Chapter 5


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.


5.10.1 Mode shapes and natural frequencies; orthogonality of mode shapes and Rayleighs Principle


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:

  1. Suppose that the beam is made to vibrate by bending it into some (fixed) deformed shape ; and then suddenly releasing it.   In general, the resulting motion of the beam will be very complicated, and may not even appear to be periodic.
  2. However, there exists a set of special initial deflections , which cause every point on the beam to experience simple harmonic motion at some (angular) frequency , so that the deflected shape has the form .
  3. The special frequencies  are called the natural frequencies of the system, and the special initial deflections  are called the mode shapes.  
  4. A continuous system always has an infinite number of mode shapes and natural frequencies. The vibration frequencies and their modes are conventionally ordered as a sequence  with .  The lowest frequency of vibration is denoted . The mode shapes for the lowest natural frequencies tend to have a long wavelength; the wavelength decreases for higher frequency modes.  If you are curious, the exact mode shapes and natural frequencies for a vibrating beam are derived in Section 10.4.1.
  5. In practice the lowest natural frequency of the system is of particular interest, since design specifications often prescribe a minimum allowable limit for the lowest natural frequency.


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

  1. External forces do not influence the natural frequencies of a linear elastic solid, so we can assume that the body force acting on the interior of the solid is zero.
  2. Part of the boundary  may be subjected to prescribed displacements.  When estimating vibration frequencies, we can assume that the displacements are zero everywhere on  
  3. The remainder of the boundary  can be assumed to be traction free.


By definition the mode shapes and natural frequencies have the following properties:

  1. The displacement field associated with this vibration mode is  
  2. The displacement field must satisfy the equation of motion for a linear elastic solid given in Section 5.1.2, which can be expressed in terms of the mode shape and natural frequency as


  1. The mode shapes must satisfy  on  to meet the displacement boundary condition, and  on  to satisfy the traction free boundary condition.


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  




5.10.2 Estimate of natural frequency of vibration for a beam using Rayleigh-Ritz method


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
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