Optimal Design for Drive Accuracy of the Space-Focus Control Mechanism Using a Simplified System Design Method

Abstract

A focus-control mechanism is essential for maintaining the optical performance of spaceborne telescopes, the mirror alignment of which is degraded by gravity release, moisture desorption, and thermal distortion in orbit. Achieving submicrometer-level drive accuracy is challenging because bearing deformation and bolted-joint hysteresis introduce nonlinear behavior, which must be addressed in ultraprecision mechanisms. In this study, the 1D Computer-Aided Engineering (1DCAE) approach was applied to the early-phase design of a spaceborne focus-control mechanism for developing practical design equations that accurately represent the stiffness and deformation characteristics of key components. Modification functions derived from finite element analysis (FEA) and the indirect fictitious boundary integral method (IFBIM) were incorporated into the equations for a linear guide, rectangular spring, and bearing deformation. These equations showed excellent agreement with analytical solutions, numerical simulations, and experimental data, achieving accuracies within 3% and 2.5% for the linear guide and rectangular spring, respectively, and close correspondence with the IFBIM-based bearing deformation reference values. Integrating the equations into the 1DCAE model enabled accurate prediction of the nonlinear drive characteristics of the mechanism and improved the overall drive accuracy to one-fortieth that of the initial design. In conclusion, 1DCAE provides an effective and computationally efficient framework for optimizing ultraprecision mechanisms used in space applications.

1. Introduction

The design process during spacecraft development typically involves conceptual, trial, preliminary, critical, and sustainment design stages. Although detailed analyses using three-dimensional (3D) models are conducted during the preliminary and critical design phases, the issues identified at these stages result in substantial setbacks. In the absence of explicit problems, the resulting design is often suboptimal. Moreover, it is widely recognized that the majority of a product’s life-cycle cost is determined by the time the design phase is completed, as any modifications made after prototyping, procurement, manufacturing, or assembly result in substantial redesign efforts, significantly impacting both cost and schedule. It is thus essential to identify technical issues and cost-related risks during the early design phase and incorporate the necessary considerations into the design. Such efforts reduce the likelihood of downstream revisions, thereby lowering overall development costs through reduced workload and shortened delivery schedules. For this reason, understanding the overall system behavior at an earlier phase becomes crucial.

To improve design quality, optimization techniques such as the Taguchi method have been applied [1,2], and the globally convergent method of moving asymptotes (GCMMA) has also been employed for parameter optimization [3]. Although these methods are effective for optimizing specific parameters, they are insufficient for comprehensively understanding the characteristics of the entire system. As an analytical method, a bilinear neural network capable of obtaining exact solutions to nonlinear equations has also been reported [4]. However, this method cannot be applied to partial differential equations that do not admit exact solutions, making it difficult to utilize for system design. Furthermore, from the perspective of designing high-precision instruments, an optimal design method for 6-degree-of-freedom (6-DOF) vibration isolation systems has been proposed [5]. Although this method formulates a rigorous nonlinear 6-DOF model based on Hamilton’s principle, the model relies on numerous assumptions, which may limit its ability to accurately reproduce the behavior of actual hardware. Consequently, it is not suitable for practical design applications.

The structures of precision space equipment are commonly analyzed and evaluated using finite element analysis (FEA) [6,7]. Although FEA is an indispensable design tool, it is constrained by several inherent limitations:

• FEA models are generally created after detailed drawings are completed, making it difficult to perform analyses in the early design stages when the shapes and dimensions remain undefined.
• The solution obtained from FEA represents only a single design state, and the underlying design implications are often unclear because of the “black-box” nature of the analysis.
• Any change in geometry or dimensions requires remodeling, which makes iterative analyses of parameter exploration time consuming.
• FEA alone cannot adequately address certain problems, such as preload behavior, purely contact-based interactions without bonding, and friction-dominated phenomena.

Thus, although FEA is highly effective, its capabilities are not fully utilized during the initial design phase, which is arguably the most critical stage of development. Analytical approaches, such as the matrix method and pseudo-rigid-body model (PRBM), have also been used [8,9]; however, these methods primarily address stiffness characteristics and are insufficient for evaluating the behavior of an entire system. In comparison, as an analytical approach for controlling the attitude constraints of satellites, a time-synchronized control method that incorporates the inertia matrix into the sliding-surface design has been reported [10]. However, introducing the inertia matrix inevitably reduces robustness, and the method is inherently meaningless for devices that operate in a quasi-static manner. In addition, as a technique for emergency satellite tracking, a tracking algorithm that integrates Automated Machine Learning (AutoML) with the Emergency Task Three-Phase Scheduling Framework (ETTSF) has been proposed [11]. Nevertheless, this approach exhibits a high degree of black-box behavior, making it impossible to explain why the obtained solution is optimal, and therefore, it cannot provide guarantees for parameter optimization.

To address this issue, a simplified system design method that enables overall optimization by representing the entire system or product in a simple form has been proposed [12,13]. This approach, originally developed by Toshiba and referred to in Japan as 1D Computer-Aided Engineering (1DCAE), was intended to clarify the influence of individual design parameters on system-level behavior using simple, intuitive models rather than complex design equations. Here, “1D” does not denote one dimension in the literal sense; instead, it refers to capturing the essential functions of a design in a concise and easily interpretable format. “CAE,” originally used in computer-aided engineering, encompasses not only computer-based simulations but also analytical calculations performed using tools such as spreadsheet software [12,13]. By simplifying the design equations, 1DCAE facilitates the understanding of how the design parameters affect the system characteristics.

Despite these advantages, 1DCAE has not been previously applied to ultraprecision space mechanisms. The optical performance of telescopes onboard Earth-observation satellites is highly sensitive to their mirror alignment. However, gravity release, moisture desorption, and thermal distortion in orbit alter the distances between the mirrors, leading to focal misalignment. Consequently, an on-orbit focus control mechanism is required, which must achieve extremely high drive accuracy on the order of submicrometers.

The novelty of this study lies in applying highly accurate practical design equations formulated using fractional expressions to simulate the drive accuracy of an ultraprecise space-focus control mechanism, an application not previously addressed within the 1DCAE framework, and demonstrating that drive accuracy can be significantly improved using the 1DCAE approach. In this paper, “parameter design” refers to design based on system parameters and does not denote the quality-engineering concept involving signal-to-noise (SN) ratios. Furthermore, the term “practical design equations” refers to design equations that are directly applicable to real-world engineering design.

2. Methods

2.1. Overview of the Space Focus Control Mechanism

A key function of optical satellites is the use of onboard optical telescopes to capture images of the Earth’s surface using sunlight as the illumination source, similar to the operation of digital cameras. Accurate focusing is essential to obtain high-quality images. However, although Figure 1 illustrates the structure around the primary and secondary mirrors, the support structure undergoes deformation once the telescope is placed in orbit due to the release of gravity. In optical telescopes, carbon-fiber-reinforced plastic (CFRP) with low thermal expansion is commonly used for the support structure to reduce mass and minimize thermal deformation. While CFRP is a high-performance material, it absorbs moisture on the ground and expands slightly, whereas in orbit, it releases this moisture and contracts. In addition, thermal deformation of the support structure caused by the orbital thermal environment further alters the alignment of the optical mirrors, with these effects leading to focal shifts that result in image blurring.

The image quality of an optical telescope is governed by its resolution and clarity, with clarity strongly depending on mirror alignment. Because alignment errors are dominated by the deformation between the primary and secondary mirrors, adjusting the position of the secondary mirror is the most effective means of compensating for these deformations, with adjustment commonly performed using a focus control mechanism. In this study, we present the design results for the drive accuracy of the focus control mechanism, which critically affects the image clarity of the optical telescope, using 1DCAE-based design methodology.

A cross-sectional view of the focus control mechanism is shown in Figure 2 [14]. The mechanism consists of an actuator, a screw mechanism, a rectangular spring, linear guides, bearings, and housing. The rotational motion of the actuator was converted into a linear displacement through the screw mechanism. To achieve an ultrahigh drive accuracy, the bearing deformation and hysteresis at bolted joints should be accounted for. The linear guide, which incorporates an elastic hinge, not only provides play-free linear motion but also forms a structural reduction mechanism when combined with a rectangular spring.

2.2. Simplified System Design Method

Figure 3 illustrates the 1DCAE model of the focus control mechanism. A key feature of the 1DCAE approach is that it represents the system in the simplest possible form while capturing the essential characteristics of the overall behavior. In the present 1DCAE model for evaluating the drive accuracy of the focus control mechanism, the housing is assumed to be sufficiently rigid compared to the other components. Consequently, the rectangular spring and linear guide can be modeled as linear springs, whereas the bearing and bolted joint hysteresis are treated as nonlinear springs. The focus control mechanism compensates for focal shifts caused by launch-induced variations and long-term structural changes in the optical telescope support system, and it is only operated a few times per year. Consequently, the actuation of the focus control mechanism can be treated as quasi-static motion.

The input displacement generated by the screw mechanism is determined via Equation (1), which relates the motor rotation angle, the reduction ratio of the strain-wave gearing, and the screw lead. The deformations of the bearings and hysteresis at the bolted joints were subtracted from this linear drive displacement. The resulting displacement is then multiplied by the reduction effect of the spring reducer to yield the final output displacement of the I/F plate. Here, the linear guide (the stiffer spring) and the rectangular spring (the softer spring) form a parallel-spring configuration, and because the displacement of the rectangular spring is reduced according to the stiffness ratio relative to the linear guide, this configuration is referred to as a “spring reducer”.

$${\delta }_A\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{n_{G}L}{360}\theta ,$$

(1)

where δ_A is the drive amount, n_G is the reduction ratio, L is the screw lead, and θ is the motor drive angle.

When the relationship between bolt axial force and microslip in a bolted joint is considered at the macroscopic level, Koizumi et al. reported that the amount of microslip is proportional to a power of the friction coefficient μ, as expressed in Equation (2), with the exponent n = 2–3.5 [15]. When the I/F plate is pushed upward by the rectangular spring, the plate undergoes a slight convex deformation, which in turn induces a microslip δ_S at the bolted interface. Because the vertical micro-displacement δ_V associated with this microslip δ_S produces hysteresis during the actuation of the I/F plate, the hysteresis can be expressed as shown in Equation (3). In this study, we adopt n = 2. It should be noted that the purpose of 1DCAE in the initial design phase is to evaluate the influence of design parameters on system characteristics. Therefore, to maintain simplicity and capture the overall system behavior, complex phenomena such as the Dahl effect (elastic frictional lag) and pre-sliding displacement (elastic deformation of surface asperities) are not considered in the hysteresis model. Accordingly, the displacement output of the I/F plate is given by Equation (4).

$${\delta }_S\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_0\hspace{0.17em}\hspace{0.17em}{\mu }^n,$$

(2)

$${\delta }_H\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_H\hspace{0.17em}\hspace{0.17em}{P}^2,$$

(3)

$${\delta }_L\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{1}{1\hspace{0.17em}+\hspace{0.17em}{k}_L/k_R\hspace{0.17em}}}\hspace{0.17em}\left\{\hspace{0.17em}{\delta }_A\hspace{0.17em}-\hspace{0.17em}{\delta }_B\hspace{0.17em}\mp \hspace{0.17em}{\delta }_H\right\},$$

(4)

where δ_S is the microslip displacement at the joint, δ_H is the hysteresis in the vertical direction of the joint, C₀ and C_H are the displacement coefficients of the bolted joint, P is the vertical load on the bolted joints, k_L is the stiffness of the linear guide, and k_R is the stiffness of the rectangular spring.

A block diagram illustrating the drive–displacement flow within the mechanism is presented in Figure 4. In this diagram, P_R denotes the rectangular spring load, and G_R, G_B, G_H, and G_S represent the transfer functions of the rectangular spring, the bearing, the hysteresis element, and the spring reducer, respectively. The actuator displacement is transmitted to the I/F plate through the rectangular spring, bearings, and the linear guide, whereas hysteresis at the bolted joints introduces nonlinear effects. Consequently, accurately characterizing the stiffness of these components is essential for determining the overall drive accuracy of the focus control mechanism.

As shown in Equation (4) and Figure 4, the output displacement of the focus control mechanism can be obtained through a remarkably simple calculation. However, in order to evaluate the drive accuracy required for the ultra-precision spaceborne focus control mechanism, the design equations used for each component must be highly reliable. In response, practical design equations capable of accurately determining the stiffness characteristics of the linear guide, rectangular spring, and bearing are presented in Section 2.3, Section 2.4 and Section 2.5.

2.3. Design Equation for Deflection of the Linear Guide

A conceptual diagram of the linear guide is shown in Figure 5a. Points A₁B₁C₁D₁ and A₂B₂C₂D₂ move in parallel. The linear guide consists of three or four sets of parallel leaf springs arranged circumferentially and functions as a straight guide by combining two parallel elastic hinges. The value of the guide is equivalent to that of a single beam with zero angular deflection at its tip, as illustrated in Figure 5b. In this model, P_L is the applied load, δ_L is the displacement, M₀ is the moment acting on the tip, s₁ and s₂ are the notch distances, s₃ is the beam length, r_n is the notch radius, and 2t is the bottom thickness of the notch. Because the stiffness of the non-notched region was significantly higher than that of the notched region, the overall stiffness was dominated by the stiffness of the hinge portion. Therefore, the hinge stiffness was evaluated first, as shown in Figure 6. The stiffness of the hinge is summarized below; further information is provided in a previous report [16].

The angular deflection θ_B of an elastic hinge subjected to a moment M₀, as illustrated in Figure 6a, is given by Equation (5), which is equivalent to the Paros Equation [17], derived through partial integration:

$${\theta }_B\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{3\hspace{0.17em}{M}_0}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}{\displaystyle {\int }_{-r_n}^{r_n}\frac{1\hspace{0.17em}\hspace{0.17em}}{\hspace{0.17em}\hspace{0.17em}{\left\{\hspace{0.17em}{a}_n\hspace{0.17em}-\hspace{0.17em}\sqrt{r_n{}^2\hspace{0.17em}-\hspace{0.17em}{u}^2\hspace{0.17em}}\hspace{0.17em}\right\}}^3\hspace{0.17em}}\hspace{0.17em}d\hspace{0.17em}u}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{3\hspace{0.17em}{M}_0}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E}}\hspace{0.17em}{\displaystyle \frac{1}{\hspace{0.17em}2\hspace{0.17em}{c}_n^5}}\hspace{0.17em}\left\{6\hspace{0.17em}{a}_n\hspace{0.17em}{r}_n{}^2\hspace{0.17em}{\tan }^{-1}\hspace{0.17em}{\displaystyle \frac{r_n}{c_n}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{2\hspace{0.17em}{c}_n\hspace{0.17em}{r}_n\hspace{0.17em}}{a_n}}\hspace{0.17em}(2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}+\hspace{0.17em}{r}_n{}^2)\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}3\hspace{0.17em}\pi \hspace{0.17em}{a}_n{}^2\hspace{0.17em}{r}_n{}^2\hspace{0.17em}}{a_n}}\right\},$$

(5)

where E is Young’s modulus, w_L is the hinge width, x is the distance from the origin of the coordinate system, x − r_n = u, r_n + t = a_n, and a_n² − r_n² = c_n².

By expanding Equation (5) and expressing the small terms as a correction function using fractional expressions, we obtained a practical design equation for the deflection angle, as shown in Equation (6):

$${\theta }_{Bmd}\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}{f}_B\left({\displaystyle \frac{t}{r_n}}\right)\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}9\hspace{0.17em}\pi \hspace{0.17em}{M}_0\hspace{0.17em}}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}1\hspace{0.17em}}{\hspace{0.17em}{r}_n{}^2\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \frac{r_n}{\hspace{0.17em}2\hspace{0.17em}t\hspace{0.17em}}}\right)}^{\hspace{0.17em}{\displaystyle \frac{5}{2}}},$$

(6)

where f_B(t/r_n) is the modification function.

Similarly, the practical design equation for lateral deflection, as illustrated in Figure 6b, is obtained from the lateral load deflection in Equation (7), leading to the modified expression in Equation (8).

$$\begin{array}{l}{\delta }_T\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{3\hspace{0.17em}{P}_L}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}{\displaystyle {\int }_{-r_n}^{r_n}\frac{(u\hspace{0.17em}+\hspace{0.17em}{r}_n)}{\hspace{0.17em}{\left\{\hspace{0.17em}{a}_n\hspace{0.17em}-\hspace{0.17em}\sqrt{r_n{}^2\hspace{0.17em}-\hspace{0.17em}{u}^2\hspace{0.17em}}\hspace{0.17em}\right\}}^3\hspace{0.17em}}\hspace{0.17em}d\hspace{0.17em}u\hspace{0.17em}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\left({\displaystyle \frac{P_L}{\hspace{0.17em}{M}_0\hspace{0.17em}}}\hspace{0.17em}{\theta }_B\right)\hspace{0.17em}\hspace{0.17em}{r}_n{}^2\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{3\hspace{0.17em}{P}_L}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}\left\{{\displaystyle \frac{a_n}{\hspace{0.17em}{c}_n^3}}\hspace{0.17em}\left(-\hspace{0.17em}2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}+\hspace{0.17em}3\hspace{0.17em}{r}_n^2\right)\hspace{0.17em}\hspace{0.17em}{\tan }^{-1}\hspace{0.17em}{\displaystyle \frac{r_n}{c_n}}\right.\left.\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{a_n{}^3\hspace{0.17em}{r}_n\hspace{0.17em}}{c_n^2}}\hspace{0.17em}{\displaystyle \frac{2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}-\hspace{0.17em}{r}_n^2}{\hspace{0.17em}{(c_n{}^2\hspace{0.17em}+\hspace{0.17em}{r}_n{}^2)}^2\hspace{0.17em}}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{\pi \hspace{0.17em}{a}_n{}^4\hspace{0.17em}}{2\hspace{0.17em}{c}_n^2}}\hspace{0.17em}{\displaystyle \frac{-\hspace{0.17em}2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}+3\hspace{0.17em}{r}_n^2}{\hspace{0.17em}{(c_n{}^2\hspace{0.17em}+\hspace{0.17em}{r}_n{}^2)}^{3/2}\hspace{0.17em}}}\right\}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\end{array},$$

(7)

$${\delta }_{Tmd}\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}{f}_T\left({\displaystyle \frac{t}{r_n}}\right)\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}9\hspace{0.17em}\pi \hspace{0.17em}{P}_L\hspace{0.17em}}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \frac{r_n}{\hspace{0.17em}2\hspace{0.17em}t\hspace{0.17em}}}\right)}^{\hspace{0.17em}{\displaystyle \frac{5}{2}}},$$

(8)

where f_T(t/r_n) is the modification function.

Here, we use the results of the aforementioned studies to derive a practical design equation for the linear-guide stiffness. The approximate deflection of the linear-guide beam illustrated in Figure 5b is given by Equation (9). By neglecting the infinitesimal terms, Equation (10), which is a function of (r_n/2t)^5/2, is obtained. The derivation from Equation (9) to Equation (10) is provided in Appendix A. By comparing this approximation with the finite element analysis (FEA) results and applying a fractional correction, the practical design equations in Equations (11) and (12) are obtained. The specific form of the modification function is described in Section 3.1.

$${\delta }_L\hspace{0.17em}\hspace{0.17em}\approx \hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{3}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}{\displaystyle {\int }_{-r_n}^{r_n}\frac{P_L\hspace{0.17em}{(u\hspace{0.17em}+\hspace{0.17em}{s}_1)}^2\hspace{0.17em}-\hspace{0.17em}{M}_0\hspace{0.17em}(u\hspace{0.17em}+\hspace{0.17em}{s}_1)\hspace{0.17em}\hspace{0.17em}}{\hspace{0.17em}\hspace{0.17em}{\left\{\hspace{0.17em}{a}_n\hspace{0.17em}-\hspace{0.17em}\sqrt{r_n{}^2\hspace{0.17em}-\hspace{0.17em}{u}^2\hspace{0.17em}}\hspace{0.17em}\right\}}^{\hspace{0.17em}3}\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}d\hspace{0.17em}u}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{3}{\hspace{0.17em}2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}{\displaystyle {\int }_{-r_n}^{r_n}\frac{P_L\hspace{0.17em}{(u\hspace{0.17em}+\hspace{0.17em}{s}_2)}^2\hspace{0.17em}-\hspace{0.17em}{M}_0\hspace{0.17em}(u\hspace{0.17em}+\hspace{0.17em}{s}_2)\hspace{0.17em}\hspace{0.17em}}{\hspace{0.17em}\hspace{0.17em}{\left\{\hspace{0.17em}{a}_n\hspace{0.17em}-\hspace{0.17em}\sqrt{r_n{}^2\hspace{0.17em}-\hspace{0.17em}{u}^2\hspace{0.17em}}\hspace{0.17em}\right\}}^{\hspace{0.17em}3}\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}d\hspace{0.17em}u}$$

$$\begin{array}{l}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}3\hspace{0.17em}{P}_L\hspace{0.17em}{(s_1\hspace{0.17em}-s_2)}^2\hspace{0.17em}}{\hspace{0.17em}4\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}{\displaystyle \frac{1}{\hspace{0.17em}2\hspace{0.17em}{c}_n{}^5\hspace{0.17em}}}\hspace{0.17em}\left\{\hspace{0.17em}6\hspace{0.17em}{a}_n\hspace{0.17em}{r}_n{}^2\hspace{0.17em}{\tan }^{-1}\hspace{0.17em}{\displaystyle \frac{r_n}{c_n}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{2\hspace{0.17em}{c}_n\hspace{0.17em}{r}_n\hspace{0.17em}}{a_n}}\hspace{0.17em}(2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}+\hspace{0.17em}{r}_n{}^2)\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}3\hspace{0.17em}\pi \hspace{0.17em}{a}_n{}^2\hspace{0.17em}{r}_n{}^2\hspace{0.17em}}{a_n}}\right\}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{3\hspace{0.17em}{P}_L}{2\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}{\displaystyle \frac{(s_1{}^2\hspace{0.17em}+s_2{}^2)\hspace{0.17em}}{r_n^2}}\hspace{0.17em}\left\{{\displaystyle \frac{a_n}{\hspace{0.17em}{c}_n^3}}\hspace{0.17em}\left(-2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}+\hspace{0.17em}3\hspace{0.17em}{r}_n^2\right)\hspace{0.17em}\hspace{0.17em}{\tan }^{-1}\hspace{0.17em}{\displaystyle \frac{r_n}{c_n}}\right.\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left.{\displaystyle \frac{\hspace{0.17em}{a}_n{}^3\hspace{0.17em}{r}_n}{c_n^2}}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}-\hspace{0.17em}{r}_n^2}{\hspace{0.17em}{(c_n{}^2\hspace{0.17em}+\hspace{0.17em}{r}_n{}^2)}^2\hspace{0.17em}}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}\pi \hspace{0.17em}{a}_n{}^4\hspace{0.17em}}{2\hspace{0.17em}{c}_n^3}}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}-2\hspace{0.17em}{a}_n{}^2\hspace{0.17em}+\hspace{0.17em}3\hspace{0.17em}{r}_n^2}{{(c_n{}^2\hspace{0.17em}+\hspace{0.17em}{r}_n{}^2)}^{3/2}}}\right\}\end{array}$$

(9)

$${\delta }_L\hspace{0.17em}\hspace{0.17em}\approx \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}9\hspace{0.17em}\pi \hspace{0.17em}{P}_L\hspace{0.17em}\hspace{0.17em}}{\hspace{0.17em}4\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}{\left(s_2\hspace{0.17em}-\hspace{0.17em}{s}_1\right)}^2\hspace{0.17em}{\displaystyle \frac{1}{\hspace{0.17em}{r}_n{}^2\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \frac{\hspace{0.17em}{r}_n\hspace{0.17em}}{\hspace{0.17em}2\hspace{0.17em}t\hspace{0.17em}}}\right)}^{5/2},$$

(10)

$${\delta }_{Lmd}\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}{f}_L\left({\displaystyle \frac{t}{r_n}}\right)\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}9\hspace{0.17em}\pi \hspace{0.17em}{P}_L\hspace{0.17em}\hspace{0.17em}}{\hspace{0.17em}4\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}{\left(s_2\hspace{0.17em}-\hspace{0.17em}{s}_1\right)}^2\hspace{0.17em}{\displaystyle \frac{1}{\hspace{0.17em}{r}_n{}^2\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \frac{\hspace{0.17em}{r}_n\hspace{0.17em}}{\hspace{0.17em}2\hspace{0.17em}t\hspace{0.17em}}}\right)}^{5/2},$$

(11)

$$k_{Lmd}\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}{f}_L{\left({\displaystyle \frac{t}{r_n}}\right)}^{-1}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}4\hspace{0.17em}{w}_L\hspace{0.17em}E\hspace{0.17em}\hspace{0.17em}}{\hspace{0.17em}9\hspace{0.17em}\pi \hspace{0.17em}{\left(s_2\hspace{0.17em}-\hspace{0.17em}{s}_1\right)}^2\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}{r}_n{}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left({\displaystyle \frac{\hspace{0.17em}2\hspace{0.17em}t\hspace{0.17em}}{r_n}}\right)}^{5/2},$$

(12)

where δ_L is the approximate linear guide displacement, δ_Lmd is the modified displacement, k_Lmd is the modified stiffness, and f_L(t/r_n) is a modification function.

2.4. Design Equation for the Stiffness of the Rectangular Spring

The stiffness of the rectangular spring is summarized below; further information is provided in a previous study [18]. In this study, the rectangular wire helical spring illustrated in Figure 7 was considered. A uniform pressure p acts on the spring seat, producing the total load P_R. Among the four deformation components, that is, compression, shear, torsion, and bending, the torsional moment dominates the displacement of the rectangular spring. Therefore, the displacement can be approximated using Equation (13), and stiffness k_R can be expressed using Equation (14).

$${\delta }_R\hspace{0.17em}\hspace{0.17em}\approx \hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}{P}_R\hspace{0.17em}{R}^2\hspace{0.17em}l\hspace{0.17em}}{\hspace{0.17em}G\hspace{0.17em}J\hspace{0.17em}}}\hspace{0.17em}{\cos }^2\hspace{0.17em}{\alpha }_R\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}{P}_R\hspace{0.17em}{n}_e\hspace{0.17em}\pi \hspace{0.17em}{D}_R{}^3\hspace{0.17em}}{4\hspace{0.17em}\hspace{0.17em}G\hspace{0.17em}J\hspace{0.17em}}}\hspace{0.17em}\cos \hspace{0.17em}{\alpha }_R,$$

(13)

$$k_R\hspace{0.17em}\hspace{0.17em}\approx \hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}4\hspace{0.17em}\hspace{0.17em}G\hspace{0.17em}J\hspace{0.17em}}{\hspace{0.17em}{n}_e\hspace{0.17em}\pi \hspace{0.17em}{D}_R{}^3\hspace{0.17em}\cos \hspace{0.17em}{\alpha }_R\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}4\hspace{0.17em}\hspace{0.17em}G\hspace{0.17em}{f}_{stiff}\hspace{0.17em}{w}_R\hspace{0.17em}{h}^3\hspace{0.17em}}{\hspace{0.17em}{n}_e\hspace{0.17em}\pi \hspace{0.17em}{D}_R{}^3\hspace{0.17em}\cos \hspace{0.17em}{\alpha }_R\hspace{0.17em}}},$$

(14)

where α_R is the spring pitch angle (°), n_e is the number of effective turns, D_R is the mean coil diameter, R is the mean coil radius, G is the shear modulus, f_stiff is the torsional moment coefficient, w_R is the rectangular wire width, h is the wire thickness, and J is the St. Venant torsional constant.

To improve the accuracy of the approximate equation, a modification function was derived from the ratio between the stiffness obtained from Equation (16) and the stiffness k_FEA obtained from FEA. Incorporating this modification yields the practical design equation shown in Equation (15) [18]:

$$k_{\mathit{Rmd}}\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}{f}_R\left({\alpha }_R\right)\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}4\hspace{0.17em}\hspace{0.17em}G\hspace{0.17em}{f}_{stiff}\hspace{0.17em}{w}_R\hspace{0.17em}{h}^3\hspace{0.17em}}{\hspace{0.17em}{n}_e\hspace{0.17em}\pi \hspace{0.17em}{D}_R{}^3\hspace{0.17em}\cos \hspace{0.17em}{\alpha }_R\hspace{0.17em}}},$$

(15)

where f_R(α_R) is the modification function.

This practical design equation provides a simple and accurate alternative to the classical Lieseke Equation [19], which requires reading the coefficients from a graph.

2.5. Design Equation for Deformation of the Bearing

First, we considered the deformation that occurs when two equivalent spheres come into contact with one another. The stiffness of the sphere is summarized below; further information is provided in a previous paper [20]. Figure 8 illustrates the definition of the spherical contact deformation analyzed in this study. Here, P_C is the contact load, r₀ is the sphere radius, δ is the displacement of one sphere (half of the total approach), and a is the contact radius. Hertz’s classical solution introduces a slight error because it approximates the ellipsoidal contact region using an elliptical paraboloid. Therefore, in this study, the indirect fictitious boundary integral method (IFBIM) developed by Kishida et al. [21] was used to obtain a more accurate reference deformation. The Neuber–Papkovich solution provides a general representation (16) for linear elastic problems and can be expressed in the form of Equation (17) using potential theory [22,23].

As illustrated in Figure 9, the IFBIM performs integration not on the real boundary S (black line) but on a fictitious boundary S* (red line) located at a distance t_B (=α_B h_B) from the real boundary. Quantities defined on the fictitious boundary are denoted by asterisks (*). In this analysis, the pressure distribution on the real boundary was prescribed as the boundary condition, and the displacement on the real boundary was computed by solving a system of equations for the discretized density function on the fictitious boundary.

$$\overrightarrow{u}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\nabla \hspace{0.17em}(B_0\hspace{0.17em}+\hspace{0.17em}\overrightarrow{r}\cdot \overrightarrow{B}\hspace{0.17em})\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\kappa \hspace{0.17em}\overrightarrow{B}, \hspace{0.17em}\overrightarrow{r}\cdot \overrightarrow{B}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}x\hspace{0.17em}{B}_x\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}y\hspace{0.17em}{B}_y\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}z\hspace{0.17em}{B}_z\hspace{0.17em}, \kappa \hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}4\hspace{0.17em}(1\hspace{0.17em}-\hspace{0.17em}\nu ),$$

(16)

$$B_0\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}, \overrightarrow{B}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{S^{*}}{\mu }^{*}(Q^{*})/\hspace{0.17em}R}\hspace{0.17em}d\hspace{0.17em}{S}^{*}\hspace{0.17em}, R\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}\sqrt{{(x_Q\hspace{0.17em}-x)}^2\hspace{0.17em}+\hspace{0.17em}{(y_Q\hspace{0.17em}-\hspace{0.17em}y)}^2\hspace{0.17em}+\hspace{0.17em}{(z_Q\hspace{0.17em}-\hspace{0.17em}z)}^2\hspace{0.17em}},$$

(17)

where $\overrightarrow{u}\hspace{0.17em}(u_x,u_y,u_z)$ is the displacement vector, $\overrightarrow{r}\hspace{0.17em}(x,y,z)$ is the position vector, B₀ is the scalar harmonic function, $\overrightarrow{B}$ is the vector harmonic function, and κ is a constant defined by Equation (16).

Hertz’s classical formula for spherical contact deformation is expressed by Equation (18) [24]. By applying a modification function derived from the IFBIM results, the practical design equation for spherical deformation is obtained, as shown in Equation (19) [20].

$$\delta \hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\sqrt[3]{{\displaystyle \frac{\hspace{0.17em}9\hspace{0.17em}}{\hspace{0.17em}\hspace{0.17em}16\hspace{0.17em}\hspace{0.17em}}}\hspace{0.17em}{\left({\displaystyle \frac{\hspace{0.17em}1\hspace{0.17em}-\hspace{0.17em}{\nu }^2\hspace{0.17em}}{E}}\right)}^2\hspace{0.17em}\left({\displaystyle \frac{1}{\hspace{0.17em}{r}_0\hspace{0.17em}}}\right)\hspace{0.17em}\hspace{0.17em}{P}_c{}^2\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}{\delta }_{HZ},$$

(18)

$${\delta }_{md}\hspace{0.17em}\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{BE0}\left(P_C\right)\hspace{0.17em}\hspace{0.17em}{\delta }_{HZ},$$

(19)

where δ is the displacement of the sphere, δ_md is the modified displacement, E is the Young’s modulus, ν is Poisson’s ratio, and f_BE0(P_C) is the modification function. The specific form of the modification function is described in Section 3.3.

In the present study, we use the results of existing studies to derive a practical design equation for the thrust stiffness of bearings. Palmgren’s bearing-stiffness formulation was originally derived from the Hertzian contact theory [25,26]. Equation (20) yields the thrust-stiffness expression, which applies the Hertzian theory to the inner ring, outer ring, and rolling elements. Figure 10 illustrates the relationship between the thrust bearing loads and corresponding displacements of the bearing. By incorporating these relationships and applying the same correction function used for the spherical contact, a practical design equation for the bearing thrust deformation is obtained, as shown in Equation (21).

$${\delta }_n\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}0.00044\hspace{0.17em}\hspace{0.17em}{\left\{{\displaystyle \frac{\hspace{0.17em}\hspace{0.17em}{P}_C{}^2\hspace{0.17em}}{\hspace{0.17em}{D}_w}}\hspace{0.17em}\right\}}^{1/3},$$

(20)

$${\delta }_B\hspace{0.17em}\equiv \hspace{0.17em}\hspace{0.17em}{f}_{BE}\left(F_a\right)\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{0.00044}{\hspace{0.17em}{Z}^{2/3}\hspace{0.17em}{D}_w{}^{1/3}\hspace{0.17em}{(\sin \hspace{0.17em}{\alpha }_C)}^{5/3}\hspace{0.17em}}}\hspace{0.17em}\hspace{0.17em}{F}_a{}^{2/3},$$

(21)

where δ_B is the modified bearing thrust deformation, f_BE(F_a) is the modification function for the deformation, F_a is the thrust load, Z is the number of rolling elements, D_W is the diameter of the rolling elements, and α_C is the contact angle.

3. Results

3.1. Stiffness of the Linear Guide

Figure 11 illustrates the modification function for the bending stiffness, which was fitted using the fractional expression given in Equation (22). As shown in Figure 12, the bending stiffness of the circular notch hinge was evaluated by comparing the proposed design in Equation (6) with the existing approximate equations proposed by Tabata, Schotborgh et al., Smith et al., and Tseytlin, in addition to comparison with the experimental results reported by Xu et al. [16,27,28,29,30,31]. The bending stiffness of a rectangular prism without a circular notch was nondimensionalized using k_B0. The proposed equation exhibited the closest agreement with the exact analytical solution, particularly in the region where the other approximations diverged significantly.

Similarly, Figure 13 illustrates the modification function for the lateral stiffness, fitted using the fractional expression in Equation (23). As shown in Figure 14, the design proposed in Equation (8) demonstrates better agreement with the exact analytical solution than Paros’s approximation. The lateral stiffness of a rectangular prism without a circular notch was nondimensionalized using k_T0.

$$f_B\left({\displaystyle \frac{t}{r_n}}\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}3\hspace{0.17em}{r}_n\hspace{0.17em}+\hspace{0.17em}t\hspace{0.17em}}{\hspace{0.17em}3\hspace{0.17em}{r}_n\hspace{0.17em}+\hspace{0.17em}2\hspace{0.17em}t\hspace{0.17em}}},$$

(22)

$$f_T\left({\displaystyle \frac{t}{r_n}}\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}102\hspace{0.17em}-\hspace{0.17em}15\hspace{0.17em}{\left\{\hspace{0.17em}(t/r_n)\hspace{0.17em}-\hspace{0.17em}0.24\hspace{0.17em}\right\}}^2\hspace{0.17em}}{100\hspace{0.17em}-\hspace{0.17em}(t/r_n)}},$$

(23)

To derive a practical design equation for the deflection of the linear guide, the ratio between the approximate Equation (10) and the FEA results was examined. Figure 15 illustrates the FEA model of the linear guide constructed using CalculiX 2.17 with a Young’s modulus of 205.5 GPa and Poisson’s ratio of 0.3. The number of mesh elements varied depending on the geometry. For example, the model with r_n = 5 mm, w_L = 10 mm, h = 12 mm, and s₃ = 35 mm contained 105,100 elements (155,911 nodes).

Figure 16 illustrates the modification function for the linear guide when Δs = 19 mm, fitted using the fractional expression in Equation (24). Because this function depends on the inter-notch distance Δs (=s₂ − s₁), as shown in Figure 17a, an additional modification term was introduced to account for variations in Δs, resulting in the modification function (25). Figure 17b illustrates the error of the practical design equation for linear-guide stiffness relative to the FEA results, and the maximum error is 3% (at t/r_n = 0.58).

$$f_{L0}\left({\displaystyle \frac{t}{r_n}}\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{16\hspace{0.17em}{r}_n\hspace{0.17em}+\hspace{0.17em}29\hspace{0.17em}t}{\hspace{0.17em}18\hspace{0.17em}{r}_n\hspace{0.17em}+\hspace{0.17em}10\hspace{0.17em}t\hspace{0.17em}}},$$

(24)

$$f_L\left({\displaystyle \frac{t}{r_n}}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{16\hspace{0.17em}{r}_n\hspace{0.17em}+\hspace{0.17em}29\hspace{0.17em}t}{\hspace{0.17em}18\hspace{0.17em}{r}_n\hspace{0.17em}+\hspace{0.17em}10\hspace{0.17em}t\hspace{0.17em}}}\hspace{0.17em}{\left\{1\hspace{0.17em}-\hspace{0.17em}{\displaystyle \frac{1}{\hspace{0.17em}0.77\hspace{0.17em}}}\hspace{0.17em}\left(1\hspace{0.17em}-\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}{s}_2\hspace{0.17em}-\hspace{0.17em}{s}_1\hspace{0.17em}}{19}}\right)\hspace{0.17em}\left({\displaystyle \frac{\hspace{0.17em}t\hspace{0.17em}}{r_n}}\right)\hspace{0.17em}\right\}}^{-1}.$$

(25)

3.2. Stiffness of Rectangular Spring

In this study, the helical spring was analyzed using a configuration with six turns, which is consistent with the work of Shimizu et al. [31]. For further details on the analytical model, please refer to a previously reported study [18]. FEA was performed for a wide range of rectangular wire helical springs with varying pitch angles, spring indices, and aspect ratios. Although slight variations were observed depending on these parameters, the differences were sufficiently small to justify the use of the averaged values. A modification curve was then fitted to the ratio between the analytical stiffness from Equation (13) and the FEA-derived stiffness. Incorporating this modification (Equation (26)) yields the practical design in Equation (15).

Figure 18b illustrates the error of the practical design formula for rectangular spring stiffness relative to the FEA results, and the maximum error is 2.5% (at αR = 5.1°) demonstrating that it provides a simple yet accurate alternative to the classical Lieseke equation without requiring the use of graphical coefficients.

$$f_R\left({\alpha }_R\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}461\hspace{0.17em}-\hspace{0.17em}13\hspace{0.17em}{\alpha }_R}{\hspace{0.17em}433\hspace{0.17em}-\hspace{0.17em}10\hspace{0.17em}{\alpha }_R\hspace{0.17em}}}\hspace{0.17em}.$$

(26)

3.3. Deformation of Spherical Contact

The spherical contact deformation was analyzed using IFBIM. For further details on the analytical model, please refer to a previously reported study [20].

To reproduce the experimental results reported by Jacobson [32], the sphere radius was set to r₀ = 7.89 mm, the Young’s modulus to E = 208 GPa, and Poisson’s ratio to ν = 0.3. Using the IFBIM, the spherical contact deformation was computed with high accuracy by solving the discretized density function on the fictitious boundary.

In Figure 19, Hertz’s classical solution is compared with the IFBIM results. A modification function was derived from this comparison, leading to the practical design equation for spherical deformation presented in Equation (19). The modification function in Equation (27) for spherical deformation can be transformed into the modification function in Equation (28) for bearing displacement under a thrust load as follows:

$$f_{BE0}\left(P_C\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}1200\hspace{0.17em}+\hspace{0.17em}20\hspace{0.17em}{P}_C\hspace{0.17em}}{2000\hspace{0.17em}+\hspace{0.17em}21\hspace{0.17em}{P}_C}},$$

(27)

$$f_{BE}\left(F_a\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{20\hspace{0.17em}{F}_a\hspace{0.17em}+\hspace{0.17em}1200\hspace{0.17em}Z\hspace{0.17em}\sin \hspace{0.17em}{\alpha }_C}{21\hspace{0.17em}{F}_a\hspace{0.17em}+\hspace{0.17em}2000\hspace{0.17em}Z\hspace{0.17em}\sin \hspace{0.17em}{\alpha }_C}},$$

(28)

where F_a is the bearing thrust load, Z is the number of rolling elements, and α_C is the contact angle.

3.4. Drive Accuracy for the Focus Control Mechanism

The validity of the practical design equations for the linear guide, rectangular spring, and bearing—components of the focus control mechanism—was confirmed in Section 3.1, Section 3.2 and Section 3.3. In this section, we present the results of the 1DCAE design of the focus control mechanism using these design equations. As shown in Figure 20a, the experimental measurements naturally indicate that the displacement of the focus control mechanism varies almost linearly with the motor-drive angle. However, when the differential displacement—defined as the deviation from the best-fit straight line—is magnified, it exhibits a nonlinear “smiley-shaped” profile, as shown in Figure 20b, and significantly exceeds the required specification (red line). This nonlinearity originates from the bearing deformation and hysteresis at the bolted joints. Such behavior is critical for an ultraprecision focus control mechanism because it prevents the system from achieving the required submicrometer-level accuracy.

The simulated displacement characteristics of the focus control mechanism modeled using the 1DCAE approach is shown in Figure 21. These results were derived from Equation (4) in combination with the practical design equations for the displacement and configuration of each element. The 1DCAE model successfully reproduced the experimentally observed nonlinear behavior, as shown in Figure 20. In Figure 21, the purple curve corresponds to drive angles between −5000 and 5000°; in comparison, the green curve represents the range from 0 to 12,000°. The focus control mechanism requires a drive accuracy of ±100 nm (red lines) and a drive resolution of ±25 nm; however, these hysteresis curves exceed the specified limits by a wide margin.

High-precision actuation can be achieved either by returning the mechanism to a reference position before driving or by calculating and correcting the hysteresis loop. However, as shown in Figure 21, when the driving direction reverses midway, the hysteresis loop changes shape—forming the purple and green loops; thus, the entire driving history must be taken into account, making the compensation extremely complex. Therefore, in this study, the 1DCAE model was used to analyze the differential displacement (drive error) associated with various design parameters to minimize the hysteresis.

The differential displacement obtained by varying parameters w_R/h and t/r_n of the spring reducer, which consisted of a rectangular spring and linear guides, is shown in Figure 21. This value was obtained by varying the dimensional parameters in the 1DCAE model calculation Equation (4) using the linear-guide stiffness expression (14) and the rectangular spring stiffness expression (16), which are the practical design equations derived in this study. The parameter study showed that the differential displacement is most effectively minimized when both w_R/h and t/r_n are set to their smallest values. However, because reducing t/r_n degrades the resolution, the parameters w_R/h = 1.2 and t/r_n = 0.25 were selected.

The differential displacement as a function of the bearing preload and contact angle is presented in Figure 22b. This figure was plotted by applying the practical design Equation (22) for the bearing thrust displacement derived in this study to the 1DCAE model calculation Equation (4) and by varying the parameters related to the contact angle and preload. It was found that increasing the bearing preload consistently reduces the differential displacement over all contact angles. However, to ensure sufficient thrust-load capacity for the bearing, a contact angle of 30° was adopted, and a preload of 2200 N was selected.

By applying a ceramic coating to the mating surfaces of the bolted joint, the hysteresis was further reduced: compared with the initial friction coefficient of 0.2, the ceramic-coated surface with a friction coefficient of 0.5 decreases the hysteresis to 0.16 times its original value (0.2/0.5)², thereby improving the performance of the focus control mechanism. Figure 23 illustrates the results of a parameter survey in which the differential displacement was minimized using the 1DCAE model calculation Equation (4) when each practical design equation was applied, demonstrating that the drive accuracy (differential displacement) sufficiently satisfied the required specification lines and that the displacement error was reduced to the level at which hysteresis correction was no longer required. The drive stroke refers to the converted displacement δ_L of the I/F plate corresponding to the motor rotation angle and is defined as the peak-to-peak value calculated using Equation (4). The drive accuracy was defined as the peak-to-peak value of the differential displacement obtained by subtracting the best-fit straight line from the drive stroke. In addition, the resolution was defined as the output displacement per step angle of the motor. The measurement results of the initial model and the improved design are summarized in

Table 1. Comparison between initial results and improved results.

Specification Items	Specification	Initial Results	Improved Results
Stroke	+/−100 μm	+/−100 μm	+/−150 μm
Drive resolution	+/−25 nm	+/−24 nm	+/−12 nm
Drive accuracy	+/−100 nm	+/−1410 nm	+/−35 nm
Remarks	N/A	Calculation required	Calculation not required

. As shown in Figure 20, the drive accuracy of the initial model significantly exceeded the required specification lines; in comparison, the improved design, as shown in Figure 23, exhibited a substantial enhancement in drive accuracy, achieving an improvement by a factor of 40 compared with the initial model.

4. Discussion

4.1. Validity of the Practical Design Equations

The practical design equations formulated using fractional expressions for the linear guide, rectangular spring, and spherical contact deformations exhibited excellent agreement with the analytical solutions, FEA results, and IFBIM calculations.

In contrast to conventional approximation formulas whose accuracy deteriorates outside narrow parameter ranges, the proposed equations maintain high fidelity across a significantly broader design domain. This robustness confirms their suitability for practical engineering design, particularly in the early development phase, where rapid yet reliable evaluation is essential and detailed 3D-CAE models are not yet available.

4.2. Implications for the Focus Control Mechanism

The 1DCAE model clarified the individual contributions of the linear guide, rectangular spring, bearing deformation, and bolted-joint hysteresis to the overall drive accuracy of the focus control mechanism.

The following insights were obtained:

• Reducing the stiffness of a rectangular spring effectively reduces the differential displacement (drive error).
• A higher bearing preload enhances stiffness and effectively suppresses nonlinear displacement errors.
• Larger contact angles improve thrust-load resistance and contribute to more stable displacement characteristics.
• Reducing joint hysteresis through surface treatment significantly improves repeatability and minimizes nonlinear behaviors.

These findings highlight the advantage of 1DCAE: the system-level parameter sensitivity can be evaluated efficiently, enabling design optimization that is difficult to achieve using 3D-CAE alone.

4.3. Improvement of Drive Accuracy

The optimized design achieved a drive accuracy of ±35 nm without requiring hysteresis compensation, surpassing the required accuracy of ±100 nm.

This result demonstrates that parameter optimization based on the simplified system design method can dramatically improve performance while simultaneously reducing the complexity of the control algorithm because no drive-history-dependent correction is necessary.

Overall, the results confirm that the simplified system design method is a powerful and practical tool for the early-stage design of ultraprecision space mechanisms, enabling highly accurate prediction and optimization without relying on detailed FEA models.

5. Conclusions

The results presented herein demonstrate that the 1DCAE approach combined with practical design equations can be used to accurately evaluate and significantly improve the drive accuracy of an ultraprecision space focus control mechanism. Practical design equations were formulated using fractional expressions to determine the stiffness of the linear guide, rectangular spring, and bearing deformation. These equations were constructed by introducing modification functions derived from finite element analysis (FEA) and the indirect fictitious boundary integral method (IFBIM), thereby enabling highly accurate predictions across a wide range of design parameters.

The proposed practical design equations showed excellent agreement with analytical solutions, numerical simulations, and experimental data. In particular, the linear-guide stiffness equation achieved an accuracy within 3% of the FEA results, that of the rectangular spring stiffness equation was within 2.5%, and the bearing deformation equation closely matched the IFBIM-based reference values. By incorporating these accurate component-level models into the 1DCAE framework, the overall drive accuracy of the focus control mechanism was improved to one fortieth that of the initial design.

These results confirm that the 1DCAE method is highly effective for the early-phase design of ultraprecision space mechanisms, where conventional FEA-based approaches are difficult to apply owing to modeling complexity and computational cost. The methodology presented in this study provides a practical and efficient tool for system-level optimization and can be extended to the design of optical mirror bipod support structures, optical telescope structures, space robotic mechanisms, and so forth. Moreover, in future studies, we plan to conduct experiments using the improved design model and compare the results with the design predictions, thereby further validating the effectiveness of the proposed design methodology.