Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 64

If we ignore the effects of axial loads, rotary inertia, and shear deformation, the equation governing the free transverse vibrations w(x,t) of a uniform beam is given by the Euler–Bernoulli beam theory as [10, 13]

(2.51)

where E is the Young's modulus, ρ is the mass density, I is the cross‐sectional moment of inertia, and S is the cross‐sectional area. Assuming harmonic vibrations in the form

(2.52)

and substituting w(x,t) from Eq. (2.52) into Eq. (2.51) we get

(2.53)

The solution of Eq. (2.53) is

(2.54)

where λ = (ω² ρS/EI)^1/4 and the C's are arbitrary constants that depend upon the boundary conditions (the deflections, slope, bending moment, and shear force constraints). Classical boundary conditions for a beam are

(2.55)

(2.56)

(2.57)

A very important practical case is a cantilever beam (clamped‐free beam) of length L. In this case, the deflection and slope are zero at the clamped end, while the bending moment and shear force are zero at the free end, i.e.

(2.58)

(2.59)

Substituting the boundary conditions Eq. (2.58) and Eq. (2.59) into Eq. (2.54), we find that C₂ = −C₄, and we obtain the equation

(2.60)

The roots of the transcendental Eq. (2.60) can be obtained numerically. The first four roots are λ₁ L = 1.875, λ₂ L = 4.694, λ₃ L = 7.855, and λ₄ L = 10.996. For large values of n, the roots can be calculated using the equation

(2.61)

Noting that , we can solve for ω_n so that the first four natural frequencies of the cantilever beam are

The mode shapes are given by [10, 13]

(2.62)

where A_n is an arbitrary constant. Thus, the total solution for the free transverse vibration of the cantilever beam is

(2.63)

Figure 2.16 shows the first four mode shapes for a cantilever beam.