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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  Â say a slender beam that is free at both
ends, as illustrated in the figure.Â
The physical significance of the mode shapes and natural frequencies
of the vibrating beam can be visualized as follows:
- Suppose that the
beam is made to vibrate by bending it into some (fixed) deformed shape ;
and then suddenly releasing it. Â
In general, the resulting motion of the beam will be very
complicated, and may not even appear to be periodic.
- 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 .
- The special
frequencies  are called the natural frequencies
of the system, and the special initial deflections  are called the mode shapes. Â
- 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.
- 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
Then
,
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
- 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.
- Part of the boundary
 may be subjected to prescribed displacements. When estimating vibration frequencies,
we can assume that the displacements are zero everywhere on
- The remainder of the
boundary  can be assumed to be traction free.
By
definition the mode shapes and natural frequencies have the following
properties:
- The displacement
field associated with this vibration mode is
- 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
- 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
  Â
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 .
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