Basic Kinematics Concepts and Kutzbach Equation PDF

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This document contains the material related to the Engineering Dynamics Lab. The theory is comprehensively written and the calculations are accurate and precise....

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Experiment # 1 To understand and demonstrate the concept of degree of freedom in a plane types of links, joints, and lower and higher pairs. Objectives: 1. To form sound basis of kinematics concepts like those described in the title. 2. Analyze mechanism and calculate their degree of freedom. Apparatus: Not applicable. Theory: In this lab session, basic concepts of kinematics have been extensively discussed. The major concepts are as follows: Degree of freedom: Degrees of freedom in mechanics indicate or refer to the number of variable required to specify the position or state of a mechanism. Considering a body in three-dimensional space, it can be translated along the three axis and also can be rotated about any of them. So it can be assumed that it have three translation degrees of freedom and three rotational degree of freedom. So in total rigid body have six Degrees of freedom, if it is unconstrained in the space. Following figure presents the motion available for a rigid body in space.

Figure 1:Motion of Rigid Body in Space. Figure 2:DOF of a Rigid Body in Plane. Now, we consider a body in a plane and calculate its degree of freedom by inspecting. We can see that the body can translate along the two axes of plane and it may also be rotated about the third axis. So, the degrees of freedom of a rigid body in the plane is 3. These three degrees of freedom are shown in the following figure.

Degrees of freedom for complex systems cannot be calculated by mere inspection, which demands the need of an equation or some sort of a mathematical formula and here kutzbach equation comes into play.[1] Kinematic link: Each element or part of machine or mechanism is regarded as the kinematic link. Kinematic link is simply a rigid body, which combines with others to form mechanisms, machines and structures.

Figure 3:Labelled Links in Mechanism. There are three variants of Link available in mechanics. Each of them is briefly over viewed as follows: Rigid Link: Rigid link is the one which while operating in mechanisms, machines or structures do not deform. Deformation here, means that it's dimensions do not change even when acted upon

Figure 4: Rigid Links. by stresses. One thing to note is that in reality every link undergo some sort of deformation to some extent but these deformations are not appreciable as compared with the dimensions of the body. So such link may be called rigid links. The piston, crank and cylinder are the rigid link as shown in the figure below. Flexible links:

Flexible links undergo some sort of deformation while working. The sort of deformation, which the link undergo, do not alter the working of the assembly, sometimes that this deformation aids in carrying out the operation and an example of flexible link is the belt in the motor which undergoes deformation during motion. The following figure shows a belt assembly with a motor and we can well imagine that when the motor rotates the belt is stretched.

Figure 5: Flexible Links. Fluid links: As the name suggests fluid link is composed of a fluid. The fluid may be gas or some sort of liquid and important thing to note in this regard is that the motion through this link can be transferred, only by compression and expansion. And pressure is the element that plays an important role. We are well aware of the assembly used to lift heavy cars, and machinery. It works on principle that pressure through some fluid will remain the same. In these assemblies, as the one shown in the following figure, the intermediate transmission link is fluid.

Figure 6: Fluid Links. Kinetic pair:

Simple pair is said to be formed when two links come in contact with each other. It's a common observation that whenever the bodies are in contact, their relative motion is of concerned, rather than individual motion. If it appears that the relative motion is completely or successfully constrained, we say that pair to be a kinematic pair.[2] Now, what does completely or successfully constrained motion means, completely constrained motion is the one in which the link moves in a specific direction relative to other and its motion is unchanged to altering the direction of the force being applied. Similarly, successfully constrained motion means that the motion itself is not constrained. However, if some external forces or load is applied, the motion become constrained, So if a pair is exhibiting any of the above-mentioned scenario, it is said to be kinmetic pair. However, if the motion between pairs is not dependent on each other and takes place in more than one direction, we call it a simple pair and the motion is said to be incompletely constrained. The following figure shows an example of each type of motion.[3]

Figure 7: Types of Relative Motion. Classification of kinematic pairs: Kinematic pairs can be classified into different categories based on certain classification standards. According to relative motion: Kinematic pairs are divided into five types, according to the relative motion between the elements of the pair. These types with a brief are as follows. Sliding pair: A kinematic pair is said to be sliding pair if one element can slide only relative to the other element. You find a handful example of Such pairs. Some of them are square bars in a square hole. The Piston cylinder device in the reciprocating engine and the lathe bed and tailstock of the lathe machine. We can well imagine that the motion in these examples is only sliding of one pair or one part relative to other. Sliding pair has completely constrained motion and Degrees of Freedom is equal to 1.

Figure 8:Sliding Pair.

Turning pair: In the Turning pair, one link has the leverage to only rotate or revolve around a fixed axis of the other part of the pair. Such pair also have a completely constrained motion and degrees of freedom is equal to 1. A simple example of the turning pair is the wheel of bicycle. Another one is the crankshaft of reciprocating engine. As we look around ourselves, we may find many other example of turning pair.

Figure 9:Turning Pair. Rolling Pair: Rolling pair is specification of such an assembly in which one part of the link is rolling over, in or on the other elements. Simplest example is of ball bearing and roller bearing. Another example is machines on wheels. The bottom floor and the wheel make a rolling pair. Belt and pulley system is yet another example of the rolling pair.

Figure 10: Rolling Pair. Screw Pair: Screw pair is the one in which link move about the other link by virtue of threads, such pair also have one degrees of freedom, the relative motion of the screw pair, is a combination of both turning and sliding motion.

Figure 11:Screw Pair.

Spherical pair: A spherical pair is formed when one of the two links is in the form of a sphere and it swirls around the other link. The best example of this pair is ball and socket joint. And we can, from mere observation guess the degrees of the freedom of such pair to be 3. The other member of the pair forms cup or the valley in which the first member of the pair swirls. The pen stand with a ball and socket joint is perfect example of such link.

Figure 12:Spherical Pair. According to the type of contact between links: On the basis of contact, we have two types of pair which are as follows. Lower pair: In lower pair, the two elements have a complete surface contact and during their relative motion, the surfaces of the elements are sliding or turning over each other relatively. Example of lower pair are turning pair, sliding pair, and screw pair. We can find the example

Figure 13:Lower Pair. of lower pair in the real life in the form of a piston-cylinder device and shaft rotating in the bearing. The following figure shows, one of the assembly with lower pair spotted. Higher pair: The pair is known as higher pair when the relative motion between them is partially turning or sliding and during their relative motion the type of contact is a line or a point contact. Another important thing to note here is that either of the elements of the higher pair should have a curve in its shape. Common example of higher pair is cam and the follower.

Figure 14:Higher Pair.

As evident from the figure in every assembly at least one of the members is having curved shape. Another thing is to note that whenever the cam is rotating a point, contact is maintained. According to the type of mechanical constraints: Based on the type of closure between elements of the pair, we have two types. Self closed pair: Self closed pairs are such pairs in which the assembly is such that only the required type of motion can occur. For example, we know that in piston-cylinder device, the piston can move only in the cylinder so it forms a self closed pair. And an intresting thing to know is that all the lower pairs are also self closed pairs. Force closed pair: In force closed pair, as the name indicates, the elements are not held together rather an external force is constantly acting to maintain the contact. An example of this type of pair is cam and spring loaded follower.[2] 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

Figure 15:Kinematic Chain. say that when the last link of the combination of pairs is joined with the first link of the combination we get kinematic chain. A simple kinematic chain maybe visualized from following figure. Types of joints: A kinematic chain is exclusively a combination of links tied together by joints. There are different types of joints found in the kinematic chains, which may be defined as follow. Binary:

This is the basic type of joint. Two kinematic links are joined with the help of a binary joint. The kinematic chain shown schematically in the figure contains links that are all binary joints.

Figure 16: Binary Joint. Ternary: As the name indicates, it is such a link through which three links of the kinematic chain are joined together.

Figure 17: Ternary Joint. Quaternary: At quaternary joint four links are joined together. In the following figure 5th, joint is quaternary, and rest of all are ternary joints.

Figure 18: Quaternary Joint. 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 part 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. The following figure shows a common mechanism called the slider crank mechanism.

Figure 19: 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. Structures: A structure is an assembly of kinematic pairs joined in such a way that there is no relative motion. In other words, their degree of freedom is either 0 or less than 0. Structures are usually used to bear loads or transfer forces. We find many examples of structure around us. However, the most common of them is truss. The trusses are used to bear loads and are used in the making of bridges etc. Following figure shows a structure.

Figure 20: Structure.

Formula of calculating degree of freedom: Simple mechanisms are easy to analyze, and we can tell about their degrees of freedom to great accuracy by mere inspection. However, in the case of complex mechanism, we cannot rely on mere inspection for the calculation of degree of freedom. So we need a generic equation for the calculation of the degree of freedom of any planer mechanism, and this equation is known as Kutzbach equation. The Kutzbach equation may be derived as follows. We know, that any rigid body have 3 degrees of freedom in a plane. We assume that the mechanism have “N” number of links. As we know that one of the link in the mechanism is fixed, so it's degree of freedom is 0. So the combined degree of freedom of system is then, DOF=3(N −1) Now, this equation give DOF when the bodies are not in the form of pairs. If the links are joined by a lower pair, we can tell that reduction of two DOF in the whole system occurs. Similarly when joined by a higher pair reduction of 1 DOF occurs. So the above equation becomes Kutzbach equation which is as follows. DOF=3 ( N−1 )−2 LP−HP Where, N= Number of links. LP= Number of Lower Pairs. HP= Number of Higher Pairs. Degree of freedom and it’s interpretation: We can have a variety of degree of freedom for different assemblies of kinematics pair, they are interpreted as follows. DOF < 0: The kinematic chain is statically indeterminant structure. DOF = 0: The kinematic chain is a statically determinant structure. DOF = X, where X > 0: The kinematic chain is a mechanism and requires “X” number of inputs to drive it. Observations and Calculations: We need to calculate the degree of freedom of mechanism in the following figure.

Figure 21: DOF Calculation. For A: We have N=3, LP=2 (between 1&3 and 1&2) , and HP=1 (between 2&3), so putting these values in the Kutzbach equation we have: DOF = 3(3-1) -2(2) -1= 1 For B: We have N=4, LP= 3, and HP= 1 (between 1&4),so from Kutzbach equation we have: DOF= 3(4-1) -2(3) -1= 2 Now the following figure have another mechanism for which we need to calculate degree of freedom.

Figure 22: DOF Calculation. In this figure we have N=8, LP= 10 (All the pairs are lower pairs and are shown by red dots), HP= 0, so by Kutzbach equation we have: DOF= 3(8-1) -2(10) -0=1. Now the last figure for the calculation of degree of freedom is as follows:

Figure 23: DOF Calculation. In this figure we have N=5, LP= 7, HP=0, and we have double joints at 2&3, so from the Kutzbach equation the degree of freedom is calculated as follows. DOF= 3(5-1) -2(7) -0= -3.

So this is a statically indeterminant structure. Comments: The basic concepts form a sound basis of the subject and related courses in the future. In this lab session the fundamentals of dynamics lab are comprehensively gone over....