Page 397 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

384 Classical Methods Chap. 12

If the columns arc of equal stiffness, the preceding equation reduces to
1 2h^ 3h^
24 El ^ - 24 Ei ^24EI

12.3 RAYLEIGH-RITZ METHOD

W. Ritz developed a method that is an extension of Rayleigh’s method. It not only
provides a means of obtaining a more aeeurate value for the fundamental fre­
quency, but it also gives approximations to the higher frequencies and mode
shapes.
The Ritz method is essentially the Rayleigh method in which the single shape
function is replaced by a series of shape functions multiplied by constant coeffi­
cients. The coefficients are adjusted by minimizing the frequency with respect to
each of the coefficients, which results in n algebraic equations in The solution
of these equations then gives the natural frequencies and mode shapes of the
system. As in Rayleigh’s method, the success of the method depends on the choice
of the shape functions that should satisfy the geometric boundary conditions of the
problem. The method should also be differentiable, at least to the order of the
derivatives appearing in the energy equations. The functions, however, can disre­
gard discontinuities such as those of shear due to concentrated masses that involve
third derivatives in beams.
We now outline in a general manner the procedure of the Rayleigh-Ritz
method, starting with Rayleigh’s equation:
2 ^max

CO = (12.3-1)
where the kinetic energy is expressed as In the Rayleigh method, a single
function is chosen for the deflection; Ritz, however, assumed the deflection to be a
sum of several functions multiplied by constants, as follows:
y{x) = C > ,(x ) + + ••• (12.3-2)
where (f),{x) are any admissible functions satisfying the boundary conditions.
and 7^.,^ are expressible in the form of Eqs. (7.4-1) and (7.4-2):

/ ,/
(12.3-3)
7-* = i E L m „ c,c,

where k-j and m^j depend on the type of problem. For example, for the beam, we
have
k,^ = f dx and