c NE ¼
1
8
px 0
l
2 ¼
1
8
p
l
6 EI
qA
;
c MS ¼
1
4
E
q
p
l
4 ;
ð3:176Þ
which gives the following simple ratio of the two nonlinear coefficients:
c NE
c MS
¼
1
2
p
s
2 ¼
1
2
g crit ;
ð3:177Þ
where s and η crit are defined with (3.174). For slender beams, where Bernoulli–
Euler theory applies, one has s ! 20 − 200, giving ratios of the nonlinear coefficients in the range 10
–2
–10
–4 or smaller. Thus at similar levels of modal response
amplitude u, the midplane stretching nonlinearity is much stronger.
For the nonlinear forces (cu
3 ) to be of similar magnitude for the two types of
nonlinearity, the deformations u should thus be of order magnitude (10
–2
–10
–4 )
–1/3
% 4–20 times larger for nonlinear elasticity than for midplane stretching.
So, if axial boundary motion is restricted so as to create midplane stretching, at
least for slender beams one can usually neglect nonlinear elasticity in comparison
with midplane stretching nonlinearity.
3.7.2 Primary Resonance, Weak Excitations
We now turn to obtaining approximate solutions for the Duffing system (3.152).
When X % x 0 , then the excitation frequency is close to the linear natural frequency
of the system, and we are dealing with a case of primary external resonance. At
primary resonance one can expect weak excitations to cause comparatively large
responses. Consequently we assume for (3.152) that if q = O(e) where e ( 1, then
u = O(1). Further, the influence of the damping term and the nonlinear term is
supposed to be weak as compared to the terms describing linear inertia and stiffness.
Assuming damping and nonlinearity to be both O(e), the corresponding terms will
enter the analysis at the same level of approximation as the external forcing. This
magnitude ordering is expressed by writing (3.152) in the form:
€ u þ x
2
0 u ¼ e q cos Xt À 2bx 0 _
u À cu
3
À
Á :
ð3:178Þ
Heading on for a multiple scales approximation, we assume a solution in form of
a uniformly valid expansion:
uðt; eÞ ¼ u 0 ðT 0 ; T 1 Þ þ eu 1 ðT 0 ; T 1 Þ þ Oðe
2
Þ;
ð3:179Þ
3.7 Externally Excited Duffing Systems
153
1
8
px 0
l
2 ¼
1
8
p
l
6 EI
qA
;
c MS ¼
1
4
E
q
p
l
4 ;
ð3:176Þ
which gives the following simple ratio of the two nonlinear coefficients:
c NE
c MS
¼
1
2
p
s
2 ¼
1
2
g crit ;
ð3:177Þ
where s and η crit are defined with (3.174). For slender beams, where Bernoulli–
Euler theory applies, one has s ! 20 − 200, giving ratios of the nonlinear coefficients in the range 10
–2
–10
–4 or smaller. Thus at similar levels of modal response
amplitude u, the midplane stretching nonlinearity is much stronger.
For the nonlinear forces (cu
3 ) to be of similar magnitude for the two types of
nonlinearity, the deformations u should thus be of order magnitude (10
–2
–10
–4 )
–1/3
% 4–20 times larger for nonlinear elasticity than for midplane stretching.
So, if axial boundary motion is restricted so as to create midplane stretching, at
least for slender beams one can usually neglect nonlinear elasticity in comparison
with midplane stretching nonlinearity.
3.7.2 Primary Resonance, Weak Excitations
We now turn to obtaining approximate solutions for the Duffing system (3.152).
When X % x 0 , then the excitation frequency is close to the linear natural frequency
of the system, and we are dealing with a case of primary external resonance. At
primary resonance one can expect weak excitations to cause comparatively large
responses. Consequently we assume for (3.152) that if q = O(e) where e ( 1, then
u = O(1). Further, the influence of the damping term and the nonlinear term is
supposed to be weak as compared to the terms describing linear inertia and stiffness.
Assuming damping and nonlinearity to be both O(e), the corresponding terms will
enter the analysis at the same level of approximation as the external forcing. This
magnitude ordering is expressed by writing (3.152) in the form:
€ u þ x
2
0 u ¼ e q cos Xt À 2bx 0 _
u À cu
3
À
Á :
ð3:178Þ
Heading on for a multiple scales approximation, we assume a solution in form of
a uniformly valid expansion:
uðt; eÞ ¼ u 0 ðT 0 ; T 1 Þ þ eu 1 ðT 0 ; T 1 Þ þ Oðe
2
Þ;
ð3:179Þ
3.7 Externally Excited Duffing Systems
153