- Rayleigh-Ritz method
In
applied mathematics andmechanical engineering , the Rayleigh-Ritz method is a widely used, classical method for the calculation of the natural vibrationfrequency of a structure in the second or higher order. It is adirect variational method in which the minimum of a functional defined on anormed linear space is approximated by a linear combination of elements from that space. This method will yield solutions when an analytical form for the true solution may be intractable.The method is also widely used in Quantum Chemistry.
Typically in
mechanical engineering it is used for finding the approximate real resonant frequencies of multi degree of freedom systems, such asspring mass system s orflywheel s on a shaft with varying cross section. It is an extension ofRayleigh's method . It can also be used for findingbuckling loads for columns, as well as more esoteric uses.The following discussion uses the simplest case, where the system has two lumped springs and two lumped masses, and only two mode shapes are assumed. Hence M= [m1,m2] and K= [k1,k2] .
A
mode shape is assumed for the system, with two terms, one of which is weighted by a factor B, eg Y= [1,1] +B* [1,-1] .Simple harmonic motion theory says that thevelocity at the time when deflection is zero, is theangular frequency w times the deflection (y) at time of maximum deflection. In this example thekinetic energy (KE) for each mass is 1/2*w^2*Y_1^2*m_1 etc, and thepotential energy (PE) for each spring is 1/2*k_1*Y_1^2 etc . Forcontinuous system s the expressions are more complex.We also know, since no
damping is assumed, that KE when y=0 equals the PE when v=0 for the whole system. As there is no damping all locations reach v=0 simultaneously.so, since KE=PE
sum_{i=1}^2 (1/2*w^2*Y_i^2*M_i)=sum_{i=1}^2 (1/2*K_i*Y_i^2)
Note that the overall amplitude of the mode shape cancels out from each side, always. That is, the actual size of the assumed deflection does not matter, just the mode "shape".
A bit of mathematical skullduggery then reveals a solution for w, in terms of B. Then
differentiate w with respect to B, and find the minimum, i.e. when dw/dB=0. This gives the value of B for which w is lowest. This is an upper bound solution for w if w is hoped to be the predicted fundamental frequency of the system because the mode shape is "assumed", but we have found the lowest value of that upper bound, given our assumptions, because B is used to find the optimal 'mix' of the two assumed mode shape functions.There are many tricks with this method, the most important is to try and choose realistic assumed mode shapes. In the case of
beam deflection problems it is wise to use aquartic (?check) , as the naturally deformed shape of a uniform beam is a quartic. The springs and masses do not have to be discrete, they can be continuous (or a mixture), and this method can be easily used in aspreadsheet to find the natural frequencies of quite complex distributed systems, if you can describe the distributed KE and PE terms easily, or else break the continuous elements up into discrete parts.This method could be used
iteratively , adding additional mode shapes to the previous best solution, or you can build up a long expression with many Bs and many mode shapes, and then differentiate them partially.Timeline
*J. W. Rayleigh did his relevant paper before 1900
*W. Ritz did his after 1900
*E. T. Whittaker and G. Robinson worked with it in the 1960s
*G. Arfken did work concerning it in the 1980sSee also
*
Ritz_method External links
*http://www.samparker.co.uk - Guide to Dunkerley and Rayleigh Method.
*http://ctr.stanford.edu/Summer98/eyink.pdf
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