Solution to CA-3 G1

8/3/2019 Solution to CA-3 G1

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Kinematics

and

Dynamics

ofMachines

Group 1Ravi Agarwal

Himanshu Dewangan

Sumateja

Tarun Rai

[ANALYTICAL SYNTHESIS OF

LINKAGES]


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Analytical synthesis of Linkages :

Problem statement:

Design a 4-bar linkage which will move a line on its coupler such that a point P moves from P1 to

P2while coupler rotates through 2.

Find out the lengths and angles of all links and coupler dimensions.


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SOLUTION:- Shown below is the carmetal construction diagram for the analytical synthesis of

linkages for parts (a),(b) and (c)

(a) Design a four bar mechanism to move the link CD from position 1 to position 2. Ignorethe third position and the fixed pivots O2 and O4 shown. Build a scale model and add a

driver dyad to limit its motion to the range of positions designed, making it a six bar.

Solution:

In analytical synthesis move point P1 to P2 while rotating the coupler by 2. For our case of problem

we will take point P1 on C1 and P2 on C2. We are using Nortons Four Bar software to synthesis the

module.

Steps:

Choose any appropriate coordinate system X-Y Draw vector P21 inclined at angle 2. Define position vector R1 and R2. Draw an arbitrary vector Z1 and then form vector Z2 with the same magnitude but with

angle 2 with Z1. Note that Z1 = Z2.

Draw vector W1 and W2 such that W1=W2 and both meet at O2 which is the pivot Write down the vector loop equation for the loop The diagram is shown above


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Now substitute the complex number equivalents for vectors

The sum of angles can be re-written as:-

Simplifying

(a) Now we have a complex equation, one real part and one imaginary.(b) There are 8 variables(c) The last three are known to us.(d) We are assuming values of z,,2.

P21 = 488.45

2 = -19.1055

2 = 22.89

P21x = 450; P21y= 190

We have assumed value of Z, and for simpler calculations:

Z= 0; = 33.7; = 213.7;2=-58.803;2=-69.75


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Fig :- Graphical construction in carmetal software

Graphical method Analytical Method

Link No. Length (m) Length (m)

1 0.3485 0.3484

2 0.4975 0.4975

3 0.4500 0.4500

4 0.4494 0.4494

(b) Design a four bar mechanism to move the link from position 2 to position 3. Ignore thefirst position and the fixed pivots O2 and O4 shown. Build a scale model and add a driver

dyad to limit its motion to the range of positions designed, making it a six bar.

Steps:

Choose any appropriate coordinate system X-Y

Draw vector P32 inclined at angle 2. Define position vector R1 and R2. Draw an arbitrary vector Z1 and then form vector Z2 with the same magnitude but with


Draw vector W1 and W2 such that W1=W2 and both meet at O2 which is the pivot Write down the vector loop equation for the loop (same as in part(a)).

P32 = 322.8

2 = 14.6

2 = -16.19

P32x = -310; P32y= 90


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We have assumed value of Z, and for simpler calculations:

Z= 0; = 14.95; = 194.595; 2=-49.27; 2=--53.26


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1 0.3454 0.3455

2 0.3872 0.3872

3 0.4500 0.4500

4 0.4272 0.4272

(c) Design a four bar mechanism to give the three positions. Ignore the fixed pivots O2 andO4 shown. Build a scale model and add a driver dyad to limit its motion to the range of

positions designed, making it a six bar.

Steps:



Draw vector W1 and W2 such that W1=W2 and both meet at O2 which is the pivot Write down the vector loop equations.

Substituting the complex number equivalents for the vectors

Rewriting as the sum of angles

Simplifying and rearranging

We have 12 variables, out of which 6 are known and we have 2 as free choices. Use values from (a) for P21, 2, 2.


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P21 = 488.45

2 = 19

2 = 22.89

P21x = 450; P21y= 190

P31 = 767.18

3 = -33.70

3 = 7

P31x = 760; P31y= 100; 2=-47.37;2=-43.85 3=-78.16;3=-76.16


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1 0.2619 0.2617

2 0.6079 0.6080

3 0.4500 0.4495

4 0.6882 0.6878


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(d) Design a four bar mechanism to give the three positions using the fixed pivots O2 and O4shown. Build a scale model and add a driver dyad to limit its motion to the range of

positions designed, making it a six bar.

SOLUTION:- Shown below is the carmetal construction diagram for the analytical synthesis

of linkages for part (d).

Steps:



Draw vector W1 and W2 such that W1=W2 and both meet at O2 which is the pivot Write down the vector loop equation for the loop


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Use values from (a) for P21, 2, 2.P21 = 488.45

2 = 19

2 = 23

P21x = 450; P21y= 190

P31 = 767.18

3 = 34

3 = 7

P31x = 760; P31y= 100; 2=-47.85;2=-54.98 3=-76.15;3=-92.87

Provide the coordinates of O2 and O4. We assumed O2 at origin and O4 at (300,0).

And point P1 as (-290,759.7)


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1 0.3000 0.30002 0.5931 0.5925

3 0.5271 0.5266

4 0.4084 0.4087

TOGGLE ANGLES

Figures of graphical toggle angles.


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Analytical transmission angle calculation


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Comparison of Graphical and analytical Methods:-


Question

No.

Right

toggle

angle

Left Toggle

angle

Right

toggle

angle

Left Toggle

angle

1 46.53 97.17 46.53 97.16

2 31.29 69.9 31.295 69.966

3 41.76 84.2 41.83 84.30

4 74.91 134.11 74.98 134.57

Contribution of each member:

Graphical Synthesis:1. All the members of the group discussed the problem and came up with the solution.2. Tarun Rai and Sumateja worked on making the CaRMetal diagrams and the power point

presentation.

3. Himanshu Dewangan and Ravi Agarwal built the physical model from the links obtained fromthe diagram.

Analytical Synthesis:

1) Analytical Solution was calculated by all four of us.a) First problem was solved by Tarun Rai.b) Second part was solved by SumaTeja.c) Third and Fourth part was done by Ravi and Himanshu together.

2) Car metal and Four Bar linkage was simultaneously used.3) Toggle angles was calculated altogether and compared.

Documents

Solution to CA-3 G1