Problem 5.3 Consider a beam with hinged-hinged supports that are fixed a given
distance apart (Fig. P5.3). A single-mode approximation describing freely damped
transverse oscillations takes the form (cf. (3.169)):
€ u þ _
u þ ku þ u
3
¼ 0:
ð5:70Þ
The stiffness k is positive, zero, or negative when the initial separation of supports corresponds to, respectively, pre-critical, critical or post-critical loading.
(a) Recast the system into first-order form, determine the singular points, and
show that k = 0 is a bifurcation value. Examine the stability of singular points as
k is varied near zero and sketch locally the bifurcation diagram.
(b) Show that the bifurcating system can be described on a two-dimensional
center manifold. Compute an approximation to the center manifold, reduce the
system to it, and examine the bifurcations near k = 0 for the reduced system.
Problem 5.4 Consider a system subjected to nonlinear dissipative forces:
€ u þ 2b þ u
2
À
Á _
u þ x
2 u þ u
3
¼ 0:
ð5:71Þ
(a) Show that Hopf bifurcations occur on the lines L 1 and L 2 that are defined by
L 1 : {b = 0, x
2 > 0} and L 2 : {2b = x
2 , x
2 < 0}.
(b) Show that the bifurcations on L 1 occur at the singular point (u, _
u) = (0, 0)
and are supercritical, whereas the bifurcations on L 2 occur at the singular points
ðu; _
uÞ ¼ ðÆ
ffiffiffiffiffiffiffiffiffi ffi
Àx 2
p
; 0Þ and are subcritical.
Problem 5.5 Consider the system (Nayfeh and Balachandran 1995):
€ h þ 2b _
h þ x
2
À X
2 cos h
À
Á
sin h ¼ 0
ð5:72Þ
describing the position h (t) of a mass sliding on a rotating hoop (Fig. P5.5). Here b
is the coefficient of viscous damping, X the angular velocity of the hoop, and
x
2 = g/R where g is the acceleration due to gravity and R the hoop radius.
(a) For b = 0, determine the equilibrium positions of the system and sketch the
phase plane orbits when X < x, X = x and X > x, respectively.
(b) For b > 0, examine and sketch the local bifurcations of equilibriums as X is
increased from zero and beyond x.
Fig. P5.3
5.15 Problems
319
distance apart (Fig. P5.3). A single-mode approximation describing freely damped
transverse oscillations takes the form (cf. (3.169)):
€ u þ _
u þ ku þ u
3
¼ 0:
ð5:70Þ
The stiffness k is positive, zero, or negative when the initial separation of supports corresponds to, respectively, pre-critical, critical or post-critical loading.
(a) Recast the system into first-order form, determine the singular points, and
show that k = 0 is a bifurcation value. Examine the stability of singular points as
k is varied near zero and sketch locally the bifurcation diagram.
(b) Show that the bifurcating system can be described on a two-dimensional
center manifold. Compute an approximation to the center manifold, reduce the
system to it, and examine the bifurcations near k = 0 for the reduced system.
Problem 5.4 Consider a system subjected to nonlinear dissipative forces:
€ u þ 2b þ u
2
À
Á _
u þ x
2 u þ u
3
¼ 0:
ð5:71Þ
(a) Show that Hopf bifurcations occur on the lines L 1 and L 2 that are defined by
L 1 : {b = 0, x
2 > 0} and L 2 : {2b = x
2 , x
2 < 0}.
(b) Show that the bifurcations on L 1 occur at the singular point (u, _
u) = (0, 0)
and are supercritical, whereas the bifurcations on L 2 occur at the singular points
ðu; _
uÞ ¼ ðÆ
ffiffiffiffiffiffiffiffiffi ffi
Àx 2
p
; 0Þ and are subcritical.
Problem 5.5 Consider the system (Nayfeh and Balachandran 1995):
€ h þ 2b _
h þ x
2
À X
2 cos h
À
Á
sin h ¼ 0
ð5:72Þ
describing the position h (t) of a mass sliding on a rotating hoop (Fig. P5.5). Here b
is the coefficient of viscous damping, X the angular velocity of the hoop, and
x
2 = g/R where g is the acceleration due to gravity and R the hoop radius.
(a) For b = 0, determine the equilibrium positions of the system and sketch the
phase plane orbits when X < x, X = x and X > x, respectively.
(b) For b > 0, examine and sketch the local bifurcations of equilibriums as X is
increased from zero and beyond x.
Fig. P5.3
5.15 Problems
319