Unformatted text preview: ider a cantilever beam (not shown) of length with a force of applied to the end of the
beam. The elastic modulus of the beam is and the beam’s area moment of inertia is . The
goal is to obtain an upper bound on the first natural frequency of the beam using Rayleigh’s
Quotient.
Rayleigh’s method involves assuming a ‘mode’ of deflection. Though many different functions
can be chosen for such a mode, we will use the static deflection of the beam, as the static
deflection satisfies the geometrical boundary conditions (no deflection at the cantilevered end of
the beam). The static deflection of a cantilevered beam with a force applied at the end of the
beam is found from many mechanics of materials references to be
3 6 The magnitude of the deflection at the end of the beam, at
3 This deflection is used to normalize the deflection of the beam,
1
2
which can be rewritten as , is therefore: 3 : 3 2 Recall that the general form of the kinetic energy for an elastic bar is
1
2
Taking the derivative of yields
3 2 where the velocity at the end of the beam is designated
becomes:
1
2 1
2 3 1
2 1
2 3 1
2
However, the density of the bar 33 140
/: is equal to
1
2 The effective mass of the beam, . Therefore, the kinetic energy 33
140 , is therefore
33
140 To find the effective stiffness of the beam, an expression for the potential energy stored in the
beam must be found. A general expression for the potential energy is
1
2
where a spatial derivative with respect to is designated with the prime (‘) symbol. Taking the
derivate of with respect to , and integrating, yields:
1
2 3 From which the equivalent stiffness is found to be
3
The natural frequency is thus
3 140
33 3.5675 From vibration of continuous systems theory, the exact value of is 3.5160
So, RQ method provides an upper bound on the natural frequency (and a fairly good estimate, at
that)....
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 Fall '12
 I.Green
 Energy, Force, Kinetic Energy, Strain

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