TABLE OF CONTENTS Chapter 4 is devoted to special functions that are used for the dynamical calculation of different kind of beams and frames. Analytical expressions, properties and fundamental relationships, as well as tables of numerical values, are presented.
| A | Cross-sectional area | |
| E | Young's modulus | |
| EI | Bending stiffness | |
| I z | Moment of inertia of a cross-section | |
| i=EI/l | Bending stiffness per unit length | |
| k | Frequency parameter, | |
| l | Length of a beam | |
| r ik | Unit reactions | |
| S, T, U, V | Krylov Duncan functions | |
| t | Time | |
| X(x) | Mode shape | |
| x | Spatial coordinate | |
| y | Transversal displacement | |
| ?, m | Density of material and mass per unit length | |
| ? i | Displacement influence functions | |
| ? | Dimensionless coordinate, ? = x/l | |
| ? | Frequency parameter, ? = kl | |
| ?( t), ?( t) | Harmonic angular and linear displacement | |
| ?(?), ?(?) | Zal'tsberg functions | |
| ? | Natural frequency, |
The transverse vibration of the uniform Bernoulli Euler beam is described by the partial differential equation
where
| y = y(x, t) | = transverse displacement of a beam; |
| ? | = mass density; |
| A | = cross-sectional area; |
| E | = modulus of elasticity; |
| I | = moment of inertia of the cross-section about the neutral axis. |
The travelling wave method. D'Alembert's solution. A solution of differential equation (4.1) may be presented in the form
where
| A | = amplitude of vibration; |
| ? | = frequency of free vibration; |
| k | = propagation constant; |
| t |
