The
Rayleigh's quotient
In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as:
R(M,x) = .
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the co ...
represents a quick method to estimate the natural
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is ...
of a multi-degree-of-freedom vibration system, in which the
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
and the stiffness matrices are known.
The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
problem for a general system of the form
in absence of damping and external forces reduces to
The previous equation can be written also as the following:
where
, in which
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
,
that satisfy the equation
By multiplying both sides of the equation by
and dividing by the scalar
, it is possible to express the eigenvalue problem as follow:
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)
is known. For academic interests, if the modal vectors are not known, we can repeat the foregoing process but with
and
taking the place of
and
, respectively. By doing so we obtain the scalar
, also known as Rayleigh's quotient:
Therefore, the Rayleigh's quotient is a scalar whose value depends on the vector
and it can be calculated with good approximation for any arbitrary vector
as long as it lays reasonably far from the modal vectors
, ''i'' = 1,2,3,...,''n''.
Since, it is possible to state that the vector
differs from the modal vector
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
, that generally works well for most structures (even though is not guaranteed), is to assume
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:
Thus, the trial vector will become
that allow us to calculate the Rayleigh's quotient:
Thus, the lowest natural frequency, calculated by means of Rayleigh's quotient is:
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:
that has led to an error of
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
{{reflist
Abstract algebra
Linear algebra
Mathematical physics
Mechanical vibrations