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 $$M\,{\ddot {\textbf {q}}}(t)+C\,{\dot {\textbf {q}}}(t)+K\,{\textbf {q}}(t)={\textbf {Q}}(t)$$ in absence of damping and external forces reduces to $$M\,{\ddot {\textbf {q}}}(t)+K\,{\textbf {q}}(t)=0$$

The previous equation can be written also as the following: $$K\,{\textbf {u}}=\lambda \,M\,{\textbf {u}}$$ 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 $\lambda _{m}$, ${\textbf {u}}_{m}$ that satisfy the equation $$K\,{\textbf {u}}_{m}=\lambda _{m}\,M\,{\textbf {u}}_{m}$$

By multiplying both sides of the equation by ${\textbf {u}}_{m}^{T}$ and dividing by the scalar ${\textbf {u}}_{m}^{T}\,M\,{\textbf {u}}_{m}$, it is possible to express the eigenvalue problem as follow: $$\lambda _{m}=\omega _{m}^{2}={\frac {{\textbf {u}}_{m}^{T}\,K\,{\textbf {u}}_{m}}{{\textbf {u}}_{m}^{T}\,M\,{\textbf {u}}_{m}}}$$ 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) ${\textbf {u}}_{m}$ is known. For academic interests, if the modal vectors are not known, we can repeat the foregoing process but with $\lambda =\omega ^{2}$ and ${\textbf {u}}$ taking the place of $\lambda _{m}=\omega _{m}^{2}$ and ${\textbf {u}}_{m}$, respectively. By doing so we obtain the scalar $R({\textbf {u}})$, also known as Rayleigh's quotient: $$R({\textbf {u}})=\lambda =\omega ^{2}={\frac {{\textbf {u}}^{T}\,K\,{\textbf {u}}}{{\textbf {u}}^{T}\,M\,{\textbf {u}}}}$$

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

Since, it is possible to state that the vector ${\textbf {u}}$ differs from the modal vector ${\textbf {u}}_{m}$ 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 $(u_{1})$, 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: $$M={\begin{bmatrix}1&0&0\\0&1&0\\0&0&3\end{bmatrix}}\;,\quad K={\begin{bmatrix}3&-1&0\\-1&3&-2\\0&-2&2\end{bmatrix}}$$

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: $${\textbf {F}}=k{\begin{bmatrix}m_{1}\\m_{2}\\m_{3}\end{bmatrix}}=1{\begin{bmatrix}1\\1\\3\end{bmatrix}}$$

Thus, the trial vector will become $${\textbf {u}}=K^{-1}{\textbf {F}}={\begin{bmatrix}2.5\\6.5\\8\end{bmatrix}}$$ that allow us to calculate the Rayleigh's quotient: $$R={\frac {{\textbf {u}}^{T}\,K\,{\textbf {u}}}{{\textbf {u}}^{T}\,M\,{\textbf {u}}}}=\cdots =0.137214$$

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

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: $w_{\text{real}}=0.369308$ 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.