Perturbation theory and the Rayleigh quotient

• Published in: Journal of sound and vibration Vol. 330; no. 9; pp. 2073 - 2078
• Main Authors: Chan, K.T., Stephen, N.G., Young, K.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 25.04.2011; Elsevier
• Subjects: Eigenfunctions; Eigenvalues; Error analysis; Errors; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Quotients; Shear; Solid mechanics; Structural and continuum mechanics; Theorems; Vibration; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Elastic constant; Perturbation method; Rayleigh quotient; Vibration; Shear modulus; Deformation modulus; Eigenfunction; Timoshenko beam; Eigenvalue problem; Modeling; Small parameter method
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2010.11.001

The characteristic frequencies ω of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter ε, it is shown that when an approximate eigenfunction is known with an error O(εN), the Rayleigh quotient gives the frequency with an error O(ε2N), a gain of N orders. This result generalizes a well-known theorem for N=1. A non-trivial example is presented for N=4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theory.