Design and Analysis of A Nonrational B-Spline Profiled Horn for High Displacement Amplification Huu Tu Nguyen, Dung-An Wang Graduate Institute of Precision Engineering National Chung Hsing Universtiy Taichung, Taiwan daw@dragon.nchu.edu.tw Abstract—A new horn for high displacement amplification is developed. The profile of the horn is a nonrational B-spline curve with an open uniform knot vector. The ultrasonic actuation of the horn exploits the first longitudinal displacement mode of the horn. The horn is designed by an optimization scheme. Performances of the proposed horn have been evaluated by experiments. The displacement amplification of the proposed horn is 41.4% and 8.6% higher than that of the traditional catenoidal horn and a Bézier-profile horn, respectively, with the same length and end surface diameters. The developed horn has a lower displacement amplification than the nonuniform rational B-spline profiled horn but a much smoother stress distribution. Keywords—B-spline horn; optimization; displacement amplification; stress distribution I. INTRODUCTION Parametric curves, such as Bezier and B-spline curves, may be used to describe the profiles of horns for high displacement amplification and ease of machining of ultrasonic horns. Bezier curve is generated by using the vertices of a defining polygon and the Bernstein basis functions. From a mathematical point of view, the flexibility of Bezier curve is limited by its two characteristics. First, the number of polygon vertices fixes the order of the polynomial which defines the curve. Second, a local change within a curve is inhibited due to the global nature of the Bernstein basis. Therefore, a parametric curve, such as B-spline curve, defined by a nonglobal basis which allows the degree of the curve to be changed without changing the number of the defining polygon vertices can be used to describe the profile of horns for higher displacement amplification. In this paper, we develop a new horn with a nonrational B- spline profile for high displacement amplification. The optimal designs of the horns are sought by a multiobjective optimization algorithm. Prototypes of horns are fabricated by a numerical control machining process. Displacements and resonant frequencies of the horns are obtained by a laser displacement sensor. Performance of the developed horn is compared with traditional horns and other parameter curve profiled horns. Fig. 1. A profile of a B-spline horn and its five control points 1 Q , 2 Q , 3 Q , 4 Q and 5 Q . II. DESIGN The design of the horn is based on an optimization procedure where the profile of the horn is optimized via the parameters of an open uniform nonrational B-spline (OUNBS) curve to meet the requirement of displacement amplification. The OUNBS curve is determined by a five-point polygon 5 4 3 2 1 Q Q Q Q Q as shown in Fig. 1. The nonrational B-spline curves are included as a special case of rational B-spline curves. The parametric rational B-spline curve is given by [1]        1 1 , 1 1 , ) ( ) ( ) ( n i k i i n i k i i Q t N h t N h P t P i  where t is the parameter, i Q P the position vector of the point i Q . The homogeneous coordinates i h are weights of their corresponding polygon vertex, i Q P , and k i N , the normalized B-spline base functions. The i th normalized B-spline basis function of order k (degree 1  k ), ) ( , t N k i , are defined as [1]