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.

Python script for static deflection of a beam using finite elements

Below we present a simple script for calculating the static deflection of a beam with a variety of boundary conditions and load types.  The finite element method is implemented using Python with the numpy library and plot are made using matplotlib.  This code can be easily modified for other boundary conditions or loads.


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Quick calculations for radiation heat transfer

In certain cases, radiation heat transfer is important to include in one’s calculations.  Radiation heat transfer is nonlinear because the heat flux is proportional to the temperature to the fourth power.  I find two frequent simplifications for radiation problems quite useful.  The first approximation is to linearize and create an effective convection coefficient.  The second approximation is the effective emissivity for two bodies which are transferring heat between each other via radiation.

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Optimal 3-point mounts for square and rectangular plates

Below is a table of support positions for three-point mounts of rectangular plates to maximize the first natural frequency of vibration.  If one has a plate with an aspect ratio not listed, then one can interpolate between the values.  We will publish a more detailed post giving methods and results at a later date.


Young’s modulus for silicon

Silicon is one of the most commonly encountered materials in precision engineering (including silicon wafers and MEMS), but it poses a challenge in modeling as its material properties are anisotropic (orthotropic to be exact), meaning that the stiffness varies depending on the direction of loading relative to the crystal orientation.

A great reference for its elastic properties including when isotropic approximations are appropriate is the paper by Hopcroft et al (M. A. Hopcroft, W. D. Nix, and T. W. Kenny, “What is the Young’s Modulus of Silicon?,”
Journal of Microelectromechanical Systems, vol. 19, pp. 229-238, 2010.  link to researchgate  Additionally Hopcroft’s course notes are useful



Identical natural frequencies of beams with different boundary conditions

One somewhat surprising result in the vibration of beams is that the natural frequencies of a free-free beam are identical to those of a clamped-clamped beam. Our intuition may tell us that the fixed-fixed beam is stiffer and should have higher natural frequencies but while the fixed beam is indeed statically stiffer, the natural frequencies are identical. As we show below, fixed and free boundary conditions result in the equivalent characteristic equations to find the natural frequency because they differ simply by two differentiations. Similarly, pinned and sliding boundary conditions also have the same equivalence. As a result many pairs of boundary conditions can result in identical sets of natural frequencies but with clearly different mode shapes.

Fixed-fixed and free-free beams

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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, 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  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.

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