NATURAL FREQUENCES OF LINEAR SYSTEMS
101
5.1
NATURAL FREQUENCIES OF LINEAR SYSTEMS
In this section, two methods for calculating the natural frequency of linear single
degree of freedom structural models are presented: the Direct Method, which is
based on the équation of motion; and the Rayleigh Method, which is based on
energy principles. Several example problems are presented that illustrate the
relationships between these two methods.
Direct Method
The direct method of obtaining the natural frequency of a single degree
of freedom structure is to use its équation of motion. The structure’s natural
frequency u>o is defined as the frequency compatible with an undamped structure
of constant mass, with a restraint force that varies linearly with the displacement
coordinate, and with no external excitation force. Under these conditions, the
governing équation of motion, written in terms of the displacement coordinate
v, is
mi) + kiv = 0
(5.1)
which is a spécial case of équation (2.2). When this linear structure is given an
arbitrarily small displacement amplitude vq and then released, then v exhibits
free harmonie oscillations of the form
v — vq sin
When équation (5.2) and its second dérivative are substituted into équation
(5.1), the resuit is
(—mujg + ki)vo sin wqU 0
(5.3)
In the latter équation, the term sin u>ot is not zéro for ail time t and tfaus the
term in brackets must be zéro. This leads the following équation for the natural
frequency of the structure:
wo = v'(5-4)
V m
It is noted that the natural frequency is independent of the initial displacement
amplitude vq, which is characteristic of linear Systems.
Example Problem 5.1. Shown in Figure 5.1a is a spherical buoy of mass m,
which is half submerged in water in its static equilibrium. The buoy is depressed
vertically by a small initial amplitude t'o and released, after which it undergoes
free oscillations along the vertical coordinate v. Dérivé the équation of motion
for the buoy, and from that compute its natural frequency in units of rad/sec
and in Hz. The buoy’s radius is R = 3.5 ft and the water density is yw = 64
lb/ft3. Neglect System damping and the fluid drag forces. Let Cm = 0.
101
5.1
NATURAL FREQUENCIES OF LINEAR SYSTEMS
In this section, two methods for calculating the natural frequency of linear single
degree of freedom structural models are presented: the Direct Method, which is
based on the équation of motion; and the Rayleigh Method, which is based on
energy principles. Several example problems are presented that illustrate the
relationships between these two methods.
Direct Method
The direct method of obtaining the natural frequency of a single degree
of freedom structure is to use its équation of motion. The structure’s natural
frequency u>o is defined as the frequency compatible with an undamped structure
of constant mass, with a restraint force that varies linearly with the displacement
coordinate, and with no external excitation force. Under these conditions, the
governing équation of motion, written in terms of the displacement coordinate
v, is
mi) + kiv = 0
(5.1)
which is a spécial case of équation (2.2). When this linear structure is given an
arbitrarily small displacement amplitude vq and then released, then v exhibits
free harmonie oscillations of the form
v — vq sin
When équation (5.2) and its second dérivative are substituted into équation
(5.1), the resuit is
(—mujg + ki)vo sin wqU 0
(5.3)
In the latter équation, the term sin u>ot is not zéro for ail time t and tfaus the
term in brackets must be zéro. This leads the following équation for the natural
frequency of the structure:
wo = v'(5-4)
V m
It is noted that the natural frequency is independent of the initial displacement
amplitude vq, which is characteristic of linear Systems.
Example Problem 5.1. Shown in Figure 5.1a is a spherical buoy of mass m,
which is half submerged in water in its static equilibrium. The buoy is depressed
vertically by a small initial amplitude t'o and released, after which it undergoes
free oscillations along the vertical coordinate v. Dérivé the équation of motion
for the buoy, and from that compute its natural frequency in units of rad/sec
and in Hz. The buoy’s radius is R = 3.5 ft and the water density is yw = 64
lb/ft3. Neglect System damping and the fluid drag forces. Let Cm = 0.