5.1 Steps in synthesis process Type number and dimensional synthesis

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TOM2

Unit – 5

Synthesis of Mechanism

5.1 Steps in synthesis process: Type, number and dimensional synthesis

Kinematics synthesis

Synthesis process may be accomplished, in general, in three interrelated faces:

The form or type synthesis of mechanism,

The number synthesis, and

The dimensional synthesis.

1. Type synthesis

Type synthesis refers to the kind of mechanism selected.

The option can go in favor of gear combination, a belt Pulley combination, a linkage, or a cam mechanism.

During this early stage of synthesis, one may have to consider design aspects like space consideration, safety consideration, overall economics, manufacturing process, material selection, etc.

Type synthesis is thus an involved process and makes it very difficult to lay down a systematic procedure aimed at unique determination of mechanism that assures specified motion characteristics.

2. Number synthesis

Number synthesis is based on the most obvious external characteristics of a kinematics chain, namely, the number of links together with the number and type of joints required for a specified motion.

3. Dimensional synthesis

Dimensional synthesis aims at determining significant dimensions of links and the starting position of links in a mechanism, to accomplish the specified task and prescribed motion characteristics.

Hinge pin to hinge pin distances on binary, ternary, quaternary links angles between arms of bell crank levers, cam contour dimension, the diameter of the roller of follower, gear ratios, eccentricity are just a few examples of significant dimensions.

5.2 Tasks of Kinematic synthesis: Path, function, and motion generation (Body guidance).

1. Path generation

In path generation, a point on the coupler link is a constraint to describe a path regarding a fixed frame of reference.

Generally, the portion of the path is an Arc of the circle, an ellipse, or a straight line.

Sometimes the tracing point May be required to follow a path that crosses over itself in the form of the figure of eight.

A linkage mechanism whose points C on coupler is required to follow a part y = f(x) as the crank rotates, is an example of the past generation.

Other typical examples include the thread guiding eye on a sewing machine and mechanism to advance the film on camera.

2. Function generation

A function generator represents a major category of synthesis problems.

It is frequently required in a kinematic synthesis that an output link must rotate, oscillate or reciprocate according to a prescribed function of time or input motion.

This is called function generation.

For instance in a four-bar mechanism, for generating a function y=f(x), x would represent the motion of input crank and the proportions of link length of the mechanism are to be so chosen that the motion of output link represents the function y=f(x) approximately.

Another example of function generation can be seen in linkage mechanism to correlate steering angles of axes of front wheels of an all-terrain vehicle with the relative speed at which each driving wheel should rotate avoiding skidding or access wear.

3. Motion generation

Motion generation or rigid body guidance requires that an entire body be guided through a pre-selected sequence of motion.

Such a body is guided usually as a part of the coupler.

For example, in the construction industry, heavy parts such as buckets and blades of bulldozer must be moved through a series of prescribed positions.

In the case of buckets of bucket loader mechanism, the tip of the bucket on coupler must have a prescribed path.

Path to be followed by the tip of the bucket is very important because the tip is supposed to have a scooping trajectory followed by lifting and dumping trajectory.

In past generation one is concerned with the path of tracer., In motion generation, the entire motion of the coupler link is of concern.

5.3 Precision Positions

Function generation and path generation problems can be solved either by approximate or exact approach.

The difference between the two approaches can be understood if we compare Watt approximate straight-line motion mechanism with Hart and Peaucelliar straight-line mechanism.

Being an approximate straight-line motion mechanism, the tracing point of Watt's mechanism will have a wavy path, intersecting the desired path of straight-line y=mx adds a finite number of points.

Thus except for the points of intersection, all other points raised by the coupler do not lie on the exact straight path.

The points of intersection of the desired path with the actual path are defined as accuracy or precision points.

The theme of the approximate approach of synthesis should be:

There should be as many accuracy points as possible, and

There should be a minimum deviation between the desired path and the actual path.

5.4 Chebychev spacing

When a mechanism is designed to generate specified function or trace a given curve y=F(x) over a given range

the function is exactly generated at a finite number of points only.

These are known as accuracy points and are represented in figure a by points of intersection of generated function y=f(x) with desired function y=F(x).

These points are indicated as

in the same figure.

The number of these accuracy points is equal to the number of fixed parameters that may be used for synthesis & varies between three & six.

Let x1 x2 x3……….. xn be the value of independent variables at precision points

, working range

The difference between the two functions is called a structural error.

The structural error

is shown plotted against independent variable x in Fig (b).

It must be clearly understood that structural error is inherent in an approximate synthesis.

Best approximation between the function to be generated and function generated can be obtained when the absolute value of maximum structural error between any two precision points equal corresponding value at the ends

Quite often Chebyshev spacing of precision points is used as a first guess to minimize the structural error.

It must be remembered, however, that this approach is applicable in special cases when the function is symmetric.

For the first guess, the best spacing of accuracy points, called Chebyshev's spacing, are given for the range

Where are the accuracy points. Thus, for first, second, and third accuracy points, j is equal to 1,2, and 3 respectively.

n = total number of accuracy points.

For instance, let us assume that using Chebyshev's spacing, three precision points are to be obtained for generating a function

in the range

Here.

And n= number of accuracy points=3

First, second and third accuracy points are

And

Corresponding values of the dependent variable by, as obtained by substituting values (j=1,2,3) in the function

The Chebychev accuracy points can also be obtained using a simple geometric construction.

The method consists of drawing a circle on a diameter equal to the range of the function of the independent variable. Thus,

The next step is to inscribe in a regular polygon of sides 2n. Where n is the number of accuracy points.

The polygon must be so inscribed that at least two sides are perpendicular to the x-axis.

Feet of perpendiculars dropped from the corners of this polygon on the x-axis then give accuracy points.

Chebyshev accuracy points are used for solving problems on function generation.

5.5 Mechanical and structural errors

Accuracy is an important factor in designing mechanisms.

It is possible to design a mechanism that will generate a given motion theoretically.

The difference between the required motion & actual motion is called a structural error.

The errors producing because of manufacturing problems like tolerances in the length of link & bearing clearances are called mechanical error.

Structural errors are present even there are no mechanical errors.

Structural errors are affected by precision point selection.

5.6 Three-position synthesis of the four-bar mechanism using Freudenstein’s equation.

Consider a four-bar mechanism ABCD as shown in figure a, in which AB=a, BC=b, CD=c, and DA=d.

The link AD is fixed and lies along the x-axis .

Let the links AB (input links), BC (coupler), and DC (output link) make angles

respectively along the x-axis or fixed link AD.

The relation between the angles and link lengths may be developed by considering the links as a vector.

The expressions for displacement, velocity, and acceleration analysis are derived as discussed below:

Displacement analysis

For equilibrium of the mechanism, the sum of the components along the x-axis & along the y-axis must be equal to zero. First of all, taking the sum of the components along the x-axis as shown in figure b we have

Squaring both sides

Now taking the sum of the components along y-axis we have

Squaring both sides

Adding equations (ii) and (iv)

Let

Equation (v) can be written as

The equation (vii) is known as the Freudenstein equation.

5.7 Analytical synthesis using kinematic coefficient in the four-bar mechanism

Consider a four-bar mechanism as shown in the figure.

The synthesis of the four-bar mechanism, when input and output angles are specified, is discussed below:

Let the three positions, angular displacement (

of the input link AB and the three positions (

of the output links as shown in figure B are known and we have to determine the dimensions a,b,c, and d of four-bar mechanism.

The Freudenstein's equation

Where

For the three different positions of the mechanism, the equation (i) may be written as

The equation (iii),(iv) and (v) are 3 simultaneous equation and maybe solve for

using Cramer's rule of determinants as discussed below:

Now the values of are given by

Once the values of

are known, then the link lengths a, b,c, and d are determined by using equation (ii).

In actual practice either the value of a or d is assumed to be unity to get the proportionate values of other links.

Numericals

1. Synthesize a four-bar linkage as shown in Fig using Freudenstein’s equation to satisfy in one of its positions. The specification of position velocity and accelerations are as follows:

Given

The four-bar linkage is shown in Figure Let

AB = Input Link=a

BC=Coupler=b

CD=Output link =c and

AD=Fixed link =d

The Freudenstein’s equation is given by

Substituting the value of in equation (i)

Differentiating equation (i) with respect to time

Now differentiating equation(iii) with respect to time

= -

=-[sin(60-90)(2-7)+(5-2)2 cos(60-90)]

From equation (iv) and (v)

From equation (v)

And from equation (ii)

Assuming the length of one of the links say an as one unit we get the length of the links as follows

We know that

units

b=35.12 units

2. Synthesize a four-bar mechanism to generate a function y = sinx for . The range of the output crank may be chosen as 60 while that of input crank be 1200. Assume three precision points which are to be obtained from Chebyshev spacing. Assume fixed link to be 52.5mm long and

Solution Given

The three values of x corresponding to three precision points according to Chebyshev spacing are given by

where j=1,2,3

Since y = sinx, therefore corresponding values of y are

Also

And

The relation between the input angle and x is given by

where j=1,2,3

The above expression may be written as

The three values of corresponding to three precision points are given by

From equations (i) (ii) and (iii)

Since

The relation between the output angle and y is given by

The expression may be written as

The three values of corresponding to three precision points are given by

From equations (vii),(viii) and (ix)

Since therefore

We have calculated above the three positions.

Let a=Length of the input crank

b=Length of the coupler

c=Length of the output crank, and

d= Length of the fixed crank=52.5mm

We know that Freudenstein displacement equation is

Where

The equation (xiii) for the first position of input and output crank may be written as

Similarly, for the second position

And for the third position (when

Solving the three simultaneous equations (xiv), (xv) and (xvi) we get

Since the length of the fixed link (i.e. d=52.5mm) is known therefore we get the length of other links as follows:

We know that

b=75.88mm

3. Design a slider-crank mechanism to co-ordinate three positions of the input link and the slider for the following angular and linear displacement of the input link and the slider respectively,

Solution The required slider-crank mechanism can be designed as follows:

Draw two parallel lines

20 mm apart from each other fig a.

Take an arbitrary point A on the line

for the fixed ground pivot.

Locate the relative pole

by rotating a vertical line through A about A through an angle of 20° = (

counterclockwise and drawing a vertical line at 90 mm =

to the left of A. Similarly, locate the relative pole

by rotating vertical line through A about A through an angle 60° = (

/2) counterclockwise and drawing a vertical line at 150mm =

to the left of A.

At point

construct an angle of 20° = (

and at point

construct an angle of 60° = (

/2) in such a way that the intersection of their arms locates the points B and C at suitable positions.

Join AB and BC.

Then, ABC is the required slider-crank mechanism. The figure shows the same in three positions.

(a)

(b)

4. Design a slider-crank mechanism to co-ordinate three positions of the input and the slider for the following data by inversion method

Eccentricity = 20 mm.

For the given to angular displacement of the input link and the two linear displacements of the slider along with the eccentricity e, the required slider-crank mechanism is obtained as follows:

Draw two parallel lines

a distance of 20 mm apart. Take an arbitrary point A on the line

For the fixed pivot and three points

on the line

at distances 40mm and 96mm apart for the initial and subsequent positions of the slider.

Rotate the point

about A through an angle 30° in the contour clockwise direction to obtain the point

similarly rotate the point

about A through an angle 60° to obtain the point

Join

and draw their mid normal to intersect at point B.

Then is the required slider-crank mechanism. Figure b shows the mechanism in the required three positions.

(a)

(b)

5. Design of four-link mechanism to co-ordinate four positions of the input and output links for the following angular displacement of the input link and the output link respectively:

Solution. Make the following construction:

Draw a line segment AD of suitable length to be the distance between the fixed pivots. Fig a

Locate the position of the relative pole by rotating AD about A through angle 25° (=

and about D through an angle 15° (=

/2) both in the counter-clockwise direction. Take this as point

Draw the input link AB in four positions

its angular displacement is known.

Locate the points

about D through angles

respectively in the counter-clockwise direction such that the location of

is at

The intersection of the mid normal of

point C. Then ABCD is the required mechanism. Figure b shows the mechanism in the required four positions.

(a)

(b)

6. A four-bar mechanism is to be designed by using three precision points to generate the function

Assuming starting position and finishing position for the input link and starting position and finishing position for the output link, find the values of x, y, corresponding to the three precision points.