1. 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
FB 361
Prepared by:
Abhishek Sharma
Final Year Dual Degree Student
Department of Mechanical Engineering
Indian Institute of Technology Kanpur
FB369

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

3. 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.
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

4. 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
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

5. 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:
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

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

7. 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.
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

8. 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!
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

9. 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
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

10. 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
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

11. 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.
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

12. 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
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

13. 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
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur

14. 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
Dr. Anupam Saxena, Abhishek Sharma IIT Kanpur