## 5 Experimentation
The input stiffness of the mechanism is approximately \(0.0146\) N/\(\upmu\)m. Between \(0\) and \(2000\)\(\upmu\)m, the input stiffness of the mechanism remains approximately constant, which is consistent with the previous optimization objectives. The ratio of output stiffness to input stiffness is approximately \(47.94\). This study compares the reduction ratio, structural type, constant reduction ratio range, lumped mass, and energy transfer efficiency of the displacement reducers shown in Table 5. Among the selected mechanisms, the proposed displacement reducer demonstrates the largest constant reduction ratio range and energy transfer efficiency. Simultaneously, this mechanism significantly reduces the moving lumped mass. The reducers in the second structure [19] and the fifth structure [1] in Table 5 adopt lumped compliance, which concentrates the load on a few compliant hinges, leading to higher stress concentration factors. In contrast, the other reducers employ distributed compliance, where the load is spread across multiple compliant beams, reducing the stress on each unit and minimizing the likelihood of stress concentration by ensuring a more even distribution of the load [28]. Compared to other reducers in Table 5, the proposed reducer provides a larger constant reduction ratio range and higher energy transfer efficiency. Overall, the displacement reducer presented in this paper exhibits the largest constant reduction ratio range, alleviates the concentrated stress problem, and achieves higher energy transmission efficiency, avoiding energy loss. Figure 20: Reduction ratio under different loads. simulation results exhibit differences of less than 1.8%. Subsequently, the PSO algorithm is employed for the further optimization of the structure. Under the imposed constraints, the optimization of geometric relationships and beam width resulted in a 149.5% increase in the reduction ratio. This optimization approach is also applicable to other structural designs. The experimental reduction ratio was measured to be 11.03, while the constant reduction ratio ranges from 0 to 2000 \(\upmu\)m. The main innovations of this study are as follows:  
* Grounded in the principles of over-constrained design, it presents a distributed compliant reducer capable of maintaining a nearly constant reduction ratio over a wide range of input displacements, simultaneously minimizing the moving lumped mass;
* The analytical model of the mechanism is developed using the compliance matrix method, which is employed to conduct a detailed structural sensitivity analysis;
* An integrated optimization approach combining the PSO algorithm with the analytical model is applied to achieve a comprehensive optimization of the structure. This methodology is applicable for the structure optimization of most distributed compliant mechanisms;
* In the field of precision operations, such as micro-grippers, biological manipulation, and micro-alignment mechanisms, closed-loop systems with stringent environmental requirements and complex structures are commonly used to achieve high-precision positioning and operation. In contrast, the mechanism proposed in this study features a simple structure and easy fabrication, capable of achieving stable, high-precision output displacements under complex operating conditions, thus providing a more straightforward and reliable solution for precision operations. Our subsequent main task is to design a compliant mechanism with high output stiffness and resolution, while also considering both input and output strokes. Building on the foundation of this study, further structural design and optimization enable it to be suitable for micro-/nano-operations. Additionally, the design content of this stage will serve as the first phase for future adjustable-resolution compliant mechanism designs. Conceptualization, Y.T. and B.H. ; Methodology, Y.T. and B.H. ; Software, Y.T. ; Validation, Y.T. and S.L. ; Formal analysis, Y.T., S.L. and P.Y. ; Investigation, Y.T. and Z.Y. ; Resources, Y.T. and Z.Y. ; Data curation, Y.T. and P.Y.