Page 497 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

484 Nonlinear Vibrations Chap. 14

14-8 For the undamped spring-mass system with initial conditions j(0) = A and i(0 ) = 0,
determine the equation for the state speed V and state under what condition the
system is in equilibrium.
14-9 The solution to a certain linear differential equation is given as
jr = cos 7Tt + sin Int
Determine y = x and plot a phase plane diagram.
14-10 Determine the phase plane equation for the damped spring-mass system
X -f 2^(O^^X -h OJ^^jX —0

and plot one of the trajectories with v = y/(o,j and x as coordinates.
14-11 If the potential energy of a simple pendulum is given with the positive sign
U(0) = + j cos 6
determine which of the singular points are stable or unstable and explain their
physical implications. Compare the phase plane with Fig. 14.4-2.
14-12 Given the potential U{x) = 8 - 2 c o s 7t x/ 4 , plot the phase plane trajectories for
E = 6,1, 8,10, and 12, and discuss the curves.
14-13 Determine the eigenvalues and eigenvectors of the equations
X = 5x - y
y = 2x 2y
14-14 Determine the modal transformation of the equations of Prob. 14-13, which will
decouple them to the form
i = A,^
V =
14-15 Plot the ri phase plane trajectories of Prob. 14-14 for Aj/A^ = 0.5 and 2.0.
14-16 For Aj/A2 = 2.0 in Prob. 14-15, plot the trajectory y vs. x.
14-17 If A, and A2 of Prob. 14-14 are complex conjugates -a ± ifi, show that the equation
in the u,u plane becomes
dv _ Pu -y ar

du au —pv
14-18 Using the transformation u = p cos 6 and r = p sin 6, show that the phase plane
equation for Prob. 14-17 becomes
dp
13
with the trajectories identified as logarithmic spirals
p =
14-19 Near a singular point in the xy-plane, the trajectories appear as shown in Fig. P14-19.
Determine the form of the phase plane equation and the corresponding trajectories
in the fi7-plane.