Analytical techniques and solutions for linear elastic solids
5.10 Rayleigh-Ritz method for estimating natural frequency of an elastic solid
We conclude this chapter by describing an energy based method for estimating the natural frequency of vibration of an elastic solid.
It is helpful to review the definition of natural frequencies and mode shapes for a vibrating solid. To this end, consider a representative elastic solid say a slender beam that is free at both ends, as illustrated in the figure. The physical significance of the mode shapes and natural frequencies of the vibrating beam can be visualized as follows:
We will derive two important results below, which give a quick way to estimate the lowest natural frequency:
1. The mode shapes are orthogonal, which means that the displacements associated with two different vibration modes and have the property that
2. We will prove Rayleigh’s principle, which can be stated as follows. Let denote any kinematically admissible displacement field (you can think of this as a guess for the mode shape), which must be differentiable, and must satisfy on . Define measures of potential energy and kinetic energy associated with as
, and if and only if
The result is useful because the fundamental frequency can be estimated by approximating the mode shape in some convenient way, and minimizing .
Orthogonality of mode shapes
We consider a generic linear elastic solid, with elastic constants and mass density . Note that
By definition the mode shapes and natural frequencies have the following properties:
Orthogonality of the mode shapes can be seen as follows.
1. Let and be two mode shapes, with corresponding vibration frequencies and . Since both mode shapes satisfy the governing equations, it follows that
2. Next we show that
To see this, integrate both sides of this expression by parts. For example, for the left hand side,
where we have used the divergence theorem, and noted that the integral over the surface of the solid is zero because of the boundary conditions for and . An exactly similar argument shows that
Recalling that shows the result.
3. Finally, orthogonality of the mode shapes follows by subtracting the second equation in (1) from the first, and using (2) to see that
If m and n are two distinct modes with different natural frequencies, the mode shapes must be orthogonal.
Proof of Rayleigh’s principle
1. Note first that any kinematically admissible displacement field can be expressed as a linear combination of mode shapes as
To see the formula for the coefficients , multiply both sides of the first equation by , integrate over the volume of the solid, and use the orthogonality of the mode shapes.
2. Secondly, note that the mode shapes satisfy
To see this, note first that because satisfies the equation of motion, it follows that
Next, integrate the first term in this integral by parts (see step (2) in the poof of orthogonality of the mode shapes), and use the orthogonality of the mode shapes to see the result stated.
3. We may now expand the potential and kinetic energy measures and in terms of sums of the mode shapes as follows
where we have used the result given in step (2) and orthogonality of the mode shapes.
4. Finally, we know that for , which shows that
We see immediately that , with equality if and only if for
The figure illustrates the problem to be solved: an initially straight beam has Young’s modulus and mass density , and its cross-section has area A and moment of area . The left hand end of the beam is clamped, the right hand end is free. We wish to estimate the lowest natural frequency of vibration.
The deformation of a beam can be characterized by the deflection of its neutral section. The potential energy of the beam can be calculated from the formula derived in Section 5.7.4, while the kinetic energy measure T can be approximated by assuming the entire cross-section displaces with the mid-plane without rotation, which gives
The natural frequency can be estimated by selecting a suitable approximation for the mode shape , and minimizing the ratio , as follows:
1. Note that the mode shape must satisfy the boundary conditions . We could try a polynomial , where C is a parameter that can be adjusted to get the best estimate for the natural frequency.
2. Substituting this estimate into the definitions of V and T and evaluating the integrals gives
3. To get the best estimate for the natural frequency, we must minimize this expression with respect to C. It is straightforward to show that the minimum value occurs for . Substituting this value back into (2) gives
4. For comparison, the a formula for exact natural frequency of the lowest mode is derived in Section 10.4.1, and gives .
(c) A.F. Bower, 2008