TABLE OF CONTENTS Chapter 15 considers the vibration of arches. Fundamental relationships for uniform and non-uniform arches are presented-they are the differential equations of vibrations, strain and kinetic energy, as well as the governing functional. Eigenvalues for arches with different equations of the neutral line, different boundary conditions, uniform, continuously and discontinuously varying cross-sections are presented.
| A | Cross-sectional area | |
| E, ?; | Young's modulus and the density of the material of the arch | |
| EI | Bending stiffness | |
| f, l | Rise and span of an arch | |
| I 0 | Second moment of inertia with respect to the neutral line of the cross-sectional area | |
| m | Mass per unit length of an arch | |
| M, N, Q | Bending moment, normal force and shear force | |
| n | Integer number | |
| r | Radius of gyration, r 2 A = I | |
| R( ?) | Radius of curvature | |
| t | Time | |
| U, T | Potential and kinetic energy | |
| ? | Tangential displacement of an arch | |
| V 0( ?), W 0( ?) | Tangential and rotational amplitude displacement | |
| w | Radial displacement of an arch | |
| x | Spatial coordinates | |
| x, y, z | Cartesian coordinates | |
| ? | Slope | |
| ? 0 | Angle of opening | |
| ? | Angle of rotation of the cross-section | |
| ? | Natural frequency |
This section presents the geometric parameters for arches with various equations of the neutral line (i.e. different shapes of arches) as well as differential equations of vibrations of arches.
