Page 498 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 14 Problems 485

Figure P14-19.

14-20 The phase plane trajeetories in the vicinity of a singularity of an overdamped system
(¿' > 1) are shown in Fig. P14-20. Identify the phase plane equation and plot the
corresponding trajectories in the ^77-plane.

Figure P14-20.

14-21 Show that the solution of the equation
y
dx X + 3y

is X" + 2xy -(- 3y^ == C, which is a family of ellipses with axes rotated from the x, y
coordinates. Determine the rotation of the semimajor axis and plot one of the
ellipses.
14-22 Show that the isoclines of the linear differential equation of second order are straight
lines.
14-23 Draw the isoclines for the equation

^ = A ry(y-2)
14-24 Consider the nonlinear equation
X + iO,^X + /xx^ = 0
Replacing x by yidy/dx), where y = x, gives the integral
y^ + (o^x^ + {fjix^ = 2E
With y = 0 when x = A, show that the period is available from
rA dx
- A A) p [ E - U { x ) ]