Mechanical PE Exam Topic Concepts

Four bar Mechanism Kinematics Shortcut

Four bar Mechanism Kinematics Shortcut to save time on the PE Exam

For mechanical engineers the PE exam will definitely contain some dynamics problems.  If you choose to take the Mechanical Systems and Materials depth session (PM portion of the exam) you will also have problems related to machine analysis.  Both dynamics and machine analysis commonly contain problems related to the kinematics of four-bar mechanisms.  This blog will give some details on a cool four bar mechanism kinematics shortcut for calculating velocities for four bar mechanisms.  It is not a method that is frequently taught in dynamics or machine analysis courses, but it can definitely save some time on the PE exam.

When can I use this shortcut?

A common type of problem has a four-bar mechanism with a given input angular velocity of the driving link.  The problem requires you to determine the angular velocity of the output link.  Even though this may seem fairly basic, the problem can be time consuming… and you don’t have a lot of time to waste during the PE exam.  This shortcut method is great for this type of problem!  This method also allows for a very quick estimation of the answer… which may be all you need on a multiple choice exam like the PE exam.

What is the method?

OK… let’s look at this method.  The method is not new and has actually been around for a long time.  However, it is not commonly taught.  The method is based on the fact that the velocity vector of both coupler endpoints must have the same component along the long axis of the coupler.  Confusing?  Probably…  Let’s look at this in more detail with some illustrations.

Terminology Review

Before I get into the method let me review some quick terminology for four-bar mechanisms.  The figure below shows a typical four-bar mechanism.  Link 1 is the ground link, which does not move.  We will call link 2 the driving link (or input link).  Link 3 is the coupler, and link 4 is the output link.

Four bar Mechanism Kinematics Shortcut 1

Velocity Calculations

Now that we have reviewed the names of the links, let’s discuss how to calculate velocities using this method.  Say we know the input rotational velocity of the driving link (link 2).  The velocity of point A (part a of the figure below) can be calculated.  Link 2 is in pure rotation, so the velocity of point A is the product of the length of link 2 and the rotational velocity in radians per second.  Look now at part b of the figure.  Take the component of the velocity vector at A along the length of the coupler (parallel component).  That must also be the length of the component at B (equal parallel components).  The velocity at point B can easily be found as the vector perpendicular to link 4 with that parallel component.  The rotational velocity of the output link can be determined from that velocity if needed.

Four bar Mechanism Kinematics Shortcut 2

Final Thoughts

The nice thing about this method for the PE exam is that it is a quick graphical method.  Drawing a simple sketch (roughly to scale) can allow you to estimate the velocity with minimal work.  Because the PE exam is a multiple choice exam, that quick estimate may be all that is required to get the answer.

What are your thoughts?

I hope this quick introduction to this method helped you work mechanism problems more efficiently.  More details on this method, along with examples, are provided in my textbook Machine Analysis with Computer Applications (you can get it on Amazon here).  As a general note, the figures used in this post are from that textbook.

As always, comments are welcomed and encouraged.  Also, please share this blog if you found it useful.  Please head over to the Mechanical PE Academy Facebook page and give it a like!  It is under development, but it will have updates and exclusive content to help you prepare for the Mechanical PE exam.


3 comments on “Four bar Mechanism Kinematics Shortcut

  1. This is essentially graphical approach to solving velocity or similar to velocity polygon. I personally find graphical methods very quick and easy. I teach machine dynamics to juniors and it is extremely difficult to get the graphical method across to the students. Also in this age of computer technology they find analytical methods more appealing.

  2. Old school computation is fast compared to entering the data into a computer or spreadsheet. Close enough calculation for many problems unless you are doing a space shot.

  3. It?¦s really a great and useful piece of info. I am happy that you shared this helpful info with us. Please stay us up to date like this. Thank you for sharing.

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