Analytical Modeling and Structural Optimization of Slender Variable Cross-Section Rod for High-Speed Chip Placement

The cantilever pick-and-place arm of the high-speed placement machine is susceptible to micro-vibration and elastic deformation under high-acceleration motion, thereby degrading chip placement accuracy. To address this issue, this paper presents an analytical study on the natural frequency characteristics and structural optimization of slender variable-cross-section rods. First, based on the thin-walled shell theory, a displacement field model of the thin-walled cantilever rod is established. Second, combining the energy method and Hamilton’s principle, the undamped free vibration equation is derived. Using the Rayleigh–Ritz method with Chebyshev polynomials as the basis functions, an analytical calculation model for the natural frequency of the variable-section thin-walled rod is constructed. The model is validated against finite element simulations, and the relative errors of the low-order natural frequencies are controlled within 5%, confirming its favorable accuracy and robustness. Furthermore, the four-factor three-level orthogonal experiment is designed with the objective of maximizing natural frequency to conduct parameters sensitivity analysis. Accordingly, the optimal structural parameter combination (${\varphi }_3$ = 8 mm, L₁ = 10 mm, L₂ = 50 mm, and L₃ = 5 mm) is determined. Finally, the maximum dynamic deformation under high-acceleration motion decreases from 0.066 mm to 0.021 mm, a reduction of 68.2%, which effectively suppresses residual vibration and end displacement deviation. The analytical method proposed in this study provides a theoretical basis for the rapid dynamic performance evaluation of flexible components in high-speed precision equipment, and the optimization strategy can offer engineering references for the high-stiffness design of key components in chip placement machines.