All posts by Justin Verdirame

Justin received his BS, MS, and PhD all in mechanical engineering all from MIT. He has worked on the design, modeling, and analysis of precision machines primarily in the semiconductor industry.

Quick and easy way to calculate the area of any polygon — the shoelace formula

In my spare time, I enjoy watching a number of math channels on youtube, such as Numberphile, PBS infinite series,  and standupmaths.  Typically, most of the videos are about number theory or prime numbers and are not very useful to a mechanical engineer.  However, this video from mathologer discusses the shoelace formula for calculating the area of a polygon, which an engineer may find useful for calculating the area of a fluid channel or a beam section (see also the wikipedia entry for the shoelace formula).  It works for any polygon that does not intersect itself.  It may be useful in doing quick hand calculations, and it is easily scripted into a function for a computer to calculate.  One could also make a straight line approximation of a shape with curved lines to estimate its area.

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Good Reference for Fracture Mechanics of Ceramics

Ceramics are an important material for precision engineers.  Ceramics are often used for their low coefficients of thermal expansion, high Young’s modulus, as well as other properties.  The text Ceramics: Mechanical Properties, Failure Behaviour, Materials Selection by Dietrich Munz and Theo Fett (google books link, amazon.com link) is a very good reference for ceramics and for their failure in particular.  The text derives formulas for lifetime (time to failure) under constant and cyclic loads.  The text also provides extensive information on materials testing, statistical methods (Weibull distributions), and probability of fracture for ceramics.

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Low Reynolds Number Flow

Low Reynolds number flow can be a very interesting topic.  Low Reynolds number flow (Re <<1) is also called Stokes flow.  At very low Reynolds number, the Navier-Stokes equations can be greatly simplified.  Fluid mechanics at human length scales, such as swimming, is generally not very low Reynolds number.  Developments in microfluidics, nanotechnology, and biomimicry has increased the frequency with which engineers encounter low Reynolds flows problems.  Because humans often encounter fluids at moderate of high Reynolds numbers, our intuition can deceive us.  Two of the most basic results of low Reynolds flow is that it is fully reversible and independent of time.

As a refresher, the Reynolds number is the ratio of inertial to viscous forces and is given by

    \[ \mbox{Re}=\frac{\rho v d}{\mu} \]

where \rho is the fluid density, v is the velocity, d is a characteristic length such as a diameter, and mu is the viscosity.  The Reynolds number can also be thought of as the ratio of the momentum diffusion rate to the viscous diffusion rate.  At Reynolds numbers less than approximately 2000, the flow is laminar.  For Reynolds numbers greater than approximately 4000, the flow is turbulent. Continue reading Low Reynolds Number Flow

Classic Fluid Mechanics Lecture Series by Ascher Shapiro

A great set of video lectures by Ascher Shapiro of MIT on Fluid Mechanics is available on youtube.  The videos are old, but fluid mechanics hasn’t changed.   Each video is presented by a world renowned fluids expert such as Ascher Shapiro (wikipedia link) or G.I. Taylor (wikipedia link).  An accompanying set of notes is available at http://web.mit.edu/hml/notes.html.  He uses experiments to explain the topics in a way which helps develop one’s intuition for understanding and solving fluids problems.

 

An Incomplete History of the Hindle Mount, Whiffletree, Swingletree, Swingles, and the Grubb Telescope

Mechanical supports for mirrors and other optical components and substrates to maintain their initial undeformed shape is a common engineering problem.   Ideally a mirror or similar substrate can be supported on three points if the mirror or substrate is stiff enough.  However in many cases, the deflections are too large and more support is required.  One of the earliest areas where this problem arose was for the mirrors in early telescopes.  Irishman Howard Grubb came up with a novel solution by supporting the mirror on a set of levers known as a whiffletree.  For a historical bio of Howard Grubb see Biographical Encyclopedia of Astronomers or the Museum Victoria (Australia) bio or a history of the Armagh Observatory and Grubb’s telescope.

Whiffletree mirror support developed by Howard Grubb in 1835.
Whiffletree mirror support developed by Howard Grubb in 1835.  This sort of mirror mount is also commonly known as a Hindle mount in some optomechanics literature.

Continue reading An Incomplete History of the Hindle Mount, Whiffletree, Swingletree, Swingles, and the Grubb Telescope

Good References for Vacuum Flow Calculations

High vacuum systems are becoming more common and a number of semiconductor processes already operate in high vacuum.  The following references are ones that I have found useful in performing vacuum system calculations.

Pfeiffer Vacuum, which manufactures vacuum pumps and other components, has a nice PDF handbook which is free to download.  It covers the basics of vacuum as well as operating principles of various pumps and as well as a number of practical issues.
https://www.pfeiffer-vacuum.com/filepool/File/Vacuum-Technology-Book/Vacuum-Technology-Book-II-Part-2.pdf?referer=1456&request_locale=en_US

The Handbook of Vacuum Technology edited by Karl Jousten is a thorough reference with detailed calculations for wide variety of problems in vacuum systems.
amazon.com link

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Tuned-Mass Damper on a Cantilever Beam

In this article, we compare the performance of a tuned-mass damper mounted at the end of a cantilever beam to the Lanchester damper which was shown in the previous article.  The classic single-degree-of-freedom (SDOF) tuned-mass damper is sketched in the figure below.  The design approach is to find the equivalent SDOF system for the cantilever beam’s mode of interest and then use the design formulas for an optimal SDOF TMD to determine the stiffness and damping of the absorber.

SDOF tuned-mass damper
SDOF tuned-mass damper

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Lanchester Damper on a Cantilever Beam

In this article, we show the robust and broadband performance  of a Lanchester damper applied to a cantilever beam and how it achieves good performance without tuning and good performance over a number of modes, not just the primary mode.

lanchester damper
Lanchester damper

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Writing Constraint Equations for Two-Point and Three-Point Mounts

We describe how to obtain the constraint equations for a two point pivot and three point pivot.  Designing a mechanism which can obtain a desired set of constraints is often an important step in kinematic or exact constraint machine design.

We begin with the simple lever mechanism shown in the figure below constraining the motion of two points A and C using the pivot at O.

two point constraint lever

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Airy Points, Bessel Points, Minimum Gravity Sag, and Vibration Nodal Points of Uniform Beams

Beams are often used in precision engineering applications.  One common question is “what are the optimal support locations for a beam?”  The answer depends on the desired objective.  Below we describe some of the most common support locations:  Airy points, Bessel points, minimum deflection, and nodal points.  It turns out that these points are relatively close to each other for the uniform beam.  The basic problem is sketched in the figure below.  A uniform beam is supported on two points and the objective is the determine the placement of the supports in the presence of gravity.

beam support locations

Continue reading Airy Points, Bessel Points, Minimum Gravity Sag, and Vibration Nodal Points of Uniform Beams