B.1.4 Local Bifurcation Analysis for a Balanced Shaft
a) For the single-mode approximation (B.7) with p = 0, introduce a new variable
v = _
y and set up the associated system of first-order autonomous ODEs. Determine
the singular points, and examine their stability for the case b > 0, x
2
2 R.
b) Classify the singular points (saddle, node, center, focus), and sketch in the (y,
v) plane the local flow of orbits near the singular points when b > 0 and
1. x
2 > 0, j 1 > 0.
2. x
2 > 0, j 1 < 0.
3. x
2 < 0, j 1 > 0.
4. x
2 < 0, j 1 < 0.
c) Show that (y, v, x
2 ) = (0, 0, 0) is a bifurcation point, and sketch the bifurcation
in a (x
2 , y) plane when, respectively, j 1 > 0 and j 1 < 0. Do you recognize this
type of bifurcation? Describe (in words) the pre- and post-bifurcation states of
the controlled shaft.
B.1.5 Quantitative Analysis of the Controlled System
This section concerns quantitative analysis of the dynamics of the unbalanced and
controlled shaft.
a) Assume that x
2 > 0, so that the static component Q 0 of the axial control force
either causes the beam to be tensioned when u(x, t) = 0, or causes compressive
forces that are less than the critical buckling load. Assume further that the
unbalance p is sufficiently small that the shaft performs small but finite oscillations about the statical equilibrium u(x, t) = 0. For the single-mode approximation (B.7), determine a first-order multiple scales approximation
y = y 0 + ey 1 + O(e
2 ) for the response of the controlled shaft near primary
external resonance X % x, and b, p, j 1,2 = O(e), e ( 1. Also, set up the frequency response equation governing stationary oscillations, and sketch the stationary amplitudes as a function of shaft speed X. Briefly discuss the changes in
beam response during slow passage of resonance, as compared to the linear case
(j 1 = j 2 = 0). How does the closed-loop control qualitatively affect stationary
oscillations of the shaft? (Compare to the uncontrolled case, j 1,2 = 0). How do
the effects of the two elements of control (j 1 and j 2 ) differ qualitatively?
b) Assume now that x
2 < 0, so that when u(x, t) = 0 the static control force is
compressive and exceeds the critical buckling load. Assume that the shaft
performs small but finite oscillations about the buckled state of equilibrium u(x,
t) 6 ¼ 0, and that the control law has no velocity feedback (j 2 = 0). For the
single-mode approximation (B.7), determine a zero-order multiple scales
approximation y = y 0 + O(e) for the shaft response near the primary external
resonance X %
ffiffiffiffiffiffiffiffiffiffiffi ffi
À2x 2
p
, when terms describing damping, excitation, and
Appendix B: Major Exercises
461
a) For the single-mode approximation (B.7) with p = 0, introduce a new variable
v = _
y and set up the associated system of first-order autonomous ODEs. Determine
the singular points, and examine their stability for the case b > 0, x
2
2 R.
b) Classify the singular points (saddle, node, center, focus), and sketch in the (y,
v) plane the local flow of orbits near the singular points when b > 0 and
1. x
2 > 0, j 1 > 0.
2. x
2 > 0, j 1 < 0.
3. x
2 < 0, j 1 > 0.
4. x
2 < 0, j 1 < 0.
c) Show that (y, v, x
2 ) = (0, 0, 0) is a bifurcation point, and sketch the bifurcation
in a (x
2 , y) plane when, respectively, j 1 > 0 and j 1 < 0. Do you recognize this
type of bifurcation? Describe (in words) the pre- and post-bifurcation states of
the controlled shaft.
B.1.5 Quantitative Analysis of the Controlled System
This section concerns quantitative analysis of the dynamics of the unbalanced and
controlled shaft.
a) Assume that x
2 > 0, so that the static component Q 0 of the axial control force
either causes the beam to be tensioned when u(x, t) = 0, or causes compressive
forces that are less than the critical buckling load. Assume further that the
unbalance p is sufficiently small that the shaft performs small but finite oscillations about the statical equilibrium u(x, t) = 0. For the single-mode approximation (B.7), determine a first-order multiple scales approximation
y = y 0 + ey 1 + O(e
2 ) for the response of the controlled shaft near primary
external resonance X % x, and b, p, j 1,2 = O(e), e ( 1. Also, set up the frequency response equation governing stationary oscillations, and sketch the stationary amplitudes as a function of shaft speed X. Briefly discuss the changes in
beam response during slow passage of resonance, as compared to the linear case
(j 1 = j 2 = 0). How does the closed-loop control qualitatively affect stationary
oscillations of the shaft? (Compare to the uncontrolled case, j 1,2 = 0). How do
the effects of the two elements of control (j 1 and j 2 ) differ qualitatively?
b) Assume now that x
2 < 0, so that when u(x, t) = 0 the static control force is
compressive and exceeds the critical buckling load. Assume that the shaft
performs small but finite oscillations about the buckled state of equilibrium u(x,
t) 6 ¼ 0, and that the control law has no velocity feedback (j 2 = 0). For the
single-mode approximation (B.7), determine a zero-order multiple scales
approximation y = y 0 + O(e) for the shaft response near the primary external
resonance X %
ffiffiffiffiffiffiffiffiffiffiffi ffi
À2x 2
p
, when terms describing damping, excitation, and
Appendix B: Major Exercises
461