TABLE OF CONTENTS Chapter 14 is devoted to nonlinear transverse vibrations of beams. Static, physical and geometrical nonlinearities are discussed. In many cases, a method of reduction of nonlinear problems to linear ones with modified parameters is applied. This method is developed and presented by Bondar (1971). Some of the examples from the above mentioned book, are presented in this chapter. These are beams in magnetic fields, beams on nonlinear foundations, pipelines under moving liquids and internal pressure, etc. The frequency equations and the fundamental modes of vibrations are presented.
| A | Cross-sectional area | |
| A F | Cross-sectional area of the rods | |
| A 0 | Open area of the pipeline | |
| E, ? | Young's modulus and density of the beam material | |
| E F, ? F | Young's modulus and density of the foundation material | |
| EI | Bending stiffness | |
| I 2 I 4 | Cross-sectional area moments of inertia of order 2 and 4 | |
| k | Magnetic field parameter | |
| l | Length of the beam | |
| m, m L | Mass per unit length of the beam and liquid | |
| M | Bending moment | |
| P | Internal pressure | |
| r | Radius of gyration, r 2 A = I | |
| t | Time | |
| u | Longitudinal displacement of the rod | |
| v, v cr | Velocity and critical velocity of the moving liquid | |
| w 0 | Uniformly distributed force due to self-weight | |
| x | Spatial coordinate | |
| x, y, z | Cartesian coordinates | |
| X(x) | Mode shape | |
| y | Transversal displacement of the beam | |
| ?, ? |
