Page 92 - Marine Structural Design

68 Part I Structural Design Principles

3.8 Appendix A: Elastic Vibrations of Beams
In order to conduct fatigue assessment and the control of vibrations and noises, it is usually
necessary to estimate natural frequency and vibration modes of a structure. In this section,
basic dynamics is described on the vibration of beams and plates.
3.8.1 Vibration of A SpringMass System
Consider a system with a mass m, and spring constant k. When the system does not have
damping and external force, the equilibrium condition of the system may be expressed as
following,
mii+ku=O (3.40)
where u is the displacement of the mass. The free vibration may be expressed as the solution
of Eq.(3.40),
u = u, cos(o,t + a) (3.41)

where the natural frequency 01 may be expressed as,

(3.42)

and where u,, and a are determined by the initial condition at time b.
Assuming a cyclic force, Focosot, is applied to the mass, the equilibrium condition of the mass
may be expressed as following,
mu + ku = F, cos& (3.43)
and the above equation has a special solution as expressed in the following,

U= Ellk COS(&-p) (3.44)
1 - (w/w,)2
where the value of I$ may be taken as 0 ( if olol ) or R ( if o>ol ). The general solution is the
sum of the special solution and the solution to the free vibration. When o+o1, the value of u
will be far larger than that due to FO alone that is Fa. This phenomenon is called “resonance”.
In reality, the increase of vibration displacement u may take time, and damping always exists.
Assuming the damping force is proportional to velocity, we may obtain an equilibrium
condition of the system as,
mii+cu+ku = F,coswt (3.45)
The general solution to the above equation is

(3.46)

<=- C (3.47)
2m w,