Analytical techniques and solutions for
linear elastic solids
5.10 Rayleigh-Ritz method for estimating
natural frequency of an elastic solid
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
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:
- 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.
- 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.
- 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
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
- 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.
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
Orthogonality of mode shapes
consider a generic linear elastic solid, with elastic constants and mass density .
- 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.
definition the mode shapes and natural frequencies have the following
- 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
Orthogonality of the mode
shapes can be seen as follows.
Let and be two mode shapes, with corresponding
vibration frequencies and .
Since both mode shapes satisfy the governing equations, it follows that
Next we show that
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
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
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
that the mode shapes satisfy
see this, note first that because satisfies the equation of motion, it follows
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.
We may now expand
the potential and kinetic energy measures and in terms of sums of the mode shapes as follows
we have used the result given in step (2) and orthogonality of the mode shapes.
Finally, we know
that for ,
which shows that
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
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.
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
natural frequency can be estimated by selecting a suitable approximation for
the mode shape ,
and minimizing the ratio ,
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.
estimate into the definitions of V
and T and evaluating the integrals
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 .