THOERY OF MECHANISMS AND MACHINES Module-04 Displacement Analysis & Grashoff’s Criteria Instructed by: Dr. Anupam Saxena Associate Professor Department of Mechanical Engineering Indian Institute of Technology Kanpur anupams@iitk.ac.in FB 361 Prepared by: Abhishek Sharma Final Year Dual Degree Student Department of Mechanical Engineering Indian Institute of Technology Kanpur abhishr@iitk.ac.in FB369

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur M03- Recap Instantaneous Center Aronhold-Kennedy Theorem Example 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur Kinematic Analysis Displacement Analysis Determining position of various points on links, joints etc. Velocity Analysis Determining velocities of various points on links, joints etc. Acceleration Analysis Determining acceleration of various points on links, joints etc. 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Displacement Analysis We can analyse displacements and related parameters in any mechanism using two methods which will be demonstrated using several examples: Graphical (might need a tracing paper in several cases) Analytical 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur Graphical Method A VARIATION WHITWORTH’S QUICK RETURN MECHANISM Quick Return Ratio of a Whitworth’s Quick Return Mechanism is defined by the ratio between time taken by the mechanism for forward motion to time taken by it for backward motion. Since forward and backward are relative terms, QRR is usually taken to be greater than 1, by taking the ratio between the greater of the two times with the smaller one. Graphically determine QRR in the following: 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Quick Return Mechanism (G) 𝜃 1 𝜃 2 Since our crank rotates at constant angular velocity, 𝑄𝑅𝑅= 𝜃 2 𝜃 1 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Displacement Analysis (Graphical) RRRR MECHANISM Given all link lengths ( 𝑙 1 to 𝑙 4 ), can you find out final state of the mechanism when, 1. Initial state of 𝑙 2 is given? 2. Orientation of coupler ( 𝑙 3 ) is given (say 170° from horizontal)? For Case 2: Arcs are drawn with centres on fixed hinges & radii equal to link lengths. Now coupler is drawn on a tracing paper and it’s feasible position is found out. 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Displacement Analysis (Analytical) RRRR MECHANISM Given all link lengths and 𝜃 2 , find 𝜃 3 and 𝜃 4 . We’ll use vectors. 𝒍 𝟏 𝒍 𝟐 𝒍 𝟑 𝒍 𝟒 𝜃 2 𝜃 3 𝜃 4 𝒍 𝟏 + 𝒍 𝟐 + 𝒍 𝟑 + 𝒍 𝟒 =0 Get two equations by projecting along 𝒊 and 𝒋 axis or by taking real and imaginary parts in complex analysis (by using 𝒍 𝒋 = 𝑙 𝑗 𝑒 𝑖 𝜃 𝑗 ). −𝑙 1 + 𝑙 2 𝑐𝑜𝑠 𝜃 2 + 𝑙 3 𝑐𝑜𝑠 𝜃 3 + 𝑙 4 𝑐𝑜𝑠 𝜃 4 =0 𝑙 2 𝑠𝑖𝑛 𝜃 2 + 𝑙 3 𝑠𝑖𝑛 𝜃 3 + 𝑙 4 𝑠𝑖𝑛 𝜃 4 =0 Two equations, two unknowns. Bingo! 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur Grashoff’s Criterion 4 links together can create a wide range of possibilities for motion. Let’s introduce them to you. Double Rocker When both the links 2 & 4 “rock” Crank Rocker When one of them can make complete rotations Double Crank When both of them (links 2 & 4) make complete rotations Here, link 1 is grounded and link 2 is given input motion The input link is commonly known as crank, the grounded link is known as frame, link 3 is called coupler and link 4 is called follower 1 2 3 4 𝝎 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur Grashoff’s Criterion Let us denote lengths of various links as follows: 𝑙 = length of longest link 𝑠 = length of shortest link 𝑝 & 𝑞 = lengths of remaining links Now, following cases arise: When 𝑠+𝑙 < 𝑝+𝑞 If s is frame, then double crank If s is connected to frame, crank rocker If s is coupler, double rocker When 𝑠+𝑙 > 𝑝+𝑞 Only double rocker possible When 𝑠+𝑙 = 𝑝+𝑞 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur Grashoff’s Criterion Grashoff’s criteria is 𝑠+𝑙 < 𝑝+𝑞 If this criteria is satisfied then all kinematic inversions are possible, otherwise only double rockers are possible. Linkages satisfying Grashoff’s criterion are called Grashoff’s linkages. 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur In Class Exercise Find out what needs to be the relation between the link lengths of a slider crank mechanism if we want it to have a crank (like in the piston cylinder mechanism)? Also find which link should be placed where. 𝒍 𝟏 and 𝒍 𝟒 are clearly the longest links with almost equal lengths, the former is grounded. 𝒍 𝟏 ∞ 𝒍 𝟐 𝒍 𝟑 𝒍 𝟒 If there needs to be a crank, there could only be one. There is no possibility of double crank (why?). For a crank rocker smallest link should be connected to the frame, i.e. 𝑠 = 𝒍 𝟐 . Let 𝑙 = 𝒍 𝟏 ,𝑝 = 𝒍 𝟑 and 𝑞 = 𝒍 𝟒 Now, 𝑠+𝑙 < 𝑝+𝑞 for a crank (why?) 𝑠 < 𝑝, since 𝑙 =𝑞 Therefore 𝒍 𝟐 < 𝒍 𝟑 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur Pantograph (A) Given ∆𝐶𝐵𝐸 ~∆𝐴𝐷𝐵 find magnification and rotation of copying when the initial sketch is traced by D and E draws the resultant (copied) sketch O A B C D E Let the scale be 𝑚= |𝑶𝑬| |𝑶𝑫| 𝑶𝑬=𝑶𝑪+𝑪𝑬 & 𝑶𝑫=𝑶𝑨+𝑨𝑫 𝑶𝑬=𝑨𝑩+𝑪𝑬 & 𝑶𝑫=𝑪𝑩+𝑨𝑫 𝑶𝑬=𝑨𝑩+𝑚𝑨𝑩 𝑒 𝑖𝜃 & 𝑶𝑫=𝑚𝑨𝑫 𝑒 𝑖𝜃 +𝑨𝑫 Where, 𝜃 is the angle between 𝑨𝑩 & 𝑪𝑬 at any given time 𝑶𝑬=(1+𝑚 𝑒 𝑖𝜃 )𝑨𝑩 & 𝑶𝑫=(1+𝑚 𝑒 𝑖𝜃 )𝑨𝑫 Now, 𝑚= |𝑶𝑬| |𝑶𝑫| = |𝑨𝑩| |𝑨𝑫| = 𝐴𝐵 𝐴𝐷 And angle of rotation between 𝑶𝑬 and 𝑶𝑫 (angle of rotation of copying) is the same as angle between 𝑨𝑩 and 𝐀𝑫 i.e. ∠BAD 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur Assignment #2 For a Whitworth mechanism given in slide#5, write down all loop closure equations governing the variables of motion. (The equations we write using one closed loop of vectors are called loop closure equations, the vector equation in slide#8 is one such example. Variables of motion means the parameter which varies during motion, in a revolutary joint angle of rotation is the variable of motion whereas in a prismatic joint displacement is a variable of motion) In the following picture, link 1 & 2 are welded together, link 4 is welded to the moving shuttle. Find out if there is possibility of having a crank? If yes, what are the conditions? What are the extreme positions of AB? Prove the validity of Grashoff’s criterion for Grashoff and non-Grashoff linkages. A B C D E 13-08-2015 Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur