Rayleigh's quotient in vibrations analysis

The Rayleigh's quotient represents a quick method to estimate the natural frequency of a multi-degree-of-freedom vibration system, in which the mass and the stiffness matrices are known.

The eigenvalue problem for a general system of the form
$M\,\ddot(t) + C\,\dot(t) + K\,\textbf(t) = \textbf(t)$
in absence of damping and external forces reduces to
$M\,\ddot(t) + K\,\textbf(t) = 0$

The previous equation can be written also as the following:
$K\,\textbf = \lambda\,M\,\textbf$
where $\lambda=\omega^2$, in which $\omega$ represents the natural frequency, M and K are the real positive symmetric mass and stiffness matrices respectively.

For an ''n''-degree-of-freedom system the equation has ''n'' solutions $\lambda_m$, $\textbf_m$ that satisfy the equation
$K\,\textbf_m = \lambda_m\,M\,\textbf_m$

By multiplying both sides of the equation by $\textbf_^$ and dividing by the scalar $\textbf_m^T\,M\,\textbf_m$, it is possible to express the eigenvalue problem as follow:
$\lambda_m = \omega_m^2 = \frac$
for .

In the previous equation it is also possible to observe that the numerator is proportional to the potential energy while the denominator depicts a measure of the kinetic energy. Moreover, the equation allow us to calculate the natural frequency only if the eigenvector (as well as any other displacement vector) $\textbf_$ is known. For academic interests, if the modal vectors are not known, we can repeat the foregoing process but with $\lambda = \omega^2$ and $\textbf$ taking the place of $\lambda_ = \omega_^2$ and $\textbf_$, respectively. By doing so we obtain the scalar $R(\textbf)$, also known as Rayleigh's quotient:
$R(\textbf) = \lambda = \omega^2 = \frac$

Therefore, the Rayleigh's quotient is a scalar whose value depends on the vector $\textbf$ and it can be calculated with good approximation for any arbitrary vector $\textbf$ as long as it lays reasonably far from the modal vectors $\textbf_$, ''i'' = 1,2,3,...,''n''.

Since, it is possible to state that the vector $\textbf$ differs from the modal vector $\textbf_m$ by a small quantity of first order, the correct result of the Rayleigh's quotient will differ not sensitively from the estimated one and that's what makes this method very useful. A good way to estimate the lowest modal vector $(u_1)$, that generally works well for most structures (even though is not guaranteed), is to assume $(u_1)$ equal to the static displacement from an applied force that has the same relative distribution of the diagonal mass matrix terms. The latter can be elucidated by the following 3-DOF example.

Example – 3DOF

As an example, we can consider a 3-degree-of-freedom system in which the mass and the stiffness matrices of them are known as follows:
$M = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end \; , \quad K = \begin 3 & -1 & 0 \\ -1 & 3 & -2 \\ 0 & -2 & 2 \end$

To get an estimation of the lowest natural frequency we choose a trial vector of static displacement obtained by loading the system with a force proportional to the masses:
$\textbf = k\begin m_1 \\ m_2 \\ m_3 \end = 1 \begin 1 \\ 1 \\ 3 \end$

Thus, the trial vector will become
$\textbf = K^\textbf = \begin 2.5 \\ 6.5 \\ 8 \end$
that allow us to calculate the Rayleigh's quotient:
$R = \frac = \cdots = 0.137214$

Thus, the lowest natural frequency, calculated by means of Rayleigh's quotient is:
$w_\text = 0.370424$

Using a calculation tool is pretty fast to verify how much it differs from the "real" one. In this case, using MATLAB, it has been calculated that the lowest natural frequency is: $w_\text = 0.369308$ that has led to an error of $0.302315 \%$ using the Rayleigh's approximation, that is a remarkable result.

The example shows how the Rayleigh's quotient is capable of getting an accurate estimation of the lowest natural frequency. The practice of using the static displacement vector as a trial vector is valid as the static displacement vector tends to resemble the lowest vibration mode.

References

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Abstract algebra
Linear algebra
Mathematical physics
Mechanical vibrations