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.
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.
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.
In a previous post, we built a quick model for eddy currents in a plate stationary in a time-varying magnetic field. Here we examine the induced currents and damping force that result from motion of the plate relative to the magnetic field.
In electric machines, there are two common causes of eddy currents: (1) time-varying currents in coils, and (2) motion of conductors relative to sources of magnetic field. In this post, we show how to estimate the current density arising from a time-varying magnetic field passing through a plate.
Low Reynolds number flow can be a very interesting topic. Low Reynolds number flow (Re ) 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
where is the fluid density, is the velocity, is a characteristic length such as a diameter, and 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
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.
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.
The Handbook of Vacuum Technology edited by Karl Jousten is a thorough reference with detailed calculations for wide variety of problems in vacuum systems.
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.
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.