TABLE OF CONTENTS Chapter 8 describes the different mathematical models of an elastic foundation. A mechanical model of the Winkler model is discussed and natural frequencies of vibration of Bernoulli Euler uniform and stepped one-span beams with different boundary conditions on the elastic foundation are presented.
| A | Cross-sectional area of the beam | |
| d | Viscous damping coefficient of foundation | |
| E 0 | Elastic constant of the foundation material | |
| E, G | Modulus of elasticity and shear modulus of the beam material | |
| EI | Bending stiffness | |
| G 0 | Foundation modulus of rigidity (Pasternak model) | |
| l | Moment of inertia of a cross-sectional area of the beam | |
| k n | Frequency parameter, | |
| k | Shear factor | |
| k slope, D 0 | Elastic sloping stiffness of medium | |
| k tilt | Elastic tilting (transverse rotating) stiffness of medium [Nm/m] | |
| k tr, k 0 | Elastic transverse translatory stiffness of medium (Winkler foundation modulus) | |
| l | Length of the beam | |
| M | Lumped mass | |
| p | Foundation reaction | |
| t | Time | |
| V i | Puzyrevsky functions | |
| x | Spatial coordinate | |
| X( x) | Mode shape | |
| x, y, z | Cartesian coordinates | |
| y( x, t), w | Lateral displacement of the beam | |
| ? | Frequency parameter, k 4 = ?4 ? 4 | |
| ? | Frequency parameter, ? 2 = k 2 l 2 | |
| ? | Slope | |
| ?, m | Density of material and mass per unit length of beam, m = ? A | |
| ? | Natural frequency... |
