Grashof's Law and Inversion

Experiment#3 To demonstrate the Grashof’s condition and Gruebler's equation for four bar mechanism. Objectives: 1) Gain a knowhow of Grashof’s conditions. 2) Apply it to determine the type of four bar mechanism. 3) Apply the Gruebler's equation and deduce results from it. Apparatus: Four bar mechanism. Theory: Four bar mechanism: Four bar mechanism is simplest form of mechanism. It consists of the minimum number of links required to make a mechanism that is to say four. Most of the four bar mechanisms are of degree of freedom equal to unity. In planar four-bar, mechanisms links from a closed chain, and each link moves in parallel plane to the other, the four links of the mechanism may act as the ground link that is fixed. Rocker, the link carrying out the reciprocating motion. Crank, the link completing the revolution. And an intermediate link connecting the rocker and crank. Four bar mechanisms may act differently according to the length of the links and the link grounded. Different variants of four bar mechanisms are discussed as the document follows. Grashof’s criteria: We know that by adjusting the length of the links of four bar mechanism, we may change the motion of it but by adjusting we are uncertain about the output that's where Grashof's analysis and its formula comes into play. According to Grashof if s +l≤ p + q Where “s=length of the shortest link”, “l=length of the longest link”, “p= length of the either of the other two links”, and “q= length of the either of the other two links”. Then the mechanism will have at least one crank, or in other words, at least one of the links of the mechanism will be able to rotate completely[4]. There are further cases, which are discussed below. s +l≤ p + q Case 1: When the shortest link is adjacent to the fixed link. In such scenario, the shortest link will act as a crank and will rotate with complete revolution, as the mechanism will operate. Example of such link is windshield wiper. The motor is attached to the shortest link and rotates. Whereas, the wiper oscillates on the windscreen. The following figure indicates the motion of the four links of the crank-rocker mechanism. Figure 1: Crank-Rocker Mechanism. Case 2: When the shortest link is fixed, we have a mechanism known as double crank mechanism. Such mechanism would have two links adjacent to the fixed link that is going to rotate completely. Such a mechanism is also known as drag link mechanism. Following figure shows the motion of the mechanism. Figure 2: Double-Crank Medchanism. Case 3: We encounter case 3, when we have the shortest link opposite to the fixed Link in such condition. The output is a double-rocker mechanism in this mechanism both the links adjacent to the fixed link act as rockers whereas the shortest link act as the 360-degree rotating link in some scenarios. The following figure shows case 3’s configuration. Figure 3: Double-Rocker Mechanism. s +l= p+q In this scenario we can have all the above-mentioned mechanisms in somewhat similar way. The interesting thing about the case is when the sums are equal, during the mechanism we have a time when all the lengths of the mechanism become collinear and due to this property, this mechanism is also known as change point mechanism. In this case we can have further two specifications which are as follows. Lengths are unique: When the lengths of the all the members are unique, we don't have some specific geometry to name but the mechanism works as change point and can become double crank, crank-rocker or double-rocker mechanism depending upon the scenario of the fixed link. Lengths of two links are same: When the length of two links is same, the length of the other two must also be the same to satisfy the equation. In this way we get two pairs of links of equal lengths. In this case, we have two geometrical variants, which are as follows: Parallelogram linkage: When the equal length links are parallel are placed opposite to each other. We get a configuration called parallelogram linkage. As evident from the mechanism is that we get a shape of a parallelogram as shown in the following figure. Deltoid linkage: The equal length members are adjacent to each other and they seem to form a shape of a delta, both parallelogram and deltoid linkage can work as different mechanisms depending upon the link that is fixed. Both configurations are shown in the following figure. Figure 4: Change Point Mechanisms. s +l> p+q When this inequality holds then none of the links of the four bar mechanism acts as a crank. All three links act as rocker and the mechanism is known as triple rocker mechanism. This mechanism is rarely useful. It works and looks like as follows. Figure 5: Triple-Rocker Mechanism. Practical examples: Beam engine: In beam engine, crank-rocker mechanism is used, and the purpose of it is to convert rotational motion into oscillating motion, this sort of mechanism is also used in car wipers. Also, the mechanism is shown in the following figure. Figure 6: Beam Engine. Coupling rod of locomotives: The mechanism used for connecting the wheels of the Railway is a parallelogram linkage which is already discussed during the document. The mechanism is a double crank mechanism and the crankshafts are connected to the wheels. Figure 7: Parallelogram Linkage in Locomotive Wheels. Inversion: Inversion of a kinematic chain is a phenomenon in which the overall motion of the mechanism is changed and it is achieved by choosing different link as the grounding link. The relative motions of the links remain somewhat similar, but the overall motion is disturbed. As the overall motion is different, the use and the look of the mechanism must also be different. Each of the achieved mechanism is therefore known as an inversion of a kinematic chain. From the Grashof’s law we have seen that there are four inversions of a four bar mechanism. It's not just a coincidence. In fact, the number of inversions of a kinematic chain is always equal to the number of kinematic links in the mechanism, and this doesn’t need any proof. As the inversions depend on the number of times the grounding length can be changed, which is exactly equal to the total number of the links[7]. During the case studies of Grashof’s law inversion was also indirectly discussed. Gruebler’s Criterion: Gruebler's criteria are the modification of Kutzbach equation. According to Gruebler, if we want to make a mechanism with degree of freedom equal to one and it contains no higher pair, it must fulfil or satisfy the following equation. 3 N−2 J =4 We can tell at a glance that Gruebler derived this equation from Kutzbach equation, as follows. 3 ( N−1 ) − 2 LP−HP = DOF In the above equation we substitute DOF=1, HP=0, and LP=J, where J is the no of joints in the mechanism. So we get. 3 ( N−1 ) −2 J −0=1 3 N−2 J =4 This is the same as Gruebler’s equation. Significance of Gruebler's equation: Gruebler's equation can prove to be very much important. The first thing we may use Gruebler's equation is for checking if certain mechanism have degree of freedom equal to unity or not. And we know that if a mechanism has no higher pair, it must satisfy the above mentioned Gruebler's equation in order to have unity degree of freedom. Another important insight that Gruebler's equation gives us is that the mechanisms with unity degree of freedom and no higher pair should have even number of links. This take away from Gruebler’s equation is welcomed and cherished unanimously as it tells us and researchers that in order to make useful mechanisms we must confine ourselves to even number of links. How we deduced this? We have Gruebler's equation as follows. 3 ( N−1 ) −2 J =1 3 N=2 ( J + 2 ) Anything multiplied by 2 becomes even, so the right hand side is always even and to make left hand side even also, the N must be even because the product of odds gives another odd and product of an odd and even gives an even number. So the links should be even. The first even number we have is 2, but a mechanism with 2 links cannot be made. The next one is four, so the simplest mechanism can be made with the help of four links. And some of the common four bar mechanisms that are going to be discussed in detail in the subsequent lab sessions are as follows: Slider crank mechanism. Quick return mechanism. Double crank mechanism. Double-rocker mechanism.[6] Observations and calculations: In this part, various mechanisms are judged on the basis of grounding linkage and the length of the other ones. Figure 8: Mechanisms to Judge Their Type. Mechanism 1: In this mechanism we have s=3, l=7, p=6, and q=5. And from Grashof’s law we have. s +l=10<11= p+q So, at least one link will be rotary. Now, smallest link is adjacent to the fixed link. So the mechanism is a crank-rocker mechanism. Mechanism 2: In this mechanism, we have s=4, l=8, p=5, q=6, so by Grashof’s law we have. s +l=12>11= p+q So we will have no rotating link, and this mechanism is regarded as triple rocker mechanism. Mechanism 3: In this mechanism we have, s=4, l=8, p=7, and q=6, so by Grashof’s law we have. s +l=12<13=p +q So the mechanism will have at least one rotating link as the shortest link is grounded link, both the links adjacent to the ground link will rotate and we'll get our double crank mechanism. Comments: 1) As evident from the Gruebler's equation, four bar mechanism is the simplest mechanism. 2) Four bar mechanism’s different inversions have different useful applications. For example, crank-rocker mechanism is used in car wipers, and double crank parallelogram linkage is used in locomotives.