ETBX Uniform Beam Frequencies

Uniform Beam Frequencies

An EngineersToolbox Calculation Module

Background Information:

This module solves for the natural frequencies and mode shapes for transverse free vibration of uniform beams with specified boundary conditions. It uses closed-form solutions to the differential equation:

where:

and:

x -	axial position of a point on the beam
v -	vertical displacement at x
A -	cross-sectional area of the beam
I -	cross-sectional moment of inertia
r -	mass density of the material
E -	elastic modulus of the material

The closed-form solutions are derived by considering the general solution:

The constants A₁, A₂, A₃, and A₄ are calculated by a evaluating set of four equations given by the beam boundary conditions at x=0 and x=L. Some simple boundary conditions and assumed end constraints are listed in Table 1.

Table 1: Typical beam boundary (end) conditions

End Condition:	Fixed	Guided	Pinned	Free
Boundary assumptions:	No displacement or slope.	No slope or shear.	No displacement or moment.	No moment or shear.
Equations:

Substituting for A₁ through A₄ leads to a characteristic equation with multiple roots C_i that are the eigenvalues l_i times the beam length L. In most cases, no simple expression for the roots of the characteristic equation is available and a numerical solution is required.

The natural frequencies of the jth mode of the beam are given by the expression:

and the mode shapes corresponding to the jth natural frequency are given by:

where:

Input:

The Uniform Beam Frequencies module input form is shown in Figure 1. The module verifies that all input parameters are positive real values.

Figure 1. Uniform Beam Frequencies module input form

Output:

The module calculates the natural frequencies for transverse free vibration of the uniform beam with the specified boundary conditions. Results are tabulated using the standard ETB output window shown in Figure 2.

Figure 2. Uniform Beam Frequencies tabulated results.

References:

Craig, R. R. (1981), Structural Dynamics, John Wiley & Sons (New York).

Thomson W.T., and Dahleh, M. D. (1997) Theory of Vibrations with Applications (5th Edition), Prentice Hall (New Jersey).

Woods, R. L., Lawrence, K. L. (1997), Modeling and Simulation of Dynamic Systems, Prentice-Hall Inc. (New Jersey).

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