Rayleigh's quotient in vibrations analysis

rdf:langString Rayleigh's quotient in vibrations analysis

rdf:langString 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 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 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) 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.