Vibration Analysis of Non-Uniform Rods Considering Lateral Inertia Effects Under Ultrasonic Excitation

Non-uniform rods are widely present in ultrasonic vibration systems, and the accuracy of their resonance design significantly impacts system performance. Traditional design methods often treat non-uniform rods as one-dimensional members, neglecting the influence of lateral inertia, which results in lower resonance design accuracy. This paper establishes a vibration model for non-uniform rods under ultrasonic excitation that accounts for lateral inertia effects. Subsequently, the natural frequencies and mode shapes of the rod are obtained using the transfer matrix method. The modal superposition method is then employed to derive the internal displacement distribution function and stress distribution function of the rod. The validation of the tapered rod and the ultrasonic fatigue specimen demonstrates that the proposed method is closer to the Finite Element Method (FEM) results than the traditional method. The results show that the relative deviation between the resonant length of the tapered rod and the FEM calculation is less than 0.16%, and the harmonic response analysis results of the two are also in good agreement; the relative deviation between the first-order natural frequency of the ultrasonic fatigue specimen and the FEM calculation is less than 0.315%, and the predicted maximum stress is highly consistent with the FEM. The research findings presented herein can serve as a universal methodology for resonance design and performance evaluation of longitudinal vibration components in ultrasonic vibration systems.