Dme 5201 tom lecture 3 4 constrained motion, types of joints, kutzback, joint, grubler,dof
Download as PPTX, PDF0 likes650 views
This document provides an overview of kinematic chains and concepts related to their motion analysis including: 1. It defines constrained motion and classifies it as completely, incompletely, and successfully constrained. 2. It introduces kinematic chains, distinguishing between open and closed chains. Formulas are provided to analyze chains using the number of links, joints, and pairs. 3. Common criteria for analyzing kinematic chains are described, including Kutzbach's criterion for determining degrees of freedom and Grubler's criterion for minimum links required for single degree of freedom chains.
Dme 5201 tom lecture 3 4 constrained motion, types of joints, kutzback, joint, grubler,dof
- 1. THEORY OF MACHINES (DME 5201) UNIVERSITY POLYTECHNIC GLA UNIVERSITY, MATHURA LECTURE 3-4
- 2. MODULE -1_Lecture 3- 4 : Contents UNIVERSITY POLYTECHNIC GLA UNIVERSITY, MATHIURA 2 • Constrained Motion • Types of joints (otb connectivity) • Kinematic chain and conditions • Kutzback criteria • Grubler’s criteria • Degree of freedom
- 3. Constrained Motion • Constrained motion results when an object is forced to move in a restricted way. – For example a fan rotates on its axis, it is not allowed to translate – Wheel of car is made to rotate only – Piston is made to slide inside cylinder • Constrained motion can be classified as- 1. Completely constrained motion 2. Incompletely constrained motion 3. Successfully constrained motion UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 3
- 4. 1. Completely Constrained Motion When the motion between a pair is limited to a definite direction irrespective of the direction of force applied, then the motion is said to be a completely constrained motion. • Example- 1. The triangular bar can translate only 2. The round bar can rotate only and translation is restricted by using collars UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 4
- 5. 2. Incompletely Constrained Motion When the motion between a pair can take place in more than one direction, then the motion is called an incompletely constrained motion. • Example- 1. The round bar can rotate as well as translate inside round hole UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 5
- 6. 3. Successfully Constrained Motion A kinematic pair is said to be partially or successfully constrained if the relative motion between its links occurs in a definite direction, not by itself, but by some other means. Example- 1. Footstep bearing 2. Piston UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 6
- 7. Kinematic Chain UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 7
- 8. In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained (or desired) motion. Kinematic chains can be classified as: • Open Kinematic Chain: There are bodies in the chain with only one associated kinematic joint • Closed Kinematic Chain: Each body in the chain has at least two associated kinematic joints The above concept is enough to study some criterions and nature of kinematic chains. K’Chains will be continued in lecture 5 and onwards. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 8
- 9. Finding out Nature of Kinematic Chain Consider, L= Number of links J= Number of Joints P= Number of Pairs If each link is assumed to form two pairs with two adjacent links, then , L=2P- 4 …...……………..(1) Relationship between number of links and joints can be given as, J= (3/2)L - 2 ………………(2) Now these two equations (1) and (2) will be applied to find our nature of kinematic chains. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 9
- 10. CASE I: 3 Links o Adjacent arrangement has three links connected to each other in the manner shown. o Here, Number of Links = L = 3 Number of Joints = J = 3 Number of Pairs = P = 3 From Equation (1), L=2P-4 3=(2*3)-4 3=2 L.H.S. > R.H.S. Conclusion: Since the arrangement does not satisfy equation (1) and (2) and L.H.S. is greater than R.H.S. therefore its not a Kinematic Chain and hence no relative motion is possible. These chains are called Locked Chains and form structure. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 10 Link 1 Link 3 Link 2 From Equation (2), J=(3/2)L-2 3=(3/2)3-2 3=2.5 L.H.S. > R.H.S.
- 11. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 11 CASE II: 4 Links o Adjacent arrangement has four links connected to each other in the manner shown. o Here, Number of Links = L = 4 Number of Joints = J = 4 Number of Pairs = P = 4 From Equation (1), L=2P-4 4=(2*4)-4 4=4 L.H.S. = R.H.S. Conclusion: Since the arrangement satisfy equation (1) and (2) and L.H.S. is equal R.H.S., therefore its a Kinematic Chain with one d.o.f. By fixing any one link we find definite motion, hence it may be called constrained kinematic chain. Link 1 Link 3 Link 4 Link 2 From Equation (2), J=(3/2)L-2 4=(3/2)4-2 4=4 L.H.S. = R.H.S.
- 12. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 12 CASE III: 5 Links o Adjacent arrangement has three links connected to each other in the manner shown. o Here, Number of Links = L = 5 Number of Joints = J = 5 Number of Pairs = P = 5 From Equation (1), L=2P-4 5=(2*5)-4 5=6 L.H.S. < R.H.S. Conclusion: Since equation (1) and (2) show L.H.S. is smaller than R.H.S. i.e. does not satisfy, hence is called unconstrained chain, means relative motion is not completely constrained. But they have some practical importance. Link 1 Link 3 Link 4 Link 2 From Equation (2), J=(3/2)L-2 5=(3/2)5-2 5=5.5 L.H.S. < R.H.S. Link 5
- 13. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 13 CASE IV: 6 Links (Series) o Adjacent arrangement has three links connected to each other in the manner shown. o Here, Number of Links = L = 6 Number of Joints = J = 6 Number of Pairs = P = 6 From Equation (1), L=2P-4 6=(2*6)-4 6=8 L.H.S. < R.H.S. Conclusion: Since equation (1) and (2) show L.H.S. is smaller than R.H.S. i.e. does not satisfy, hence is called unconstrained chain. Hence, Series connection beyond 4 links gives unconstrained chain Link 1 Link 3Link 5 Link 2 From Equation (2), J=(3/2)L-2 6=(3/2)6-2 6=7 L.H.S. < R.H.S. Link 4 Link 6
- 14. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 14 CASE V: 6 Links o Adjacent arrangement has three links connected to each other in the manner shown. o Here, Number of Links = L = 6 Number of Joints = J = 7 Number of Pairs = P = 5 From Equation (1), L=2P-4 6=(2*5)-4 6=6 L.H.S. = R.H.S. Conclusion: Since equation (1) and (2) are satisfied, hence it is a kinematic chain. From Equation (2), J=(3/2)L-2 7=(3/2)6-2 7=7 L.H.S. = R.H.S. 1 3 6 5 2 4 NOTE- The chain having more than four links are known as Compound K’Chain.
- 15. TYPES OF LINKS AND JOINTS (On the basis of connectivity) UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 15
- 16. Types of Links UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 16 Remarks •Connected with two links •Can connect with three links •It is equivalent to two binary links •Can connect with four links •It is equivalent to three links
- 17. Types of Joints Sr. No. Name of Joint Illustration Remarks 1 Binary Joint 2 Ternary Joint One ternary joint is equivalent to two binary joints 3 Quaternary Joint One quaternary joint is equivalent to three binary joint UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 17
- 18. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 18 If there are n number of links connected at one junction, then at that junction, No. of Binary Joints = n-1
- 19. Klein’s Equation • To determine the nature of chain following equation is used, which was given by A.W. Klein, Where, j = number of binary joints h = number of higher pairs L= number of links If L.H.S = R.H.S, kinematic or constrained chain If L.H.S > R.H.S, chain is locked chain UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 19
- 20. Number of Degree of Freedom and Kutzbach Criterion • Degree of freedom can be defined as how many possible moves can a mechanism or kinematic chain may have. • If a mechanism has L number of links, J is number of binary joints or lower pairs and h is number of higher pairs, then according to Kutzbach criterion, d.o.f. can be given by for plane mechanisms- n= 3 (L-1) – 2 J - h UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 20
- 21. Grubler’s Criteria • It is derived from Kutzbach criteria where d.o.f. is taken as one and total higher pairs as zero. From Kutzbach criteria, n= 3 (L-1) – 2 J – h Putting n=1, 1= 3 ( L-1) – 2 J Case1: For, L=1, J=0 LHS > RHS Case 2: For, L=2, J=1 LHS = RHS Case 3: For, L=3, J=3 LHS > RHS Case 4: For, L=4, J=4 LHS = RHS Simplest possible mechanism is possible with least four links. Single d.o.f. is not possible for mechanisms having odd number of links. UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 21
- 22. Grashoff’s Law • The Grashof's law states that for a four-bar linkage system, the sum of the shortest and longest link of a planar quadrilateral linkage is less than or equal to the sum of the remaining two links, then the shortest link can rotate fully with respect to a neighboring UNIVERSITY POLYTECHNIC, GLA UNIVERSITY, MATHIURA 22