Bryn Lewis: KKM Experiment 3 - 4-Bar Planar Mechanism Analysis Report

This report details an experiment on a 4-bar planar mechanism, focusing on the determination of its kinematic properties. The experiment involved recording data from incremental changes in input and output angles to verify the mechanism's behavior against analytical displacement equations. The procedure included tracing the path of a point on the coupler link, measuring angular changes, and calculating velocity and acceleration. Results are presented with displacement, velocity, and acceleration values at various input angles, along with the percentage error. The analysis aims to understand the relationship between input and output angles and the mechanism's ability to transfer loads and displace simultaneously, as evidenced in various real-world applications.

KKM experiment 3 Bryn Lewis, 19421230
Displacement, Velocity and Acceleration Analysis of a
4-Bar Planar Mechanism
Introduction
The experiment undertaken considered the known dimensions of a 4-bar planar mechanism
and how to integrate this provided information with the analytically derived kinematic
properties to verify the mechanism. This verification involved taking readings provided by
specific values of data across the various quantities of the device. Throughout the world, 4-
bar mechanisms are found in various areas involving machines due to the specific functions
and operations preformed by 4-bar mechanisms. A significant example of this real-life use is
the Ackerman steering mechanism in cars, this is where the input and output links conform
to a certain functional relationship to allow both wheels to turn without any slipping, this is
called synthesising the mechanism. Comparatively, this experiment asks for an analytical
response to a 4-bar mechanism, this is the opposite s a synthesis and calls for the
determination of the angular relationships with given dimensions of the mechanism.
Through plotting the path of a point in the known configuration of the coupler system,
numerical calculation can be undertaken to find velocity and acceleration throughout the
mechanism. Since this is not a motorised system, to calculate the angular velocity, there
must be a time increment equal to each angular increment of the input crank. This will
provide base for the velocity and acceleration equations.
Once the link lengths are known, positions of A, B and C can be calculated in terms of the
angles shown. Once the velocities and acceleration is found, displacement can be found by
differentiating the equations.
Abstract
This experiment is preformed to determine the angular relationship between the input and
the output angles of the 4-bar mechanism provided, this is completed by recording data
found from the incremental changes of each of these angles for a complete cycle and then
compare the results to the analytical displacement equation. Once the analytical equations
are complete, confirmation of the equations can be found graphically at a point of the
mechanism.

KKM experiment 3 Bryn Lewis, 19421230
Materials and Equipment
 4-bar 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 links respectively
 A pen fixed to the coupler ink for tracing the path of a pint in the coupler
 Given dimensions of the mechanism (A=150mm, B=290mm, C=270mm, D=380mm).
the slot position between A and B is symmetrical with a length of 150mm and the
distance between the centre of the slot and line AB is 80mm
Procedure
1. Line up input angle to 0 and record the output angle
2. Increase the input angles by increments of 10 degrees for a full rotation and record
data placed by output angles
3. At increments of 30, 130, 210 degrees respectively, fix paper behind the mechanism
and trace the movement preformed by it for increments of 5 degrees with two
above and below the desired angle (eg: 20, 25, 30, 35, 40). Repeat for all angles of
30, 130, 210
4. By approximation with a ruler, determine the length of these increments between
each 5 degrees, this will be used to determine the velocity later
5. Record the x and y displacements of the points noted down from an agreed origin
Equations used
The equations can be found here in order in provide insight into how the results were
obtained and going through all working out would be many pages long.
By substituting the respective values for these numbers which will
later be used in the derivation of the equations to find velocity and
acceleration analytically.

KKM experiment 3 Bryn Lewis, 19421230
Results
Coupler links around input = 30 degrees
Input
angle
Output
angle
Displacement(d) d (x) d (y)
40 39 N+2 201 165
35 36 N+1 206 155
30 33 N 215 145
25 31 N-1 220 136
20 28 N-2 226.5 123.5
Coupler links around input = 130 degrees
Input
angle
Output
angle
Displacement
(d)
d (x) d
(y)
140 85 N+2 37 222
135 84 N+1 41 227
130 83 N 44 232
125 82 n-1 50 236
120 80 n-2 55 236
Coupler links around input = 210 degrees
Input
angle
Output
angle
Displacement
(d)
d (x) d (y)
220 45 N+2 179 61
215 47 N+1 169 74
210 51 N 158 84
205 52 n-1 146 96
200 58 n-2 135 102
195 200 205 210 215 220 225 230
0
50
100
150
200
X vs Y couplers at 30 degrees
36 38 40 42 44 46 48 50 52 54 56
215
220
225
230
235
240
X vs Y couplers at 130 degrees
130 140 150 160 170 180 190
0
50
100
150
X vs Y couplers at 210 degrees

KKM experiment 3 Bryn Lewis, 19421230
Results of increments of 10 degrees for full rotation
input
angle
output
angle
percent
error
(%)
0 20 2.2
10 23 2.2
20 28 1.8
30 33 1.1
40 39 1.5
50 45 2.3
60 51 2.4
70 56 1.7
80 63 1.6
90 68 1.8
100 73 2.1
110 77 2.3
120 80 2.1
130 83 1.7
140 85 1.1
150 85 1.1
160 82 1.4
170 77 1.6
180 72 1.7
190 65 2.1
200 58 1.9
210 51 1.3
220 45 1.1
230 39 1.6
240 35 1.3
250 30 1.7
260 27 1.3
270 24 1.9
280 22 2.1
290 20 2.2
300 18 1.7
310 17 2.4
320 17.5 1.7
330 16 2.1
340 16 2
350 18 1.6
360 20 1.3

KKM experiment 3 Bryn Lewis, 19421230
Conclusion
As seen in the above equations, there is a finite equation for velocity and acceleration
respectively where the top values (delta[s] and delta[v]) are values of the coupler point for
one increment. The graphs of the incremental values under the results subheading show
slight curve (though only experimental), provide an accurate representation of the path
made by the mechanism’s coupler arm. Inaccuracies in the experiment halt the results from
becoming as precise as an analytical value produced but still re-create that curve,
predominantly seen in graph 3 (X vs Y couplers at 210 degrees). Due to a 4-bar mechanisms
ability to transfer applied loads across its body as well as move and displace simultaneously,
this mechanism can be found in many worldwide applications such as rock-climbing
harnesses, bicycles, hedge clippers, the list goes on.