Kinematic Diagrams and Gruebler's Criterion

Experiment To understand and demonstrate kinematic diagrams, difference between a kinematic chain, mechanism, and machine and deriving the gruebler's equation for the degree of freedom of mechanism. Objectives: 1. Learn to make kinematic diagrams of complex mechanisms. 2. Be able to differentiate the kinematic chain, mechanism, and machine. 3. Brief overview about gruebler's equation and deriving it. Theory: Kinematic diagrams: Kinematic diagrams are the schematic diagrams, used to simplify the mechanisms, structures or machine for analysis. Kinematic diagrams usually show the skeleton of the links and joints with which the links are joined together. Kinematic diagrams are the counterpart of free body diagrams in dynamics. As free body diagrams made the analysis simpler and convenient in statics, kinematic diagrams aid in dynamics. While working with kinematic diagrams there are certain standard representations that may be followed to avoid any sort of inconveniences in the analysis. These standard representations are shown in the following table. Table 1: Kinematic Diagrams Representations. Name Slider joint. Real diagram Kinematic representation Screw joint. Pin in a slot. Cam joint. Gear. And also another convention that followed is, links are to be numbered and lettering is used to denote the joints. Numbering of links start from the one that is fixed, that link also serves as frame of reference for the motion of the mechanism[4]. Now, we would also see how useful kinetic diagram could be. Scotch yoke mechanism: This is a mechanism that converts rotational motion into linear sliding motion. Visually it seems very complex, but as shown, in the following figure, the kinetic diagram made, it look simpler and easier to be analyzed simultaneously.[5] Figure 1: Kinematic Diagram of Scotch Yoke Mechanism. Four bar window wiper mechanism: The wiper for cleaning the windscreen of automobiles is also a four bar mechanism. Its kinematic diagram is as follows.[5] Figure 2: Kinematic Diagram of Windscreen Wiper. Mechanism with complex link: Mechanisms with complex links are tedious to even look at. Here kinematic diagrams, come into play and make it look a lot simpler. Figure 3: Kinematic Diagram of Complex Link Mechanism. Some of the important things to note while making kinematic diagrams are as follows: We don't need to bother about exact dimensions of the links. A simple schematic diagram will suffice for the calculation of degree of freedom. However, if we wish to determine position, displacement, rotation or speed etc. in our analysis, we must scale our diagrams. Anything, irrelevant to the mechanism is disregarded straightforward. All the fix links are considered as one or we may put it as “All the fixed links are counted once”. The only formula for making the right kinematic diagram is right identification of links and types of joints. The point where input force is applied has no significance in kinematic diagrams. Pin joints form lower Pairs. Ternary and quaternary joints are assumed to form two and three kinematic pairs at the same spot respectively. Any link, not taking part in transmission of motion is not a part of mechanism. And hence, not a part of kinematic diagram also. Mechanism: Mechanism is essentially a kinematic chain having at least one of the links that is fixed. By fixing a link we make it possible that a mechanism may be used to transfer motion. Mechanism maybe thought of as a system containing constrained rigid bodies. As the rigid bodies are constrained we may expect a desired output motion by carefully analyzing and combining the rigid bodies, which is the sole purpose of mechanisms. Mechanisms are really crucial and play a very important role in the design of various parts of mechanical systems. For example, mechanism is used in reciprocating engines, which convert the translatory motion to rotary motion. There are three basic type of mechanism. Simple mechanism is the one containing four links. If the links are more than four we call it as compound mechanism. Machine: Machine is a sort of mechanism that is used to transfer motion as well, as power. For example, we have slider crank mechanism, which converts the reciprocating motion to rotary motion. While the same mechanism act as machine in piston cylinder device, where it is also used to transfer power. Kinematic Chain: A system of kinematic pairs may be regarded as a kinematic chain if each link of the system is simultaneously a part of two pairs. Moreover, the relative motion between the links is either completely constrained or successfully constrained. To put it in a brief form we may say that when the last link of the combination of pairs is joined with the first link of the combination we get kinematic chain. 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 section, kinematic diagrams of various mechanisms are made and degree of freedom is calculated. In the entire kinematic diagram standard is followed and lower pairs are shown by red dots and higher pairs are shown by green dots. Mechanism 1: Following figure shows the actual mechanism and its kinetic diagram. Figure 4: DOF Calculation and Kinematic Diagram. In this mechanism the variables as from the kinematic diagram are, N=5, LP=5, HP=1. The calculation of DOF is as follows. 3 ( 5−1 ) −2 ( 5 )−1=DOF DOF=1 Mechanism 2: The kinematic diagram and actual mechanism are as follows: Figure 5: DOF Calculation and Kinematic Diagram. The variables for this mechanism as evident from the kinematic diagram are N=6, LP=7 and HP=0, the degree of freedom of the mechanism is calculated as follows: DOF=3 ( 6−1 ) −2 ( 7 )−0 DOF=1 Mechanism 3: We will view such mechanism as 2D by considering the side view and then making the kinematic diagram for that. Figure 6: DOF Calculation and Kinematic Diagram. For this mechanism we have N=4, LP=4, Hp=0. Its DOF is calculated as follows: DOF=3 ( 4−1 ) −2 ( 4 ) −0 DOF=1 Mechanism 4: The kinematic diagram of actual mechanism is shown in the following figure. Figure 7: DOF Calculation and Kinematic Diagram. The links of the mechanism are, N=3, lower pairs, LP=2, and higher pairs, HP=1. Its degree of freedom is calculated as follows: DOF=3 ( 3−1 ) −2 ( 2 )−1 DOF=1 Comments: 1) The Kutzbach equation, have one drawback that it gives us misleading results in some cases. These cases, no doubt are rare. As we know that Kutzbach equation pays no attention to the geometry of mechanism. So some special geometries as the one shown in the following figure make Kutzbach equation face failure. Figure 8: Failure of Kutzbach Equation. Its degree of freedom from Kutzbach equation is calculated is as following. DOF=3 ( 5−1 ) −2 ( 6 )−0 DOF=0 However, from just viewing and practically checking we have it's degree of freedom equal to unity.[5] 2) The degree of freedom calculated tells us the mobility of the mechanism. In other words, it tells us the amount of inputs required to run the mechanism. The inputting device is known as actuator. 3) Gruebler’s criteria defined the path of developers of mechanisms. Actually he halved the work of them by telling the world that to make mechanisms of unity mobility we should use even number of Links.