Design of an Ultrasonic Steel Horn with a Bézier Profile

A novel ultrasonic steel horn with a Bézier profile is developed. The first longitudinal displacement mode of the horn is exploited for high displacement amplification. An optimization scheme and finite element analyses are used to design the horn. The displacement amplification and stress distribution characteristics of the Bézier horn and catenoidal horn are examined. Prototypes of the horns are manufactured by a laser cutting process. Performances of the proposed horn are evaluated by experiments. Experimental results of the harmonic response of the fabricated horn confirm the effectiveness of the design method. The displacement amplification of the proposed horn is 62% higher than that of the traditional catenoidal horn with the same length and end surface widths. INTRODUCTION Ultrasonic horns have been widely used in atomizers [1], ophthalmic surgery [2], welding devices [3], wire bonding [4,5], ultrasonic motor [6], ultrasonic lubrication [7] and ultrasonic bistoury [8], etc. Ultrasonic horn of different profiles such as Gaussian [9], Fourier, exponential [10], stepped [11,12], sinusoidal [13], conical, catenoidal [14] and spline [15], have been proposed and investigated by many researchers. Salmon [16] synthesized a horn where the profile is a perturbation from the exponential contour. In comparing with the traditional horns, new horns with non-straight structures may offer higher displacement amplification. Sherrit et al. [17] presented a folded horn in order to reduce the length of the resonator. Iula et al. [17] proposed an ultrasonic horn vibrating in a flexural mode. Conical, exponential, catenoidal, stepped and Gaussian are the most commonly used horns [18]. Abromov [18] points out that the displacement amplification of catenoidal horns is greater than that of exponential or conical ones and less than that of stepped horns. Gaussian horns may possess high displacement amplification, but numerical methods might be needed for the design of Gaussian horns. Parametric curve based geometry is flexible enough to give a much better control over the profile of horns for design purpose. In parametric form each coordinate of a point on a curve is represented as a function of a single parameter [19]. Therefore, it has more potential to find higher displacement amplification while keeping the stress in the horns low. Because the parametric curve has more freedom to define the horn profile, it is a more difficult problem to optimize the performance of the horn. Finite element method (FEM) has been used to study and analyze behaviors of horns [17,20]. Using FEM, detailed stress and displacement distributions can be obtained. Fu et al. [21] discussed the design of a piezoelectric transducer with a stepped horn via multiobjective optimization. They formulated the optimization problem using Pareto-based multiobjective genetic algorithms [21]. In order to design horns with conflicting design objectives, the genetic algorithms capable of finding multiple optimal solutions in a single optimization run may be used. In this investigation, design and analysis of a steel horn for high displacement amplification are presented. The design is based on a cubic Bézier curve. The optimal designs of the horns are sought by a multiobjective optimization algorithm. Prototypes of horns are fabricated and tested. The experimental results are in good agreement with those based on the optimization design procedure. DESIGN Fig. 1 schematically shows a horn driven by a Langevin transducer. The horn is a displacement amplifier designed to work in a longitudinal mode. The Langevin transducer is composed of a couple of piezoelectric disks poled along y direction but with opposite polarities. The flange allows the mounting of the Langevin transducer at the longitudinal nodes. The horn is actuated by the transducer at the designated frequency, which is set to be the working frequency, 28.0 kHz, of the Langevin transducer used in the experiment. The nearly uniformly distributed displacement of the Langevin transducer is transformed into a longitudinal deformation of the horn. A typical displacement distribution curve is