In this section, case studies and discussions are presented. Firstly, we propose a finite element modeling solution for an elliptical notched hinge. Then, the validity and generalizability of the proposed parametric analytical model are verified using the finite element analysis (FEA) method. Finally, the necessity of the proposed improvements is demonstrated by analyzing the parametric design of multiple sets of thrust stands.
5.1. The Improved and Corrected Analytical Model
The FEA technique is a powerful tool for research in the field of structural mechanics. The well-modeled finite element model (FEM) has very high accuracy in the problem domain of this study. In this research, the FEM results are adopted as a benchmark to verify the validity of the proposed analytical model for thrust analysis. Reasonably, we are interested in the elastic deformation of the flexure hinge. Hence, the bending and tensile deformations of the pendulum arm are neglected by assuming the pendulum arm to be rigid. Moreover, the proposed analytical model points out that the mechanical properties of hinges not previously considered are the core link of the research. Therefore, the focus of finite element modeling is the flexure hinge.
COMSOL Multiphysics 6.0 as a very mature FEA simulation software will be applied. In the FEA, the accuracy of the solution is related to the mesh size. In principle, a finer custom meshing brings higher accuracy. However, it is difficult to be adopted in real applications due to the limitation of time and computing resources. Nevertheless, unstructured meshing algorithms represented by tetrahedral elements are prone to low-quality meshes with distorted tetrahedral shapes when meshing structures with large aspect ratios such as flexure hinges. Consequently, a solution for meshing is given to make the FEM results convincing.
In the finite element modeling, as shown in
Figure 9a, the left side of the hinge is the fixed end, and the right side is the free end. A shear force
is applied at the middle position of the free end to generate torque with respect to the assumed center of rotation. Based on Saint-Venant’s principle, the length of the non-hinged part is designed to be
to avoid the effect of fixed restraint and load on the hinge part with inhomogeneous stress. For the building of the hinge mesh, we divide it into three parts: the fine region A, the transition region B, and the coarse region C, which are described as follows,
- (1)
Region A is the thinnest region of the hinge, has a length about 1/3 of the entire one, and has the characteristics of a large aspect ratio. The stresses and strains in both bending and tensile deformation are large. It is a key concern in force analysis. Therefore, a controlled structured hexahedral mesh is used instead of an unstructured one for the meshing (see
Figure 9b). A three-level hexahedral mesh is established in the thickness direction, and 30 and 100 elements are divided in the axial and width directions, respectively, using “mapped” and “swept” techniques to accomplish the above operations. Accordingly, region A is equivalent to a large number of healthy micro-cantilevers.
- (2)
Region B contains the part of the hinge root with larger curvature, the hexahedral element is no longer applicable, and the physical field-controlled tetrahedral element is used to build the mesh. Furthermore, in order to avoid a poor-quality mesh in the narrow region of the hinge root, a virtual mesh technique is applied to its root to supplement a circular arc-shaped region (see
Figure 9a); this region is only used to distinguish the difference between the meshes, and does not have an actual physical partitioning function (i.e., the machining of the hinge shown in
Figure 9a is shaped in one piece).
- (3)
Region C is the part outside the hinge, which is not the focus of attention, so it is subjected to a coarser free-division tetrahedral mesh.
The above meshing scheme for the finite element modeling of the flexure hinge can provide a feasible FEA solution strategy not only for thrust stands but also for flexure hinge structures in other devices.
5.2. Results and Discussion
In this section, we present comparisons between the parametric equations and FEA in order to evaluate the accuracy and generalizability of the proposed parametric equations to describe the mechanical behavior of the hinge. Beryllium copper material is used in the flexure hinge due to its advantages of high mechanical strength and high temperature stability.
Table 2 gives the initial geometric parameters and mechanical parameters of the hinge.
The primary concern is the reliability of the equations describing the assumed rotational center offset of the hinge in the proposed model. It is noticeable that the length change in the hinge under axial tension can also be interpreted as an axial offset of the assumed rotational center (see
Figure 7), except that this offset does not lead to uncertainty in the displacement measurement. The linear stiffness
and
obtained from the analytical model are
and
(refer to Equation (14) and Equation (22)), respectively, which are compared with the FEM results to verify the stiffness equations under a small deformation.
Figure 10 shows a schematic of the access of FEM results. Five points are selected at equal intervals at the front (fixed end), middle, and rear (free end) of the hinge section, distinguished by black, green, and red in turn (see
Figure 10). The mean values of their respective displacements are calculated to represent the displacements at the front, middle, and rear of the hinge. Thus, the bending and tensile deformations caused by the black points are subtracted from the total deformation obtained from the red points or green points to obtain the pure deformation of the flexure hinge itself.
In real applications, the minimum thickness of the flexure hinge frequently becomes the point of penetration for customizing the mechanical properties of the hinge. Therefore, the variation range of the minimum thickness
is set to be 0.1~0.4 mm, and the hinge height
is kept constant. As formula
needs to be satisfied, the short axis
matches 1.3~1.45 mm. The other parameters are listed in
Table 2. The comparisons within this parameter range are given in
Figure 11.
Figure 11a,b present the comparative results of tension line stiffness and offset line stiffness under the stiffness equation and FEM, respectively. It can be obtained that the result curves calculated by the stiffness equations agree well with the result curves of FEM, and they also have the same trend of change. In addition, the hinge tension line stiffness is much greater than the offset line stiffness by about two orders of magnitude over a range of variations in parameter
. This is because the function of the hinge is to provide bending rather than axial elongation. The comparative results seen for an incomplete fit mainly come from the assumptions made in those theoretical derivations. It can be substantiated that the equations (Equations (14) and (22)) describing the hinge rotational center offset in the proposed analytical model are valid when the thickness
is within the allowed variation range.
The accuracy of the hinge bending stiffness equation is central to the persuasiveness of the proposed model. With reference to
(see Equation (8)), we define the bending line stiffness of the hinge as
. It is important to point out that the actual displacement produced by the force
in the FEM should be the displacement of the free end of the hinge minus the offset of the middle of the hinge due to the offset of the assumed rotational center. The comparison is plotted in
Figure 12. The results of the equation calculation and the FEM results also have small discrepancies and the same trends, which verifies the stiffness equation (Equation (8)). Comparing
Figure 11b, the offset line stiffness is only about 10 times higher than the bending line stiffness for the hinge itself. Even if the thrust is applied at the end of the pendulum arm rather than at the end of the hinge in practice, this multiplier is magnified again. However, the system still has significant measurement uncertainty due to the offset of the assumed rotational center. The subsequent analysis will give a detailed confirmation of this point.
In order to further investigate the influence of dimensional changes on the bending stiffness under a gravity-induced extension effect in the hinge proposed in the analytical model (refer to Equation (18)), an axial tensile force is first applied to the hinge in the FEM. Then, the changing dimensional parameters are recorded and modified before applying force
, which provides torque. It is significant to emphasize that the axial tensile force needs to be removed after the correction of the hinge dimensional parameters; otherwise, an additional stiffening effect will be introduced. Meanwhile, the offset of the assumed rotational center is also considered in the FEA. Based on the thrust stand we designed, the whole pendulum load needs to reach about 10 kg. Thus, we choose
. The results of the bending line stiffness variation obtained using both approaches are shown in
Figure 13. Such results show that the overall effect of the dimensional change is a decrease in the bending line stiffness. It can be observed that the stiffness equation successfully predicts the variation in the bending line stiffness under the axial tensile force, and it coincides very well with the FEM fitting (the green curve in
Figure 13). Among them, the best prediction is achieved when the thickness
lies in the range of 0.1 mm to 0.2 mm.
In addition, combined with Equation (10), it can be seen that the variation in stiffness is equivalent to in the formula, and the difference between the two is a conversion multiple of linear stiffness and torsional stiffness. In other words, the presented analytical model can accurately give the numerical solution of the bending stiffness shift in the flexure hinge under a certain pendulum load, indicating that it makes sense.
To demonstrate the generalizability of the improved analytical model for different notch boundary sizes and dimensions of the hinge, two additional sets of studies (see
Table 3) are carried out as follows.
Figure 14 shows the comparisons for Case I. It is clear from
Figure 14a–c those stiffness parametric equations give a precise description of the corresponding stiffness even under the condition of a wide range of variation in the elliptic long axis parameter a. Since there are several outlier points in the FEM result, the first fitting (the green curve) is not convincing, as shown in
Figure 14d. The fitting result of the post-processing FEM (the yellow curve) is highly consistent with that of the stiffness equation. The comparisons under Case II are plotted in
Figure 15. The results demonstrate the predictive performance of these parametric equations. Notably, the bending line stiffness variation under an axial extension effect is independent of the hinge width.
Furthermore,
Figure 16 demonstrates the comparisons under three typical flexure hinges. The three hinges maintain a consistent minimum thickness of
, and their key boundary parameters are provided by the table in the figure. The results again verify the improved analytical model. In the meantime, it can be summarized that the elliptical notched hinge has both the advantage of the circular notched hinge, which has a clear center of rotation, and the advantage of the leaf-type hinge, which has less bending line stiffness. This is because with a very large bending line stiffness, the displacement from a micro-thrust may be not perceptible.
In summary, these case studies on the flexure hinge reflect, on the one hand, the validity and generalizability of the proposed new quasi-static thrust analytical model. On the other hand, the variation laws of those stiffness properties of the hinge with the changes in its geometrical parameters are shown in those comparative analyses. Therefore, based on the parametric equation of the bending stiffness, the appropriate can be easily found for the thrust stand with the selection of its independent variables. In the next section, we demonstrate the necessity of improvements to the conventional model analytically.
5.3. Discussion for Thrust Measurement
In the previous sections, we verified the accuracy of the improved analytical model in characterizing the neglected parts in conventional thrust analytical models, including the shift in the bending stiffness and the offset of the assumed rotational center during the actual operation. In this section, we illustrate that the presence of these non-ideal factors can introduce large uncertainties into the theoretical analysis of thrust measurements, which is worthy of attention by designers of thrust stands.
On the basis of what is studied in
Figure 12 and
Figure 13 (the parameter
changes with
(
, and
remains constant), attention is paid to the impact of the change in the hinge bending stiffness on the thrust analysis. We rewrite Equation (24) as follows:
where
is the revised thrust calculated by the improved analytical model. And
Figure 12 and
Figure 13 show that the equivalent spring stiffness
introduced by the gravitational force component
is a negative number and its magnitude is small with respect to the original stiffness
(about 0.02% to 0.06% of the total). Thus, if the pendulum is designed so that the pendulum load gravity component
provides the main torsional return stiffness, then the effect of the shift in
will be very slight. On the other hand, a problem with this is that the heavy load requirements of the pendulum often make the equivalent stiffness
too large, making displacement measurements much more difficult. Therefore, the designer will adjust the counterweight to make
as small as possible, or even have a negative equivalent stiffness
to weaken the stiffness effect of
, so as to obtain a lower total stiffness coefficient
. This is exactly what our team is trying to undertake. In such a case, the effect of the shift in
on the thrust will be magnified, and needs to be seriously considered.
To quantify the above analyses, several case studies are carried out, with the parameters for the compound pendulum given in
Table 4. The six designs are divided into three groups: A, B, and C. The minimum thickness of the hinge is
in both groups A and B, while
is 0.3 mm in group C. The other geometric parameters of the hinge are provided in
Table 2. The difference between groups A and B lies in the different thrusters and counterweight masses; the total stiffness coefficient
is adjusted by changing the height of the counterweight bar within each group. The position of the thruster is fixed in different groups, and the length and mass of the pendulum arm vary with the height of the counterweight bar. The acceleration of gravity is chosen as
. The distance from the measurement point to the assumed rotational center is
, and the true value of the displacement at the measurement point is assumed to be
. The results of the stiffness and thrust obtained with these three groups of parameters are reported in
Table 5, where
is the absolute error of thrust and
is the relative error.
From these results, it can be observed that the original bending stiffness is determined by its own dimensions (here ). In addition, the shift () in is determined by both the pendulum load gravity and the hinge dimensions. A negative value of the equivalent stiffness may cancel most of the hinge’s own stiffness , making the effect of the stiffness shift () amplified, leading to large uncertainties in the thrust calculations, such as 1.57% in A1, 0.29% in B1, and 7.84% in C1. However, the high-precision metrology standard generally needs to reach an error level of 0.05% or even higher, and only the designs in A2, B2, and C2 can reach this error level.
Table 6 shows the effect of the offset of the assumed rotational center on the theoretical calculation of thrust under several sets of different parameters. The geometric parameters of the hinge, except for the minimum thickness
, are shown in
Table 2, and the structural parameters of the compound pendulum are shown in group A2 in
Table 4. Again,
and
are set. The thrust calculation error due to the bending stiffness shift is not considered at this point. From the relative error results (see
Table 6), it can be obtained that the offset of the assumed rotational center introduces non-negligible uncertainty into the thrust calculation. And this relative error depends mainly on the geometric parameters of the hinge, independent of the measured displacement.
To sum up, it is necessary to always pay attention to the influence of when the designer of the compound pendulum tries to customize the stiffness coefficient by changing the weight or height of the counterweight. The shift () of the hinge bending stiffness can be regarded as the systematic error of the thrust measurement device together with the offset of the assumed rotational center. The systematic error of the equipment needs to be taken seriously in the design or measurement process. Ultimately, an improved analytical model that takes these factors into account is given in Equation (25).