4.2 Camshafts 123 FIGURE 4.11 Mechanism for contour grinding. Views of contours produced when settings of parts 1 and 3 are changed. 4.2 Camshafts It is not always possible to satisfy the desired position function by means of the mechanisms discussed in the previous section. The requirements dictated by the timing diagram (see Chapter 2) vary, but they can often be met by using cam mechanisms. The idea underlying such mechanisms is clear from Figure 4.12a), in which a disc cam is presented schematically. A linearly moving follower has a roller to improve friction and contact stresses at the cam profile-follower contact joint. It is easy to see that, by rotating the cam from positions 0 to 11, the follower will be forced to move vertically in accordance with the radii of the profile. Graphical interpretation of the position function has the form shown in Figure 4.12b). During cam rotation through the angle 0! (positions 0-1-2-3-4-5) the follower climbs to the highest point; during the angle 0 2 (6-7-8-9-10-11) it goes down, and during the angle 0 3 the follower dwells (because this angle corresponds to that part of the profile where the radius is constant). Changing the profile radii and angles yields various position functions, which in turn produce different speeds and acceleration laws for the follower movements. Figure 4.13 illus- trates the cosine acceleration law of follower movement. The analytical description of this law is given by the following formulas: TEAM LRN 124 Kinematics and Control of Automatic Machines FIGURE 4.12 a) Disc cam mechanism; b) The follower motion law. To provide the desired sequence and timing of actions, it is convenient to mount all cams needed for the machine being designed on one shaft, thus creating a camshaft, for example, as shown in Figure 4.14. One rotation of this shaft corresponds to one cycle of the machine, and thus one revolution lasts one period or T seconds. As can be seen from Figure 4.14, a camshaft can drive some mechanisms by means of cams (mechanisms A, B, C, and D), some by cranks (mechanism E), and some by gears (mech- anism F). Sometimes a single straight shaft is not optimum for a given task. Then the solution shown in Figure 4.15 can be useful. Here, motor 1, by means of belt drive 2, drives camshaft 3, which is supported by bearings 4. A pair of bevel gears 5 drive shaft 6. The ratio of transmission of the bevel gears is 1:1; thus, both shafts complete one revolution in the same time and all cams and cranks (7 and 8) complete their tasks at the same time. (Figure 4.14a) shows a photograph of a specific camshaft controlling an automa- tive assembly machine serving the process of dripping irrigation devices production). FIGURE 4.13 Cosine acceleration law carried out by link driven by cam mechanism. TEAM LRN 4.2 Camshafts 125 FIGURE 4.14 Camshaft as a mechanical program carrier. Main camshaft. This solution is convenient when the tools handle the product from more than one side. However, this example also illustrates the disadvantage of mechanical types of kinematic solutions, namely, difficulties in transmission of motion from one point to another. For instance, Figure 4.16 shows what must be done to transform the rota- tional motion of shaft 1 into the translational motion of rack 2. (Note that the guides of follower 3, bearings of intermittent shaft 4, and guides of rack 2 are not shown, although they belong to the design and contribute to its cost.) FIGURE 4.l4a) General view of a camshaft serving a specific production machine for automative assembly of dripping irrigation units. (Netafim, Kibbutz Hatzerim, Israel.) TEAM LRN 126 Kinematics and Control of Automatic Machines FIGURE 4.15 Generalized concept of a main camshaft. FIGURE 4.16 Complexity of motion transforma- tions carried out by purely mechanical means; see text for explanation. To make the structure more flexible, some kinds of transmission can be adopted. Figure 4.17a) presents a ball transmission for very lightly loaded mechanisms. The action of this transmission is obvious—it is a rough model of a hydraulic transmission. For larger loads it is recommended that cylindrical inserts 1 be used between the balls. A purely hydraulic transmission (Figure 4.17b)) can also be used. BeUows 1 on the cam- follower's side transmits pressure through connecting pipe 3 to bellows 2, located on the tool side, and actuates the latter. A third possibility for transmitting motion in a flexible manner by mechanical means is shown in Figure 4.18. This device consists of guide 1 made of some flexible metal in which plastic ribbon 2, which possesses open- ings, is borne. The friction between the ribbon and the guide is reduced by proper FIGURE 4.17 Models of a "flexible" trans- mission of the camfollower's motion: a) Ball transmission; b) Hydraulic transmission. TEAM LRN 4.2 Camshafts 127 FIGURE 4.18 Flexible toothed rack for motion transmission. choice of materials and coatings, so that the guide can be folded in various ways and still provide satisfactory transmission of the motion. Cam mechanisms, being a kind of mechanical program carrier, operate under certain restrictions which must be known to the designer, together with the means to reduce the harm these restrictions cause. The main restriction is the pressure angle. This is the angle between the direction of follower movement and a line normal to the profile point in contact with the follower at a given moment. Figure 4.19 illustrates the situation at the follower-cam meeting point. The profile radius r= OA makes angle j with the direction of follower motion KA. Angle ft, between the tangent at contact point A and speed vector V al (perpendicular to the radius vector r) may be termed the profile slope. The same angle ft appears between radius vector r and the normal Nat contact point A. Thus, we can express the pressure angle a as follows: FIGURE 4.19 Pressure angle in cam mechanism. TEAM LRN 128 Kinematics and Control of Automatic Machines Calling the follower speed V a , we obtain from the sine law From Figure 4.19 we have where r 0 = the constant radius of the dwelling profile arc. Obviously, and Substituting (4.22) into (4.21) and taking into account (4.20), we can rewrite Expres- sion (4.21) as follows: or which gives From Figure 4.19 it follows that Thus, we obtain Remembering that we finally obtain, from (4.23) TEAM LRN 4.2 Camshafts 129 For the central mechanism, where e = 0, we obtain a simpler expression for (4.23), i.e.: or The larger the pressure angle a, the lower the efficiency of the mechanism. When this angle reaches a critical value, the mechanism can jam. The critical value of the pressure angle depends on the friction conditions of the follower in its guides, on the geometry of the guides, on the design of the follower (a flat follower always yields a = 0 but causes other restrictions), and on the geometry of the mechanism. To reduce the pressure angle, we must analyze Expressions (4.23) and (4.24). It follows from them that the pressure angle decreases as: 1. The value of a or r 0 increases; 2. The IT(0) function that describes the slope of the profile decreases. Taking advantage of the first conclusion is impractical since it involves enlarging the dimensions of the mechanism. Thus, we usually recommend use of the second conclusion, that is, to "spread" the profile over a wider profile angle. However, to stay within the limits determined by the timing diagram, we must increase the rotating speed of the cam. This can be done by introducing the concept of an auxiliary camshaft. (See Figure 4.20a)) The main camshaft 1 is driven by a worm reducer and controls three mechanisms by means of cams I, II, and III. Cam III has a special function, namely, to actuate the auxiliary camshaft. This shaft is driven by a separate motor 4 and belt drive 5. The latter brings into rotation one-revolution mechanism 6, which is controlled by FIGURE 4.20 Concept of an auxiliary camshaft, a) Mechanical layout; b) Timing diagram. TEAM LRN 130 Kinematics and Control of Automatic Machines cam III. This cam actuates follower 8 by pressing against spring 7. The one-revolution mechanism is then switched on, carries out one revolution, and stops. Obviously, the speed of shaft 3 can be considerably higher than the rotational speed of camshaft 1. Thus, cam iy which is mounted on shaft 3, completes its revolution much faster than those on shaft 1. The timing diagram shown in Figure 4.20b) is helpful here. According to this diagram, auxiliary shaft 3 rotates four times faster than main camshaft 1; that is, during one- quarter of a revolution of the main shaft, the auxiliary shaft completes a full revolu- tion, then rests until shaft 1 finishes its revolution. This makes it possible to design the profile of cam IV with a more gradual slope, thus reducing the pressure angle. In our example, the profile of cam IV is extended over an angle of 300°, providing the needed displacement s for the follower. Obviously, without the auxiliary shaft the same profile must extend over an angle less than 90°, and the pressure angle would be much larger. It is worthwhile to study the operation of the one-revolution mechanism. In Figure 4.21 we show a possible design of this mechanism, consisting of: permanently rotat- ing part 1 (in Figure 4.20 motor 4 and belt 5 drive this part); driven part 2, key 3, and stop 4. The rotating part is provided with a number of semicircular slots (say 6). The driven part 2 has one slot. As is clear from cross section A-A, in a certain position of key 3 (frontal view a)), part 1 can rotate freely around driven part 2 (i.e., key 3 does not hinder this rotation). However, when key 3 takes the position shown in frontal view b), parts 1 and 2 are connected and rotate together as one body. When not actuated, key 3 is usually kept in the disconnecting position by stop 4, which presses lever 3a of the key. When the command to actuate the mechanism is given (cam III actuates follower 8 in Figure 4.20), stop 4 is removed from its position, freeing lever 3a. Thus, key 3 rotates FIGURE 4.21 One-revolution mechanism: a) Disengaged state; b) Engaged state; c) Key. TEAM LRN 4.2 Camshafts 131 into the connecting position, due to spring 5, when the slots of part 1 permit this. From this moment on, parts 1 and 2 move together, as mentioned earlier. However, if, during this first revolution, stop 4 returns to its previous position (cam III in Figure 4.20 ensures this), then before the revolution is completed lever 3a meets the stop, rotates key 3 into the disconnecting position, and frees part 1 from part 2. Pins 6 restrict the angular motion of the key. This kind of one-revolution mechanism has extensive applications. At this point it is interesting to consider the problem of the flexibility of camshafts and other mechanisms for carrying out position functions in hardware. Indeed, from our description of the action of main and auxiliary camshafts, it would appear that, once designed, manufactured, and assembled, these mechanisms cannot be changed. This is, of course, true, and is completely adequate for "bang-bang" robotic systems (type 5 in Figure 1.5). However, there are ways of introducing some flexibility even into this seemingly stiff, purely mechanical approach. Figure 4.22 shows the design of cam 1 and shaft 2 together with special lock 3 which permits rapid cam change on the shaft. To some extent, such cam change is like reprogramming a programmable machine. Another design with the same purpose is shown in Figure 4.23a). Here, cam 1 is fixed on shaft 2 by means of nut 3. In Figure 4.23b) cam 1 is fixed by means of tooth-like coupling 2. There are other ways in which the position function may be realized in a relatively flexible way by mechanical means. For instance, the cam shown in Figure 4.24 is built so that profile piece 2 can be fastened by bolts 3 and 4 at any angle on the circular base 1, which has a circular slot (it can be moved along this slot), thus yielding a wide range of 5 values. Another example, shown in Figure 4.25, allows easy adjustment of the cam FIGURE 4.22 Arrangement for rapid cam exchange on a camshaft. FIGURE 4.23 Another arrangement for a) Cam exchange; b) Change of fixation angle. TEAM LRN 132 Kinematics and Control of Automatic Machines FIGURE 4.24 Arrangement for rapid cam profile exchange. FIGURE 4.25 Arrangement for rapid timing change. profile. Here the cam is composed of parts A and B, which can be fixed by bolts in dif- ferent relative positions. Use of a spatial cam introduces flexibility, as in Figure 4.26. Here, drum 1, having a system of openings, is fastened onto the shaft, while pieces 2 of profiles of appro- priate shapes and sizes are mounted on the drum by bolts. A diagram of how the pro- files are located on the drum surface is also shown. The spatial approach to cams offers additional solutions to the flexibility problem. The common feature of the examples illustrated in Figures 4.27-4.29 is the use of the third dimension, namely the z-axis, for adjusting the position function. To change the s value, one can use the solutions shown FIGURE 4.26 Spatial cam with an arrangement for rapid profile exchange. TEAM LRN [...]... following relationships for the damping forces and torques, respectively: The dissipated forces and torques are proportional to the linear and angular speeds, respectively For the mechanism described, the motion function relating the rotation of the drive shaft to the motion 5 of the follower link has the following form: Here e is the eccentricity of the cam (To simplify the example, the cam is circular... upon the position of the edge of the nozzle D relative to the inlets of the channels (The diameter of the nozzle output is about 0 .5 mm.) The air flow from the nozzle is divided by the partition dividing the channel's input and brought to valve ports 3 and 4, through ports 1 and 2, moving the piston in accordance with the pressure difference The subsequent action of the system is as described above The. .. of the valve The pressure entering the left volume of the cylinder causes a leftward movement and equivalent displacement of the slide valve housing The movement of this housing relative to the valve piston closes all ports and therefore stops the cylinder To continue the movement of the cylinder, the valve piston must again be displaced leftward, and so on To reverse the motion of the cylinder, the. .. 4. 45) was obtained with an experimental device in the Mechanical Engineering Department of Ben-Gurion University Here, a section of a cosine-type acceleration carried out by a cam-driven follower is presented The thicker line shows the calculated curve, while the thin line shows the real data for the acceleration of the follower Obviously, the higher the rotational speed of the cam, the larger are the. .. 4.35a) the situation of the valve corresponds to the resting state of the hydraulic motor When, due to rotation of motor 17, the piston begins to move rightward (see arrow in Figure 4.35a)) and thereby connects port 1 with port 4 (see Figure 4.35b)), the pressure reaches port 6 of the hydraulic motor, while port 7 connects with ports 2, 3, and 5, permitting drainage of the oil into the reservoir The. .. output due to the limited stiffness of the shaft, so that Lastly, the model in Figure 4.47d) takes the stiffness of both the drive shaft and the connecting rod into consideration Then the equations are or For cases c) and d), s does not equal s* because, at the input of the mechanism, the rotation of the drive shaft equals 0 = 0* + ql and the driven mass moves in accordance with y = s + q2 where 5 = n(0*... has the form of an angular lever One arm of this lever actuates a group of electric contacts 4 Sometimes a spring 5 is used to keep the armature at a certain distance from the core When energized, the coil induces a magnetic flux and the armature overpowers the spring, closing the gap and simultaneously actuating the contacts In the general case, some of the contacts can be normally open while the others... cylinder Obviously, the speed of the slider's rotation determines the time during which the cylinders are under pressure, and this time must be longer than that needed for the piston to be displaced as required (Opening 10 in the slider connects the cylinders with the atmosphere to permit the pistons to return to their initial positions, assuming that there are springs for this purpose in the cylinders.)... is engaged with another wheel z2 which drives a cam (the eccentricity of which is e), and the latter, in turn, drives a follower The motion s(f) of the follower is described by the motion function and equals 11(0) By means of a connecting rod, the follower drives mass m and overcomes external force E The rod connecting the follower to the mass has stiffness c2 The damping effects in the system are described... mechanism, with their deformation by external and inertial forces These deformations are usually too small to significantly alter the shape of the displacement s(f) However, the first- and second-order derivatives, namely, the velocity s and especially the acceleration s, can (and usually do) acquire significant deflections or errors Sometimes the errors in acceleration reach the order of magnitude of the nominal . 1 and 2. The pressure difference between these channels depends upon the position of the edge of the nozzle D relative to the inlets of the channels. (The diameter of the nozzle. Camshafts 129 For the central mechanism, where e = 0, we obtain a simpler expression for (4.23), i.e.: or The larger the pressure angle a, the lower the efficiency of the mechanism. . value, the mechanism can jam. The critical value of the pressure angle depends on the friction conditions of the follower in its guides, on the geometry of the guides, on the