end, i.e.
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
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
The mode shapes are given by [10, 13]
(2.62)
where An 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.
Example 2.10
Determine the natural frequencies of a uniform beam which is simply supported at both ends.
Solution
Applying the boundary conditions w = 0 and
Applying the boundary conditions w = 0 and
C2 sin(λL) + C4 sinh(λL) = 0 and −λ2 C2 sin(λL) + λ2 C4 sinh(λL) = 0. Therefore, nontrivial solutions are obtained when C4 = 0 and sin(λL) = 0, so λ = nπ/L (for n = 1,2,…). Since λ = (ω2 ρS/EI)1/4, we find that the natural frequencies ωn are given by
Example 2.11
Determine the lowest natural frequency for a cantilever steel beam of thickness a = 6 mm, width b = 10 mm, and length L = 0.5 m. Repeat the calculation when the beam is simply supported at both ends.
Solution
For steel we have E = 210 × 109 N/m2 and ρ = 7800 kg/m3. The cross‐sectional moment of inertia of a rectangular beam is I = ab3/12 = 0.006 × (0.01)3/12 = 5 × 10−10 m4. The cross‐sectional area of the beam is S = 0.006 × 0.01 = 6 × 10−5 m2. Then,
If the beam is now simply supported, we use Eq. (2.64) with n = 1,
Figure 2.16 First four mode shapes of a cantilever beam.
2.5.2 Vibration of Thin Plates
The present section is concerned with systems possessing two dimensions which are large compared with the third, e.g. plates whose lengths and widths are much greater than their thicknesses. There are numerous applications of vibrating plates in electroacoustical equipment such as loudspeakers, microphones, earphones, ultrasonic transducers, etc. In addition, plates can be found as constituting elements in several mechanical systems such as cars, trains, aircraft, and machinery. Plates, considered as plane systems, are a particular case of shells, whose surface may be of any shape. The general theory of vibrations of shells, which constitutes a large branch of mechanics, has been discussed by Leissa [14]. In 1828, Poisson and Cauchy established, for the first time, an approximate differential equation for flexural vibrations of a plate of infinite extent, and Poisson obtained an approximate solution for a particular case of the vibration of a circular plate. The development by Ritz in 1909 of the method for calculating the bending of plates on the basis of the energy balance was an important advance. This method made it possible to solve more complicated cases of the vibration of plates having different shapes [11]. Further advances were accomplished by the accurate calculation of the vibration distribution in rectangular plates with uniform and mixed boundary conditions [15].
a) Free Vibrations of a Rectangular Plate
For the requirements of acoustical engineering, it is sufficient to consider the flexural vibrations of thin plates (λ>>h, where h is the thickness of the plate) to which we can apply an approximate method analogous to that employed for bar [11].
The
