Transverse Vibrations in Viscoelastic Beams

As has been shown in preceding paragraphs, the classical theory of Euler and Bernoulli describing transverse vibrations in elastic beams can be generalized for the viscoelastic case. According to Eq. (17.13b), and based on the same grounds as Eq. (16.6b), we can write 

Free vibration, the motion that persists after the excitation is removed, is governed by Eq. (17.85), in which the applied transverse force has been made zero. Let us assume a solution of the form Uy(x, t) = /(x)exp(io)0 where f x) specifies the lateral displacement and is the angular frequency of the motion. For low loss viscoelastic materials, the free vibrations can be assumed to be quasi-harmonic, and therefore the complex modulus in the equation of motion can be used. The Laplace transform of Eq. (17.85) gives 

Equation (17.89) has the trivial solution C = Ci the values of X satisfying the secular equation 

This equation is obtained by equalizing to zero the determinant formed by the coefficients C, Cj, C3, and C4 in Eq. (17.89) under the four boundary conditions given by Eq. (17.90). This eigenvalue equation has infinite solutions for XI. The six first values are given in the following table. 

Each eigenvalue determines, through Eq. (17.91), a natural frequency of free vibration of the beam, while Eq. (17.89) gives the mode shape function f (x). 

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