TABLE OF CONTENTS Chapter 13 is devoted to the optimal design of vibrating one-span beams. Two main problems are discussed.
The volume-frequency problem: find a configuration of the cross-sectional area A(x) along the beam for the minimum (for maximum) frequency ? of a beam, if the volume of the beam V 0 is given.
The frequency-volume problem: find a configuration of the cross-sectional area A(x) for the minimum (or maximum) volume V of a beam, if frequency ? = ? 0 is given.
The Bernoulli Euler and Timoshenko beam theories are applicable. Analytical and numerical results for a beam with classical boundary conditions are presented. The maximum principle of Pontryagin has been applied.
| A(x) | Cross-sectional area of a beam | |
| E | Modulus of elasticity of the beam material | |
| EI | Bending stiffness | |
| h, b, r | Geometrical dimensions of the cross-section of the beam | |
| H | Hamiltonian | |
| I(x) | Moment of inertia of a cross-sectional area of a beam | |
| k 1, k 2 | Lagrange multipliers | |
| l | Length of the beam | |
| M, Q | Bending moment and shear force | |
| t | Time | |
| V | Volume of a beam | |
| V _, V + | Lower and upper limit of the volume of a beam | |
| x | Spatial coordinate | |
| x, y, z | Cartesian coordinates | |
| X(x) | Mode shape | |
| y(x) | Lateral displacement of a beam | |
| ?, m | Density of material and mass per unit length of beam, m... |
