Page 32 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 2.2 Equations of Motion: Natural Frequency 19

Eq. (2.2-2) can be written as
X + o)lx = 0 (2.2-4)
and we conclude by comparison with Eq. (1.1-6) that the motion is harmonic.
Equation (2.2-4), a homogeneous second-order linear differential equation, has the
following general solution:
X = Asin (oj + B cos o)J (2.2-5)

where A and B are the two necessary constants. These constants are evaluated
from initial conditions jc(0) and i(0), and Eq. (2.2-5) can be shown to reduce to
^(0) •
X = Sin (jjj + jc(0) cos (oJ (2.2-6)
V /
«

n
The natural period of the oscillation is established from io„T = 2tt, or
(2.2-7)
and the natural frequency is
. _ 1 _ 1 (2.2-8)
r 2tt V m
These quantities can be expressed in terms of the statical deflection A by observing
Eq. (2.2-1), /:A = mg. Thus, Eq. (2.2-8) can be expressed in terms of the statical
deflection A as
f _ 1 / X (2.2-9)
2irV A
Note that r, /„, and depend only on the mass and stiffness of the system, which
are properties of the system.
Although our discussion was in terms of the spring-mass system of Fig. 2.2-1,
the results are applicable to all single-DOF systems, including rotation. The spring
can be a beam or torsional member and the mass can be replaced by a mass
moment of inertia. A table of values for the stiffness k for various types of springs
is presented at the end of the chapter.
Example 2.2-1
A ;|-kg mass is suspended by a spring having a stiffness of 0.1533 N/mm. Determine
its natural frequency in cycles per second. Determine its statical deflection.
Solution: The stiffness is
k = 153.3 N/m
By substituting into Eq. (2.2-8), the natural frequency is

LIT y m LTT y 0.25