Mechanical Wave Vibrations: Analysis and Control by Mei Chunhui;

Author:Mei, Chunhui; [Mei, Chunhui]
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2023-07-03T19:51:22+00:00

Figure 9.2 Wave propagation relationships.

where

(9.6)

are the wave vectors and

(9.7)

is the bending propagation matrix for a distance x.

9.1.2 Shear Bending Vibration Theory

For a uniform beam subjected to no external force, as shown in Figure 9.1, the governing equations of motion for free bending vibration by the Shear bending vibration theory are

(9.8a)

(9.8b)

where is the slope due to bending, is the slope of the centerline of the beam, and is the shear angle. G and are the shear modulus and shear coefficient, respectively.

Equations (9.8a) and (9.8b) are coupled through the bending slope and the bending deflection of the structure.

Eliminating from Eqs. (9.8a) and (9.8b), the equation of motion of the bending deflection of the centerline of the beam is

(9.9a)

Similarly, eliminating from Eqs. (9.8a) and (9.8b), the equation of motion of the bending slope is

(9.9b)

which is governed by the same differential equation as Eq. (9.9a).

Assuming time harmonic motion and using separation of variables, the solutions can be written in the form and , where i is the imaginary unit, ω is the circular frequency, and k is the wavenumber. and are the amplitudes of the bending deflection of the centerline of the beam and the bending slope, respectively. Substituting these expressions for y(x,t) and ψ(x,t) into Eqs. (9.8a) and (9.8b) and putting the equations into matrix form,

(9.10)

Setting the determinant of the coefficient matrix of Eq. (9.10) to zero gives the dispersion equation, from which the bending wavenumbers are solved,

(9.11)

where and subscript s indicates that coefficient is related to the shear effect. An infinite shear rigidity, that is, , corresponds to zero shear deformation.

The bending wavenumbers obtained in Eq. (9.11) are functions of the circular frequency as well as the material and geometrical properties of the structure. The signs outside the brackets of Eq. (9.11) indicate that waves in the beam travel in both the positive and negative directions. The waves are dispersive because the phase velocity is frequency dependent. Equation (9.11) shows that, in the absence of damping, one pair of wavenumbers is always real, which corresponds to positive- and negative-going propagating waves. The other pair of wavenumbers is always imaginary, which corresponds to positive- and negative-going decaying waves.

With time dependence suppressed, the solutions to Eqs. (9.8a) and (9.8b) can be written as

(9.12a)

(9.12b)

where the wavenumbers are

(9.13)

The wave amplitudes a of and of are related and their relationships are found from Eq. (9.10),

(9.14)

where

(9.15)

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