Page 93 - Marine Structural Design

Chapter 3 Loads and Dynamic Response for wshore Structures 69

(3.48)

The displacement at resonance (@=a,) is
F lk
u = -cos(wt - 4) (3.49)
26

3.8.2 Elastic Vibration of Beams
The elastic vibration of a beam is an important subject for fatigue analysis of pipelines, risers
and other structures such as global vibration of ships. The natural fiequency of the beam may
be written as
E

mi = ai - (rad Isec) (3.50)

where
El = bending stiff5ess of the beam cross-section
L = length of the beam

m = mass per unit length of the beam including added mass
ai = a coefficient that is a function of the vibration mode, i
The following table gives the coefficient a,for the determination of natural frequency for
alternative boundary conditions.

Clamped-free Pin-Pin Free-freee Clamped- Clamped-pin
Beam beam B~~ clamped beam beam
IS'mode a, I 3.52 1 r2=9.87 I 22 1 22 I 15.4

2ndmode a2 1 22 I 4r2=39.5 1 61.7 I 61.7 I 50

3d mode a3 61.7 9n2 =88.9 121 121 104
4Ih mode a4 121 16z2 =158 200 200 I78
5thm~dea, 1 200 I 25n2=247 I 298.2 I 298.2 I 272