0 | Page School of Engineering (Mechanical and Automotive) Major Assessment Course: Mechanics of Machines (MIET1077) Experiment: Six Bar Linkage Mechanism Course Coordinator: Prof. Firoz Alam ([email protected]) Student Information Family Name Given Name Pace Samuel Student Number: S3659265 Due Date: 25/10/20
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School of Engineering (Mechanical and Automotive)
Major Assessment Course: Mechanics of Machines (MIET1077)
Related Theory .................................................................................................................................................................................................. 2
Calculating Lengths of Links .............................................................................................................................................................................. 3
Graphical Position Analysis ................................................................................................................................................................................ 4
Graphical Instantaneous Centre Analysis ........................................................................................................................................................ 16
Comlete Instantaneous Centre Diagram .................................................................................................................................................... 22
Results for Instantaneous Centre of Zero Velocity Method ....................................................................................................................... 23
Analytical Method for Position, Velocity and Acceleration ............................................................................................................................. 23
Position Analysis ........................................................................................................................................................................................ 23
Part A - Crank ........................................................................................................................................................................................ 23
Part B β Coupler & Rocker .................................................................................................................................................................... 24
Part C β Slider ....................................................................................................................................................................................... 29
Part A - Crank ........................................................................................................................................................................................ 30
Part B β Coupler & Rocker .................................................................................................................................................................... 30
Part C - Slider ........................................................................................................................................................................................ 32
Part A β Crank ....................................................................................................................................................................................... 34
Part B β Coupler & Rocker .................................................................................................................................................................... 34
Part C β Slider ....................................................................................................................................................................................... 37
Summary of Analytical Values .................................................................................................................................................................... 38
Equations of Motion .................................................................................................................................................................................. 39
Working Model Simulation .............................................................................................................................................................................. 45
Balancing Strategy Development ...................................................................................................................................................................... 0
Summary This report details the methods for calculating the position, velocity and acceleration of a 6 bar linkage mechanism. The graphical method of calculating the position, velocity, acceleration and instantaneous centres of zero velocity use vector polygons and position diagrams to calculate and display graphically the magnitude and direction of each component which can be measured to provide an accurate result. The values are then cross checked using the analytical method for calculating each component. A simulation was made using Working Model to provide further analysis and verification of the results. Each of these methods provided results that would be used to calculate the dynamic forces of the system and generate a balancing strategy for reducing the vibration in the system. From this report, the positions, velocity, and accelerations of the linkages have been accurately calculated and verified using multiple methods of calculating the results.
Objectives The objectives of this major assessment are to:
β’ Apply the understanding of Mechanics of Machines course content β’ Calculate position, velocity, acceleration, and instantaneous centres of the system using
graphical methods β’ Calculate position, velocity and acceleration using analytical methods β’ Develop and analyse a working model simulation to evaluate the outputs β’ Calculate the dynamic forces on the system and develop a balancing strategy development
Related Theory A linkage mechanism has a grounded link, driver link (crank) and a slave link (rocker). The grounded (link g) link throughout the whole motion of the system will always have no velocity or acceleration. This link is the link which the system revolves around and defines the joints which the links rotate around. The crank (a and Ξ±) is what drives the motion of the system and determines the limits of motion for each link and joint. The rocker (b and Ξ²) has its motion driven by the crank and that motion is dependent on the size of the crank. The floating link (link f) has complex motion dependant on the crank and rocker and is what connects them together.1
Figure 1 4 Bar Linkage Mechanism Source: http://dynref.engr.illinois.edu/aml.html
Many different linkage mechanisms can be seen in our daily life, some examples can be seen below.
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Figure 2 Bike Pedalling Source:
http://dynref.engr.illinois.edu/aml.html
Figure 3 Knee Joint Source
:http://dynref.engr.illinois.edu/aml.html
Figure 4 Suspension with Watt's linkage Source: http://dynref.engr.illinois.edu/aml.html
These different linkages seen in our daily lives are made up of different types of 4 bar linkage mechanisms and can have the same principles applied to finding out their function and ranges of movement.
Calculating Lengths of Links To begin the following values were given to begin calculating the links.
OA (m) 0.2 B (deg) 12 ΞΈ1 (deg) 6 n2 (rpm) 480
The following lengths are calculated using the provided equations
Since both these statements are true, the system may continue with the following values used for each of the links.
Graphical Position Analysis
Figure 5 Graphical Position Analysis
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The positions of 16 different angles of ππ2 and the subsequent positions of each of the other positions. The paths are πππ΄π΄, πππ΅π΅, πππΆπΆ and πππ·π· with the limits of the motion for B being π΄π΄πΏπΏ and π΄π΄π π and the limits of D being πΆπΆπΏπΏ and πΆπΆπ π .
Graphical Velocity Analysis To begin the graphical velocity analysis, the velocity of point A is calculated as it has a known rotational speed and direction and so can be found.
This equation produces the following vector polygon.
Figure 6 Finding VB
The same technique is used to find πππΆπΆ , but there are 2 equations which need to be used. πππΆπΆ = πππ΄π΄β₯OA + πππΆπΆ/π΄π΄β₯AC
The following vector polygon is produced, and the connection points define the vectors of the velocities.
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Figure 9 Finding VD
Figure 10 Solving for VD
Complete Vector Polygon Now that all the individual components are found, the entire velocity vector polygon can be seen below.
Figure 11 Velocity Vector Polygon
The program which the vector polygon was drawn in had a scale which was used at πππ΄π΄ of 1 point = 100mm/s. Using this scale, the following magnitudes and angles for the rest of the vectors can be solved.
Graphical Acceleration Analysis To begin the graphical acceleration analysis, first the known values are calculated to calibrate the scale and distance. The following equation is used to find the magnitude of πππ΄π΄ππ.
πππ΄π΄ππ =πππ΄π΄2
πππ΄π΄=
(πππ΄π΄πππππ΄π΄)2
πππ΄π΄=
(50.265 β 200)2
200= 505323.6 ππππ/ππ2
Since it is known that the direction is perpendicular to link OA, πππ΄π΄ππ can be sketched as follows. Since the velocity is known to be constant, the tangential acceleration of the support is known to be 0. Because there is no tangential component to the acceleration, the normal component of the acceleration is equal to the overall acceleration of the link.
From this graph, the connection between the vector between the tangential and normal components of the vector can be resolved to find the full vectors for πππ΅π΅ and πππ΅π΅/π΄π΄.
Figure 17 Finding aB and aB/A
Now the acceleration at C is to be calculated starting with πππΆπΆ/π΄π΄. First πππΆπΆ/π΄π΄ππ is to be calculated using
the following.
πππΆπΆ/π΄π΄ππ =
πππΆπΆ/π΄π΄2
πππ΄π΄πΆπΆ=
5731.242
300= 109490.7 ππππ/ππ2
πππΆπΆ/π΄π΄ππ can be sketched from πππ΄π΄ with πππΆπΆ/π΄π΄
π‘π‘ being perpendicular to the normal component as follows.
From here, the πππΆπΆ/π΄π΄ and πππΆπΆ/π΅π΅ values are combined to find the tangential components using the following equation and vector polygon.
The lines can be connected accordingly and the vector component of πππΆπΆ/π΅π΅, πππΆπΆ/π΄π΄ and πππΆπΆ can be found as follows.
Figure 21 Calculating aC/B and aC/A
The final acceleration to be found is πππ·π· which required πππΆπΆ and since the directions are known the following vector polygon can be created using the same principles as before.
πππ·π·/πΆπΆππ =
πππ·π·/πΆπΆ2
πππΆπΆπ·π·=
570.132
800= 406.31 ππππ/ππ2
The πππ·π·/πΆπΆππ is sketched with πππ·π·/πΆπΆ
π‘π‘perpendicular and πππ·π· parallel to the x-axis.
This vector polygon can be solved for πππ·π·/πΆπΆπ‘π‘ and πππ·π·.
Figure 23 Calculating aD
Complete Vector Polygon All the vector polygons are combined to produce the overall vector polygon for the acceleration.
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Figure 24 Acceleration Vector Polygon
Graphical Acceleration Results Using the initial scale, the vector polygon can be resolved to find the rest of the components in the system. The magnitudes for each of the accelerations calculated using the graphical method can be observed in the table below.
Width (pts)
Height (pts)
Length (pts)
Acceleration (mm/s2)
Angle (deg)
Aa 252.662 437.623 505.323635 505323.635 239.999976
Graphical Instantaneous Centre Analysis For the graphical instantaneous centre of zero velocities analysis, we will be using Kennedyβs theorem. To begin Kennedyβs theorem, the number of ICβs must be found using the following.
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πΆπΆ =ππ(ππ β 1)
2 =
6(6 β 1)2
= 15
From observation of the model, the position of the following instantaneous centres can be found.
The instantaneous centres are found in the following order using the lines between the respective instantaneous centres.
πΌπΌπΆπΆ13 is found using the intersection of lines produced from πΌπΌπΆπΆ12 to πΌπΌπΆπΆ23 and πΌπΌπΆπΆ14 to πΌπΌπΆπΆ34.
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Figure 26 Finding IC13
πΌπΌπΆπΆ15 is found using the lines πΌπΌπΆπΆ13 to πΌπΌπΆπΆ35 and πΌπΌπΆπΆ16 to πΌπΌπΆπΆ56.
Figure 27 Finding IC15
πΌπΌπΆπΆ24 is found using the lines πΌπΌπΆπΆ12 to πΌπΌπΆπΆ14 and πΌπΌπΆπΆ23 to πΌπΌπΆπΆ34.
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Figure 28 Finding IC24
πΌπΌπΆπΆ45 is found using the lines πΌπΌπΆπΆ14 to πΌπΌπΆπΆ15 and πΌπΌπΆπΆ35 to πΌπΌπΆπΆ34.
Figure 29 Finding IC54
πΌπΌπΆπΆ46 is found using the lines πΌπΌπΆπΆ14 to πΌπΌπΆπΆ16 and πΌπΌπΆπΆ45 to πΌπΌπΆπΆ56.
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Figure 30 Finding IC64
πΌπΌπΆπΆ36 is found using the lines πΌπΌπΆπΆ13 to πΌπΌπΆπΆ16 and πΌπΌπΆπΆ35 to πΌπΌπΆπΆ56.
Figure 31 Finding IC63
πΌπΌπΆπΆ25 is found using the lines πΌπΌπΆπΆ12 to πΌπΌπΆπΆ15 and πΌπΌπΆπΆ24 to πΌπΌπΆπΆ45.
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Figure 32 Finding IC25
πΌπΌπΆπΆ26 is found using the lines πΌπΌπΆπΆ12 to πΌπΌπΆπΆ16 and πΌπΌπΆπΆ25 to πΌπΌπΆπΆ56.
Figure 33 Finding IC26
All the instantaneous centres can be seen placed onto the 6 bar linkage mechanism as seen below.
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Comlete Instantaneous Centre Diagram
Figure 34 Graphical Inspection of All Instantaneous Centres
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πΌπΌπΆπΆ15 is off the top of the graph but can be seen below.
Figure 35 Graphical IC15
Using these graphs, the velocities can be calculated with πππ΄π΄ calibrating π·π·1 using πΌπΌπΆπΆ13. Once π·π·1 is calculated πππ΅π΅ and πππΆπΆ can be calculated. Using πππΆπΆ and πΌπΌπΆπΆ15, π·π·2 can be calibrated and used to calculated πππ·π·. From the graph, the following values were calculated
Results for Instantaneous Centre of Zero Velocity Method Height
Analytical Method for Position, Velocity and Acceleration Position Analysis Part A - Crank To start the analytical method, the calculations are completed for the crank (link OA). The position of the crank is first calculated.
Since the link is below the x-axis, the link will not work for ππ3 = 41.514 πππππΌπΌ so therefore ππ3 =β64.346 πππππΌπΌ.
From this the X and Y position of B can be solved using the following equations.
Since the link is above the x-axis, the link will not work for ππ3 = β68.458 πππππΌπΌ so therefore ππ3 =49.670 πππππΌπΌ.
From this the X and Y position of B can be solved using the following equations.
After visually producing each of the scenarios, it can be deduced that the angle used for the system will be whenππ4 = 128.32 πππππΌπΌ and ππ3 = 49.670 πππππΌπΌ. For calculating the πΆπΆ position, the Ξ² angle will need to be used along with the previous results.
Part B β Coupler & Rocker To begin finding the velocities at each point, first the rotational velocity for link 3 and 4 must be found (ππ3 and ππ4). Using the closed vector loop method, these values can be found.
Observing these values obtained from the graphical and analytical methods shows that the numbers obtained from the calculations are within 3% of each other at every calculation step. This agreement between both methods proves that both the methods are effective in evaluating the numbers.
Acceleration Analysis Part A β Crank To begin the acceleration analysis, the crank acceleration is found using the following equations.
Part C β Slider In order to calculate πππ·π·, firstly πΌπΌ5 must be calculated as follows. Firstly, the velocity equation is differentiated.
Comparing the graphical method for calculating acceleration to the analytical method, similar to the velocities, both the methods have very similar values.
Graphical (mm/s2)
Analytical (mm/s2)
Difference (%)
Aa 505323.6 505323.7 2.178E-05
Ab/a 289431.2 286188.6 -1.133
Ab 692624.5 696609 0.5720
Ac/a 172900.2 171713.2 -0.6913
Ac 579735.8 593949.2 2.393
Ad/c 462024.6 483864.7 4.514
Ad 187244 177402.2 -5.548 Table 6 Analytical Acceleration Results Compared to Graphical
The variation in the methods can be attributed to the compounding small errors between calculations and difference in velocity values to calculate acceleration.
Summary of Analytical Values
Joint X Component (mm)
Y Component (mm)
A 100 173.21
B 423.67 554.42
C 337.46 356.54
D 1026.47 -50
E 795.62 83.62 Table 7 Analytical Position Results
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Angle Value (degrees)
ΞΈ1 6 ΞΈ2 60 ΞΈ3 49.66962 ΞΈ4 128.31 ΞΈ5 -30.5421
Table 8 Analytical Angles
Joint Linear Velocity (mm/s)
Linear Acceleration (mm/s2)
A 10053.1 505323.7
B 1838.78 696609
C 5236.74 593949.2
D 5508.55 177402.2 Table 9 Analytical Linear Velocity and Acceleration
Link Angular Velocity Ο (rad/s)
Angular Acceleration Ξ± (rad/s2)
2 50.26548 0
3 -19.0559 442.4406
4 3.064637 1151.876
5 -0.72779 605.2945 Table 10 Analytical Angular Velocity and Acceleration
In order to reduce the shaking moment transferred to the ground, some actions could be made.
Firstly, if the shaking moment isnβt too great, a damper can be used to absorb the energy being transferred to the ground to help reduce the overall effect on the ground.
If the shaking moment is too great or a damper is unable to be installed, another solution is to remove or add mass to the system at points which can reduces the overall moments at each of the ground points.
The shaking force, shaking torque and shaking moment at ππ2 = 60 degrees were found as seen below:
π΄π΄π π = 2979.34ππ
πππ π = β147.883ππππ
π΄π΄π π = β580.02ππππ
Discussion The results at every stage of the process produced values which were within an acceptable tolerance of other values found using different methods. Most values were within 5% of other calculated values for other methods, however, the values which were outside this calculation were still relatively close to the other methods calculations.
This discrepancy between some of the values can be attributed to measurement inaccuracy for the different methods. Some of the errors in the graphical method can be attributed to rounding of numbers and measurement tools resolution being unable to 100% accurately determine values. For the Working Model results, the snapshot which the results were taken may not have been at exactly when ππ2 = 60 degrees causing a slightly different situation to be analysed. Across all methods, compounding differences and rounding errors in the methods will attribute to some of the larger differences in results obtained towards the end of each section. Since all the results have relatively
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similar magnitude and results, it can be assumed that these values are correct for if this mechanism was to be created in the real world.
Conclusion Overall the values that were produced from all the methods may not be completely accurate but most of the values are close enough that any of them can be used as an approximate used for a system and thus confirms that the methods are relatively accurate in determining the values for the position, velocity, acceleration and forces on the system.
References 1. Matthew West 2015, Four-Bar Linkages, Dynamics, Viewed September 12 2020,
<http://dynref.engr.illinois.edu/aml.html> 2. Course material of MIET1077: Mechanics of Machines provided by Prof. Firoz Alam
Attachments Attached to this document is a zip folder containing all the Working Model files used.