1 00:00:10,300 --> 00:00:16,560 In today’s lecture, we shall discuss four link planar mechanisms. 2 00:00:16,560 --> 00:00:21,650 As we shall see such mechanisms have versatile applications. 3 00:00:21,650 --> 00:00:28,930 In fact, a major portion of this course will be devoted to the study of four link planar 4 00:00:28,930 --> 00:00:49,800 mechanisms. 5 00:00:49,800 --> 00:00:56,270 Before getting into the four link planar mechanisms, let us recall the technical, that is, kinematic 6 00:00:56,270 --> 00:00:58,649 definition of a mechanism. 7 00:00:58,649 --> 00:01:16,730 We have already defined mechanism as a closed kinematic chain with one of its links fixed. 8 00:01:16,730 --> 00:01:31,620 It is obvious, that to get a closed chain, we need at least three links. 9 00:01:31,620 --> 00:01:38,970 In the next figure, we shall see a three-linked closed chain with three revolute pairs. 10 00:01:38,970 --> 00:01:45,220 In such a three-linked closed chain, it is obvious that if one of the links, say, number 11 00:01:45,220 --> 00:01:51,890 1 is held fixed, then there cannot be any relative movement between these three bodies. 12 00:01:51,890 --> 00:01:59,570 In fact, if we apply an external load there is no relative movement between various links, 13 00:01:59,570 --> 00:02:04,070 in fact this load can be supported by this assembly. 14 00:02:04,070 --> 00:02:09,259 Such an assembly with zero degree of freedom is called a structure. 15 00:02:09,259 --> 00:02:15,569 Hence, we see the four link planar mechanism is the simplest mechanism that we can think 16 00:02:15,569 --> 00:02:16,569 of. 17 00:02:16,569 --> 00:02:23,250 To start with, we consider a four link planar mechanism with four revolute pairs. 18 00:02:23,250 --> 00:02:30,530 As we see in this four-link planar mechanism, there are four revolute pairs: one at O2, 19 00:02:30,530 --> 00:02:40,610 one at O4, one at A and the other at B. The four links are numbered, 1 as the fixed link 20 00:02:40,610 --> 00:02:44,410 and 2, 3, 4 as the moving links. 21 00:02:44,410 --> 00:02:52,069 The two hinges at O2 and O4 which are connected to the fixed link are called fixed pivots 22 00:02:52,069 --> 00:02:58,900 and the revolute pairs at A and B are referred to as moving hinges. 23 00:02:58,900 --> 00:03:06,730 Link 2 and link 4, which are connected to the fixed link are normally used as the input 24 00:03:06,730 --> 00:03:08,969 and the output link. 25 00:03:08,969 --> 00:03:15,740 Link 3, which connects the links 2 and 4, is called the coupler, which is the motion 26 00:03:15,740 --> 00:03:19,780 transfer link between 2 and 4. 27 00:03:19,780 --> 00:03:32,250 Such a 4R mechanism has four kinematic dimensions namely, O2O4 which we represent by l1 and 28 00:03:32,250 --> 00:03:36,069 O2A which represent by l2. 29 00:03:36,069 --> 00:03:49,170 Similarly, AB is equal to l3 and O4B as l4. 30 00:03:49,170 --> 00:03:54,670 If one of the input links which is connected to the fixed link makes a complete rotation, 31 00:03:54,670 --> 00:04:00,420 as is normally the case, when the mechanism is driven by an electric motor, then that 32 00:04:00,420 --> 00:04:07,140 link is referred to as a crank. 33 00:04:07,140 --> 00:04:20,269 The output link 4, which is the output link, is called follower and the intermediate connecting 34 00:04:20,269 --> 00:04:29,520 link is called the coupler. 35 00:04:29,520 --> 00:04:37,039 Let us consider a 4R planar mechanism where one of these kinematic pairs at B goes to 36 00:04:37,039 --> 00:04:40,400 infinity along a vertical direction. 37 00:04:40,400 --> 00:04:47,160 Consequently, that kinematic pair is converted to a prismatic pair and what we get is a three 38 00:04:47,160 --> 00:04:57,759 revolute one prismatic pair, referred to as the 3R-1P mechanism. 39 00:04:57,759 --> 00:05:05,960 So we get a 3R-1P mechanism where there are three revolute pairs at O2, A and B and the 40 00:05:05,960 --> 00:05:13,009 kinematic pair between link 4 and link 1 is a prismatic pair. 41 00:05:13,009 --> 00:05:20,330 Such a mechanism has three kinematic dimensions namely O2A which is called the crank length 42 00:05:20,330 --> 00:05:30,310 is represented by l2, AB which is called the connecting rod whose link is represented by 43 00:05:30,310 --> 00:05:39,909 l3 and the line of reciprocation of B is at a distance from O2 and this perpendicular 44 00:05:39,909 --> 00:05:43,039 distance, which is called the offset. 45 00:05:43,039 --> 00:05:50,839 The other kinematic dimension is the offset which is equal to e. 46 00:05:50,839 --> 00:05:57,620 This is also known as an offset slider crank mechanism which is used to convert uniform 47 00:05:57,620 --> 00:06:05,439 rotation of link 2 in to rectilinear to and fro oscillation of the slider 4. 48 00:06:05,439 --> 00:06:12,729 Let us consider different kinematic inversions which can arise out of such a 3R-1P chain 49 00:06:12,729 --> 00:06:14,719 as shown in this figure. 50 00:06:14,719 --> 00:06:25,499 Here, as we see there are three revolute pairs namely at O2, O4 and at A. Comparing to the 51 00:06:25,499 --> 00:06:33,960 previous 3R-1P chain, here the link 1 which is fixed has revolute pairs at both ends, 52 00:06:33,960 --> 00:06:35,800 that is at O2 and O4. 53 00:06:35,800 --> 00:06:41,990 Whereas, in the previous mechanism, the fixed link had a revolute pair at one end and a 54 00:06:41,990 --> 00:06:44,180 prismatic pair at the other end. 55 00:06:44,180 --> 00:06:52,080 Here again, we have three kinematic dimensions namely O2O4 - that is the fixed link length, 56 00:06:52,080 --> 00:06:58,719 O2A that may be the crank length and the distance of this line of reciprocation from these fixed 57 00:06:58,719 --> 00:06:59,719 hinges. 58 00:06:59,719 --> 00:07:08,659 Such a mechanism with this offset e equal to 0 is used in a quick return mechanism that 59 00:07:08,659 --> 00:07:10,460 is used in a shaper. 60 00:07:10,460 --> 00:07:17,149 We consider again yet another kinematic inversion of the same 3R-1P chain. 61 00:07:17,149 --> 00:07:23,449 Here, we have three revolute pairs namely, at O2, A and O4. 62 00:07:23,449 --> 00:07:33,909 Here again the fixed link has revolute pairs at both ends namely O2 and O4. 63 00:07:33,909 --> 00:07:39,569 Such a mechanism is called an oscillating cylinder mechanism which is used for example 64 00:07:39,569 --> 00:07:41,930 in a bicycle foot pump. 65 00:07:41,930 --> 00:07:50,560 The thing to note is that here the input member is link 3 which is given an oscillatory motion. 66 00:07:50,560 --> 00:07:56,339 This (piston) link 3 moves within this oscillating cylinder (4) which has the same angular velocity 67 00:07:56,339 --> 00:07:57,339 as link 3. 68 00:07:57,339 --> 00:07:59,229 (Refer slide Time 07:59) 69 00:07:59,229 --> 00:08:05,349 We see another kinematic inversion from the same 3R-1P chain where we have kinematic pair 70 00:08:05,349 --> 00:08:15,279 at O2, A and B. The link 4 which has a revolute pair at B has a prismatic pair in the vertical 71 00:08:15,279 --> 00:08:18,889 direction with respect to the fixed link 1. 72 00:08:18,889 --> 00:08:24,669 Such a mechanism, as we know is used in a hand pump mechanism and here again it is link 73 00:08:24,669 --> 00:08:31,899 3 which is given the input motion such that the piston which is link 4 moves vertically 74 00:08:31,899 --> 00:08:36,580 up and down within this fixed link. 75 00:08:36,580 --> 00:08:41,310 So far, we have considered 3R-1P chain. 76 00:08:41,310 --> 00:08:47,209 Let us now have another revolute pair converted into a prismatic pair and consequently we 77 00:08:47,209 --> 00:08:51,210 get what we call a 2R-2P chain. 78 00:08:51,210 --> 00:08:59,360 However, unlike in a 3R-1P chain, there can be two varieties that is the sequential order 79 00:08:59,360 --> 00:09:01,279 of this kinematic pairs. 80 00:09:01,279 --> 00:09:08,000 For example, we can have RRPP or RPRP. 81 00:09:08,000 --> 00:09:16,800 Here, the two revolute pairs are connected by one link, then another link has a revolute 82 00:09:16,800 --> 00:09:24,140 pair at one end and a prismatic pair at the other end, another link has prismatic pair 83 00:09:24,140 --> 00:09:31,589 at both ends and this link has a P pair at one end and a revolute pair at other end. 84 00:09:31,589 --> 00:09:40,110 This is what we call RRPP chain, whereas here these kinematic pairs appear alternately as 85 00:09:40,110 --> 00:09:46,130 RPRP. 86 00:09:46,130 --> 00:09:50,709 So we see all the four links has a revolute pair at one end and a prismatic pair at the 87 00:09:50,709 --> 00:09:55,620 other end that is same for all these four links. 88 00:09:55,620 --> 00:10:02,529 It may be emphasized that in a 3R-1P chain whichever way we go, we start from one prismatic 89 00:10:02,529 --> 00:10:07,430 pair then it is followed by three sequential R pairs. 90 00:10:07,430 --> 00:10:11,550 So, PRRR is the only possibility. 91 00:10:11,550 --> 00:10:16,390 Let me now consider some inversions from this 2R-2P chain. 92 00:10:16,390 --> 00:10:24,069 Here, we consider a four link planar mechanism with a revolute pair at O2, another revolute 93 00:10:24,069 --> 00:10:32,360 pair at O, which I called as O4, that is link 2 has two revolute pairs at its either ends, 94 00:10:32,360 --> 00:10:38,830 and link number 3 has a revolute pair at O and a prismatic pair with 4 along the vertical 95 00:10:38,830 --> 00:10:39,830 direction. 96 00:10:39,830 --> 00:10:45,540 Similarly, link 1 has a revolute pair with 2 and a prismatic pair with 4 in the horizontal 97 00:10:45,540 --> 00:10:48,750 direction. 98 00:10:48,750 --> 00:10:54,230 We consider a special case when these two prismatic pairs have an angle of 90 degree 99 00:10:54,230 --> 00:10:56,290 between them. 100 00:10:56,290 --> 00:11:03,029 From such a 2R-2P, that is RRPP chain, we can have different mechanism by the process 101 00:11:03,029 --> 00:11:04,310 of kinematic inversion. 102 00:11:04,310 --> 00:11:10,019 For example, let us hold this link number 1 fixed. 103 00:11:10,019 --> 00:11:15,430 So we can see that link 4, the output link can have horizontal movement with respect 104 00:11:15,430 --> 00:11:21,640 to the fixed link as link 2 undergoes rotary motion with respect to link 1, that is the 105 00:11:21,640 --> 00:11:23,990 fixed link. 106 00:11:23,990 --> 00:11:25,160 This is the kinematic sketch. 107 00:11:25,160 --> 00:11:31,629 Let me show the physical construction of such a mechanism, which is known as the scotch 108 00:11:31,629 --> 00:11:33,370 yoke mechanism. 109 00:11:33,370 --> 00:11:41,000 As we see, such a mechanism will convert uniform rotary motion of link 2 into simple harmonic 110 00:11:41,000 --> 00:11:46,949 translatory motion in the horizontal direction for link 4. 111 00:11:46,949 --> 00:11:52,730 So again we have RRPP chain. 112 00:11:52,730 --> 00:11:58,940 There is one revolute pair at O2, one revolute pair at A, there is a vertical prismatic pair 113 00:11:58,940 --> 00:12:04,300 between 3 and 4 and a horizontal prismatic pair between 4 and 1. 114 00:12:04,300 --> 00:12:19,880 If we consider the kinematic dimension O2A as L2 and this angle if I call Theta, assuming 115 00:12:19,880 --> 00:12:26,970 at ‘t’ equal to zero is Theta is equal to 0, then I can represent Theta = Omega2t, 116 00:12:26,970 --> 00:12:34,339 where Omega2 is the constant angular speed of link 2. 117 00:12:34,339 --> 00:12:42,389 Then the position of the link 4 which can be represented by the point is given by x. 118 00:12:42,389 --> 00:12:57,090 It is easy to see that x is nothing but L2 cosTheta, that is L2 cosOmegat. 119 00:12:57,090 --> 00:13:04,839 So, we have produced a simple harmonic motion out of continuous rotation. 120 00:13:04,839 --> 00:13:11,769 It may be emphasized that continuous uniform rotary motion would have been translated into 121 00:13:11,769 --> 00:13:16,770 to and fro rectilinear oscillation by a slider crank mechanism that we have seen earlier. 122 00:13:16,770 --> 00:13:23,490 However, the to and fro rectilinear oscillation of the piston of a slider in a slider crank 123 00:13:23,490 --> 00:13:28,839 mechanism is periodic, but not purely harmonic. 124 00:13:28,839 --> 00:13:34,760 This periodic motion tends to be harmonic as the ratio of the connecting rod length 125 00:13:34,760 --> 00:13:37,670 to the crank radius keeps on increasing. 126 00:13:37,670 --> 00:13:44,690 In fact, if the connecting rod length in a slider crank mechanism tends to infinity, 127 00:13:44,690 --> 00:13:48,430 then the slider motion becomes purely harmonic. 128 00:13:48,430 --> 00:13:55,160 As soon as the connecting rod length becomes infinity, one of the kinematic pair, which 129 00:13:55,160 --> 00:14:00,689 was the revolute pair for a connecting rod, gets converted into this prismatic pair, because 130 00:14:00,689 --> 00:14:06,139 we have already seen a prismatic pair is nothing but a revolute pair at infinity. 131 00:14:06,139 --> 00:14:09,449 I will now show this through a model. 132 00:14:09,449 --> 00:14:11,110 (Refer Slide Time 14:10) 133 00:14:11,110 --> 00:14:14,699 This is the model of that scotch yoke mechanism. 134 00:14:14,699 --> 00:14:20,670 This is the link 2 that is the crank, which has a revolute pair with the fixed link at 135 00:14:20,670 --> 00:14:23,180 this point. 136 00:14:23,180 --> 00:14:29,180 Another revolute pair between link 2, that is the crank and this block is here. 137 00:14:29,180 --> 00:14:36,189 This block link 3 has a prismatic pair in the vertical direction with link 4, and link 138 00:14:36,189 --> 00:14:40,870 4 has a horizontal prismatic pair with link 1. 139 00:14:40,870 --> 00:14:49,190 So, if we rotate link 2 at a constant angular speed, as we see, the red link that is link 140 00:14:49,190 --> 00:14:55,920 4 is performing simple harmonic oscillation in the horizontal direction. 141 00:14:55,920 --> 00:15:03,110 Let us now consider another kinematic inversion of the same RRPP chain. 142 00:15:03,110 --> 00:15:10,579 As shown in this mechanism, the link which has revolute pair at its both ends at O2 and 143 00:15:10,579 --> 00:15:14,410 O4 is held fixed. 144 00:15:14,410 --> 00:15:22,370 Now link 3 has a prismatic pair in the horizontal direction with link 2 and a prismatic pair 145 00:15:22,370 --> 00:15:25,870 in a vertical direction with link 4. 146 00:15:25,870 --> 00:15:34,199 So, thus if we rotate link 2 that is in translation with link 3 which is again in translation 147 00:15:34,199 --> 00:15:41,809 with link 4, thus link 2 and link 4 has only relative translatory motion. 148 00:15:41,809 --> 00:15:45,860 In other words, they have the same angular motion. 149 00:15:45,860 --> 00:15:53,769 Thus, this mechanism known as Oldham’s coupling can be used to connect two parallel shafts: 150 00:15:53,769 --> 00:16:02,319 one at O2 and the other at O4 and transmitting unity angular velocity ratio. 151 00:16:02,319 --> 00:16:04,380 This is the kinematic diagram. 152 00:16:04,380 --> 00:16:10,160 The physical construction of this Oldham’s coupling is shown in the above slide. 153 00:16:10,160 --> 00:16:17,149 As we see, there is a revolute pair between link 1 and link 2, there is a prismatic pair 154 00:16:17,149 --> 00:16:23,509 between link 2 and link 3, another prismatic pair between link 3 and link 4. 155 00:16:23,509 --> 00:16:28,949 The direction of these two prismatic pairs are at right angles to each other. 156 00:16:28,949 --> 00:16:32,720 Link 4 again has a revolute pair with the fixed link 1. 157 00:16:32,720 --> 00:16:39,569 Thus link 4, that is, this shaft and link 2, that is the other shaft which are parallel 158 00:16:39,569 --> 00:16:45,149 can be connected by such a coupling and it will transmit unity angular velocity ratio. 159 00:16:45,149 --> 00:16:46,870 (Refer Slide Time 16:47) 160 00:16:46,870 --> 00:16:50,329 I will now show this through a model. 161 00:16:50,329 --> 00:16:53,470 This is the model of Oldham’s coupling. 162 00:16:53,470 --> 00:17:01,060 As we see, the rotation of this shaft is converted into the rotation of that shaft at the same 163 00:17:01,060 --> 00:17:07,400 speed through the intermediate member which has a prismatic pair here and a prismatic 164 00:17:07,400 --> 00:17:08,720 pair at the top. 165 00:17:08,720 --> 00:17:15,580 So, there are two prismatic pairs at 90 degrees to each other. 166 00:17:15,580 --> 00:17:23,810 This intermediate member moves in and out in this prismatic pair and also at this prismatic 167 00:17:23,810 --> 00:17:29,910 pair. 168 00:17:29,910 --> 00:17:36,070 Of course, this coupling is good enough to transmit power, only when the offset between 169 00:17:36,070 --> 00:17:42,051 these two shafts is not very large, because a lot of power is wasted in fiction at this 170 00:17:42,051 --> 00:17:46,100 two prismatic pairs. 171 00:17:46,100 --> 00:17:53,130 Now let us consider another kinematic inversion from the same RRPP chain. 172 00:17:53,130 --> 00:17:59,940 In this mechanism, we consider the link which has prismatic pair at both ends fixed. 173 00:17:59,940 --> 00:18:07,010 For example, this link 1 has a prismatic pair with link 4 and a prismatic pair with link 174 00:18:07,010 --> 00:18:10,320 2 at right angles to each other. 175 00:18:10,320 --> 00:18:13,650 This mechanism is known as elliptic trammel. 176 00:18:13,650 --> 00:18:19,630 It will be obvious now why this name elliptic trammel? 177 00:18:19,630 --> 00:18:25,310 Let us consider the physical construction of this mechanism rather than this kinematic 178 00:18:25,310 --> 00:18:27,030 representation. 179 00:18:27,030 --> 00:18:34,530 As we see, this link has two prismatic pairs, one in the horizontal direction and the other 180 00:18:34,530 --> 00:18:37,310 in the vertical direction. 181 00:18:37,310 --> 00:18:46,990 The rod ABCD has revolute pair with the block at C and another revolute pair with the block 182 00:18:46,990 --> 00:18:56,360 at A. As the rod moves, let us look at the coordinates of point D on the moving rod. 183 00:18:56,360 --> 00:19:00,740 This is x-axis and this is y-axis. 184 00:19:00,740 --> 00:19:09,900 So the x-coordinate of the point D, when the rod is making an angle Theta with the vertical 185 00:19:09,900 --> 00:19:19,450 is easily seen to be ADsinTheta. 186 00:19:19,450 --> 00:19:28,700 Similarly, the y coordinate of this moving point D is CD and this angle is Theta. 187 00:19:28,700 --> 00:19:30,530 So it is CDcosTheta. 188 00:19:30,530 --> 00:19:37,650 So, yD is CDcosTheta. 189 00:19:37,650 --> 00:19:53,620 If we eliminate Theta from these two coordinates, we can easily see that {(xD/AD)2 + (yD/CD)2 190 00:19:53,620 --> 00:19:57,470 = 1}. 191 00:19:57,470 --> 00:20:05,670 That is, as Theta changes, this point D moves on an ellipse with semi major axis given by 192 00:20:05,670 --> 00:20:07,690 AD and CD. 193 00:20:07,690 --> 00:20:10,440 That is why it is called an elliptic trammel. 194 00:20:10,440 --> 00:20:16,640 It must be pointed out that there are three points on this rod AB which are exceptions 195 00:20:16,640 --> 00:20:22,800 namely, this point A which generates a vertical straight line because of this prismatic pair. 196 00:20:22,800 --> 00:20:28,410 Similarly, this point C which generates a horizontal straight line because of this horizontal 197 00:20:28,410 --> 00:20:35,580 prismatic pair and for the mid-point B which is the midpoint of the distance AC, that is, 198 00:20:35,580 --> 00:20:38,990 AB is equal to BC. 199 00:20:38,990 --> 00:20:47,140 If AB and BC are equal, then the point B generates a circle. 200 00:20:47,140 --> 00:20:50,650 If I call this point O2, then the radius of that circle is O2B. 201 00:20:50,650 --> 00:20:58,100 We can say that these are nothing but the degenerated cases of the same ellipse. 202 00:20:58,100 --> 00:21:04,810 Now, we show you a model to generate this ellipse through an elliptic trammel. 203 00:21:04,810 --> 00:21:07,740 This is the model of the elliptic trammel. 204 00:21:07,740 --> 00:21:12,880 As we see, there are two perpendicular slots in the fixed link. 205 00:21:12,880 --> 00:21:22,540 This is rod AB, which has a revolute pair with two blocks and these two blocks move 206 00:21:22,540 --> 00:21:26,310 along these two prismatic pairs. 207 00:21:26,310 --> 00:21:34,520 If we move the rod, as we can see this particular point of the rod generates the ellipse which 208 00:21:34,520 --> 00:21:40,680 has been drawn with the green line. 209 00:21:40,680 --> 00:21:50,190 Now, if we consider the mid point of these two revolute pairs, then as we see the rod 210 00:21:50,190 --> 00:22:03,220 moves this particular mid-point generates this red circle, as mentioned earlier. 211 00:22:03,220 --> 00:22:09,960 Now that we have discussed different inversions from a RRPP chain. 212 00:22:09,960 --> 00:22:20,120 Let us go back to another 2R-2P chain namely, RPRP chain. 213 00:22:20,120 --> 00:22:26,900 We consider here only the portion covered by these dashed lines. 214 00:22:26,900 --> 00:22:32,970 This is the part of an automobile steering gear known as Davis steering gear. 215 00:22:32,970 --> 00:22:40,540 Here, as we see there is a revolute pair between link 4 and link 1, then there is a prismatic 216 00:22:40,540 --> 00:22:47,070 pair between link 4 and link 3, then there is a revolute pair between link 3 and link 217 00:22:47,070 --> 00:22:52,570 2 and then there is a prismatic pair between link 1 and link 2. 218 00:22:52,570 --> 00:22:55,610 These are the two wheels of the automobile. 219 00:22:55,610 --> 00:23:04,470 By moving the steering wheel, we move this link 2 in this prismatic pair between 1 and 220 00:23:04,470 --> 00:23:05,660 2. 221 00:23:05,660 --> 00:23:08,790 Consequently, these two wheels will rotate. 222 00:23:08,790 --> 00:23:14,890 So, here is an RPRP chain. 223 00:23:14,890 --> 00:23:21,140 Now that we have discussed 4 linked planar mechanisms which is two revolute and two prismatic 224 00:23:21,140 --> 00:23:28,380 pairs, let me now increase one more prismatic pair instead of a revolute pair, that is, 225 00:23:28,380 --> 00:23:33,480 can we talk of 3P-1R chain? 226 00:23:33,480 --> 00:23:39,900 As we see, there is a P pair and there is a link 1 connecting to these two prismatic 227 00:23:39,900 --> 00:23:40,900 pairs. 228 00:23:40,900 --> 00:23:48,050 Similarly, link 2 connects to two prismatic pairs and link 3 connects a prismatic pair 229 00:23:48,050 --> 00:23:53,080 and a revolute pair and a link 4 connects a revolute pair and a prismatic pair. 230 00:23:53,080 --> 00:24:02,300 It will be easy to show that we cannot have a mechanism with such 3P-1R chain. 231 00:24:02,300 --> 00:24:10,090 For example, we can see there is a prismatic pair between link 1 and 2, there cannot be 232 00:24:10,090 --> 00:24:11,870 any relative rotation. 233 00:24:11,870 --> 00:24:17,530 Same goes between link 2 and 3 because they are connected between a prismatic pair. 234 00:24:17,530 --> 00:24:22,540 There cannot be any relative rotation between link 2 and link 3. 235 00:24:22,540 --> 00:24:29,730 There is a prismatic pair between link 1 and link 4, so there cannot be any relative rotation 236 00:24:29,730 --> 00:24:32,030 between link 1 and link 4. 237 00:24:32,030 --> 00:24:39,320 Thus, if we follow these three prismatic pairs, we conclude there cannot be any relative rotation 238 00:24:39,320 --> 00:24:41,880 between link 4 and link 3. 239 00:24:41,880 --> 00:24:48,660 Thus, this revolute pair which allows only relative rotation between link 3 and link 240 00:24:48,660 --> 00:24:51,940 4 cannot permit any relative motion. 241 00:24:51,940 --> 00:24:59,130 Thus, we can conclude that such a 3P-1R chain cannot give rise to any mechanism. 242 00:24:59,130 --> 00:25:04,520 Now, can we have a mechanism with 4 prismatic pairs? 243 00:25:04,520 --> 00:25:08,260 A four-link planar mechanism with all pairs prismatic. 244 00:25:08,260 --> 00:25:16,470 We will see later that such a 4P mechanism is not a constrained mechanism. 245 00:25:16,470 --> 00:25:24,500 In fact, three links connected by three prismatic pairs having the relative translation at various 246 00:25:24,500 --> 00:25:32,740 angles itself constitutes a planar mechanism which is constant, but that we shall see later. 247 00:25:32,740 --> 00:25:38,780 So far we have discussed mechanisms only with revolute and prismatic pairs. 248 00:25:38,780 --> 00:25:41,000 Let me now change the topic a little bit. 249 00:25:41,000 --> 00:25:46,760 Can we consider mechanisms involving higher pairs as well? 250 00:25:46,760 --> 00:25:53,590 As we shall show, now that a mechanism with higher pair can be replaced equivalently by 251 00:25:53,590 --> 00:26:00,210 a mechanism having only lower pairs that is revolute pair and prismatic pairs. 252 00:26:00,210 --> 00:26:07,540 Of course, this equivalence is only instantaneous and holds good for velocity and acceleration 253 00:26:07,540 --> 00:26:13,950 analysis at a particular configuration. 254 00:26:13,950 --> 00:26:20,100 So we are talking of an equivalent lower pair linkage for a higher pair mechanism, i.e., 255 00:26:20,100 --> 00:26:25,490 how a higher pair can be replaced equivalently by lower pairs. 256 00:26:25,490 --> 00:26:33,110 As an example, let us look at this higher pair mechanism which consists of 3 links namely, 257 00:26:33,110 --> 00:26:38,020 one the fixed link and 2 and 3. 258 00:26:38,020 --> 00:26:46,090 There is a revolute pair between link 1 and 2 at O2 and a revolute pair between link 1 259 00:26:46,090 --> 00:26:56,560 and 3 at O3, but there is a higher pair at the point C between link 2 and link 3. 260 00:26:56,560 --> 00:27:03,590 Our objective is to replace this higher pair mechanism by a kinematically equivalent lower 261 00:27:03,590 --> 00:27:07,860 pair linkage consisting only of lower pairs. 262 00:27:07,860 --> 00:27:17,040 I repeat again that this equivalence is only instantaneous, that means, only for this particular 263 00:27:17,040 --> 00:27:18,040 configuration. 264 00:27:18,040 --> 00:27:25,350 Towards this end, let us consider that the centre of curvature of body 3 at the point 265 00:27:25,350 --> 00:27:35,020 C is at B. Similarly, the centre of curvature of the body 2 at this contact point C is at 266 00:27:35,020 --> 00:27:43,140 A. Due to the property of centre of curvature, that is circle of curvature or osculating 267 00:27:43,140 --> 00:27:52,460 circle, we can consider that for three infinitesimally separated time instances, these points A and 268 00:27:52,460 --> 00:27:54,560 B do not change. 269 00:27:54,560 --> 00:28:03,840 Thus, we can replace this higher pair by having a lower pair at the point A which is a revolute 270 00:28:03,840 --> 00:28:09,690 pair and a lower pair at the point B, which is again a revolute pair and because the distance 271 00:28:09,690 --> 00:28:17,480 between A and B are not changing for three infinitesimally separated time intervals, 272 00:28:17,480 --> 00:28:23,400 I can join these two points A and B by a rigid additional link. 273 00:28:23,400 --> 00:28:33,120 Thus, this higher pair mechanism has been replaced by a 4R-linkage which has all 4 revolute 274 00:28:33,120 --> 00:28:44,020 pairs at O2, O3, A and B. As I said earlier, because as these two bodies 2 and 3 move the 275 00:28:44,020 --> 00:28:50,450 centers of curvature also move, so for every instance, we have a different equivalent lower 276 00:28:50,450 --> 00:28:51,450 pair linkage. 277 00:28:51,450 --> 00:28:56,820 Later on, we will explain this equivalence through a model. 278 00:28:56,820 --> 00:29:05,420 But we should see that a higher pair at C has been replaced by an additional link that 279 00:29:05,420 --> 00:29:15,270 is link 4 and two additional lower pairs, one at A and the other at B, where A and B 280 00:29:15,270 --> 00:29:21,630 are respectively the centre of curvatures of the contacting surfaces between 2 and 3 281 00:29:21,630 --> 00:29:29,310 at the contact point C. Of course, this equivalence can be permanent if the centers of curvature 282 00:29:29,310 --> 00:29:30,600 do not change. 283 00:29:30,600 --> 00:29:38,040 That is, one of these contacting surfaces, say circular or cylindrical or coinstantaneous 284 00:29:38,040 --> 00:29:43,000 of curvature or flat that is again of infinite (constant) radius of curvature. 285 00:29:43,000 --> 00:29:44,120 (Refer Slide Time 29:32) 286 00:29:44,120 --> 00:29:50,210 For example, let us look at this higher pair mechanism which is a cam with a flat face 287 00:29:50,210 --> 00:29:51,380 follower. 288 00:29:51,380 --> 00:30:01,330 The cam that is this rigid link 2 has revolute pair with the fixed link at O2 and this follower 289 00:30:01,330 --> 00:30:09,540 3 has a prismatic pair with fixed link 1 and there is a higher pair at this contact point. 290 00:30:09,540 --> 00:30:19,770 So, the rotary motion of this cam 2 is converted into rectilinear motion of this follower 3. 291 00:30:19,770 --> 00:30:25,420 Let the centre of curvature of the cam surface at this contact point is at C2. 292 00:30:25,420 --> 00:30:32,141 The centre of curvature of the follower surface at the contact point is at infinity, that 293 00:30:32,141 --> 00:30:38,640 is, the revolute pair at the centre of curvature is converted to an equivalent prismatic pair 294 00:30:38,640 --> 00:30:40,960 in the horizontal direction. 295 00:30:40,960 --> 00:30:46,970 If we recall that a prismatic pair is nothing but a revolute pair at infinity. 296 00:30:46,970 --> 00:30:53,270 Consequently, this cam follower mechanism with a flat face follower is replaced by this 297 00:30:53,270 --> 00:30:55,970 equivalent lower pair linkage. 298 00:30:55,970 --> 00:31:02,890 This link 2 represents the cam, link 3 represents the follower and there is an additional link 299 00:31:02,890 --> 00:31:10,460 4 which has a revolute pair with link 2 at C2 and a prismatic pair with link 3 in the 300 00:31:10,460 --> 00:31:11,970 horizontal direction. 301 00:31:11,970 --> 00:31:17,450 This is the equivalent lower pair linkage of this cam follower mechanism. 302 00:31:17,450 --> 00:31:24,570 That means, the input-output characteristics of this cam follower mechanism can be carried 303 00:31:24,570 --> 00:31:31,900 on by analyzing this lower pair linkage, because C2 keeps on changing if the cam surface is 304 00:31:31,900 --> 00:31:33,230 not circular. 305 00:31:33,230 --> 00:31:39,370 That’s why, for every instance we have to have separate equivalent lower pair linkage, 306 00:31:39,370 --> 00:31:44,620 because this link length C2 will keep on changing. 307 00:31:44,620 --> 00:31:51,630 This equivalence is valid only up to acceleration analysis, velocity and acceleration because 308 00:31:51,630 --> 00:31:59,210 the centre of curvature or osculating circle is in contact with the surface only for three 309 00:31:59,210 --> 00:32:01,700 infinitesimally separated positions. 310 00:32:01,700 --> 00:32:06,620 Higher order derivatives do not match and consequently, higher order time derivatives 311 00:32:06,620 --> 00:32:08,360 cannot be calculated. 312 00:32:08,360 --> 00:32:15,370 Now I will show these two equivalence lower pair linkage by models. 313 00:32:15,370 --> 00:32:21,780 Let us consider this model where we have a higher pair mechanism and the link 2 and link 314 00:32:21,780 --> 00:32:25,510 3 is having a higher pair at this point. 315 00:32:25,510 --> 00:32:32,480 The centre of curvature for the body 2 at this point of contact is here at C2. 316 00:32:32,480 --> 00:32:40,630 Similarly, the centre of curvature of the surface of body 3 is at this point C3. 317 00:32:40,630 --> 00:32:47,080 Now as we said earlier, we can have an equivalent lower pair linkage by having a revolute pair 318 00:32:47,080 --> 00:33:00,990 at C2 and another at C3 and connected by an additional rigid link. 319 00:33:00,990 --> 00:33:12,700 Thus, we have a 4R mechanism which is instantaneously equivalent to the original higher pair mechanism. 320 00:33:12,700 --> 00:33:21,350 I have connected these two bodies, that is, this red link and the original link 2, rigidly. 321 00:33:21,350 --> 00:33:28,270 As we shall see that I can move this mechanism, the same motion is transmitted at least around 322 00:33:28,270 --> 00:33:35,231 this contact region between body 2 and body 3, whether, it is through the higher pair 323 00:33:35,231 --> 00:33:38,800 mechanism or through this equivalent lower pair linkage. 324 00:33:38,800 --> 00:33:45,280 To distinguish the movement of this body 3 in the lower pair linkage and this body 3 325 00:33:45,280 --> 00:33:50,890 in the higher pair mechanism, let us notice these two lines, there is a red line on this 326 00:33:50,890 --> 00:33:55,220 body and there is a blue line on this body. 327 00:33:55,220 --> 00:34:05,890 Around this point, as we see these two lines move almost the same way because velocity 328 00:34:05,890 --> 00:34:12,120 and acceleration relationship between the original higher pair mechanism and the equivalent 329 00:34:12,120 --> 00:34:15,860 lower pair linkage is just the same. 330 00:34:15,860 --> 00:34:20,700 However, when there is a lot of movement these two lines separate out. 331 00:34:20,700 --> 00:34:25,190 This blue line and this red line are not the same any more. 332 00:34:25,190 --> 00:34:37,929 But around this point, it is only here that these two lines separate out because the centers 333 00:34:37,929 --> 00:34:44,510 of curvature are very different from what it was at this configuration. 334 00:34:44,510 --> 00:34:52,600 This is what we mean by instantaneously equivalent lower pair linkage. 335 00:34:52,600 --> 00:34:56,490 We can show another model with one surface flat. 336 00:34:56,490 --> 00:35:02,650 That is, radius of curvature is infinity and the other curve is a circle, that is the radius 337 00:35:02,650 --> 00:35:04,510 of curvature is constant. 338 00:35:04,510 --> 00:35:10,330 Under such a situation of course the equivalence will hold good for the entire cycle of motion. 339 00:35:10,330 --> 00:35:20,440 A higher pair mechanism between link 1, 2 and 3 and the centre of curvature of body 340 00:35:20,440 --> 00:35:28,280 2 at the contact point is here, whereas, body 3 has a flat or a straight contact in surface. 341 00:35:28,280 --> 00:35:38,150 Consequently, the extra link that is having here gets a prismatic pair between body 3 342 00:35:38,150 --> 00:35:44,930 and that extra link and a revolute pair between body 2 and that extra link. 343 00:35:44,930 --> 00:35:52,221 If we connect these lower pair linkage link 2 with the link 2 of the original higher pair 344 00:35:52,221 --> 00:35:59,270 mechanism rigidly, then we can see that same motion is transmitted by the both lower pair 345 00:35:59,270 --> 00:36:12,760 linkage and the higher pair mechanism. 346 00:36:12,760 --> 00:36:18,200 Let me now summarize, what we have talked so far in this lecture. 347 00:36:18,200 --> 00:36:24,680 What we have seen is 4R planar mechanisms of different varieties consisting of four 348 00:36:24,680 --> 00:36:33,520 revolute pairs or three revolute and one prismatic pair or two revolute and two prismatic pairs. 349 00:36:33,520 --> 00:36:38,980 When we have two revolute and two prismatic pairs, the order of the sequence of the pairs 350 00:36:38,980 --> 00:36:40,650 becomes important. 351 00:36:40,650 --> 00:36:49,400 We have talked of two varieties namely, RRPP or RPRP. 352 00:36:49,400 --> 00:36:55,800 At the end, we have also seen we have a three-link mechanism with a higher pair and that also 353 00:36:55,800 --> 00:37:05,400 can be equivalently represented by a four-link planar linkage having only R pairs or R and 354 00:37:05,400 --> 00:37:06,400 P pairs. 355 00:37:06,400 --> 00:37:06,401 14