a) Using the method of multiple scales, show that a first-order uniformly valid
expansion for the response is:
u ¼ a cosðx 0 T 0 þ uÞ þ OðeÞ:
ð3:340Þ
b) For the case of primary resonance (x 0 % 1), show that the slow modulations of
amplitudes and phases are governed by:
a
0
¼ À
a
2x 0
sin w À
4
3p
bx 0 a
2
;
aw
0
¼ 2ra À
a
x 0
cos w;
ð3:341Þ
where er = 1– x 0 and w = 2rT 1 – 2u.
c) Obtain the frequency response equation and sketch the frequency response
(stationary amplitude versus r).
d) Examine the stability of solutions.
Problem 3.18 (Back to basics …) You have been beached on a distant sandbank,
with no textbooks surviving. Written in the sand you find the below puzzle. With
nothing else to do, you find yourself solving it, using a very basic approach.
Given the equation of motion for a linear single-degree-of-freedom oscillator:
€ u þ 2c _
u þ ku ¼ p;
ð3:342Þ
where
0\c\
ffiffiffiffiffi
k
j j
p ; k 2 R; p 2 R:
ð3:343Þ
a) Show that u = p/k is a static equilibrium
b) Show that this equilibrium is stable when k > 0, and unstable when k < 0
Problem 3.19 Consider a system with a fifth-order nonlinear restoring term:
€ u þ u þ u
5
¼ 0:
ð3:344Þ
Using the method of multiple scales, obtain a two-term approximate solution for
the motions u(t), and set up a relationship between the amplitude and the frequency
of the motion.
Problem 3.20 Make a MATLAB (or similar) program for numerical simulation
of a pendulum on a vertically vibrating support, cf. Fig. 3.5 and eqs. (3.11)–(3.12).
The program should be able to display time series and animate solutions. Extra
options could be, e.g., phase plane plots and frequency spectra. It should be
204
3 Nonlinear Vibrations: Classical Local Theory
expansion for the response is:
u ¼ a cosðx 0 T 0 þ uÞ þ OðeÞ:
ð3:340Þ
b) For the case of primary resonance (x 0 % 1), show that the slow modulations of
amplitudes and phases are governed by:
a
0
¼ À
a
2x 0
sin w À
4
3p
bx 0 a
2
;
aw
0
¼ 2ra À
a
x 0
cos w;
ð3:341Þ
where er = 1– x 0 and w = 2rT 1 – 2u.
c) Obtain the frequency response equation and sketch the frequency response
(stationary amplitude versus r).
d) Examine the stability of solutions.
Problem 3.18 (Back to basics …) You have been beached on a distant sandbank,
with no textbooks surviving. Written in the sand you find the below puzzle. With
nothing else to do, you find yourself solving it, using a very basic approach.
Given the equation of motion for a linear single-degree-of-freedom oscillator:
€ u þ 2c _
u þ ku ¼ p;
ð3:342Þ
where
0\c\
ffiffiffiffiffi
k
j j
p ; k 2 R; p 2 R:
ð3:343Þ
a) Show that u = p/k is a static equilibrium
b) Show that this equilibrium is stable when k > 0, and unstable when k < 0
Problem 3.19 Consider a system with a fifth-order nonlinear restoring term:
€ u þ u þ u
5
¼ 0:
ð3:344Þ
Using the method of multiple scales, obtain a two-term approximate solution for
the motions u(t), and set up a relationship between the amplitude and the frequency
of the motion.
Problem 3.20 Make a MATLAB (or similar) program for numerical simulation
of a pendulum on a vertically vibrating support, cf. Fig. 3.5 and eqs. (3.11)–(3.12).
The program should be able to display time series and animate solutions. Extra
options could be, e.g., phase plane plots and frequency spectra. It should be
204
3 Nonlinear Vibrations: Classical Local Theory