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International Journal of Machine Tools received in revised form 2 October 2000; accepted 6 October 2000 Abstract Gears are crucial components for modern precision machinery as a means for the power transmission mechanism. Due to their complexity and unique characteristics, gears have been designed and manufactured by a special type of machine tools, such as gear hobbing and shaping machines. In this paper, we attempt to manufacture the spiral bevel gear (SBG: the most complex type among the gear products) by a three- axis CNC milling machine interfaced with a rotary table. This consists of (a) geometric modeling of the spiral bevel gears, (b) process planning for NC machining, (c) a tool path planning and execution algorithm for both 4-axis and 3/4-axis (three out of four axes) controls. Experimental cuts were made to ascertain the validity and effectiveness of the presented method with a CNC milling machine controlled by the 3/4- axis control mode. 2001 Published by Elsevier Science Ltd. Keywords: Gear manufacturing; Spiral bevel gear; Geometric modeling of gears; Sculptured surface machining; Rotary table application; Additional-axis machining technology 1. Introduction Gears are efficient and precision mechanisms for industrial machinery as a means for power transmission. Among the various types of gears (Fig. 1), the spiral bevel gears (SBG) are the most complex type and are used to transmit the rotational motion between angularly crossed shafts. Previous studies on gears have been mainly concerned with the design and analysis of gears. The geometric characteristics and design parameters of SBGs have been studied extensively * Corresponding author. Fax: +82-54-279-5998. E-mail address: shspostech.ac.kr (S.H. Suh). 0890-6955/01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S0 890-6955(00)00104-8 834 S.H. Suh et al. / International Journal of Machine Tools (2) by the SSM method, a broad range Fig. 2. Special machine tools and cutters for manufacturing SBGs. 835S.H. Suh et al. / International Journal of Machine Tools (3) a special type of gear, for instance “huge” gears of diameter over 1000 mm, and “crown” gears can be machined by the SSM method, which cannot be done by the dedicated gear machine tools, except in very limited cases. In view of the above, special attention is given to the capability of the SSM method in terms of geometric accuracy and surface quality together with machining time. If the performance is comparable, except for the production rate, the SSM method can be applied in industrial practice for NC machining of huge SBGs, while the production rate is not emphasized. This paper presents comprehensive technology including geometric modeling, process planning, tool path algorithms, and experimental validation. 2. Geometric modeling of the SBGs Typically, geometric specification of SBGs is given by a set of parameters. These parameters are provided with an engineering drawing, as shown in Fig. 3. Some parameters (principal parameters) are required for defining the geometry, while others (auxiliary parameters) can be derived based on a formula. Table 1 summarizes some of the crucial parameters including relation- ships among parameters 2. Using the parameters, the surface model can be derived as follows. As illustrated in Fig. 4, the surface between two teeth is modeled by the large section curve swept along the spiral curve. The section curve is decomposed into five fragments; S i (u i , i P 1:5, where u i P 0, u max i ) is the parameter for fragment i. Denoting w by the parameter along the spiral curve, the surface model can be represented by S i (u, w), i P 1:5 as shown in Fig. 4. S i and S 5 are the involute surfaces, and S 2 and S 4 the filleted surfaces, and S 3 the bottom surface. Fig. 3. An engineering drawing. 836 S.H. Suh et al. / International Journal of Machine Tools (b) only three axes out of the four axes can be controlled simultaneously. The latter is called the additional-axis machine system, which can often be found in industrial practice where the rotary table is controlled by the fourth axis of the three-axis machine tool controller (see 11 for details). In this paper, we present a tool path algorithm for both configurations. 3.2. Machining strategy The workpiece is premachined as a conic form by turning operation. The volume to be removed is the swept volume of the cross-section CUIW along the spiral curve defined by S i (u, w), i P 1:5). The volume is removed by several processes: (1) rough cut with several flat endmills; (2) semi-finished cut with several ball endmills; and (3) finish cut with a ball endmill. To minimize the machining time, a larger tool is desired for the rough cut and semi-finish cut. The finish cut allowance is set (for instance 0.3 mm), and the semi-finish cut removes the uneven surface (resulting from the rough cut). During the finish cut, the whole surface is machined by a single ball endmill of diameter D=0.8I (this is based on a heuristic), where I is the chodal length between the two points defining the S 3 curve in the small section curve (see Fig. 4). This is to prevent any cutter marks on the surface due to tool change. Tool path algorithms for rough and semi-finished cuts are omitted for the brevity of the paper. 3.3. Tool path planning for finish cut The surface model S 1 (u, w) is machined by a ball endmill of radius R. As mentioned earlier, the involute surfaces S 1 and S 5 are the most important surfaces, the accuracy of which should be strictly controlled. Our method is based on the CC-parametric scheme, where the CC-points are sampled from the parametric surface model, equally distanced on the parametric plane. For the sake of machining efficiency, tool motion along the w direction is chosen. In what follows, inter- ference-free CL-data for S 5 (u, w) is presented, as the same can be applied for S 1 (u, w). 841S.H. Suh et al. / International Journal of Machine Tools and (b) finding a feasible range of the interference-free tool axis, followed by selecting an interference-free axis. We take the second approach. In what follows, we present a computationally efficient method for finding a feasible range defined by the two bounding axes A 1 and A r for the given CC-point S 1 (u, w) as shown in Fig. 8. Suppose the tool center point and its unit normal vector are given by C and N C . Then, the tool motion in the four-axis configuration is defined on the CL-plane: P C =PuP x =C x , where C x is the x value of C. Let C 1i , i P 1:m, C 2j , j P 1:n be the offset points on the CL-plane (Fig. 9). Noting that T 1 P C 1i , T 2 P C 2i , consider the problem of finding T i . Define the reference axis (Fig. 9(a) as V5CC 11 3N C (11) Fig. 7. TBI in the boundary region. 842 S.H. Suh et al. / International Journal of Machine Tools A5 uCT 2 uCT 1 +uCT 1 uCT 2 uCT 2 uCT 1 +uCT 1 uCT 2 (16) In general, the tool axis vector is not aligned with the spindle axis (Z). In the four-axis con- figuration where the workpiece is oriented by the rotary table, it is necessary to align the tool axis vector with the spindle axis. The rotation angle to access a CC-point S 5 (u, w) is determined such that the tool axis vector A(u, w) is parallel to the XZ-plane. Decomposing the tool axis vector into A x , A y , A z the rotation angle (Fig. 10) is Fig. 10. Tool-axis determination for four-axis control. 844 S.H. Suh et al. / International Journal of Machine Tools 2. convert the rotational angle range into the y i value y min i , y max i by Eq. (17). Step 2: Determine y* as follows: set I=1, where I is the number of groups in the CC-path 1. find y min i , y max i =intersection of y min i , y max i , for i P I; 2. if the intersection range is found, then y =y min +y max /2, and exit; 3. otherwise, divide the CC-points into I+1 groups and go to Step 2.2. 845S.H. Suh et al. / International Journal of Machine Tools (b) Adjacent pitch error, (c) Accumulated pitch error. 5. Concluding remarks In this paper we attempted to manufacture the spiral bevel gear using the CNC milling machine based on the sculptured surface machining method. For such a purpose, we investigated surface modeling and the tool path computing algorithm. The surface model accepts the gear parameters as input and outputs the bi-parametric surface model so that it can be directly used for deriving the tool path via the CC-parametric scheme. Note that almost all previous works have been con- cerned with the design aspect, and the bi-parametric surface model has not been explicitly derived. The tool path algorithm presented in this paper was based on the CC-parametric scheme. In developing the tool path algorithm, geometric accuracy and surface quality were addressed, 850 S.H. Suh et al. / International Journal of Machine Tools & Manufacture 41 (2001) 833850 together with machine tool configuration. By the tool path algorithm. the involute surface can be accurately machined without tool interference due to tool size and the tool axis for both four- axis and 3/4-axis control. Also, we tried to reduce the computational complexity by exploiting the geometric characteristics of the gear. The validity of the surface model was verified via comparison with the genuine product (manufactured by dedicated machine tools). The result showed good conformity, as the bi-para- metric model was derived based on the definition of the gear parameters. Even if there is slight unconformity, the performance of a pair of gears is practically acceptable. Also, the machined pair of gears was tested via a gear-mating machine, showing a smooth motion without noise. Distinguished from the conventional method by the dedicated machine tools, the presented method can produce any type and size of SBGs, so long as the geometric model is provided. Hence, it can be practically applied, especially to produce huge gears with crowns which cannot be machined by the dedicated machine tool. Elaboration of the machining strategy and feedrate optimization for the reduction of machining time is left for further study. 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