Page 476 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 476
Sec. 14.2 Conservative Systems 463
Dividing, we obtain
dx
Separating variables and integrating
+ o)^x^ = C
which is a series of ellipses, the size of which is determined by C. The preceding
equation is also that of conservation of energy:
\mx^ + ^kx^ = C
Because the singular point is at x = y = 0, the phase plane plot appears as in Fig.
14.1-1. If y/o) is plotted in place of y, the ellipses of Fig. 14.1-1 reduce to circles.
Figure 14.1-1.
14.2 CONSERVATIVE SYSTEMS
In a conservative system the total energy remains constant. Summing the kinetic
and potential energies per unit mass, we have
U{x) = E = constant (14.2-1)
Solving for y = i, the ordinate of the phase plane is given by the equation
y = i = ±y/2[E - U(x)] (14.2-2)
It is evident from this equation that the trajectories of a conservative system must
be symmetric about the x-axis.
The differential equation of motion for a conservative system can be shown
to have the form
X =f { x ) (14.2-3)
Because x = x(dx/dx), the last equation can be written as
Xdx —f {x) dx = 0 (14.2-4)
