Written by Dr John Gal Kinematics\KKM\lab3 Page 1 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
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Written by Dr John Gal Kinematics\KKM\lab3 Page 1
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
Written by Dr John Gal Kinematics\KKM\lab3 Page 2
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.
Written by Dr John Gal Kinematics\KKM\lab3 Page 3
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)
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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.
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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.
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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)