Frudenstin’s Mechanism

 

 Authors : Yash Gandhi, Gaurav Khadke, Shruti Gavare, Chaitanya Gawali, Shreyas Gaware

Mechanisms-

A combination of a number of bodies (usually rigid) assembled in such a way that the motion of one causes constrained and predictable motion to the others is known as a mechanism. Thus the function of a mechanism is to transmit and modify a motion.

 

Link –

A mechanism is made of a number of resistant bodies out of which some may have motions relative to the others. A resistant body or a group of resistant bodies with rigid connections preventing their relative movement is known as a link. A link may also be defined as a member or a combination of members of a mechanism, connecting other members and having motion relative to them. Links can be classified into binary, ternary and quaternary depending upon their ends on which revolute or turning pairs can be placed.

  

Kinematic pairs –

A kinamatic pair or simply a pair is a joint of two links having relative motion between them.

 

Kinematic Chain-

A kinematic chain is an assembly of links in which the relative motions of the links is possible and the motion of each relative to the other is definite.

 

Linkage-

Linkage is obtained if one of the link of a kinematic chain is fixed to the ground. If motion of any of the moveable links results in definite motions of the others, the linkage is known as a mechanism. However, this distinction between a mechanism and a linkage is hardly followed and each can be refered in place of the other.

 

What is the Four Bar mechanism?

It is the planer mechanism consisting of the four rigid members:

These members are connected by the four revolute pairs forming a closed-loop kinematics chain.

Four bar mechanisms have 1 degree of freedom

Types of four Bar mechanism

Grashof’’s Law-

S + L <= P +Q

S = length of shortest link

L = length of longest link

P = length of the 1 remaining  link

Q= length of other remaining link

             Double Rocker

Rocker and Crank 

Double Rocker

Frudenstein Mechanism Design Approach

The Freudenstein Mechanism is an analytical approach to the analysis and design of four-bar mechanisms, along with their variants.

So before Frudenstin’s Mechanism let’s discuss what is four-bar mechanisms or four-link mechanisms?

Before Ferdinand Freudenstein

Before Ferdinand, there were two methods to dram the mechanism.

1.     Graphical Method

2.   Inversion Method

Graphical method: Steps to Draw the mechanism is:

1.     Draw an Arc with A as the center of radius b

2.   Mark the point B on the circular arc such that the line AB makes the given angle Φ with the fixed frame

3.    With D as the center, draw a circular arc of radius d.

4.   With point B as the center draw a circular arc of radius c.

5.    The two circular arcs centered at B and D can intersect at most at two possible points — let they be denoted by C and C’

                                              Source: Reference [1]

By this technique, we can draw the Four bar mechanism but the problem here was that in this method there can be manual errors during drawings.

Inversion Method:

In this method, desired characteristics could be satisfied at a finite number of configurations, also called precision points, in the range of motion of the mechanism, also it has the advantage that there is no error during this method. This method was typically designed for the 3–4 precession points.

Steps to draw the mechanism using Inversion Method:

1.     Let the 4 bar mechanisms ABCD where

a) AD is a Fixed link

b) AB Input link (Crank)

c) BS is Coupler

d) CD Output link

2. Take 3 positions of crank AB1, AB2, and AB3 at different angles from the fixed link AB

3. Join B2D and B3D

4. Draw a circle with D as the center and Radius B2

5. Do the same As step 4, but change the radius to B3

6. Rotation Of Output link: rotate DB2 at an angle DB2B`2 will rotate the input link is at B3A

7. Joint B`2 And D

8. Rotation Of Output link: rotate DB3 at an angle DB3B`3 will rotate the input link is at B3A

9. Joint B`2 and B1

10. Join B`2 and B`3

11. Draw perpendicular bisector of B`2B1 and B`2B3

12. Intersection of above two Bicester will give A point

13. Joint A and D to get the output link

14. Join the AB crank link to point A to get the coupler link.

After Ferdinand Freudenstein

Ferdinand Freudenstein has developed the analytical method for the analysis and design of the four-bar mechanism. His thesis published in 1954 mentioned the analytical method. In this study, he has given the equation between the θ and ɸ in terms of the link length a, b, c, and d.

Freudenstein derived a detailed formulation to design four-link mechanisms when only one precision point is together unlike the inversion method which uses 3 precision to construct 4-bar mechanisms.

In the graphical approach, the geometry constructions become very complex compared to

Freudenstein’s approach can be easily programmed into a computer.

The Sum of the position of the vector along the horizontal direction will be zero

The Sum of the position of the vector along the vertical direction will be zero

aCOSθ + bCOSβ - cCOSɸ -d =0

bCOSβ = d + cCOSɸ - aCOSθ      ...(1)

aSINθ + bSINβ - cSINɸ =0

bSINβ =  cSINɸ - aSINθ      ...(2)

Now, we will square and add equations 1 and 2 i.e.

Where,

Eq.3 is Ferdinand Freudenstein Equation

 

REFERENCES-

[1] The Freudenstein Equation and Design of Four-link Mechanisms Paper by Ashitava Ghosal.

[2] S S RATTAN