Rayleigh's quotient in vibrations analysis explained

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

The previous equation can be written also as the following:$$K\backslash ,\backslash textbf\; =\; \backslash lambda\backslash ,M\backslash ,\backslash textbf$$where

λ= \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

λ _{m} , bf{u} _{m} that satisfy the equation$$K\backslash ,\backslash textbf\_m\; =\; \backslash lambda\_m\backslash ,M\backslash ,\backslash textbf\_m$$

By multiplying both sides of the equation by

bf{u} ^{T} _{m} and dividing by the scalar bf{u} ^{TM bf{u}} _{m} , it is possible to express the eigenvalue problem as follow:$$\backslash lambda\_m\; =\; \backslash omega\_m^2\; =\; \backslash frac$$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)

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

Therefore, the Rayleigh's quotient is a scalar whose value depends on the vector

bf{u} and it can be calculated with good approximation for any arbitrary vector bf{u} as long as it lays reasonably far from the modal vectors bf{u} _{i} , i = 1,2,3,...,n.

Since, it is possible to state that the vector

bf{u} differs from the modal vector bf{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:

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:$$\backslash textbf\; =\; k\backslash begin\; m\_1\; \backslash \backslash \; m\_2\; \backslash \backslash \; m\_3\; \backslash end\; =\; 1\; \backslash begin\; 1\; \backslash \backslash \; 1\; \backslash \backslash \; 3\; \backslash end$$

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

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

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 _{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.

Notes and References

1. Book: Meirovitch. Leonard. Fundamentals of Vibration. 2003. McGraw-Hill Education. 9780071219839. 806.