The tool SD-01 computes the natural frequencies of a uniform Euler-Bernoulli beam in transverse free vibration, and plots and animates associated mode shapes of vibration. The tool also gives analytical expressions of modes shapes. The determined natural frequencies and mode shapes are eigensolutions (eigenvalues and eigenfunctions) of an eigenvalue problem in beam free vibration; see the subject review below.
Consider an example as shown in the following figure.
The input in the tool is
After clicking "Run", the results will be shown in another page. The first four modes of the system are
The eigenvalues and the analytical expression of eigenvectors are also shown as follows.
w1 = 22.56994213 rad/s; f1 = 3.59211785 Hz
w2 = 73.14106874 rad/s; f2 = 11.64076263 Hz
w3 = 152.60300186 rad/s; f3 = 24.28752208 Hz
w4 = 260.96016303 rad/s; f4 = 41.5330999 Hz
Flexible Modes: u(x) = a1*exp(beta*x)+a2*exp(-beta*x)+a3*cos(beta*x)+a4*sin(beta*x)
Mode #1: a1=0.00002287, a2=-0.05886029, a3=0.05883742, a4=-0.05888316 ;
Mode #2: a1=0.00000001, a2=-0.01089782, a3=0.01089781, a4=-0.01089782 ;
Mode #3: a1=0, a2=-0.00368794, a3=0.00368794, a4=-0.00368794 ;
Mode #4: a1=0, a2=-0.00166198, a3=0.00166198, a4=-0.00166198 ;
where beta = (wk2*ρ/(E*I))1/4
Eigenvalue problems play an essential role in dynamic analysis of flexible structures. In this note, the eigenvalue problem of Euler-Bernoulli Beams and related characteristic equation for natural frequencies are derived.
Figure 1. A uniform Euler-Bernoulli beam in transverse free vibration
A uniform Euler-Bernoulli beam in Fig. 1 is in transverse free vibration, which is governed by the partial differential equation
with the boundary conditions
Here, by “uniform beam”, the beam cross section area and bending stiffness are assumed to be constant along the beam length (x). Four types of beam boundary conditions are given in Table 1.
Table 1. Beam boundary conditions
To derive the eigenvalue problem for the Euler-Bernoulli beam, let the beam displacement be
Substituting Eq. (3) into Eqs. (1) and (2) yields
and the boundary conditions
Equation (4) is known as an eigen equation because it is a homogeneous equation with an unknown parameter (eigenvalue) and an unknown function (eigenfunction). A solution of the equation, namely, an eigenfunction, can be written as
where a1, a2, a3 and a4 are constants to be determined by the boundary conditions (5). Plugging Eq. (6) into the boundary conditions (5) yields the homogenous matrix equation
For a nonzero solution (vector) of Eq. (7), the determinant of the matrix must be zero. This gives the characteristic equation of the beam
Equation (8) is transcendental, and thus has an infinite number of roots
It follows that the natural frequencies of the beam are given by
Once the natural frequencies are determined by solving the characteristic equation (8), the associated mode shapes can be obtained by solving Eq. (7) for a nonzero vector and by using Eq. (6).
As an example, consider a beam with clamped-pinned ends, whose eigenfunctions satisfy the boundary conditions
where u' = du/dx. Substituting Eq. (6) into the above boundary conditions eventually leads to
Hence, the characteristic equation of the beam by Eq. (8) is
The mode shapes are given by
Table 2 lists eigensolutions of Euler-Bernoulli beams in several cases of boundary conditions.
Table 2. Eigensolutions of Euler-Bernoulli beams with certain boundary conditions
1. Den Hartog, J. P., 1985, Mechanical Vibrations, Dover Publications, New York.
2. Timoshenko, S., 1990, Vibration Problems in Engineering, John Wiley & Sons; 5th edition.
3. Inman, D. J., 2000, Engineering Vibrations, 2nd edition, Prentice Hall, Inc., Upper Saddle River, New Jersey.
4. Meirovitch, L., 1967, Analytical Methods in Vibrations, Macmillan, New York
5. Rao, S. S., 2003, Mechanical Vibrations, 4th edition, Prentice Hall, Inc., Upper Saddle River, New Jersey..
6. Yang, B., 2005, Stress, Strain, and Structural Dynamics: An Interactive Handbook of Formulas, Solutions, and MATLAB toolboxes, Elsevier Science.