TABLE OF CONTENTS In this chapter, free vibration analyses of non-uniform one-span beams with different boundary conditions are presented. Continuous and stepped beams are investigated.
| A | Cross-sectional area of the beam | |
| E | Modulus of elasticity of the beam material | |
| EI | Bending stiffness | |
| h, d, b | Geometrical dimensions of the cross-sectional of the beam | |
| I | Moment of inertia of a cross-sectional area of the beam | |
| k | Stiffness coefficient of a transversal spring | |
| k F | Stiffness coefficient of the elastic foundation | |
| l | Length of the beam | |
| t | Time | |
| x | Spatial coordinate | |
| x, y, z | Cartesian coordinates | |
| X(x) | Mode shape | |
| y(x, t) | Lateral displacement of the beam | |
| ? | Taper parameter | |
| ? | Frequency parameter | |
| ?, m | Density of material and mass per unit length of beam, m = ? A | |
| ? | Flexibility constant of the rotational spring | |
| ? | Circular natural frequency of the transverse vibration of the beam |
A wedge and truncated wedge of length l are presented in Figs. 12.1(a) and (b).
The differential equation for the Bernoulli Euler theory
where r( x) is the radius of gyration of a cross-section about an axis through its centre parallel to the z axis.
Wedge. The natural frequency of vibration is
where A 0 = cross-sectional area in the root section
I 0 =...
