The Rayleigh-Ritz Method for Structural Analysis by Ilanko Sinniah Monterrubio Luis Mochida Yusuke

Author:Ilanko , Sinniah, Monterrubio , Luis, Mochida , Yusuke
Language: eng
Format: epub
Publisher: Wiley
Published: 2014-11-24T16:00:00+00:00

9.2. Theoretical derivations of the eigenvalue problems

In this chapter, the set of admissible functions φi (x) is taken as a combination of a complete quadratic polynomial and a cosine series, described in equation [8.2], that is:

[8.2a]

[8.2b]

[8.2c]

[8.2d]

where x is the coordinate along the beam axis, L is the beam length and n is the maximum number of terms included in the set of admissible functions.

The deflection of the beam w(x) is defined as:

[9.1]

where ω is the frequency of oscillation, t is time and W(x) is the deflected shape of the neutral line of the beam that can be expressed in terms of the set of admissible functions as

[9.2a]

or in non-dimensional form:

[9.2b]

where ξ = x / L is the non-dimensional coordinate and ai are arbitrary coefficients. Thus,

[9.3a]

[9.3b]

[9.3c]

[9.3d]

These functions combined with penalty parameters allow us to model beams with all possible combinations of boundary conditions: the classical free (F), simply supported (S) and clamped (C), as well as sliding (G) conditions.

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