Rayleigh-Ritz and finite difference methods

Rayleigh-Ritz method

It improves Rayleigh's quotient by using a trial function with several adjustable constants.

1. Assume the mode shape as a series of admissible functions (satisfying the geometric boundary conditions):

$$y(x) = c_1\phi_1(x) + c_2\phi_2(x) + \cdots + c_n\phi_n(x)$$

2. Write the Rayleigh quotient $\omega^2 = \dfrac{U_{max}}{T^*_{max}}$, where $U_{max}$ is the maximum strain energy and $T^*_{max}$ is the kinetic energy expression with $\omega^2$ taken out.
3. Make $\omega^2$ stationary with respect to every constant:

$$\frac{\partial}{\partial c_i}\left(U_{max} - \omega^2T^*_{max}\right) = 0, \qquad i = 1,2,\ldots,n$$

This gives $\left([K] - \omega^2[M]\right)\{c\} = 0$, an eigenvalue problem of order $n$. 4. The $n$ roots approximate the lowest $n$ natural frequencies, always as upper bounds. More terms give better accuracy, and the higher modes are less accurate.

Finite difference method

It replaces the derivatives in the differential equation by differences between displacements at equally spaced stations.

For a beam divided into stations a distance $h$ apart:

$$\left.\frac{d^2y}{dx^2}\right|_i \approx \frac{y_{i+1} - 2y_i + y_{i-1}}{h^2}, \qquad \left.\frac{d^4y}{dx^4}\right|_i \approx \frac{y_{i+2} - 4y_{i+1} + 6y_i - 4y_{i-1} + y_{i-2}}{h^4}$$

Procedure:

1. Divide the system into $n$ segments and write the governing equation, e.g. $EI\dfrac{d^4y}{dx^4} = \omega^2\rho Ay$, at every interior station.
2. Apply the boundary conditions (e.g. $y = 0$ and $y'' = 0$ at a simple support) using extra fictitious stations if necessary.
3. The result is $[A]\{y\} = \omega^2[B]\{y\}$, solved as an eigenvalue problem.

Accuracy improves as $h$ decreases, but the order of the matrix grows. For a uniform beam, 4 to 6 segments already give the first frequency within a few percent.