Structural Dynamics Characteristic Roots

Finding the the natural frequencies of a continuous beam, plate, membrane or even an acoustical volume requires solving for the eigenvalue roots of a characteristic equation, which may contain trigonometric or Bessel functions.

Matlab provides an fzero function which can be used for this purpose. The fzero command is a function file. The algorithm, created by T. Dekker, uses a combination of bisection, secant, and inverse quadratic interpolation methods.

Here is a script which demonstrates this function: vibration_roots.m

The script has options for the following equations:

f(x)=cos(x)cosh(x)+1
f(x)=cos(x)cosh(x)-1
f(x)=tan(x)+tanh(x)
f(x)=tan(x)-tanh(x)
f(x)=Bessel Jo
f(x)=Bessel J1
f(x)=Bessel J2
f(x)=Bessel (Jn+1(x)/Jn(x)) + (In+1(x)/In(x)) -2x/(1-mu)

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The roots may also be found via the Newton-Raphson method as shown in: Newrap.pdf

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See also:

Beam Natural Frequencies

Bessel Functions in Vibration Analysis

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– Tom Irvine

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