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

$${\displaystyle M\,{\ddot {\textbf {q}}}(t)+C\,{\dot {\textbf {q}}}(t)+K\,{\textbf {q}}(t)={\textbf {Q}}(t)}$$

in absence of damping and external forces reduces to

$${\displaystyle M\,{\ddot {\textbf {q}}}(t)+K\,{\textbf {q}}(t)=0}$$

The previous equation can be written also as the following:

$${\displaystyle K\,{\textbf {u}}=\lambda \,M\,{\textbf {u}}}$$

where ${\displaystyle \lambda =\omega ^{2}}$, in which ${\displaystyle \omega }$ 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 ${\displaystyle \lambda _{m}}$, ${\displaystyle {\textbf {u}}_{m}}$ that satisfy the equation

$${\displaystyle K\,{\textbf {u}}_{m}=\lambda _{m}\,M\,{\textbf {u}}_{m}}$$

By multiplying both sides of the equation by ${\displaystyle {\textbf {u}}_{m}^{T}}$ and dividing by the scalar ${\displaystyle {\textbf {u}}_{m}^{T}\,M\,{\textbf {u}}_{m}}$, it is possible to express the eigenvalue problem as follow:

$${\displaystyle \lambda _{m}=\omega _{m}^{2}={\frac {{\textbf {u}}_{m}^{T}\,K\,{\textbf {u}}_{m}}{{\textbf {u}}_{m}^{T}\,M\,{\textbf {u}}_{m}}}}$$

for m = 1, 2, 3, ..., n.

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) ${\displaystyle {\textbf {u}}_{m}}$ is known. For academic interests, if the modal vectors are not known, we can repeat the foregoing process but with ${\displaystyle \lambda =\omega ^{2}}$ and ${\displaystyle {\textbf {u}}}$ taking the place of ${\displaystyle \lambda _{m}=\omega _{m}^{2}}$ and ${\displaystyle {\textbf {u}}_{m}}$, respectively. By doing so we obtain the scalar ${\displaystyle R({\textbf {u}})}$, also known as Rayleigh's quotient:[1]

$${\displaystyle R({\textbf {u}})=\lambda =\omega ^{2}={\frac {{\textbf {u}}^{T}\,K\,{\textbf {u}}}{{\textbf {u}}^{T}\,M\,{\textbf {u}}}}}$$

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

Since, it is possible to state that the vector ${\displaystyle {\textbf {u}}}$ differs from the modal vector ${\displaystyle {\textbf {u}}_{m}}$ 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 ${\displaystyle (u_{1})}$, that generally works well for most structures (even though is not guaranteed), is to assume ${\displaystyle (u_{1})}$ 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.