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Kinematic and Dynamic Analysis of Linkage Mechanisms | Maths

Home Recent Questions Kinematic and Dynamic Analysis of Linkage Mechanisms | Maths

Answer the following questions:

1. Determine all possible positions of links and joints by graphical position analysis. Draw to scale all positions of joints for sixteen subsequent positions of link 2 and determine the limits of motion where appropriate. Identify and outline the paths of each moving joint.

2. Determine linear and angular velocities by graphical velocity analysis for the given position 2 = 60 of the mechanism, as shown above. Draw to scale the velocity vector diagram encompassing all linear velocities to scale and present the results in a tabular form.

3. Determine all linear and angular accelerations by graphical acceleration analysis for the given position 2 = 60 of the mechanism. Draw to scale the acceleration vector diagram encompassing all linear accelerations to scale and present the results in tabular form.

4. Determine all instantaneous centres of velocity for the given mechanism using Kennedy’s rule, and velocities of joints A, B, C and D using identified instantaneous centres. Draw all instantaneous centres and velocities to scale on a separate diagram of the mechanism.

5. Obtain analytic solutions for positions, velocities and accelerations by vector loop equations and complex number notation and present the results in a tabular form. Compare these results with those obtained using the graphical approach. You should have good correlation.

6. Determine all dynamic forces at the joints for the given position of the mechanism using the analytical matrix method. Assume, for link 2: m2 = 1 kg, CG at OA/2, I2 = 0.002 kgm2; for link 3: m3 = 2.5 kg, CG at /2 and AB/2, I3 = 0.008 kgm2; for link 4: m4 = 1.5 kg, CG at EB/2, I4 = 0.005 kgm2; for link 5: m5= 1.8 kg, CG at CD/2, I5 = 0.006 kgm2; for slider 6: m6 = 0.9 kg and CG at D.

7. Determine the shaking force and moment, and work out analytically an optimal strategy for balancing of the given mechanism. Discuss the results.

8. Develop an equivalent computer model capable of simulating the motion of the given mechanism using software Working Model and compute and plot the paths, velocities, and accelerations over one revolution of the crank at the given angular velocity . Also, find the pin forces, slider side loads and driving torque over one revolution. How do the results compare with those obtained using the graphical and analytical approaches?

The Assur-Artobolevsky Method

The aim of the analytical solution is to determine the position, velocity and acceleration function for each joint (A, B, C and D) as well as the angular velocity and angular acceleration functions for the coupler, the slider and each linkage. The resulting functions are to be evaluated at a number of points and the output compared with that from the graphical method.

When highly accurate results are necessary or the kinematic analysis has to be repeated for a large number of configurations the analytical approach is best. The Assur-Artobolevsky method is based on the principle of analysing the mechanism as a set of kinematic pairs. This is essential for our example because it includes a coupler and a slider.

This is a complex mechanism in terms of its motion.

In order to solve the complex six-bar linkage we have to decompose the mechanism into three kinematic pairs:

Part 1 Crank (point A)

Since the underlying four-bar linkage is Grashof, the crank OA will complete a full revolution. Hence the position of the endpoint A is given by:

Part 2 Coupler and Rocker (points B and C)

Because the coupler undergoes complex motion we will represent the links as position vectors. The planar vectors for position, velocity and acceleration can then be represented in complex number form using Euler’s identity:

Part 3 Slider (point D)

The final part of the analysis concerns the position of the slider.

Obtaining a Complete Matrix

Once the accelerations about the centre of gravity for each link have been obtained, the masses of each link, together with the angular acceleration, moment of inertia and the position vector values can be inserted into the matrix.

Shaking Force and Moment

Shaking Forces

The net effect of all the dynamic forces experienced by the ground of the mechanism translates to vibrations in the structure that supports the mechanism. The sum of all the forces acting on the ground plane is referred to as the shaking force, which for this particular case is represented as:

The shaking force will tend to move the structure (ground) in the transverse direction, but due to the influence of the driving link with respect to the entire mechanism shaking torque is induced to the system. The shaking torque is simply the reaction torque felt by the ground that tends to rock the structure about its driveline axis. The shaking torque is represented as:

with –T12 representing the negative of the source torque which is delivered by the driving link.


TS = - ? Nm

Shaking Moments

The shaking moment of a linkage mechanism is the summation of the reaction torque T21 and the shaking couples introduced by the connection to the ground, which is represented as:



T21 = the negative of the driving torque

R1 = the position vector from ground of driver (O) to ground of link 4

F41 = force of link 4 onto ground

R2 = the position vector from ground of driver (O) to ground of slider

F61 = force of slider onto the ground

Computational Model with Working Model 2D

Configuring the Software

• Turn on the grid, axes and ruler in the View ï‚® Workspace menu.

• Customise the units in the View ï‚® Numbers and Units menu. Set the rotational velocity to revs/min and the frequency to Hertz.

• Specify a frequency of 1000 Hz in the World - Accuracy - Animation Step.

• The parameters of each link can be customised in the Window - Properties/Appearance menus.

Model Construction

• Know each link's dimension to its COG (X, Y, , L).

• Sketch three rectangles using the Rectangle tool (around 10 mm thick) in the positive X, Y plane. Specify the required dimensions, ensuring each is the correct position.

• Place Point Elements on both ends of each link with the exception of the ground end of link 4 and the end of link 5 that connects to the slider. Place Pin Joints at these ends.

• Sketch the main body using the Polygon tool. Click on the appropriate ends of the three links.

• Drag the body to the left and place Point Elements on each edge. Reposition the body and connect it to the links using the Join tool, selecting an element on the body and link.

• Sketch the slider using the Rectangle tool. Place a pin joint at its COG. Connect the slider to the bottom edge of link 5.

• Create a Slot Joint near the slider. Grab its pin joint and place it over the sliders pin joint (drop it when an ‘x’ appears).

• Place a motor at the appropriate location using the Motor tool.

• Rotate the main body to check if the model is connected.

• Customise each link using the Appearance menu. Uncheck Track outline and check Show center of mass.

• Specify the correct masses and motor speed using the Properties menu.

Monitoring the Links

Link velocities and accelerations:

• Select the appropriate link and choose the appropriate output variable to monitor using the Measure menu (i.e. Centre of mass acceleration).

• Each output box can be configured as desired.

Joint forces and motor torque:

• Select the appropriate joint and check the Measure ï‚® Force.

• Select the motor and check Measure ï‚® Torque.

Running the Simulation

• Check each frame in the World ï‚® Tracking tab.

• Specify a frequency of 1000 Hz in the World ï‚® Accuracy ï‚®

Animation Step. Exporting Results:

• Select File ï‚® Export, choosing an appropriate last frame.

• Open the file in Excel as Tab-delimited.

Example of Working Model Results

The model represents the mechanism consisting of a motor, linkages, pin joints, slider and a slot for the slider to travel along. In order to achieve one revolution the time incrementation was set for one frame of sixteen using the ‘Accuracy’ command, and then by setting the ‘Pause Control’ for sixteen frames the desirable stopping location was obtained.

Velocity/Acceleration of Joints

In order to determine the correlation between the analytical and computational results for the velocity at a computational analysis needs to be done for the joints of the mechanism. The results obtained from the software are shown in the figures.

The results were obtained by running the mechanism for the velocity and acceleration at joints A, B, C and D. The results were exported from Working Model into Excel while the model was running.

Note: The charts below were completed for one revolution of the crank (link OA).

Force Analysis

In order to determine the forces present in the model, the forces at the linkages need to be investigated using Working Model,

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