Page 522 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Normal Modes

of Uniform Beams

W e assume the free vibrations of a uniform beam to be governed by E u le r’s
differential equation.

d ‘*y d^y
£/ ^ + m ^ = 0 (D -1)
dx* dt^
To determ ine the norm al modes of vibration, the solution in the form

y { x , t ) (D -2 )
is substituted into Eq. (D -1) to obtain the equation

- n M x ) = o (D -3 )
dx^
where
4>„{x) = characteristic function describing the deflection of the nth mode
m = mass density per unit length
= m w l / E I
ù)„ = { P J Ÿ ^ f E ^ / n ^ í * - natural frequency of the nth mode
The characteristic functions and the norm al mode frequencies
depend on the boundary conditions and have been tabulated by Young and Felgar.
A n abbreviated sum m ary taken from this work^ is presented here.

^D. Young, and R. P. Felgar, Jr., Tables of Characteristic Functions Representing Normal Modes
of Vibration of a Beam. The University of Texas Publication No. 4913, July 1, 1949.

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