Abstract of the problem:

In 1890 G.H. Bryan observed that when a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotates at a rate proportional to the inertial rate of rotation. This effect, called “Bryan’s effect”, as well as the proportionality constant, called “Bryan’s factor”, have numerous navigational applications. Using a computer algebra system, we present a numerically accurate method for determining fundamental eigenvalues (and some of the overtone eigenvalues) as well as the corresponding eigenfunctions for a linear ordinary differential equation (ODE) boundary value problem (BVP) associated with a slowly rotating vibrating disc. The method provides easy and accurate calculation of Bryan’s factor, which is used to calibrate the resonator gyroscopes used for navigation in deep space missions, stratojets and submarines. Bryan used “thin shell theory” to calculate Bryan’s factor for fundamental vibrations. Apart from the high accuracy achieved, the numerical routine used here is more robust than “thin shell theory” because it determines (at least for low frequencies) the fundamental as well as the first three overtone frequencies and each Bryan’s factor associated with these vibration modes. The theory involved and the calculation of results with this numerical method are quick, easy and accurate and might be applied in other disciplines that need to solve suitable eigenvalue problems. Indeed, results are obtained directly using commercial software to numerically solve a system of linear ODE BVPs without having to formulate the extremely technical solution that is traditionally used (viz: solve the governing system of partial differential equations via Helmholtz potential functions and the necessary numerical calculation of Bessel and Neumann functions).

• Simulating gyroscopes as used in NASA spaceflights: Need to use numerical calculations.
• Found a new method that hasn’t been used before. Need several decimal places of accuracy in order to arrive at specific place in the solar system.
• Helps students to understand about eigenvalues and eigenfunctions

Present an easy four step algorithm for finding eigenvalues of a vibrating cylindrical bar

• First step: determine a formula in terms of an eigenfunction for an eigenvalue of vibration. This involves finding kinetic and potential energy of elements of the bar and the Euler Lagrange equations
• Second step: to determine the partial differential equations that model the eigenvalue problem
• Third step: assume that a harmonic solution exists
• Fourth step: numerical solve the associated ODE

Numerical routine can be found in the paper here.

Use an iterative approach to find a sequence of $\omega$ accurate to six digits using Mathematica.

Use the Rayleigh-Love model.

Student exercise:

• Determine the fundamental eigenvalue
• Using Mathematica, solve the classical bar ODE Boundary value problem
• Use the expression to calculate the frequencies and use the iteration procedure until convergence is observed.
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