and show that the modal amplitudes y i (t) are solutions of the following set of
nonlinear ODEs:
€ y i þ 2b_ y i þ x
2
i þ c i
X n
k¼1
k
2 y
2
k
!
y i ¼ p i X
2 cos Xt; i ¼ 1; n
ðB:15Þ
where x i denote the linear natural frequencies given by (B.13), and:
c i
i
2 Kp
4
4qAL
[ 0; p i
2me
qAL 2 sin
ip
2
:
ðB:16Þ
b) With n = 1 (B.15) take the form
€ y þ 2b_ y þ x
2 y þ cy
3
¼ pX
2 cos Xt
ðB:17Þ
where x x 1 , c c 1 = Kp
4 /(4qAL) and p p 1 = 2me/(qAL
2 ). At certain
conditions this single-mode approximation adequately describes the dynamic
response of the beam. Which conditions?
c) Write down the two-mode approximation corresponding to (B.15) with n = 2.
What are the conditions for external resonance? Can internal resonance occur
between the two modes? If yes, for which values of the tension force KZ 0 ?
B.2.4 Local Bifurcation Analysis for the Unloaded System
In this section the engine mounted on the beam is assumed to be perfectly balanced,
that is, p = me = 0.
a) For the single-mode approximation (B.17) with p = 0, introduce a new variable
v = _
y and set up the associated system of first-order autonomous ODEs.
Determine the singular point of this system, and examine their stability for the
case b > 0, c > 0 and 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 c > 0 and
1. x
2 > 0, 0 < b ( x
2. x
2 < 0, 0 < b (
ffiffiffiffiffiffiffiffiffiffiffi ffi
À2x 2
p
c) Show that (y, v, x
2 ) = (0, 0, 0) is a bifurcation point, and sketch the bifurcation
in a (x
2 , y) plane. Do you recognize this type of bifurcation? Describe the preand post-bifurcation states in terms of the physical system.
466
Appendix B: Major Exercises
nonlinear ODEs:
€ y i þ 2b_ y i þ x
2
i þ c i
X n
k¼1
k
2 y
2
k
!
y i ¼ p i X
2 cos Xt; i ¼ 1; n
ðB:15Þ
where x i denote the linear natural frequencies given by (B.13), and:
c i
i
2 Kp
4
4qAL
[ 0; p i
2me
qAL 2 sin
ip
2
:
ðB:16Þ
b) With n = 1 (B.15) take the form
€ y þ 2b_ y þ x
2 y þ cy
3
¼ pX
2 cos Xt
ðB:17Þ
where x x 1 , c c 1 = Kp
4 /(4qAL) and p p 1 = 2me/(qAL
2 ). At certain
conditions this single-mode approximation adequately describes the dynamic
response of the beam. Which conditions?
c) Write down the two-mode approximation corresponding to (B.15) with n = 2.
What are the conditions for external resonance? Can internal resonance occur
between the two modes? If yes, for which values of the tension force KZ 0 ?
B.2.4 Local Bifurcation Analysis for the Unloaded System
In this section the engine mounted on the beam is assumed to be perfectly balanced,
that is, p = me = 0.
a) For the single-mode approximation (B.17) with p = 0, introduce a new variable
v = _
y and set up the associated system of first-order autonomous ODEs.
Determine the singular point of this system, and examine their stability for the
case b > 0, c > 0 and 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 c > 0 and
1. x
2 > 0, 0 < b ( x
2. x
2 < 0, 0 < b (
ffiffiffiffiffiffiffiffiffiffiffi ffi
À2x 2
p
c) Show that (y, v, x
2 ) = (0, 0, 0) is a bifurcation point, and sketch the bifurcation
in a (x
2 , y) plane. Do you recognize this type of bifurcation? Describe the preand post-bifurcation states in terms of the physical system.
466
Appendix B: Major Exercises