Design of Cam Mechanisms with Swinging Roller Follower: The Modern Algorithm-Based Approach

Abstract

Modern mechanical engineering and design of the mechanisms and machines are closely connected with development of numerical methods and algorithms. Education is the useful target to apply these methods. Development of the analytical and algorithmic competences is possible only using high technologies through whole course. The modern numerical method on the metric synthesis of cam mechanism is demonstrated in this paper using new algorithmical approach and MathCAD realization.

1 Introduction

In Bauman Moscow State Technical University the generalized methods of design and exploration of machines are expounded within Theory of Mechanisms and Machines (TMM) course [2, 12]. In the past the approximate graphical methods [19, 20] were mostly used. Now the available calculation resources made it possible to use almost any method [3], including numerical and analytical [16].

The followers of cam mechanisms move under strictly determined trajectory [11] and motion laws [4]. They require precise calculation of the profile’s coordinates for successful manufacturing [17, 18]. The modern calculation software products like MathCAD [14, 15] provide such capabilities. They are easy to use, so development and realization of new approaches and algorithms are good challenges for engineering students [7, 10]. Here and below the typical cam mechanism with swinging follower and four-phased (rise, return and two dwellings) cycle will being considered as an example.

Fig. 1.

Cam mechanism and follower’s motion law

2 Target Setting

The mechanism (Fig. 1a) is constrained with following parameters:

• Maximal displacement $h_B$ of the follower’s contact point B which determines rising angle $\beta$;

• Length $l_2$ of the rocker 2;

• Phase angles: rise angle $\phi _{1y}$, return angle $\phi _{1c}$ and dwelling angle $\phi _{1\partial }$;

• Motion law of the follower defined graphically;

• Upper limit of the pressure angle $\left[ \vartheta \right]$;

• Rotation direction defined with sign of the $\omega _1$ value.

The whole task of designing of the cam mechanism can be splitted into three stages [8]:

• 1. Building of kinematical diagrams and calculating of geometrical characteristics;

• 2. Metrical synthesis constrained by $\left[ \vartheta \right]$;

• 3. Kinematical synthesis—building of theoretical cam profile and envelope surface.

The whole motion law of the follower for the considered mechanism is presented on Fig. 1b. The motion law here defined with coefficients $a_1$, $a_2$, $a_3$, $a_4$, scales $\mu$ and ratio of their values.

3 Kinematical Diagrams

According to Fig. 1b the acceleration analog function $a_{qB}$ has discontinuity points of type I at $f_3$, $f_4$ and $f_5$. To satisfy requirements of algorithmical approach the Haeviside function $\varPhi \left( x\right)$ [9] used to determine discontinuities. Now the following expressions can be written for the followers’s motion law on the rising phase:

$$\begin{aligned} a_{qBy}(\varphi ) = {\left\{ \begin{array}{ll} \frac{\varphi }{\varphi _1} \sin \left( \pi ,\frac{\varphi }{f_1}\right) ,\text { where } 0 \le \varphi< f_1 \\ \frac{-\pi }{f_2 - f_1} \sin \left[ \pi \frac{f_2-\varphi }{f_2 - f_1}\right] ,\text { where } f_1 \le \varphi \le f_2 \\ 0,\text { where } f_2 < \varphi \le f_3 \end{array}\right. } \end{aligned}$$

(1)

... and on returning phase where $f_3 < \varphi \le f_5$ (Fig. 1b):

$$\begin{aligned} a_{qBc}(\varphi ) = \left[ -a_3 \cdot \varPhi \left( \varphi - f_3 \right) + (a_3 + a_4) \cdot \varPhi \left( \varphi - f_4\right) - a_4 \cdot \varPhi \left( \varphi - f_5 \right) \right] \end{aligned}$$

(2)

Now the kinematical characteristics: speed analog function $v_{qB}(\varphi )$ and follower displacement function $S_B(\varphi )$ could be obtained.

4 Metric Synthesis

The main constraint of the metric synthesis task is the pressure angle limitation $\left[ \vartheta \right]$. The formalities for algorithmical approach are the following:

• 1. All schemes considered as they are built in right Cartesian coordinates $S_{xOy}$ with origin placed into the fixed point $O_1$ of the cam (Fig. 1a).

• 2. Rotation angle $\varphi$ of the cam directed conterclockwise is generalized coordinate;

• 3. Angular velocity analog function $\omega _{q1}$ of the cam determines rotation direction (1 for counterclockwise direction).

Let the cam 1 (Fig. 1a) rotate and the follower 2 moves by arbitrary trajectory and vector $\overline{v_B} = \overline{v_2}$ of the contact point B’s absolute speed is known. Let the Y axis of the $S_{xOy}$ coordinate system with origin in $O_1$ is parallel to $\overline{v_2}$. Now the pressure angle can be defined as the following function:

$$\begin{aligned} \tan \vartheta (\varphi ) = \frac{v_{qB}(\varphi ) - x_B(\varphi )}{y_B(\varphi )} \end{aligned}$$

(3)

Now we reinterpret (3) for case of swinging follower:

$$\begin{aligned} \tan \vartheta (\varphi ) = \frac{\omega _{q1} \cdot v_{qB}(\varphi ) - l_2 + a_w\cos \left( \varphi _{20} + \varphi _2(\varphi )\right) }{a_w\sin \left( \varphi _{20} + \varphi _2(\varphi )\right) } \end{aligned}$$

(4)

where

$$\begin{aligned} \cos \varphi _{20} = \frac{l_2^2 + a_w^2 - r_0^2}{2l_0 \cdot a_w} \end{aligned}$$

and $\varphi _{20}$ is initial angle [5] between $l_2$ and $O_1 O_2$ line at the lower dwelling phase (Fig. 1a).

With known dependence between $\vartheta$ and $\varphi$ the phase diagram $S_B\left[ v_{qB}(\varphi )\right]$ can be built [17] in right Cartesian coordinate system $S_{xAy}$ with origin in A point which coincides with center $O_2$ of the rocker (Fig. 1). On the diagram (Fig. 2) the $O_1$ and $O_2$ points demonstrate possible positions of the cam center [5]. The metric synthesis of the cam mechanism includes obtaining values for $a_w$ and $r_0$ parameters with known $l_2$ and $\left[ \vartheta \right]$. With known direction of rotation of the cam the upper limit of the pressure angle [6, 12] can be determined on rising phase $\left[ \vartheta _y \right]$, returning phase $\left[ \vartheta _c \right]$ or on both phases for reversible mechanism. Now the lines defined by $\left[ \vartheta \right]$ have to be determined in $S_{xAy}$. If coordinates of i-point of the diagram are $X_i$, $Y_i$ the line passing this point is defined as $Y_i = k_i \cdot X_i + b_i$ equation. The extreme position of such line is determined on the rising phase with angle $\varphi _2(\varphi ) + \pi / 2 - \left[ \vartheta _y \right]$ and on returning phase with angle $\varphi _2(\varphi ) + \pi / 2 + \left[ \vartheta _c \right]$. Now:

$$\begin{aligned} \begin{aligned} k_{yi}&= \tan \left( \varphi _2(\varphi ) + \pi / 2 + \omega _{q1}\left[ \vartheta \right] \right) \\ k_{ci}&= \tan \left( \varphi _2(\varphi ) + \pi / 2 - \omega _{q1}\left[ \vartheta \right] \right) \end{aligned} \end{aligned}$$

(5)

Fig. 2.

Phase diagram

The intersection point $O_i$ between lines on Fig. 2 is defined by equation $k_y \cdot X + b_y = k_c \cdot Y + b_c$. After substitutions we have:

$$\begin{aligned} \begin{aligned} X&= -\frac{b_c - b_y}{k_c - k_y} \\ Y&= \frac{k_c b_y - k_y b_c}{k_c - k_y} \end{aligned} \end{aligned}$$

(6)

Now the point $O_1$ with coordinates $X_1$, $Y_1$ is placed where the lines defined by (6), maximal and minimal values of $v_{qB}(\varphi )$ intersect. For each point of the diagram the line could be drawn through $O_1$:

$$\begin{aligned} X_i = \frac{Y_1 - b_i}{k_i} - X_1 \end{aligned}$$

(7)

Now the function $f(x) = X_i(\varphi )$ can be obtained and the lines which constrain the zone of possible location for the cam center are placed above Ax axis for rising phase and below it for returning. The algorithm of cam center point calculation is now determined within the three stages:

• Extremum analysis of the function $X_i(\varphi )$ for coordinates $\varphi _y$ and $\varphi _c$ of the most distant lines (Fig. 2);

• Substitution of $\varphi _y$ and $\varphi _c$ into (6) for $X_2$ and $Y_2$. Now the levels $y_1$ and $y_2$ can be placed;

• Obtaining of the pressure angle function $\vartheta (\varphi )$ from (4).

Realization of the considered approach were performed by I. Safronoff using MathCAD. Here the fragment of the source code¹ is presented:

Initial approximations: left extremum

Right extremum:

Intersection point:

$a_w$ and $R_{min}$:

Fig. 3.

Phase diagram obtained using MathCAD and cam profile with coordinate systems applied

Phase diagram obtained from MathCAD is presented on Fig. 3a. The coordinate system described in Sect. 4 is presented on Fig. 3b as $x_K O y_K$ axis. The coordinate system $x_T A y_T$ is connected to the follower. Angle $\alpha _{\partial }$ represents pressure angle $\vartheta$ from Fig. 2. From phase diagram the minimal cam radius $R_{min}$ now can be obtained.

5 Kinematical Synthesis

Matrix equation of coordinate transform between cam and follower (Fig. 3) is presented below. The transition performs over coordinate systems $S_{x_T A y_T}$ and $S_{x_{KC} A y_{KC}}$:

$$\begin{aligned} \left[ A_{KcT}(\varphi ) \right] = \left[ A_{KcK} \right] \cdot \left[ A_{KK^*}(\varphi ) \right] \cdot \left[ A_{K^* K_0} \right] \cdot \left[ A_{K_0 B} \right] \left[ A_{BT}(\varphi ) \right] \end{aligned}$$

(8)

The final equation of theoretical cam profile with argument $\varphi$ looks as:

(9)

where $l_2$ is rocker length (Fig. 1a). MathCAD realization of matrix calculations described here is obtained by the following code for initial circle $R_0$:

theoretical cam profile:

...and for the envelope curve:

The final cam profiles calculated and drawn in MathCAD are presented on Fig. 4. The dotted lines represent initial circle $R_{K0}$ and theoretical profile from Fig. 3. The solid ones represent envelope curve (manufacturing-ready profile) and minimal radius $R_K$. The roller radius $r_p$ is fixed [11]. The algorithm developed by I. Safronoff allows to easily connect the numerical methods which MathCAD implements to both highly-constrained calculation task of the metric synthesis and geometrical task of profiling.

Fig. 4.

Cam profiles drawn by MathCAD

6 Conclusion

The algorithm-based approach demonstrated above allows the student to use his skills of software development and toolchain building. In Bauman University it links traditional design with new techniques. Theoretical knowledge of the theory of mechanisms and machines provided with lectures could be successfully applied by student using continuous integration with modern software and technologies [1]. The case of MathCAD usage quoted in this paper allows to export data into simple format supported by manufacturing solutions using available built-in functions. In the modern system of engineering education the scripts, datasheets and programs developed by students themselves is the way to provide possibility to obtain industrial experience within training. Also the networks and Web-based programming techniques [13] allow university to build open educational space with free distributed workflow.