资源描述
英文资料翻译
英文原文：
Design and machining of cylindrical cams with translating conical followers
By DerMin Tsay and Hsien Min Wei
A simple approach to the profile determination and machining of cylindrical cams with translating conical followers is presented .On the basis of the theory of envelopes for a 1-parameter family of surfaces,a cam profile with a translating conical follower can be easily designed once the follower-motion program has been given .In the investigation of geometric characteristics ,it enables the contact line and the pressure angle to be analysed using the obtained analytical profile expressions .In the process of machining ,the required cutter path is provided for a tapered endmill cutter ,whose size may be identical to or smaller than that of the conical follower .A numerical example is given to illustrate the application of the procedure .
Keywords : cylindrical cams, envelopes , CAD/CAM
A cylindrical cam is a 3D cam which drives its follower in a groove cut on the periphery of a cylinder .The follower, which is either cylindrical or conical, may translate or oscillate. The cam rotates about its longitudinal axis, and transmits a transmits a translation or oscillation displacement to the follower at the same time. Mechanisms of this type have long been used in many devices, such as elevators, knitting machines, packing machines, and indexing rotary tables.
In deriving the profile of a 3Dcam, various methods have used. Dhande et al.1 and Chakraborty and dhande2 developed a method to find the profiles of planar and spatial cams. The method used is based on the concept that the common normal vector and the relative velocity vector are orthogonal to each other at the point of contact between the cam and the follower surfaces. Borisov3 proposed an approach to the problem of designing cylindrical-cam mechanisms by a computer algorithm. By this method, the contour of a cylindrical cam can be considered as a developed linear surface, and therefore the design problem reduces to one of finding the centre and side profiles of the cam track on a development of the effective cylinder. Instantaneous screw-motion theory4 has been applied to the design of cam mechanisms. Gonzalez-Palacios et al.4 used the theory to generate surfaces of planar, spherical, and spatial indexing cam mechanisms in a unified framework. Gonzalez-Palacios and Angeles5 again used the theory to determine the surface geometry of spherical cam-oscillating roller-follower mechanisms. Considering machining for cylindrical cams by cylindrical cutters whose sizes are identical to those of the followers, Papaioannou and Kiritsis6 proposed a procedure for selecting the cutter step by solving a constrained optimization problem.
The research presented in this paper shows q new, easy procedure for determining the cylindrical-cam profile equations and providing the cutter path required in the machining process. This is accomplished by the sue of the theory of envelopes for a 1-parameter family of surfaces described in parametric form7 to define the cam profiles. Hanson and Churchill8 introduced the theory of envelopes for a 1-parameter family of plane curves in implicit form to determine the equations of plate-cam profiles Chan and Pisano9 extended the envelope theory for the geometry of plate cams to irregular-surface follower systems. They derived an analytical description of cam profiles for general cam-follower systems, and gave an example to demonstrate the method in numerical form. Using the theory of envelopes for a 2-parameter family of surfaces in implicit form, Tsay and Hwang10 obtained the profile equations of camoids. According to the method, the profile of a cam is regarded as an envelope for the family of the follower shapes in different cam-follower positions when the cam rotates for a complete cycle.
THEORY OF ENVELPOES FOR 1-PARAMETER FAMILY OF SURFACES IN PARAMETRIC FORM
In 3D xyz Cartesian space , a 1-parameter family of surfaces can be given in parametric form as
(1)
where ζ is the parameter of the family, and u1, u 2, are the parameters for a particular surface of the family. Then, the envelope for the family described in Equation 1 satisfies equation 1 and the following Equation:
(2)
where the right-hand side is a constant zero7. Litvin showed the proving process of the theorem in detail.
If we can solve Equation2 and substitute into equation1to eliminate one of the three parameters u1, u 2, and ζ , we may obtain the envelope in parametric form. However, one important thing should be pointed out here. Equations 1 and 2 can also be satisfied by the singular points of surfaces described below I the family, even if they do not belong to the envelope. Points which are regular points of surfaces of the family and satisfy Equation 2 lie on the envelope.
The condition for the singular points of a surface is discussed here.. a parametric representation of a surface is
(3)
where u1 and u 2 are the parameters of the surface. A point of the surface that corresponds to
in a given parameterization is called a singular point of the parameterization. A point of a surface is called singular if it is singular for every parameterization of the surface7. A point that is singular in one parameterization of a surface may not be singular in other parameterizations.
For a fixed value of ζ, equations 1 and 2 represent, in general, a curve on the surface which corresponds to this value of the parameter. If this is not a line of singular points, the curve slso lies on the envelope. The surface and the envelope are tangent to each other along this curve. Such curves are called characteristic lines of the family7. they can be used to find the contact lines between the surfaces of the cylindrical cam and the follower.
THEORY OF ENVELOPES FOR DETERMINATION OF CYLINDRICAL-CAM PROFILES
On the basis of the theory of envelopes, the profile of a cylindrical cam can be regarded as the envelope of the family of follower surfaces in relative positions between the cylindrical cam and the follower while the motion of cam proceeds. In such a condition, the input parameters of the cylindrical cam serve as the family parameters. Because the cylindrical or conical follower surface can be expressed in parametric form without difficulty, the theory of envelopes for a 1-parameter of surfaces represented in parametric form (see equations 1 and 2) is used in determining the analytical equations of cylindrical-cam profiles. As stated in the last section, a check for singular points on the follower surface is always needed.
Figure 1a shows a cylindrical-cam mechanism with a translating conical follower. The axis which the follower translates along is parallel to the axis of rotation of the cylindrical cam. a is the offset, that is, the normal distance between the longitudinal axis of the cam and that of the follower. R and L are the radius and the axial length of the cam, respectively.
The rotation angle of the cylindrical cam is Ф2 about its axis. The distance traveled by the follower is s1 , which is a function of parameter Ф2 ,as follows:
(4)
The displacement relationship (see equation 4 ) for the translating follower is assumed to be given.
In figure 1b, the relative position of the follower when the follower moves is shown. The follower is in the form of a frustum of a cone. The semicone angle is α, and the smallest radius is r. δ1 is the height, and μ is the normal distance from the xz plane to the base of the cone. The fixed coordinate system Oxyz is located in such a way that the z axis is along the rotation axis of the cam, and the y axis is parallel to the longitudinal axis of the conical follower. the unit vectors of the x axis, y axis and z axis are i , j and k, respectively.
By the use of the envelope technique to generate the cylindrical-cam profile, the cam is assumed to be stationary. The follower rotates about the dam axis in the opposite direction. It is assumed that the follower rotates through an angle Ф2 about the axis. At the same time, the follower is transmitted a linear displacement s1 by the cam, as shown in Figure 1b. Consequently using the technique, if we introduce θ and δ as two parameters for the follower surface, the family of the follower surfaces can be described as
(5)
where
0≤θ＜2pi
And ф2 is the independent parameter of the cam motion.
Referring to theory of envelopes for surfaces represented in parametric form (see equations 1and 2), we proceed with the solving process by finding
(6)
There are no singular points on the family of surfaces, since(r +δtanα) ＞0 in actual applications. The profile equation satisfies equation 5 and the following equation:
(7)
Where or
(8)
Where
Substituting equation 8 into equation 5, and eliminating θ, we obtain the profile equation of the cylindrical cam with a translating conical follower, and denote it as
(9)
As shown in equation 8,θ is a function of the selected follower-motion program and the dimensional parameters. As a consequence, the cylindrical-com profile can be controlled by the chosen follower-motion curves and the dimensional parameters. Two values of θ correspond to the two groove walls of the cylindrical cam.
Now the profile of the cylindrical cam with a translating conical follower is derived by the new proposed method. As stated above, Dhande et al.1 and Chakraborty and Dhande2 have derived the profile equation of the same type of cam by the method of contact points. A comparison of the result is carried out here. Since the same fixed coordinate system and symbols are used, one can easily see that the profile equation is identical although the methods used are different. Moreover, we find that the process of finding the cam profile is significantly reduced by this method.
CONTACT LINE
At every moment, the cylindrical cam touches the follower along space lines. The contact lines between the cylindrical cam and its follower are discussed in this section.
The concept of characteristic lines in the theory of envelopes for a 1-parameter family of surfaces mentioned above could be applied to finding the contact lines in a cylindrical com. The profile of a cylindrical cam with a translating conical follower is given by Equation9. Then, the contact line at a specific value of ф2, say ф20, is
(10)
Where, in Equation 10, the value of θ is a function of δ defined by Equation 8.
The contact lines between the surfaces of the cam and the follower at each moment is determined by Equation 10. we see that the relationship between the two parameters θ and δ of the follower surface is given by Equation 8, a nonlinear function. Thus, one can easily find that the contact line is not always a straight line on the conical follower surface.
PRESSURE ANGLE
The angle that the common normal vector of the cam and the follower makes with the path of the follower is called the pressure angle12. the pressure angle must be considered when designing a cam, and it is a measure of the instantaneous force-transmission properties of the mechanism13. The magnitude of the pressure angle in such a cam-follower system affects the efficiency of the cam. The smaller the pressure angle is, the higher its efficiency because14.
In figure2, the unit normal vector which passes through the point of contact between the cylindrical cam and the translating conical follower in the inversion position, i.e. point C, is denoted by n. The path of the follower labeled as the unit vector p is parallel to the axis of the follower. from the definition, the pressure angle Ψ is the angle between the unit vectors n and p.
Since, at the point of contact, the envelope and the surface of the family possesses the same tangent plane, the unit normal of the cylindrical-cam surface is the same as that of the follower surface. Referring to the family equation Equation5 and Figure2, we can obtain the unit vector as
(11)
where the value of θ is given by equation 8, and the unit vector of the follower path is
(12)
By the use of their inner product, the pressure angle Ψ can be obtained by the following equation:
(13)
The pressure angle derived here is identical to that used in the early work carried out by Chakraborty and Dhande2.
CUTTER PATH
In this section, the cutter path required for machining the cylindrical cam with a translating conical follower is found by applying the procedure described below.
Usually, with the considerations of dimensional accuracy and surface finish, the most convenient way to machine a cylindrical cam is to use a cutter whose size is identical to that of the conical roller. In the process of machining, the cylindrical blank is held on a rotary table of a 4-axis milling machine. As the table rotates, the cutter, simulating the given follower-motion program, moves parallel to the axis of the cylindrical blank. Thus the cutter moves along the ruled surface generated by the follower axis, and the cam surface is then machined along the contact lines step by step. If we have no cutter of the same shape, an available cutter of a smaller size could also be sued to generate the cam surface. Under the circumstances, the cutter path must be found for a general endmill cutter. Figure 3 shows a tapered endmill cutter machining a curved surface. The front portion of the tool is in the form of a cone. The smallest radius is R, and the semicone angle is β.
If the cutter moves along a curve δ =δ0 on the surface X=X（δ，ф2），the angle σ between the unit vector of the cutter axis ax and the unit common normal vector n at contact point C is determined by
(14)
Thus the path of the point ό on the cutter axis that the vector n passes through is
(15)
and the tip centre T follows the path
(16)
Figure 4 shows a tapered endmill cutter machining the groove wall of a cylindrical cam. The axis of the tapered endmill is parallel to the y axis. Note that the two conditions
（17）
(18)
for the geometric parameters of the cutter and the roller follower must hold, or otherwise the cutter would not fit the groove. The unit vector of the cutter axis is
(19)
For the profile of the cylindrical cam with a translating conical follower given by equation 9, the angle σ is determined by the inner product:
(20)
Thus, by using the results obtained earlier, the position of the tip centre of the cutter can be derived as
(21)
where
NUMERICAL EXAMPLE
The procedures developed are applied in this section to determine the
展开阅读全文
浙ICP备2021020529号-1 | 浙B2-20240490 
