Radius of curvature analysis of oscillating follower cylindrical cam’s expansion pitch curve

Abstract

The currently used method to calculate radius of curvature of a cylindrical cam’s planar expansion pitch curve produces errors. Therefore, there is an urgent need to find an accurate method for cylindrical cam design. This research utilizes the three-dimensional expansion curve coordinates of the follower’s motion track and establishes the planar expansion pitch curve equation with the consideration of deviation angle, which can be easily used to analyze and design an oscillating follower cylindrical cam. The deduced equation of radius of curvature is based on motion law of the mechanism and prime cylinder radius of the cam. The relation curve between minimum radius of curvature and prime cylinder radius, which takes into consideration that planar expansion pitch curve varies with prime cylinder radius in the event of certain motion law, is described by MATLAB, and specific minimum radius of prime cylinder is generated for each allowable radius of curvature according to the relation curve. Thus, geometric parameters of a cylindrical cam can be designed. The feasibility of our methodology is validated with a specific example.

Introduction

Cylindrical cam mechanisms are common in machinery field. Compared with other mechanisms, cylindrical cam mechanisms have the advantages of small size, compact structure, good rigidity, high driving torque, and so on. The pitch curve of a cylindrical cam is a complex spatial curve, so that it is hard to design the pitch curve of it based on the motion law of the follower. In general, for the sake of simplifying design and manufacture process, a design of a cylindrical cam is to expand the surface of the cam and calculate the planar expansion pitch curve coordinates and also determine geometric parameters of the cam according to the engineering conditions such as allowable pressure angle and radius of curvature.

When designing an oscillating follower cylindrical cam mechanism, it is highly significant to calculate the minimum radius of curvature ${\rho }_{\min }$ of planar expansion pitch curve, both convex portion and concave portion, given the motion law between the rotation angle of the cam and the swing angle of the swing link. If the ${\rho }_{\min }$ is too small, it will be difficult to select the cutters and the roller. This is because, first, the radius of the roller $R_r$ should be smaller than ${\rho }_{\min }$ of convex portion to avoid the presence of cusps in the cam profile; ¹ second, the radius of the cutter to process the profile of the cam $R_T$ should be smaller than ${\rho }_{\min }$ of concave portion to avoid interference between the cutter and the cam profile ² and to reduce cutting force and violent vibration as well. However, if the ${\rho }_{\min }$ is too big, it will lead to an oversized cam mechanism, which implies large inertia forces that may result in high contact forces and stresses between the cam and the follower. ³ Therefore, there is an urgent need to find an appropriate method to calculate the radius of curvature of cylindrical cam’s planar expansion pitch curve. In addition, the minimum radius of curvature ${\rho }_{\min }$ relates to the prime cylinder radius of the cylindrical cam $R$ . Thus, it is also essential to determine the relation curve between ${\rho }_{\min }$ and $R$ .

Scholars all over the world have conducted a lot of researches on cylindrical cam and proposed many methods to calculate the planar expansion pitch curve coordinates as well as its radius of curvature. For example, Liu et al. ⁴ deduced the contact line equation of cutter surface revolution and workpiece spiral surface which have universal meaning, strong operability, and good use for reference value for the analysis of other cams. Hidalgo-Martínez ⁵ discussed the application of Bézier curves for designing cams and proposed a numerical method for optimizing the design of the cam profile using a Bézier ordinate as an optimization parameter. This kind of cam–follower mechanisms is recommended in applications where the space is restricted and very high forces are involved. Hung and Lai ⁶ presented a system for the design and manufacturing of cylindrical cams to cut the roller groove using small-sized tools. Hsieh and Lin ⁷ deduced profile equations of spatial cam through matrix transformation, according to pure rolling condition of the roller on the pitch curve of cam. Hsieh ⁸ and Grant and Soni ⁹ described theoretical scheme for the process of cylindrical cam. Thinh and Joong ¹⁰ proposed a method of designing flexible cam profiles using smoothing spline curves. However, with the consideration of practical engineering application, it is still necessary to find a method which has the virtues of uncomplicated, high precision and obvious geometric property. Shi and Wu ¹¹ and Norton ¹² discussed the techniques of plotting the planar expansion pitch curve, obtaining its coordinate equations, and deducing the radius of curvature. However, these methods generated errors when dealing with the offset between the trace point and the rotating axis of cam because they incorrectly assumed that the moving locus of the trace point on the expansion surface is an arc. Three-dimensional (3D) expansion method is an error-free method to expand the pitch curve of a cylindrical cam, which was proposed by Yong, ¹³ Chen and Wu, ¹⁴ and so on. However, they have not performed further research on the radius of curvature.

The aim of this research is to establish the planar expansion pitch curve equation with the consideration of deviation angle by analyzing the 3D expansion curve coordinates of the follower’s motion track and thus to propose an accurate design method for cylindrical cam with oscillating follower.

Existed errors in commonly used methods

A cylindrical cam mechanism is composed of a cylindrical cam and a follower which is driven by the rotary motion of the cam. Followers can be divided into two categories, namely, translating follower and oscillating follower. Cylindrical cam mechanisms with translating follower are always designed with graphic methods, which have been studied well these years and will not be discussed here. We will mainly analyze the design errors of cylindrical cams with oscillating follower caused by “Δ,” as shown in Figure 1.

Figure 1.

Cylindrical cam with oscillating follower.

The frequently used method to analyze an oscillating follower cylindrical cam is called plane expansion method. To do this, one should first expand the cylindrical surface of the cam to a rectangular plane and then calculate the pressure angle, radius of curvature, and so on of the planar expansion pitch curve. This method is feasible when the axis of the follower and the axis of the cam intersect or, in other words, when the axis of the follower is perpendicular to the surface of the cam. As for oscillating follower cylindrical cam mechanism, when we consider the arc motion of the follower, there is an offset between the follower and the axis of cylindrical cam, so that the axis of the follower is not perpendicular to the surface of the cam. In this case, the plane expansion method would produce errors. ¹⁵

Shi and Norton discussed the expansion of cylindrical cam pitch curve in accordance with its prime cylinder; details are shown in Figure 2. $B_0$ is the starting point of the trace point, which is a point on the follower that corresponds to the contact point of a fictitious knife edge follower ¹⁶ and the center of roller in this case, when both the rotation angle of the cam and the swing angle of the swing link are zero. $\Delta$ is the distance between the trace point and the axis of cylindrical cam. $B_1$ and $B_2$ are two positions of the trace point at the moments when the rotation angles of cylindrical cam are ${\phi }_1$ and ${\phi }_2$ , and the swing angles of the swing link are ${\psi }_1$ and ${\psi }_2$ . $\Delta \prime$ is the distance between the trace point and the axis of cylindrical cam at the position $B_2$ . The radius of the arc on the expansion surface equals the length of the swing link $l$ (as shown in Figure 2).

Figure 2.

Expansion pitch curve in accordance with prime cylinder.

The moving locus of the trace point is arc, which is projected on the surface of prime cylinder, as shown in Figure 3. Thus, we can see that the locus, which is expanded in accordance with prime cylinder shown in Figure 2, is definitely not an arc. The distance between the trace point and the axis of cylindrical cam is $\Delta \prime$ as shown in Figure 1. Also, the distance on the expansion surface in accordance with prime cylinder, which corresponds to the expansion length of the arc $B_2{A}_2$ , is obviously longer than $\Delta \prime$ , and the error is proportional to the distance between the trace point and the axis of cylindrical cam. Thus, the expansion method introduced in Shi and Wu ¹¹ and Norton ¹² would generate errors, and therefore, it is not suitable for calculating radius of curvature of the planar expansion pitch curve.

Figure 3.

Diagram of the corresponding arc.

Design oscillating follower cylindrical cam’s planar expansion pitch curve and determine its radius of curvature

Here, we will assess the working processes and working conditions of cylindrical cam mechanisms. According to the motion law of relationship between the rotation angle of the cam $\phi$ and the swing angle of the swing link $\psi$ , the 3D expansion curve of motion track of the follower can be obtained. The radius of curvature of each point can be calculated based on the planar expansion curve equation. Thus, the relation curve between the minimum radius of curvature and the prime cylinder radius can be drawn by MATLAB.

The 3D expansion curve of follower’s motion track

The follower moves through the groove, driven by the rotary motion of the cylindrical cam. The circular motion of the follower in the rotation plane can be divided into two orthogonal linear motions. So, the overall motion of a cylindrical cam mechanism is composed of the rotation of the cam and two linear motions of the follower, which is a spatial motion. ¹⁷ Moreover, the spatial motion of cylindrical cam with oscillating follower can be transformed into a surface motion. The surface is a cylinder whose radius is equal to the length of swing link, called 3D expansion surface, ¹⁸ as shown in Figure 4. According to the motion law, the relationship between the swing angle of the swing link and the rotation angle of cylindrical cam can be defined as equation (1)

$$\psi =f\left(\phi \right)$$

(1)

Figure 4.

The 3D expansion surface of cylindrical cam.

According to equation (1), corresponding points can be found on the 3D expansion surface if it is picked up by random points on the pitch curve, which is the path of the trace point, to form the expansion pitch curve. The system equations of expansion pitch curve in Cartesian coordinates are shown in equation (2)

$$\{\begin{array}{c}\hfill X=a-l\times \cos \psi \hfill \\ \multicolumn{1}{c}{Y=\phi }\\ \multicolumn{1}{c}{Z=l\times \sin \psi }\end{array}$$

(2)

Equation of the planar expansion pitch curve

In the motion of an oscillating follower cylindrical cam mechanism because of the offset between the trace point and the axis of cylindrical cam, the center line (Y-axis) and the line connected by the trace point with the center of the prime cylinder on the surface of prime cylinder form an angle, which is called deviation angle $\delta$ . ¹⁹ The expression of $\delta$ is shown in equation (3)

$$\delta =\arcsin \frac{-x}{R}=\arcsin \frac{l\times \cos \psi -a}{R}$$

(3)

$\delta$ is positive when it deviates from the center line along the cylindrical cam’s rotation direction. $\delta$ is negative when it deviates from the center line against the cylindrical cam’s rotation direction. When the rotation angle of cylindrical cam is $\phi$ , the above-mentioned intersection point rotates behind $\phi$ by an angle $\delta$ . If we define the circumference angle as the angle between the intersection point and the starting point of the cylindrical cam, the circumference angle at that moment is $\phi -\delta$ . With due consideration of deviation angle $\delta$ , we can obtain equation (4) if the pitch curve is expanded on YZ plane according to the circumference angle of cam. Thus, the expansion pitch curve is transformed into a planar curve

$$\{\begin{array}{c}\hfill Y\prime =\phi -\arcsin \frac{l\times \cos \psi -a}{R}\hfill \\ \multicolumn{1}{c}{Z=l\times \sin \psi }\end{array}$$

(4)

According to proportional transformation of the circumference length, equation (4) is transformed into equation (5) to make the units identical

$$\{\begin{array}{c}\hfill x=R(j-\arcsin \frac{l\times \cos y-a}{R})\hfill \\ \multicolumn{1}{c}{y=l\times \sin y}\end{array}$$

(5)

Expression of the radius of curvature

We assume that parametric equation

$$\{\begin{array}{c}\hfill x=x\left(\phi \right)\hfill \\ \multicolumn{1}{c}{y=y\left(\phi \right)}\end{array}$$

(6a)

is second differentiable, then

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\frac{\mathrm{d}y}{\mathrm{d}\phi }}{\frac{\mathrm{d}x}{\mathrm{d}\phi }},\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=\frac{\frac{\mathrm{d}x}{\mathrm{d}\phi }.\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{\phi }^2}-\frac{\mathrm{d}y}{\mathrm{d}\phi }.\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{\phi }^2}}{{\left(\frac{\mathrm{d}x}{\mathrm{d}\phi }\right)}^3}$$

(6b)

Thus, we can deduce the expression of the radius of curvature as follows

$$\rho =\frac{{[1+{\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)}^2]}^{\frac{3}{2}}}{\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}}=\frac{{[{\left(\frac{\mathrm{d}x}{\mathrm{d}f}\right)}^2+{\left(\frac{\mathrm{d}y}{\mathrm{d}f}\right)}^2]}^{\frac{3}{2}}}{\frac{\mathrm{d}x}{\mathrm{d}f}\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{f}^2}-\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{f}^2}\frac{\mathrm{d}y}{\mathrm{d}f}}$$

(7)

As for the expression of the planar expansion pitch curve, according to equation (5)

$$\frac{\mathrm{d}x}{\mathrm{d}\phi }=R\{1+l\times \sin \psi \frac{\mathrm{d}\psi }{\mathrm{d}\phi }{[R^2-{(l\times \cos \psi -a)}^2]}^{-\frac{1}{2}}\}$$

(8a)

$$\begin{array}{cc}\multicolumn{1}{c}{}& \frac{{\mathrm{d}}^{2}x}{\mathrm{d}{\phi }^2}=R\{[l\times \cos \psi {\left(\frac{\mathrm{d}\psi }{\mathrm{d}\phi }\right)}^2+l\times \sin \psi \frac{{\mathrm{d}}^2\psi }{\mathrm{d}{\phi }^2}]\hfill \\ \multicolumn{1}{c}{}& {[R^2-{(l\times \cos \psi -a)}^2]}^{-\frac{1}{2}}+(l\times \cos \psi -a)l\hfill \\ \multicolumn{1}{c}{}& \times \sin \psi \frac{\mathrm{d}\psi }{\mathrm{d}\phi }{[R^2-{(l\times \cos \psi -a)}^2]}^{-\frac{3}{2}}\}\hfill \end{array}$$

(8b)

$$\frac{\mathrm{d}y}{\mathrm{d}\phi }=l\times \cos \psi \frac{\mathrm{d}\psi }{\mathrm{d}\phi }$$

(8c)

$$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{\phi }^2}=-l\times \sin \psi {\left(\frac{\mathrm{d}\psi }{\mathrm{d}\phi }\right)}^2+l\times \cos \psi \frac{{\mathrm{d}}^2\psi }{\mathrm{d}{\phi }^2}$$

(8d)

It is known that a convex portion of the cam pitch curve has a positive $\rho$ , while a concave portion has a negative $\rho$ . For the determination of minimum $\rho$ , the expression of radius of curvature can be obtained as its absolute value

$$\begin{array}{cc}\multicolumn{1}{c}{\rho }& =\frac{{[{\left(\frac{\mathrm{d}x}{\mathrm{d}\phi }\right)}^2+{\left(\frac{\mathrm{d}y}{\mathrm{d}\phi }\right)}^2]}^{\frac{3}{2}}}{|\frac{\mathrm{d}x}{\mathrm{d}\phi }·\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{\phi }^2}-\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{\phi }^2}·\frac{\mathrm{d}y}{\mathrm{d}\phi }|}\hfill \\ \multicolumn{1}{c}{}& =\frac{{\{R^2{\{1+l\times \sin \psi \frac{\mathrm{d}\psi }{\mathrm{d}\phi }{[R^2-{(l\times \cos \psi -a)}^2]}^{-\frac{1}{2}}\}}^2+l^2\times {\cos }^2\psi {\left(\frac{\mathrm{d}\psi }{\mathrm{d}\phi }\right)}^2\}}^{\frac{3}{2}}}{R|-l\times \sin \psi {\left(\frac{\mathrm{d}\psi }{\mathrm{d}\phi }\right)}^2+l\times \cos \psi \frac{{\mathrm{d}}^2\psi }{\mathrm{d}{\phi }^2}-l^2{\left(\frac{\mathrm{d}\psi }{\mathrm{d}\phi }\right)}^3{[R^2-{(l\times \cos \psi -a)}^2]}^{-\frac{1}{2}}+(l\times \cos \psi -a)l^2\sin \psi \cos \psi {\left(\frac{\mathrm{d}\psi }{\mathrm{d}\phi }\right)}^2{[R^2-{(l\times \cos \psi -a)}^2]}^{-\frac{3}{2}}|}\hfill \\ \multicolumn{1}{c}{}\end{array}$$

(9)

Relation curve between the minimum radius of curvature and the prime cylinder radius

In equation (9), $f$ is the function of $\phi$ , $f\prime$ is the derivative of $f$ with respect to $\phi$ , $f″$ is the second derivative of $f$ , and $l$ and $a$ are constants. The minimum radius of curvature ${\rho }_{\min }$ can be calculated when $R$ is known. Furthermore, the corresponding ${\rho }_{\min }$ can be calculated when the value of $R$ varies, and thus, the relation curve between the value of $R$ and the value of minimum radius of curvature can be plotted.

An example for design using this newly developed method

Let us consider an example to compare the new method proposed above with the former method. In this example, we will calculate the equations of coordinates of planar expansion pitch curve with the two different methods in same conditions and give the contrast analysis of the minimum prime cylinder radius.

We assume that the equation of the follower’s motion law $f(\phi )$ is as equation (10), which is shown in Figure 5

$$\begin{array}{cc}\multicolumn{1}{c}{}& \psi =f\left(\phi \right)=\hfill \\ \multicolumn{1}{c}{}& \{\begin{array}{cc}\hfill f_1\left(\phi \right)=-10\cos \left(2\phi \right)\hfill & \hfill (0\le \phi <90°)\hfill \\ \multicolumn{1}{c}{f_2\left(\phi \right)=10}& \hfill (90°\le \phi <180°)\hfill \\ \multicolumn{1}{c}{f_3\left(\phi \right)=10\sin (2\phi +90°)}& \hfill (180°\le \phi <270°)\hfill \\ \multicolumn{1}{c}{f_4\left(\phi \right)=-10}& \hfill (270°\le \phi <360°)\hfill \end{array}\hfill \end{array}$$

(10)

Figure 5.

Motion curve.

$\psi =f_1(\phi )$ is the equation of the curve of lift segment, and $\psi =f_3(\phi )$ is the equation of the curve of return travel segment. Considering that they are symmetrical, we take only $\psi =f_1(\phi )$ into consideration.

Thus, the relation curve between the minimum radius of curvature and the prime cylinder radius according to this motion law can be described by MATLAB. We can also obtain the value of minimum prime cylinder radius when allowable radius of curvature is 20 mm. The known conditions are $l=253\mathrm{mm}$ and $a=251\mathrm{mm}$ .

The relation curve between the minimum radius of curvature and the prime cylinder radius is shown in Figure 6, and the minimum prime cylinder radius is 59.4 mm when the allowable radius of curvature is 20 mm.

Figure 6.

Relation curve between the minimum radius of curvature and R (3D expansion).

According to equations (5) and (10), the system equations of planar expansion pitch curve are obtained, where R = 59.4 mm is the prime cylinder radius, l = 253 mm is the length of the swing link, and a = 251 mm is the displacement between the rotation axis of swing link and the axis of cylindrical cam

$$\{\begin{array}{cc}\multicolumn{1}{c}{}& x=\frac{59.4\pi }{180}(\phi -\arcsin \frac{253\cos f-251}{59.4})\hfill \\ \multicolumn{1}{c}{}& y=253\sin f\hfill \end{array}$$

(11)

The planar expansion pitch curve is shown in Figure 7.

Figure 7.

Planar expansion pitch curve.

By modifying allowable radius of curvature into some specific values in the programs, the corresponding minimum radii of prime cylinder are shown in Table 1.

Table 1.

Relation table between allowable radius of curvature and minimum R (3D expansion).

Allowable radii of curvature (mm)	10	15	20	25	30
Minimum radii of prime cylinder (mm)	42.0	51.5	59.4	66.5	72.8

If expanded according to Shi and Wu ¹¹ and Norton, ¹² the obtained system equations of coordinates of planar expansion pitch curve are

$$\{\begin{array}{c}\hfill x=a+R\phi +l\cos (\pi -\psi )\hfill \\ \multicolumn{1}{c}{y=l\sin (\pi -\psi )}\end{array}$$

(12)

According to equation (12), by deriving expression of the radius of curvature and writing MATLAB programs, the relation curve between the minimum radius of curvature and the prime cylinder radius can be plotted as Figure 8, and the minimum prime cylinder radius is 60.2 mm when the allowable radius of curvature is 20 mm.

Figure 8.

Relation curve between the minimum radius of curvature and R (plane expansion).

By modifying allowable radius of curvature into some specific values in the program, the corresponding minimum radii of prime cylinder are shown in Table 2.

Table 2.

Relation table between allowable radius of curvature and minimum R (plane expansion).

Allowable radii of curvature (mm)	10	15	20	25	30
Minimum radii of prime cylinder (mm)	42.6	52.1	60.2	67.2	73.6

The contrast of relation curves between the minimum radius of curvature and the prime cylinder radius based on these two methods is shown in Figure 9.

Figure 9.

Contrast of relation curves between the minimum radius of curvature and R.

We can see from Tables 1 and 2 and Figure 9 that the difference gradually increases between minimum radii of prime cylinder based on these two methods when allowable radius of curvature increases. When allowable radius of curvature reaches above 30 mm, the difference is 0.8 mm or more, so that the difference between the result obtained according to Hsieh ⁸ and Grant and Soni ⁹ and that obtained according to 3D expansion method is obvious.

Besides, it is worth mentioning that considering the error existed in plane expansion, a method is described in Grant and Soni ⁹ to correct the pitch cylinder radius $R$ for each position of the follower as

$$R\prime =\sqrt{R^2+{(l\cos \psi -a)}^2}$$

(13)

The derived system equations of coordinates of planar expansion pitch curve are

$$\{\begin{array}{c}\hfill x=a+\phi \sqrt{R^2+{(l\cos \psi -a)}^2}+l\cos (\pi -\psi )\hfill \\ \multicolumn{1}{c}{y=l\sin (\pi -\psi )}\end{array}$$

(14)

The relation curve between the minimum radius of curvature and $R$ obtained through this method approximates to the one obtained through 3D expansion method, the contrast of them are shown in Figure 10, which is enlarged in order to facilitate comparison.

Figure 10.

Contrast of relation curves between the minimum radius of curvature and R.

We can see from Figure 10 that when allowable radius of curvature is 30 mm, the difference between minimum radii of prime cylinder based on these two methods is less than 0.1 mm. Nevertheless, errors still cannot be eliminated through this correction. Because it reduces errors by broadening the motion distance of the rotation axis of swing link on the expansion surface, instead of polishing the difference in length between arc $B_2{A}_2$ and $\Delta \prime$ , which are shown in Figure 3.

Above all, in view of the virtues of uncomplicated and error free, the expression of radius of curvature of cylindrical cam’s planar expansion pitch curve, as well as relation curve between the minimum radius of curvature and $R$ obtained through 3D expansion method, is very effective to design a cylindrical cam with oscillating follower.

Conclusion

1. By analyzing the working process of cylindrical cam mechanisms, we have discussed the errors in the existing methods for the planar expansion pitch curve of cylindrical cam with oscillating follower and the calculation of its radius of curvature.

2. According to motion law of cylindrical cam mechanism, we have deduced error-free equations for surface expansion pitch curve and methods for calculating radius of curvature by applying 3D expansion formula of the follower’s motion track.

3. The relation curve between the minimum radius of curvature and the prime cylinder radius can be drawn by MATLAB, and the minimum prime cylinder radius can be determined when radius of curvature is greater than allowable value. The feasibility of our theory is validated by a specific example with comparison.