mechatronic and mechanical engineering

Written by Dr John Gal Kinematics\KKM\lab3

Mechatronic and Mechanical Engineering

300035 – Kinematics and Kinetics of Machines

(Kinematics Half)

EXPERIMENT 3

DISPLACEMENT, VELOCITY & ACCELERATION

ANALYSIS OF A 4-BAR PLANAR MECHANISM

Aim:

To determine the angular relationship between the input and output angles of a

4-bar mechanism by recording the incremental changes of each of these angles

for a complete cycle of the input crank and compare with the analytical

displacement equation. Another task is to verify graphically the velocity and

acceleration of a point on the coupler at a certain configuration of the

mechanism and compare it to that obtained by using the velocity polygon

method for the velocity and numerical approximation method for the

acceleration.

Introduction:

In this experiment, the dimensions of a planar 4-bar mechanism are known and

the main task is to verify the analytically derived kinematic properties of the

mechanism by obtaining actual values for various quantities from the device.

Many machines found in industry in such applications as packaging, car

manufacturing and in consumer products incorporate a form of the common 4-

bar mechanism to perform a specific function or operation. The Ackerman

steering mechanism in cars is an example where the input-output links conform

to a certain functional relationship to allow both wheels to turn without

slipping. In such an application the designer would need to synthesise a 4-bar

mechanism that has the desired input-output relationship and determine the

lengths of the links of the mechanism. For the current experiment, the reverse

is required, namely to determine the angular relationship given the dimensions

of the mechanism. This is called analysis and is a simpler task than synthesis.

By plotting the path of a point in the coupler for a small displacement about an

arbitrary, given configuration of the mechanism a numerical calculation can be

performed to determine both the velocity and the acceleration of the coupler

point at the given position. The method is similar to the one used in

B

B

B


Experiment A1 and uses the same numerical approximations for obtaining a

value for the velocity and acceleration.

Since the mechanism is not driven by a motor, the method of obtaining the

rotational speed, i.e. the angular velocity of the input crank, is to assign an

equal time interval to each equal angular increment of the input crank. This will

provide a time base for calculating velocities and accelerations.

.

In the schematic diagram of a general 4-bar mechanism below, the link ABC is

called the coupler link and is one rigid body. For this mechanism the functional

relationship between the input angle, , and the output angle, , can be written

as follows:

)cos(coscos 321

RRR (1)

where,

c

dR

1 ,

a

dR

2 ,

2ac

bdcaR

2222

3

and the link lengths are a, b, c and d.

If the link lengths are known then the positions of points A, B and C can be

calculated in terms of the angles and and the velocities and accelerations

can be determined by differentiating the resulting displacement equations of

these points with respect to time.

Apparatus:

A 4-link, planar mechanism with a coupler link that has a pen-holder for

tracing coupler curves.

Two protractors for reading the input and output angles of the input and

output links respectively.

A pen fixed to the coupler link for tracing the path of a point in the

coupler.


The dimensions for the mechanism are a=150mm, b=290mm, c=270mm

and d=380mm. The position of the slot in the coupler is symmetrical

between the A and B with a length of 150mm. The distance between the

centre of the slot and the line AB is 80mm.

Method:

1. Starting with the input angle at 0o read the output angle from the protractor.

2. Increment by a constant amount of say 10o around the complete cycle (or as

far as the mechanism allows) and record the corresponding value of . These

can be plotted later and compared to the values obtained by substituting the

values of into equation (1). Use Excel to do this task.

3. Use a setting for the input angle of say 30o as the nominal configuration of the

mechanism. Fix a piece of paper to the frame under the coupler link for tracing

the movement.

4. With the pen fixed to the coupler link at a known location in the slot, trace the

coupler curve for 2 increments of 5o of the input angle on either side of the

nominal position. Mark each increment on the coupler curve.

5. Determine the length of each incremental movement by measuring with a ruler.

This is an approximation of the incremental displacement for each step of the

input link and can be used in the approximation of velocity in equation (2).

6. For the polynomial approximations in equations (4) and (5) you will need to

measure the x as well as the y components of the displacement at each step

from an arbitrary origin. [The details of this method is explained at the end of

these notes; see note (e) on page 8 below]

7. Repeat steps 3, 4, 5 and 6 for two other configurations (different values 300,

1300, 210

0) of the mechanism.

8. Use the method below, this is the same as for experiment A1, to calculate the

linear velocity and acceleration of the point on the coupler represented by the

pen for each configuration of the 4-bar linkage.

Background:

The velocity between adjacent configurations of the mechanism at which the

displacement of the coupler point was recorded can be calculated by using a finite

difference method:

t

sv

. (2)


Here s is the displacement of the coupler point for one increment of the input

link and t is the time for the input link to move one increment. You can assume

that the angular speed of the input link is one increment per second i.e. 5o/sec or

you can use a more realistic rotational speed if you wish.

The acceleration can likewise be calculated to be:

av

t

. (3)

In the above calculations, the velocities and accelerations are approximated over

single time increments by constant functions.

A more accurate method passes a polynomial through a series of adjacent values

and then differentiates the polynomial. For example if a polynomial of degree

four is passed through five evenly spaced data points ( xn-2, xn-1, xn, xn+1, and xn+2 )

and then differentiated, the velocity at the central data point is given by

vx x x x

tn

n n n n 2 1 1 28 8

12. (4)

The acceleration is given by

ax x x x x

tn

n n n n n 2 1 1 2

2

16 30 16

12( ). (5)

Note that in the above formulae, the xi are the distances from the nominal position

along the coupler curve [See note (e) on page 8 below regarding formula for

acceleration].

Calculations

1. After the laboratory session enter your data into a spreadsheet package such as

Microsoft Excel.

2. Tabulate the input-output angles for both the experimentally obtained values

and by using equation (1) and create a chart for both.

3. Calculate the velocities and accelerations by using the numerical approximation

techniques for each of the 3 values of that you used in the experiment.

Compare the velocities with those obtained by the velocity polygon method in

(4) below.

4. Draw a velocity polygon for one of the configurations of the mechanism that

you used in the experiment.


Report:

Your report should start by describing, in your own words, the aim behind this

experiment.

Also, describe briefly, in your own words what you did and how you did it.

Draw a diagram of the apparatus used and any other relevant illustrations.

Include tables of all recorded and calculated results and all the graphs described in

the calculation sections (a), (b) and (c).

Discuss how you may be able to use the knowledge that you gained in this

experiment to design a 4-bar linkage for a practical application.


Kinematic Calculations for the Displacement & Velocity of Coupler point C

(a) Derivation of equation (1)

Establish a coord system x-y as shown. Contsruct a vertical from B to meet DO in P, a vertical from A to meet

DO in R, and a horizontal from A to meet BP in Q.

Then, the length DR and BP can be expressed as:

DR = DO + OR = DP + QA (1)

and

BP = c sinφ = QP + QB (2)

The two sets of expressions above can be written as two equations:

d + a cosθ = c.cosφ + b.cosα (where α is the angle BAQ) (3)

c sinφ = a.sinθ + b.sinα (4)

We can re-arrange (3) and (4) as follows:

b.cosα = (d + a.cosθ) - c.cosφ (5)

b.sinα = c.sinφ – a.sinθ (6)

If we now square both (5) and (6) and add, we get

b2 = a

2 + c

2 + d

2 + 2ad.cosθ – 2cd.cosφ –2ac.cosθ.cosφ – 2ac.sinφ.sinθ (7)

Combining the last two terms, dividing both sides by 2ac and re-arranging equation (7), we get:

c

dcosθ -

a

d cosφ +

2ac

bdca 2222 = cos(θ – φ) (8)

The above equation is the same as the one on page 2 of the notes, namely;

)cos(coscos 321 RRR (9)

P

Q

x

y

R

α

S

β


(b) Solving equation (9) for φ in terms of a known input value of θ

We expand the right-hand-side of (9) and collect terms as follows:

sin φ.[sin θ] + cos φ.[R2 + cos θ] = R3 + R1cos θ (10)

and this can be written as :

A. sin φ + B. cos φ = C (11)

where A = [sin θ]; B = [R2 + cos θ]; and C = R3 + R1cos θ

Now let A = U.sinδ (12)

and B = U.cosδ (13)

then equation (11) can be written as

U.sinδ.sin φ + U. cosδ.cos φ = C (14)

which can then be simplified as: cos(φ - δ) = U

C (15)

giving φ = cos-1

(U

C) + δ (16)

and from (12) and (13) we can determine that

U = 22 BA and δ = tan-1

(B

A) (17)

Hence, in equation (16) all the parameters on the right-hand-side, namely, C, U and δ are functions of θ only and

the link lengths, a, b, c and d.

The above equations can be used in an Excel spreadsheet to calculate the theoretical values of φ for each value of

θ and compare them to the corresponding values obtained through the experimental process.

(c) Position coordinates of the coupler point C

First, construct a horizontal from B and a vertical line from C to meet in S. Then the angle CBS is given by β. We

now write down the coordinates of C as follows:

XC = DP + BS and YC = PB + SC (18)

This can then be written as

XC = c.cos φ + BC.cos β (19)

YC = c.sin φ + BC.sin β (20)

Now, BC and CA are known lengths and can be represented by e and f, respectively. From this the angle β can be

determined using the cosine rule. Let the whole angle CBA be ξ then using the cosine rule


cos ξ = 2eb

fbe 222 , giving ξ = cos

-1(

2eb

fbe 222 ) and therefore (21)

β = ξ – α (22)

From equations (5) and (6), an expression for α can be obtained which is dependent on θ and φ and by using

(16) only θ is the independent parameter, viz.;

α = tan-1

(

c.cosa.cosd

a.sinc.sin) (23)

Again, by implementing (19) to (23) and the previous equations relating θ and φ to each other, in an Excel

spreadsheet we can calculate the position of C for each value of θ.

(d) Velocity of the coupler point C

By differentiating, with respect to time, equations (19) and (20) we can get the velocity of C i.e.

VC = 2C

2C YX (24)

The differentiation process requires the application of the chain rule to equations (19) and (20) as follows:

dt

d

d

d

dt

d

d

d

dt

d

e.cosc.cosXC (25)

which can be re-written together with the differential of (20) as

.. e.sinc.sinXC (26)

.. e.cosc.cosYC (27)

Now, we need to obtain expressions for and in terms of the one independent velocity parameter, namely, .

This done by performing similar differentiations on equations (9), (6) and (22).

Differentiating (22) w.r.t. time:

, where 0 since it does not change with time (28)


b.cosα. = c.cosφ. - a.cosθ. and re-arranging gives (29)

=

b.cos

a.cosc.cos .. (30)


)).((.. sinsinRsinR- 21 , which can be re-arranged as (31)


)(

)(

sinsinR

sinsinR

2

1 (32)

Now, we substitute (32) into (30) to get an expression for in terms of and then using (28) we can get an

expression for CX and CY

in terms of only and the position parameters θ and φ.

Working with the Excel spreadsheet that was developed for the calculations in sections (a), (b) and (c) the

velocity of C, that is VC, can be calculated for every value of θ. In the report for this experiment you do not need

to implement the velocity calculations, it is optional.

(e) Using the polynomial approximation for velocity & acceleration

On page 4, the equations (4) and (5) represent the approximation to velocity and acceleration if the movement

was in a straight line. Since in this case the coupler point, C, moves in both the x and the y directions we need to

apply the formula in equation (4) and (5) twice, once for the x-direction and once for the y-direction. The

diagram below represents a portion of the coupler curve and C is the nominal position where we want to

determine the velocity. Thus, VC is given by VC = 2C

2C YX where CX (or x for short) is determined by

equation (4) and CY (or y for short) is also determined by equation (4) as follows:

t

xxxxx nnnn

n

12

88 2112 and t

yyyyy nnnn

n

12

88 2112

Similarly, the acceleration, ac, is given by ac=22CC YX where CX (or x for short) and CY (or y for short) are

both determined by equation (5) as follows:

22112

12

163016

)( t

xxxxxx nnnnn

n

and

22112

12

163016

)( t

yyyyyy nnnnn

n

C

y

x xn+2

yn+1

yn+2

xn+1 xn xn-1 xn-2

yn

yn-1

yn-2

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