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

Summary
  • Airy points — zero slope at ends — supports located 0.21132 L from each end
  • Bessel points — minimum change in length — supports located 0.2203 L from each end
  • Minimum deflection due to gravity — supports located 0.2232 L from each end
  • Nodal points of first vibration mode — zero deflection at these points during free vibration — supports located 0.2242 L from each end
Airy points

Airy points are support points such that the ends of the beam have zero local slope.  A nice derivation is given in a few different references including

The end result is that the spacing s between the supports is s=L/sqrt(3)=0.57735 L.  The distance of each supports from the ends is 0.21132 L.  

Yoder and Vukobratovich in Opto-mechanical Systems Design  give the result for a beam with multiple supports as s=L/sqrt(n^2-1).

 

Bessel points

Bessel points are the support points which minimize the change in length of the beam.  A beam will undergo a length change as it bends given by

    \[ \Delta L=\frac{1}{2} \int_0^L \left( \frac{ \partial v}{\partial x} \right)^2 dx \]

The Bessel points are located at 0.2203 L from each end.

A good discussion of Airy points, Bessel points, and minimum gravity sag is given in the thesis by Nijsse at TU Delft, Linear motion systems: a modular approach for improved straightness performance.    He makes a good point about accuracy of results by comparing the locations and distortions calculated from various beam models including Euler-Bernoulli, Timoshenko, and plane strain elasticity.

A brief mention of Bessel and Airy points is also given in the Mitutoyo catalog.

Minimum deflection

The minimum deflection occurs when the deflection at each end is the same as the deflection in the middle of the beam.  For a Euler-Bernoulli beam, this minimum deflection occurs with supports placed 0.2232 L from each end.

Nodal points

Nodal points refer to the locations which have no displacement during vibration.  In general we are interested in the first vibration mode.  By supporting at these points, the first natural frequency of the beam on supports is maintained at its maximum value, which is the same frequency as the first flexible mode of the free-free beam.

The boundary conditions of a free-free beam are no moment and no force on the end.  The mode shapes are given by

    \begin{equation*} W_n(x)=C_n \left[ \sin \beta_n x + \sinh \beta_n x + \alpha_n \left(  \cos \beta_n x + \cosh \beta_n x  \right) \right] \end{equation*}

where n is the vibration mode number and

    \begin{equation*} \alpha_n= \frac{\sin \beta_n L - \sinh \beta_n L }{ \cosh \beta_n L - \cos \beta_n L} \end{equation*}

The natural frequency can be obtained from \omega_n^2=\beta_n^4  EI/\rho A by solving (in the case of a free-free beam)

    \begin{equation*} \cos \beta_n L  \cosh \beta_n L = 1 \end{equation*}

The nodal points are found by solving W_1(x)=0.  For a free-free beam, one finds that \beta_1 L=4.73004074486270 and that the nodal points are located at 0.22415752270 L from each end.

 

 

2 thoughts on “Airy Points, Bessel Points, Minimum Gravity Sag, and Vibration Nodal Points of Uniform Beams”

  1. Thanks for this reference; I vaguely remember these things from engineering school 25 years ago!

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