The anelastic effects were investigated with an interferometric material testing facility that was developed at Technische Universität Ilmenau and utilizes a static method to determine the magnitude of anelastic effects [17]. As schematically depicted in Figure 3, it consists of an interferometer with a symmetric measurement and reference beam path, which are both reflected by mirrors attached to the upper face of the load cell specimen (see Figure 4). This results in a nominal optical path difference of zero and a good rejection of influences on the refractive index of air. The influence of ambient temperature and humidity is further reduced by a temperature controlled climate chamber and placement of desiccant inside the chamber. The coolant flows through channels inside all sides of the chamber to produce a homogeneous temperature field inside the chamber and reduce air convection. A temperature sensor is placed next to a load cell to allow a correction of the temperature dependence of Young’s modulus if necessary. The typical observed peak-to-peak temperature drifts and fluctuations over time period of two weeks were less than 50 mK.

Figure 3:

Schematic representation of the used measurement setup; the complete climate chamber with all its components is located in an air-conditioned and vibration-damped room.

Figure 4:

Load cell specimen built into the testing facility with mirrors attached.

The loading system can apply different weights in the range of 50 g–1,000 g in order to apply different forces to the test load cell via a cantilever arm. The cantilever arm is designed such, that the weights are placed underneath the point where the load cell is fixed to the measurement frame in order to reduce the torque and the resulting deformation at this point. Otherwise, the load cell would tilt around this fixing point, resulting in an additional path difference that is not produced by the deformation of the load cell and can not be separated from the deformation of the load cell.

The meteorological frame of the test facility as well as the mounting plate of the load cell are manufactured from Invar, while the load cantilever arm and the clamping elements of the mirror, which are not part of the measurement chain of the deformation measurement, are made from aluminum. To ensure a comparable stress in the load cells due to the mounting, the load cells are fixed to the mounting plate with the same stainless steel M3-sized screws that are tightened with a torque of 1.5 Nm, which is ensured by the use of a measuring torque wrench.

The measurement system and the mass loading system are fully automatized, so there is no need for manual interaction of the operator with the system once the load cell is mounted and the chamber is closed. It has to be noted, that a stable, homogeneous air temperature is only present after a settling time of approximately 6 h. Even more time is necessary for initial mechanical drift effects to be reduced to a reasonable amount. These effects are caused by the anelastic effects due to the weight of the load cell, the cantilever arm and the clamping elements as well as mechanical stress that is introduced during the mounting and the adjustment of the system.

Since not all of these processes can be done exactly in the same way when a load cell is replaced, it was decided to wait 24 h until the actual measurements are taken. The measurements were taken in at least two sets of data with different loading masses depending on the thickness of the flexure hinges of each load cell. Each set of measurement data consists of six loading and unloading cycles, where the loading period was 4 h and the unloading period was 8 h. Another 6 h additional waiting time was inserted between two measurement cycles with different load masses.

3 FEM simulation

To be able to estimate the stresses occurring in the load cell, we build an FEM model that simulates the mechanical conditions in the measuring setup in the best possible way. The software ANSYS^® Workbench™ 2023 R2 is used for FEM analyzes. We perform structural mechanic simulations in ANSYS Mechanical. The Young’s modulus of the load cell is set to 105 GPa, the density of the titan alloy is set to 4.51 g cm⁻³. The load cell is clamped on the left-hand side as described in Section 2. Since we are not interested in the stresses that arise due to the screw connection on both sides of the load cell, we are using the surfaces for the boundaries. Like in the measurement setup, the support surface is a fixed clamping; see Figure 5, depicted by the green panel. On the right side, we use the contact surface of the cantilever arm to model the contact (red panel). The cantilever arm is modeled as a rigid body, as the deformation is not engaging here either. Material parameters define the mass of the cantilever arm, which is 74.3 g. In the measurement setup, a cylinder hangs from the cantilever arm and grips the masses (see Figure 3). Its weight of 23.2 g is concentrated into a point mass at the connection point of the cantilever arm.

Figure 5:

Schematic representation of the conducted FEM-model. Presentation of the surfaces used for clamping and contact (green and red), placement of deformation-probe, point-mass for the mass of the cylinder and force application point (blue). Additional detailed view on the mesh refinement in the thin sections.

The earth’s gravitational field creates the actual basic load of the system and is set to 9.806 ms⁻². Individual external forces in the direction of the gravitational field realize the individual masses attached to the cylinder in the measurement setup. The respective forces are calculated based on the masses and the value of the gravitational field. The results are obtained using several load steps, each varying the external force. The first load step only calculates the basic load without an external force. This includes the dead weight of the load cell and the cantilever arm, as well as the point mass for the cylinder. Further load steps apply the forces resulting from the attached masses.

The load cell itself is meshed with SOLID186 elements. This type of element represents 3D elements and has freedom in all spatial directions in order to obtain the most precise information possible about the stresses that occur. The mesh is refined in the areas of the thin sections to obtain convergence of the stresses as presented in Figure 5 in detail. We use the same model regarding the boundaries, contacts, and meshing. Only the thickness of the thin sections varies for the three geometries concerned. As a result, we get the deformation w in the y-direction at a desired point, which is also the measurement point in the measurement setup and the overall Von Mises stress. Table 3 presents the resulting stresses and deflections for the different load cells and loads.

4 Results

Preliminary measurements had shown, that even several days of waiting time after the mounting of the load cell were not sufficient to reduce the drift to the order of measurement noise. Since this drift is caused by the dead load of ∼130 g, which is composed of the weight of the load cell, the loading system and the mirror clamping, the load cell with thinner hinges are showing a much greater magnitude of drift. To allow a comparison of the different load cells without employing unreasonably long waiting time, a drift correction was introduced to the data analysis. The correction consists of the determination of the linear drift by calculating the slope of the last 10 s time spans of each of the six unloading steps of a measurement data set over time via linear regression. This assumed linear base line drift was subtracted from the raw measurement data and the detection of steps and the calculation of the relative elastic aftereffects (or “creep”) that is described by Kühnel et al. [17] was performed. As a first approximation it is usually expected that the relative anelastic effects, which are calculated from the absolute anelastic effects Δs_ae by dividing them by the instantaneous deformation Δs_inst, are independent from the load. However, in greater ranges of applied forces, the observed effects can differ from a linear dependence of the absolute effects on instantaneous deformation [20].

Figure 6 shows the relative anelastic effects of the load cell specimen with a hinge thickness of 1 mm and load weight of 200 g. The loading curves are plotted with a red color tone, while the unloading is plotted in blue. The absolute anelastic effects Δs_ae have the same direction as the preceding instantaneous deformation Δs_inst. that is used to calculate the relative magnitude. Therefore, the sign of the magnitude cancels out and both types of curves are positive. The darkness of the color tone corresponds to the ordinal number of the loading or unloading process within the measurement data set. The darkest curves depict the last steps. The curves of loading and unloading approach each other during the progression of the measurement cycle, but a remaining difference can be observed. This effect was consistent for all tested load cells and the process restarted after the additional waiting time that is mentioned in Section 2. The magnitude of the effects after a time span of 4 h is shown in Table 4 and it can be observed that the values do not progress strictly, since there are remaining effects from temperature variations. However, the measured magnitudes become more consistent over the progression of the loading cycles. The following comparison of load cell specimen will focus on the sixth loading cycle of each set of measurement data, but the conclusions could comparably be drawn from a comparison of the other curves.

Figure 6:

Relative elastic aftereffects of six loading and unloading steps. Loading curves are plotted in red, unloading curves in blue. The darkness of the color tone represents the number of steps.

Figure 7 shows the loading curves of all three load cell specimen with two different loads respectively. The loads were chosen such, that the difference of the overall Von Mises stress between loaded and unloaded state in the same order of magnitude. The differences, which are given in the legend of the plot, are calculated from the simulated stresses that are highlighted in Table 3 in Section 3. An exact matching of the stress was not viable due to the manufacturing tolerances and the availability of test masses. Each specimen is loaded with two masses, called “low load” and “high load”.

Figure 7:

Relative anelastic effects of Ti grade 2 load cells with different hinge thickness after applying load.

The magnitudes of the relative elastic anelastic effects after a time span of 4 h are given in Table 5 and show a significant dependence not only from the hinge thickness, but also from the load mass. The sensitivity towards the mass increases with the magnitude of the effect, which, in turn, increases with smaller hinge thickness. The time constants seem to be largely independent from hinge thickness and load, besides a greater tendency for long term drift at high stress.

A comparison with the curves in Figure 8 that are associated with load cell specimen made from the aluminum alloy EN AW 2017, which is a common material with good machinability, shows a greater sensitivity towards both parameters of the Ti grade 2 alloy. The aluminum load cells feature the same geometry as the titanium load cells and are also machined with a EDM process in a single batch. While the dependence on the hinge thickness is only reduced, the effect of the load is effectively absent. Furthermore, the magnitudes of the effects are approximately only one third compared to titanium. Interestingly, the time constants are quite similar.

Figure 8:

Relative anelastic effects of aluminum load cells with different hinge thickness after applying load.

The thickness effect of the aluminum alloy is most likely caused by the aluminum oxide layer that naturally builds up on any aluminum workpiece that is in contact with air. Since the thickness of the oxide layer depends only on the material and its surface treatment, the ratio of oxide layer thickness to bulk material thickness increases with decreasing hinge thickness. Titanium usually does not exhibit a natural oxide layer, but a study on the effects of the EDM process on the characteristics of titanium alloys shows the fabrication with different concentration of the additives of the alloy in the process [21]. Even though a different alloy was investigated in that study, it can be suspected that similar effects occur also in Ti grade 2. Further investigations need to be conducted to identify the cause of the observed thickness dependence.

5 Conclusion and outlook

The testing facility for properties of materials for compliant mechanisms [17] has been adapted for investigations on EDM processed aluminum and Ti grade 2 load cells. The measurement procedure and data evaluation have been improved for materials that show a high magnitude of drift due to being loaded with their own weight and the weight of the testing assembly. The Ti grade 2 specimen showed a significant dependence on the magnitude of elastic aftereffects on the hinge thickness and load. These effects surpass the magnitude of similar effects in a common aluminum alloy that has been produced with the same process.

Comparing our results for the anelastic behavior of titanium to those of aluminum, we observe that titanium grade 2 exhibits relative anelastic effects approximately four times greater than those of the tested aluminum alloy. This indicates that titanium grade 2 is less suitable for applications in weighing technologies.

Future investigations will improve and refine the understanding of the observed effects by utilizing new load cell specimens produced using different processes, such as milling and different titanium alloys. Due to their prevalent usage in force metrology and positioning applications, it is also promising to extend the investigations also to different aluminum alloys. Additionally, a new testing facility will be set up to increase the flexibility regarding the specimen geometry that can be used for the investigations. Furthermore, the investigations will be extended to mechanical hysteresis effects as described by ISO 376 [22].